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, Universität Osnabrück, FB Mathematik/Informatik, 49069 Osnabrück, Germany,\
, Universität Oldenburg, FB Mathematik, 26111 Oldenburg, Germany,
Let $R$ be a noetherian ring and $M$ a finite $R$-module. With a linear form $\chi$ on $M$ one associates the Koszul complex $K(\chi)$. If $M$ is a free module, then the homology of $K(\chi)$ is well-understood, and in particular it is grade sensitive with respect to $\Im\chi$.
In this note we investigate the case of a module $M$ of projective dimension $1$ (more precisely, $M$ has a free resolution of length $1$) for which the first non-vanishing Fitting ideal $\I_M$ has the maximally possible grade $r+1$, $r=\rank M$. Then $h=\grade \Im\chi\le r+1$ for all linear forms $\chi$ on $M$, and it turns out that $H_{r-i}(K(\chi))=0$ for all even $i<h$ and $H_{r-i}(K(\chi))\iso \SS^{(i-1)/2}(C)$ for all odd $i<h$ where $\SS$ denotes symmetric power and $C=\Ext_R^1(M,R)$, in other words, $C=\Cok\psi^*$ for a presentation $$0\to F\stackrel{\psi}{\to} G \to M\to 0.$$ Moreover, if $h\le r$, then $H_{r-h}(K(\chi))$ is neither $0$ nor isomorphic to a symmetric power of $C$, so that it is justified to say that $K(\chi)$ is grade sensitive for the modules $M$ under consideration.
We furthermore show that the maximally possible value $\grade \Im\chi=r+1$ can only occur in two extreme cases: (i) $r=1$ or (ii) $\rank F=1$ and $r$ is odd.
The note was motivated by a result of Migliore, Nagel, and Peterson (see \[MNP\], Proposition 5.1). They implicitly prove the result on $K(\chi)$ for Gorenstein rings $R$, using local cohomology. Our method allows more general assumptions. (Even the assumption that $R$ is noetherian is superfluous if one uses the correct notion of grade.) It is based on results in \[BV1\] and has a predecessor in \[HM\]. The case in which $\rank F=1$ has been treated in \[BV3\].
The situation in which the Fitting ideal $\I_M$ of $M$ has only grade $r$ is also of interest. For example, it occurs for the Kähler differentials of complete intersections with isolated singularities. While our method also yields results in this case, we have restricted ourselves to the case of grade $r+1$ for the sake of clarity.
The detailed account of the linear algebra of $M$ and its exterior powers has been given in \[BV2\], Section 2.
For technical reasons we start with a situation dual to the above one. So let $F$, $G$ be finite free $R$-modules of rank $m,n$ and $\psi: G \to F$ an $R$-homomorphism. Set $\hat G=G\otimes \SS(F)$ where $\SS(F)$ denotes the symmetric algebra of $F$. Then we may consider $\psi$ an $\SS(F)$-linear form on $\hat G$ and can define the Koszul antiderivation $$\partial_\psi:\bigwedge\hat G\longrightarrow \bigwedge\hat G$$ with respect to $\psi$ in the usual way, i.e. $$\partial_\psi(x_1\wedge\ldots\wedge x_i)=\sum_{j=1}^i(-1)^{j+1}\psi(x_j)x_1
\wedge \ldots{\widehat x}_j\ldots\wedge x_i$$ for $x_1\ldots x_i\in \hat G$. We use the term Koszul complex also for the complex $$0\to
R\stackrel{\phi}{\to}G\stackrel{\phi(1)\wedge}{\to}\bigwedge^2
G\stackrel{\phi(1)\wedge}{\to}\bigwedge^3 G\stackrel{\phi(1)\wedge}{\to}\dots$$ associated with a linear map $\phi:R\to G$. Suppose that $\psi\phi=0$ and let $$d_\phi:\bigwedge\hat G\longrightarrow \bigwedge\hat G$$ be the differential of the Koszul complex associated with $\phi\otimes \SS(F)$, i. e. $$d_\phi(x)=(\phi(1)\otimes 1)\wedge x$$ for $x\in \bigwedge\hat G$. Since $\psi\phi=0$, obviously $\partial_\psi d_\phi+d_\phi\partial_\psi =0$. Thus we obtain the *Koszul bicomplex* $\KK$ $$\begin{CD}
&&0&&0&&0&&0&&0&&\\
&& @VVV @VVV @VVV @VVV @VVV \\
0@>>> R@>\phi>> G@>>> \cdots \bigwedge^{p-1} G@>d_\phi>>\bigwedge^p G
@>>> \bigwedge^{p+1} G && \cdots\\
&& @VVV @V\psi VV @VVV @V\partial_\psi VV @VVV \\
&&0@>>> F@>>> \cdots M^{p-1,1}@>>> M^{p,1}@>>> M^{p+1,1}&& \cdots\\
&&&& @VVV @VVV @VVV @VVV \\
&&&&0@>>> \cdots M^{p-1,2}@>>> M^{p,2}@>>> M^{p+1,2}&& \cdots\\
&&&&&& @VVV @VVV @VVV\\
&&&&&& \vdots && \vdots && \vdots\\
&&&&&& 0@>>> M^{p,p}@>>> M^{p+1,p}&& \cdots\\
&&&&&&&& @VVV @VVV \\
&&&&&&&& 0@>>> M^{p+1,p+1} &&\cdots \\
&&&&&&&&&& @VVV\\
&&&&&&&&&& 0 &&\cdots
\end{CD}$$ where $$M^{p,q}=\bigwedge^{p-q}G\otimes\SS^q(F)$$ for all integers $p$, $q$, and $\SS^q$ means $q$th symmetric power. The row homology of $\KK$ at $M^{p,q}$ is denoted by $H_\phi^{p,q}$, the column homology by $H_\psi^{p,q}$. Thus $H_\phi^{p,0}$ is the $p$th homology module $H^p$ of the Koszul complex associated to $\phi$. Set $N^p= \Ker
\partial_\phi^{p,0}$. The canonical injections $N^p\to \bigwedge^p G$ yield a complex homomorphism $$\begin{CD}
0@>>> R@>\bar\phi>>N^1 @>>>\cdots && N^p @>d_{\bar\phi}>> N^{p+1}&&\cdots \\
&& \parallel && @VVV && @VVV @VVV \\
0@>>> R@>\phi>> G@>>>\cdots && \bigwedge^p G@>d_\phi >> \bigwedge^{p+1}G&&\cdots
\end{CD}$$ where the maps $\bar\phi$, $d_{\bar\phi}$ are induced by $\phi$, $d_\phi$. The homology of the first row at $N^p$ is denoted by $\bar{H}^p$. We are now ready to state the key proposition. Here as in the following $^*$ means $\Hom_R(\;\; ,R)$. Moreover, $\I_\Psi$ denotes the ideal of $m$-minors of (a matrix representing) $\psi$.
Set $g=\grade\I_M$, $C=\Cok\psi$, and let $h=\grade\Im
\phi^*$. Assume that $r=n-m\ge 1$ and $g=r+1$. Then
1. $\Im\phi^*\subset \I_M$ and, in particular, $h\le g$;
2. $\bar H^i=0$ for $0\le i\le\min(2,h-1)$;
3. $$\bar{H}^i= \begin{cases} \SS^{\frac{i-1}2}(C) &\quad\textrm{ if }\quad 3\le
i< h,\; i\not\equiv 0\ (2)\\ 0 &\quad\text{if}\quad 3\le i< h,\; i\equiv 0\
(2);
\end{cases}$$
4. moreover, for $h\ge 3$ there is an exact sequence $$\begin{aligned}
{2}
&0\to \SS^{\frac{h-1}2}(C)\to \bar{H}^h \to H^h\quad&&
\text{if } h\not\equiv 0\ (2),\\
&0\to \bar{H}^h\to H^h&&\text{if } h\equiv 0\ (2).\end{aligned}$$
Let $M=\Cok\psi^*$. We choose a basis $e_1,\dots,e_m$ of $F^*$ and define the linear map $\Psi:G^*\to \bigwedge^{m+1}G^*$ by $\Psi(x)=\psi^*(e_1)\wedge\ldots\wedge\psi^*(e_m)\wedge x$. Then one obtains a complex $$0\to F^*\stackrel{\psi^*}{\to} G^*\stackrel{\Psi}{\to} \bigwedge^{m+1}G^*$$ whose dual is the head of the Buchsbaum–Rim complex resolving $C=\Cok \psi$. It follows that $M^*=\Im \Psi^*$, and obviously $\Im\Psi^*\subset \I_M G$. Since $\phi\in M^*$, one has $\Im\phi\subset \I_M$. This shows (a).
We quote some well-known facts about the homology of $\KK$. Let $0\le p\le g$. Then $H_\psi^{p,q}=0$ for $q\ne 0,p$ and $H_\psi^{p,p}=\SS^p(C)$. (See \[BV1\], Proposition 2.1 for the general statement.) Furthermore $H_\phi^{p,q}=0$ for $p-q<h$ by the grade sensitivity of the Koszul complex for $\phi$.
Claim (b) on $\bar H^i$ for $0\le i\le \min(2,h-1)$ is easily proved from the long exact (co)homology sequence.
For (c) we modify the complex $\KK$ to the complex $\tilde\KK$ by setting (i) $M^{p,p+1}=\SS^p(C)$ and (ii) $M^{p,-1}=N^p$. The maps to be added are the natural surjection $M^{p,p}\to \SS^p(C)$, the zero map $\SS^p(C)\to
M^{p+1,p+1}$, and those induced by the canonical injections $N^p\to\bigwedge^p
G$. The truncation of $\tilde \KK$ to the “rectangle” $0\le p\le h$, $0\le
q\le g$ has exact columns. Moreover, row homology for indices $<h$ can only occur at $M^{p,-1}$, namely $\bar H^p$, $0\le p\le h$, and at $M^{p,p+1}$, namely $\SS^p(C)$.
For an inductive argument we let $\RR_q$, $q\ge -1$, be the $q$th row of $\tilde \KK$ and $\BB_{q+1}$ be the image complex of $\RR_q$ in $\RR_{q+1}$. Then we have a series of exact sequences $$0\to \BB_q\to \RR_q \to \BB_{q+1}\to 0.$$ Thus we can use the long exact (co)homology sequence for each $q$. With $E^{p,q}=H^p(\BB_q)$ one therefore obtains the “southwest” isomorphisms $$\bar H^i=E^{i,0}\iso E^{i-1,1}\iso\dots\iso E^{\frac i2,\frac i2}$$ if $i$ is even, and $$\bar H^i=E^{i,0}\iso E^{i-1,1}\iso\dots\iso E^{\frac {i+1}2,\frac {i-1}2}$$ if $i$ is odd, $0\le i<h$. In fact, there is an exact sequence $$H^{i-(j+1),j} \to E^{i-(j+1),j+1}\to E^{i-j,j} \to H^{i-j,j}$$ and the extreme terms in this sequence are $0$ for all $j$ under consideration.
Let now $i$ be even, and $j=i/2$. Since the map $M^{j,j}\to M^{j+1,j}$ is injective, the same holds true for its restriction $\BB^j_j\to \BB^{j+1}_j$ in $\BB_j$, and so $\bar H^p=E^{j,j}=0$.
In the case in which $i$ is odd we can go one further step southwest, and obtain the isomorphism $\SS^j(C)=E^{j,j+1}\iso E^{j+1,j}$ for $j=(i-1)/2$.
Since $H^h\neq 0$, we only have an exact sequence $$0\to E^{h-1,h}\to \bar H^h\to H^h,$$ but the arguments above can be applied to $E^{h-1,1}$; it is zero if $h$ is even, and isomorphic to $\SS^{(h-1)/2}(C)$ if $h$ is odd.–
shows that roughly the first half of the symmetric powers $\SS^i(C)$, $i\le h$, can be interpreted as homologies of a Koszul complex. It is also possible to interpret the other half in a similar way. To this end we consider the column complexes $\CC_0,\dots,\CC_h$ of $\KK$ ([*not*]{} of $\tilde\KK$) and set $\CC_{h+1}=\Cok(\CC_{h-1}\to\CC_h)$. Then we obtain an exact sequence $$0\to\CC_0\to\dots\to\CC_h\to\CC_{h+1}\to 0$$ of complexes, and the only nonzero (co)homology can occur along $\CC_{h+1}$ and at $H^p(\CC_p)\iso S_p(C)$ for $p>0$. If one applies arguments similar to those in the proof of Proposition 1 (now proceeding in northwestern direction), then one obtains $$H^i(\CC_{h+1})\iso\begin{cases}
\SS^{\frac{h+i}2}(C)& \quad\text{if } h+i\equiv 0\ (2),\\
0 & \quad\text{if } h+i\not\equiv 0\ (2).
\end{cases}$$ for $0\le i\le h$. Note that the module $\CC_{h+1}^i$ is resolved by $\RR_i$, and $\RR_i$ is just a truncated and shifted version of $\RR_0\tensor \SS^i(F)$. The truncations of $\RR_0$ resolve the exterior powers $\bigwedge^j M_\phi$ where $M_\phi=\Cok(\phi)$. Thus $\CC_{h+1}^i=\bigwedge^{h-i} M_\phi\tensor
\SS^i(F)$.–
As in the proof of Proposition 1 let $M=\Cok\psi^*$. Then it is easy to see that $N^p=(\bigwedge^p M)^*$ for all $p$. In fact, $\psi^*$ induces a presentation $$\bigwedge^{p-1}G^*\otimes F^*\to \bigwedge^p G^*\to \bigwedge^p M\to 0$$ for all $p$, and the exact sequence $0\to N^p\to\bigwedge^p G\to
\bigwedge^{p-1}G\otimes F$ is obtained by dualizing. It follows that $N^r$ is free of rank $1$ (provided $\grade \I_M\ge 2$), and $N^p=0$ for $p>r$.
Let $\psi:G\to F$ be as above, and assume that $g=\grade\I_M=r+1$. Then the following conditions are equivalent.
1. There is a homomorphism $\phi:R\to G$ such that $\psi\phi=0$ and the ideal $\Im\phi^*$ has grade $r+1$;
2. [(i)]{} $r=1$ or [(ii)]{} $m=1$ and $r$ is odd.
The implication $(2)\Rightarrow (1)$ is an easy exercise. (See also the considerations at the end of this note.)
For the converse observe that $N^p=0$ for $p>r$ and that $N^r$ is free of rank $1$. So $\bar{H}^r$ must be cyclic. If $r$ were even, then $\bar{H}^r=0$ by Proposition 1, and $\Im\phi^*=R$. Thus $r$ must be odd. In this case $\bar{H}^r=\SS^{(r-1)/2}(C)$ by Proposition 1 where $C=\Cok\psi^*$. So if $r\ge
3$, then $C$ must be cyclic, which in turn means $m=1$.–
We now return to our original purpose. Therefore let $\bar\chi$ be a linear form on $M=\Cok\psi^*$. The induced linear form on $G^*$ is denoted by $\chi$; note that $\chi\psi^*=0$. Set $\phi=\chi^*$, and, as above, $r=n-m$. We want to connect the truncated Koszul complex $$0 \to \bigwedge^r M \to \cdots \bigwedge^i M\stackrel{\partial_{\bar\chi}}{\to}
\bigwedge^{i-1} M\cdots \to M \to R\to 0\tag{1}$$ with the complex $$0\to R\stackrel{\bar\phi}{\to}N^1 \to\cdots N^p\stackrel{d_{\bar\phi}}{\to}
N^{p+1}\cdots\tag{2}$$ considered above. We have already observed that $N^p=(\bigwedge^p M)^*$.
With notation from above, let $g=\grade\I_M\ge r+1$. Then there are maps $\mu_p:\bigwedge^p
M\to N^{r-p}$, $p=0,\ldots, r$, such that $d_{\bar\phi}\mu_p=\pm
\mu_{p-1}\partial_{\bar\chi}$ and which are isomorphisms for $p=0,\ldots, r-1$ and injective for $p=r$. If, in particular, $g=r+1\ge 2$, then we obtain the following diagram of maps, the columns of which are exact and with commutative or anticommutative rectangles: $$\begin{CD}
&& && 0 && 0\\
&& && @VVV @VVV\\
0@>>>\bigwedge^r M@>>>\bigwedge^{r-1} M@>>> \bigwedge^{r-2} M\\
&& @V\mu_r VV @V\mu_{r-1} VV @VVV \\
0 @>>> R@>\bar\phi>>N^1 @>>> N^2&&\\
&& @VVV @VVV \\
0@>>> R/\I_M@>>> 0\\
&& @VVV\\
&& 0
\end{CD}$$
As in \[BV2\], p. 17, we choose isomorphisms $\gamma:\bigwedge^m
F^*\to R$, $\delta:\bigwedge^n G^*\to R$, and define maps $$\nu_p:\bigwedge^p G^* \to (\bigwedge^{r-p} G^*)^*,$$ $p=0,\ldots, r$, by $$\nu_p(x)(y)=\delta(x\wedge y\wedge\bigwedge^m\psi^*(z)),$$ $x\in \bigwedge^p G^*,\ y\in\bigwedge^{r-p} G^*,\ z=\gamma^{-1}(1)$. Via the natural isomorphism $(\bigwedge^{r-p} G^*)^*\cong \bigwedge^{r-p} G$ we regard $\nu_p$ as a map $\bigwedge^p G^* \to \bigwedge^{r-p} G$. One has $\Im\nu_p\subset N^{r-p}$, and it is an easy exercise to show that the diagram $$\begin{CD}
\bigwedge^p G^*@>\partial\chi>> \bigwedge^{p-1} G^* \\
@V\nu_pVV @V\nu_{p-1}VV\\
\bigwedge^{r-p} G@>d_{\chi^*}>> \bigwedge^{r-p+1} G
\end{CD}$$ is commutative or anticommutative (see for example \[HM\], proof of Theorem 3.1). Consequently the same is true for $$\begin{CD}
\bigwedge^p M @>\partial\bar\chi>>\bigwedge^{p-1} M\\
@V\rho_pVV @V\rho_{p-1}VV\\
\Im\nu_p@>{d_{\chi^*}|\Im\nu_p}>>\Im\nu_{p-1}
\end{CD}$$ where $\rho_p$ and $\rho_{p-1}$ are induced by $\nu_p$ and $\nu_{p-1}$. Now let $\mu_p$ be the composition of $\rho_p$ and the canonical injection $\Im\nu_p\to
N^{r-p}$. Then the equation asserted in the proposition obviously holds. In case $p<\grade\I_M-1$, $\Im\nu_p=N^{r-p}$. This proves the remaining statements.–
If we look at the homology of the truncated Koszul complex (1) associated to $\bar\chi$ instead of the homology of (2), then a somewhat smoother assertion than Proposition 1 is possible.
Let $M$ be module with a finite free resolution of length $1$, $M=\Cok\psi^*$ where $\psi:G\to F$ is as above, and assume that $g=\grade\I_M =r+1$. Let $\bar\chi$ be a linear form on $M$. Then $\Im\bar\chi\subset \I_M$, and for the homology $\bar H_p$ at $\bigwedge M^p$ of the truncated Koszul complex (1) associated to $\bar\chi$ the following holds: $$\bar H_{r-i}=
\begin{cases}
\SS^{\frac{i-1}2}(C) &\quad\text{if}\ \ 0\le i< h,\; i\not\equiv 0\ (2)\\
0 &\quad\text{if}\ \ 0\le i< h,\; i\equiv 0\ (2),
\end{cases}$$ where $\SS^0(C)=R/\I_M$.
Furthermore there is a $\bar\chi$ with $\grade\Im\bar\chi=g$ if and only if [(i)]{} $r=1$ or [(ii)]{} $m=1$ and $r$ is odd. In this case we have necessarily $\Im\chi=\I_M$.
Consider the diagram in Lemma 4. Since $\mu_r$ is injective, $\bar H_r$ must be zero if $h>0$. Next let $h>1$. Then we obtain $R/\I_M\cong
\bar H_{r-1}$, since $\bar{H}^0=\bar{H}^1=0$. Finally if $h>2$, then in addition $\bar{H}^2=0$, so $\bar H_{r-2}=0$ as stated. The remaining assertions concerning the homology of (1) are contained in Proposition 1.
Instead of $\bar\chi$ we can consider the induced linear form $\chi$ on $G^*$. Corollary 3 yields the statement about the existence of such a linear form $\chi$ satisfying $\grade\Im\chi=g$. Assume that such a $\chi$ exists. If $m=1$, then $\Im\chi=\I_M$ by Proposition 2 in \[BV3\]. If $r=1$, then, under our assumptions, $M=\Cok\psi^*$ is an ideal in $R$ which must be isomorphic to $\Im\chi$. Using the Hilbert-Burch Theorem, we have $\Im\chi=a\I_M$ with an element $a\in R$. Since $\grade\Im\chi=2$, $a$ must be a unit.–
1. It is a noteworthy fact that the homology $\bar
H_{r-i}$ of the truncated Koszul complex (1) associated to $\bar\chi$ is independent of $\chi$ for $i<h$.
2. The Koszul complex associated to a linear form $\chi$ on a free module of rank $r$ is grade sensitive: its homology vanishes for $j>r-\grade
\Im\chi$, and does not vanish at $r-\grade\Im\chi$.
In a sense, this is also true for the linear form $\bar\chi$ considered in Theorem 5: of course, “vanishes” must be replaced by “vanishes if $i$ is even and is isomorphic to $\SS^{(i-1)/2}(C)$ if $i$ is odd”. Then Theorem 5 covers all $i=0,\dots,h-1$, but the analogy also persists if $i=h<g$. In fact, let $\pp$ be a prime ideal of grade $h$ such that $\Im \chi\subset\pp$. The module $M_\pp$ is free and $\SS^j(C_\pp)=\SS^j(C)_\pp=0$ for all $j$. Therefore one can apply the grade sensitivity of the Koszul complex for a free module, and $\bar H_h$ can be neither $0$ nor isomorphic to a non-vanishing symmetric power of $C$: otherwise its localization would vanish.
3. That we have truncated the Koszul complex of $\bar\chi$ is inessential. In fact, $\bigwedge^r M$ is torsionfree of rank $1$, and $\bigwedge^{r+1} M$ has rank $0$. Hence $\Hom_R(\bigwedge^{r+1} M, \bigwedge^r
M)=0$, and the homology of the full and of the truncated Koszul complexes at $\bigwedge^r M$ coincide.
Let $\psi:G\to F$ be as above, $r=n-m$, and $g=\grade\I_M=r+1$. In case $r>1$, the existence of a linear form $\chi$ on $G^*$ with $\grade\Im\chi = g$ can be described equivalently and independently of the last theorem.
With the assumptions on $\psi$ and $\chi$ from Theorem 5, assume in addition that $r>1$. Then $\grade\Im\chi= g$ is possible if and only if there exists a submodule $U$ of $M=\Cok\psi^*$ with the following properties:
1. $\rank U=r-1$;
2. $U$ is reflexive, orientable, and $U_{\frak p}$ is a free direct summand of $M_{\frak p}$ for all primes $\frak p$ of $R$ such that $\grade\frak p\le r$.
Let $\bar\chi$ be a linear form on $M$ such that $\grade\Im\bar\chi= r+1$. Set $U=\Ker\bar\chi$. Then (1) and the last property in (2) are obvious. Since $\Im\bar\chi$ is torsionfree and $M$ is reflexive, $U$ must be reflexive. Because $\Im\bar\chi$ has grade $\ge 2$, it is orientable. $M$ being orientable, the orientability of $U$ follows from Proposition (2.8) in \[B\].
Conversely let $U$ be a submodule of $M$ which satisfies (1) and (2). Then $I=M/U$ is torsionfree of rank 1 and therefore an ideal in $R$ which is orientable since $U$ and $M$ are orientable. Consequently $\grade I\ge 2$. So for a prime $\frak p$ in $R$ which contains $I$, the localization $IR_{\frak
p}$ cannot be free. On the other hand $I=R$ is impossible since $g=r+1$. In view of the last condition in (2), $I$ must have grade $r+1$.–
>From Theorem 5 we know that the hypothesis of Proposition 7 can only be satisfied with $m=1$ and $r$ odd. The submodule $U$ of $M$ in Proposition 7 has projective dimension $r-1$. In fact, the ideal $\I_M=\Im \chi$ is generated by $r+1$ elements and has grade $r+1$. Therefore it is perfect, i. e. $\projdim
R/I=r+1$. This implies $\projdim U=r-1$ via the exact sequence $0\to U\to M\to
I\to 0$. Dualizing this exact sequence, we obtain an exact sequence $0\to R \to
M^* \to U^*\to 0$. Since $M^*\cong \bigwedge^{r-1}M$ has projective dimension $r-1$, it follows that $U^*$ has projective dimension $r-1$.
The dualization argument just used amounts to interchanging the roles of $\psi^*$ and $\chi$, so that $U^*$ plays the same role for $\chi^*$ and $\psi$ as $U$ for $\psi^*$ and $\chi$. If we choose $\chi$ in a special way, then we can even achieve that $U$ and $U^*$ are isomorphic in a skewsymmetric way: for the isomorphism $\sigma:U\to U^*$ one has $\sigma^*=-\sigma$ upon the identification of $U$ and $U^{**}$ via the natural isomorphism.
As we mentioned at the beginning of the proof of Corollary 3, it is easy to see that there is a linear form $\chi$ on $G^*$ such that $\psi^*(1)\in \Ker\chi$ and $\grade\Im\chi=r+1=n$: fix a basis $z_1,\ldots,z_n$ of $G^*$ and let $\psi^*(1)=\sum_{i=1}^n x_iz_i$; the map $\chi: \sum_{i=1}^n a_iz_i \mapsto
\sum_{i=1}^n(-1)^ia_ix_{n+1-i}$ is an appropriate linear form. Let $\bar\chi$ be the induced form on $M=\Cok\psi^*$. The submodule $U=\Ker\bar\chi$ has the properties (1) and (2) of Proposition 7 (see the first part of the proof). Consider the commutative diagram $$\begin{CD}
0@>>> \Ker\chi @>>> G^* @>\chi>> R\\
&& @V\rho_1 VV @V\rho VV \parallel\\
0@>>> N @>>> G @>\psi >> R,
\end{CD}$$ with exact rows. The isomorphism $\rho$ is defined by $\rho(z_i)=(-1)^iz^*_{n+1-i}$ where $z_1^*,\ldots,z_n^*$ is the basis of $G$ dual to $z_1,\ldots,z_n$, and $\rho_1$ is induced by $\rho$. Since $\rho_1(\psi^*(1))=-\bar\chi$, we obtain a second commutative diagram $$\begin{CD}
0@>>> R^* @>(\psi^*)_1>> \Ker\chi @>>> U @>>> 0\\
&& @VVV @V\rho_1 VV @V\bar\rho_1VV\\
0@>>> R^* @>\bar\chi^*>> N @>>> \phantom{*}U^* @>>> 0
\end{CD}$$ with exact rows. $(\psi^*)_1$ is induced by $\psi^*$ and the first vertical arrow maps $1$ to $-1$. As $\rho_1$ is an isomorphism, the same is true for $\bar\rho_1$, so $U$ turns out to be self-dual. Moreover $\bar\rho_1$ is skew-symmetric, i.e. $(\bar\rho_1)^*=-\bar\rho_1$ since the same is true for $\rho$.
Suppose that $R=K[X_1,\dots,X_n]$ is the polynomial ring in $n$ indeterminates over a field $K$. If we define $\psi:R^n\to R$ by $\psi(e_i)=X_i$, then the module $U$ is associated with a rank $n-2$ vector bundle on $\PP^{n-1}(K)$. Such bundles exist however also for odd $n$; see \[V\]. The module $V$ assocíated with the the construction in \[V\] is self-dual only for $n=4$.
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W. Bruns and U. Vetter. [*Length formulas for the Local Cohomology of Exterior Powers*]{}. Math Z. [**191**]{} (1986), 145–158.
W. Bruns, U. Vetter. [*Determinantal rings*]{}. Lect. Notes Math. [**1327**]{}, Springer 1988.
W. Bruns and U. Vetter. [*A Remark on Koszul Complexes*]{}. Beitr. Algebra Geom. [**39**]{} (1998), 249–254.
J. Herzog and A. Martsinkovsky. [*Glueing Cohen-Macaulay modules with applications to quasihomogeneous complete intersections with isolated singularities*]{}. Comment. Math. Helv. [**68**]{} (1993), 365–384
J.C. Migliore, U. Nagel, and C. Peterson. [*Buchsbaum-Rim sheaves and their multiple sections*]{}. J. Algebra [**219**]{} (1999), 378–420.
U. Vetter. [*Zu einem Satz von G. Trautmann über den Rang gewisser kohärenter analytischer Garben*]{}. Arch. Math. [**24**]{} (1973), 158–161.
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---
abstract: |
TensorFlow is a popular deep learning framework used by data scientists to solve a wide-range of machine learning and deep learning problems such as image classification and speech recognition. It also operates at a large scale and in heterogeneous environments — it allows users to train neural network models or deploy them for inference using GPUs, CPUs and deep learning specific custom-designed hardware such as TPUs. Even though TensorFlow supports a variety of optimized backends, realizing the best performance using a backend may require additional efforts. For instance, getting the best performance from a CPU backend requires careful tuning of its threading model. Unfortunately, the best tuning approach used today is manual, tedious, time-consuming, and, more importantly, may not guarantee the best performance.
In this paper, we develop an automatic approach, called [[<span style="font-variant:small-caps;">TensorTuner</span>]{}]{}, to search for optimal parameter settings of TensorFlow’s threading model for CPU backends. We evaluate [[<span style="font-variant:small-caps;">TensorTuner</span>]{}]{} on both Eigen and Intel’s MKL CPU backends using a set of neural networks from TensorFlow’s benchmarking suite. Our evaluation results demonstrate that the parameter settings found by [[<span style="font-variant:small-caps;">TensorTuner</span>]{}]{} produce 2% to 123% performance improvement for the Eigen CPU backend and 1.5% to 28% performance improvement for the MKL CPU backend over the performance obtained using their best-known parameter settings. This highlights the fact that the default parameter settings in Eigen CPU backend are not the ideal settings; and even for a carefully hand-tuned MKL backend, the settings are sub-optimal. Our evaluations also revealed that [[<span style="font-variant:small-caps;">TensorTuner</span>]{}]{} is efficient at finding the optimal settings — it is able to converge to the optimal settings quickly by pruning more than 90% of the parameter search space.
author:
-
bibliography:
- 'ms.bib'
title: |
Auto-tuning TensorFlow Threading Model for\
CPU Backend[^1]
---
Auto-tuning; HPC; Deep Learning; Black-box Optimization; Software Optimization;
[^1]: Presented at ML-HPC workshop, held along with SuperComputing’18, Dallas, TX. DOI 10.1109/MLHPC.2018.000-7
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In [@go2] we proposed an analog of the classical Riemann hypothesis for characteristic $p$ valued $L$-series based on the work of Wan, Diaz-Vargas, Thakur, Poonen, and Sheats for the zeta function $\zeta_{\Fr[\theta]}(s)$. During the writing of [@go2], we made two assumptions that have subsequently proved to be incorrect. The first assumption is that we can ignore the trivial zeroes of characteristic $p$ $L$-series in formulating our conjectures. Instead, we show here how the trivial zeroes influence nearby zeroes and so lead to counter-examples of the original Riemann hypothesis analog. We then sketch an approach to handling such “near-trivial” zeroes via Hensel’s and Krasner’s Lemmas (whereas classically one uses Gamma-factors). Moreover, we show that $\zeta_{\Fr[\theta]}(s)$ is not representative of general $L$-series as, surprisingly, all its zeroes are near-trivial, much as the Artin-Weil zeta-function of $\mathbb{P}^1/\Fr$ is not representative of general complex $L$-functions of curves over finite fields. Consequently, the “critical zeroes” (= all zeroes not effected by the trivial zeroes) of characteristic $p$ $L$-series now appear to be quite mysterious. The second assumption made while writing [@go2] is that certain Taylor expansions of classical $L$-series of number fields would exhibit complicated behavior with respect to their zeroes. We present a simple argument that this is not so, and, at the same time, give a characterization of functional equations.'
address: |
Department of Mathematics\
The Ohio State University\
231 W. $18^{\text{th}}$ Ave.\
Columbus, Ohio 43210
author:
- David Goss
date: 'May 25, 2001'
title: 'The impact of the infinite primes on the Riemann Hypothesis for Characteristic $p$ $L$-series'
---
[^1]
Introduction {#intro}
============
In the paper [@go2] (whose notations etc., we generally follow here) we defined a possible Riemann hypothesis for the zeroes of characteristic $p$ $L$-functions. This work is based on the results of Wan [@w1], Diaz-Vargas and Thakur [@dv1], Poonen, and Sheats [@sh1] for the zeta function $\zeta_{\Fr[\theta]}(s)$ ($s=(x,y)\in S_\infty$; see Section \[trivial\] for the definitions). In particular the zeroes of $\zeta_{\Fr[\theta]}(s)$ were found to be both simple and lie on the line $\Fr((1/T))$ just as the completed Riemann zeta function $\pi^{-s/2}\Gamma(s/2)\zeta(s)$, $s=1/2+it$, is conjectured to have only real zeroes in $t$. This finite characteristic Riemann hypothesis has two components to it. The first, Conjecture 4 of [@go2], is a basic finiteness statement about the extension fields generated by $L$-series zeroes (obviously no such conjecture need be made classically). The second, Conjecture 5 of [@go2], focuses on counting the number of zeroes of a given absolute value; based on the above examples, we postulated that, outside of finitely many exceptions, the absolute value determined the zero. (Both conjectures are recalled in Section \[review\] below; therefore we will drop the reference to [@go2] from now on.) We pointed out similarities between Conjecture 5 and the classical Riemann hypothesis for number fields.
Now in classical theory we work, of course, with the completed $L$-function including the Gamma-factors arising from the infinite primes as above for $\zeta(s)$. These Gamma-factors have poles at some of the negative integers (e.g., $\Gamma(s/2)$ has poles at the non-positive even integers) and so, by analytic continuation, they force the $L$-series to have associated “trivial zeroes.” The functional equation then assures us that all other zeroes must lie in the critical strip. On the other hand, in the characteristic $p$ case presented in [@go2] we ignored input from the infinite primes and their associated trivial zeroes. Indeed, our conjectures allow for finitely many exceptional cases for each $y$, and, for $y$ a negative integer, there are only finitely many trivial zeroes; thus ignoring them appeared innocuous. However, since the publication of [@go2], we have realized that the infinite primes must also be taken into account in the characteristic $p$ Riemann hypothesis. It is our goal here to explain how the trivial zeroes lead to counter-examples for Conjecture 5 (though Conjecture 4 still appears to be valid). More precisely, the topology of our space $S_\infty$, which is the natural domain of definition for the characteristic $p$ $L$-series, permits us to inductively construct counter-examples using zeroes sufficiently close to trivial zeroes (we call such zeroes “near-trivial zeroes”). We also suggest some possible replacements for Conjecture 4 by sketching a procedure based on Hensel’s Lemma to isolate the near-trivial zeroes. The original conjectures may then work for the remaining zeroes (which, following classical precedent, we call the “critical zeroes”).
However, there is now a great surprise. Upon searching for the critical zeroes of $\zeta_{\Fr[\theta]}(s)$, $s\in S_\infty$, we find that they do not exist! (More precisely, we prove this fact for $\Fp[\theta]$; the general $\Fr[\theta]$ case seems very likely to follow from Sheats’ techniques — see also Corollary \[evenval\] of Section \[basic\].) That is, the case of $\zeta_{\Fr[\theta]}(s)$ may not be representative of general characteristic $p$ $L$-series, much as the Artin-Weil zeta-function of $\mathbb{P}^1$ over a finite field is not typical of the $L$-series of general curves. Indeed, the Artin-Weil zeta-function of the projective line also has no critical zeroes. We will exhibit here a few examples of critical zeroes in the characteristic $p$ theory and we hope to have more examples in the near future. However, as of this writing, they are a complete mystery. For instance where they lie, or even if there are infinitely many of them (for a given $L$-series and interpolation place) is not known. It is certainly possible that there may be more surprises ahead.
Let $K$ be a complete, algebraically closed, non-Archimedean field and let $f(x)=\sum a_nx^n$ be an entire power series with coefficients in $K$. Let $L$ be a complete subfield of $K$ which contains the zeroes of $f(x)$. It is then a standard fact that there exists $\alpha\in K^\ast$ such that the coefficients of $\alpha f(x)$ lie in $L$; i.e., the coefficients of $f(x)$ lie on an $L$-line through the origin in $K$. As is well-known, in complex analysis the relationship between the zeroes and the Taylor coefficients of an entire function is more complicated; e.g., $e^{2\pi i x}-1$. (However, as Keith Conrad pointed out, a similar statement is true in complex analysis if we allow ourselves to multiply by an invertible entire function; in non-Archimedean analysis such entire functions are constant.) During the writing of [@go2], we had assumed that Taylor expansions of classical $L$-series (of number fields) would exhibit similar complicated behavior. This assumption has also proved to be incorrect. Indeed, in our last section we give a simple argument which shows that the associated Taylor coefficients in $t$ (where $s=1/2+it$) lie on a line through the origin in $\mathbb C$. This argument also characterizes the functional equation of the $L$-series via the description of complex conjugation given in Equation \[funceq\].
We now describe in more detail the contents of this paper. In Section \[trivial\] we review the definitions and interpolations of characteristic $p$ $L$-series. We recall how the analogue of Artin $L$-series has trivial zeroes (essentially from the classical theory of Artin and Weil) at the negative integers. We also present a reasonable approach to trivial zeroes for the general $L$-series of a Drinfeld module; it is our belief that the work of Boeckle and Pink [@bp1], [@b1] will ultimately flesh out this construction.
In Section \[krasner\] we recall Krasner’s Lemma in order to put it into a form more useful for calculations in characteristic $p$. In Section \[review\] we review the statements of Conjectures 4 and 5.
In Section \[counter\] we construct counter-examples to Conjecture 5. Along the way we establish that the family of Newton polygons associated to an $L$-series possesses certain invariance properties, see Lemma \[newtonclose\]. More precisely, the first $n$-segments of the Newton polygon will, generically, be an invariant of a natural group of translations. This leads us to believe that the family of Newton polygons itself will serve to distinguish between $L$-series (see Question \[langlands\] in this section). We also sketch the beginnings of an approach to remove these counter-examples by isolating the offending near-trivial zeroes via Hensel’s Lemma. We give some examples of how this will work (as well as find a few critical zeroes) but total success here will need many more results on the structure of the zeroes.
In Section \[basic\] we use the techniques of Diaz-Vargas and Sheats to study the effects of the trivial zeroes for $\zeta_{\Fr[\theta]}(s)$, $s\in S_\infty$, and we establish in Proposition \[jandn\] that all zeroes for $\zeta_{\Fp[\theta]}(s)$ are near-trivial. It is our belief that ultimately there should be some massive generalization of the techniques of Diaz-Vargas and Sheats to general characteristic $p$ $L$-series enabling us, at least, to specify exactly where the near-trivial zeroes lie. Indeed, Boeckle has established the first of such results [@b1] by showing the logarithmic growth of the degrees of special polynomials in general. For $\Fr[T]$, this logarithmic growth is a first corollary of the techniques of [@dv1] and [@sh1]. As one sees by simple examples (such as in the discussion directly after Corollary \[log\]), this logarithmic growth is analogous to the formula giving the growth of the number of zeroes of classical $L$-series in the critical strip.
Finally, let $L(\chi,s)$, $s=1/2+it\in \C$, be the $L$-series associated to a number field and abelian character $\chi$. Let $\Lambda(\chi,s)$ be the completed $L$-series which includes the Gamma-factors. In our last section, Section \[taylor\], we present an elementary argument that the Taylor coefficients of $\Lambda(\chi,s)$ about $t=0$ are, up to multiplication be a non-zero constant, real numbers. This is contrary to what was expected during the writing of [@go2].
It is my pleasure to thank Gebhard Boeckle, Keith Conrad, Mike Rosen, and Dinesh Thakur for their comments on an earlier version of this manuscript.
Trivial zeroes {#trivial}
==============
As mentioned in the introduction, we will follow the notation of [@go2]. We begin by briefly reviewing the definition of $L$-functions of Galois characters, Drinfeld modules, etc. Let $\mathcal X$ be a smooth, projective, geometrically connected curve over the finite field $\Fr$ where $r=p^m$ and $p$ is prime. Let $\infty\in \mathcal X$ be fixed closed point of degree $d_\infty$ over $\Fr$. Let $\k$ be the function field of $\mathcal X$ and let $\A$ be the subring of those functions which are regular outside of $\infty$. In the theory, one views $\k$ as the analogue of $\Q$ and $\A$ as the analogue of $\Z$; indeed, $\A$ is a Dedekind domain with finite class group and group of units $\Fr^\ast$.
Let $w$ be an arbitrary place of $\k$. We let $\vert x\vert_w$ be the normalized absolute value (= multiplicative valuation) at $w$ with associated additive valuation $\nu_w(x)$; by definition $\nu_w(t)=1$ if $t\in \k$ has a simple zero at $w$. We let $\k_w$ be the completion of $\k$ with respect to $w$ and we denote its finite field of constants by $\F_w$. We let $d_w:=[\F_w\colon \Fr]$ be the degree of $w$ over $\Fr$. We let $\bfC_w$ be the completion of a fixed algebraic closure of $\k_w$ equipped with the canonical extension of $\vert x\vert_w$. In particular we put $\K:=\k_\infty$. Thus $\K, {\mathbf C}_\infty$ are the analogues of $\R, \C$, respectively, whereas $\k_v$, for a finite $v$, is the analogue of $\Qp$ etc.
We now let $A,k,K$ etc., be another copy of these rings. There is an obvious isomorphism $\theta$ from the “bold” to the “non-bold” which makes $k,K, C_\infty$ into $\A$-fields. As in [@go2], we view the non-bold fields as being the “scalars” over which we can define Drinfeld modules etc., equipped with an action by the “operators” in $\A$. The basic example is the Carlitz module $C$ defined for $\A=\Fr[T]$. Here we put $\theta=\theta(T)\in k$ and $$C_T:=\theta \tau^0+\tau,$$ where $\tau\colon C_\infty\to C_\infty$ is the $r$-th power mapping of the field $C_\infty$; thus the Carlitz module is obviously defined over $k=\Fr(\theta)$. In fact, it is easy to see that $C$ gives rise to an obvious family of Drinfeld modules over ${\rm Spec}(A)$. (It will always be clear to the reader when “$C$” is being used to denote a field or the Carlitz module.)
More generally, let $L\subset C_\infty$ be a finite extension of $k$ and let $\omega$ be a place of $L$. We say $\omega$ is a “finite place” (or finite prime) if it lies over a prime of $A$; otherwise it is an “infinite place” (or infinite prime). This notion carries over to $\k$ etc., in the obvious fashion.
Let $\psi$ be a Drinfeld $\A$-module of rank $d$ defined over $L$. Let $\mathfrak P$ be a finite prime of $L$ lying over the prime $\mathfrak p$ of $A$ with finite residue class fields $\F_\mathfrak P$ and $\F_\mathfrak p$ respectively. As usual, we define the norm $n\mathfrak P$ of $\mathfrak P$ to be ${\mathfrak p}^f$ where $f$ is the residue field degree. All but finitely many such primes $\mathfrak P$ are good for $\psi$ in that one can reduce $\psi$ modulo $\mathfrak P$ to obtain a Drinfeld module $\psi^\mathfrak P$ over $\mathbb{F}_\mathfrak P$ with the same rank as $\psi$. Associated to $\mathbb{F}_\mathfrak P$ there is the Frobenius endomorphism ${\rm Fr}_\mathfrak P$ of $\psi^\mathfrak P$, and one sets $$P_\mathfrak P(u):=\det(1-u{\rm Fr}_{\mathfrak P}
\mid T_v(\psi^{\mathfrak P}))\,;$$ here $v\subset \A$ is a non-trivial prime such that $\theta(v)\neq \mathfrak p$ and $T_v$ is the $v$-adic [*Tate module*]{} of $\psi^\mathfrak P$. It is easy to see that the $\A_v$-module $T_v(\psi)$ is free of rank $d$. In complete accordance with classical theory, $P_{\mathfrak P}(u)$ has $\A$-coefficients which are independent of the choice of $v$. Moreover, as Drinfeld has shown, its roots inside $\bfC_\infty$ satisfy the local Riemann hypothesis in terms of their absolute values, see, e.g., §4.12 of [@go1].
Now assume that we have chosen a notion of “sign” on $\K$; that is, a homomorphism ${\rm sgn}\colon \K^\ast \to \F_\infty^\ast$ which is the identity on $\F_\infty^\ast\subset \K^\ast$. Elements $x$ with ${\rm sgn}(x)=1$ are called “positive.” Let $\pi\in \K^\ast$ be a fixed positive uniformizer and let $a\in\K$ be a positive element with a pole of order $e$ at $\infty$. We set $$\langle a \rangle=\langle a \rangle_\pi:=\pi^e a\,.$$ It is clear that $\langle a \rangle\in U_1$, where $U_1$ is the multiplicative group of $1$-units in $\K^\ast$, and that $\langle ab\rangle=\langle a\rangle \langle b \rangle$ for positive $a$ and $b$. Note that the binomial theorem makes $U_1$ into a $\Zp$-module.
Let $a$ be a positive element of $\A$. We set $\langle (a)\rangle:=\langle a \rangle$ thereby giving a homomorphism from the group of principal and positively generated $\A$-fractional ideals to $U_1$. Let $\hat{U}_1 \supset U_1$ be the group of all $1$-units in $\bfC_\infty$. As $\hat{U}_1$ may be seen to be a $\Qp$-vector space ($p$-th roots may be uniquely taken), a simple argument implies that $\langle ?\rangle$ extends uniquely to a homomorphism from the group of all fractional ideals to $\hat{U}_1$. Let $\K_{\mathbf V}\subset \bfC_\infty$ be the subfield obtained by adjoining $\langle I\rangle$ to $\K$ where $I$ ranges over all fractional ideals of $\A$. As $U_1$ is a $\Zp$-module and the class number of $\A$ is finite, we conclude that $\K_{\mathbf V}$ is a finite, totally inseparable, extension of $\K$.
Let $S_\infty:=\bfC^\ast_\infty\times
\Zp$ and let $s=(x,y)\in S_\infty$. For a non-zero ideal $I\subseteq \A$ we put $$I^s:=x^{\deg I}\langle I\rangle^y\,.$$ Let $\pi_\ast\in \bfC_\infty$ be a fixed $d_\infty$-th root of $\pi$ and let $j$ be an integer. It is then easy to see that $$(i)^{s_j}=i^j$$ for positive $i\in \A$ and $s_j:=(\pi_\ast^{-j},j)\in S_\infty$. We frequently write “$j$” for “$s_j$.”
By abuse of language, we also write $n\mathfrak P$ for $\theta^{-1}(n\mathfrak P)$ whenever no confusion will arise. Thus, finally, the $L$-series $L(\psi,s)$, $s\in S_\infty$, of $\psi$ over the field $L$ is defined by $$L_\infty(\psi,s)=L(\psi,s):=
\prod_{{\mathfrak P}~\rm good}P_{\mathfrak P}(\theta^{-1}(n{\mathfrak P })^{-s})^{-1}=
\prod_{{\mathfrak P}~\rm good}P_{\mathfrak P}(n{\mathfrak P }^{-s})^{-1}\,.$$ The local Riemann hypothesis assures us that these $L$-series converge on the “half-plane” of $S_\infty$ defined by $\{(x,y)\mid \vert x \vert_\infty
> t\}$, for some positive real $t$, to a ${\mathbf C}_\infty$-valued function. In a similar fashion one can construct $L$-series of pure $T$-modules etc.
Ultimately, as with classical theory, Euler factors at the finitely many bad primes ought to be added into the definition, see Remark \[whybadfactors\]. Such factors should be given by recent work of F. Gardeyn [@ga1]. In any case, unlike classical theory, there are also many examples where all finite primes are good.
Now let $L^{\rm sep} \subset C_\infty$ be the separable closure of $L$. Let $\Qpbar$ be a fixed algebraic closure of $\Qp$ and let $V$ be a finite dimensional $\Qpbar$-vector space. Let $\rho\colon \Gal (L^{\rm sep}/L)\to \Aut_{\Qpbar} (V)$ be a representation of Galois type (i.e., which factors through the Galois group of a finite Galois extension of $L$). As explained in §8 of [@go1], the classical definition of Artin $L$-series is easily modified to define a ${\mathbf C}_\infty$-valued $L$-series $L(\rho,s)$, $s\in S_\infty$, whose Euler product converges on the half-plane $\{(x,y)\mid \vert x\vert_\infty>1\}$.
We will refer to $L(\psi,s)$, for $\psi$ a Drinfeld module, as an $L$-series of “Drinfeld type,” and $L(\rho,s)$, for a Galois representation $\rho$, as an $L$-series of “Galois type.” Both types of $L$-series will be referred to as “$L$-series of arithmetic type” from now on.
Let $L(s)$, $s=(x,y)\in S_\infty$, be an $L$-series of arithmetic type. For fixed $y\in \Zp$, $L(x,y)$ is an power series in $x^{-1}$ with coefficients in a finite extension $\K_L(y)$ of $\K$. (To avoid any possibility of confusion we will always use a subscripted “$L$” to refer to a particular $L$-series and not a field $L$.) For instance, if $L(s)=L(\psi,s)$ is an $L$-series of Drinfeld type, then $\K_L(y)$ is a subfield of $\K_{\bf V}$. In fact, let $u$ be the order of the strict class group of $\A$ (which is the quotient of the group of all $\A$-fractional ideals modulo the subgroup of principal and positively generated ideals). Then if $y\in u\Zp$, one sees easily that $\K_L(y)=\K$. If $L(s)$ is an $L$-series of Galois type, then $\K_L(y)/\K$ may contain a finite extension of constant fields.
Let $\pi_\ast\in \bfC_\infty$ be a fixed $d_\infty$-th root of our fixed parameter $\pi$, and let $j$ be a non-negative integer. As is standard, we put $$z_L(x,-j):=L(x\pi^j_\ast,-j)\,.$$ The finiteness of the class number of $\A$ implies that $z_L(x,-j)$ is a power series whose coefficients lie in a finite extension of $\k$ and are integral over $\A$. For $L(\psi,s)$ of Drinfeld type, recent work [@bp1], [@b1] expresses these power series in terms of the cohomology of certain “crystals” and thus establishes that they are actually polynomials in $x^{-1}$. A similar statement in the case of an $L$-series of Galois type had previously been shown using elementary estimates and the classical theory of Weil.
\[specialpolys\] The polynomials $z_L(x,-j)$ are the [*special polynomials*]{} of $L(s)$.
The cohomological description of the special polynomials is critical in the analytic continuation of an $L$-series $L(\psi,s)$ of Drinfeld type to an essentially algebraic entire function on all of $S_\infty$ (as defined in §8.5 of [@go1]). Indeed, as in [@b1], one may use the cohomological description to give a [*logarithmic*]{} bound on the growth of the degrees (in $x^{-1}$) of $z_L(x,-j)$ as a function of $j$ from which the analytic continuation is readily deduced. For the Carlitz module $C$, whose $L$-series is easily seen to be $\zeta_{\Fr[\theta]}(s-1)$, this bound was originally shown by H. Lee [@le1] using elementary methods; see Th. VIII of [@th1] for a statement of related results. This bound also follows from the work of Diaz-Vargas and Sheats as in Section \[basic\]. In fact, the logarithmic bound on the degrees of the special polynomials for any $L$-series of Galois type may be established by elementary, non-cohomological means.
Let $L(\rho,s)$ be an $L$-series of Galois type. In Sections 8.12 and 8.17 of [@go1] there is a “double congruence” relating $z_L(x,-j)$ to the incomplete characteristic $0$-valued $L$-function $\hat{L}(\rho\otimes \omega^{-j},t)$ of $\rho$ twisted by powers of the Teichmüller character; this incomplete $L$-series is defined by the usual Euler product but taken only over the finite primes. By Weil’s Theorem (= the Artin Conjecture in this context), this incomplete $L$-series is a polynomial in $t=r^{-s}$ which is divisible by the finite Euler product taken only over the infinite primes. By using such double congruences infinitely often, one deduces that $z_L(x,-j)$, and thus $L(\rho,(x,-j))$, have a number of canonical zeroes.
\[trivialzero\] The zeroes just described are the called the trivial zeroes of $L(\rho,s)$, and $z_L(x-j)$, at $y=-j$.
We will use the expression “trivial zeroes” to refer to the union of the trivial zeroes at $-j$ for all $j$. It is very easy to see that the trivial zeroes belong to a finite extension of $\K$.
Note that as $L(\rho, (x,-j))$ is a polynomial in $x^{-1}$, there are obviously only finitely many trivial zeroes for each $j$. As $\rho$ is of Galois type, the trivial zeroes for $z_L(x,-j)$ are easily seen to be in the algebraic closure of $\Fr\subset \bfC_\infty$ and so have $\infty$-adic absolute value $1$.
\[trivialzeta\] In Example 3 of [@go2] we discussed the basic example $\zeta_A(s)$, $s\in S_\infty$, of the zeta-function of $A=\Fr[\theta]$. Note obviously that $d_\infty=1$ for such an $A$ and a “positive” polynomial is just a monic polynomial. Let $s=(x,y)$; one finds easily that $$\label{zetaform}
\zeta_A(s)=\sum_{e=0}^\infty x^{-e}\left(\sum_{\substack{n{\rm ~monic}\\
\deg(n)=e}}\langle n\rangle^{-y}\right)\,.$$ For a non-negative integer $j$ one then has $$\label{zetaformneg}
z_{\zeta_A}(x,-j)=\sum_{e=0}^\infty x^{-e}\left( \sum_{\substack{n{\rm
~ monic}\\\deg(n)=e}} n^j\right)\,.$$ For $e\gg 0$, the sum in parentheses vanishes (in fact, Lee shows that one can choose $e> l_r(j)/(r-1)$ where $l_r(j)$ is the sum of the $r$-adic digits of $j$). When $j$ is positive and divisible by $r-1$, then $z_{\zeta_A}(x,-j)$ has a simple zero at $x=1$. Thus $\zeta_A(s)$ has a simple trivial zero at $s_{-j}=(\pi^j,-j)$, just as the Riemann zeta function has a simple trivial zero at negative even integers. A very similar story happens for general $L(\rho,s)$ of Galois type.
For $r$ and small positive $j$ not divisible by $r-1$, computer calculations have shown that $z_{\zeta_A}(x,-j)$ is a polynomial in $x^{-1}$ which is irreducible over $\Fr(T)$ and has associated Galois group equal to the full symmetric group. If $j\equiv 0\pmod{r-1}$, a similar statement is true computationally once $1-x^{-1}$ is factored out.
For an $L$-series $L(\psi,s)$ of Drinfeld type, as well as other more general $L$-functions, one should find trivial zeroes in a very similar fashion. For simplicity let $\A=\Fr[T]$. Then, for a Drinfeld module (and, more generally, for pure, uniformizable $T$-modules), the factors at the infinite primes (whose zeroes are then the trivial zeroes) should conjecturally arise in a fashion completely analogous to the one used above for $L(\rho,s)$. Indeed, as mentioned before, the theory of Boeckle and Pink [@bp1] computes $$Z_L(x,-j)=\prod_{\mathfrak P~{\rm good}}P_{\mathfrak P}(n{\mathfrak P}^jx^{-\deg_{\Fr}
\mathfrak P})^{-1}$$ via the cohomology of the crystal associated to the $\A$-module $\psi\otimes C^{\otimes j}$ (where $C$ is the Carlitz module) through an associated trace formula. Each local factor $P_{\mathfrak P}(n{\mathfrak P}^j u)$ at a good prime $\mathfrak P$ may be computed via the canonical Galois action on the Tate module $T_v(\psi \otimes C^{\otimes j})$.
Suppose that $\psi$ is defined over a finite extension $L$ of $k$. Let $\sigma \colon L \to C_\infty$ be a $k$-embedding and let $L_\sigma$ be the completion of $L$ under the induced absolute value. Let $\psi^\sigma\otimes C^{\otimes j}$ be the $T$-module defined over $L_\sigma$ obtained by applying $\sigma$ to the coefficients of $\psi\otimes C^{\otimes j}$ (note that $\sigma$ acts as the identity on the coefficients of $C$ and its tensor powers). Via a fundamental result of Anderson [@a1], the module $\psi^\sigma\otimes C^{\otimes j}$ is [*uniformizable*]{} and arises from a lattice $M_{\sigma,j}$. The action of the decomposition group at the infinite place defined by $\sigma$ can then be computed via the Galois action on this lattice. Therefore it must factor through a finite Galois extension precisely because the lattice generates a finite extension of $L_\sigma$. Consequently the associated characteristic polynomial of Frobenius, which should conjecturally impart trivial zeroes to $L(\psi,s)$, will have constant (i.e., in the algebraic closure of $\Fr\subset \bfC_\infty$) coefficients just as it does for $L(\rho,s)$. The product of such characteristic polynomials over all infinite primes should then give all the trivial zeroes at $y=-j$. A very similar description is expected for general $\A$.
\[carlitzmodule\] As mentioned above, the $L$-series $L(C,s)$ of the Carlitz module $C$ over $\Fr(\theta)$ is $\zeta_{\Fr[\theta]}(s-1)$. From Example \[trivialzeta\] we see then that $L(C,s)$ has a trivial zero at $-j+1$ where $j$ runs over the positive integers divisible by $r-1$. This is what is predicted by the above prescription.
Let $v$ be a closed point in ${\rm Spec}(\A)$ and let $L(s)$ be an $L$-series of arithmetic type. Then the logarithmic growth of the degrees of the special polynomials also allows one to establish the $v$-adic interpolation $L_v(x,y)$ of the $L$-functions given above. The function $L_v(x,y)$ is naturally defined on the space $\bfC_v^\ast\times S_v$, where $\bfC_v$ is the completion of an algebraic closure of the local field $\k_v$ and $S_v$ is the completion of $\Z$ with respect to a certain topology (one sees easily that $S_v$ is isomorphic to the product of a finite cyclic group $H_v$ with $\Zp$; see §8.3 of [@go1]). We will continue to use $x$ for the first variable (now in $\bfC_v^\ast$) and $y$ for the second variable (now in $S_v$); thus, in this case, our notation here differs slightly from that of [@go2]. These functions have analytic properties completely similar to those possessed by the original functions on $S_\infty$; so we again have a $1$-parameter family of entire power series in $x^{-1}$ with very strong continuity properties in the variable $y$.
Notice that the zeroes of all these entire functions are algebraic over the base completion of $\k$ (i.e., $\K$ or $\k_v$) by standard non-Archimedean function theory.
\[vadiczetafunction\] Let $\A=\Fr[T]$ and let $v$ be a prime of degree $d$. The $v$-adic interpolation of $\zeta_A(s)$ will be denoted $\zeta_{A,v}(x,y)$. One has $$\label{vadiczetasum}
\zeta_{A,v}(x,y)=\sum_{e=0}^\infty x^{-e}
\left(\sum_{\substack{n {\rm ~monic}\\ \deg(n)=e\\ (n,v)=1 }}n^y \right)\,,$$ where $x\in \bfC_v^\ast$ and $y\in S_v$.
\[nonimpact\] Let $\rho:\Gal (L^{\rm sep}/L)\to \Aut_{\Qpbar}(V)$ be a representation of Galois type, as above, with $L$-series $L(\rho,s)$. Let $j$ be a non-negative integer. It is important to note that as the trivial zeroes of $z_L(x,-j)$ are constants, they also have $v$-adic absolute value $1$; thus their effect $v$-adically is very limited. Under the above conjectures on the Galois modules associated to Drinfeld modules, etc., a similar remark should ultimately hold in complete generality.
Notice that the very act of interpolating $L(\rho,s)$ $v$-adically also removes the Euler factors at the primes above $v$ in $z_L(x,-j)$. In other words, let $$z_L(v;x,-j):=
z_L(x,-j)\hat{z}_L(v;x,-j)\,$$ where $$\hat{z}_L(v;x,-j):=
\prod_{\substack{{\mathfrak P} \mid v\\{\mathfrak P}~ {\rm good}}}P_{\mathfrak P}(n{\mathfrak P}^{-(x\pi_\ast^j,-j)})\,.$$ Then $z_L(v;x,-j)=L_v(x,-j)$ where $-j\in S_v$ (and is obviously also a polynomial in $x^{-1}$).
\[vadictrivial\] The zeroes of $\hat{z}_L(v;x,-j)$ are the [*$v$-adic trivial zeroes*]{} of $L(\rho,s)$ at $-j$.
The impact of Remark \[nonimpact\] is precisely that we can ignore $v$-adically the $\infty$-adic trivial zeroes (i.e., the trivial zeroes of $z_L(x,-j)$) and work as above. As usual, the union over all $j$ of these zeroes is the set of all $v$-adic trivial zeroes. They lie in a finite extension of $\k_v$.
For an $L$-series $L(\psi,s)$ of Drinfeld type, the $v$-adic trivial zeroes are given in exactly the same way. The main difference is that the existence of the $\infty$-adic trivial-zeroes and their $v$-adic influence is conjectural for such $L$-series at this moment.
\[vadiczeta\] We continue examining the basic case of Example \[trivialzeta\]. Let $v$ be a finite prime of degree $d$ in $\Fr[T]$ associated to a monic irreducible $f(T)$ and let $j$ be a non-negative integer. Then the $v$-adic trivial zeroes of $\zeta_A(s)$ are the elements $x\in \bfC_v$ with $0=1-f^jx^{-d}$; these are considered with the obvious multiplicities when $d$ is divisible by $p$.
Note the remarkable similarity between the $\infty$-adic and $v$-adic trivial zeroes. For instance, let $\zeta_A(s)$ be as in Examples \[trivialzeta\] and \[vadiczeta\], and let $v=(f)$ be a prime of degree $1$. Then the $\infty$-adic trivial zeroes occur at $(\pi^j,-j)\in S_\infty$ for $j>0$ and divisible by $(r-1)$, while the $v$-adic trivial zeroes occur at $(f^j,-j)\in \bfC_v^\ast \times S_v$ for non-negative $j$. Obviously both $\pi$ and $f$ are parameters in their respective local fields.
We finish this section by using the above ideas to factor the special polynomials. We begin at $\infty$ and let $L(s)$, $s\in S_\infty$, be an $L$-series of arithmetic type. Let $j$ be a non-negative integer. Then, as we have seen, $L(x,-j)$ is a polynomial in $x^{-1}$ and, conjecturally (in the case $L=L(\psi,s)$), there is a polynomial factorization $$\label{inftyfactorization}
L(x,-j)=L_{\rm triv}(x,-j)L_{\rm nontriv}(x,-j)\,,$$ where $L_{\rm triv}(x,-j)$ is the product of the factors arising from the infinite primes and whose zeroes are the trivial zeroes at $-j$. It is important to note that these polynomials may be trivial (i.e., the constant polynomial $1$). The zeroes of $L_{\rm nontriv}(x,-j)$ are referred to as the “non-trivial zeroes at $-j$.” Now let $v$ be a finite prime and view $-j$ as lying in $S_v$. Let $$L_{v,\rm triv}(x,-j)=\prod_{\substack{{\mathfrak P}\mid v\\
{\mathfrak P}~ {\rm good}}}P_{\mathfrak P}(n{\mathfrak P}^jx^{-\deg n{\mathfrak P}})\,,$$ and “rename” $z_L(x,-j)$ as $L_{v,{\rm nontriv}}(x,-j)$. Then by the $v$-adic construction we have the factorization $$\label{vadicfactorization}
L_v(x,-j)=L_{v,{\rm triv}}(x,-j)L_{v, \rm nontriv}(x,-j)\,.$$ The zeroes of $L_{v, \rm nontriv}(x-j)$ are then called the “$v$-adic non-trivial zeroes at $-j$,” etc. Again, it is possible that these polynomials will be identically $1$. Both the factorization at $\infty$ and at finite primes can be put in exactly the same form by setting $L_\infty(x,-j):=L(x,-j)$, $L_{\infty, \rm triv}(x,-j):=L_{\rm triv}(x,-j)$, etc.
In Section \[counter\] we will see that there is, conjecturally, a further decomposition of these polynomials.
\[whybadfactors\] The reader may well wonder why, besides the obvious classical analogies, one would want to have Euler factors at the finitely many bad primes in the definition of an arithmetic $L$-series $L(s)$. However, we have seen how removing Euler factors adds zeroes to an $L$-series. These zeroes might then unnecessarily enlarge the splitting field associated to $L(s)$ and $y$ (i.e., the algebraic extension $\K_L^{\rm tot}(y)$ of $\K_L(y)$ obtained by adjoining the zeroes of $L(x,y)$). So, from the viewpoint of splitting fields at least, having such local factors is quite desirable.
Krasner’s Lemma {#krasner}
===============
In this section we recall Krasner’s Lemma and put it in a form which is particularly useful in characteristic $p$.
Let $\mathcal K$ be an arbitrary field which is complete under a general (not necessarily discrete) non-trivial non-Archimedean absolute value $\vert?\vert$. The characteristic of $\mathcal K$ may also be completely arbitrary. Let $\overline{\mathcal K}$ be a fixed algebraic closure of $\mathcal K$ equipped with the canonical extension of $\vert?\vert$. Let ${\mathcal F}$ be a subfield of $\overline{\mathcal K}$ with maximal separable (over $\mathcal K$) subfield ${\mathcal F}_s$. In particular, $\overline{\mathcal K}_s=\mathcal{K}^{\rm sep}$= the separable closure of $\mathcal K$ in $\overline{\mathcal K}$.
Let $\alpha\in \overline{\mathcal K}$.
\[distance\] If $\alpha$ is not totally inseparable over $\mathcal K$ then we set $$\delta(\alpha)=\delta_{\mathcal K}(\alpha):=\min_{\sigma\neq id}\{\vert
\sigma (\alpha)-\alpha\vert\}\,,$$ where $\sigma$ runs over the non-identity $\mathcal K$-injections of $\mathcal{K}(\alpha)$ into $\overline{\mathcal K}$. If $\alpha$ is purely inseparable over $\mathcal K$, then we set $\delta(\alpha)=0$.
Notice that if ${\rm char}({\mathcal K})=p>0$ then $$\delta(\alpha^{p^i})=\delta(\alpha)^{p^i}$$ for $i\geq 0$.
Now let $\beta$ be another element in $\overline{\mathcal K}$. Krasner’s Lemma is then stated as follows.
\[Krasnerlemma\] Suppose that $\alpha$ is separable over ${\mathcal K}(\beta)$ and that $\vert \beta-\alpha\vert< \delta (\alpha)$. Then ${\mathcal K}(\alpha)\subseteq {\mathcal K}(\beta)$.
By the separability assumption, the result follows if one knows that there are no non-trivial embeddings of $\mathcal{K}(\alpha,\beta)$ over $\mathcal{K}(\beta)$. But if $\tau$ is any such injection then one has $$\vert \tau(\alpha)-\alpha\vert =\vert (\tau(\alpha)-\beta)+(\beta -\alpha)\vert \leq \vert \beta-\alpha\vert<\delta(\alpha)$$ as $\vert \tau(\alpha)-\beta\vert=\vert \tau(\alpha-\beta)\vert=\vert \beta-\alpha\vert$. Thus $\tau=id$.
\[pthpowers\] Let $\alpha$ be any element in $\overline{\mathcal K}$ and suppose that $\vert \beta-\alpha\vert<\delta(\alpha)$. Then ${\mathcal K}(\alpha)_s\subseteq {\mathcal K}(\beta)_s
\subseteq {\mathcal K} (\beta)$.
Suppose that $\mathcal K$ has characteristic $p>0$. Now, for some $i\geq 0$ one knows that $\alpha^{p^i}$ is separable over $\mathcal K$. As $$\vert \beta^{p^i}-\alpha^{p^i}\vert=\vert\beta -\alpha\vert^{p^i}<
\delta(\alpha)^{p^i}=\delta(\alpha^{p^i})\,,$$ the result follows from the proposition.
\[betaandalpha\] Suppose that $\vert \beta-\alpha\vert <\delta (\alpha)$. Then $\delta(\beta)\leq \delta (\alpha)$ with equality if and only if ${\mathcal K}(\beta)_s={\mathcal K}(\alpha)_s$.
Let $\sigma$ be an injection of ${\mathcal K}(\beta)$ into $\overline{\mathcal K}$ over $\mathcal K$. Then $$\beta-\sigma (\beta)=( \beta - \alpha) +(\alpha -\sigma (\alpha))+
(\sigma (\alpha)-\sigma (\beta))\,.$$ The first and third terms on the right have the same absolute value. Moreover, by assumption, if the second term is non-zero then its absolute value is the greatest of the three; thus it is also the absolute value of $\beta-\sigma(\beta)$. The result now follows.
Let $\mathcal K$ have characteristic $3$ and let $\lambda\in \mathcal K$ with $\vert \lambda\vert >1$. Using $\alpha:=\lambda^{1/2}$ and $\beta:=\alpha+\lambda^{1/{3^i}}$, for some $i>0$, one sees that the above corollary cannot be strengthened to an equality between ${\mathcal K}(\alpha)$ and ${\mathcal K}(\beta)$ in general.
Finally, the reader may trivially establish an Archimedean analogue of Krasner’s Lemma upon defining $\delta (\alpha):=\vert \alpha-\bar{\alpha}
\vert/2$ for a complex number $\alpha$.
Review of some conjectures from [@go2] {#review}
======================================
Since they are used so often in this paper, we recall Conjectures 4 and 5. Let $L(s)$, $s=(x,y)\in S_\infty$, be an $L$-function of arithmetic type. We write $$L(x,y)=\sum_{e=0}^\infty a_e(y)x^{-e}\,.$$ For each $y\in \Zp$, this power series has coefficients in the finite extension $\K_L(y)$ of $\K$. As in Remark \[whybadfactors\], we let $\K_L^{\rm tot}(y)$ be the extension of $\K_L(y)$ obtained by adjoining the zeroes of $L(x,y)$; we let $\K_{L,s}^{\rm tot}(y)$ be its maximal separable (over $\K_L(y)$) subfield.
The essential part of an algebraic extension of function fields in $1$-variable over a finite field, whether local or global, is the maximal separable subfield. Indeed, well-known arguments show that totally-inseparable extensions are defined uniquely by their degree (see Corollary 8.2.13 of [@go1]).
[**Conjecture 4 of [@go2]**]{}. The field $\K_{L,s}^{\rm tot}(y)$ is a finite extension of $\K$.
The obvious $v$-adic analogue of the above conjecture is also postulated in [@go2]
Viewed as power series in $x^{-1}$ for fixed $y$, $L(x,y)$ has an associated Newton polygon in ${\mathbb R}^2$. (To distinguish between the characteristic $p$ theory, we use $X$ and $Y$ for the coordinates of ${\mathbb R}^2$.)
In $\bfC_\infty$ we may write $$L(x,y)=\prod_i (1-\beta_i^{(y)}/x)\,.$$ Obviously only non-zero $\beta_i^{(y)}$ are of interest, in which case we set $\lambda_i^{(y)}:=1/\beta_i^{(y)}$. The valuation (using $\nu_\infty$) of $\lambda_i^{(y)}$, and so $\beta_i^{(y)}$, is computed by the Newton polygon of $L(x,y)$. Standard theory shows that the $\beta_i^{(y)}$ tend to $0$ as $i$ tends to $\infty$, whereas the $\lambda_i^{(y)}$ also tend to $\infty$; in fact, with a little thought one sees that this can be made uniform with respect to $y$. We call the $\beta_i^{(y)}$ (resp. $\lambda_i^{(y)}$) the “zeroes in $x$” of $L(s)$ (resp. “zeroes in $x^{-1}$”). An advantage of using $\beta_i^{(y)}$ as opposed to $\lambda_i^{(y)}$ is that the slope of a side of the Newton polygon equals the valuation of the corresponding element $\beta_i^{(y)}$; for $\lambda_i^{(y)}$ one needs to multiply by $-1$.
[**Conjecture 5 of [@go2]**]{}. There exists a positive real number $b=b(y)$ such that if $\delta\geq b$, then there exists at most one zero in $x^{-1}$ of $L(x,y)$ of absolute value $\delta$.
In other words, outside of finitely many anomalous cases, zeroes are uniquely determined by their absolute values. The conjecture is also formulated $v$-adically.
Conjecture 5 is based on the examples of Wan, Sheats, etc., and appears to play a role similar to the classical Generalized Riemann Hypothesis. Indeed in [@go2] we showed how it leads to a variant of the classical Generalized Riemann Hypothesis for number fields. It implies Conjecture 4 simply because one can then easily show that almost all zeroes of $L(s)$ are totally inseparable over $\K_L(y)$. To show that $\K^{\rm tot}_L(y)$ is itself a finite extension of $\K_L(y)$ (and so of $\K$), one factors $L(x,y)$ into the $L$-series of “simple motives” and then applies the Generalized Simplicity Conjecture (Conjecture 7 of [@go2]).
Our next section explains how to use the trivial zeroes to find counter-examples to Conjecture 5. We also suggest a reasonable modification of Conjecture 5.
The counter-examples {#counter}
====================
Let $L(s)$, $s=(x,y)\in S_\infty$, be an arithmetic $L$-series which we continue to write as $\sum_{e=0}^\infty a_e(y)x^{-e}$. Let $n$ be a positive integer and let $y_0\in \Zp$. Suppose that the first $n$ slopes of the Newton polygon of $L(x,y_0)$ (as a function of $x^{-1}$) are finite.
\[newtonclose\] There is an non-trivial open neighborhood $U(y_0,n)$ of $y_0$ such that if $y\in U(y_0,n)$, then the first $n$ segments of the Newton polygon in $x^{-1}$ of $L(x,y)$ are the same as those for $L(x,y_0)$.
The functions $a_e(y)$ are continuous. Moreover, let $\nu_\infty$ be the additive valuation associated to $\infty$. Then, from [@b1], one also has exponential lower bounds on $\nu_\infty(a_e(y))$ which are independent of $y$. The result follows directly.
A completely analogous $v$-adic result follows in the same way.
\[equivrelations\] a. We can use the Newton polygons of $L(x,y)$ to define equivalence relations on $\Zp$ (or its $v$-adic analogue $S_v=\Zp\times H_v$ where $H_v$ is a finite abelian group etc.) in the following fashion. Let $n$ be a fixed positive integer. Let $y_i\in \Zp$, $i=1,2$, be such that the Newton polygon of $L(x,y_i)$ has $n$ finite slopes for each $i$. We then say that $y_1 \sim_n y_2$ if and only if the Newton polygons of both $L(x,y_1)$ and $L(x,y_2)$ have the same first $n$ segments. If $y\in \Zp$ does not have $n$ finite slopes, then, by definition, $y$ will only be equivalent to itself. It is clear that $\sim_n$ is an equivalence relation which only depends on $L(s)$ and $n$.\
b. The impact of Lemma \[newtonclose\] is precisely that an equivalence class of $\sim_n$ consisting of more than one element is then open in $\Zp$.\
c. Let $y\in \Zp$ belong to an open equivalence class $E_y$ under $\sim_n$ and let $m$ be the least non-negative integer such that $U:=y+p^m\Zp\subseteq E_y$. Thus, on $U$, the first $n$-segments of the Newton polygon are an invariant of the maps $z\mapsto z+\beta$ where $\beta\in p^m\Zp$. We believe that such statements may be viewed as possible “micro-functional-equations” for (the Newton polygon of) $L(x,y)$. See Section \[basic\] for an example worked out in detail.
It seems reasonable that the family of Newton polygons associated to $L$-series actually determine the $L$-series itself. We state this more succinctly in the following question.
\[langlands\] Let $A=\Fr[T]$ and let $\phi_1$ and $\phi_2$ be two non-isogenous Drinfeld modules over $\Fr(\theta)$ of the same degree. Does the family of Newton polygons serve to distinguish between $L(\phi_i,s)$ ($s\in S_\infty$) for $i=1,2$?
Obviously, there are many variants of Question \[langlands\] that may also be formulated.
We can now construct the counter-examples.
\[counter1\] Let $\A$ be arbitrary but where $d_\infty>1$. If $j$ is a positive integer divisible by $r^{d_\infty}-1$ then $\zeta_A(s)$ has trivial zeroes at $(\zeta\pi_\ast^j,-j)$, where $\zeta$ runs over the $d_\infty$-th roots of $1$ with multiplicity. Thus there is a segment of the Newton polygon of $\zeta_A(x,-j)$ (in $x^{-1}$) which has slope $j/d_\infty$ and whose projection to the $X$-axis has length $\geq d_\infty$. Lemma \[newtonclose\] now assures us that all $y$ sufficiently close to $-j$ will possess this property. We now construct a counterexample to Conjecture 5 inductively. Let $y_0=r^{d_\infty}-1$. Let $y_1=y_0+p^{t_1}(r^{d_\infty}-1)$ where $t_1$ is a non-negative integer chosen (in accordance with Lemma \[newtonclose\]) so that the first $n$ segments of the Newton polygons at $-y_0$ and $-y_1$ are the same and where these segments include the one associated to the trivial zeroes at $-y_0$. Now construct $y_2$ in the same fashion but where we choose $t_2$ to also be greater than $t_1$ etc. The sequence $\{y_i\}$ clearly converges to a $p$-adic integer $\hat y$. The Newton polygon in $x^{-1}$ of $\zeta_A(x,-\hat{y})$ will have infinitely many segments whose projection to the $X$-axis will have lengths $\geq d_\infty$. There are then two cases to discuss:\
1. $d_\infty$ is a pure power of $p$. In this case we cannot directly conclude that there are infinitely many zeroes of $\zeta_A(x,-\hat{y})$ which are not uniquely determined by their absolute value simply because we do not know a-priori that the zeroes of $\zeta_A(x,-\hat{y})$ are not totally inseparable. However, if one also assumes the Generalized Simplicity Conjecture (Conjecture 7 of [@go2]), then almost all such zeroes cannot be totally inseparable and so Conjecture 5 must now be false.\
2. $d_\infty$ is not a pure $p$-th power. In this case, there are at least two distinct $d_\infty$-roots of unity. One can then choose the $t_i$ sufficiently large so that the distinct trivial zeroes separate the nearby zeroes. In this case, one obtains a counter-example unconditionally.
One can often use Krasner’s Lemma to obtain similar constructions as in the following example.
\[counter2\] Let $\A=\Fr[T]$. Let $f$ be a prime of degree $d>1$ with associated place $v$ and assume that $d$ is not a pure $p$-th power. Then the $v$-adic trivial zeroes of $\zeta_A(s)$ at $-j$ are the roots of $1-f^jx^{-d}$. Let $j\not \equiv 0\pmod{d}$ and let $\alpha$ be one such root. Then $\alpha\not\in \k_v$. Moreover, it is easy to see that, in the notation of Section \[krasner\], we have $$\delta_{\k_v}(\alpha)=\vert \alpha\vert_v\, .$$ Let $y\in S_v$ be sufficiently close to $-j$ so that $\zeta_{A,v}(x,y)$ has a zero $\beta$ with $\vert \beta-\alpha\vert_v<\vert \alpha\vert_v$. By Corollary \[pthpowers\], the separable degree of $\k_v(\beta)$ is greater than $1$. As such this $\beta$ possesses a non-trivial Galois conjugate $\beta^\prime$ which is also a zero of $\zeta_{A,v}(x,y)$ of the same absolute value. One can now proceed as in Example \[counter1\] to obtain a counter-example to the $v$-adic version of Conjecture 5.
In the above example, it is easy to see that all $v$-adic trivial zeroes belong to a finite extension of $\k_v$. Thus, Krasner’s Lemma does not allow us to deduce a counter-example to Conjecture 4.
Conjecture 5 may still remain valid in its original form in the much more limited case where there exists only one (including multiplicity!) trivial zero of a given absolute value. Indeed, the techniques used in the above counter-examples do not work in his case.
Ideally, one would like to negate the effects of the trivial zeroes which permit the above counter-examples. Classically one removes the effects of the trivial zeroes through the use of the Gamma-factors (as in the introduction), which are the Euler factors arising from the infinite primes, and the functional equation. Indeed, the functional equation assures us that the trivial zeroes are quite far from the critical zeroes (= all “non-trivial” zeroes).
In the characteristic $p$ case that we are studying, it has been known for a long time that the Gamma-functions do not seem to be related to the trivial zeroes of $L$-series. A philosophical explanation for this phenomenon comes from the “two $T$’s” approach; indeed, $L$-series have values in the field $\bfC_\infty$ whereas Gamma-functions, as they are related to exponential functions and their periods, must take values in $C_\infty$.
In any case, we need to find other methods in the characteristic $p$ theory. We now sketch an approach to removing the “harmful” effects of trivial zeroes based on Hensel’s Lemma. The idea is simply to isolate those zeroes which are influenced by the trivial zeroes so that they can be removed from the conjectures and handled separately. Whether the definition of the $L$-series should be altered, as in the classical case, to physically remove these zeroes is unknown.
In order to isolate those zeroes which are sufficiently close to trivial zeroes, an affirmative answer to the following question about trivial zeroes would give the nicest situation. This question seems reasonable in view of examples and ramification considerations. So let $w$ be a place of $\k$ (either $\infty$ or a finite place) and consider the $w$-adic interpolation of an $L$-series $L(s)$ of arithmetic type. Recall that, from Equations \[inftyfactorization\] and \[vadicfactorization\], we have a factorization $$L_w(x,-j)=L_{w,\rm triv}(x-j)L_{w,\rm nontriv}(x,-j)\,.$$ Let $e>0$ be a real number. Then, standard non-Archimedean analysis leads to a rational factorization $$\label{absolutevaluefactorization}
L_w(e;x,-j)=L_{w,\rm triv}(e;x,-j)L_{w,\rm nontriv}(e;x,-j)$$ where $L_w(e;x,-j)$ is the product of $1-\beta/x$ where $\beta$ runs through all zeroes in $x$ of $L_w(x,-j)$ with $\nu_w(\beta)=e$; etc. (Recall that the zeroes in $x$ of $L_w(x,y)$ uniformly tend to $0$ so that their valuations tend to $\infty$.)
\[trivialabsolutevalue\] Let $j$ be a non-negative integer. Does there exists a constant $C>0$ (depending only on $L(s)$ and $w$) so that for $e>C$ the polynomials $L_{w,\rm triv}(e;x-j)$ and $L_{w, \rm nontriv}(e;x,-j)$ are relatively prime polynomials in $x^{-1}$?
Let us assume that the above question may be answered in the affirmative and let $e$ be as in its statement. As $L_{w, \rm triv}(e;x,-j)$ and $L_{w,\rm nontriv}(e;x,-j)$ are relatively prime, Hensel’s Lemma now applies to polynomials which are close to $L_w(e; x,-j)$ (see, eg., Thm. 4.1 of [@dgs1]).
Now let $y$ be chosen sufficiently close to (but [*not*]{} equal to) $-j$ so that the first $m$ segments of the Newton polygons are the same, where $m$ is large enough so that the segment associated to $e$ is among the first $m$ chosen. It is reasonable to assume that $L_w(e;x,y)$, with the obvious definition, is also then close enough to $L_w(e;x,-j)$ for Hensel’s Lemma to apply. Thus, under these assumptions, $L_w(e;x,y)$ will inherit a rational factorization $$\label{yadictrivcrit}
L_w(e;x,y)=L_{w,\rm triv}(e;x,y)L_{w, \rm nontriv}(e;x,y)\,.$$ In other words, outside of finitely many exceptional $e$, we would then be able to isolate those zeroes of $L_w(x,y)$ which are influenced by the trivial zeroes.
\[henselstill\] Even if Question \[trivialabsolutevalue\] is answered in the negative, one can still proceed as follows. Let $d_w(e;x,-j)=1+\cdots$ be the greatest common divisor of $L_{w,\rm triv}(e;x,-j)$ and $L_{w,\rm nontriv}(e;x,-j)$ as polynomials in $x^{-1}$. We can then apply Hensel’s Lemma to the relatively prime factorization $$\label{henselalso}
L_w(e;x,-j)=L_{w,\rm triv}(e;x,-j)d_w(e;x,-j)\times
L_{w,\rm nontriv}(e;x,-j)/d_w(e;x,-j)\,.$$ By construction, the trivial zeroes are only associated to the factor on the left in Equation \[henselalso\].
The zeroes of $L_{w,\rm triv}(e;x,y)$ are called the “near-trivial zeroes associated to $e$ and $y$,” etc. They are precisely the zeroes which are influenced by the original trivial zeroes. (N.B.: If $y$ is actually a negative integer itself, there is nothing a-priori to rule out having a near-trivial zero also being an actual trivial zero for $y$.) The rest of the zeroes are called the “critical zeroes” (in analogy with classical theory) and these are the ones Conjecture 5 may indeed apply to.
It remains to deal with Conjecture 4. Assuming that Conjecture 5 is established somehow for critical zeroes, the only issue that remains is to somehow establish that the field generated by all the near-trivial zeroes for a given $y$ is also finite over $\k_w$. However, the degree of $L_{w,\rm triv}(e;x,y)$ is bounded (the example of $L(\rho,s)$ will suffice to convince the reader that this is so). Thus it would suffice to bound the discriminants of the maximal separable subfield of the splitting field of $L_{w,\rm triv}(e;x,y)$ (as there are only finitely many separable extensions of a local function field of bounded degree and discriminant, see Prop. 8.23.2 of [@go1]). Needless to say, such a problem never comes up in classical theory. However, the following examples give some evidence in favor of such bounds in the characteristic $p$ theory.
\[f5\] Let $\A=\F_3[T]$ and let $v$ correspond to a monic prime $f$ of degree $2$. Let $z_{\zeta_A}(x,-j)$ be as in Equation \[zetaformneg\]; one computes easily that $z_{\zeta_A}(x,-5)=1+(T-T^3)x^{-1}$. Thus $$z_{\zeta_A}(v;x,-5)=\zeta_{A,v}(x,-5)=(1-f^5x^{-2})(1+(T-T^3)x^{-1})\,.$$ The first factor gives the trivial zeroes and the second gives the non-trivial zeroes. Clearly these two factors are relatively prime. Thus for $y\in S_v$ sufficiently close to $-5$, Hensel’s Lemma may be used. Note also that $f^{5/2}$ is obviously a separably algebraic element. Thus, if $y$ is also close enough to $-5$ so that Krasner’s Lemma applies, then we find that the near-trivial zeroes at $y$ associated to $5/2$ generate $\k_v(\sqrt{f})$. Indeed, if $\beta$ is a near-trivial zero associated to $f^{5/2}$ then $\beta$ will also satisfy a quadratic equation over $\k_v$ (and so we deduce equality of fields as opposed to merely inclusion as in Lemma \[Krasnerlemma\]). Thus Corollary \[betaandalpha\] implies that $\delta_{\k_v}(\beta)=
\delta_{\k_v}(f^{5/2})=\vert f^{5/2}\vert_v$.
\[f4\] We continue with the set-up of Example \[f5\]. One has $z_{\zeta_A}(x,-4)=(1-x^{-1})$. Thus $$z_{\zeta_A}(v;x,-4)=\zeta_{A,v}(x,-4)=(1-f^4x^{-2})(1-x^{-1})\,.$$ In this case, Hensel’s Lemma implies that near-trivial zeroes associated to $\pm f^2$ are in $\k_v$.
Simple considerations of Newton polygons imply that both $(T^3-T,-5)$ and $(1,-4)$ are critical zeroes for $\zeta_{A,v}(x,y)$.
The analytic behavior of $\zeta_{\Fp[\theta]}(s)$, $s\in S_\infty$ {#basic}
==================================================================
We will use the techniques and results of Diaz-Vargas [@dv1] (see also §8.24 of [@go1]) and Sheats [@sh1] to describe the influence of the trivial zeroes for $\zeta_{A}(s)$, $A=\Fr[\theta]$ and $s\in S_\infty$. We will see that, contrary to what we first expected, all zeroes of $\zeta_{\Fp[\theta]}(s)$ are near-trivial. In fact, examples lead us to expect this to hold for all $r$; the proof will take a detailed analysis of Sheats’ method which we hope to return to in later works.
Our first result along these lines concerns the valuation at $\infty$ of the zeroes of $\zeta_A(s)$. Let $s=(x,y)\in S_\infty$ and write $$\label{zetasumform}
\zeta_A(s)=\sum_{i=0}^\infty a_i(y)x^{-i}\,.$$ As before, let $\nu_\infty$ be the normalized valuation at $\infty$ with $\nu_\infty(1/T)=1$.
\[missed\] We have $\nu_\infty(a_i(y))\equiv 0\pmod{r-1}$ for all $i$ and $y$.
Let $j$ be a non-negative integer and (in the notation of [@sh1]) $$S_k^\prime(j)=\sum_{\substack{n\in \Fr[T]\\n~{\rm monic}\\\deg(n)=k}}
n^j\,.$$ The main result in [@sh1] is to establish a formula originated by Carlitz for $\deg(S_k^\prime(j))$ (this formula is then used to compute $\nu_\infty(a_i(y))$ and the Newton polygon of $\zeta_A(x,y)$). The formula expresses $\deg(S_k^\prime(j))$ in terms of a certain $k+1$-tuple, called the “greedy element,” $(x_1,\ldots,x_{k+1})$ of non-negative integers such that $\sum x_t=j$ in such a way that there is no carry-over of $p$-adic digits and such that the first $k$ elements are both positive and divisible by $r-1$. From this formula one obtains a formula for $\nu_\infty(a_i(y))$ by choosing $j$ sufficiently close to $-y$ (see Equation 2.2 and Lemma 2.1 of [@sh1]). The result follows simply by noting that this formula is linear and involves only the first $k$-terms of the given $k+1$-tuple.
\[evenval\] Let $\alpha\in \K$ be a zero of $\zeta_A(x,y)$ for $y\in \Zp$. Then $\nu_\infty(\alpha)$ is positive and divisible by $r-1$.
The fact that $\nu_\infty(\alpha)$ is positive follows easily from general theory. To see that it is divisible by $r-1$, note that Sheats’ work shows that the Newton polygon of $\zeta_A(x,y)$ only has segments of vertical length $1$ (i.e., their projection to the $X$-axis has unit length). Thus the divisibility follows immediately from the proposition.
We now set $r=p$ in order to use Diaz-Vargas’ simple techniques to compute the greedy element and to avoid problems involving carry-over of $p$-adic digits (this carry-over is what makes the general $\Fr$-case so subtle). Let $n$ be a positive integer and let $\sim_n$ be as in Part a of Remark \[equivrelations\]. Our first goal is to describe explicitly the equivalence classes of $\sim_n$ in $\Zp$. Let $y\in \Zp$ be a non-negative integer which we write $p$-adically as $\sum_{t=0}^w c_tp^t$ where $0\leq c_t <p$ for all $t$. We set $l(y)=l_p(y):=\sum_t c_t$ as usual. If $y\in \Zp$ is not a non-negative integer then we set $l(y)=\infty$.
\[diazcor\] a. Let $j$ be a non-negative integer. Then the degree in $x^{-1}$ of $\zeta_A(x,-j)$ is $[l(j)/(p-1)]$ (where $[?]$ is the standard greatest integer function).\
b. Let $y\in \Zp$. Then $\zeta_A(x,y)$ has at least $n$ distinct slopes if and only if $[l(-y)/(p-1)]\geq n$.
The first part follows immediately from Diaz-Vargas’ construction of the greedy element (see e.g., the proof of Lemma 8.24.11 of [@go1]). The second part follows from the first part and the fact that all segments of the Newton polygon of $\zeta_A(x,y)$ are known to have projections to the $X$-axis of unit length.
Note that Part a of the proposition allows one to compute explicitly the zeroes in $y$ of $a_i(y)$ (as defined in Equation \[zetasumform\]).
Now let $y\in \Zp$ be chosen so that $[l(-y)/(p-1)]\geq n$ and expand $-y$ $p$-adically as $\sum_{t=0}^\infty c_i p^i$ (where it may happen that all but finitely many of the $c_i$ vanish). Set $$y_n=\sum_{i=0}^e c_ip^i$$ where $\sum_{i=0}^e c_i=n(p-1)$ and $c_e\neq 0$. Clearly $y_n\equiv 0\pmod{p-1}$.
\[diazcor2\] a. We have $-y_n\sim_n y$.\
b. $y_n$ is the smallest element in the set of positive integers $i$ with $-i\sim_n y$.\
c. Let $y$ and $z$ be in $\Zp$. Then $y\sim_n z$ if and only if $y_n=z_n$.
This again follows from Diaz-Vargas’ construction of the greedy element.
Thus the open equivalence classes under $\sim_n$ are in one to one correspondence with negative integers $-j$ with $l(j)=n(p-1)$ (and, in particular, $j$ is divisible by $p-1$). Let $E$ be the equivalence class of one such $-j$ and write $j$ $p$-adically as $\sum_{t=0}^u c_tp^t$ where $c_u\neq0$. Let $\beta\in p^{u+1}\Zp$. It is then clear that $E$ is stable under the mapping $z\mapsto z+\beta$.
We finish this section by reworking the above results in a way which makes more transparent the close connection all zeta zeroes (at $\infty$!) have with trivial zeroes. Thus let $y\in \Zp$ be arbitrary and let $\alpha$ be a zero (in $x$) of $\zeta_A(x,y)$. From Corollary \[evenval\] we know that $\nu_\infty(\alpha)$ is both positive and divisible by $p-1=r-1$. Set $j:=\nu_\infty(\alpha)$.
\[jandn\] Let $n(j,y)$ be the number of zeroes $\beta$ of $\zeta_A(x,y)$ with $\nu_\infty(\beta)\leq j$.\
a. We have $n(j,y)=l(j)/(p-1)$.\
b. We have $-j \sim_n y$.\
c. The zero of $\zeta_A(x,-j)$ corresponding to $\alpha$ is precisely the trivial zero of $\zeta_A(x,-j)$.
Clearly the trivial zero of $\zeta_A(x,-j)$ has valuation $j$ and it is easy to see that this is the unique zero of $\zeta_A(x,-y)$ of highest valuation. The result now follows as before.
\[log\] We have $n(j,y)=O(\log (j))\,.$
Let $y=-1$. The $i$-th slope of the Newton polygon of $\zeta_A(x,-1)$ is $p^{i+1}-1$ and it is easy to see that $n(p^{i+1}-1,-1)=i$ is asymptotic to $\log_p(p^{i+1}-1)$. Thus the number of zeroes of $\zeta_A(x,-1)$ of valuation $\leq x$, for a positive real $x$, is asymptotic to $\log_p(x)$. Of course many other such examples may be worked out.
In any case, one sees that all zeroes of $\zeta_{\Fp[\theta]}(s)$ are near-trivial. For general $\Fr[T]$, calculations indicate that Parts b and c of Proposition \[jandn\] should remain valid. If so, then Part c of Proposition \[jandn\] may ultimately afford an explanation why the results of Wan, Diaz-Vargas, Thakur, Poonen and Sheats were obtainable by elementary means. Moreover, it also shows that, as of this writing, we have had precious little experience with critical zeroes.
It is also reasonable to expect that Corollary \[log\] will be true for all arithmetic $L$-series at all primes. A much more interesting question is whether some version of Part a of Proposition \[jandn\] will be true. That is, is the analogue of $n(j,y)$ independent of $y$?
Taylor expansions of classical $L$-series {#taylor}
=========================================
Let $\A=\Fr[T]$ and consider $\zeta_A(s)$, $s=(x,y)\in S_\infty$ as in Example \[trivialzeta\]. It is clear from Equation \[zetaform\] that for all $y\in \Zp$, $\zeta_A(x,y)$ is a power series in $x^{-1}$ with coefficients in $\K=\k_\infty$. In this section we will establish in great generality a very similar result for the complex analytic functions $\Xi(\chi,t)$ of [@go2] (the definition of $\Xi(\chi,t)$ will be recalled below). Consequently, as mentioned in the introduction, these Taylor expansions reflect the (conjectured!) rationality of their zeroes in a simpler fashion than one finds for arbitrary entire complex functions.
Let $p(t)=\sum c_jt^j$ be a non-zero complex power series. We say that $p(t)$ is [*almost real*]{} if and only if $$p(t)=\alpha h(t)$$ where $\alpha\in \mathbb{C}^\ast$ and where $h(t)$ is a non-zero power series with real coefficients.
\[galfe\] A complex power series $p(t)=\sum c_j t^j$ is almost real if and only if the coefficients $c_j$ satisfy the “Galois functional equation” $$\overline{c_j}=w c_j$$ for a fixed complex number $w$ of absolute value $1$.
Suppose that $p(t)=\alpha h(t)$ is almost real, where $\alpha$ is non-zero and $h(t)\in \mathbb{R}[[t]]$. Put $w:=\overline{\alpha}/
\alpha$; it is simple to check that with this $w$ the Galois functional equation holds. Conversely, assume the Galois functional equation and let $j_1$ and $j_2$ be two non-negative integers such that $c_{j_1}\neq 0$. Then $$\overline{c_{j_2}/c_{j_1}}=\overline{c_{j_2}}/\overline{c_{j_1}}=(w c_{j_2})/
(w c_{j_1})=c_{j_2}/c_{j_1}\,;$$ thus $c_{j_2}/c_{j_1}$ is real. Now let $j_0$ be the smallest non-negative integer with $c_{j_0}\neq 0$. Then $$p(t)=c_{j_0}\times t^{j_0}(1+\sum_{i=1}^\infty b_it^i)$$ with $b_i$ real, and the result is established.
Now let $\chi$ be a non-trivial finite abelian character associated to a Galois extension of number fields $L/k$. Let $L(\chi,s)$ be the classical (complex) $L$-series and let $\Lambda (\chi, s)$ be the completed $L$-function with the Euler factors at the infinite primes. As is standard $\Lambda (\chi,s)$ is entire and there is a functional equation connecting $\Lambda (\chi,s)$ and $\Lambda (\overline{\chi},1-s)$. In particular, $$\Lambda (\overline{\chi},1-s) =w(\chi)\Lambda (\chi, s) \,,$$ where $w(\chi)$ has absolute value $1$. We then set $\Xi(\chi,t):=\Lambda(\chi, 1/2+it)$ following Riemann. Let $$\Xi(\chi,t)=\sum_{n=0}^\infty a_n t^n\,,$$ be the Taylor expansion of $\Xi(\chi,t)$ about the origin.
\[anfe\] We have $$\Lambda (\overline{\chi},1-s) =w(\chi)\Lambda (\chi,s)$$ if and only if the coefficients $\{a_n\}$ satisfy $$\overline{a_n}=w(\chi) a_n\,,$$ for all $n$.
We know that $\overline{\Lambda (\chi,s)}=
\Lambda (\overline{\chi}, \overline{s})$. Thus we see $$\overline{\Xi(\chi,t)}=\overline{\Lambda (\chi,1/2+it)}=
\Lambda (\overline{\chi},1/2-i\overline{t})=
\Xi(\overline{\chi},-\overline{t})\,;$$ consequently, $\overline{\Xi(\chi,\overline{t})}=\Xi(\overline{\chi},-t)$. On the other hand, the functional equation immediately gives us $$\begin{aligned}
\Xi(\overline{\chi},-t)&=&\Lambda(\overline{\chi},1/2-it)\\
&=&\Lambda(\overline{\chi},1-(1/2+it))\\
&=&w(\chi)\Lambda(\chi,1/2+it)=w(\chi) \Xi(\chi,t)\,.\end{aligned}$$ Consequently we deduce that $$\label{funceq}
\overline{\Xi(\chi,\overline{t})}=w(\chi)\Xi(\chi,t)\,.$$ The only if part now follows upon substituting in the power series for $\Xi(\chi,t)$. The if part follows since these calculations are reversible.
\[main\] The existence of a classical style functional equation for $\Lambda (\chi,s)$ is equivalent to the Taylor expansion at the origin $t=0$ of $\Lambda (\chi,1/2+it)$ being an almost real power series.
This follows directly from Propositions \[anfe\] and \[galfe\].
For Dedekind zeta functions a completely similar result may easily be established along with some vanishing of the Taylor coefficients. In many instances it is known that classical $L$-series may be factored as infinite products over their zeroes. Such a factorization gives another approach to showing that the Taylor expansion of $\Xi(\chi,t)$ is almost real.
[33334]{} $t$-motives, [*Duke Math. J.*]{} [**53**]{} (1986), 457-502.
Global $L$-functions over function fields, (preprint).
A cohomological theory of crystals over function fields, (in preparation).
On certain functions connected with polynomials in a Galois field, [*Duke Math. J.*]{} [**1**]{} (1935), 137–168.
Elliptic modules, Math. Sbornik [**94**]{} (1974), 594-627, English transl.: [*Math. U.S.S.R. Sbornik*]{} [**23**]{} (1976), 561-592.
Riemann hypothesis for $\Fp[T]$, [*J. Number Theory*]{} [**59**]{} (1996), 313-318.
, Ann. Math. Study 133, Princeton Univ. Press, 1994.
Models of $\tau$-sheaves, (preprint, available at
http://www.math.ethz.ch/~fgardeyn/FG/index.shtml).
On power sums of polynomials over finite fields, [*J. Number Theory*]{} [**30**]{} (1988), 11-26.
: [*Basic Structures of Function Field Arithmetic*]{}, Springer-Verlag, Berlin, 1996.
: A Riemann hypothesis for Characteristic $p$ $L$-functions, [*J. Number Theory*]{} [**82**]{} (2000), 299-322.
: Power sums of polynomials in a Galois field, [*Duke Math. J.*]{} [**10**]{} (1943), 277-292.
The Riemann hypothesis for the Goss zeta function for $\Fq[T]$, [*J. Number Theory*]{} [**71**]{} (1998), 121-157.
: Zeta-measure associated to $\Fq[t]$, [*J. Number Theory*]{} [**35**]{} (1990), 1-17.
$L$-functions of $\varphi$-sheaves and Drinfeld modules, [*J. Amer. Math. Soc.*]{} [**9**]{} (1996), 755-781.
Entireness of $L$-functions of $\varphi$-sheaves on affine complete intersections, [*J. Number Theory*]{} [**63**]{} (1997), 170-179.
On the Riemann hypothesis for the characteristic $p$ zeta function, [*J. Number Theory*]{} [**58**]{} (1996), 196-212.
[^1]: This paper is dedicated to Ram with great respect and affection on his $70$-th birthday
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'A space is [*$n$–arc connected*]{} ($n$–ac) if any family of no more than $n$–points are contained in an arc. For graphs the following are equivalent: (i) $7$–ac, (ii) $n$–ac for all $n$, (iii) continuous injective image of a closed sub–interval of the real line, and (iv) one of a finite family of graphs. General continua that are $\aleph_0$–ac are characterized. The complexity of characterizing $n$–ac graphs for $n=2,3,4,5$ is determined to be strictly higher than that of the stated characterization of $7$–ac graphs.'
address:
- |
Department of Mathematics\
University of Pittsburgh at Greensburg\
236 Frank A. Cassell Hall\
150 Finoli Drive\
Greensburg, PA 15601\
USA
- |
Department of Mathematics\
University of Pittsburgh\
508 Thackeray Hall\
Pittsburgh, PA 15260\
USA
- |
Department of Mathematics\
University of Pittsburgh\
301 Thackeray Hall\
Pittsburgh, PA 15260\
USA
author:
- 'Benjamin Espinoza, Paul Gartside, Ana Mamatelashvili.'
title: '$n$–Arc Connected Spaces'
---
Introduction
============
A topological space $X$ is called [*$n$–arc connected*]{} ($n$–ac) if for any points $p_1, p_2, \dots, p_n$ in $X$, there exists an arc $\alpha$ in $X$ such that $p_1, p_2, \dots p_n$ are all in $\alpha$. If a space is $n$–ac for all $n\in {\mathbb{N}}$, then we will say that it is [*$\omega$–ac*]{}. Note that this is equivalent to saying that for any finite $F$ contained in $X$ there is an arc $\alpha$ in $X$ containing $F$. Call a space $\aleph_0$–ac if for every countable subset, $S$, there is an arc containing $S$. Evidently a space is arc connected if and only if it is $2$–ac, and ‘$\aleph_0$–ac’ implies ‘$\omega$–ac’ implies ‘$(n+1)$–ac’ implies ‘$n$–ac’ (for any fixed $n$).
Thus we have a family of natural strengthenings of arc connectedness, and the main aim of this paper is to characterize when ‘nice’ spaces have one of these strong arc connectedness properties. Secondary aims are to distinguish ‘$n$–ac’ (for each $n$), ‘$\omega$–ac’ and ‘$\aleph_0$–ac’, and to compare and contrast the familiar arc connectedness (i.e. $2$–ac) with its strengthenings.
Observe that any Hausdorff image of an $n$–ac (respectively, $\omega$–ac, $\aleph_0$–ac) space under a continuous injective map is also $n$–ac (respectively, $\omega$–ac, $\aleph_0$–ac). Below, unless explicitly stated otherwise, all spaces are (metrizable) continua.
It turns out that ‘sufficiently large’ (in terms of dimension) arc connected spaces tend to be $\omega$–ac. Indeed it is not hard to see that manifolds (with or without boundary) of dimension at least $2$ are $\omega$–ac. Thus we focus on curves ($1$–dimensional continua) and especially on graphs (those connected spaces obtained by taking a finite family of arcs and then identifying some of the endpoints).
To motivate our main results consider the following examples.
[\[exs\]]{}
- The arc (the closed unit interval, $I=[0,1]$) is $\aleph_0$–ac.
- The open interval, $(0,1)$; and ray, $[0,1)$, are $\omega$–ac.
- From (A) and (B), all continua which are the continuous injective images of the arc, open interval and ray are $\omega$–ac. It is easy to verify that these include: (a) the arc, (b) the circle, (c) figure eight curve, (d) lollipop, (e) dumbbell and (f) theta curve.
[cccccc]{} (a) & (b) & (c) & (d) & (e) & (f)\
\[vertex/.style=[circle,draw=blue!50,fill=blue!20,thick, inner sep=0mm, minimum size=1mm]{}, point/.style=[coordinate]{}\]
at (0,0) ; at (0,2) ;
(arc\_bot) at (0,.5) ; (arc\_top) \[above=of arc\_bot\] ; (arc\_bot) – (arc\_top);
&
\[vertex/.style=[circle,draw=blue!50,fill=blue!20,thick, inner sep=0mm, minimum size=1mm]{}, point/.style=[coordinate]{}\] at (0,0) ; at (0,2) ;
at (0,1) (circle\_pt) ; (pt) \[right=of circle\_pt\] ;
(circle\_pt.center) to \[out=90, in=90,looseness=2.5\] (pt.center); (pt.center) to \[out=270, in=270, looseness=2.5\] (circle\_pt.center);
at (circle\_pt) ;
&
\[vertex/.style=[circle,draw=blue!50,fill=blue!20,thick, inner sep=0mm, minimum size=1mm]{}, point/.style=[coordinate]{}\]
(center\_pt) ; (top\_pt) \[above=of center\_pt\] ; (bot\_pt) \[below=of center\_pt\] ;
(center\_pt.center) to \[out=-10, in=-10,looseness=1.5\] (top\_pt); (top\_pt.center) to \[out=170, in=170, looseness=1.5\] (center\_pt.center);
(center\_pt.center) to \[out=-10, in=-10,looseness=1.5\] (bot\_pt); (bot\_pt.center) to \[out=170, in=170, looseness=1.5\] (center\_pt.center);
&
\[vertex/.style=[circle,draw=blue!50,fill=blue!20,thick, inner sep=0mm, minimum size=1mm]{}, point/.style=[coordinate]{}\]
(lollipop\_bot) ; (lollipop\_mid) \[above=of lollipop\_bot\] ; (pt) \[above=of lollipop\_mid\] ;
(lollipop\_bot.north) – (lollipop\_mid.south); (lollipop\_mid.east) to \[out=10,in=10,looseness=1.5\] (pt); (pt) to \[out=170, in=170,looseness=1.5\] (lollipop\_mid.west);
&
\[vertex/.style=[circle,draw=blue!50,fill=blue!20,thick, inner sep=0mm, minimum size=1mm]{}, point/.style=[coordinate]{}\] at (0,0) ; at (0,2) ;
at (0,1) (dumbbell\_left) ; (dumbbell\_right) \[right=of dumbbell\_left\] ; (pt1) \[left=of dumbbell\_left\] ; (pt2) \[right=of dumbbell\_right\] ;
(dumbbell\_left.east) – (dumbbell\_right.west); (dumbbell\_left.north) to \[out=90, in=90,looseness=1.5\] (pt1.north); (pt1.south) to \[out=270, in=270,looseness=1.5\] (dumbbell\_left.south); (dumbbell\_right.north) to \[out=90, in=90,looseness=1.5\] (pt2.north); (pt2.south) to \[out=270, in=270,looseness=1.5\] (dumbbell\_right.south);
&
\[vertex/.style=[circle,draw=blue!50,fill=blue!20,thick, inner sep=0mm, minimum size=1mm]{}, point/.style=[coordinate]{}\]
at (0,0) ; at (0,2) ;
at (0,1) (theta\_left) ; (theta\_right) \[right=of theta\_left\] ;
(theta\_left.east) to (theta\_right.west) ; (theta\_left.north) to \[out=90, in=90,looseness=2.5\] (theta\_right.north); (theta\_right.south) to \[out=270, in=270,looseness=2\] (theta\_left.south);
- The Warsaw circle; double Warsaw circle; Menger cube; and Sierpinski triangle, are $\omega$–ac.
- The simple triod is $2$–ac but not $3$–ac. It is minimal in the sense that no graph with strictly fewer edges is $2$–ac not $3$–ac.
The graphs (a), (b) and (c) below are: $3$–ac but not $4$–ac, $4$–ac but not $5$–ac, and $5$–ac but not $6$–ac, respectively. All are minimal.
[ccc]{} (a) & (b) & (c)\
(0,0) ellipse (8mm and 8mm); (v1) at (180:12mm) ; (v2) at (0:12mm) ; (v3) at (180:8mm) ; (v4) at (0:8mm) ;
(v1) – (v3); (v2) – (v4);
&
at (-12mm,0) ; at (12mm,0) ;
(0,0) circle (8mm); (v1) at (60:8mm) ; (v2) at (180:8mm) ; (v3) at (300:8mm) ; (v1) – (v2) – (v3);
&
at (-12mm,0) ; at (12mm,0) ;
(0,0) circle (8mm);
(v1) at (0:8mm) ; (v2) at (120:8mm) ; (v3) at (240:8mm) ;
(0,0) – (v1); (0,0) – (v2); (0,0) – (v3);
(v0) at (0,0) ;
- The Kuratowski graph $K_{3,3}$ is $6$–ac but not $7$–ac. It is also minimal.
- The graphs below are all $6$–ac and, by Theorem \[main1\], none is $7$–ac. Unlike $K_{3,3}$ all are planar. It is unknown if the first of these graphs (which has $12$ edges) is minimal among planar graphs. A minimal example must have at least nine edges.
[cccc]{}
at (-12mm,0) ; at (12mm,0) ;
(0,0) circle (8mm); (0,0) circle (4mm); (v1) at (0:8mm) ; (v2) at (90:8mm) ; (v3) at (180:8mm) ; (v4) at (270:8mm) ; (v1i) at (0:4mm) ; (v2i) at (90:4mm) ; (v3i) at (180:4mm) ; (v4i) at (270:4mm) ;
(v1i) – (v1); (v3i) – (v3); (v2i) – (v2); (v4i) – (v4);
&
at (-12mm,0) ; at (12mm,0) ;
(0,0) circle (8mm); (0,0) circle (4mm); (v1) at (0:8mm) ; (v2) at (72:8mm) ; (v3) at (144:8mm) ; (v4) at (216:8mm) ; (v5) at (288:8mm) ; (v1i) at (0:4mm) ; (v2i) at (72:4mm) ; (v3i) at (144:4mm) ; (v4i) at (216:4mm) ; (v5i) at (288:4mm) ;
(v1i) – (v1); (v3i) – (v3); (v2i) – (v2); (v4i) – (v4); (v5i) – (v5);
&
at (-12mm,0) ; at (12mm,0) ;
(0,0) circle (8mm); (0,0) circle (4mm); (v1) at (0:8mm) ; (v2) at (60:8mm) ; (v3) at (120:8mm) ; (v4) at (180:8mm) ; (v5) at (240:8mm) ; (v6) at (300:8mm) ; (v1i) at (0:4mm) ; (v2i) at (60:4mm) ; (v3i) at (120:4mm) ; (v4i) at (180:4mm) ; (v5i) at (240:4mm) ; (v6i) at (300:4mm) ;
(v1i) – (v1); (v3i) – (v3); (v2i) – (v2); (v4i) – (v4); (v5i) – (v5); (v6i) – (v6);
&
at (0,8mm) ; (0,0) – (0.5,0); at (0,-8mm) ;
The diversity of examples in (D) of $\omega$–ac curves suggests that no simple characterization of these continua is likely. The authors, together with Kovan–Bakan, prove that there is indeed no characterization of $\omega$–ac curves any simpler than the definition, see [@egkm] for details.
This prompts us to restrict attention to the more concrete case of graphs, and leads us to the following natural problems.
- Characterize the $\omega$–ac graphs.
- Characterize the $\aleph_0$–ac graphs.
- Characterize, for each $n$, the graphs which are $n$–ac but not $(n+1)$–ac.
In Section \[omegag\] below we show that the list of $\omega$–ac graphs given in (C) is complete, answering Problem (1).
[\[main1\]]{} For a graph $G$ the following are equivalent:
- $G$ is $7$–ac,
- $G$ is $\omega$–ac,
- $G$ is the continuous injective image of a sub–interval of the real line,
- $G$ is one of the following graphs: the arc, simple closed curve, figure eight curve, lollipop, dumbbell or theta curve.
In Section \[aleph0\] we go on to characterize the $\aleph_0$–ac continua, giving a very strong solution to Problem (2).
[\[main2\]]{} For any continuum $K$ (not necessarily metrizable) the following are equivalent:
- $K$ is $\aleph_0$–ac,
- $K$ is a continuous injective image of a closed sub–interval of the long line,
- $K$ is one of: the arc, the long circle, the long lollipop, the long dumbbell, the long figure eight or the long theta-curve.
From Theorem \[main1\] we see that there are no examples of graphs that are $n$–ac but not $(n+1)$–ac, for $n \ge 7$, solving Problem (3) in these cases. Of course Examples (E) and (F) show that there are $n$–ac not $(n+1)$–ac graphs for $n=2,3,4,5$ and $6$. But the question remains: can we characterize these latter graphs? Informally our answers are as follows.
[\[main3\]]{}
- The characterization of $\omega$–ac graphs given in Theorem \[main1\] is as simple as possible,
- there exist reasonably simple characterizations of $n$–ac not $(n+1)$–ac graphs for $n \le 7$,
- for $n=2,3,4$ and $5$, there is no possible characterization of $n$–ac not $(n+1)$–ac graphs which is as simple as that for $\omega$–ac graphs.
In Section \[cplxty\] we outline some machinery from descriptive set theory which allows us to formalize and prove these claims. The situation with $n=6$ — the complexity of characterizing graphs which are $6$–ac but not $7$–ac — remains unclear. This, and other remaining open problems, are discussed in Section \[probs\].
Characterizations
=================
In this section we prove the characterization theorems stated in the Introduction. First Theorem \[main1\] characterizing $\omega$–ac graphs. Second Theorem \[main2\] characterizing $\aleph_0$–ac continua.
$\omega$–ac Graphs {#omegag}
------------------
As noted in Example (C) the graphs listed in part 4) of Theorem \[main1\] are all the continuous injective image of a closed sub–interval of the real line, giving 4) implies 3), and all such images are $\omega$–ac, yielding 3) implies 2) of Theorem \[main1\]. Clearly $\omega$–ac graphs are $7$–ac, and so 2) implies 1) in Theorem \[main1\].
It remains to show 1) implies 4) in Theorem \[main1\], in other words that any $7$–ac graph is one of the graphs listed in 4). This is established in Theorem \[ins\] below. We proceed by establishing an ever tightening sequence of restrictions on the structure of $7$–ac graphs.
We note that Lelek and McAuley, [@lelek], showed that the only Peano continua which are continuous injective images of the real line are the figure eight, dumbbell and theta–curve. Their proof can be modified to establish the equivalence of 3) and 4) in Theorem \[main1\].
Let $G$ be a finite graph, and let $H \subset G$ be a subgraph of $G$ such that $G-H$ is connected and $\overline{G-H}\cap H = r$ is a branch point of $G$. If $G$ is $n$–ac, then $\overline{G-H}$ is $n$–ac. \[complement\]
First note that $\overline{G-H} = (G-H)\cup {\left\{r\right\}}$. Hence every connected set intersecting $G-H$ and $H-{\left\{r\right\}}$, must contain $r$.
Let $\mathcal{P}={\left\{p_1, p_2, \dots , p_n\right\}}$ be a set of $n$ points in $\overline{G-H}$. Then, since $G$ is $n$–ac, there exists an arc $\alpha$ in $G$ containing $\mathcal{P}$. If $\alpha \subset \overline{G-H}$, we are done. So assume $\alpha$ intersects $H-{\left\{r\right\}}$. Let $t_0, t_1\in [0,1]$ such that $\alpha(t_0)\in G-H$ and $\alpha(t_1)\in H-{\left\{r\right\}}$, assume without loss of generality that $t_0< t_1$. Hence there exists $s\in [t_0, t_1]$ such that $\alpha(s)=r$. Then $\alpha([0,s])$ is an arc in $\overline{G-H}$ containing $\mathcal{P}$, otherwise $r\in \alpha((s,1])$ which is impossible since $\alpha$ is an injective image of $[0,1]$. This proves that $\overline{G-H}$ is $n$–ac.
The reverse implication of Proposition \[complement\] does not hold. To see this, let $G$ be a simple triod and $H$ be one of the edges of $G$. Clearly $G$ is not $3$–ac but $\overline{G-H}$ (an arc) is $3$–ac.
Let $G$ be a finite graph. An edge $e$ of $G$ is called a [*terminal edge*]{} of $G$ if one of the vertices of $e$ is an end–point of $G$. \[isolated\]
Let $G$ be a finite graph, and let $I={\left\{e_1, e_2, \dots e_m\right\}}$ be the set of terminal edges of $G$. Let $G^*$ be the graph given by $\overline{G-I}$. Clearly this operation can be applied to $G^*$ as well. We perform this operation as many times as necessary until we obtain a graph $G'$ having no terminal edges. We called the graph $G'$ the [*reduced graph of*]{} $G$. \[reduced\]
The following is a corollary of Proposition \[complement\].
Let $G$ be an $n$–ac finite graph. Then the reduced graph of $G$ is an $n$–ac finite graph containing no terminal edges. \[nwacreduced\]
Observe that the reduced graph of $G$ can also be obtained by removing terminal edges one at a time.
Now, from Proposition \[complement\], if $G$ is an $n$–ac finite graph and $e$ is a terminal edge of $G$, then $\overline{G-e}$ is $n$–ac. This implies that each time we remove a terminal edge we obtain an $n$–ac graph. This and the observation prove the corollary.
Note that if $X$ is an $n$–ac space and ${\left\{p_1, p_2, \dots , p_n\right\}}$ are $n$ different points of $X$, then there is an arc $\alpha$ such that ${\left\{p_1, p_2, \dots , p_n\right\}}\subset \alpha$ and such that the end–points of $\alpha$ belong to ${\left\{p_1, p_2, \dots , p_n\right\}}$. To see this, let $\beta$ be the arc containing ${\left\{p_1, p_2, \dots , p_n\right\}}$, given by the fact that $X$ is $n$–ac. Let $t_0=\text{min}{\left\{\beta^{-1}(p_i) \, |\, i=1, \dots , n\right\}}$ and $t_1=\text{max}{\left\{\beta^{-1}(p_i) \, | \, i=1, \dots ,n\right\}}$. Then $\beta([t_0, t_1])$ satisfies the conditions of $\alpha$.
From now on, if $X$ is an $n$–ac space, ${\left\{p_1, \dots , p_n\right\}}$ are $n$ different points and $\alpha$ is an arc passing through ${\left\{p_1, p_2, \dots , p_n\right\}}$, then we will assume that the end–points of $\alpha$ belong to ${\left\{p_1, p_2, \dots , p_n\right\}}$. \[endpoints\]
Let $G$ be a finite graph. Assume that $G$ contains a simple triod $T=L_1\cup L_2\cup L_3$ (with ${\left\{q\right\}}=L_i\cap L_j$, for $i\neq j$) such that for each $i$, $L_i-{\left\{q\right\}}$ contains no branch points of $G$. For each $i=1, 2, 3$, let $p_i\in \text{int}(L_i)$. If $\alpha$ is an arc containing ${\left\{p_1, p_2, p_3\right\}}$, then
1. $q\in \text{int}(\alpha)$, and
2. at least one of the end points of $\alpha$ lies in $[q,p_1]\cup [q,p_2]\cup [q,p_3]$.
\[4ptsarc\]
Let $G$, $T$ and $p_1, p_2, p_3$ as in the hypothesis of the lemma. Let $\alpha \subset G$ be an arc containing ${\left\{p_1, p_2, p_3\right\}}$, and denote, for each $i=1,2,3$, by $[q, l_i]$ the arc $L_i$.
1. Assume, without loss of generality, that $\alpha(t_i)=p_i$ and that $t_1 < t_2 <t_3$. Then $p_2\in \text{int}(\alpha)$ and $\alpha = \alpha([0, t_2])\cup \alpha([t_2, 1])$.
We consider two cases: $q\not\in \alpha([0,t_2])$ and $q\in \alpha([0,t_2])$. Assume $q\not\in \alpha([0,t_2])$, then, since $L_2-{\left\{q\right\}}$ contains no branch points of $G$ and $p_2 \in \text{int}(L_2)$, we have that $l_2=\alpha(s)$ for some $s$ with $0<s<t_2$. Hence $[l_2,p_2]{\subseteq}\alpha([0,t_2])$. Therefore, since ${\left\{p_2, p_3\right\}}{\subseteq}\alpha([t_2,1])$, $p_2\in \text{int}(L_2)$, $L_2-q$ has no branch points of $G$, and $\alpha$ is a $1-1$ function, we have that $[p_2,q]{\subseteq}\alpha([t_2, 1))$. This implies that $q\in \text{int}(\alpha)$.
Now suppose that $q\in \alpha([0,t_2])$. If $q\in \alpha((0,t_2])$, then we are done. So assume that $q=\alpha(0)$, i.e. $q$ is an end-point of $\alpha$. Using the same argument as in the previous case, we can conclude that $[l_2,p_2]{\subseteq}\alpha([0,t_2])$. This implies, as before, that $[p_2,q]{\subseteq}\alpha([t_2, 1))$ wich contradicts the fact that $\alpha$ is a $1-1$ function. Hence $q\in \text{int}(\alpha)$.
2. First, assume that $\alpha(t_i)=p_i$ and that $t_1 < t_2 < t_3$. We will show that one end point of $\alpha$ lies on either $[q,p_1]$ or $[q,p_3]$. The other cases (rearrangements of the $t_i$s) are done in the same way as this case, the only difference is the conclusion: the end point lies either on $[q,p_1]$ or $[q,p_2]$, or the end point lies either on $[q,p_2]$ or $[q,p_3]$.
By (1), $q\in \text{int}(\alpha)$ and if $q=\alpha(s)$, then $s<t_3$; otherwise the arc $\alpha([0,t_3])$ would contain $p_1,p_2,p_3$ and $q\not\in\text{int}(\alpha([0,t_3]))$ which is contrary to (1). Similarly, $t_1<s$. Hence $t_1<s<t_3$.
If $s<t_2$, then $p_1, q\not\in \alpha([t_2,1])=\alpha([t_2,t_3])\cup \alpha([t_3, 1])$. Now, since $L_3-{\left\{q\right\}}$ has no branch points of $G$, $q\in \alpha([0,t_2])$, and $p_3\in\text{int}(L_3)$, we have $l_3\in \alpha([t_2,t_3])$. Thus, since $\alpha$ is a $1-1$ function, $\alpha([t_3,1])\subset (q,p_3]$. This shows that $\alpha(1)$ lies in $[q,p_3]$.
If $t_2<s$, then a similar argument using $-\alpha$ ($\alpha$ traveled in the opposite direction) shows that one of the end points of $\alpha$ lies on $[q,p_1]$.
We obtain the following corollaries.
With the same conditions as in Lemma \[4ptsarc\]. If $\alpha$ is an arc containing ${\left\{p_1,p_2,p_3\right\}}$, and $q=\alpha(s)$, $p_i=\alpha(t_i)$ for $i=1,2,3$, then $t_j<s <t_k$ for some $j,k\in{\left\{1,2,3\right\}}$. \[inbetween\]
To see this, note that if $q$ does not lie between two of the $p_i$s, then either $s<t_i$ for all $i$, or $t_i<s$ for all $i$. Then either $\alpha([s,1])$ or $\alpha([0,s])$ are arcs containing ${\left\{p_1,p_2,p_3\right\}}$ for which $q$ is an end-point, this contradicts (1) of Lemma \[4ptsarc\].
Let $G$ be a finite graph, and let ${\left\{p_1, p_2, \dots p_n\right\}}\subset G$ be $n$ different points. In addition, let $\alpha$ be one arc containing ${\left\{p_1, p_2, \dots p_n\right\}}$, with end–points belonging to ${\left\{p_1, p_2, \dots p_n\right\}}$. If there are three different indexes $i,j,k$ such that $p_i$, $p_j$ and $p_k$ belong to a triod $T$ satisfying the conditions of (Lemma \[4ptsarc\]), and such that $([q,p_i]\cup [q,p_j]\cup [q,p_k])\cap {\left\{p_1, p_2, \dots p_n\right\}} ={\left\{p_i, p_j, p_k\right\}}$, then either $p_i$, $p_j$ or $p_k$ is an end–point of $\alpha$. \[triodendpt\]
By (2) of Lemma \[4ptsarc\], at least one of the end points of $\alpha$ lies in $[q,p_i]\cup [q,p_j]\cup [q,p_k]$. Hence, since the end-points of $\alpha$ belong to ${\left\{p_1, p_2, \dots p_n\right\}}$ and $([q,p_i]\cup [q,p_j]\cup [q,p_k])\cap {\left\{p_1, p_2, \dots p_n\right\}} ={\left\{p_i, p_j, p_k\right\}}$, one of $p_i$, $p_j$ or $p_k$ is an end-point of $\alpha$
Let $G$ be a finite graph. If $G$ is $5$–ac, then $G$ has no branch point of degree greater than or equal to five. \[degree5\]
Assume, by contradiction, that $G$ contains at least one branch point, $q$, of degree at least $5$. Then, since $G$ is a finite graph, $G$ contains a simple $5$-od, $T=L_1\cup L_2 \cup L_3\cup L_4\cup L_5$, such that ${\left\{q\right\}}=L_i\cap L_j$ for $i\neq j$, and such that $L_i-{\left\{q\right\}}$ contains no branch points of $G$.
For each $i=1, \dots, 5$, let $p_i\in \text{int}(L_i)$. Then, since $G$ is $5$–ac, there exists an arc $\alpha \subset G$ such that ${\left\{p_1, p_2, \dots, p_5\right\}}\subset \alpha$. Note that $T$ contains a triod satisfying the conditions of Lemma \[4ptsarc\], hence $q\in \text{int}(\alpha)$. Let $t_0\in (0,1)$ be the point such that $\alpha(t_0)=q$. Then $\alpha-{\left\{q\right\}}=\alpha([0,t_0))\cup \alpha((t_0,1])$, and either $\alpha([0,t_0])$ or $\alpha([t_0,1])$ contains three points out of ${\left\{p_1, p_2, p_3, p_4, p_5\right\}}$, note that $q$ is an end–point of $\alpha([0,t_0])$ and of $\alpha([t_0,1])$. Without loss of generality, suppose that $p_1, p_2, p_3 \subset \alpha([0,t_0])$; then $L_1, L_2, L_3$ and the corresponding $p_i$s satisfy the conditions of Lemma \[4ptsarc\] implying that any arc containing those points contains $q$ in its interior, a contradiction, since $q$ is an end point of $\alpha([0,t_0])$. This shows that $G$ does not contain a branch point of degree greater than or equal to five.
From Proposition \[degree5\] we obtain the following corollaries.
Let $G$ be a finite graph. If $G$ is $n$–ac, for $n\geq 5$, then $G$ has no branch point of degree greater than or equal to five. \[nodegree5n\]
The following proposition is easy to prove.
Let $G$ be a finite connected graph. If $G$ has at least three branch points, then there is an arc $\alpha$ such that the end-points of $\alpha$ are branch points of $G$ and all the points of the interior of $\alpha$, except for one, are not branch points of $G$. So $\alpha$ contains exactly three branch points of $G$. \[p3branchpts\]
A finite graph with three or more branch points cannot be $7$–ac. \[no7wac\]
Let $G$ be a finite graph with at least three branch points.
By Proposition \[p3branchpts\], there is an arc $\alpha$ in $G$ containing exactly three branch points of $G$ such that two of them are the end–points of $\alpha$. Denote by $q_1$, $q_2$, and $q_3$ these branch points, and assume without loss of generality that $q_1$ and $q_3$ are the end-points of $\alpha$.
Let $p_3$ be a point between $q_1$ and $q_2$, and let $p_5$ be a point between $q_2$ and $q_3$. Since $G$ is a finite graph, we can find, in a neighborhood of $q_1$, two points $p_1$ and $p_2$ such that $p_1$, $p_2$, $p_3$ belong to a triod $T_1$ satisfying the conditions of Lemma \[4ptsarc\], and such that $q_1$ is the branch point of $T_1$. Similarly, we can find a point $p_4$, in a neighborhood of $q_2$, such that $p_3$, $p_4$ and $p_5$ belong to a triod $T_2$ satisfying the conditions of Lemma \[4ptsarc\], and such that $q_2$ is the branch point of $T_2$. Finally, we can find two points $p_6$, and $p_7$, in a neighborhood of $q_3$, such that $p_5$, $p_6$ and $p_7$ belong to a triod $T_3$ satisfying the conditions of Lemma \[4ptsarc\], and such that $q_3$ is the branch point of $T_3$.
(pq2) ; (pq1) \[below left=1cm and 3cm of pq2\] ; (pq3) \[below right=1cm and 3cm of pq2\] ;
(pq1) – (pq2) node\[vertexp,midway,label=above left:$p_3$\] ; (pq2) – (pq3) node\[vertexp,midway,label=above right:$p_5$\] ;
(q1) \[label=above:$q_1$\] at (pq1) ; (q2) \[label=below:$q_2$\] at (pq2) ; (q3) \[label=above:$q_3$\] at (pq3) ;
(q21) \[above left=of q2\] ; (q22) \[above=of q2\] ; (q23) \[above right=of q2\] ;
(q2) – (q21) node\[vertexp,midway,label=left:$p_4$\] ; (q2) – (q22); (q2) – (q23);
(q11) \[left=of q1\] ; (q12) \[above left=of q1\] ; (q13) \[below left=of q1\] ; (q14) \[below right=of q1\] ;
(q1) – (q11); (q1) – (q12); (q1) – (q13) node\[vertexp,midway,label=right:$p_1$\] ; (q1) – (q14) node\[vertexp,midway,label=right:$p_2$\] ;
(q31) \[right=of q3\] ; (q32) \[above right=of q3\] ; (q33) \[below left=of q3\] ; (q34) \[below right=of q3\] ;
(q3) – (q31); (q3) – (q32) node\[vertexp,midway,label=right:$p_6$\] ; (q3) – (q33); (q3) – (q34) node\[vertexp,midway,label=right:$p_7$\] ;
We show by contradiction that there is no arc containing ${\left\{p_1, p_2, \dots , p_7\right\}}$. Suppose that there is an arc $\beta\subset G$ containing the points ${\left\{p_1, p_2, \dots , p_7\right\}}$, using the same argument from Remark \[endpoints\], we can assume that the end points of $\beta$ belong to ${\left\{p_1, p_2, \dots , p_7\right\}}$.
Now, by Corollary \[triodendpt\], one of ${\left\{p_1, p_2, p_3\right\}}$ is an end point of $\beta$. Similarly, one of ${\left\{p_3, p_4, p_5\right\}}$ is an end–point of $\beta$, and one of ${\left\{p_5, p_6, p_7\right\}}$ is an end–point of $\beta$. So, since $\beta$ is an arc with end–points in ${\left\{p_1, p_2, p_3, \dots , p_7\right\}}$, we have that either
1. $p_1$ or $p_2$ and $p_5$ are the end–points of $\beta$, or
2. $p_6$ or $p_7$ and $p_3$ are the end–points of $\beta$ or
3. $p_3$ and $p_5$ are the end–points of $\beta$,
are the only possible cases. We will prove that every case leads to a contradiction.
1. Assume that $p_1$ and $p_5$ are the end–points of $\beta$. Since the arc between $q_2$ and $q_3$ contains no branch points of $G$, we have that either $[q_2, p_5]\subset \beta$ or $[p_5, q_3]\subset \beta$.
Assume first that $[q_2, p_5]\subset \beta$. Then, by the way $p_4$ was chosen and the fact that $p_4 \in \text{int}(\beta)$, the arc $[p_4, q_2]\subset \beta$; similarly, since the arc $[q_1,q_2]$ contains no branch points of $G$ and the fact that $p_3\in\text{int}(\beta)$, the arc $[p_3,q_2]\subset\beta$. Then $\left([p_3,q_2] \cup [p_4, q_2] \cup [q_2, p_5]\right)\subset \beta$, which is a contradiction since $\left([p_3,q_2] \cup [p_4, q_2] \cup [q_2, p_5]\right)$ is a nondegenerate simple triod.
Assume that $[p_5, q_3]\subset \beta$. Then, by the way $p_6$ was chosen and the fact that $p_6\in\text{int}(\beta)$, the arc $[q_3, p_6]\subset \beta$. Using the same argument we can conclude that the arc $[q_3, p_7]\subset \beta$. Hence $\left([p_5,q_3] \cup [p_6, q_3] \cup [q_3, p_7]\right)\subset \beta$, which is a contradiction.
The case when $p_2$ and $p_5$ are the end–points of $\beta$ is similar the case we just proved. So (i) does not hold.
2. This case is equivalent to (i), therefore (ii) does not hold.
3. Assume that $p_3$ and $p_5$ are the end-points of $\beta$. Then, since the arc $[q_1,q_2]$ contains no branch points of $G$ and $p_3$ is an end-point of $\beta$, either $[q_1, p_3]\subset \beta$ or $[p_3,q_2]\subset \beta$.
Suppose that $[q_1, p_3]\subset \beta$. As in ($i$), since $p_1, p_2 \in\text{int}(\beta)$, we have that the arc $[p_1,q_1]$ and $[q_1,p_2]$ are contained in $\beta$. Implying that the nondegenerate simple triod $\left([q_1, p_3] \cup [p_1,q_1] \cup [q_1,p_2]\right) \subset \beta$, which is a contradiction.
Now assume that $[p_3,q_2]\subset \beta$. Then the arc $[p_5,q_3]\subset \beta$. Again, the same argument as in (i) leads to a nondegenerate simple triod being contained in $\beta$ since $p_6, p_7\in\text{int}(\beta)$. Hence (iii) does not hold.
This proves that there is no arc containing ${\left\{p_1, p_2, \dots , p_7\right\}}$. Therefore $G$ is not $7$–ac.
Since every $(n+1)$–ac space is $n$–ac, we have the following corollary.
A finite graph with three or more branch points cannot be a $n$–ac, for $n\geq 7$.
If $G$ is a finite graph with only $2$ branch points each of degree greater than or equal to $4$, then $G$ is not $7$–ac.
Let $q_1$ and $q_2$ be the two branch points of $G$. Then there exists at least one edge $e$ having $q_1$ and $q_2$ as vertices. Let $p_1\in \text{int}(e)$. Since $G$ is a finite graph, and $q_1$ and $q_2$ have degree at least 4, we can chose three points $p_2$, $p_3$, $p_4$ in a neighborhood of $q_1$ such that $T_1=[q_1, p_1] \cup [q_1, p_2] \cup [q_1, p_3] \cup [q_1, p_4]$ is a simple $4$-od, and three points $p_5$, $p_6$, $p_7$ in a neighborhood of $q_2$ such that $T_2=[q_2, p_1] \cup [q_2, p_5] \cup [q_2, p_6] \cup [q_2, p_7]$ is a simple $4$–od, and they are such that $T_1\cap T_2 ={\left\{p_1\right\}}$.
We show by contradiction, that there is no arc $\alpha\subset G$ containing ${\left\{p_1, p_2, \dots , p_7\right\}}$. For this suppose that there exists such an arc $\alpha$, assume further that the end–points of $\alpha$ belong to ${\left\{p_1, p_2, \dots , p_7\right\}}$. Then, since ${\left\{p_2, p_3, p_4\right\}}$ satisfy the conditions of Corollary \[triodendpt\], we can assume without loss of generality that $p_4$ is an end–point of $\alpha$. Similarly for the set ${\left\{p_5, p_6, p_7\right\}}$, so we can assume without loss of generality that $p_5$ is an end–point of $\alpha$. On the other hand, the set ${\left\{p_1, p_2, p_3\right\}}$ also satisfies the conditions of Corollary \[triodendpt\], hence $p_1$, or $p_2$ or $p_3$ is an end-point of $\alpha$ which is impossible since $\alpha$ only has two end-points. This shows that there is no arc in $G$ containing ${\left\{p_1, p_2, \dots , p_7\right\}}$. This proves that $G$ is not $7$–ac.
If $G$ is a finite graph with only $2$ branch points each of degree greater than or equal to $4$, then $G$ is not $n$–ac, for $n\geq 7$. \[no24\]
Let $G$ be a finite graph. If $G$ is $7$–ac, then $G$ is one of the following graphs: arc, simple closed curve, figure eight, lollipop, dumbbell or theta–curve. \[ins\]
Let $G$ be a finite graph. Suppose that $G$ is $n$–ac, for $n\geq 7$. We will show that $G$ is (homeomorphic to) one of the listed graphs.
Let $K$ be the reduced graph of $G$. By Corollary \[nwacreduced\] $K$ is $n$–ac and contains no terminal edges. By Theorem \[no7wac\] $K$ has at most two branch points, and by Corollary \[nodegree5n\] the degree of each branch point is at most $4$. We consider the cases when $K$ has no branch points, one branch point or two branch points.
#### [**Case 1:**]{} $K$ has no branch points.
In this case $K$ is either homeomorphic to the arc, $I$, or to the simple closed curve, $S^1$.
Assume first that $K$ is homeomorphic to $I$, then, by the way $K$ is obtained, $K=G$. Otherwise, reattaching the last terminal edge that was removed gives a simple triod which is not $7$–ac, contrary to the hypothesis. In this case $G$ is on the list.
Next assume $K$ is homeomorphic to $S^1$. If $K=G$, then $G$ is on the list. So assume $G\neq K$, and let $e$ denote the last terminal edge that was removed. Then $K\cup e$ is homeomorphic to the lollipop curve. Furthermore, $G= K\cup e$, otherwise reattaching the penultimate terminal edge will give a homeomorphic copy of the graph (a) of Example (E) which is not $7$–ac, or a simple closed curve with two arcs attached to it at the same point at one of their end points which is not $7$–ac either. Hence, again, $G$ is on the list.
#### [**Case 2:**]{} $K$ has one branch point.
Note that the only possibility for $K$ to have a single branch point of degree $3$ is for $K$ to be homeomorphic to a simple triod or to the lollipop curve, the former is not $7$–ac and the latter is not a reduced graph. Hence the degree of the branch point of $K$ is $4$. In this case $K$ is homeomorphic to either a simple $4$–od, a simple closed curve with two arcs attached to it at the same point at one of their end points, or to the figure eight curve. The first two cases are not $7$–ac. Therefore $K$ must be homeomorphic to the figure eight curve. If $G=K$, then $G$ is on the list. In fact, since attaching an arc to the figure eight curve yields a non $7$–ac curve, we must have that $G=K$.
#### [**Case 3:**]{} $K$ has two branch points.
Since the sum of the degrees in a graph is always even and $K$ has no terminal edges, then $K$ can not have one branch point of degree $3$ and another of degree $4$. Hence the only options are that $K$ has two branch points of either degree $3$ or degree $4$. However, by Corollary \[no24\], $K$ has only two branch points of degree $3$.
If $K$ has two branch points of degree $3$, then it could be homeomorphic to one of the following graphs.
[ccccc]{} (a) & (b) & (c) & (d) & (e)\
(base) ; (top) at (0,2cm) ; (base\_l) at (-0.5cm,0) ; (base\_r) at (0.5cm,0) ; (top\_l) at (-0.5cm,2cm) ; (top\_r) at (0.5cm,2cm) ;
(base.north) – (top.south); (base\_l.east) – (base.west); (base.east) – (base\_r.west); (top\_l.east) – (top.west); (top.east) – (top\_r.west);
&
at (0,-1) ; at (0,1) ;
(0,0) ellipse (6mm and 7mm); (v1) at (180:10mm) ; (v2) at (0:10mm) ; (v3) at (180:6mm) ; (v4) at (0:6mm) ;
(v1) – (v3); (v2) – (v4);
&
\[vertex/.style=[circle,draw=blue!50,fill=blue!20,thick, inner sep=0mm, minimum size=1mm]{}, point/.style=[coordinate]{}\]
(lollipop\_bot) ; (lollipop\_mid) \[above=of lollipop\_bot\] ; (pt) \[above=of lollipop\_mid\] ; (base\_1) at (-0.5cm,0) ; (base\_2) at (0.5cm,0) ;
(lollipop\_bot.north) – (lollipop\_mid.south); (lollipop\_mid.east) to \[out=10,in=10,looseness=1.5\] (pt); (pt) to \[out=170, in=170,looseness=1.5\] (lollipop\_mid.west); (base\_1.east) – (lollipop\_bot.west); (lollipop\_bot.east) – (base\_2.west);
&
\[vertex/.style=[circle,draw=blue!50,fill=blue!20,thick, inner sep=0mm, minimum size=1mm]{}, point/.style=[coordinate]{}\] at (0,-1) ; at (0,1) ;
(dumbbell\_left) ; (dumbbell\_right) at (0.5cm,0) ; (pt1) \[left=of dumbbell\_left\] ; (pt2) \[right=of dumbbell\_right\] ;
(dumbbell\_left.east) – (dumbbell\_right.west); (dumbbell\_left.north) to \[out=90, in=90,looseness=1.5\] (pt1.north); (pt1.south) to \[out=270, in=270,looseness=1.5\] (dumbbell\_left.south); (dumbbell\_right.north) to \[out=90, in=90,looseness=1.5\] (pt2.north); (pt2.south) to \[out=270, in=270,looseness=1.5\] (dumbbell\_right.south);
&
\[vertex/.style=[circle,draw=blue!50,fill=blue!20,thick, inner sep=0mm, minimum size=1mm]{}, point/.style=[coordinate]{}\] at (0,-1) ; at (0,1) ;
(theta\_left) ; (theta\_right) \[right=of theta\_left\] ;
(theta\_left.east) to (theta\_right.west) ; (theta\_left.north) to \[out=90, in=90,looseness=2.5\] (theta\_right.north); (theta\_right.south) to \[out=270, in=270,looseness=2\] (theta\_left.south);
However the graphs (a), (b), and (c) contain terminal edges. So $K$ can only be homeomorphic to the dumbbell (d) or the $\theta$–curve (e); in any case if $G=K$, then $G$ is on the list. Note that neither curve, (d) nor (e), can be obtained from a $n$–ac graph ($n\geq 7$) by removing a terminal edge since by Theorem \[no7wac\] the edge has to be attached to one of the existing branch points; it is easy to see that such a graph is not $4$–ac, just take a point on the interior of each edge. Hence $G=K$. This ends the proof of the theorem.
$\aleph_0$–ac Continua {#aleph0}
----------------------
Call a space $\kappa$–ac, where $\kappa$ is a cardinal, if every subset of size no more than $\kappa$ is contained in an arc. Note that for finite $\kappa=n$ and $\kappa=\aleph_0$ this coincides with the earlier definitions. For infinite $\kappa$ we have a complete description of $\kappa$–ac continua (not necessarily metrizable), extending Theorem \[main2\]. To start let us observe that the arc is $\kappa$–ac for every cardinal $\kappa$. We will see shortly that the arc is the only separable $\kappa$–ac continuum when $\kappa$ is infinite. In particular, the triod and circle are not $\aleph_0$–ac, and so any continuum containing a triod or circle is also not $\aleph_0$–ac. This observation will be used below.
To state the theorem precisely we need to make a few definitions. Recall that $\omega_1$ is the first uncountable ordinal, or equivalently the set of all countable ordinals, with the induced order topology. Note that a subset of $\omega_1$ is bounded if and only if the set is countable. The [*long ray*]{}, $R$, is the lexicographic product of $\omega_1$ with $[0,1)$ with the order topology. We can identify $\omega_1$ (with its usual order topology) with $\omega_1 \times \{0\}$. Evidently $\omega_1$ is cofinal in the long ray. Write $R^-$ for $R$ with each point $x$ relabeled $-x$. The [*long line*]{}, $L$, is the space obtained by identifying $0$ in the long ray, $R$, with $-0$ in $R^-$. The topology on the long ray and long line ensures that for any $x < y$ in $R$ (or $L$) the subspace $[x,y]=\{z \in R : x \le z \le y\}$ is (homeomorphic to) an arc. Note that any countable subset of the long ray, or the long line, is bounded, hence both the long ray and long line are $\aleph_0$–ac. To see this for the long ray take any countable subset $S$ then since $\omega_1$ is cofinal in $R$ the set $S$ has an upper bound, $x$ say, and then $S$ is contained in $[0,x]$, which is an arc.
Let $\alpha R$ be the one point compactification of $R$, and $\gamma L$ be the corresponding two point compactification of $L$. The [*long circle*]{} and [*long lollipop*]{} are the spaces obtained from $\alpha R$ by identifying the point at infinity to $0$, or any other point, respectively. The [*long dumbbell*]{}, [*long figure eight*]{} and [*long theta*]{} curves come from $\gamma L$ by respectively identifying the negative ($-\infty$) and positive ($+\infty$) endpoints to $-1$ and $+1$, $0$ and $0$, or $+1$ and $-1$. As continuous injective images of the $\aleph_0$–ac spaces $R$ and $L$, all the above spaces are also $\aleph_0$–ac.
\[aleph0\_ch\] Let $K$ be a continuum.
- If $K$ is separable and $\aleph_0$–ac then $K$ is an arc.
- If $K$ is non–separable, then the following are equivalent:
\(i) $K$ is $\aleph_0$–ac, (ii) $K$ is the continuous injective image of a closed sub–interval of the long line, and (iii) $K$ is one of: the long circle, the long lollipop, the long dumbbell, long figure eight, or the long theta–curve.
- If $K$ is $\kappa$–ac for some $\kappa > \aleph_0$, then $K$ is an arc.
For part 1) just take a dense countable set, then any arc containing the dense set is the whole space. Part 2) is proved in Proposition \[nonsepaleph0\] ((i) $\implies$ (ii)), Proposition \[11image\] ((ii) $\implies$ (iii)), while (iii) $\implies$ (i) was observed above with the definition of the curves in 2) (iii). For part 3) note that all non–separable $\aleph_0$–ac spaces (as listed in part 2) (iii)) have a dense set of size $\aleph_1$, and so are not $\aleph_1$–ac. Thus $\kappa$–ac continua for $\kappa \ge \aleph_1$ are separable, hence an arc, by part 1).
It is traditional to use Greek letters ($\alpha, \beta$ etcetera) for ordinals. Consequently we will use the letter ‘$A$’ and variants for arcs, and because in Proposition \[nonsepaleph0\] we need to construct a map, in this subsection by an ‘arc’ we mean any homeomorphism between a homeomorph of the closed unit interval and a given space. If $K$ is a space, then by ‘$A$ is an arc in $K$’ we mean the arc $A$ maps into $K$. When $A$ is an arc in a space $K$, then write $\mathop{im}(A)$ for the image of $A$ (it is, of course, a subspace of $K$ homeomorphic to the closed unit interval). For any function $f$, we write ${\mathop{dom}(f)}$, for the domain of $f$.
[\[nonsepaleph0\]]{} Let $K$ be an $\aleph_0$–ac non–separable continuum. Then there is a continuous bijection $A_\infty : J_\infty\rightarrow K$ where $J_\infty$ is a closed unbounded sub–interval of the long line, $L$.
We prove this by an application of Zorn’s Lemma. The following lemmas help to establish that Zorn’s Lemma is applicable, and that the maximal object produced is as required.
[\[l1\]]{} Let $K$ be an $\aleph_0$–ac non–separable continuum. If $\mathcal{K}$ is a countable collection of separable subspaces of $K$ then there is an arc $A$ in $K$ such that $\bigcup \mathcal{K}\subseteq \mathop{im}(A)$.
Let $\mathcal{K}=\{S_n : n \in \mathbb{N}\}$ be a countable family of subspaces of $K$, and, for each $n$, let $D_n$ be a countable dense subset of $S_n$. Let $D=\bigcup_n D_n$ — it is countable. Since $K$ is $\aleph_0$–ac there is an arc $A$ in $K$ such that $D \subseteq \mathop{im}(A)$. As $D$ is dense in $\bigcup \mathcal{K}$ and $\mathop{im}(A)$ is closed, we see that $\bigcup \mathcal{K} \subseteq \mathop{im}(A)$.
[\[l3\]]{} Let $K$ be an $\aleph_0$–ac non–separable continuum. Suppose $[a,b]$ is a proper closed subinterval of $L$ (or $R$), $A : [a,b] \to K$ is an arc in $K$ and $y\in K\setminus \mathop{im}(A)$. Then either (i) for every $c>b$ in $L$ there is an arc $A':[a,c]\rightarrow K$ such that $A'\restriction_{[a,b]} =A$ and $A'(c)=y$, or (ii) for every $c<a$ in $L$ there is an arc $A':[c,b]\rightarrow K$ such that $A'\restriction_{[a,b]} =A$ and $A'(c)=y$.
Fix $a,b$, the arc $A$ and $y$. Let $\mathcal{K}=\{\mathop{im}(A) , \{y\}\}$, and apply Lemma \[l1\] to get an arc $A_0:[0,1] \to K$ in $K$ such that $\mathop{im}(A_0) \supseteq \mathop{im}(A) \cup \{y\}$. Let $J=A_0^{-1} (\mathop{im}(A))$, $a'=\min J$, $b'=\max J$ and $c'=A_0^{-1} (y)$. Without loss of generality (replacing $A_0$ with $A_0 \circ \rho$ where $\rho (t)=1-t$ if necessary) we can suppose that $A_0(a')=A(a)$ and $A_0(b')=b$.
Since $y\not\in \mathop{im}(A)$, either $c'>b'$ or $c'<a'$. Let us suppose that $c' >b'$. This will lead to case (i) in the statement of the lemma. The other choice will give, by a very similar argument which we omit, case (ii). Take any $c$ in $L$ such that $c>b$. Let $A_1$ be a homeomorphism of the closed subinterval $[a,c]$ of $L$ with the subinterval $[a',c']$ of $[0,1]$ such that $A_1(a)=a'$, $A_1(b)=b'$ and $A_1(c)=c'$. Set $A_2=A_0 \circ A_1 : [a,c] \to K$. So $A_2$ is an arc in $K$ such that $A_2(a)=A(a)$, $A_2(b)=A(b)$, $A_2(c)=y$ and $A_2([a,b])= \mathop{im} (A)$. The arc $A_2$ is almost what we require for $A'$ but it may traverse the (set) arc $\mathop{im}(A)$ at a ‘different speed’ than $A$. Thus we define $A':[a,c] \to K$ to be equal to $A$ on $[a,b]$ and equal to $A_2$ on $[b,c]$. Then $A'$ is the required arc.
[**(Of Proposition \[nonsepaleph0\])**]{} Let $\mathcal{A}$ be the set of all continuous injective maps $A: J \to K$ where $J$ is a closed subinterval of $L$, ordered by: $A \le A'$ if and only if ${\mathop{dom}(A)} \subseteq {\mathop{dom}(A')}$ and $A'\restriction_{{\mathop{dom}(A)}} = A$. Then $\mathcal{A}$ is the set of all candidates for the map we seek, $A_\infty$. We will apply Zorn’s Lemma to $(\mathcal{A},\le)$ to extract $A_\infty$. To do so we need to verify that $(\mathcal{A},\le)$ is non–empty, and all non–empty chains have upper bounds.
As $K$ is $\aleph_0$–arc connected we know there are many arcs in $K$, so the set $\mathcal{A}$ is not empty. Now take any non–empty chain $\mathcal{C}$ in $\mathcal{A}$. We show that $\mathcal{C}$ has an upper bound. Let $\mathcal{J}=\{ {\mathop{dom}(A')} : A' \in \mathcal{C}\}$. Since $\mathcal{J}$ is a chain of subintervals in $L$, the set $J = \bigcup \mathcal{J}$ is also a subinterval of $L$. Define $A:J \to K$ by $A(x)=A'(x)$ for any $A'$ in $\mathcal{C}$ with $x \in {\mathop{dom}(A')}$. Since $\mathcal{C}$ is a chain of injections, $A$ is well–defined and injective. Since the domains of the functions in $\mathcal{C}$ form a chain of subintervals, any point $x$ in $J$ is in the $J$–interior of some ${\mathop{dom}(A')}$ (there is a set $U$, open in $J$ such that $x \in U \subseteq {\mathop{dom}(A')}$), where $A' \in \mathcal{C}$, and so $A$ coincides with $A'$ on some $J$–neighborhood of $x$, thus, since $A'$ is continuous at $x$, the map $A$ is also continuous at $x$. If $J$ is closed, then we are done: $A$ is in $\mathcal{A}$ and $A \ge A'$ for all $A'$ in $\mathcal{C}$.
If the interval $J$ is not closed then it has at least one endpoint (in $L$) not in $J$. We will suppose $J=(a,\infty)$. The other cases, $J=(a,b)$ and $J=(-\infty,a)$, can be dealt with similarly. We show that we can continuously extend $A$ to $[a,\infty)$. If so then $A$ will be injective, hence in $\mathcal{A}$, and an upper bound for $\mathcal{C}$. Indeed the only way the extended $A$ could fail to be injective was if $A(a)=A(c)$ for some $c>a$, and then $A([a,c])$ is a circle in $K$, contradicting the fact that $K$ is $\aleph_0$–ac.
Evidently it suffices to continuously extend $A'=A\restriction _{(a,b]}$ to $[a,b]$. Let $\mathcal{K}=\{ A( (a,b])\}$ and apply Lemma \[l3\] to see that $A'$ maps the half open interval, $(a,b]$, into (a homeomorphic copy of) the closed unit interval. So we can apply some basic real analysis to get the extension. Indeed the map $A'$ is continuous and injective, and hence strictly monotone. By the inverse function theorem, $A'$ has a continuous inverse, and so is a homeomorphism of $(a,b]$ with some half open interval, $(c,d]$ or $[d,c)$ in the closed unit interval. Defining $A(a)=c$ gives the desired continuous extension.
Let $A_\infty$ be a maximal element of $\mathcal{A}$. Then its domain is a closed subinterval of the long line, $L$. We first check that ${\mathop{dom}(A_\infty)}$ is not bounded. Then we prove that $A_\infty$ maps onto $K$.
If $A_\infty$ had a bounded domain, then it is an arc. So it has separable image. As $K$ is not separable we can pick a point $y$ in $K \setminus \mathop{im}(A_\infty)$. Applying Lemma \[l3\] we can properly extend $A_\infty$ to an arc $A'$. But then $A'$ is in $\mathcal{A}$, $A_\infty \le A'$ and $A_\infty \ne A'$, contradicting maximality of $A_\infty$.
We complete the proof by showing that $A_\infty$ is surjective. We go for a contradiction and suppose that instead there is a point $y$ in $K \setminus \mathop{im}(A_\infty)$. Two cases arise depending on the domain of $A_\infty$.
Suppose first that ${\mathop{dom}(A_\infty)}=L$. Pick a point $x$ in $\mathop{im}(A_\infty)$. Pick an arc $A$ from $x$ to $y$. Taking a subarc, if necessary, we can suppose $A:[0,1] \to K$, $A(0)=x$ and $A(t) \notin \mathop{im}(A_\infty)$ for all $t >0$. Let $x'=A_{\infty}^{-1}(x)$. Pick any $a',b'$ from $L$ such that $a' < x' < b'$. Then the subspace $A_\infty ([a',b']) \cup A([0,1])$ is a triod in $K$, which contradicts $K$ being $\aleph_0$–sac.
Now suppose that ${\mathop{dom}(A_\infty)}$ is a proper subset of $L$. Let us assume that ${\mathop{dom}(A_\infty)}=[a,\infty)$. (The other case, ${\mathop{dom}(A_\infty)}=(-\infty,a]$, follows similarly.) Pick any $b>a$, and apply Lemma \[l3\] to $A=A_{\infty}\restriction_{[a,b]}$ and $y$. If case (ii) holds then pick any $c<a$ and $A$ can be extended ‘to the left’ to an arc $A'$ with domain $[c,b]$. This gives a proper extension of $A_\infty$ defined on $[c,\infty)$ (which is $A'$ on $[c,a]$ and $A_\infty$ on $[a,\infty)$), contradicting maximality of $A_\infty$.
So case (i) must hold. Pick any $c >b$, and we get an arc $A':[a,c] \to K$ in $K$ extending $A$. Let $T=A_{\infty} ([a,c]) \cup A'([a,c])$. Observe that $T$ has at least three non cutpoints, namely $A'(a)=A_{\infty}(a)$, $A_\infty(c)$ and $A'(c)$. So $T$ is not an arc, but it is a separable subcontinuum of the $\aleph_0$–ac continuum $K$, which is the desired contradiction.
To complete the proof of Theorem \[aleph0\_ch\] it remains to identify the continuous injective images of closed sub–intervals of the long line. We recall some basic definitions and facts connected with the space of countable ordinals, $\omega_1$ (see [@Kunen], for example). A subset of $\omega_1$ is *closed and unbounded* if it is cofinal in $\omega_1$ and closed in the order topology. A countable intersection of closed and unbounded sets is closed and unbounded. The set $\Lambda$ of all limit ordinals in $\omega_1$ is a closed and unbounded set. A subset of $\omega_1$ is *non-stationary* if it is contained in the complement of a closed and unbounded set. A subset of $\omega_1$ is *stationary* if it is not non-stationary, or equivalently if it meets every closed and unbounded set. The Pressing Down Lemma (also known as Fodor’s lemma) states than if $S$ is a stationary set and $f: S \to \omega_1$ is regressive (for every $\alpha$ in $S$ we have $f(\alpha) < \alpha$) then there is a $\beta$ in $\omega_1$ such that $f^{-1}(\beta)$ is cofinal in $\omega_1$.
\[11image\] If $K$ is a non–separable continuum and is the continuous injective image of a closed sub–interval of the long line, then $K$ is one of: the long circle, the long lollipop, the long dumbbell, long figure eight, or the long theta–curve.
The closed non–separable sub–intervals of the long line are (up to homeomorphism) just the long ray and long line, itself.
Let us suppose for the moment that the $K$ is the continuous injective image of the long ray, $R$. We may as well identify points of $K$ with points in $R$. Note that on any closed subinterval, $[a,b]$ say, of $R$, (by compactness of $[a,b]$ in $R$, and Hausdorffness of $K$) the standard order topology and the $K$–topology coincide. It follows that at any point with a bounded $K$–open neighborhood the standard topology and $K$–topology agree. We will show that there is a point $x$ in $R$ such that every $K$–open $U$ containing $x$ contains a tail, $(t,\infty)$, for some $t$. Assuming this, then by Hausdorffness of $K$, every point distinct from $x$ has bounded neighborhoods, and so $x$ is the only point where the $K$–topology differs from the usual topology. Then $K$ is either the long circle or long dumbbell depending on where $x$ is in $R$ (in particular, if it equals $0$). The corresponding result for continuous injective images of the long line follows immediately.
Suppose, for a contradiction, that for every $x$ in $R$, there is a $K$–open set $U_x$ containing $x$ such that $U_x$ contains no tail. By compactness of $K$, some finite collection, $U_{x_1}, \ldots , U_{x_n}$, covers $K$. Let $S_{i}=U_{x_i} \cap \Lambda$, where $\Lambda$ is the set of limits in $\omega_1$. Then (since the finitely many $S_i$ cover the closed unbounded set $\Lambda$) at least one of the $S_i$ is stationary. Take any $\alpha$ in $S_i$, and consider it as a point of the closed subinterval $[0,\alpha]$ of $R$, where we know the standard topology and the $K$–topology agree. Since $\alpha$ is a limit point which is in $U_{x_i} \cap [0,\alpha]$, and this latter set is open, we know there is ordinal $f(\alpha)<\alpha$ such that $(f(\alpha),\alpha] \subseteq U_{x_i}$. Thus we have a regressive map, $f$, defined on the stationary set $S_i$, so by the Pressing Down Lemma there is a $\beta$ such that $f^{-1} (\beta)$ is cofinal in $\omega_1$. Hence $U_{x_i}$ contains $\bigcup \{ (f(\alpha),\alpha] : \alpha \in f^{-1}(\beta)\} = (\beta,\infty)$, and so $U_{x_i}$ does indeed contain a tail.
Complexity of Characterizations {#cplxty}
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Theorem \[main3\] from the Introduction makes certain claims about the complexity of characterizing, for various $n$, the $n$–ac graphs which are not $(n+1)$–ac. We introduce the necessary technology from descriptive set theory to make these claims precise. Then Theorem \[main3formal\] is the formalized version of Theorem \[main3\].
Recall (see [@kech]) that the Borel subsets of a space ramify into a hierarchy, $\Pi_\alpha, \Sigma_\alpha$, indexed by countable ordinals. Sets lower in the hierarchy are less complex than those found higher up. Most relevant here are: $\Pi_3$ which is the set of $F_{\sigma \delta}$ subsets, $\Sigma_3$ which is the set of all $G_{\delta \sigma}$ subsets, and $D_2(\Sigma_3)$ the set of intersections of one $\Pi_3$ and one $\Sigma_3$ set.
The complexity of a set in terms of its position in the Borel hierarchy is precisely correlated to the complexity of the logical formulae needed to define it. A $\Pi_3$ set, $S$, can be defined by a formula, $\phi$ (via $S=\{ x : \phi (x) \text{ is true}\}$), of the form $\forall p \exists q \forall r \ \text{(something simple)}$, where the quantifiers run over [*countable*]{} sets, and ‘something simple’ is boolean. A $\Sigma_3$ set, $T$, can be defined by a formula, $\psi$, of the form $\exists p \forall q \exists r \ \text{(something simple)}$. While a $D_2(\Sigma_3)$ set can be defined by a formula of the form $\phi \land \psi$, where $\phi$ and $\psi$ are as above.
For example, let $S_3^* = \{ \alpha \in 2^{\mathbb{N} \times \mathbb{N}} : \exists J \, \forall j >J \, \exists k \ \alpha (j,k)=0\}$, and $P_3 = \{ \beta \in 2^{\mathbb{N} \times \mathbb{N}} : \forall j \, \exists K \, \forall k \ge K \ \beta (j,k) =0\}$. Then $S_3^*$ is $\Sigma_3$, and $P_3$ is $\Pi_3$ in $2^{\mathbb{N} \times \mathbb{N}}$. And $S_3^* \times P_3$ is a $D_2(\Sigma_3)$ subset of $\left( 2^{\mathbb{N} \times \mathbb{N}} \right)^2$.
For a class of subsets $\Gamma$, a set $A$ is $\Gamma$–hard if $A$ is not in any proper subclass, while it is $\Gamma$–complete if it $\Gamma$–hard and in $\Gamma$. In other words, $A$ is $\Gamma$–complete if and only if it has complexity precisely $\Gamma$. It is known [@kech] that $S_3^*$ is $\Sigma_3$–complete, $P_3$ is $\Pi_3$–complete, and $S_3^* \times P_3$ is $D_2(\Sigma_3)$–complete. We can re–phrase these last two statements as follows: there is a formula characterizing $P_3$ of the form, $\forall \exists \forall$ but we can be [*certain*]{} that [*no logically simpler characterizing formula exists*]{}, and there is a formula characterizing $S^*_3 \times P_3$ of the form, $(\exists \forall \exists) \land (\forall \exists \forall)$ but no logically simpler characterizing formula exists.
Let $A \subseteq X$, $B \subseteq Y$ and $f$ a continuous map of $X$ to $Y$ such that $f^{-1}(B)=A$ (such an $f$ is a [*Wadge reduction*]{}). Note that if $B$ is in some Borel class $\Gamma$, then by continuity so is $A=f^{-1}(B)$. Hence if $A$ is $\Gamma$–hard, then so is $B$.
We work inside the hyperspace $C(I^N)$ of all subcontinua of $I^N$ with the Vietoris topology, which makes it a continuum. In light of the remarks above, it should now be clear that the following is indeed a formal version of Theorem \[main3\].
[\[main3formal\]]{} Fix $N \ge 2$. Inside the space $C(I^N)$:
- the set of graphs which are $\omega$–ac is $\mathbf{\Pi}_3$–complete,
- any family of homeomorphism classes of graphs is $\Pi_3$–hard and always $D_2(\mathbf{\Sigma}_3)$, and
- the set of $n$–ac not $(n+1)$–ac graphs is $D_2(\mathbf{\Sigma}_3)$–complete, for $n=2,3,4,5$.
Claims 1)–3) are the contents of Lemmas \[omega\_c\], \[between\] and Proposition \[n\_c\], respectively.
\[between\] Let $\mathcal{C}$ be any collection of graphs. Then $H(\mathcal{C})$, the set of all subcontinua of $I^N$ homeomorphic to some member of $\mathcal{C}$, is $\Pi_3$–hard and in $D_2(\Sigma_3)$.
That $H(\mathcal{C})$ is $\Pi_3$–hard is immediate from Theorem 7.3 of [@cdm]. It remains to show it is in $D_2(\Sigma_3)$.
For spaces $X$ and $Y$, write $X \le Y$ if $X$ is $Y$–like, $X < Y$ if $X \le Y$ but $Y \not \le X$ and $X \sim Y$ if $X \le Y$ and $Y \le X$. Further write $\mathcal{L}_X = \{ Y : Y \le Y\}$ and $Q(X)=\{ Y : X \sim Y\}$.
Let $\mathcal{C}_0$ be a maximal family of pairwise nonhomeomorphic members of $\mathcal{C}$. Up to homeomorphism there are only countably many graphs. So enumerate $\mathcal{C}_0=\{G_m : m \in \mathbb{N}\}$. According to Theorem 1.7 of [@cdm], for a graph $G$ and Peano continuum, $P$, we have that $P$ is $G$–like if and only if $P$ is a graph obtained from $G$ by identifying to points disjoint (connected) subgraphs. For a fixed graph $G$, then, there are, up to homeomorphism, only finitely many $G$–like graphs. For each $G_m$ in $\mathcal{C}$ pick graphs $G_{m,i}$ for $i=1, \ldots , k_m$ such that each $G_{m,i}$ is $<G$, and if $G'$ is a graph such that $G' < G$ then for some $i$ we have $H(G')=H(G_{m,i})$.
For a graph $G$, $H(G)=Q(G)$ ([@KaYe]). Hence, writing $\mathcal{P}$, for the class of Peano continua, we have that $H(\mathcal{C}) = \bigcup_m Q(G_m) = \mathcal{P} \cap \left( \bigcup_m R_{m} \right)$, where $R_{m}=\mathcal{L}_{G_m} \setminus \bigcup_{i=1}^{k_m} \mathcal{L}_{G_{m,i}} = \mathcal{L}_{G_m} \cap \left( C(I^N) \setminus \bigcup_{i=1}^{k_m} \mathcal{L}_{G_{m,i}}\right)$.
By Corollary 5.4 of [@cdm], for a graph $G$, the set $\mathcal{L}_G$ is $\Pi_2$. Hence each $R_{m}$, as the intersection of a $\Pi_2$ and a $\Sigma_2$, is $\Sigma_3$, and so is their countable union. Since $\mathcal{P}$ is $\Pi_3$, we see that $H(\mathcal{C})$ is indeed the intersection of a $\Pi_3$ set and a $\Sigma_3$ set.
\[omega\_c\] The set $AC_\omega$ of all subcontinua of $I^N$ which are $\omega$–ac graphs is $\Pi_3$–complete.
For a graph $G$, $H(G)$ is $\Pi_3$. By Theorem \[main1\], $AC_{\omega}$ is a finite union of $H(G)$ for graphs $G$, and so is also $\Pi_3$. Hence by Lemma \[between\], $AC_\omega$ is $\Pi_3$–complete.
\[n\_c\] For any $n$, let $AC_n$ be the set of subcontinua of $I^N$ which are $n$–ac but not $(n+1)$–ac graphs. Then for $n=2,3,4$ and $5$ the sets $AC_n$ are $D_2(\Sigma_3)$–complete.
According to Lemma \[between\] $AC_n$ is $D_2(\Sigma_3)$, so it suffices to show that $AC_n$ is $D_2(\Sigma_3)$–hard.
To show that $AC_n$ is $D_2(\Sigma_3)$–hard it suffices to show that there is a continuous map $F: \left( 2^{\mathbb{N}\times \mathbb{N}} \right)^2 \to C(I^N)$ such that $F^{-1} (AC_n) = S_3^* \times P_3$. We do the construction for $N=2$. Since $\mathbb{R}^2$ embeds naturally in general $\mathbb{R}^N$, the proof obviously extends to all $N\ge2$.
We first do the case when $n=5$. Then we will explain how to make the minor modifications needed for the other cases, $n=2,3$ and $4$.
For $x, y$ in $\mathbb{R}^2$, let ${\overline{x y}}$ be the straight line segment from $x$ to $y$. Set $O=(0,0)$, $T=(3,1)$, $B_1=(1,0)$, $B_2=(4/3,0)$, $B_3=(5/3,0)$, $B_4=(2,0)$ and $T_1=(1,1)$, $T_2=(4/3,1)$, $T_3=(5/3,1)$, $T_4=(2,1)$. Let $K_0={\overline{O B_4}} \cup {\overline{B_4 T}} \cup {\overline{T T_1}} \cup {\overline{T_1 O}} \cup {\overline{B_2 T_2}} \cup {\overline{B_3 T_3}}$. Then $K_0$ is a $5$–ac not $6$–ac graph. Define $b_j=(1/j,0)$, $t_j=(1/j,1/j)$, $t_j^k=(1/j,1/j-1/(kj))$ and $s_j^k=(1/j-1/(kj(j+1)),0)$. Then $K_J = K_0 \cup \bigcup_{j=1}^J {\overline{b_j t_j}}$ — for each $J$ — is also a $5$–ac not $6$–ac graph.
Let $K_0'$ be $K_0$ with the interior of the line from $O$ to $B_1$, and the interior of the line from $T_4$ to $T$, deleted.
We now define $F$ at some $\alpha$ and $\beta$ in $2^{\mathbb{N} \times \mathbb{N}}$. Fix $j$. If $\alpha (j,k)=1$ for all $k$, then let $R_j={\overline{b_j t_j}} \cup {\overline{b_j b_{j+1}}}$. Otherwise, let $k_0=\min \{ k : \alpha (j,k)=0\}$, and let $R_j = {\overline{b_j t_j^{k_0}}} \cup {\overline{t_j^{k_0} s_j^{k_0}}} \cup {\overline{s_j^{k_0} b_{j+1}}}$.
For any $j,k$ set $p_j=3-1/j$, $q_j^k=1-1/(j+k)$, $\ell_j=p_j + (1/8)(p_{j+1} - p_j)$ and $r_j=p_j + (7/8)(p_{j+1}-p_j)$. Fix $j$. Define $$\begin{aligned}
S_j &=&
{\overline{ (p_j,1) (p_j, q_j^1)}} \cup {\overline{(p_j,q_j^1) (\ell_j,q_j^1)}} \cup {\overline{(\ell_j,1) (p_{j+1},1)}}\\
&\cup & \bigcup \{ {\overline{(\ell_j,q_j^k) (\ell_j,q_j^{k+1}) }} : \beta (j,k)=0\} \\
& \cup & \bigcup \{ {\overline{(\ell_j,q_j^k) (r_j,q_j^k)}} \cup {\overline{(r_j,q_j^k) (\ell_j,q_j^{k+1})}} : \beta (j,k)=1\}.\end{aligned}$$
Let $F(\alpha,\beta) = K_0' \cup \bigcup_j (R_j \cup S_j)$. Then it is straightforward to check $F$ maps $\left(2^{\mathbb{N} \times \mathbb{N}}\right)^2$ continuously into $C([0,4]^2)$.
(1,0) – (2,0) – (3,1); (2,1) – (1,1) – (0,0); (4/3,0) – (4/3,1); (5/3,0) – (5/3,1);
(1,0) – (1, 1-1/2) – (1-1/4,0) – (1/2,0);
at (1,1.2) [1]{}; at (1,1.4) [0]{}; at (1,1.6) [1]{};
(1/2,0) – (1/2,1/2); (1/2,0)– (1/3,0); at (1/2,1.2) [1]{}; at (1/2,1.4) [1]{}; at (1/2,1.6) [1]{};
(1/3,0) – (1/3, 1/3-1/9) – (1/3-1/36,0) – (1/4,0); at (1/3,1.2) [1]{}; at (1/3,1.4) [1]{}; at (1/3,1.6) [0]{};
(1/4,0) – (0,0); at (1/6,1.4) \[label=above:$j$\] […]{}; at (1/2,1.8) \[label=right:$k$\] ;
at (1.25,1.4) [$\alpha$]{};
at (1.75,1.4) [$\beta$]{};
(2,1) – (2,0.5) – (2.0625, 0.5); (2.0625,1) – (2.5,1); (2.065,0.5) – (2.4375,0.5) – (2.0625,0.6666); (2.0625,0.6666) – (2.0625,0.75); (2.0625,0.75) – (2.4375,0.75) – (2.0625,0.8); (2.0625,0.8) – (2.4375,0.8) – (2.0625,0.83333);
(2.0625,0.83333) – (2.0625,1);
at (2,1.2) [1]{}; at (2,1.4) [0]{}; at (2,1.6) [1]{}; at (2,1.8) [1]{};
(2.5,1) – (2.5,0.6666) – (2.5208325, 0.6666); (2.5208325,1) – (2.6666,1); (2.5208325,0.6666) – (2.6458275,0.6666) – (2.5208325,0.75); (2.5208325,0.75) – (2.6458275,0.75) – (2.5208325,0.8); (2.5208325,0.8) – (2.6458275,0.8) – (2.5208325,0.83333); (2.5208325,0.83333) – (2.6458275,0.83333) – (2.5208325,0.857143); (2.5208325,0.857143) – (2.6458275,0.857143) – (2.5208325,0.875); (2.5208325,0.875) – (2.6458275,0.875) – (2.5208325,0.88888); (2.5208325,0.88888) – (2.6458275,0.88888) – (2.5208325,0.9);
(2.58333,0.9) – (2.58333,1);
at (2.5,1.2) [1]{}; at (2.5,1.4) [1]{}; at (2.5,1.6) [1]{}; at (2.5,1.8) \[label=left:$k$\] ;
(2.6666,1) – (2.6666,0.75)–(2.6770775, 0.75); (2.6770775,1) – (2.75,1); (2.6770775,0.75) – (2.7395825,.75) – (2.6770775,0.8); (2.6770775,0.8) – (2.6770775,1);
at (2.6666,1.2) [1]{}; at (2.6666,1.4) [0]{}; at (2.6666,1.6) [0]{};
(2.6770775,1) – (3,1); at (2.83333,1.4) \[label=above:$j$\] […]{}; at (1.5,1.1) [$F(\alpha, \beta)$]{};
Take any $\alpha$. For any $j$, the set $R_j$ connects the bottom edge ${\overline{O B_1}}$ with the diagonal edge ${\overline{O T_1}}$ if $\alpha(j,k)=1$ for all $k$, and otherwise is an arc from $b_j$ to $b_{j+1}$. Hence $\bigcup_{j>J} R_j$ is a free arc from $B_{J+1}$ to $O$ if $\alpha$ is in $S_3^*$, and otherwise can’t be a subspace of a graph (because it contains infinitely many points of order $3$).
Take any $\beta$. For any $j$, $S_j$ is an arc from $(p_j,1)$ to $(p_{j+1},1)$ if $\beta(j,k)=0$ for all but finitely many $k$, but contains a ‘topologists sine curve’ if $\beta(j,k)=1$ for infinitely many $k$. Thus $\bigcup_j S_j$ is a free arc from $T_4$ to $T$ if $\beta$ is in $P_3$, and otherwise can’t be a subspace of a graph (because it contains a ‘topologists sine curve’).
Hence if $(\alpha,\beta)$ is in $S_3^* \times P_3$, $F(\alpha,\beta)$ is homeomorphic to some $K_J$, which in turn means it is a graph which is $5$–ac but not $6$–ac. On the other hand, if either $\alpha$ is not in $S_3^*$ or $\beta$ is not in $P_3$, then $F(\alpha,\beta)$ contains subspaces which can’t be subspaces of a graph — and so is not a graph. Thus $F^{-1} (AC_5) = S_3^* \times P_3$ as required.
Let $T_1^+=(1,2)$, $T_2^+=(4/3,2)$ and $T_3^+=(5/3,2)$. Suppose now that $n=4$. Modify $K_0$ by adding a line segment from $T_1$ to $T_1^+$. Then this modified $K_0$ is $4$–ac but not $3$–ac. Further, for any $J$, the modified $K_J$ obtained by taking the modified $K_0$ as a base is also $4$–ac but not $5$–ac. Thus we get the desired reduction in the case when $n=4$.
Similarly, for $n=3$, modify $K_0$ by adding both a line segment ${\overline{T_1 T_1^+}}$ and ${\overline{T_2 T_2^+}}$. This gives a base graph, and family of $K_J$, which are all $3$-ac but not $4$–ac. Finally, by adding the three line segments, ${\overline{T_1 T_1^+}}$, ${\overline{T_2 T_2^+}}$ and ${\overline{T_3 T_3^+}}$ to $K_0$ we get $2$–ac not $3$–ac graphs. The desired reductions for $n=3$ and $n=2$ follow.
Open Problems {#probs}
=============
The main theorems, Theorem \[main1\], \[main2\] and \[main3\], raise some natural problems.
- Find examples of continua which are $n$–ac but not $(n+1)$–ac for $n \ge 7$.
Theorem \[main1\] implies that no graph is an example. Are there regular examples?
- Find a ‘simple’ (i.e. $\Pi_3$) characterization of $6$–ac graphs which are not $7$–ac. Alternatively, prove that no such characterization is possible, and show that the set of $6$–ac not $7$–ac graphs is $D_2(\Sigma_3)$.
Note (Examples \[exs\] (G)) that there are infinitely many $6$–ac not $7$–ac graphs — rather than the only finite family of $7$–ac graphs — but this does not rule out a ‘simple’ characterization.
- Characterize the $\omega$–ac regular continua.
The Sierpinski triangle is a regular $\omega$–ac continuum. The authors, with Kovan–Bakan, show in [@egkm] that there is no Borel characterization of [*rational*]{} $\omega$–ac continua. However the examples used in that argument are far from regular (not even locally connected).
Acknowledgements {#acknowledgements .unnumbered}
================
The authors thank the referee for his/her comments that improved the paper and for observing that Kuratowski’s $K_{3,3}$ graph is $6$-ac but not $7$-ac (see Example \[exs\] (F)).
[XX]{}
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We report the results of a search for pair production of scalar bottom quarks (${\tilde{b}_1}$) and scalar third-generation leptoquarks ($LQ_3$) in 5.2 fb$^{-1}$ of $p\bar{p}$ collisions at the D0 experiment of the Fermilab Tevatron Collider. Scalar bottom quarks are assumed to decay to a neutralino (${\tilde{\chi}_{1}^0}$) and a $b$ quark, and we set 95% C.L. lower limits on their production in the ($m_{{\tilde{b}_1}}, m_{{\tilde{\chi}_{1}^0}}$) mass plane such as $m_{{\tilde{b}_1}}>247$ GeV for $m_{{\tilde{\chi}_{1}^0}}=0$ and $m_{{\tilde{\chi}_{1}^0}}>110$ GeV for $160<m_{{\tilde{b}_1}}< 200$ GeV. The leptoquarks are assumed to decay to a tau neutrino and a $b$ quark, and we set a 95% C.L. lower limit of 247 GeV on the mass of a charge-1/3 third-generation scalar leptoquark.'
date: 'May 12, 2010'
title: 'Search for scalar bottom quarks and third-generation leptoquarks in collisions at = 1.96 TeV'
---
\#1[0= 0=0 1= 1=1 0>1 \#1 / ]{}
author\_list.tex
The standard model (SM) offers an accurate description of current experimental data in high energy physics but it is believed to be embedded in a more general theory. In particular, extensions of the SM to higher mass scales have been proposed that predict the existence of new particles and phenomena which can be searched for at the Tevatron.
Supersymmetric (SUSY) models provide an extension of the SM that resolves the “hierarchy problem" by introducing supersymmetric partners to the known fermions and bosons [@susy]. The supersymmetric quarks (squarks) are mixtures of the states $\tilde{q}_L$ and $\tilde{q}_R$, the superpartners of the SM quark helicity states. The theory permits a mass difference between the squark mass eigenstates, $\tilde{q}_1$ and $\tilde{q}_2$, and allows the possibility that the lighter states of top and bottom squarks have masses smaller than the squarks of the first two generations. In this analysis we consider the region of SUSY parameter space where the only decay of the lighter bottom squark is $\tilde{b}_1\rightarrow b{\tilde{\chi}_{1}^0}$, with $m_b+m_{{\tilde{\chi}_{1}^0}} <
m_{\tilde{b}_1}<m_t+m_{\tilde{\chi}_1^-}$, and the neutralino ${\tilde{\chi}_{1}^0}$ and chargino $\tilde{\chi}_1^\pm$ are the lightest SUSY partners of the electroweak and Higgs bosons. This analysis is interpreted within the framework of the minimal supersymmetric standard model (MSSM) with $R$-parity [@mssm] conservation, and under the hypothesis that the lightest, and consequently stable, SUSY particle is the ${\tilde{\chi}_{1}^0}$. We therefore search for $p\bar{p} \rightarrow \tilde{b}_1\bar{\tilde{b}}_1 \rightarrow b{\tilde{\chi}_{1}^0}\bar{b}{\tilde{\chi}_{1}^0}$.
Leptoquarks are hypothesized fundamental particles that have color, electric charge, and both lepton and baryon quantum numbers. They appear in many extensions of the SM including extended gauge theories, composite models, and SUSY with $R$-parity violation [@lqtheory]. Current models suggest that leptoquarks of each of the three generations should decay to the corresponding generation of SM leptons and quarks to avoid introducing unwanted flavor changing neutral currents. Charge-1/3 third-generation leptoquarks would decay to $b\nu$ with branching fraction $B$ or to $t\tau$ with branching fraction $1-B$.
We report on a search for the production of pairs of bottom squarks and third-generation scalar leptoquarks in data collected by the D0 Collaboration at the Fermilab Tevatron Collider. For both searches, the signature is defined to be two $b$-jets and missing transverse energy ($\etmiss$) from the escaping neutrinos or neutralinos. This topology is identical to that for $p\bar{p}\rightarrow ZH \rightarrow \nu\bar{\nu}+b\bar{b}$ production, and the two analyses are based on the same data and selection criteria [@zhpaper]. Bottom squark or leptoquark pairs are expected to be produced mainly through $q\bar{q}$ annihilation or $gg$ fusion, with identical leading order QCD production cross sections. We use the next-to-leading order (NLO) cross sections calculated by [prospino]{} 2.1 for both bottom squark [@prospino] and leptoquark [@Kramer:1997hh] pair production, and found them to agree to better than 3%. Previous measurements excluded bottom squark masses $m_{{\tilde{b}_1}} <
222$ GeV for a massless neutralino [@sboldD0], as well as charge-1/3 third-generation scalar leptoquark masses $m_{LQ}<229$ GeV for $B=1$ [@lqold].
The D0 detector [@d0nim] consists of layered systems surrounding the interaction point. The momenta of charged particles and the location of the interaction vertices are determined using a silicon microstrip tracker and a central fiber tracker immersed in the magnetic field of a 2 T solenoid. Jets, electrons, and tau leptons are reconstructed using the tracking information and the pattern of energy deposits in three uranium/liquid-argon calorimeters located outside the tracking system with a central calorimeter covering pseudorapidity $|\eta| < 1.1$, and two end calorimeters housed in separate cryostats covering the regions up to $|\eta| \approx 4.2$. Jet reconstruction uses a cone algorithm [@blazey] with radius ${\cal R} = \sqrt{(\Delta y)^2+(\Delta \phi)^2} = 0.5$ in rapidity ($y$) and azimuth ($\phi$). Muons are identified through the association of tracks with hits in the muon system, which is outside of the calorimeter and consists of drift tubes and scintillation counters before and after 1.8 T iron toroids. The $\etmiss$ is determined from the negative of the vector sum of the transverse components of the energy deposited in the calorimeter and the transverse momenta $p_T$ of detected muons. The jet energies are calibrated using transverse energy balance in events with photons and jets and this calibration is propagated to the value of $\etmiss$.
The data were recorded using triggers based on jets and the $\etmiss$ in the event. In addition to requirements on $\etmiss$ and jet energy, the vector sum of the transverse energies of all jets, defined as $\htmiss \equiv
|\sum_{\text{jets}}\vec{p}_T|$, the scalar sum of the $p_T$ of the jets ($H_T$), and the angle $\alpha$ between the two leading jets in the transverse plane, are also used for triggering. Typical requirements are $\etmiss > 25$ GeV, $\htmiss > 25$ GeV, $H_T >$ 50 GeV, and $\alpha < 169^\circ$. After imposing quality requirements, the data correspond to an integrated luminosity of 5.2 fb$^{-1}$. The previous D0 publications [@sboldD0; @lqold] used a subset of this data sample, and are superseded by the results obtained in this Letter.
Monte Carlo (MC) samples for $200< m_{LQ}<280$ GeV, and for (${\tilde{b}_1},{\tilde{\chi}_{1}^0}$) pairs with $80<m_{{\tilde{b}_1}}<260$ GeV and $m_{{\tilde{\chi}_{1}^0}}<120$ GeV, are generated with [pythia]{} [@Pythia]. Backgrounds from SM processes with significant $\etmiss$ are estimated using MC. The most important backgrounds are from $W/Z$ bosons produced in association with jets, with leptonic decays such as $Z \rightarrow \nu\bar{\nu}$ and $W\rightarrow e\nu$, and processes with $t\bar{t}$ and single top quark production. The cross sections used to estimate these contributions to the background are obtained from [@mc_xsec] and [@mc_xsec1]. At the parton level, vector boson pair production and the single-top quark events are generated with [pythia]{} and [comphep]{} [@CompHEP], respectively, while [alpgen]{} [@Alpgen] is used for all other samples. All MC events are then processed with [pythia]{}, which performs parton showering and hadronization. The resulting samples are processed using a [geant]{} [@GEANT] simulation of the D0 detector. To model the effects of multiple interactions and detector noise, data from random $p\bar{p}$ crossings are overlaid on MC events. The [cteq6l1]{} parameterization [@PDF:CTEQ6L1] is used for all parton density functions (PDF). Instrumental background comes mostly from multijet processes with $\etmiss$ arising from energy mismeasurement. This background, which we label MJ, dominates the low $\etmiss$ region and is modeled using data.
A signal sample and a sample used to model the MJ background are selected. We select events with two or three jets with $\vert\eta\vert<2.5$ and $p_T>20$ GeV, and require that the interaction vertex has at least three tracks and is reconstructed within $\pm$40 cm of the center of the detector along the beam direction so that the tracks are within the geometric acceptance of the silicon tracker. As the leading highest $p_T$ jets in the signal events are assumed to originate from decays of $b$ quarks, we require that at least two jets, including the leading jet, have at least two tracks pointing to the primary vertex in order to apply b-tagging algorithms. We also require the two leading jets satisfy $\alpha < 165^\circ$. To reduce the contribution from $W\rightarrow l\nu$ decays, we veto events with isolated electrons or muons with $p_T>15$ GeV, as well as tau leptons that decay hadronically to a single charged particle with $p_{T}>12$ GeV when there is no associated electromagnetic cluster or $p_{T}>10$ GeV if there is such a cluster [@tauid]. To suppress the MJ background, we require $\etmiss>40$ GeV and $\etmiss$ significance ${\cal S} > 5$ [@metsig]. We also remove events when the direction of the $\etmiss$ overlaps with a jet in $\phi$ by requiring $\etmiss/$GeV $> 80 - 40 \times {\Delta\phi}_{\text{min}}(\etmiss,$ jets), where $\Delta{\phi}_{\text{min}}(\etmiss$, jets) denotes the minimum of the angles between the $\etmiss$ and any of the selected jets.
The contribution from multijet processes is determined using the techniques described in [@zhpaper]. For signal events, the direction of $\etmiss$ tends to be aligned with the missing track transverse momentum, $\ptmiss$, defined as the negative of the vectorial sum of the $p_T$ of the charged particles. A strong correlation of this kind is not expected in multijet events, where $\etmiss$ originates mainly from mismeasurement of jet energies in the calorimeter. We exploit this difference by requiring ${\cal D} < \pi/2 $ for signal, where ${\cal{D}}$ is the azimuthal distance between $\etmiss$ and $\ptmiss$, $\Delta\phi(\etmiss,\ptmiss)$, and use events with ${\cal D} > \pi/2 $ to model the kinematic distributions of the MJ background in the signal sample after subtracting the contribution from SM processes. The MJ background is normalized before $b$-tagging by requiring the number of observed events in data to equal the sum of SM and MJ contributions in the ${\cal D} < \pi/2$ region. The signal contribution is assumed to be zero. Figure \[fig:met\_and\_mht\_pretag\] shows the $\etmiss$ distribution and the background contributions from SM and MJ sources after these selections.
![\[fig:met\_and\_mht\_pretag\] (color online). The $\etmiss$ distribution before $b$-tagging. The points with the error bars represent data while the shaded histograms show the contributions from background processes. Signal distributions with ($m_{{\tilde{b}_1}}$,$m_{{\tilde{\chi}_{1}^0}}$)=(130,85) GeV and $m_{LQ}=240$ GeV are shown as solid and dashed lines, respectively. ](fig1.eps)
A neural network (NN) $b$-tagging algorithm [@nnbtag] is used to identify heavy-flavor jets, and reduce the SM and MJ backgrounds that are dominated by light flavor jets. We apply $b$-tagging and use the requirements on the NN output that give one jet to be tagged with an average efficiency of $\approx$70% and the other with an average efficiency of $\approx$50%, where the corresponding probabilities of a light-flavored jet to be wrongly identified as a $b$-jet are $\approx $6.5% and $\approx$0.5%, respectively. These conditions are designed to optimize the discovery reach for a ${\tilde{b}_1}$ and $LQ_3$.
Additional selections reduce the remaining number of events with poorly measured $\etmiss$. We require ${\Delta\phi}_{\text{min}}(\etmiss, {\rm jets})>0.6$ rad, and define an asymmetry ${\cal A} = (\etmiss-\htmiss)/(\etmiss+\htmiss)$ and require $-0.1 < {\cal A} <0.2$ [@zhold]. The $\etmiss$ and $H_T$ distributions after imposing $b$-tagging and the requirements on ${\Delta\phi}_{\text{min}}(\etmiss,{\rm jets})$ and ${\cal A}$ are shown in Fig. \[fig:L3VTbtag\], along with the expectations for two possible signals which show the kinematic variation for different masses.
![\[fig:L3VTbtag\] (color online). The (a) $\etmiss$ and (b) $H_T$ distributions after $b$-tagging and additional selections. The points with the error bars represent data while the shaded histograms show the contributions from background processes. Signal distributions with ($m_{{\tilde{b}_1}}$,$m_{{\tilde{\chi}_{1}^0}}$)=(130,85) GeV and $m_{LQ}=240$ GeV are shown as solid and dashed lines, respectively ](fig2a.eps "fig:")\
![\[fig:L3VTbtag\] (color online). The (a) $\etmiss$ and (b) $H_T$ distributions after $b$-tagging and additional selections. The points with the error bars represent data while the shaded histograms show the contributions from background processes. Signal distributions with ($m_{{\tilde{b}_1}}$,$m_{{\tilde{\chi}_{1}^0}}$)=(130,85) GeV and $m_{LQ}=240$ GeV are shown as solid and dashed lines, respectively ](fig2b.eps "fig:")
--------------------------------------------------------------- -------------------------- --------------------------- -------------------------------------------- --------------------------- -----------------------------
Process Pretag $b$-tag $-0.1<{\cal A}<0.2$ $X_{jj}>0.75$ $X_{jj}>0.9$
${\Delta\phi}(\etmiss,{\text {jets}})>0.6$ $p^{\text{jet1}}_T $p^{\text{jet1}}_T >50$ GeV
>20$ GeV
$\etmiss>40$ GeV $\etmiss>150$ GeV
$H_T>60$ GeV $H_T>220$ GeV
Diboson 2,060 38 35 31 0.3
$W(\rightarrow l{\nu})$ + light jets 49,250 130 119 105 0.5
$Wc\bar{c}, Wb\bar{b}$ 7,792 353 325 261 1.9
$Z(\rightarrow ll)$ + light jets 17,663 11 9 8 0
$Zc\bar{c},Zb\bar{b}$ 4,526 256 247 217 1.9
Top 2,019 348 301 190 2.2
MJ 30,243 444 205 157 0
Total background 113,553 1,579 $\pm$ 230 1,242 $\pm$ 188 971 $\pm$ 152 6.9 $\pm$ 1.7
\# data events 113,553 1,463 1,131 901 7
Signal (acceptance, %)
($m_{{\tilde{b}_1}}$,$m_{{\tilde{\chi}_{1}^0}}$)=(240,0) GeV 145 $\pm$ 11 (38.7) 43.3 $\pm$ 6.4 (11.4) 42.0 $\pm$ 6.2 (11.1) - 10.5 $\pm$ 1.9 (2.8)
($m_{{\tilde{b}_1}}$,$m_{{\tilde{\chi}_{1}^0}}$)=(130,85) GeV 1,928 $\pm$ 158 (10.9) 544 $\pm$ 85 (3.1) 529 $\pm$ 77 (3.0) 481 $\pm$ 66 (2.7) -
--------------------------------------------------------------- -------------------------- --------------------------- -------------------------------------------- --------------------------- -----------------------------
We then apply final selections to improve the sensitivity. As our signals consist of two high-$p_T$ $b$-jets, we use $X_{jj} \equiv ( p_{T}^{\text{jet1}} + p_{T}^{\text{jet2}}
)/H_T$ as a discriminant against top-quark processes. We optimize selections on $p_{T}^{\text{jet1}}$, $\etmiss$, $H_{T}$, and $X_{jj}$ for different ($m_{{\tilde{b}_1}},m_{{\tilde{\chi}_{1}^0}}$) and $m_{LQ}$ by choosing selections that yield the smallest expected limit on the cross section. These selections are more restrictive for $LQ_3$ and ${\tilde{b}_1}$ signals with larger mass. For regions with small $m_{{\tilde{b}_1}}-m_{{\tilde{\chi}_{1}^0}}$, the average $\etmiss$ and jet energies are lower, and relaxed requirements are found to be optimal. The results of the selections, and the predicted numbers of events from background processes are listed in Table \[table-jlip-sl\], including two final signal selection examples. For a signal with high $\etmiss$, the largest backgrounds are from $W/Z$ + $b\bar{b}$ production and top quark processes. There is in addition a significant contribution from multijets for bottom squark signal points with a small value of $\etmiss$.
Systematic uncertainties include those on the integrated luminosity (6.1%), trigger efficiency (2%), and jet energy calibration and reconstruction (3% for signal and (2–7)% for background). Uncertainties associated with $b$-tagging are (6–17)% for signal and (5–11)% for background. Uncertainties on theoretical cross sections for SM processes include 10% on top quark production, and 6% on the total ($W/Z$)+jets cross section with an additional 20% uncertainty on heavy flavor content. The contribution from the MJ background is assigned a 25% uncertainty which includes the impact of possible signal events contained in the pretag sample.
We obtain limits on the pair production cross section multiplied by the branching fraction squared ($\sigma\times B^2$) using the $CL_s$ approach [@Junk]. In this technique, an ensemble of MC experiments using the expected numbers of signal and background events is compared to the number of events observed in data to derive an exclusion limit. Signal and background contributions are varied within their uncertainties taking into account correlations among their systematic uncertainties. The $LQ_3$ and ${\tilde{b}_1}$ ($m_{{\tilde{\chi}_{1}^0}}=0$) observed and expected cross section limits are given in Table \[tab::95CL\_LQ3\].
Mass (GeV) 220 240 250 260 280
--------------- ------- ------- ------- ------- -------
Observed (pb) 0.077 0.063 0.056 0.052 0.054
Expected (pb) 0.067 0.056 0.049 0.046 0.040
: \[tab::95CL\_LQ3\] Observed and expected 95% C.L. limits on the cross section for different leptoquark or bottom squark (assuming $m_{{\tilde{\chi}_{1}^0}}=0$) masses.
Figure \[fig:Lq3\_Exclusion\_final\](a) shows the 95% C.L. upper limits on the cross section as a function of $m_{LQ}$, together with the theoretical cross section $\sigma_{\text {th}}$ assuming $B=1$. The uncertainty on $\sigma_{\text {th}}$ is obtained by varying the renormalization and factorization scales by a factor of two from the nominal choice $\mu = m_{LQ}$ and incorporating the PDF uncertainties [@Kramer:1997hh]. Limits on $m_{LQ}$ are obtained from the intersection of the observed cross section limit with the central $\sigma_{\text {th}}$ and yield a lower mass limit of 247 GeV for $B=1$ for the production of third-generation leptoquarks. If the 95% C.L. experimental limit is compared with the one standard deviation lower value of $\sigma_{\text {th}}$, we obtain a mass limit of $m_{LQ}=238$ GeV. Also shown is the central value of $\sigma_{\text {th}}$ when the coupling to the $b\nu$ and $t\tau $ channels are identical, yielding $B=1-0.5\times F_{sp}$ where $F_{sp}$ is a phase space suppression factor for the $\tau t$ channel [@lqold]. The mass limit in this case is 234 GeV.
Figure \[fig:Lq3\_Exclusion\_final\](b) shows the excluded region in the plane of the bottom squark versus neutralino mass obtained using the central $\sigma_{\text {th}}$. For $m_{{\tilde{\chi}_{1}^0}}=0$, the limit is $m_{{\tilde{b}_1}}>247$ GeV. The exclusion region extends to $m_{{\tilde{\chi}_{1}^0}}=110$ GeV for $160<m_{{\tilde{b}_1}}< 200$ GeV.
 
In conclusion, in the 5.2 fb$^{-1}$ data sample studied, the observed number of events with the topology of two $b$-jets plus missing transverse energy is consistent with that expected from known SM processes. We set limits on the cross section multiplied by square of the branching fraction $B$ to the $b\nu$ final state as a function of leptoquark mass. These results are interpreted as mass limits and give a limit of 247 GeV for $B=1$ for the production of charge-1/3 third-generation scalar leptoquarks. We also exclude the production of bottom squarks for a range of values in the ($m_{{\tilde{b}_1}}, m_{{\tilde{\chi}_{1}^0}}$) mass plane such as $m_{{\tilde{b}_1}}>247$ GeV for $m_{{\tilde{\chi}_{1}^0}}=0$ and $m_{{\tilde{\chi}_{1}^0}}>110$ GeV for $160<m_{{\tilde{b}_1}}< 200$ GeV. These limits significantly extend previous results.
[99]{}
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|
{
"pile_set_name": "ArXiv"
}
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---
author:
- |
G. Blazey, J. Colston, A. Dyshkant, K. Francis, J. Kalnins, S. A. Uzunyan, V. Zutshi\
[*Department of Physics, Northern Illinois University, DeKalb, IL 60115, USA*]{};\
S. Hansen, P. Rubinov,\
[ *Fermi National Accelerator Laboratory, Batavia, IL 60510, USA;* ]{}\
E. C. Dukes, Y. Oksuzian,\
[ *University of Virginia, Charlottesville, VA 22904, USA*]{}\
M. Pankuch\
[ *Northwestern Medicine Proton Center, Warrenville, IL 60555, USA*]{}
title: |
Draft version, submitted to NIM November 8, 2018.\
**Radiation Tests of Hamamatsu Multi-Pixel Photon Counters**
---
[Key words: Hamamatsu MPPC, radiation damage, photoelectron spectrum, noise rate, gain, 200 MeV protons]{}
Abstract\[abstract\]
====================
Results of radiation tests of Hamamatsu 2.0$\times$2.0 mm$^2$ through-silicon-via (S13360-2050VE) multi-pixel photon counters, or MPPCs, are presented [@hamamatsu]. Distinct sets of eight MPPCs were exposed to four different 1 MeV neutron equivalent doses of 200 MeV protons. Measurements of the breakdown voltage, gain and noise rates at different bias overvoltages, photoelectron thresholds, and LED illumination levels were taken before and after irradiation. No significant deterioration in performance was observed for breakdown voltage, gain, and response. Noise rates increased significantly with irradiation. These studies were undertaken in the context of MPPC requirements for the Cosmic Ray Veto detector of the Mu2e experiment at the Fermi National Accelerator Laboratory.
Introduction\[intro\]
=====================
The cosmic ray veto (CRV) system of the Mu2e experiment at the Fermi National Accelerator Laboratory (Fermilab) [@Mu2eTDR] is designed to identify incoming cosmic rays with an efficiency of 99.99% in order to suppress signals from cosmic ray interactions that mimic the muon-to-electron conversion signal. Cosmic ray detection is provided by four layers of scintillation counters with embedded wave length shifting (WLS) fibers connected to multi-pixel photon counters (MPPCs) on a mounting block as shown in Fig. \[fig:sipm\_assembly\]. To meet the CRV detection efficiency requirement, summed signals from photodetectors at each end of a scintillation counter should provide a photoelectron (PE) yield of at least 25 PE/cm from a minimum ionizing particle traversing at normal incidence one meter from the counter end. The entire system will require 19,840 MPPCs, which are expected to accumulate a maximum radiation fluence of approximately 1$\times$10$^{10}$ neutrons/cm$^2$ from 1 MeV equivalent neutrons over the lifetime of the Mu2e experiment (see Fig. \[fig:neutron\_damage\]). Note that only a few percent of the MPPCs see fluence above 5$\times$10$^{9}$ neutrons/cm$^2$.
![\[fig:sipm\_assembly\] The MPPC mounting block (left), which accommodates four MPPCs mounted on carrier boards; a closeup of one 2.0$\times$2.0 mm$^2$ S13360-2050VE Hamamatsu MPPC (right) within the carrier board panel.](figure1a.eps "fig:"){height="50mm"} ![\[fig:sipm\_assembly\] The MPPC mounting block (left), which accommodates four MPPCs mounted on carrier boards; a closeup of one 2.0$\times$2.0 mm$^2$ S13360-2050VE Hamamatsu MPPC (right) within the carrier board panel.](figure1b.eps "fig:"){height="50mm"}
![\[fig:neutron\_damage\] Fraction of CRV MPPCs versus expected radiation fluences over lifetime of Mu2e experiment. Distribution of peaks is a result of the layout of the CRV. ](figure2.eps)
--------------------------- -------------------- --
Number of Pixels 1584
Pixel Pitch 50 $\mu$m
Response Range 320 to 900 nm
Peak Sensitivity 450 nm
PDE 40%
Gain 1.7$\times$10$^6$
Terminal Capacitance 140 pF
Dark count rate at 0.5 PE 300 KHz
V$_{breakdown}$ 53V $\pm$ 5V
Bias Voltage V$_{breakdown}$+3V
Reference Temperature 25 $^{\circ}$C
--------------------------- -------------------- --
: Hamamatsu specifications for the 2.0$\times$2.0 mm$^2$ (S13360-2050VE) Through-silicon-via surface mount type MPPCs [@hamamatsu].
\[tab:mppc\_specs\]
We present results of radiation tests of Hamamatsu 2.0$\times$2.0 mm$^2$ MPPCs, which are intended for the instrumentation of the CRV detector. Table \[tab:mppc\_specs\] provides detailed specifications for the devices. The mandated functionality of the CRV over the lifetime of the experiment requires that, with the expected accumulated dose, the MPPC a) breakdown voltages do not drift, b) responses do not change, and c) noise rates remain within the data acquisition system bandwidth with the application of a threshold that maintains rejection. Additionally, resolution of photoelectron spectra for as great a dose as possible is desirable for ease of [*in situ*]{} calibration and subsequent maintenance of readout thresholds. The requirements are listed in Table \[tab:mu2e\_specs\].
------------------------- -------------- --
Breakdown Voltage Drift $<\pm250$ mV
Response Stability $<\pm5\%$
DAQ Bandwidth 1 Mhz
DCR Limit $<$300 KHz
DAQ Threshold $<$6 PE
------------------------- -------------- --
: The Mu2e-CRV requirements for the MPPCs.
\[tab:mu2e\_specs\]
\[1p3tests\]MPPC Tests
======================
To carry out the performance studies, the MPPCs are mounted on small printed circuit boards called carrier board panels (see Fig. \[fig:sipm\_assembly\] right). These carrier board panels connect to the readout electronics through a pogo-board, a passive board that has an array of spring loaded contacts, or pogo-pins. The pogo-board is then mounted on an interface board which communicates with the Mu2e front-end electronics board (FEB) outside a light-tight box through HDMI cables [@FEB]. Fig. \[fig:sipm\_CRboard\] shows a 16-MPPC carrier board panel in contact with the pogo-board which in turn is mounted on the interface board. MPPCs were placed in a light-tight box to take measurements with and without LED (type LED5-UV-400-30 [@ledrref]) illumination. When using LED illumination, short (11 ns) pulses were used to obtain the PE spectrum, while long (16 ns) pulses were used to collect signals with magnitudes comparable to those expected from the CRV counters during operation.
![\[fig:sipm\_CRboard\] The 16-MPPC pogo-board with spring loaded contacts (left) and the MPPC carrier board panel in contact with the pogo-board, mounted on the interface board (right).](figure3.eps)
Radiation damage measurements\[hms\]
====================================
The photoelectron spectrum, the response to a LED pulse, the dark count rates at thresholds of up to 5.5 PE, and the MPPC current as a function of applied bias voltage (I-V scans) were measured for 40 (five sets of eight) non-irradiated MPPCs. Each set of eight MPPCs will henceforth be referred to as a “panel.” One panel was never irradiated while the other four panels were exposed to different doses of 200 MeV protons (which emulate an equivalent 1 MeV neutron dose under the NIEL approximation [@why200MeV]). The radiation doses were delivered with a 200 MeV proton beam at the Northwestern Proton Therapy Facility in Warrenville, IL with intensities in the range \[5$\times 10^{9}$ p/cm$^2$, 5$\times 10^{10}$ p/cm$^2$\]. The MPPCs were irradiated on their carrier board panels and were unbiased during irradiation as bulk damage is the primary physical process of interest [@why200MeV]. The dose uniformity across the area was measured on site at Warrenville and was within $\pm$5%. After irradiation, the performance measurements were repeated on all MPPC panels after accelerated annealing. Annealing was accelerated by holding the MPPCs at 60[$^\circ$]{}C for 80 minutes, corresponding to about ten days of room temperature annealing.
\[iv\_scans\]I-V scans
----------------------
Using the FEB, we measured the MPPC currents as a function of applied bias voltage for each panel. These measurements were done before and after irradiation of the panels, as shown in Fig. \[fig:2x2\_IV\_pk21\_pk22\]a, to determine any change in breakdown voltage and current levels.
The breakdown voltage can be determined using the inverse logarithmic derivative (ILD) of the current with respect to the voltage taken from an I-V scan [@ILD]. Breakdown, or turn-on, occurs at the voltage where the derivative rapidly changes from zero to a large value or, equivalently, where the inverse of the derivative approaches zero. As the definition of the breakdown voltage, we use the extrapolation of the ILD to zero [@klanner1]. Fig. \[fig:2x2\_IV\_pk21\_pk22\]b shows the ILD as a function of bias voltage for an irradiated MPPC. The I-V breakdown voltages, before and after irradiation, are compared in Fig. \[fig:2x2\_ivBreakdown\] and are stable within 0.2 V. As shown in Fig. \[fig:2x2\_ivBreakdown\]a and Fig. \[fig:2x2\_ivBreakdown\]b the standard deviation for each non-irradiated panel was roughly 60 mV, and 55mV for irradiated panels.
Photoelectron (PE) spectrum analysis
------------------------------------
The FEB digitizes the MPPC analog signals using a 12-bit ADC with a 12.5 ns sampling period. The FEB continuously samples the MPPC signal; for this study 128 samples of the digitized waveform were recorded as an event (see Fig. \[fig:2.0x2.0\_clean\]a). The FEB is triggered by the LED pulser. Both the time at which the waveform is initiated relative to the pulser signal and the waveform length are variable features of the FEB. For every MPPC tested, we collected 5,000-10,000 events at different bias voltages ($V_b$) in the range between the MPPC breakdown voltage ($V_{0}$) and up to approximately 3 V over-voltage ($V_+$), where over-voltage is defined as: $V_{+}=V_{b}-V_{0}$ . The MPPCs are temperature sensitive, with up to a 50 mV/[$^\circ$]{}C change in the breakdown voltage. Tests of the MPPCs are corrected according to this value. Ambient temperatures during MPPC data taking averaged 20.7 $^{\circ}$C and varied no more than 0.5 $^{\circ}$C.
Four panels were irradiated at the four different levels of 5$\times 10^{9}$ p/cm$^2$, 1$\times 10^{10}$ p/cm$^2$, 2.5$\times 10^{10}$ p/cm$^2$, and 5$\times 10^{10}$ p/cm$^2$ all with $\pm$5% dose uniformity across the panel area. The two lowest doses are the primary focus due to a lack of photoelectron peak resolution at the higher doses. Only a few percent of the MPPCs for Mu2e will be exposed to a fluence above 5$\times 10^{9}$ n/cm$^2$.
To obtain the photoelectron spectrum, a 11 ns pulse is used to trigger both the LED and FEB. Figure \[fig:2.0x2.0\_clean\]a shows a FEB waveform captured when a MPPC is illuminated with the LED. The signal maximum in a five-sample region centered on the arrival time of the MPPC signal is taken as the MPPC response. To find pedestals, we histogrammed samples selected from the waveform at a time interval preceding the LED pulse and fit the distribution with a Gaussian curve. The mean of the fit or the pedestal is then subtracted from the maximum signals. Figure \[fig:2.0x2.0\_clean\]b shows a non-irradiated MPPC PE spectrum where the first peak at zero ADC counts corresponds to the electronics pedestal. We fit the PE spectrum with a series of Gaussian curves and calculate the gain of the MPPC as the difference between the mean values of the electronics pedestal and the first PE peak.
Device gain before and after irradiation
----------------------------------------
An apparent reduction in gain after irradiation was observed. The observed gain reduction is due to an effective drop of the applied bias voltage. Since irradiation increases MPPC current as shown in the I-V scans, there is a voltage drop across the net series resistance in the external circuit, effectively decreasing the applied bias voltage. The voltage drop is estimated via the following procedure: for each MPPC the current is measured for each bias voltage step and plotted to form an I-V scan. Then the I-V current and external resistances on the external circuit (totals to 8.04 $k\Omega\;\pm 2\%$) are used to correct the voltage drop for each bias voltage in the gain plots. When this voltage drop is accounted for, the apparent behavior of the gain decreasing with increased bias voltage is eliminated.
[ ![\[fig:2x2\_gain\_adjusted\] Plot of the MPPC gain ratio in percent, irradiated to non-irradiated, as a function of the MPPC number at V$_{0}$+2.3 V for (a) 5$\times 10^{9}$ p/cm$^2$ and (b) 1$\times 10^{10}$ p/cm$^2$. ](figure8a.eps "fig:") ![\[fig:2x2\_gain\_adjusted\] Plot of the MPPC gain ratio in percent, irradiated to non-irradiated, as a function of the MPPC number at V$_{0}$+2.3 V for (a) 5$\times 10^{9}$ p/cm$^2$ and (b) 1$\times 10^{10}$ p/cm$^2$. ](figure8b.eps "fig:") ]{}
Figure \[fig:voltage\_drop\] shows the gain as a function of bias voltage before and after irradiation, without (Fig. \[fig:voltage\_drop\]a) and with (Fig. \[fig:voltage\_drop\]b) the bias voltage correction for the MPPCs irradiated with 1$\times 10^{10}$ p/cm$^2$.
Figure \[fig:2x2\_gain\_adjusted\] shows the ratio of the gain for all eight MPPCs in two of the tested panels before and after irradiation. The irradiated MPPC gains are corrected for the current draw, as in Fig. \[fig:voltage\_drop\]b, by comparing to the irradiated points at the same bias over-voltage, V$_{0}$+2.3 V. The error bars for the points include systematic uncertainties associated with extraction of the PE peaks and I-V current measurements and voltage measurements, with the majority of the uncertainty from fitting PE peaks. The gains are stable within 4%. The standard deviation prior to irradiation was roughly 60 mV, and 75 mV after irradiation.
Breakdown voltages
------------------
We also measured the breakdown voltage from the gain plots by extrapolating the gain versus $V_b$ curve to zero gain (see Fig. \[fig:voltage\_drop\]b). The non-irradiated and irradiated voltage breakdowns as determined with the gain are plotted per MPPC number in Fig. \[fig:2x2\_gainBreakdown\].
Uncertainties in the breakdown plots correspond to the standard deviation of the mean breakdown voltage for each panel. Overall, using the gain method, the average $V_0$ for the non-irradiated MPPCs is $52.2\pm0.05~V$; while the average $V_0$ for the irradiated MPPCs is $52.3\pm0.05~V$ for the 5$\times 10^{9}$ p/cm$^2$ panel and $52.4\pm0.06~V$ for the 1$\times 10^{10}$ p/cm$^2$ panel.
The two breakdown determination methods, gain and I-V, are compared in Fig. \[fig:2x2\_gain-IV-br\]. In general, breakdowns calculated with the I-V method are larger than those calculated with the gain method. This effect has been observed in other radiation damage studies [@klanner1]. Differences between the breakdown determination methods are likely a result of the turn-on (I-V) and turn-off (gain) breakdown state of the MPPCs [@MPF].
Dark count rates
----------------
LED and noise data samples for each panel of MPPCs were collected at bias voltages in the range of \[53.7, 55.7\] V, corresponding to an overvoltage above breakdown in the range \[1.4, 3.4\] V. Figure \[fig:2.0x2.0\_wf\] shows the waveforms and PE spectra, measured away from the LED pulse region. For irradiated MPPCs an increase in noise with fluence is evident. Figure \[fig:2.0x2.0\_sp\] shows the pedestal-uncorrected photoelectron spectra, with 11ns LED illumination, for panels after irradiation at the lowest three doses. At 2.5$\times10^{10}$ p/cm$^2$, the very high noise rate obscures PE peak separation.
We define a noise peak (NP) as a contiguous set of samples in a waveform with all amplitudes above a threshold level, from a 0.5 PE threshold up to a 5.5 PE threshold. The dark count rate (DCR) is then calculated as the frequency of these noise peaks in the analyzed waveforms: $$\label{dcr}
DCR = \sum NP/( 12.5~ns \times \sum NS )~,$$ where $\sum NS$ is the total number of samples searched for the noise peaks, and 12.5 ns is the time between each sample measurement.
[ ![\[fig:2.0x2.0\_sp\] PE spectra for the MPPCs, drawn from the 11 ns LED pulse region, irradiated at (a) 5$\times 10^{9}$ p/cm$^2$, (b) 1$\times 10^{10}$ p/cm$^{2}$, and (c) 2.5$\times 10^{10}$ p/cm$^{2}$. Data were taken at a bias overvoltage of V$_{0}$+2.3 V. ](figure12a.eps "fig:") ![\[fig:2.0x2.0\_sp\] PE spectra for the MPPCs, drawn from the 11 ns LED pulse region, irradiated at (a) 5$\times 10^{9}$ p/cm$^2$, (b) 1$\times 10^{10}$ p/cm$^{2}$, and (c) 2.5$\times 10^{10}$ p/cm$^{2}$. Data were taken at a bias overvoltage of V$_{0}$+2.3 V. ](figure12b.eps "fig:") ![\[fig:2.0x2.0\_sp\] PE spectra for the MPPCs, drawn from the 11 ns LED pulse region, irradiated at (a) 5$\times 10^{9}$ p/cm$^2$, (b) 1$\times 10^{10}$ p/cm$^{2}$, and (c) 2.5$\times 10^{10}$ p/cm$^{2}$. Data were taken at a bias overvoltage of V$_{0}$+2.3 V. ](figure12c.eps "fig:") ]{}
The mean DCRs for a set of eight MPPCs at a bias voltage of V$_{0}$+2.3 V are shown in Table \[tab:dcr\_rates\_2x2\]. Figure \[fig:2x2\_dark\_current\] shows the mean DCRs per panel as a function of different threshold levels at two different applied biases, for the non-irradiated MPPCs and for MPPCs irradiated with 5$\times 10^{9}$ p/cm$^2$ and 1$\times 10^{10}$ p/cm$^2$. At the maximum fluence of 1$\times 10^{10}$ p/cm$^2$ the measured DCRs are below $\sim$250 kHz for a 5.5 PE threshold.
Irradiation level (p/cm$^2$) Threshold (PE) DCR, kHz
------------------------------ ---------------- --------------
No radiation 0.5 124$\pm$4
No radiation 3.5 $<$0.1
5$\times$10$^9$ 3.5 284$\pm$58
5$\times$10$^9$ 5.5 8.5$\pm$2.4
1$\times$10$^{10}$ 3.5 2550$\pm$200
1$\times$10$^{10}$ 5.5 260 $\pm$45
: The mean DCR per panel at V$_b$=V$_{0}$+2.3 V bias voltage.
\[tab:dcr\_rates\_2x2\]
![\[fig:2x2\_dark\_current\] The dark count rates of the MPPCs as a function of the PE threshold levels for two applied biases, V$_{0}$+2.3 V and V$_{0}$+3.3 V, for non-irradiated MPPCs (bottom); MPPCs irradiated at 5$\times 10^{9}$ p/cm$^2$ (middle); MPPCs irradiated at 1$\times 10^{10}$ p/cm$^2$ (top).](figure13.eps)
MPPC Response
-------------
By exposing the MPPCs to 16 ns LED pulses, we analyzed the effect of radiation on signals with magnitudes of several tens of PE. Since the 16 ns LED illumination interval is longer than the 12.5 ns FEB sampling period, we measure the response of the SiPM by summing all five amplitudes in the five-sample region centered on the arrival time of the MPPC signal. The study was done at different irradiation levels. An example of a pedestal-subtracted, gain-converted response at V$_{0}$+2.3 V is shown in Fig. \[fig:2x2\_long\_pulse\]a. The mean of the Gaussian fit is taken as the response.
The ratios of the MPPC response at fluences of 5$\times 10^{9}$ p/cm$^2$ and 1$\times 10^{10}$ p/cm$^2$ to the non-irradiated response are shown in Fig. \[fig:2x2\_long\_pulse\]b and Fig. \[fig:2x2\_long\_pulse\]c, respectively. The ratios were put on equal footing by correcting the response and gain for the voltage drop. To within 4%, the response of the MPPCs remains unchanged after irradiation. The data are not corrected for saturation effects which are on the order of 1%.
Summary
=======
I-V curves, breakdown voltages, dark rates, gains, and LED light response of Hamamatsu 2.0$\times$2.0 mm$^2$ MPPCs were measured before and after irradiation with proton fluences relevant to the Mu2e experiment. No significant deterioration in performance in terms of breakdown voltage, gain, and response is observed. As may be expected, the dark noise rate (and current) increases significantly with radiation, which degrades the ability to identify the single photoelectron peak for fluences exceeding 1$\times 10^{10}$ 1-MeV-equivalent-n/cm$^2$.
For Mu2e, because of bandwidth limitations the higher noise rate at the highest fluences will require a higher zero-suppression threshold. This raised threshold, however, is still well within the limits required for the veto efficiency.
Acknowlegements
===============
We are grateful for the vital contributions of the Fermilab staff and the technical staff of the participating institutions. We wish to thank the staff of the Northwestern proton therapy center in Warrenville, IL for generous access to their proton beam. This work was supported by the US Department of Energy and the US National Science Foundation.
[99]{} Hamamatsu MPPCs. http://www.hamamatsu.com/us/en/product/category/3100/4004. L. Bartoszek, [*et al.*]{}, Mu2e Collaboration, “Mu2e Technical Design Report”, arXiv:1501.0524 (2014). P. Rubinov,“Front End Electronics for SiPM Readout in Mu2e CRV Detector”, IEEE Nuclear Science Symposium and Medical Imaging Conference, San Diego, CA, October 31st, 2015. Bivar. Inc., 4 Tomas, Irving, CA 92618-2593 G. Lindstrom,“Radiation Damage in Silicon Detectors”, NIM A Vol 512, Issues 1-2, pp. 30-43 (2003). E. Garutti, R. Klanner, D. Lomidze, J. Schwandt, M. Zvolsky, “Characterisation of highly radiation-damaged SiPMs using current measurements”arXiv:1709.05226. V. Chmill, E. Garutti, R. Klanner, M.Nitschke, J. Schwandt, “Study of the breakdown voltage of SiPMs”, Nuclear Instruments and Methods in Physics Research, Section A845 (2017). O. Marinov, [*et al.*]{},“Theory of Microplasma Fluctuations and Noise in Silicon Diode in Avalanche Breakdown”, Journal of Applied Physics 101, 064515 (2007); doi: 10.1063/1.2654973
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'Our Galaxy and the nearby Andromeda Galaxy (M31) form a bound system, even though the relative velocity vector of M31 is currently not well constrained. Their orbital motion is highly dependent on the initial conditions, but all the reliable scenarios imply a first close approach in the next 3$-$5 Gyrs. In our study, we simulate this interaction via direct $N$-body integration, using the HiGPUs code. Our aim is to investigate the dependence of the time of the merger on the physical and dynamical properties of the system. Finally, we study the dynamical evolution of the two Supermassive Black Holes placed in the two galactic centers, with the future aim to achieve a proper resolution to follow their motion until they form a tight binary system.'
---
Introduction
============
Although the physical and dynamical properties of the Milky Way-Andromeda system are rather uncertain, it is likely that the two galaxies will not escape the collision and the final merger. According to previous simulations in literature ([@cl2008 Cox & Loeb 2008], [@vdm2019 Van der Marel 2019]), the first close approach will likely occur in 5 Gyr from today and the final merger in about 10 Gyr; however the timing of this process is highly sensitive not only to the value of the tangential component of the M31’s relative velocity, but also to the total mass of the binary and the density of the intergalactic medium (IGM). A clear understanding of how these properties affect the dynamics of the system will allow us to define a feasible scenario of the Local Group evolution. At the same time, during the interaction at large scale, we are interested in following the motion of the two SMBHs in the centers of both galaxies. After the merger of their host galaxies the two SMBHs are expected to form a close binary, which will gradually lose orbital energy during gravitational encounters with the field stars in a dense environment. Our main purpose is to increase the resolution of our simulations to follow the dynamical evolution of the two SMBHs down to small spatial scales, so to deal with the ‘final parsec problem’, when most of the orbital energy may be lost through gravitational waves (GW) emission.
Initial conditions
==================
Initial conditions for our galaxies have been generated with the NEMO code ([@Teuben1995 Teuben 1995]), combining three different components: a disk with an exponential density profile, a bulge and a halo, both with a Hernquist’s profile. For both galaxies, the mass of the bulge is half the disk, and that of the halo about 42 times larger. In unit of disk’s scale length, the core radius of the bulge and the halo are respectively 0.2 and 3 ([@cl2008 Cox & Loeb 2008], [@Widrow-Dubinki Widrow & Dubinski 2005]). Our simulations show a significant correlation between the cut-off radius of the two halos and the time of the interaction. For this reason we have chosen a large cut-off of 70 times the disk’s scale length, which for the Milky Way corresponds to a radius of about 150 kpc, in line with the estimations of [@Shull2014 Shull (2014)]. In Fig.\[fig1\] we show a snapshot of our initial model for the Milky Way.\
In the center of both galaxies we considered a black hole with a mass of one thousandth the total mass of the galaxy ($1.0\times10^{9} M_\odot$ for the Milky Way and $1.6\times10^{9}M_\odot$ for Andromeda). Even though these masses are different from the observed ones, until now this is the most reasonable choice, taking into account our current number resolution of $N=6.5\times10^4$ particles.\
We have chosen 780 kpc as initial separation between the Milky Way and Andromeda centers of mass, with galaxy spin vectors respectively oriented at $(0^{\circ}; -90^{\circ})$ and $(241^{\circ}; -30^{\circ})$ in Galactic coordinates ([@R.L.B. Raychaudury Lynden-Bell 1989]). In our simulations, we have adopted a reference frame where the x-y plane coincides with the plane of the motion.\
The Andromeda’s current velocity vector is one of the worst constrained property of the MW-M31 system. Using the redshift we can infer only the value of the radial component ($V_{r0}\approx120$ km/s), but measuring the transverse component is very difficult. Several estimations in literature give different values: from a minimum of $V_{t0}\approx17 km/s$ ([@vdm2012 Van der Marel 2012]) to a maximum of $V_{t0}\approx164 km/s$ ([@Sal2016 Salomon 2016]). Recently, [@VdM2019 Van der Marel et al. 2019], using the GAIA DR2 data, have obtained $V_{t0}\approx57$ km/s, but the level of uncertainty is still high. We took the latter estimation as reference for fixing the orientation of the M31’s velocity vector, and we have performed three simulations varying transverse component speed ($30$ km/s, $50$ km/s, $70$ km/s), keeping constant the value of the radial part ($120$ km/s).
In Tab.\[tab1\] we show all the physical properties of the two galaxies.
![Our model of the Milky Way, with the three components (disk, bulge, halo) shown in different colors. The right panel is a zoom of the innermost region. The axis unit is 100 kpc.[]{data-label="fig1"}](galaxy.png){width="4.0in"}
[**Milky Way**]{} [**Andromeda**]{}
------------------------------------------------------- --------------------- ---------------------
[**Scale radius of the disk ($R_d$)**]{} $2.2$ $3.6$
[**Core radius of the bulge(in unit of $R_d$)**]{} $0.2$ $0.2$
[**Core radius of the halo(in unit of $R_d$)**]{} $3.0$ $3.0$
[**Cut-off radius of the bulge(in unit of $R_d$)**]{} $5.0$ $5.0$
[**Cut-off radius of the halo(in unit of $R_d$)**]{} $70.0$ $70.0$
[**Mass of the disk ($M_d$)**]{} $2.27\times10^{10}$ $3.64\times10^{10}$
[**Mass of the bulge (in unit of $M_d$)**]{} $0.5$ $0.5$
[**Mass of the halo (in unit of $M_d$)**]{} $42.5$ $42.5$
[**Total Mass**]{} $1.0\times10^{12}$ $1.6\times10^{12}$
[**Mass of the central black hole**]{} $1.0\times10^{9}$ $1.6\times10^{9}$
: Values of characteristic parameters used in our simulations. Lengths are in kpc and masses in solar masses.[]{data-label="tab1"}
Methods and Preliminary Results
===============================
The HiGPUs code ([@Capuzzo2013 Capuzzo Dolcetta et al. 2013]), used in these simulations, numerically implements the integration of the classical $N$-body problem using CPUs coupled to GPUs, in parallel. This program is based on a 6th order Hermite’s integration scheme with block time steps, and directly evaluates the particle-particle forces. To reproduce the dissipative effect of the IGM, we have modified the original version of HiGPUs by including a dynamical friction term in the equations of motion, according to the Chandrasekhar’s formula ([@binney Binney & Tremaine 1987]). Anyway, our simulations indicates that the presence of a warm ($T\approx3\times10^5 K$) medium with a uniform density, $\rho_0$, which is roughly 10 times the critical density of the universe, produces just a secondary effect on the timescale of the interaction. The main factor driving the speed of the merger are the sizes of the galactic halos: the larger they are, the faster is the process.\
Our first goal has been to investigate the dependence of the time of the merger upon the initial conditions, in particular upon the M31’s initial transverse velocity. Figure\[fig2\] shows the separation between the two galaxies as function of time, for the three chosen values of $V_{t}$, keeping the radial component fixed at $120$ km/s. The time of the merger, as expected, is highly sensitive to this parameter.\
Looking at the relative motion of the two BHs, in Fig.\[fig3\], we notice that, independently of the magnitude and orientation of the initial relative velocity, they reach the same final separation of about $50$ pc, when they seem to stall on a nearly circular orbit. Anyway, our numerical resolution is too low to further investigate the BHs motion at small scales, because, clearly, the final value of their separation depends significantly on the number of particle in the simulation.
![The separation (in unit of 100 kpc) between the two galaxies centers of mass as function of time (in Gyr), for three different tangential velocities[]{data-label="fig2"}](merger.png){width="3.0in"}
![The separation (in unit of 100 kpc) between the central BHs as function of time (in Gyr), for three different tangential velocities. The plot shows that, regardless of the initial velocity, the two BHs stalls at the same distance of about 50 pc.[]{data-label="fig3"}](distanzaBH.png){width="3.0in"}
Conclusions
===========
Owing to the uncertainty on the relative tangential velocity of Andromeda, the expected time of the MW-M31 merger spans over a wide range. Moreover, we have noticed that it is heavily dependent also on the radii of the galactic halos, and, secondarily, on the density and temperature of the IGM. We are currently performing other simulations to investigate further these dependencies, with the aim of a full description of the different scenarios of the galactic collision.
The relative motion of the two central BHs shows a regular behaviour on large scales, but we need to increase the resolution of our simulations to extend the study also to the innermost region of the galaxy formed after the merger.
2013, *Journ. of Comp. Phys.*, v. 236, p. 580-593. 1987, *Galactic Dynamics*, Princeton, NJ, Princeton University Press 2008, *MNRAS*, 386, 461–474 2012, *ApJ*, 753, 9 2016, *MNRAS*, 456, 4432 2019, *ApJ*, 872, 24 1995, *Astronomical Data Analysis Software and Systems IV*, 77, 398-401 2014, *ApJ*, 784, 142 1989, *Mon. Not. R. astr. Soc.*, 240,195-218 2005, *ApJ*, 631, 838-855
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{
"pile_set_name": "ArXiv"
}
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---
author:
- |
Zhang XiangChuan\
The National University of Singapore’s Department of Physics’\
Graduate
title: Lagrangian Density for the Vacuum
---
0.8in
[**Abstract**]{}
[In this paper, the Lagrangian density for the vacuum is mainly discussed, meanwhile, the matter field, the modified Lorentz transformation relations and the reasons for the invariance of the velocity of light in the vacuum are also discussed.]{}
The problems of the existing theories$^{[1]}$
=============================================
We know, in special relativity, the priciple of relativity and the principle of the invariance of the velocity of light in the vacuum(in this paper, $c$ represents the velocity of light in the vacuum in special relativity) are two basic assumptions; Although some successes have been achieved, some problems still exist:\
(1) are there certain relations between the two assumptions? no answer.\
(2) what is the meaning of vacuum here? according to Einstein’s meaning, the vacuum was then considered to be empty, but actually, existing theories and experiments have tell us that there is matter(for example, fields) in the vacuum, because light is a kind of matter, and there are interactions among matters, the interactions can definitely affect the velocity of light, thus we can logically obtain that the velocity of light in the vacuum is variable, and the phenomenon in which the actual velocity of light exceeds $c$ can exist in the vacuum.\
(3) the task of physics is continously to explore the laws of the natural world, let people understand and make use of them better. Even if the assumption of the invariance of the velocity of light in the vacuum is correct, it is still an assumption, not explained; Why is c invariant? behind the assumption, there are definitely physical reasons, these reasons are still not found, from the theory angle, this is, of course, a problem of special relativity; if one can explain it, it is sure that a new physics will emerge.\
In the unified theories, the Higgs fields play an important role, but in both the standard model and the supersymmetric theories, the Higgs field Lagrangian densities are given by hand, the reasons for how to obtain them are not explained, this is obviously a problem for the theories from the theory angle.\
The unified theory has been studied for many years, and many theories have emerged: the electroweak unified theory, the standard model, the string theory, M-theory, etc. the research of these theories is still in progress, at present, the research of the primary theory has begun. Why can so many theories emerge? why do different problems still exist in different theoiesy? in my opinion, one or more of the bases of these theories is(are) problematic, for example, the Yang-Mill’s gauge theory is not complete. one of the purposes foe me to write this paper is to provide a new angle(the angle of the level of disorder) for the primary theory. Besides this paper, I will discuss some problems related to the unified theory from the angle of the level of disorder in my follow-up papers. I hope that I will be able to construct an acceptable unified theory of the four basic interactions.\
Now let us discuss how to solve the above-mentioned problems.
Matter Field$^{[2]}$
====================
According to the elementary particle theory, in terms of spin, all particles in the universe can be devided into two categories: one is the boson which the spin is 0, 1 , 2 , 3 , $\cdots$, the other is the fermion which the spin is ${1}\over{2}$ ,${3}\over{2}$ ,${5}\over{2}$ ,$\cdots$ According to the quantum field theory, each kind of particle corresponds to one kind of field, and is the quantum excited by the field; the fermi field and the bose field can interact each other, this shows that the fermi field and the bose field have some common properties; While all matter in the universe is composed by fermions and/or bosons, thereby, the universe can be considered as one kind of field—the matter field which is more elementary than the fermi field and the bose field. Thus, any particle can be regarded as a matter field, and is a form of expression of the matter field; Any matter system can be treated as a matter field, and is a form of expression of the matter field. According to the quantum field theory, there are four kinds of elementary interactions: the strong interaction, the weak interaction, the electromagnetic interaction and the gravitational interaction. For the strong interaction and the weak interaction, the interaction range is very short, for the electromagnetic interaction and the gravitational interaction, the interaction range can be from zero to infinity. Hence, I think colour charges and weak charges are not the essential properties of the quanta of the matter field, meanwhile considered that the matter field itself is the source of the gravitational interaction. thereby only elertric charges are the essential properties of the quanta of the matter field( here, I also mean that the level of the gravitational interaction and the level of the electromagnetic interaction are the same, they are different from that of the weak interaction and the strong interaction, the former is more elementary), this means that there are only two kinds of the quanta of the matter field, they are $M_{p}$ with the positive electric charge and $M_{n}$ with the negative electric charge.\
The modified Lorentz Transformation Relations$^{[3]}$
=====================================================
All matter in the universe is in a certain space and time, space and time mean the space and time of matter, this implies that matter and space, time depend on and affect each other. Hence, when we investigate the physics laws of matter we must consider matter, space and time as a whole, only thus, the system considered is a complete system. Hence, when we set up a reference system, in order to meet this requirement, we must set up an abstract four dimensional reference system, and regard the time axis and the space axes as the isotopic axes, meanwhile, we stipulate that the time coordinate is imaginary(for the time axis is virtual), i. e. ikt, where k is a constant which is greater than zero, i is an imaginary symbol, t represents the real time, thus, the axes of this four dimensional reference system are ikt , x , y , z respectively. Because the dimension of kt is length, the dimension of $k$ is the velosity.\
According to the principle of relativity, all inertial reference systems are equivalent.[**Now that all inertial reference systems are equivalent, and the system composed by the four dimensional space and matter is complete, so in the four dimensional space, the four dimensional shape of an object must be the same in all inertial reference systems**]{}, this means that the distance between any two points in the four dimensional space is the same in all inertial reference systems, this is the invariability of the four dimensional interval. Thereby, from the mathematical angle, the coordinate transformation between any two inertial reference systems is a rotation or the combination of a rotation and a translation in the four dimensional space, but the translation implies that the origins of the two inertial reference systems don’t coincide at the beginning, no other meaning, so, only the rotation needs to be considered. The transformation for a four dimensional space rotation is a four dimensional orthogonal transformation.\
Considering the two inertial reference systems S , S’, S moves at a velocity $\vec{V}$ relative to S’ , for an arbitrary point P(ikt, x, y, z) in S, it is corresponding to the point P’(ikt’, x’, y’, z’) in S’ , thus we obtain:\
$$\begin{aligned}
\left(\begin{array}{c}
ikt'\\x'\\y'\\z'\end{array}\right)
& =\left(\begin{array}{llll}
a_{11} & a_{12} & a_{13} & a_{14}\\
a_{21} & a_{22} & a_{23} & a_{24}\\
a_{31} & a_{32} & a_{33} & a_{34}\\
a_{41} & a_{42} & a_{43} & a_{44}
\end{array}\right) &
\left(\begin{array}{c}
ikt\\x\\y\\z\end{array}\right)\end{aligned}$$ \[e1\] where the transformation matrix $T_{matrix}$ is an orthogonal matrix.\
Now considering a special case which the axes of S are parallel to the corresponding axes of S’ , and the corresponding axes of S and S’ have the same directions, when $t=t'=0$, the origins of the two coordinate systems coincide, the direction of $\vec{V}$ is the same as the positive direction of x axis of S , thus, the transformation matrix becomes the following form:\
$$\left(\begin{array}{llrr}
a_{11} & a_{12} & 0 & 0\\
a_{21} & a_{22} & 0 & 0\\
0 & 0 & a_{33} & 0\\
0 & 0 & 0 & a_{44}
\end{array}\right)$$\
In terms of the properties of the orthogonal transformation, we obtain the following equations:\
$$\begin{aligned}
a_{33}=a_{44} & =1 \\
a_{11}^2+a_{12}^2 & =1 \\
a_{11}a_{21}+a_{12}a_{22} & =0\\
a_{11}^2+a_{21}^2 & =1\end{aligned}$$\[e2,3,4,5\] from this, we obtain:\
$$\begin{aligned}
a_{11}^2=a_{22}^2\\
a_{12}^2=a_{21}^2\end{aligned}$$ \[e6.7\] Considering the origin of S, in S, it menas $x=0$ which is corresponding to $x'=Vt'$ in S’ . According to equation (1) and the transformation matrix a of the special case, we obtain the following equations:\
$ikt'=a_{11}ikt+a_{12}x$\
$x'=a_{21}ikt+a_{22}x$\
Hence, we obtain:\
$$a_{11}^2=a_{21}^2{k^{2}\over V^{2}}$$ \[e8\]\
According to equations (2), (3), (3), (4), (5), (6), (7), (8), we obtain:\
$$\begin{aligned}
a_{12}^2&={-V^{2}\over {k^{2}-V^{2}}}\\
a_{11}^{2}&={k^{2}\over {k^{2}-V^{2}}} \end{aligned}$$ \[e9,10\] Because the positive directions of the axes of the two reference systems are the same, $a_{11}>0$ , $a_{22}>0$. thereby we obtain:\
$a_{11}=a_{22}={1\over\sqrt{1-{V^{2}\over k^{2}}}}$ , $k>|\vec{V}|$\
The increment of t can result in the increment of x’, and $a_{11}a_{21}+a_{12}a_{22}=0$\
so, $a_{12}=-a_{21}={{iV\over k}\over\sqrt{1-{V^{2}\over k^{2}}}}$\
Thus, we obtain the transformation relations of the coordinates of the two inertial reference systems:\
$$\begin{aligned}
ikt'={ikt\over\sqrt{1-{V^{2}\over k^{2}}}}+
{{ixV\over k}\over\sqrt{1-{V^{2}\over k^{2}}}}\\
x'={-iV\over k}{ikt\over\sqrt{1-{V^{2}\over k^{2}}}}+{x\over\sqrt{1-{V^{2}\over k^{2}}}}\\
y'=y\\
z'=z\end{aligned}$$\[11,12,13,14\] This is the modified Lorentz transformation relations. Because $\vec {V}$ is arbitrary, and $k>V$, k is the maximum realizable velocity in the universe.\
Please pay attention to the following two points:\
(1) to obtain the modified Lorentz relations, we use the principle of relativity only, do not use the principle of the invariance of the velocity of light in the vacuum; this means no certain relations between the two principles. This also implies that the principle of the invariance of the velocity of light in the vacuum is not necessary in special relativity.\
(2) $k$ is not less than $c$.
The Reasons Why the Velocity of Light in the Vacuum Is Constant and Equal in All Systems Moving with Constant Velocities
========================================================================================================================
In special relativity, that the velocity of light in the vacuum is constant and equal in all systems moving with constant velocities is taken as a basic assumption, till now, this assumption has still not been explained, from the theory angle, this is a problem of special relativity. Now, let us use the matter field to explain the assumption.\
According to the disscusions in section two, the quanta of the matter field are electrically charged, when external electric charges present, the quanta will be polarized and form a certain distribution; When the external electric charges change, the distribution will also change with the change of the external electric charges; When the external electric charges oscilate, the change of the distribution will transmit in a form of wave in space, the wave is regarded as the electromagetic wave here. Now let us explain the reasons why the transmitting velocity of the electromagnetic wave(the polarization wave) in the vacuum is constant and equal in all systems moving with constant velocities.\
(60,23)(0,0) (0,0)[(1,0)[50]{}]{} (0,0)[(0,1)[22]{}]{} (0,11) (2.5,12.1)[(2,2)\[l\][B]{}]{} (25,2) (35,2) (15,4)[(1,0)[30]{}]{} (45,4)[(0,1)[11]{}]{} (15,15)[(1,0)[30]{}]{} (15,4)[(0,1)[11]{}]{} (46,11)[(1,0)[3]{}]{} (2.5,8.5)[(3,3)\[l\][sound source]{}]{} (51,0)[(2,2)\[l\][X]{}]{} (1,21)[(2,2)\[l\][Y]{}]{} (31,9)[(2,2)\[l\]]{} (33,9.9)[(2,2)\[l\][A]{}]{} (27,14.1)[(2,2)\[l\]]{} (29,15)[(2,2)\[l\][C]{}]{} (50.5,11.2)[(2,2)\[l\][$\vec{V}$]{}]{} (23.5,-5.5)[(2,4)\[l\][(Fig.1)]{}]{}
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See Fig.1, there is an enough long, enough wide and enough high carriage which has an open rear side(the left side of the carriage in Fig.1) and the other closed sides and moves at a velocity $\vec{V}$ along the positive direction of X axis, a sound source is at point B and continuously give out sound waves, the transmitting velocity of the sound waves is $\vec{V_{s}}$ in air. when the sound waves arrives at observers A , C respectively, because observer C is on the top of the carriage, observer C’s velocity relative to the air outside the carriage is also $\vec{V}$, hence, the sound velocity determined by ddobserverC is $\vec{V_{s}}-\vec{V}$(here ignoring the effects of relativity); while observer A is in the carriage, observer A is at rest relative to the air inside the carriage, thus, when the sound waves transmit into the air inside the carriage, the velocity of the sound waves is still $\vec{V_{s}}$ in the air inside the carriage. Thereby, the velocity of the sound waves determined by observer A is still $\vec{V_{s}}$, though observer A is moving with the carriage, so long as the relation of $|\vec{V_{s}}|>|\vec{V}|$ is satisfied.\
In terms of the same principle, we can explain the reasons why the velocity of light in the vacuum is constant and equal in all systems moving with constant velocities. Because all matter systems in the universe are matter fields, and are the forms of expression of the matter field; Meanwhile, because there are interactions between any two matter systems in the universe, there is a matter field surrounding any matter system, therefore, there is also a matter field in the vacuum. When the matter system moves, the matter field surrounding the matter system also moves with the matter system.\
See Fig.2, supposed that in the vacuum there is a source of light at point A and an observer(equivalent to a matter system) at point B, observer $B$ wants to measure the velocity of light given out by the source of light at point A. It is obvious that no matter how the relative motion between the source of light and observer B is(but must ensure that the light of the source A can transmit into the matter field surrounding observer B), because the matter field surrounding the source of light at point A and the matter field in the vacuum mix together as well as the matter field in the vacuum and the matter field surrounding observer B also mix together, the light waves can transmit into the matter field in the vacuum from the matter field surrounding the source of light at point A, then can transmit into the matter field surrounding observer B from the matter field in the vacuum; And the velocity of light determined by observer B is always the velocity of light in the matter field surrounding observer B itself, it is impossible for observer B to directly measure the velocity of light in the vacuum, it is only possible for observer B to directly measure the velocity of light in the matter field surrounding observer B itself, so is it for the other observers, hence, the velocity of light determined by any observer is the same(supposed the matter fields surrounding all observers are the same), and is always equal to the velocity of light in the vacuum(supposed the matter fields surrounding all observers are the same as the matter field in the vacuum), so long as the absolute value of the velocity of light in the vacuum is greater than that of the velocity of any observer. Thus we have explained the reasons why the velocity of light in the vacuum is constant and equal in all systems moving with constant velocities.In 1887, in order to prove that if there was the ether in the vacuum, Michelson-Morley did the famous Michelson-Morley experiment, from the matter field’s point of view, the experimental results are natural. Now iet us discuss the influence on the velocity of light in the vacuum given by the density of the quanta of the matter field in the vacuum. because the density can affect the velocity of light, and the densities surrounding different observers are different, the velocities of light determined by different observers are different, normally, these differences are very small(of course, if the differences of the densities are big, the differences of the velocities are big). Set the density to which $c$ corresponds as $DN_{c}$, because it is impossible for the vacuum to be empty, when the density is not equal to $DN{c}$, the actual velocity of light in the vacuum is not equal to $c$($>c$ and $<c$), thus we can obtain that the phenomenon in which the actual velocity of light exceeds $c$ can exist in the vacuum.\
(60,23) (51,0)[(2,2)\[l\][X]{}]{} (1,21)[(2,2)\[l\][Y]{}]{} (0,0)[(1,0)[50]{}]{} (0,0)[(0,1)[22]{}]{} (15,10) (35,16) (14.5,12)[(2,2)\[l\][A]{}]{} (34,19.5)[(2,2)\[l\][B]{}]{} (11.5,6.5)[(2,2)\[l\][Light Source]{}]{} (32,11.5)[(2,2)\[l\][Observer]{}]{} (19,-5.5)[(2,4)\[l\][(Fig. 2)]{}]{}
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The Lagrangian Density for the Vacuum$^{[4]}$
=============================================
According to the discussions in section two and existing experimental results, we know that there are the free quanta of the matter field and the neutral and electrically charged particles which are composed by the quanta of the matter field in the vacuum, these particles are in a dynamic equilibrium state, i. e. the vacuum state. the vacuun state is also one state of the matter system. The state of a matter system is the distribution of the number density $\rho$(here the density for space density only) of the quanta of the matter field of the matter system in the space-time. Different state of the matter system corresponds to the different distribution. In order to investigate the change laws of the state of the matter system, we normally find a few state functions of the matter system, these state functions describe the properties of the state of the matter system from different sides, the density of the level of disorder(DOLOD for short) is such a state function of the matter system. DOLOD means the grade which the quanta of the matter field of the matter system in unit space volume are in chaotic state in the space-time, it is a scalar in the space-time. The greater DOLOD, the higher the grade, vice versa. There are two kinds of factors in a matter system, one of them makes DOLOD increase, the other makes DOLOD decrease, the natural nature of the matter system always makes DOLOD of itself increase, thereby, when the matter system is in a dynamic equilibrium state, DOLOD has and reaches a maximum\
Now we discuss how to quantitatively express DOLOD. Now that the state of the matter system is the distrbution of $\rho$, and DOLOD is a state function of the matter system, then there is definitely a direct relation between DOLOD and the distribution. Meanwhile, from the mathamatical angle, if $\rho$ were negative(for example, the number density with positive charges or negative charges), DOLOD would be still the same (compared to DOLOD with $+\rho$ which has the same absolute value as $-\rho$), so DOLOD must be the function of ${\rho}^2$. When $\rho=0$, DOLOD must be zero, because $\rho=0$ implies that there is no matter in the space-time. When $\rho\neq0$(consider number only), DOLOD must be a positive value. Thus, DOLOD caused by $\rho$ can be expressed as the following form:\
$W_{1}=W_{1}(\rho^{2})$\
$W_{1}(0)=0$ & $\frac{\partial W_{1}(\rho^{2})}{\partial\rho^{2}}|_{0<\rho<\rho_{m}}>0$, where, $\rho_{m}$ is the first extremum point of $W_{1}$, and $\rho_{m}>0$\
When $\rho$ has a distribution in the space-time, there is a certain relation among the $\rho$s at all points in the matter system, the certain relation can also have contribution to DOLOD, the contribution is determined by the four dimensional gradient of $\rho$. Because for any point, no matter how the direction of the four dimensional gradient is, DOLOD is the same, DOLOD must be the function of the square of the four dimensional gradient; At the same time, the greater the four dimensional gradient, the smaller DOLOD, vice versa. So DOLOD caused by the four dimensional gradient can be expressed as:\
$W_{2}=W_{2}((\nabla_{4}\rho )^{2})$, where, $\nabla_{4}$ is the symbol of the four dimensional gradient, i. e.\
$\nabla_{4u}=(\frac{\partial}{\partial(ikt)}, {\bf\nabla}), u=1, 2, 3, 4$\
$(\nabla_{4}\rho )^{2}=(\nabla_{4}\rho )\cdot(\nabla_{4}\rho )$\
This is the case that the matter system is at rest. When the matter system is in motion, DOLOD of the matter system will decrease, because the motion along a direction reduces the grade of chaos of the matter system. DOLOD caused by the motion of the matter system will be given from the invariability of DOLOD. DOLOD caused by the irregular motion of the quanta of the matter field can be involved in DOLOD caused by $\rho$, because the irregular motion can affect the distribution of $\rho$. For a certain matter system, under the same external conditions, the relation between the distribution of $\rho$ and the irregular motion one-to-one. In fact, the level of this irregular motion is determined by the universe temperature. When a matter system is in a dynamic equilibrium state, the irregular motion has a certain level, the physical quantity for expressing the level is the universe temperature. the higher the universe temperature, the higher the level , vice versa; The relation between them is one-to-one; Thereby, the relation between the distribution of $\rho$ and the universe temperature is also one-to-one. Except the above-mentioned factors, no other factor can have contribution to DOLOD. Thus, when a matter system is at rest, its DOLOD can be expressed as the following form:\
$WX=W_{1}(\rho^{2})+W_{2}((\nabla_{4}\rho )^{2})$\
Now we discuss the detailed expressions of $W_{2}((\nabla_{4}\rho)^{2})$ and $W_{1}(\rho^{2})$. Expanding $W_{2}((\nabla_{4}\rho)^{2})$ into a power series:\
$W_{2}((\nabla_{4}\rho)^{2})=W_{2c}+k_{22}(\nabla_{4}\rho)^{2}+\cdots$\
where, $W_{2c}$ , $k_{22}$ are constants.\
Because $W_{2}((\nabla_{4}\rho )^{2})$ is DOLOD for reflecting the non-uniformity of the distribution of the number density of the quanta of the matter field inside the matter system, and the non-uniformity can completely be expressed out by $(\nabla_{4}\rho )^{2}$, so we obtain:\
$W_{2}{(\nabla_{4}\rho )^{2}}=W_{2c}+k_{22}(\nabla_{4}\rho )^{2}$, $k_{22}<0$\
Because $W_{1}(\rho^{2})$ is DOLOD for reflecting the deviation of the number density of the quanta of the matter field relative to the zero point, it is related to the nature of the space-time. Expanding $W_{1}(\rho^{2})$ into a power series:\
$W_{1}(\rho^{2})=W_{1c}+k_{0}\rho^{2}+k_{1}\rho^{4}+k_{2}\rho^{6}+\cdots$\
where, $W_{1c}$, $k_{0}$, $k_{1}$, $k_{2}$ are constants. thereby,\
$WX=W_{c}+k_{22}(\nabla_{4}\rho )^{2}+k_{0}\rho^{2}+k_{1}\rho^{4}+k_{2}\rho^{6}+\cdots$\
where, $W_{c}=W_{1c}+W_{2c}$\
Now considering the case. For the vacuum, there are not only the free quanta of the matter field but also the neutral and charged particles composed by the quanta of the matter field, so, there are four kinds of number densities:\
$\rho_{s0}$, $\rho_{sq}$, $\rho_{b0}$, $\rho_{bq}$\
their meanings are as follows respectively:\
$\rho_{s0}$: The number density of the free quanta of the matter field at an arbitrary point;\
$\rho_{sq}$: The number density of free quanta of the net charge part of the matter field at an arbitrary point;\
$\rho_{b0}$: The number density of the particles which are composed by the quanta of the matter field at an arbitrary point(maybe have many kinds of particles, take one as a representative here);\
$\rho_{bq}$: The number density of particles of the net charge part which are composed by the quanta of the matter field at an arbitrary point(maybe have many kinds of particles, take one as a representative here).\
Hence, $W_{1}$ should be the function of $\rho_{s0}^{2}$, $\rho_{sq}^{2}$, $\rho_{b0}^{2}$, $\rho_{bq}^{2}$, i. e.:\
$W_{1}=W_{1}(\rho_{s0}^{2}, \rho_{sq}^{2}, \rho_{b0}^{2}, \rho_{bq}^{2})$\
$W_{2}$ should be the function of $(\nabla_{4}\rho_{s0} )^{2}$, $(\nabla_{4}\rho_{sq})^{2}$, $(\nabla_{4}\rho_{b0} )^{2}$, $(\nabla_{4}\rho_{bq} )^{2}$, i. e.:\
$W_{2}=W_{2}((\nabla_{4}\rho_{s0} )^{2}, (\nabla_{4}\rho_{sq})^{2},
(\nabla_{4}\rho_{b0} )^{2}, (\nabla_{4}\rho_{bq} )^{2})$\
thus, according to the same principle and method as the above-mentioned ones, we can obtain:\
$W_{1}=W_{1cv}+\lambda_{1}\rho_{s0}^{2}+\lambda_{2}\rho_{sq}^{2}+\lambda_{3}\rho_{b0}^{2}
+\lambda_{4}\rho_{bq}^{2}+\lambda_{5}\rho_{s0}^{4}+\cdots+\lambda_{14}\rho_{bq}^{4}+\cdots$\
where, $W_{1cv}$, $\lambda_{1}$, $\lambda_{2}$, $\cdots$, $\lambda_{14}$ are constants.\
$W_{2}=W_{2cv}+h_{1}(\nabla_{4}\rho_{s0})^{2}+h_{2}(\nabla_{4}\rho_{sq})^{2}+
h_{3}(\nabla_{4}\rho_{b0} )^{2}+h_{4}(\nabla_{4}\rho_{bq})^{2}$\
where, $W_{2cv}$, $h_{1}$, $h_{2}$, $h_{3}$, $h_{4}$ are constants.\
thereby,\
$$WX=W_{cv}+W_{1y}+W_{2y}$$\[e15\] where, $W_{cv}=W_{1cv}+W_{2cv}$\
$W_{1y}=\lambda_{1}\rho_{s0}^{2}+\lambda_{2}\rho_{sq}^{2}+\lambda_{3}\rho_{b0}^{2}
+\lambda_{4}\rho_{bq}^{2}+\lambda_{5}\rho_{s0}^{4}+\cdots+\lambda_{14}\rho_{bq}^{4}+\cdots$\
$W_{2y}=h_{1}(\nabla_{4}\rho_{s0})^{2}+h_{2}(\nabla_{4}\rho_{sq})^{2}+
h_{3}(\nabla_{4}\rho_{b0} )^{2}+h_{4}(\nabla_{4}\rho_{bq})^{2}$\
Because $\rho_{s0}$, $\rho_{sq}$, $\rho_{b0}$, $\rho_{bq}$ are the different ingredients of the vacuun, for the vacuum, they are isotopic. Thus, they can be regarded as different components of a isovector in the isospace, i. e. :\
$$\begin{aligned}
\rho_{v}&=\left(\begin{array}{l}
f_{s0}\rho_{s0}\\f_{sq}\rho_{sq}\\f_{b0}\rho_{b0}\\f_{bq}\rho_{bq}\end{array}\right)\end{aligned}$$\
where, $f_{s0}$, $f_{sq}$, $f_{b0}$, $f_{bq}$ are constants. But in the space-time, $\rho_{v}$ is a scalar. Thus, Thus, we naturally hope that we can express DOLOD for the vacuum as the following form:\
$$WX_{v}=W_{zcv}+\beta _{0}|\nabla_{4}\rho _{v}|^{2}
+\beta_{1}|\rho_{v}|^{2}+\beta_{2}|\rho_{v}|^{4}+\beta_{3}|\rho_{v}|^{6}+\cdots$$\[e16\]\
where, $W_{zcv}$, $\beta_{0}$, $\beta_{1}$, $\beta_{2}$, $\beta_{3}$ are constants,\
$|\nabla_{4}\rho _{v}|^{2}=(\nabla_{4}\rho _{v})^{\dagger}(\nabla_{4}\rho_{v})$,\
$|\rho_{v}|^{2}={\rho_{v}}^{\dagger}\rho_{v}$\
From the mathematical angle, if we expand $WX_{v}$ into the form of the right side of equation (15), the total items of the right side of equation (16) are equal to that of equation (15), hence, so long as the coefficients of the corresponding items of the two equations are equal, equation (16) can be realized, while this is completely possible. Here, equation (8) is selected to be the form of expression of DOLOD for the vacuum, because equation (16) can expresses the entirety and harmonicity of the vacuum better.\
According to the theory of the solid state physics, the third order derivative and the above of the potential of atoms are non-harmonic items, they are the reasons for the expansion caused by heat and contraction caused by cold, if no these derivatives, then no phenomenon of expansion caused by heat and contraction caused by cold. Here, $W_{1}$ is equivalent to the potential, $\rho^{2}$ is equivalent to the distance between two atoms. Under a certain universe temperature, the quanta of the matter field in the universe are in a certain dynamic equilibrium state and have a certain average of the number density at any point. When the universe temperature changes, the instantaneous number density of the quanta of the matter field will change, if the average of the number density is different from that of before the change of the universe temperature, then the third order derivative and the above of $W_{1}$ are not zero; vice versa. Of course, the change of the universe temperature must be within a certain range, out of this range, the average of the number density for the dynamic equilibrium state will change. In actual situations, what we investigate is a part of the universe, not the whole universe, so we can encounter non-harmonic problems, if what we investigate is the whole universe, or even if a part of the universe, but we consider all of the relevant factors, we will also not encounter non-harmonic problems. Maybe this is one of the meanings that the universe is harmonic. The harmonic point of view will be sticked to in this paper. Thus, DOLOD for the vacuum can be expressed as the following form:\
$$WX_{v}=W_{zcv}+
\beta_{0}|\nabla_{4}\rho_{v}|^{2}
+\beta_{1}|\rho_{v}|^{2}+\beta_{2}|\rho_{v}|^{4}$$\[e17\]\
Now we discuss the constants of equation (17). For equation (17), in the space-time, $\rho_{v}$ is a scalar, so, according to the preceding discussion in this section, $\beta_{0}<0$; when the vacuum matter system is in dynamic equilibrium state, its DOLOD has and reaches a maximum, so $\beta_{1}>0$, $\beta_{2}<0$. For the constants in $\rho_{v}$, consider the expanding form of equation (17) which is the same as the right form of equation (15), the following conditions must be satified:\
$f_{s0}>0$, $f_{b0}>0$\
Since the greater the square of the four dimensional gradient of the number density of the net charge part, the greater DOLOD, vice versa; the greater the square of the number density of the net charge part, the smaller DOLOD, vice versa, we can also obtain the following relations:\
$f_{sq}^{\dagger}f_{sq}<0$, $f_{bq}^{\dagger}f_{bq}<0$\
so $f_{sq}$, $f_{bq}$ must be operators and commutate with $\nabla_{4}$. doing the replacement:\
$f_{sq}$ $\rightarrow$ $f_{sq}\stackrel{\wedge}{C_{N}}$, $f_{bq}$ $\rightarrow$ $f_{bq}\stackrel{\wedge}{C_{N}}$, $f_{sq}>0$, $f_{bq}>0$\
where, $\stackrel{\wedge}{C_{N}}$ is an operator and commutates with $\nabla_{4}$, and satisfies the condition:\
$\stackrel{\wedge}{C_{N}}^{\dagger}\stackrel{\wedge}{C_{N}}=-\stackrel{\wedge}{I}$, where, $\stackrel{\wedge}{I}$ is the identity operator.\
This operator is a conversion operator between the number density of a net charge part and the same neutral number density, called the charge conversion operator. According to the meaning of the charge conversion operator, it must also satisfy the following condition:\
$\stackrel{\wedge}{C_{N}}\stackrel{\wedge}{C_{N}}=\stackrel{\wedge}{I}$, $\stackrel{\wedge}{C_{N}}^{-1}=\stackrel{\wedge}{C_{N}}$\
Thus we obtain:\
$$\begin{aligned}
\rho_{v}&=\left(\begin{array}{l}
f_{s0}\rho_{s0}\\f_{sq}\stackrel{\wedge}{C_{N}}\rho_{sq}\\f_{b0}\rho_{b0}\\
f_{bq}\stackrel{\wedge}{C_{N}}\rho_{bq}\end{array}\right)\end{aligned}$$\
In terms of the discussions of section three, we can obtain that the length of a moving object will contract. Supposing the rest length of the object is L, when the object moves at a velocity $\vec{V}$ along the direction in which the length of the object is measured, the length of the object in the direction of $\vec{V}$ becomes: $L\sqrt{1-{V^{2}\over k^{2}}}$, so for the vacuum, the effect of length contraction will lead to that the four kinds of number densities in $\rho_{v}$ will increase to $1\over\sqrt{1-{V^{2}\over k^{2}}}$ as much as that of at rest Because all inertial systems are equivalent, and DOLOD is an invariable, we must modify the expression of DOLOD to meet the requirement. According to equation (17), if $\rho _{v}$ is a rest physical quantity $\rho _{vr}$, $WX_{v}$ is an invariable. Thus, when the matter system moves at a velocity $\vec{V}$, the invariable expression of DOLOD is as follows:\
$$WX_{v}=W_{zcv}+\beta_{0}(1-{V^{2}\over {k^{2}}})|\nabla_{4}\rho _{v}|^{2}
+\beta_{1}(1-{V^{2}\over {k^{2}}})|\rho _{v}|^{2}
+\beta_{2}(1-{V^{2}\over
{k^{2}}})^{2}|\rho_{v}|^{4}$$\[e18\]\
According to the meanings of Lagrangian density and DOLOD, from the angle of the physics essence, Lagrangian density and DOLOD are equivalent, so the Lagrangian density for the vacuum is:\
$$L_{v}=W_{zcv}+\beta_{0}(1-{V^{2}\over {k^{2}}})|\nabla_{4}\rho _{v}|^{2}
+\beta_{1}(1-{V^{2}\over {k^{2}}})|\rho _{v}|^{2}
+\beta_{2}(1-{V^{2}\over
{k^{2}}})^{2}|\rho_{v}|^{4}$$\[e19\]\
Please pay attention to that this expression is correct for the vacuum only. If the matter system is at rest, the Lagrangian density becomes:\
$L_{v}=W_{zcv}+\beta_{0}|\nabla_{4}\rho_{vr}|^{2}
+\beta_{1}|\rho_{vr}|^{2}+\beta_{2}|\rho_{vr}|^{4}$\
According to the meaning of Higgs field, this is the Lagrangian density for Higgs field. Thus we obtain the Lagrangian density for Higgs field. let $k=c$, then obtain:\
$-|\nabla_{4}\rho _{vr}|^{2}=(\partial_{u}\rho_{vr})^{\dagger}\partial^{u}\rho_{vr}$, $u=1, 2, 3, 4$\
where, $\partial_{u}$, $\partial^{u}$ are the covariant form derivative and the contravariant form derivative, thus we obtain:\
$$L_{v}=W_{zcv}-\beta_{0}(\partial_{u}\rho_{vr})^{+}\partial^{u}\rho_{vr}
+\beta_{1}|\rho_{vr}|^{2}+\beta_{2}|\rho_{vr}|^{4}$$\[20\]\
this is the form of the Lagrangian density of Higgs field which is used in the electroweak unified theory(the difference between them is a constant only). Though their forms are similar, the component numbers in the isospace are different(In the electroweak unified theory, how to obtain the Lagrangian density of Higgs field has not been explained, so is it in supersymmetric theories). In my follow-up papers, I will discuss the essence of time, the origin of mass, the origin of spin, the quantization of electric charge; the four basic interactions and their unification from the angle of the level of disorder.
References
==========
$[1]$: Nir Polonsky, Supersymmetry:Structure and Phenomena, Extensions of Standard Model, Springer(2001).\
Gordon Kane, Supersymmetry, Squarks, Photinos and Unveiling of the Ultimate Laws of Nature. Perseus Publishing, Cambridge Massachusetts(2000)\
$[2]$: Amitabha Lahiri, Palash B. Pal, Quantum Field Theory, CRC Press (2000)\
$[3]$: Edited and translated by Fan Dai Nian, Zhao Zhong Li, Xu Liang Ying.The Collected Papers of Einstein, Vol. 2. Published by the Commercial Press (1983).\
$[4]$: Original Author, Huang Kun, Revised by Han Qi, Solid StatePhysics, Senior Education Press (1985).\
W.Greiner B.M$\hat{u}$ller J.Rafelski, Quantum Electrodynamics of Strong Fields, Springer-Verlag,Berlin Heidelberg New York okyo(1985).\
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{
"pile_set_name": "ArXiv"
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---
abstract: |
We study phase field equations based on the diffuse-interface approximation of general homogeneous free energy densities showing different local minima of possible equilibrium configurations in perforated/porous domains. The study of such free energies in homogeneous environments found a broad interest over the last decades and hence is now widely accepted and applied in both science and engineering. Here, we focus on strongly heterogeneous materials with perforations such as porous media. To the best of our knowledge, we present a general formal derivation of upscaled phase field equations for arbitrary free energy densities and give a rigorous justification by error estimates for a broad class of polynomial free energies. The error between the effective macroscopic solution of the new upscaled formulation and the solution of the microscopic phase field problem is of order $\epsilon^{1/2}$ for a material given characteristic heterogeneity $\epsilon$. Our new, effective, and reliable macroscopic porous media formulation of general phase field equations opens new modelling directions and computational perspectives for interfacial transport in strongly heterogeneous environments.
*Keywords:* Free energies, Cahn-Hilliard/Ginzburg-Landau equations, multiscale modeling, homogenization, porous media, wetting, phase transformations
author:
- 'Markus Schmuck [^1]'
- 'Grigorios A. Pavliotis [^2]'
- 'Serafim Kalliadasis [^3]'
bibliography:
- 'effWetting7.bib'
title: Rate of Convergence of Phase Field Equations in Strongly Heterogeneous Media towards their Homogenized Limit
---
maketitle
2em
1.5em
.5em
--------
author
--------
1em [date]{}
1.5em
Introduction {#sec:Intr}
============
In Section \[sec:2Fo\], we give relevant reformulations of phase field equations and introduce basic notations and mathematical assumptions. The main theorems, which state the new effective macroscopic phase field formulation (Theorem \[thm:EfPhFi\]) and the associated error (Theorem \[thm:ErEs\]) between the solution of the upscaled problem which reliably accounts for the microstructure by homogenization and the solution of the microscopic problem fully resolving the pores in Section \[sec:MaRe\]. The justification of these results then follow in Sections \[sec:FoUp\] and \[sec:ErEs\]. Conclusions and suggestions for further work are given in Section \[sec:Concl\].
Mathematical preliminaries and notation {#sec:2Fo}
=======================================
Main results {#sec:MaRe}
============
Proof of Theorem \[thm:EfPhFi\] {#sec:FoUp}
===============================
Proof of Theorem \[thm:ErEs\] {#sec:ErEs}
=============================
Conclusions {#sec:Concl}
===========
Acknowledgements {#acknowledgements .unnumbered}
================
[^1]: `[email protected]` (corresponding author)
[^2]: `[email protected]`
[^3]: `[email protected]`
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'We give two sufficient criteria for schlichtness of envelopes of holomorphy in terms of topology. These are weakened converses of results of Kerner and Royden. Our first criterion generalizes a result of Hammond in dimension 2. Along the way we also prove a generalization of Royden’s theorem.'
address: |
Department of Mathematics\
1910 University Drive\
Boise State University\
Boise, ID 83725-1555\
USA
author:
- Zach Teitler
date: 'December 2, 2010'
title: Topological criteria for schlichtness
---
Let $\Omega \subseteq {\mathbb{C}}^n$ be a domain. The [**envelope of holomorphy**]{} of $\Omega$ is a pair $(\tilde\Omega, \pi)$ consisting of a connected Stein manifold $\tilde\Omega$ and a locally biholomorphic map $\pi \colon \tilde\Omega \to {\mathbb{C}}^n$, together with a holomorphic inclusion $\alpha \colon \Omega \to \tilde\Omega$, characterized by the following properties: $\pi \circ \alpha$ is the identity, and each holomorphic function $f$ on $\Omega$ has a unique holomorphic extension $F_f$ on $\tilde\Omega$ with $f = F_f \circ \alpha$. Let $\Omega' = \pi( \tilde \Omega )$ and let $i = \pi \circ \alpha \colon \Omega \to \Omega'$. The envelope of holomorphy $(\tilde \Omega, \pi)$ is [**schlicht**]{} if $\pi \colon \tilde\Omega \to \Omega'$ is biholomorphic. One would like to give conditions on $\Omega$ to have a schlicht envelope of holomorphy.
Two results of Kerner and Royden lead to necessary conditions. Kerner [@Kerner] has shown that $\alpha_* \colon \pi_1(\Omega) \to \pi_1(\tilde\Omega)$ is surjective. Royden [@MR0152675] has shown that $\alpha^* \colon H^1(\tilde\Omega ; {\mathbb{Z}}) \to H^1(\Omega ; {\mathbb{Z}})$ is injective. It follows trivially that if $(\tilde \Omega, \pi)$ is schlicht, so $\tilde\Omega = \Omega'$, then $i_* \colon \pi_1(\Omega) \to \pi_1(\Omega')$ is surjective and $i^* \colon H^1(\Omega' ; {\mathbb{Z}}) \to H^1(\Omega ; {\mathbb{Z}})$ is injective.
Neither of these conditions is sufficient, by a result of Fornaess and Zame [@FZ] (see [@hammond 3]). Following an idea of Hammond [@hammond] one may seek sufficient conditions by adjoining to surjectivity of $i_*$ (or injectivity of $i^*$) the assumption that $\pi \colon \tilde\Omega \to \Omega'$ is a covering space. This strong assumption is still reasonable, as covering maps certainly occur among envelopes of holomorphy — indeed, Fornaess and Zame show in [@FZ] that for any covering map $\pi \colon \tilde\Omega \to \Omega'$ there is a domain $\Omega \subseteq \Omega'$ with envelope of holomorphy $(\tilde\Omega, \pi)$.
Specifically, Hammond has shown that, in dimension $n=2$, if $i_* \colon \pi_1(\Omega) \to \pi_1(\Omega')$ is surjective and $\pi \colon \tilde\Omega \to \Omega'$ is a covering map, then $(\tilde\Omega, \pi)$ is schlicht. We give an elementary proof of Hammond’s theorem in all dimensions $n \geq 2$. In addition we give a sufficient condition for schlichtness in terms of injectivity of $i^*$ on cohomology, again assuming $\pi$ is a covering map. Along the way we give an alternative proof of Royden’s theorem which also extends it to other coefficient groups than ${\mathbb{Z}}$.
If $\pi$ is a covering map and $i_* \colon \pi_1(\Omega) \to \pi_1(\Omega')$ is surjective then $(\tilde\Omega, \pi)$ is schlicht.
This extends the theorem of Hammond for dimension $n=2$. Hammond’s proof relies on a result of Jupiter [@MR2221089] which is special to dimension $2$.
The number of sheets of the covering map $\pi$ is equal to the index of $\pi_*(\pi_1(\tilde\Omega))$ in $\pi_1(\Omega')$ (see, for example, [@MR1867354 Prop. 1.32]). The surjectivity of $i_* = \pi_* \circ \alpha_*$ implies $\pi_*$ is surjective. Hence the index of the image subgroup is $1$, so $\pi \colon \tilde\Omega \to \Omega'$ is $1$-sheeted, i.e., a homeomorphism. Since $\pi$ is a holomorphic homeomorphism it is biholomorphic and so $\tilde\Omega$ is schlicht.
Compare the more technical proof in [@hammond].
The cohomology in Royden’s result is Čech cohomology with coefficients in the sheaf of locally constant ${\mathbb{Z}}$-valued functions. Since our spaces are manifolds, Čech cohomology coincides with singular cohomology (with coefficients in ${\mathbb{Z}}$); see, for example, [@MR755006 Thm. 73.2]. Recall also that by the universal coefficient theorem, $H^1(X ; G) = \operatorname{Hom}(\pi_1(X), G)$, for a path-connected space $X$ and abelian coefficient group $G$ [@MR1867354 pg. 98].
Before we go on, observe that this proves Royden’s theorem as a consequence of Kerner’s theorem and extends it to other coefficient groups.
For any abelian group $G$, $\alpha^* \colon H^1(\Omega ; G) \to H^1(\tilde\Omega ; G)$ is injective.
Since $\alpha_* \colon \pi_1(\Omega) \to \pi_1(\tilde\Omega)$ is surjective, $\alpha^* \colon \operatorname{Hom}(\pi_1(\Omega), G) \to \operatorname{Hom}(\pi_1(\tilde\Omega), G)$ is injective and these $\operatorname{Hom}$ groups coincide with $H^1(\Omega ; G)$, $H^1(\tilde\Omega ; G)$.
Royden proves this for $G = {\mathbb{Z}}$ using Čech cohomology, in particular the exponential short exact sequence (hence the restriction to $G = {\mathbb{Z}}$).
We get the following.
If $\pi$ is a covering map, $\pi_1(\Omega')$ is nilpotent, and $i^* \colon H^1(\Omega' ; G) \to H^1(\Omega ; G)$ is injective for every abelian group $G$, then $(\tilde\Omega, \pi)$ is schlicht.
Since $i^* = \alpha^* \circ \pi^*$ is injective, $\pi^*$ is injective as well. Via $\pi_*$ we regard $\pi_1(\tilde\Omega)$ as a subgroup of $\pi_1(\Omega')$. Recall that if $H$ is any nilpotent group then every maximal proper subgroup $N$ of $H$ is normal and has prime index (see [@MR1307623 Thm. 5.40]), and in particular $H/N$ is abelian. If $\pi_1(\tilde\Omega) \subsetneqq \pi_1(\Omega')$ there is a maximal subgroup $\pi_1(\tilde\Omega) \subseteq N \subsetneqq \pi_1(\Omega')$ and hence a surjection $\pi_1(\Omega') \to G = \pi_1(\Omega') / N$ to an abelian group with $\pi_1(\tilde\Omega)$ mapping to zero. This surjection is nonzero and lies in the kernel of $$\pi^* \colon H^1(\Omega' ; G) = \operatorname{Hom}(\pi_1(\Omega'), G) \to \operatorname{Hom}(\pi_1(\tilde\Omega), G) = H^1(\tilde\Omega ; G)$$ for the abelian group $G = \pi_1(\Omega') / N$, contradicting the injectivity of $\pi^*$.
It follows that $\pi_1(\tilde\Omega) = \pi_1(\Omega')$. As before this implies $\pi$ is a degree $1$ covering map, hence a biholomorphism.
It is not necessary to assume $i^*$ is injective when coefficients are taken in any abelian group $G$. It would be enough to assume $i^*$ is injective when coefficients are taken in any finite cyclic group, in any abelian quotient $G$ of $\pi_1(\Omega')$, or even just in a single abelian quotient $G = \pi_1(\Omega') / N$ for some proper normal subgroup $N$ containing $\pi_1(\tilde\Omega)$.
If in addition $\pi \colon \tilde\Omega \to \Omega'$ is a normal covering space then $\pi_1(\tilde\Omega) \subseteq \pi_1(\Omega')$ is a normal subgroup and we can take $G$ to be an abelian quotient of $\pi_1(\Omega') / \pi_1(\tilde\Omega)$, which is the group of deck transformations.
Suppose $\pi$ is a normal covering map with deck transformation group $H$. If there exists a nonzero abelian quotient $G$ of $H$ such that $i^* \colon H^1(\Omega' ; G) \to H^1(\Omega ; G)$ is injective, then $(\tilde\Omega, \pi)$ is schlicht.
I thank Chris Hammond for explaining his theorem to me, and Emil Straube, Craig Westerland, and Jens Harlander for helpful and patient conversations.
[Ham10]{}
John Erik Forn[æ]{}ss and William R. Zame, *Riemann domains and envelopes of holomorphy*, Duke Math. J. **50** (1983), no. 1, 273–283. Chris Hammond, *Schlicht envelopes of holomorphy and topology*, Math. Z. **266** (2010), no. 2, 285–288.
Allen Hatcher, *Algebraic topology*, Cambridge University Press, Cambridge, 2002. Daniel Jupiter, *A schlichtness theorem for envelopes of holomorphy*, Math. Z. **253** (2006), no. 3, 623–633. Hans Kerner, *[Ü]{}berlagerungen und [H]{}olomorphieh[ü]{}llen*, Math. Ann. **144** (1961), 126–134. James R. Munkres, *Elements of algebraic topology*, Addison-Wesley Publishing Company, Menlo Park, CA, 1984. Joseph J. Rotman, *An introduction to the theory of groups*, fourth ed., Graduate Texts in Mathematics, vol. 148, Springer-Verlag, New York, 1995. H. L. Royden, *One-dimensional cohomology in domains of holomorphy*, Ann. of Math. (2) **78** (1963), 197–200.
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{
"pile_set_name": "ArXiv"
}
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---
author:
- 'Taushif Ahmed,'
- 'Maguni Mahakhud,'
- 'Prakash Mathews,'
- Narayan Rana
- 'and V. Ravindran'
title: 'Two-Loop QCD Corrections to Higgs $\rightarrow b + \bar{b} + g$ Amplitude'
---
Introduction
============
\[sec:intro\] The tests of the Standard Model (SM) have been going on for several decades in various experiments and most of its predictions have been tested in an unprecedented accuracy. The recent discovery of Higgs boson by ATLAS [@atlashiggs] and CMS [@cmshiggs] collaborations at the Large Hadron Collider (LHC) puts the SM on firm footing. The Higgs boson results from Higgs mechanism that provides a framework for electroweak symmetry breaking. Elementary particles such as leptons, quarks, gauge bosons and Higgs boson acquire masses through the Higgs mechanism. The mass of the Higgs boson being a parameter of the theory can not be predicted by the SM and hence its discovery provides a valuable information on this. Results from Higgs searches at LEP [@higgslep] and Tevatron [@higgstev] were crucial ingredients to the recent discovery in narrowing down the search regions for the LHC collaborations. The direct searches at the LEP excluded Higgs of mass below 114.4 GeV and the precision electroweak measurements [@lepprecis] hinted for Higgs boson in the mass less than 152 GeV at $95\%$ confidence level (CL). Tevatron on the other hand excluded Higgs of mass in the range $162-166$ GeV at $95\%$ CL.
The dominant production mechanism for the Higgs production at the LHC is gluon gluon fusion through top quark loop. The subdominant ones come from vector boson fusion, associated production of Higgs with vector bosons and top anti-top pairs and bottom anti-bottom annihilation. The inclusive production cross section for the Higgs production is known to an unprecedented accuracy due to many breakthroughs in the computation of amplitudes, loop and phase space integrals. For gluon-gluon [@gghNNLO], vector boson fusion processes [@bolzoni], and associated production with vector bosons [@Han:1991ia], the inclusive rates are known to NNLO accuracy in QCD. There are also studies related to the Higgs production in association with bottom quarks which were also motivated to study Higgs boson in certain SUSY models, namely MSSM. The coupling of bottom quarks become large in the large $\tan\beta$ region, where $\tan\beta$ is the vacuum expectation values of up and down type Higgs fields in the Higgs sector of MSSM. Such large couplings can enhance gluon fusion as well as bottom quark fusion subprocesses. Fully inclusive cross section for Higgs production in association with bottom quark to NNLO level accuracy is also known in the variable flavour scheme (VFS) [@vfs], while it is known only up to NLO level in the fixed flavour scheme (FFS) [@ffs]. In the VFS, one assumes the initial state bottom quarks inside the proton. They are there as a result of emission of collinear bottom anti-bottom states from the gluons intrinsically present inside the proton. They being collinear give large logs which need to be resummed. The resummed contribution is the source for non-vanishing bottom and anti-bottom parton distribution functions inside the proton in the VFS scheme.
The differential distributions for Higgs production and its decay to pair of photons [@Anastasiou:2005qj] or massive vector bosons [@Anastasiou:2007mz; @Grazzini:2008tf] have also been known at NNLO level in QCD in the infinite top quark mass limit. Such exclusive observables allow direct comparison of theoretical predictions with experimental results which include kinematical cuts on the final state particles. In particular, observables with jet vetos enhance the significance of the signal considerably allowing us to study the properties of Higgs boson and its coupling to other SM particles. NNLO QCD prediction [@Boughezal:2013uia] for production of Higgs with one jet through effective gluon-gluon-higgs vertex in the infinite top quark mass limit is available, thanks to various ingredients that are computed to the required accuracy by different groups [@Hggg; @Gehrmann:2011aa]. As the experimental accuracy improves, it will be important to include other subdominant production mechanisms. In this article, we provide the relevant one and two loop amplitudes for the process $H \rightarrow b + \overline b + g $ which is analytically continued also to obtain the production of Higgs boson with one jet in bottom anti-bottom annihilation, i.e., $b+\overline b \rightarrow H+g$, where Higgs couples to bottom quark through Yukawa coupling denoted by $\lambda$. We use VFS scheme throughout. This will be an important supplement to the Higgs boson with one jet at NNLO level as it includes the bottom quark effects in VFS scheme.
Beyond leading order in perturbation theory, one encounters large number of Feynman amplitudes with rich Lorentz and gauge structures. In addition, the loop integrals become increasingly complicated due to their multiple kinematic dependence. Generation of diagrams, simplification of Lorentz, Dirac and color indices can be done symbolically. Using integration by parts (IBP) and Lorentz invariant (LI) identities the large number of loop integrals can be reduced in a rather straight forward way to few master integrals (MI). The two loop MIs for four legs processes where all fields but one external leg are massless were solved by Gehrmann and Remiddi [@Gehrmann:2000zt] using an elegant method of differential equations.
In this article we present one and two loop QCD amplitudes for the process $H \rightarrow b + \overline b + g $ treating both bottom and other four light quarks massless. We do not include top quark in our analysis. To obtain infrared safe observables, we require, in addition to these two loop amplitudes, one loop corrected $H\rightarrow b+\overline b + 2~{\rm partons }$ and tree level $H \rightarrow b+\overline b + 3~{\rm partons}$ amplitudes. Note that they are individually infrared singular due to the presence of massless partons in the amplitudes. There exist several equally efficient frameworks which use these infrared sensitive contributions to combine them to obtain infrared safe observables. They go by the names sector decomposition [@secdec], $q_T$-subtraction [@qtsub] and antenna subtraction [@antsub] methods. More recently the method developed by Czakon using sector decomposition and FKS [@fks] phase space slicing, was applied to obtain top quark pair production [@topnnlo] at NNLO level and NNLO QED corrections [@nnloqed] to $Z\rightarrow e^+ e^-$. Antenna subtraction was used to obtain NNLO QCD corrections to di-jet production at the LHC. The NNLO corrections to Higgs plus one jet resulting from only gluon-gluon-Higgs effective interaction are obtained recently in [@Gehrmann:2011aa] making best use of the subtraction methods in an efficient way. The amplitudes presented in this article will constitute contributions coming from bottom-antibottom-higgs interactions to Higgs plus one jet observable at NNLO level. We have presented the amplitudes in the form suitable for easier implementation to study infra-red safe hadron level observables involving Higgs plus one jet at NNLO in QCD.
In the next section, we discuss the Lagrangian that describes coupling of Higgs boson with bottom quark, explain how the projector technique can be used to obtain the amplitudes and describe the renormalization and factorization properties of the amplitudes. Section \[sec:calc\] is dedicated to the computational details. Final results in compact form are given in Section \[sec:result\] and corresponding coefficients are given in the Appendix. In section \[sec:conc\], we conclude with our findings.
Theory
======
The interaction part of the action involving bottom quarks and Higgs boson is given by $$\begin{aligned}
S^b_{I} = - \lambda \int d^4 x \, \phi(x) \overline \psi_b(x) \psi_b(x)\end{aligned}$$ where, $\psi_b(x)$ denotes the bottom quark field and $\phi(x)$ the scalar field. $\lambda$ is the Yukawa coupling given by $\sqrt{2} m_b/\upsilon$, with the bottom quark mass $m_b$ and the vacuum expectation value $\upsilon \approx 246$ GeV. For the pseudoscalar Higgs of MSSM, we need to replace $\lambda \phi(x) \overline \psi_b(x) \psi_b(x)$ by $\tilde \lambda \tilde \phi(x) \overline \psi_b(x) \gamma_5 \psi_b(x)$ in the above equation. The MSSM couplings are $$\tilde{\lambda} = \left\{
\begin{array}{ll}
- \frac{\sqrt{2} m_b \sin\alpha}{\upsilon \cos\beta} \,,& \qquad \tilde{\phi} = h\,,\\
\phantom{-} \frac{\sqrt{2} m_b \cos\alpha}{\upsilon \cos\beta} \,,& \qquad \tilde{\phi}=H\,,\\
\phantom{-} \frac{\sqrt{2} m_b \tan\beta}{\upsilon } \,, & \qquad \tilde{\phi}=A\,
\end{array}
\right.$$ respectively. The angle $\alpha$ is the measure of mixing of weak and mass eigenstates of neutral Higgs bosons. In the VFS scheme, except in the Yukawa coupling, $m_b$ is taken to be zero like other light quarks in the theory. The number of active flavours is taken to be $n_f=5$. We work in Feynman gauge throughout.
Notation and kinematics {#subsec:defs}
-----------------------
We consider the decay of Higgs boson to a bottom quark, anti-bottom quark and a gluon $$H(q) \longrightarrow b(p_1) + \bar{b} (p_2) + g(p_3) \, .$$ The associated Mandelstam variables are defined as $$s \equiv (p_1 + p_2)^2, \hspace{1cm} t \equiv (p_2 + p_3)^2, \hspace{1cm} u \equiv (p_1 + p_3)^2$$ which satisfy $$s >0,~ t > 0, ~ u > 0, ~~ s + t + u = M_H^2 \equiv Q^2 > 0$$ where, $M_H$ is the mass of the Higgs boson. We also define the following dimensionless invariants which appear in harmonic polylogarithms (HPL) [@remiddi] and 2dHPL [@Gehrmann:2000zt] as $$x \equiv s/Q^2, \hspace{1cm} y \equiv u/Q^2, \hspace{1cm} z \equiv t/Q^2$$ satisfying $$0 < x < 1,~ 0 < y < 1,~ 0 < z < 1,~ ~\text{and}~ x + y + z = 1.$$
### Analytical continuation {#sec:ancont .unnumbered}
In order to compute the Higgs + 1 jet production at hadron colliders, the decay amplitudes must be analytically continued to the appropriate kinematical regions. The corresponding processes are $$\begin{aligned}
\nonumber
1. ~~~~ \overline{b} (-p_1) + b(-p_2) &\rightarrow g(p_3) + H (p_4)
\\ \nonumber
2. ~~~~ b (-p_2) + g(-p_3) &\rightarrow b (p_1) + H (p_4)
\\ \label{subprocess}
3. ~~~~ \overline{b}(-p_1) + g(-p_3) &\rightarrow \overline{b} (p_2) + H (p_4)\end{aligned}$$
For the process 1, $Q^2 = M_H^2 > 0$, $s > 0 ,~ t < 0 $ and $u < 0$. Hence we introduce the dimensionless parameters $u_1$ and $v_1 $ with the following definitions $$u_1 \equiv - \frac{u}{s}, \hspace{1cm} v_1 \equiv \frac{Q^2}{s}$$ such that $0 < u_1 < 1$ and $0< v_1 < 1$.
Similarly, for the process 2, $Q^2 = M_H^2 > 0$, $s < 0 ,~ t > 0 $ and $u < 0$ and the dimensionless parameters are $u_2$ and $v_2 $ with the following definitions $$u_2 \equiv - \frac{u}{t}, \hspace{1cm} v_2 \equiv \frac{Q^2}{t}$$ such that $0 < u_2 < 1$ and $0< v_2 < 1$. The last one is trivially related to the second one.
The general structure of the amplitude {#sec:gentensor}
--------------------------------------
In this section, we describe how the amplitude for $H \rightarrow b+\overline b + g$ can be obtained using projector technique. Since the amplitude contains one external gluon, it can be expressed as $$| \M \rangle = {\cal S}_{\mu} (b,\bar{b};g) \varepsilon^{\mu}$$ where, $\varepsilon^{\mu}$ is the gluon polarization vector.
We observe the amplitude has the following general structure in terms of the coefficients $A',A''$ and $A_2$: $${\cal S}_{\mu} (b,\bar{b};g) = \bar{u} (p_1) \Big\{
A'~ p_{1 \mu} + A''~ p_{2 \mu} + A_2~\slashed p_3 \gamma_{\mu}
\Big\} v (p_2)$$ where, we have used $p_3.\varepsilon = 0$. QCD Ward identity gives $$A'~ p_{1}. p_3 + A''~ p_{2} . p_3 = 0
~~ \Rightarrow ~~
A' = - A''~ \frac{p_{2} . p_3}{p_{1}. p_3} \equiv A_1~ p_2.p_3 \, .$$ Hence, the amplitude takes the following form: $$\begin{aligned}
{\cal S}_{\mu} (b,\bar{b};g) ~\varepsilon^{\mu} &= \bar{u} (p_1) \Big\{
A_1~( p_2.p_3 ~ p_{1 \mu} - p_1 . p_3 ~ p_{2 \mu} )
+ A_2~\slashed p_3 \gamma_{\mu}
\Big\} v(p_2) ~\varepsilon^{\mu}
\nonumber\\ \label{tencoeff}
&\equiv A_1~ {\rm T_1} + A_2~ {\rm T_2} \; .\end{aligned}$$ The coefficients $A_m ~ (m = 1, 2)$ can be obtained from the amplitude $| \M \rangle$ using appropriate projectors ${\cal P}(A_m)$ $$A_m=\sum_{\rm spins} {\cal P} (A_m) {\cal S}_{\mu} (b,\bar{b};g) ~\varepsilon^{\mu} =\sum_{\rm spins} {\cal P} (A_m) | \M \rangle$$ where, in $d$ space-time dimensions, the projectors are found to be $$\begin{aligned}
\label{projectors}
{\cal P} (A_1) &= \frac{2 (d-2)}{s^2 \, t \, u \, (d-3)} {\rm T_1}^{\dagger}
+ \frac{1}{s \, t \, u \, (d-3)} {\rm T_2}^{\dagger} \, ,
\nonumber\\
{\cal P} (A_2) &= \frac{1}{s \, t \, u \, (d-3)} {\rm T_1}^{\dagger}
+ \frac{1}{2 \, t \, u \, (d-3)} {\rm T_2}^{\dagger} \, . \end{aligned}$$ Expanding the coefficients $A_m$ in powers of strong coupling constant $a_s = g_s^2/16 \pi^2$, we obtain $$\label{coeffexpnd}
A_m = \frac{\lambda}{\mu_R^\epsilon} ~4\pi \sqrt{a_s} T^a_{ij} \Big\{
A_m^{(0)} + a_s A_m^{(1)}
+ a_s^2 A_m^{(2)} + {\cal O}(a_s^3)
\Big\}$$ where, $T^a$ are the Gell-Mann matrices, $a$ is adjoint and $i$, $j$ are fundamental indices of SU(3) and $\mu_R$ is the renormalization scale. These coefficients $A_m^{(l)}$ completely specify the amplitude order by order in perturbation theory.
As described in section \[sec:ancont\], for Higgs + 1 jet production, the above amplitudes have to be suitably crossed and the coefficients $A_{m}$ will be expressed in terms of corresponding $u_i$ and $v_i$.
Ultraviolet renormalization
---------------------------
The Feynman amplitudes for the process $H\rightarrow b+\overline b+g$ beyond leading order develop ultraviolet divergences in QCD. We have used dimensional regularization to regulate them taking space-time dimension to be $d=4+\epsilon$. The scale $\mu_0$ is introduced to scale the mass dimension of the dimension-full strong coupling constant in $d$ dimensions. If we denote the dimensionless strong coupling constant by $\hat g_s$ in $d$ dimensions, then the unrenormalized amplitude can be expanded in terms of $\hat a_s=\hat g_s^2/16 \pi^2$ as $$\label{unm}
|{\cal M} \rangle = \frac{\hat{\lambda}}{\mu_0^\epsilon} S_\epsilon \unas^{\frac{1}{2}} \left\{ |\unM^{(0)} \rangle + \unas |\unM^{(1)} \rangle + \unas^2 |\unM^{(2)} \rangle + {\cal O}(\hat{a}_s^3) \right\}$$ where, $S_{\ep} = \exp[\frac{\ep}{2} (\gamma_E - \ln 4\pi)]$ with Euler constant $\gamma_E = 0.5772 \ldots$ , results from loop integrals beyond leading order. $|\unM^{(i)} \rangle$ is the unrenormalized color-space vector which represents the $i^{th}$ loop amplitude. In $\overline {MS}$ scheme, the renormalized coupling constant $a_s \equiv a_s (\mu_R^2)$ at the renormalization scale $\mu_R$ is related to unrenormalized coupling constant $\hat{a}_s$ by $$\begin{aligned}
\label{renas}
\frac{\hat{a}_s}{\mu_0^{\epsilon}} S_{\epsilon} &= \frac{a_s}{\mu_R^{\epsilon}} {Z}(\mu_R^2)
\nonumber\\[1ex]
&= \frac{a_s}{\mu_R^{\epsilon}} \left[ 1 + a_s \left( \frac{1}{\ep} r_{a_{1;1}} \right)
+ a_s^2 \left( \frac{1}{\ep^2} r_{a_{2;2}} + \frac{1}{\ep} r_{a_{2;1}} \right) + {\cal O}(a_s^3) \right] \end{aligned}$$ where, $$r_{a_{1;1}} = 2 \b0 \;, \quad
r_{a_{2;2}} = 4\b0^2 \;, \quad
r_{a_{2;1}} = \beta_1 \;, \quad$$
$$\b0 = \left(\frac{11}{3} C_A - \frac{4}{3} T_F n_f\right) , \hspace{0.5cm}
\beta_1 = \left(\frac{34}{3} C_A^2 - \frac{20}{3} C_A T_F n_f - 4 C_F T_F n_f\right)$$
with $C_A = N$, $C_F = (N^2 -1)/{2 N} $, $T_F = 1/2$ and $n_f$ is the number of active quark flavors. The bare coupling constant $\hat \lambda$ is renormalized using $$\begin{aligned}
\label{renl}
\frac{\hat{\lambda}}{\mu_0^\epsilon} S_\epsilon &= \frac{\lambda}{\mu_R^\epsilon} {Z}_{\lambda}(\mu_R^2)
\nonumber\\[1ex]
&= \frac {\lambda}{\mu_R^\epsilon} \left[ 1 + a_s \left( \frac{1}{\ep} r_{\lambda_{1;1}} \right)
+ a_s^2 \left( \frac{1}{\ep^2} r_{\lambda_{2;2}}
+ \frac{1}{\ep} r_{\lambda_{2;1}} \right)
+ {\cal O}(a_s^3) \right] \; ,\end{aligned}$$ with $\lambda=\lambda(\mu_R^2)$ and $$r_{\lambda_{1;1}} = 6 C_F \,, ~
r_{\lambda_{2;2}} = \Big(18 C_F^2 + 6 \b0 C_F\Big)\, , ~
r_{\lambda_{2,1}} = \left( \frac{3}{2} C_F^2
+ \frac{97}{6} C_F C_A
- \frac{10}{3} C_F T_F n_f \right).$$ Using the eqn.(\[renas\]) and eqn.(\[renl\]), we now can express $|{\cal M } \rangle$ (eqn.(\[unm\])) in powers of renormalized $a_s$ with UV finite matrix elements $|\rnM^{(i)} \rangle$ $$\label{renamp}
|\rnM \rangle = \frac{\lambda}{\mu_R^\epsilon} \rnas^{\frac{1}{2}} \Bigg( |\rnM^{(0)} \rangle + a_s |\rnM^{(1)} \rangle + a_s^{2} |\rnM^{(2)} \rangle + {\cal O}({a}_s^3) \Bigg)$$ where, $$\begin{aligned}
\label{rln}
|\rnM^{(0)} \rangle &= \left( \frac{1}{\mu^{\ep}_R} \right)^{\frac{1}{2}} |\unM^{(0)} \rangle \; ,
\nonumber\\[1ex]
|\rnM^{(1)} \rangle &= \left( \frac{1}{\mu^{\ep}_R} \right)^{\frac{3}{2}}
\left[ ~ |\unM^{(1)} \rangle + \mu^{\ep}_R \Big( \frac{r_{a_1}}{2} + r_{\lambda_1} \Big) |\unM^{(0)} \rangle ~ \right] \; ,
\nonumber\\[1ex]
|\rnM^{(2)} \rangle &= \left( \frac{1}{\mu^{\ep}_R} \right)^{\frac{5}{2}}
\Big[ ~ |\unM^{(2)} \rangle
+ \mu^{\ep}_R \Big( \frac{3r_{a_1}}{2} + r_{\lambda_1} \Big) |\unM^{(1)} \rangle
\nonumber\\
& \qquad \qquad \quad
+ \mu^{2\ep}_R \left( \frac{r_{a_2}}{2} - \frac{r_{a_1}^2}{8} + \frac{r_{a_1}}{2} r_{\lambda_1}
+ r_{\lambda_2} \right) |\unM^{(0)} \rangle ~ \Big]\end{aligned}$$ with $$\begin{aligned}
r_{a_1} &= \left(\frac{1}{\ep} r_{a_{1;1}}\right) \;, \quad \quad
r_{a_2} = \left(\frac{1}{\ep^2} r_{a_{2;2}} + \frac{1}{\ep} r_{a_{2;1}}\right) \;,
\nonumber\\
r_{\lambda_1} &= \left(\frac{1}{\ep} r_{\lambda_{1;1}}\right) \;, \quad \quad
r_{\lambda_2} = \left(\frac{1}{\ep^2} r_{\lambda_{2;2}} + \frac{1}{\ep} r_{\lambda_{2;1}} \right) \,.\end{aligned}$$ We describe the computation of unrenormalized amplitudes $|\unM^{(l)} \rangle, l=0,1,2$ in section \[sec:calc\].
Infrared factorization {#sec:infrared}
----------------------
In addition to UV divergences, the amplitude suffers from soft and collinear divergences beyond leading order due to the presence of soft gluons and collinear massless partons in the loops. According to KLN theorem [@Kinoshita:1962ur; @Lee:1964is], to obtain infrared safe observables, we need to include appropriate contributions coming from real emission processes along with mass factorization counter terms and to perform sum over degenerate configurations. Thanks to factorization properties of QCD amplitudes, the infrared divergence structure of the amplitudes is well understood. The earliest account on two loop QCD amplitudes was by Catani [@catani1], who predicted the infrared poles in $\epsilon$ of multi-parton QCD amplitudes in dimensional regularization excluding two loop single pole. In [@sterman], Sterman and Tejeda-Yeomans demonstrated the connection of single pole in $\epsilon$ to a soft anomalous dimension matrix, later computed in [@Aybat:2006wq; @Aybat:2006mz] using factorization properties of the scattering amplitudes along with infrared evolution equations. The decomposition of single pole term into universal collinear and soft anomalous dimensions at two loop level in QCD was first observed in electromagnetic and Higgs form factors [@Ravindran:2004mb]. Becher and Neubert [@Becher:2009cu], using soft collinear effective theory, derived the exact formula for the infra-red divergences of scattering amplitudes with an arbitrary number of loops and legs in massless QCD including single pole in dimensional regularization. Gardi and Magnea also arrived at, a similar all order result [@Gardi:2009qi] using Wilson lines for hard partons and soft and eikonal jet functions in dimensional regularization. Following Catani, we express the renormalized amplitudes $|{\cal M}^{(i)}\rangle$ in terms of the universal subtraction operators ${\bf I}^{(i)}_b(\epsilon)$ as follows[^1]
$$\begin{aligned}
\label{catani1}
|\rnM^{(1)} \rangle &=& 2 \hspace{0.1cm} {\bf{I}}_{b}^{(1)} (\ep) \hspace{0.1cm} |\rnM^{(0)} \rangle
+ |\rnM^{(1)fin} \rangle \, ,
\nonumber\\[2ex]
|\rnM^{(2)} \rangle &=& 2 \hspace{0.1cm} {\bf{I}}_{b}^{(1)} (\ep) \hspace{0.1cm} |\rnM^{(1)} \rangle
+ 4 \hspace{0.1cm} {\bf{I}}^{(2)}_{b} (\ep) \hspace{0.1cm} |\rnM^{(0)} \rangle
+ |\rnM^{(2)fin} \rangle \end{aligned}$$
$$\begin{aligned}
\text{where,} ~~~~
{\bf{I}}^{(1)}_{b} (\ep) &= \frac{1}{2} \frac{e^{- \frac{\ep}{2} \gamma_E}}{\Gamma(1+\frac{\ep}{2})}
\Bigg\{
\Big( \frac{4}{\ep^2} - \frac{3}{\ep} \Big) (C_A - 2 C_F) \Big[ \Big( - \frac{s}{\mu^2_R} \Big)^{\frac{\ep}{2}} \Big]
\nonumber\\[1ex]
& +
\Big( - \frac{4 C_A}{\ep^2} + \frac{3 C_A}{ 2 \ep} + \frac{\b0}{2 \ep} \Big)
\left[ \Big( - \frac{t}{\mu^2_R} \Big)^{\frac{\ep}{2}} + \Big( - \frac{u}{\mu^2_R} \Big)^{\frac{\ep}{2}} \right]
\Bigg\} \; ,
\nonumber\\[1ex]
{\bf{I}}^{(2)}_{b} (\ep) &= - \hspace{0.1cm} \frac{1}{2} {\bf{I}}^{(1)}_{b} (\ep) \Big[ {\bf{I}}^{(1)}_{b} (\ep) - \frac{2 \b0}{\ep} \Big]
+\hspace{0.1cm} \frac{e^{\frac{\ep}{2} \gamma_E} \hspace{0.1cm} \Gamma(1+\ep)}{\Gamma(1+\frac{\ep}{2})} \Big[ -\frac{\b0}{\ep} + K \Big] \hspace{0.1cm} {\bf{I}}^{(1)}_{b} (2\ep) \hspace{1.4cm}
\nonumber\\[1ex]
& +\hspace{0.1cm} \left( 2 {\bf{H}}^{(2)}_{q} (\ep) + {\bf{H}}^{(2)}_{g} (\ep) \right)\end{aligned}$$
with $$K = \left(\frac{67}{18}-\frac{\pi^2}{6} \right) C_A - \frac{10}{9} T_F n_f \, ,$$
$$\begin{aligned}
{\bf{H}}^{(2)}_q (\ep) &= \frac{1}{\ep}
\Bigg\{
C_A C_F \Big( - \frac{245}{432} + \frac{23}{16} \zeta_2 - \frac{13}{4} \zeta_3 \Big)
+ C_F^2 \Big( \frac{3}{16} - \frac{3}{2} \zeta_2 + 3 \zeta_3 \Big)
\nonumber\\[1ex]
& \quad
+ C_F n_f \Big( \frac{25}{216} - \frac{1}{8} \zeta_2 \Big)
\Bigg\} \; ,
\nonumber\\[1ex]
{\bf{H}}^{(2)}_g (\ep) &= \frac{1}{\ep}
\Bigg\{
C_A^2 \Big( - \frac{5}{24} - \frac{11}{48} \zeta_2 - \frac{1}{4} \zeta_3 \Big)
+ C_A n_f \Big(\frac{29}{54} + \frac{1}{24} \zeta_2 \Big) - \frac{1}{4} C_F n_f -\frac{5}{54} n_f^2
\Bigg\} \, .\end{aligned}$$
The born amplitude $|\rnM^{(0)}\rangle$ and the finite parts $|\rnM^{(l)fin}\rangle, l=1,2$ are process dependent and hence they are determined by explicit computation.
Calculation of the amplitudes {#sec:calc}
=============================
![Planar topologies of master integrals[]{data-label="fig:planar"}](planar.jpg){width="80.00000%"}
![Non-planar topologies of master integrals[]{data-label="fig:nonplanar"}](nplanar.jpg){width="88.00000%"}
We now describe how we compute the coefficients $A_m$ from the amplitudes $|\hat {\cal M}^{(l)}\rangle$ for the process $H \rightarrow b+\overline b+g$ up to two loop level in QCD perturbation theory. QGRAF [@qgraf] is used to generate the Feynman amplitudes for this process. There are 2 diagrams at tree level, 13 at one loop and 251 at two loops excluding tadpole and self energy corrections to the external legs.
Using FORM [@Vermaseren:2000nd] and Mathematica, output of the QGRAF is converted to a form suitable for further symbolic manipulation. Using the projectors given in eqn.(\[projectors\]), we have projected out unrenormalized $\hat{A}_i$ from these amplitudes. They contain only scalar products among internal and external momenta. For the external on-shell gluon leg the physical polarization sum is done using $$\sum_s \varepsilon^{\mu}(p_3,s) \varepsilon^{\nu *}(p_3,s) = -g^{\mu\nu} + \frac{p_3^{\mu} q^{\nu} + q^{\mu} p_3^{\nu}}{p_3.q}$$ where, $p_3$ is the gluon momentum and $q$ is an arbitrary light-like 4-vector for which we choose $q = p_1$. The Lorentz contractions and Dirac algebra are done in $d = 4 +\epsilon$ dimensions. The next step involves the evaluation of one and two loop tensor and scalar integrals. This is done by first reducing them to an irreducible set of MIs using IBP identities and LI identities and substituting the MIs evaluated to desired accuracy in $\epsilon$. We have used a Mathematica package LiteRed [@litered] to use IBP [@chet] and LI identities [@gr] in an efficient manner. The MIs for the kinematic configuration of the problem at hand are analytically known from the seminal works of Gehrmann and Remiddi [@Gehrmann:2000zt]. We use them to obtain the unrenormalized coefficients in a Laurent series in $\epsilon$. In order to optimize the use of LiteRed, we have reduced all the one and two loop integrals to belong to few integral sets. This is done by shifting the loop momenta suitably using an in-house algorithm which uses FORM. We find that the sets for both one and two loop integrals are exactly same as those given in [@Ahmed:2014gla] for the case of massive spin-2 resonance $\rightarrow$ 3 gluons. The topologies of the appearing planar and non-planar master integrals are shown in fig.(\[fig:planar\]) and fig.(\[fig:nonplanar\]) respectively. For one-loop diagrams, the integral belongs to one of the following sets: $$\begin{aligned}
\label{onebasis}
\{ \cD, \hspace{0.1cm} \cD_{1}, \hspace{0.1cm} \cD_{12}, \hspace{0.1cm} \cD_{123} \} \, ,
\{ \cD, \hspace{0.1cm} \cD_{2}, \hspace{0.1cm} \cD_{23}, \hspace{0.1cm} \cD_{123} \} \, ,\{ \cD, \hspace{0.1cm} \cD_{3}, \hspace{0.1cm} \cD_{31}, \hspace{0.1cm} \cD_{123} \}\end{aligned}$$ where, $$\begin{aligned}
\cD = k_1^2, \hspace{0.1cm} \cD_{i} = (k_1 - p_i)^2,
\hspace{0.1cm} \cD_{ij} = (k_1 - p_i - p_j)^2,
\hspace{0.1cm} \cD_{ijk} = (k_1 - p_i - p_j - p_k)^2 \; .\end{aligned}$$ At two loops, we have nine independent Lorentz invariants involving loop momenta $k_1$ and $k_2$, namely $\{ ( k_\alpha \cdot k_\beta ),
(k_\alpha \cdot p_i) \}, \alpha,\beta = 1, 2;\hspace{0.1cm} i = 1,...,3 $. Shifting of loop momenta allows us to express each two loop Feynman integral to contain terms belonging to one of the following six sets: $$\begin{aligned}
\label{twobasis}
&&\{ \cD_0, \hspace{0.1cm} \cD_1, \hspace{0.1cm} \cD_2, \hspace{0.1cm} \cD_{1;1}, \hspace{0.1cm} \cD_{2;1}, \hspace{0.1cm}
\cD_{1;12}, \hspace{0.1cm} \cD_{2;12}, \hspace{0.1cm} \cD_{1;123}, \hspace{0.1cm} \cD_{2;123} \} \,,
\nonumber\\[1.5ex]
&&\{ \cD_0, \hspace{0.1cm} \cD_1, \hspace{0.1cm} \cD_2, \hspace{0.1cm} \cD_{1;2}, \hspace{0.1cm} \cD_{2;2}, \hspace{0.1cm}
\cD_{1;23}, \hspace{0.1cm} \cD_{2;23}, \hspace{0.1cm} \cD_{1;123}, \hspace{0.1cm} \cD_{2;123} \} \,,
\nonumber\\[1.5ex]
&&\{ \cD_0, \hspace{0.1cm} \cD_1, \hspace{0.1cm} \cD_2, \hspace{0.1cm} \cD_{1;3}, \hspace{0.1cm} \cD_{2;3}, \hspace{0.1cm}
\cD_{1;31}, \hspace{0.1cm} \cD_{2;31}, \hspace{0.1cm} \cD_{1;123}, \hspace{0.1cm} \cD_{2;123} \} \,,
\nonumber\\[1.5ex]
&&\{ \cD_0, \hspace{0.1cm} \cD_1, \hspace{0.1cm} \cD_2, \hspace{0.1cm} \cD_{1;1}, \hspace{0.1cm} \cD_{2;1}, \hspace{0.1cm}
\cD_{0;3}, \hspace{0.1cm} \cD_{1;12}, \hspace{0.1cm} \cD_{2;12}, \hspace{0.1cm} \cD_{1;123} \} \,,
\nonumber\\[1.5ex]
&&\{ \cD_0, \hspace{0.1cm} \cD_1, \hspace{0.1cm} \cD_2, \hspace{0.1cm} \cD_{1;2}, \hspace{0.1cm} \cD_{2;2}, \hspace{0.1cm}
\cD_{0;1}, \hspace{0.1cm} \cD_{1;23}, \hspace{0.1cm} \cD_{2;23}, \hspace{0.1cm} \cD_{1;123} \} \,,
\nonumber\\[1.5ex]
&&\{ \cD_0, \hspace{0.1cm} \cD_1, \hspace{0.1cm} \cD_2, \hspace{0.1cm} \cD_{1;3}, \hspace{0.1cm} \cD_{2;3}, \hspace{0.1cm}
\cD_{0;2}, \hspace{0.1cm} \cD_{1;31}, \hspace{0.1cm} \cD_{2;31}, \hspace{0.1cm} \cD_{1;123} \}\end{aligned}$$ where, $$\begin{aligned}
&& \cD_0 = (k_1 - k_2)^2, \hspace{0.1cm} \cD_{\alpha} = k_{\alpha}^2, \hspace{0.1cm} \cD_{\alpha;i} = (k_{\alpha} - p_i)^2,
\hspace{0.1cm} \cD_{\alpha; ij} = (k_{\alpha} - p_i - p_j)^2,
\nonumber\\[1ex]
&& \cD_{0;i} = (k_1 - k_2 - p_i)^2, \hspace{0.1cm} \cD_{\alpha;ijk} = (k_{\alpha} - p_i - p_j - p_k)^2 \; .\end{aligned}$$ The UV singularities present in the bare coefficients are systematically removed using eqns.(\[renas\] & \[renl\]). The resulting UV finite coefficients do contain divergences from soft and collinear partons. In the next section we will demonstrate that our results correctly reproduce divergences described in the section \[sec:infrared\] at one and two loop level. We will also present the finite parts of the coefficients $A_m$ up to two loop level.
Results {#sec:result}
=======
In this section we present the results up to two loop level in QCD for the amplitude $H \rightarrow b +\overline b+g$ in the ${\overline {MS}}$ scheme. The results are presented after subtracting the one and two loop universal subtraction operators $I_b^{(i)}(\epsilon),i=1,2$ as described in the section \[sec:infrared\]. Following the eqns.(\[tencoeff\], \[coeffexpnd\] & \[renamp\]), the $l^{th}$ loop amplitude can be written as $$\label{rnMxpnd}
|\rnM^{(l)} \rangle = 4 \pi ~ T^a_{ij} \Big\{ A_1^{(l)} {\rm T_1} + A_2^{(l)} {\rm T_2} \Big\} \;.$$ The renormalised coefficients $A_m^{(l)}$ are related to their bare counterparts $\hat A_m^{(l)}$ through (see eqn.(\[rln\])): $$\begin{aligned}
\label{rln2}
A_m^{(0)} &= \left( \frac{1}{\mu^{\ep}_R} \right)^{\frac{1}{2}} \hat{A}_m^{(0)} \; ,
\nonumber\\[1ex]
A_m^{(1)} &= \left( \frac{1}{\mu^{\ep}_R} \right)^{\frac{3}{2}}
\left[ ~ \hat{A}_m^{(1)} + \mu^{\ep}_R \Big( \frac{r_{a_1}}{2} + r_{\lambda_1} \Big) \hat{A}_m^{(0)} ~ \right] \; ,
\nonumber\\[1ex]
A_m^{(2)} &= \left( \frac{1}{\mu^{\ep}_R} \right)^{\frac{5}{2}}
\Big[ ~ \hat{A}_m^{(2)}
+ \mu^{\ep}_R \Big( \frac{3r_{a_1}}{2} + r_{\lambda_1} \Big) \hat{A}_m^{(1)}
\nonumber\\
& \qquad \qquad \quad
+ \mu^{2\ep}_R \left( \frac{r_{a_2}}{2} - \frac{r_{a_1}^2}{8} + \frac{r_{a_1}}{2} r_{\lambda_1}
+ r_{\lambda_2} \right) \hat{A}_m^{(0)} ~ \Big] \;.\end{aligned}$$ Using the procedure discussed in the previous section, we first compute the bare coefficients $\hat A_m^{(l)}$ and the eqns.(\[rln2\]) give the renormalized coefficients. The finite parts of the coefficients $A_m^{(l)}$ are defined after subtracting terms proportional to universal subtraction terms $I_b^{(l)}$ as follows $$\begin{aligned}
\label{catani2}
A_m^{(1)} &= 2 \hspace{0.1cm} {\bf{I}}_{b}^{(1)} (\ep) \hspace{0.1cm} A_m^{(0)}
+ A_m^{(1)fin} \,,
\nonumber\\[2ex]
A_m^{(2)} &= 2 \hspace{0.1cm} {\bf{I}}_{b}^{(1)} (\ep) \hspace{0.1cm} A_m^{(1)}
+ 4 \hspace{0.1cm} {\bf{I}}^{(2)}_{b} (\ep) \hspace{0.1cm} A_m^{(0)}
+ A_m^{(2)fin}\end{aligned}$$ where, we have used eqns.(\[tencoeff\] & \[catani1\]).
Expanding the right sides of eqns.(\[rln2\] & \[catani2\]) in powers of $\ep$, we find that the infrared poles agree exactly, providing a crucial test on the correctness of our computation. The finite parts of the coefficients have the following expansions: $$\begin{aligned}
& A_m^{(l)fin} = \sum_{n=0}^l ~ A_m^{(0)} ~ {\cal B}^{(l)}_{m ; n} \ln^n \Big( - \frac{Q^2}{\mu^2} \Big) \end{aligned}$$ where, $$A_1^{(0)} = - \frac{4 i}{t ~ u} \hspace{1cm} \text{and} \hspace{1cm} A_2^{(0)} = i \Big( \frac{1}{t} + \frac{1}{u} \Big)$$ and the remaining coefficients ${\cal B}^{(l)}_{m;n}$ are given in the appendix. We also performed an independent computation of $\langle {\cal M}^{(0)}|{\cal M}^{(l)}\rangle$ for $l=1,2$ without using any projectors and then compared against one obtained using the projectors, i.e using the coefficients $A_m^{(l)}$. We find both give the same result, providing an independent check on our computation.
Following [@Gehrmann:2002zr] [^2], we have obtained results for the crossed reactions given in eqn. (\[subprocess\]) relevant for Higgs+1 jet production at hadron colliders. The corresponding finite coefficients $A_m^{(l)fin}$ are attached with the arXiv submission.
Conclusions {#sec:conc}
===========
We have presented the amplitudes for the partonic subprocess $H \rightarrow b+\overline b+g$ and other subprocesses related by crossing, up to two loop level in QCD that contribute to exclusive observables involving Higgs boson and a jet. The dominant one is from gluon gluon fusion which is already known to this accuracy. We have used dimensional regularization to perform our computation. Using appropriate projectors, the amplitude is expressed in terms of two scalar coefficients $A_m$. We have found that the infrared structure of the amplitude is according to Catani’s prediction on QCD amplitudes upto two loop level. Also, the coefficient of single pole term is found to be in agreement with predictions based on the observation of the universal behavior of poles in the multi-parton QCD amplitudes.
Acknowledgments {#acknowledgments .unnumbered}
===============
TA, MM and NR thank the Institute of Mathematical Sciences (IMSc) for the hospitality during the course of the work. We thank the staff of IMSc computer center for their help. We sincerely thank T. Gehrmann for providing us the master integrals and analytically continued HPLs and 2d-HPLs required for our computation. We thank R. N. Lee for his help with LiteRed. Finally, we would like to thank K. Hasegawa, M. K. Mandal and L. Tancredi for useful discussions. The work of TA, MM and NR has been partially supported by funding from RECAPP, Department of Atomic Energy, Govt. of India.
Harmonic polylogarithms
=======================
Here, we provide the definition of HPL and 2dHPL. HPL is represented by $H(\vec{m}_w;y)$ with a $w$-dimensional vector $\vec{m}_w$ of parameters and its argument $y$. The elements of $\vec{m}_w$ belong to $\{ 1, 0, -1 \}$ through which we define the following rational functions $$f(1;y) \equiv \frac{1}{1-y}, \qquad f(0;y) \equiv \frac{1}{y}, \qquad f(-1;y) \equiv \frac{1}{1+y} \, .$$ The weight 1 $(w = 1)$ HPLs are $$H(1, y) \equiv - \ln (1 - y), \qquad H(0, y) \equiv \ln y, \qquad H(-1, y) \equiv \ln (1 + y) \, .$$ For $w > 1$, the definition of $H(m, \vec{m}_{w};y)$ is $$\label{1dhpl}
H(m, \vec{m}_w;y) \equiv \int_0^y dx ~ f(m, x) ~ H(\vec{m}_w;x), \qquad \qquad m \in 0, \pm 1 \, .$$ The 2dHPLs are defined in the same way as eqn.(\[1dhpl\]) with the new elements $\{ 2, 3 \}$ in $\vec{m}_w$ representing a new class of rational functions $$f(2;y) \equiv f(1-z;y) \equiv \frac{1}{1-y-z}, \qquad f(3;y) \equiv f(z;y) \equiv \frac{1}{y+z}$$ and correspondingly with the weight 1 $(w = 1)$ 2dHPLs $$H(2, y) \equiv - \ln \Big(1 - \frac{y}{1-z} \Big), \qquad H(3, y) \equiv \ln \Big( \frac{y+z}{z} \Big) \, .$$
One-loop coefficients
=====================
[B]{}\^[(1)]{}\_[1 ; 1]{} = (-11 C\_A - 18 C\_F + 2 n\_f)
[B]{}\^[(1)]{}\_[1 ; 0]{} = ( -6 H(0,y) H(0,z)-6 H(0,y) H(1,z)-6 H(2,y) H(0,z)+12 H(3,y) H(1,z)-10 H(0,y)-9 H(2,y)-6 H(0,2,y) -6 H(2,0,y)+12 H(3,2,y)-10 H(0,z)-9 H(1,z)+6 H(0,1,z)-6 H(1,0,z)- 6 \_2 ) + C\_F ( 2 H(0,y) H(1,z)- 4 H(3,y) H(1,z) + 2 H(2,y) H(0,z) + 3 H(2,y)+12 H(0,2,y)- 2 H(1,0,y)+ 2 H(2,0,y)- 4 H(3,2,y)+ 3 H(1,z)- 2 H(0,1,z)- 2 ) + n\_f ( H(0,y)+H(0,z) )
[B]{}\^[(1)]{}\_[2 ; 1]{} = (-11 C\_A-18 C\_F + 2 n\_f)
[B]{}\^[(1)]{}\_[2 ; 0]{} = ( - 6 H(0,y) H(0,z)-6 H(0,y) H(1,z)-6 H(2,y) H(0,z)+12 H(3,y) H(1,z)-10 H(0,y)-9 H(2,y)-6 H(0,2,y) -6 H(2,0,y) +12 H(3,2,y)-10 H(0,z)-9 H(1,z)+6 H(0,1,z)-6 H(1,0,z) - 6 \_2 + 6 ) + C\_F ( 2 H(0,y) H(1,z)-4 H(3,y) H(1,z) +2 H(2,y) H(0,z) +3 H(2,y)+2 H(0,2,y)-2 H(1,0,y)+2 H(2,0,y)-4 H(3,2,y)+3 H(1,z)-2 H(0,1,z)-3 ) + n\_f ( H(0,y)+H(0,z) )
Two-loop coefficients
=====================
[B]{}\^[(2)]{}\_[1 ; 2]{} = ( 121 C\_A\^2 + 44 C\_A ( 6 C\_F - n\_f) + 4 ( 27 C\_F\^2 - 12 C\_F n\_f + n\_f\^2 ) )
[B]{}\^[(2)]{}\_[1 ; 1]{} = C\_A\^2 ( 594 H(0,y) H(0,z)+594 H(0,y) H(1,z)+594 H(2,y) H(0,z)-1188 H(3,y) H(1,z)+990 H(0,y)+891 H(2,y) +594 H(0,2,y)+594 H(2,0,y)-1188 H(3,2,y)+990 H(0,z)+891 H(1,z)-594 H(0,1,z)+594 H(1,0,z)+495 \_2-108 \_3 -702 ) + C\_A C\_F ( -1188 H(0,y) H(1,z)+54 H(0,y) (6 H(0,z)+6 H(1,z)+10)-1188 H(2,y) H(0,z)+54 (6 H(2,y)+10) H(0,z)+1728 H(3,y) H(1,z)-1296 H(2,y)-864 H(0,2,y)+1188 H(1,0,y)-864 H(2,0,y)+1728 H(3,2,y)-1296 H(1,z)+864 H(0,1,z)+324 H(1,0,z)+1566 \_2-2808 \_3-1048 ) + C\_F\^2 ( -648 H(0,y) H(1,z)-648 H(2,y) H(0,z)
[=]{} +1296 H(3,y) H(1,z)-972 H(2,y)-648 H(0,2,y)+648 H(1,0,y)-648 H(2,0,y)+1296 H(3,2,y)-972 H(1,z)+648 H(0,1,z)-1296 \_2 + 2592 \_3 + 648 ) + C\_A n\_f ( - 2 (-27 (4 H(3,y)-3) H(1,z)+81 H(2,y)+54 H(0,2,y)+54 H(2,0,y)-108 H(3,2,y)-54 H(0,1,z)+54 H(1,0,z) + 45 \_2 - 206 ) -9 H(0,y) (12 H(0,z)+12 H(1,z)+31)-9 (12 H(2,y)+31) H(0,z) ) + C\_F n\_f ( 108 H(0,y) H(1,z)+108 H(2,y) H(0,z)-216 H(3,y) H(1,z)-27 H(0,y)+162 H(2,y)+108 H(0,2,y)-108 H(1,0,y)+108 H(2,0,y)-216 H(3,2,y)-27 H(0,z)+162 H(1,z)-108 H(0,1,z) - 54 \_2 + 32 ) + n\_f\^2 ( 9 H(0,y)+9 H(0,z)-20 )
[B]{}\^[(2)]{}\_[1 ; 0]{} = C\_A\^2 { \_4 ( 39/8 ) + \_3 ( - H(0,y) - 6 H(1,y) + 11 H(2,y) - H(0,z) + 5 H(1,z) + 6 /s + 407/36 ) + \_2 ( 108 s\^2 H(0,y) H(0,z)
[=]{} +72 s\^2 H(1,y) H(0,z)+108 s\^2 H(0,y) H(1,z)+72 s\^2 H(1,y) H(1,z)+36 s\^2 H(2,y) H(0,z)-72 s\^2 H(3,y) H(1,z)+147 s\^2 H(0,y)+453 s\^2 H(2,y)+108 s\^2 H(0,2,y)+72 s\^2 H(1,2,y)+36 s\^2 H(2,0,y)-72 s\^2 H(3,2,y)+147 s\^2 H(0,z)+237 s\^2 H(1,z)+36 s\^2 H(0,1,z)+36 s\^2 H(1,0,z)+72 s\^2 H(1,1,z)-216 s H(1,y)-108 s t H(0,z)-108 s u H(0,y)-216 s u H(1,z)-40 s\^2+96 t u ) / (36 s\^2) + ( - H(2,y) H(1,z)+ H(3,y) H(1,z)+7 H(0,0,y) H(1,z)- H(0,2,y) H(1,z)-7 H(0,3,y) H(1,z)- H(2,0,y) H(1,z)+ H(2,3,y) H(1,z)-7 H(3,0,y) H(1,z)+ H(3,2,y) H(1,z)+ H(3,3,y) H(1,z)+2 H(0,0,2,y) H(1,z)+2 H(0,2,0,y) H(1,z)+4 H(0,3,0,y) H(1,z)+4 H(0,3,3,y) H(1,z)+2 H(1,0,3,y) H(1,z)-2 H(1,2,3,y) H(1,z)+2 H(2,0,0,y) H(1,z)+2 H(2,0,3,y) H(1,z)+2 H(2,1,0,y) H(1,z)+4 H(2,2,3,y) H(1,z)+2 H(2,3,0,y) H(1,z)-8 H(2,3,3,y) H(1,z)+4 H(3,0,3,y) H(1,z)+4 H(3,3,0,y) H(1,z)-16 H(3,3,3,y) H(1,z)
[=]{} - H(1,z)- H(2,y)+ H(0,0,y)+7 H(2,y) H(0,0,z)+2 H(0,0,y) H(0,0,z)+ H(0,0,z)+ H(2,y) H(0,1,z)+ H(3,y) H(0,1,z)+2 H(0,0,y) H(0,1,z)+ H(0,1,z)+2 H(0,0,z) H(0,2,y)-2 H(0,1,z) H(0,2,y)- H(0,2,y)-2 H(0,1,z) H(0,3,y)+++6 H(2,y) H(1,0,z)-7 H(3,y) H(1,0,z)+2 H(0,0,y) H(1,0,z)-6 H(0,3,y) H(1,0,z)- H(1,0,z)+ H(3,y) H(1,1,z)+2 H(0,0,y) H(1,1,z)- H(1,1,z)-2 H(0,1,z) H(1,2,y) [}]{}
[=]{} [{]{} +2 H(1,0,z) H(1,2,y)+2 H(0,0,z) H(2,0,y)+2 H(0,1,z) H(2,0,y)+2 H(1,0,z) H(2,0,y)- H(2,0,y)+2 H(0,0,z) H(2,2,y)+4 H(0,1,z) H(2,2,y)+4 H(1,0,z) H(2,2,y)- H(2,2,y)-6 H(0,1,z) H(2,3,y)+2 H(1,0,z) H(2,3,y)+2 H(0,1,z) H(3,0,y)-2 H(1,0,z) H(3,0,y)+ H(3,2,y)-12 H(0,1,z) H(3,3,y)+4 H(1,0,z) H(3,3,y)-2 H(1,y) H(0,0,1,z)-2 H(2,y) H(0,0,1,z)-8 H(3,y) H(0,0,1,z)+ H(0,0,1,z)+7 H(0,0,2,y)+++2 H(1,y) H(0,1,0,z)+6 H(2,y) H(0,1,0,z)+4 H(3,y) H(0,1,0,z)+ H(0,1,1,z)+7 H(0,2,0,y)- H(0,2,2,y)-7 H(0,3,2,y)+2 H(2,y) H(1,0,0,z)+7 H(1,0,0,z)-2 H(1,y) H(1,0,1,z)+6 H(2,y) H(1,0,1,z)+ H(1,0,1,z)--+2 H(1,y) H(1,1,0,z)+6 H(2,y) H(1,1,0,z)+H(0,y) (-++(-) H(0,z)- H(1,z)+7 H(0,0,z)- H(0,1,z)+ H(1,0,z)- H(1,1,z)-2 H(0,0,1,z)-2 H(0,1,1,z)+2 H(1,0,0,z)+2 H(1,1,0,z)-)+7 H(2,0,0,y)- H(2,0,2,y)+ H(2,1,0,y)- H(2,2,0,y)+ H(2,3,2,y)-7 H(3,0,2,y)-7 H(3,2,0,y)+ H(3,2,2,y)+ H(3,3,2,y)+H(0,z) (- H(2,y)+7 H(0,0,y)+ H(0,2,y)-+ H(2,0,y)- H(2,2,y)-7 H(3,2,y)+2 H(0,0,2,y)+2 H(0,2,0,y)-2 H(0,2,2,y)-6 H(0,3,2,y)-2 H(1,0,2,y)+2 H(2,0,0,y)+2 H(2,2,0,y)+2 H(2,3,2,y)-2 H(3,0,2,y)-2 H(3,2,0,y)+4 H(3,3,2,y)-)+8 H(0,0,1,0,z)+2 H(0,0,1,1,z)+2 H(0,0,2,2,y)+2 H(0,1,0,1,z)+6 H(0,1,1,0,z)+2 H(0,2,0,2,y)
[=]{} +4 H(0,2,1,0,y)+2 H(0,2,2,0,y)+4 H(0,3,0,2,y)+4 H(0,3,2,0,y)+4 H(0,3,3,2,y)+6 H(1,0,1,0,z)+2 H(1,0,3,2,y)+2 H(1,1,0,0,z)+4 H(1,1,0,1,z)+6 H(1,1,1,0,z)-2 H(1,2,3,2,y)+2 H(2,0,0,2,y)+2 H(2,0,1,0,y)+2 H(2,0,2,0,y)+2 H(2,0,3,2,y)+2 H(2,1,0,2,y)+2 H(2,1,2,0,y)+2 H(2,2,0,0,y)+4 H(2,2,1,0,y)+4 H(2,2,3,2,y)+2 H(2,3,0,2,y)+2 H(2,3,2,0,y)-8 H(2,3,3,2,y)+4 H(3,0,3,2,y)+4 H(3,3,0,2,y)+4 H(3,3,2,0,y)-16 H(3,3,3,2,y)- ) }
[=]{} + C\_A C\_F { \_4 ( ) + \_3 ( 22 H(1,y)-18 H(2,y)+4 H(1,z)++-+ ) + \_2 ( ----++++-------2 H(1,y) H(0,z)-6 H(0,y) H(1,z)-2 H(1,y) H(1,z)-4 H(2,y) H(0,z)+ H(1,y)+2 H(0,2,y)+6 H(1,0,y)-2 H(1,2,y)-4 H(2,0,y)-8 H(2,1,y)+8 H(2,2,y)+2 H(0,1,z)+2 H(1,0,z)-2 H(1,1,z)---- ) + ( --++
[=]{} ++-++-+--+----++-++++ H(1,z)+2 H(1,z) H(2,y)+ H(2,y)- H(1,z) H(3,y)-14 H(1,z) H(0,0,y)-14 H(2,y) H(0,0,z)- H(2,y) H(0,1,z)- H(3,y) H(0,1,z)-4 H(0,0,y) H(0,1,z)- H(0,1,z)- H(1,z) H(0,2,y)-4 H(0,0,z) H(0,2,y)+6 H(0,1,z) H(0,2,y)+ H(0,2,y)+20 H(1,z) H(0,3,y)+4 H(0,1,z) H(0,3,y)+-+3 H(1,z) H(1,0,y)
[=]{} -2 H(0,1,z) H(1,0,y)---+20 H(3,y) H(1,0,z)-4 H(0,0,y) H(1,0,z)+8 H(0,2,y) H(1,0,z)+16 H(0,3,y) H(1,0,z)+2 H(1,0,y) H(1,0,z)+ H(3,y) H(1,1,z)-8 H(0,0,y) H(1,1,z)+4 H(0,3,y) H(1,1,z)+2 H(1,1,z)+6 H(0,1,z) H(1,2,y)-2 H(1,0,z) H(1,2,y)- H(1,z) H(2,0,y)-4 H(0,0,z) H(2,0,y)-2 H(0,1,z) H(2,0,y)-6 H(1,0,z) H(2,0,y)+ H(2,0,y)-8 H(0,0,z) H(2,2,y)-16 H(0,1,z) H(2,2,y)-8 H(1,0,z) H(2,2,y)+2 H(2,2,y)- H(1,z) H(2,3,y)+12 H(0,1,z) H(2,3,y)-4 H(1,0,z) H(2,3,y)+20 H(1,z) H(3,0,y)-4 H(0,1,z) H(3,0,y)+4 H(1,1,z) H(3,0,y)+ H(1,z) H(3,2,y)-4 H(0,1,z) H(3,2,y)-4 H(1,0,z) H(3,2,y)- H(3,2,y)- H(1,z) H(3,3,y)+28 H(0,1,z) H(3,3,y) -12 H(1,0,z) H(3,3,y)-8 H(1,1,z) H(3,3,y)+6 H(1,y) H(0,0,1,z)+16 H(3,y) H(0,0,1,z)- H(0,0,1,z) -8 H(1,z) H(0,0,2,y)-14 H(0,0,2,y)+4 H(1,z) H(0,0,3,y)++-+2 H(1,z) H(0,1,0,y)+-2 H(1,y) H(0,1,0,z)-12 H(2,y) H(0,1,0,z)-4 H(3,y) H(0,1,0,z)
[=]{} -4 H(3,y) H(0,1,1,z)+ H(0,1,1,z)-8 H(1,z) H(0,2,0,y)-14 H(0,2,0,y)- H(0,2,2,y)+2 H(1,z) H(0,2,3,y)-8 H(1,z) H(0,3,0,y)+4 H(1,z) H(0,3,2,y)+20 H(0,3,2,y)-12 H(1,z) H(0,3,3,y)+4 H(1,z) H(1,0,0,y)+14 H(1,0,0,y)-4 H(2,y) H(1,0,0,z)+6 H(1,y) H(1,0,1,z)-18 H(2,y) H(1,0,1,z)-4 H(3,y) H(1,0,1,z)- H(1,0,1,z)+3 H(1,0,2,y)-6 H(1,z) H(1,0,3,y)+ H(1,1,0,y)+H(0,y) ((-++2) H(0,z)+(+) H(1,z)+ (---+-+13 H(0,1,z)--+-14 H(1,1,z)+12 H(0,0,1,z)+6 H(0,1,0,z)+24 H(0,1,1,z)+6 H(1,0,1,z)-12 H(1,1,0,z)+))-++-2 H(1,y) H(1,1,0,z)-18 H(2,y) H(1,1,0,z)-4 H(3,y) H(1,1,0,z)+3 H(1,2,0,y)+6 H(1,z) H(1,2,3,y)-8 H(1,z) H(2,0,0,y)-14 H(2,0,0,y)- H(2,0,2,y)-4 H(1,z) H(2,0,3,y)
[=]{} +-2 H(1,z) H(2,1,0,y)- H(2,2,0,y)-16 H(1,z) H(2,2,3,y)-4 H(1,z) H(2,3,0,y)- H(2,3,2,y)+16 H(1,z) H(2,3,3,y)+4 H(1,z) H(3,0,2,y)+20 H(3,0,2,y)-12 H(1,z) H(3,0,3,y)+4 H(1,z) H(3,2,0,y)+20 H(3,2,0,y)+ H(3,2,2,y)-8 H(1,z) H(3,2,3,y)-12 H(1,z) H(3,3,0,y)+H(0,z) ((+) H(2,y)+ (----+-47 H(0,2,y)+(-++28) H(1,0,y)--14 H(2,2,y)+60 H(3,2,y)-24 H(0,0,2,y)+6 H(0,1,0,y)+12 H(0,2,2,y)+48 H(0,3,2,y)+12 H(1,0,0,y)+12 H(1,0,2,y)+6 H(1,2,0,y)-12 H(2,0,0,y)+6 H(2,0,2,y)-6 H(2,1,0,y)-12 H(2,2,0,y)-12 H(2,3,2,y)+24 H(3,0,2,y)+12 H(3,2,2,y)-36 H(3,3,2,y)+))-8 H(1,z) H(3,3,2,y)- H(3,3,2,y)+40 H(1,z) H(3,3,3,y)+4 H(0,0,1,0,y)-8 H(0,0,1,0,z)-8 H(0,0,1,1,z)-8 H(0,0,2,2,y)+4 H(0,0,3,2,y)-8 H(0,1,0,1,z)+2 H(0,1,0,2,y)-12 H(0,1,1,0,z)+2 H(0,1,2,0,y)-8 H(0,2,0,2,y)-2 H(0,2,1,0,y)-8 H(0,2,2,0,y)+2 H(0,2,3,2,y)-8 H(0,3,0,2,y)-8 H(0,3,2,0,y)+4 H(0,3,2,2,y)-12 H(0,3,3,2,y)-2 H(1,0,0,1,z)+4 H(1,0,0,2,y)+4 H(1,0,1,0,y)-4 H(1,0,1,0,z)+4 H(1,0,2,0,y)-6 H(1,0,3,2,y)-12 H(1,1,0,1,z)-14 H(1,1,1,0,z)+4 H(1,2,0,0,y)+4 H(1,2,1,0,y)+6 H(1,2,3,2,y)-8 H(2,0,0,2,y)-8 H(2,0,2,0,y)-4 H(2,0,3,2,y)+4 H(2,1,0,0,y)-2 H(2,1,0,2,y)-8 H(2,1,1,0,y)-2 H(2,1,2,0,y)-8 H(2,2,0,0,y)-8 H(2,2,1,0,y)-16 H(2,2,3,2,y)-4 H(2,3,0,2,y)-4 H(2,3,2,0,y)+16 H(2,3,3,2,y)+8 H(3,0,1,0,y)+4 H(3,0,2,2,y)-12 H(3,0,3,2,y)+4 H(3,2,0,2,y)-8 H(3,2,1,0,y)+4 H(3,2,2,0,y)-8 H(3,2,3,2,y)-12 H(3,3,0,2,y)-12 H(3,3,2,0,y)-8 H(3,3,2,2,y)+40 H(3,3,3,2,y)---+-+- ) }
[=]{} + C\_F\^2 { \_4 (-22) + \_3 ( -16 H(1,y)+8 H(2,y)-8 H(1,z)--+-30 ) + \_2 ( +++----++++++-8 H(2,y) H(1,z)+8 H(0,1,y)-8 H(0,2,y)+8 H(1,1,y)+8 H(2,1,y)-16 H(2,2,y)++++16 ) + ( -+-+--+-++-++-++-+---
[=]{} -18 H(1,z)+9 H(1,z) H(2,y)-18 H(2,y)+8 H(1,z) H(3,y)+12 H(3,y) H(0,1,z)+4 H(0,1,z)+12 H(1,z) H(0,2,y)-4 H(0,1,z) H(0,2,y)-4 H(0,2,y)-12 H(1,z) H(0,3,y)-+-6 H(1,z) H(1,0,y)
[=]{} +4 H(0,1,z) H(1,0,y)++-12 H(3,y) H(1,0,z)-8 H(0,2,y) H(1,0,z)-8 H(0,3,y) H(1,0,z)-24 H(3,y) H(1,1,z)+8 H(0,0,y) H(1,1,z)-8 H(0,3,y) H(1,1,z)+9 H(1,1,z)-4 H(0,1,z) H(1,2,y)+12 H(1,z) H(2,0,y)-4 H(0,1,z) H(2,0,y)-4 H(2,0,y)+8 H(0,0,z) H(2,2,y)+16 H(0,1,z) H(2,2,y)+9 H(2,2,y)-12 H(1,z) H(2,3,y)-12 H(1,z) H(3,0,y)-8 H(1,1,z) H(3,0,y)-24 H(1,z) H(3,2,y)+8 H(0,1,z) H(3,2,y)+8 H(3,2,y)+24 H(1,z) H(3,3,y)-8 H(0,1,z) H(3,3,y)+8 H(1,0,z) H(3,3,y)+16 H(1,1,z) H(3,3,y)-4 H(1,y) H(0,0,1,z)+8 H(2,y) H(0,0,1,z)+8 H(1,z) H(0,0,2,y)-8 H(1,z) H(0,0,3,y)-++-4 H(1,z) H(0,1,0,y)+4 H(2,y) H(0,1,0,z)+8 H(3,y) H(0,1,1,z)-12 H(0,1,1,z)+8 H(1,z) H(0,2,0,y)+12 H(0,2,2,y)-4 H(1,z) H(0,2,3,y)
[=]{} -8 H(1,z) H(0,3,2,y)-12 H(0,3,2,y)+8 H(1,z) H(0,3,3,y)-8 H(1,z) H(1,0,0,y)+2 H(0,y) (--+++(--2) H(1,z)+6 H(1,1,z)-4 H(0,1,1,z)-2 H(1,0,1,z))-4 H(1,y) H(1,0,1,z)+12 H(2,y) H(1,0,1,z)+8 H(3,y) H(1,0,1,z)-6 H(1,0,1,z)-6 H(1,0,2,y)+4 H(1,z) H(1,0,3,y)+--+4 H(2,y) H(1,1,0,z)-6 H(1,2,0,y)-4 H(1,z) H(1,2,3,y)+8 H(1,z) H(2,0,0,y)+12 H(2,0,2,y)--4 H(1,z) H(2,1,0,y)+12 H(2,2,0,y)+16 H(1,z) H(2,2,3,y)-12 H(2,3,2,y)-8 H(1,z) H(3,0,2,y)-12 H(3,0,2,y)+8 H(1,z) H(3,0,3,y)-8 H(1,z) H(3,2,0,y)-12 H(3,2,0,y)-24 H(3,2,2,y)+16 H(1,z) H(3,2,3,y)+8 H(1,z) H(3,3,0,y)+16 H(1,z) H(3,3,2,y)+24 H(3,3,2,y)+2 H(0,z) (++++(--2) H(2,y)+6 H(0,2,y)-
[=]{} ++6 H(2,2,y)-6 H(3,2,y)+4 H(0,0,2,y)-4 H(0,3,2,y)-2 H(2,0,2,y)-4 H(3,0,2,y)-4 H(3,2,2,y)+4 H(3,3,2,y))-16 H(1,z) H(3,3,3,y)+8 H(0,0,1,1,z)+8 H(0,0,2,2,y)-8 H(0,0,3,2,y)+8 H(0,1,0,1,z)-4 H(0,1,0,2,y)+8 H(0,1,1,0,y)+8 H(0,1,1,0,z)-4 H(0,1,2,0,y)+8 H(0,2,0,2,y)-4 H(0,2,1,0,y)+8 H(0,2,2,0,y)-4 H(0,2,3,2,y)-8 H(0,3,2,2,y)+8 H(0,3,3,2,y)+4 H(1,0,0,1,z)-8 H(1,0,0,2,y)+4 H(1,0,1,0,y)-8 H(1,0,2,0,y)+4 H(1,0,3,2,y)+8 H(1,1,0,0,y)+8 H(1,1,0,1,z)+8 H(1,1,1,0,y)+8 H(1,1,1,0,z)-8 H(1,2,0,0,y)-4 H(1,2,1,0,y)-4 H(1,2,3,2,y)+8 H(2,0,0,2,y)-4 H(2,0,1,0,y)+8 H(2,0,2,0,y)-8 H(2,1,0,0,y)-4 H(2,1,0,2,y)+8 H(2,1,1,0,y)-4 H(2,1,2,0,y)+8 H(2,2,0,0,y)+16 H(2,2,3,2,y)-8 H(3,0,1,0,y)-8 H(3,0,2,2,y)+8 H(3,0,3,2,y)-8 H(3,2,0,2,y)+8 H(3,2,1,0,y)-8 H(3,2,2,0,y)+16 H(3,2,3,2,y)+8 H(3,3,0,2,y)+8 H(3,3,2,0,y)+16 H(3,3,2,2,y)-16 H(3,3,3,2,y)+6++-+-+ ) }
[=]{} + C\_A n\_f { \_3 ( - ) + \_2 ( - H(0,y)+ H(2,y)- H(0,z)+ H(1,z)+- ) + ( H(0,y) (H(0,z) (+20)+31 H(1,z)-36 H(0,0,z)+24 H(0,1,z)-36 H(1,0,z)+24 H(1,1,z)++86)+++H(0,z) ( H(2,y)-H(0,0,y)- H(0,2,y)-H(2,0,y)+ H(2,2,y)+H(3,2,y)++)+H(2,y) H(1,z)- H(3,y) H(1,z)-H(0,0,y) H(1,z)+ H(0,2,y) H(1,z)+H(0,3,y) H(1,z)+ H(2,0,y) H(1,z)- H(2,3,y) H(1,z)+H(3,0,y) H(1,z)- H(3,2,y) H(1,z)- H(3,3,y) H(1,z)-H(2,y) H(0,0,z)- H(2,y) H(0,1,z)- H(3,y) H(0,1,z)+H(3,y) H(1,0,z)- H(3,y) H(1,1,z)+ H(2,y)- H(0,0,y)+ H(0,2,y)+ H(2,0,y)+H(2,2,y)- H(3,2,y)-H(0,0,2,y)-H(0,2,0,y)+ H(0,2,2,y)+H(0,3,2,y)-H(2,0,0,y)+ H(2,0,2,y)- H(2,1,0,y)+ H(2,2,0,y)- H(2,3,2,y)+H(3,0,2,y)+H(3,2,0,y)- H(3,2,2,y)- H(3,3,2,y)+ H(1,z)- H(0,0,z)- H(0,1,z)+ H(1,0,z)+H(1,1,z)- H(0,0,1,z)- H(0,1,1,z)-H(1,0,0,z)- H(1,0,1,z)+ ) }
[=]{} + C\_F n\_f { \_3 ( - ) + \_2 ( (4 H(1,y)-5 H(2,y)-H(1,z)+1) ) + ( - H(0,y) H(1,z)-108 H(0,y) H(0,1,z)+27 H(0,y) H(1,0,z)-108 H(0,y) H(1,1,z)- H(2,y) H(0,z)-162 H(2,y) H(1,z)+360 H(3,y) H(1,z)+162 H(0,0,y) H(1,z)+162 H(2,y) H(0,0,z)+108 H(2,y) H(0,1,z)+54 H(3,y) H(0,1,z)+54 H(0,2,y) H(0,z)-108 H(0,2,y) H(1,z)-162 H(0,3,y) H(1,z)-27 H(1,0,y) H(0,z)-108 H(2,y) H(1,0,z)-162 H(3,y) H(1,0,z)+216 H(3,y) H(1,1,z)+54 H(2,0,y) H(0,z)-108 H(2,0,y) H(1,z)-108 H(2,2,y) H(0,z)+216 H(2,3,y) H(1,z)-162 H(3,0,y) H(1,z)-162 H(3,2,y) H(0,z)+216 H(3,2,y) H(1,z)+216 H(3,3,y) H(1,z)-27 H(0,y)-102 H(2,y)- H(0,2,y)+180 H(1,0,y)- H(2,0,y)-162 H(2,2,y)+360 H(3,2,y)+162 H(0,0,2,y)-27 H(0,1,0,y)+162 H(0,2,0,y)-108 H(0,2,2,y)-162 H(0,3,2,y)-162 H(1,0,0,y)+108 H(1,1,0,y)+162 H(2,0,0,y)-108 H(2,0,2,y)-108 H(2,2,0,y)+216 H(2,3,2,y)-162 H(3,0,2,y)-162 H(3,2,0,y)+216 H(3,2,2,y)+216 H(3,3,2,y)-27 H(0,z)-102 H(1,z)+ H(0,1,z)+ H(1,0,z)-162 H(1,1,z)+54 H(0,0,1,z)-27 H(0,1,0,z)+108 H(0,1,1,z)+108 H(1,0,1,z)+200 ) / 81 }
[=]{} + n\_f\^2 { ( H(0,y) (3 H(0,z)-20)+15 H(0,0,y)-20 H(0,z)+15 H(0,0,z)+6 \_2 ) }
[B]{}\^[(2)]{}\_[2 ; 2]{} = (121 [C\_A]{}\^2+44 [C\_A]{} (6 [C\_F]{}-[n\_f]{})+4 (27 [C\_F]{}\^2-12 [C\_F]{} [n\_f]{}+[n\_f]{}\^2))
[B]{}\^[(2)]{}\_[2 ; 1]{} = C\_A\^2 { ( 594 H(0,y) H(0,z)+594 H(0,y) H(1,z)+594 H(2,y) H(0,z)-1188 H(3,y) H(1,z)+990 H(0,y)+891 H(2,y)+594 H(0,2,y)+594 H(2,0,y)-1188 H(3,2,y)+990 H(0,z)+891 H(1,z)-594 H(0,1,z)+594 H(1,0,z)+495 \_2-108 \_3 - 1296 ) } + C\_A C\_F { (-1188 H(0,y) H(1,z)+54 H(0,y) (6 H(0,z)+6 H(1,z)+10)-1188 H(2,y) H(0,z)+54 (6 H(2,y)+10) H(0,z)+1728 H(3,y) H(1,z)-1296 H(2,y)-864 H(0,2,y)+1188 H(1,0,y)-864 H(2,0,y)+1728 H(3,2,y)-1296 H(1,z)+864 H(0,1,z)+324 H(1,0,z)+1566 [\_2]{}-2808 [\_3]{}-778 ) } + C\_F\^2 { ( -648 H(0,y) H(1,z)-648 H(2,y) H(0,z)+1296 H(3,y) H(1,z)-972 H(2,y)-648 H(0,2,y)+648 H(1,0,y)-648 H(2,0,y)+1296 H(3,2,y)-972 H(1,z)+648 H(0,1,z)
[=]{} -1296 [\_2]{}+2592 [\_3]{}+972 ) } + C\_A n\_f { ( -2 ( -27 (4 H(3,y)-3) H(1,z)+81 H(2,y)+54 H(0,2,y)+54 H(2,0,y)-108 H(3,2,y)-54 H(0,1,z)+54 H(1,0,z)+45 [\_2]{}-260 )-9 H(0,y) (12 H(0,z)+12 H(1,z)+31)-9 (12 H(2,y)+31) H(0,z) ) } + C\_F n\_f { ( -2 (54 ( 4 H(3,y)-3 ) H(1,z)-162 H(2,y)-108 H(0,2,y)+108 H(1,0,y)-108 H(2,0,y)+216 H(3,2,y)+108 H(0,1,z)+54 [\_2]{}+22 )-9 H(0,y) (6-24 H(1,z))-9 (6-24 H(2,y)) H(0,z) ) } + n\_f\^2 { H(0,y)+ H(0,z)- }
[B]{}\^[(2)]{}\_[2 ; 0]{} = C\_A\^2 { \_4 ( ) + \_3 ( -H(0,y)-6 H(1,y)+11 H(2,y)-H(0,z)+5 H(1,z)- ( 25 s\^4 +2 s\^3 (79 t\^2+266 t u+79 u\^2)+s\^2 (133 t\^3+1097 t\^2 u+1097 t u\^2+133 u\^3)+2 s t u (241 t\^2+806 t u+241 u\^2)+457 t\^2 u\^2 ) / ( 36 \^2 \^2 ) ) + \_2 ( ( H(0,z) ( s\^2 (109 t+49 u)+2 s t (115 t+61 u)+t\^2 (121 t+85 u))) / ( 12 \^2 ) + ( H(0,y) (s\^2 (49 t+109 u)+2 s u (61 t+115 u)+u\^2 (85 t+121 u)) ) / ( 12 \^2 ) +( H(2,y) (-144 s\^5-257 s\^4 -2 s\^3 (35 t\^2+202 t u+35 u\^2)+s\^2 (43 t\^3-25 t\^2 u-25 t u\^2+43 u\^3)+2 s t u (55 t\^2+74 t u+55 u\^2)+79 t\^2 u\^2) ) / ( 12 \^2 \^2 ) +( H(1,z) (-72 s\^5-s\^4 (113 t+41 u)+2 s\^3 (t\^2+14 t u+73 u\^2)+s\^2 (43 t\^3+191 t\^2 u+407 t u\^2+115 u\^3)+2 s t u (55 t\^2+182 t u+127 u\^2)+t\^2 u\^2 (79 t+151 u)) ) / ( 12 \^2 \^2 ) ++3 H(0,y) H(0,z)+2 H(1,y) H(0,z)+3 H(0,y) H(1,z)+2 H(1,y) H(1,z)+H(2,y) H(0,z)-2 H(3,y) H(1,z)+3 H(0,2,y)+2 H(1,2,y)+H(2,0,y)-2 H(3,2,y)+H(0,1,z)+H(1,0,z)+2 H(1,1,z) -( 28 s\^3 +s\^2 (28 t\^2+71 t u+28 u\^2)+34 s t u-3 t\^2 u\^2 ) / ( 9 s ) )
[=]{} + ( - + (-17) H(0,z) H(0,y) -( (179 u\^2+161 s u+107 t u+107 s t) H(1,z) H(0,y) ) / ( 18 ) +7 H(0,0,z) H(0,y) -( ((13 t+25 u) s\^2+2 u (19 t+28 u) s+31 u\^2) H(0,1,z) H(0,y)) / ( 6 \^2 ) +( (17 t+23 u) H(1,0,z) H(0,y) ) / ( 2 ) - H(1,1,z) H(0,y)-2 H(0,0,1,z) H(0,y)-2 H(0,1,1,z) H(0,y)+2 H(1,0,0,z) H(0,y)+2 H(1,1,0,z) H(0,y) - - H(1,z) -( (179 t\^2+161 s t+107 u t+107 s u) H(0,z) H(2,y) ) / ( 18 ) - H(1,z) H(2,y) - H(2,y) +( (188 s\^2+197 s+206 t u) H(1,z) H(3,y) ) / (9 ) + 7 H(0,z) H(0,0,y)+7 H(1,z) H(0,0,y)
[=]{} + H(0,0,y)+7 H(2,y) H(0,0,z)+2 H(0,0,y) H(0,0,z)+ H(0,0,z) +( ((215 t+269 u) s\^2+(215 t\^2+520 u t+287 u\^2) s+t u (233 t+305 u)) H(0,1,z)) / ( 18 ) +( (18 s\^5+59 s\^4+(61 t\^2+152 u t+61 u\^2) s\^3+4 (5 t\^3+31 u t\^2+31 u\^2 t+5 u\^3) s\^2+34 t \^2 u s+11 t\^2 u\^2) H(2,y) H(0,1,z) ) / ( 3 \^2 \^2 ) + H(3,y) H(0,1,z)+2 H(0,0,y) H(0,1,z) -( (179 u\^2+161 s u+107 t u+107 s t) H(0,2,y)) / ( 18 ) + ( ((17 t+29 u) s\^2+2 t (14 t+23 u) s+11 t\^2 ) H(0,z) H(0,2,y) ) / ( 6 \^2 ) - H(1,z) H(0,2,y)+2 H(0,0,z) H(0,2,y)-2 H(0,1,z) H(0,2,y) -( ((5 t+9 u) s\^4+(9 t\^2+28 u t+19 u\^2) s\^3+2 (2 t\^3+14 u t\^2+21 u\^2 t+5 u\^3) s\^2+2 t u (5 t\^2+14 u t+9 u\^2) s+7 t\^2 u\^2) H(1,z) H(0,3,y) ) / ( \^2 \^2 ) -2 H(0,1,z) H(0,3,y)+( u (6 s\^2-2 t s+9 u s+t u) H(1,0,y)) / ( 3 s ) - - ( 3 (s+t+u) H(1,z) H(1,0,y) ) / ( ) -( ((125 t+107 u) s\^2+t (125 t+119 u) s-6 t\^2 u) H(1,0,z) ) / ( 18 s ) +( (-3 s\^3-(t-3 u) s\^2+8 t s+6 t\^2 ) H(2,y) H(1,0,z) ) / ( \^2 ) -7 H(3,y) H(1,0,z)+2 H(0,0,y) H(1,0,z)-6 H(0,3,y) H(1,0,z)
[=]{} + H(3,y) H(1,1,z)+2 H(0,0,y) H(1,1,z)- H(1,1,z)-2 H(0,1,z) H(1,2,y)+2 H(1,0,z) H(1,2,y) -( (179 u\^2+161 s u+107 t u+107 s t) H(2,0,y) ) / ( 18 ) + H(0,z) H(2,0,y)- H(1,z) H(2,0,y)+2 H(0,0,z) H(2,0,y)+2 H(0,1,z) H(2,0,y)+2 H(1,0,z) H(2,0,y)- H(0,z) H(2,2,y)+2 H(0,0,z) H(2,2,y)+4 H(0,1,z) H(2,2,y)+4 H(1,0,z) H(2,2,y)- H(2,2,y) +( (18 s\^5+61 s\^4+5 (13 t\^2+32 u t+13 u\^2) s\^3+2 (11 t\^3+67 u t\^2+67 u\^2 t+11 u\^3) s\^2+38 t \^2 u s+13 t\^2 u\^2) H(1,z) H(2,3,y) ) / ( 3 \^2 \^2 )-6 H(0,1,z) H(2,3,y)+2 H(1,0,z) H(2,3,y)-7 H(1,z) H(3,0,y)+2 H(0,1,z) H(3,0,y)-2 H(1,0,z) H(3,0,y)+ ( (188 s\^2+197 s+206 t u) H(3,2,y) ) / ( 9 ) -7 H(0,z) H(3,2,y)+ H(1,z) H(3,2,y)+ H(1,z) H(3,3,y)-12 H(0,1,z) H(3,3,y)+4 H(1,0,z) H(3,3,y)+ ( ((7 u-5 t) s\^4+(-13 t\^2+4 u t+17 u\^2) s\^3-2 (4 t\^3+8 u t\^2-13 u\^2 t-5 u\^3) s\^2+2 t u (-5 t\^2+2 u t+7 u\^2) s+t\^2 u\^2) H(0,0,1,z) ) / ( 3 \^2 \^2 ) - 2 H(1,y) H(0,0,1,z)-2 H(2,y) H(0,0,1,z)-8 H(3,y) H(0,0,1,z)+2 H(0,z) H(0,0,2,y)+2 H(1,z) H(0,0,2,y)+7 H(0,0,2,y) - ( u (s\^2+(u-2 t) s-3 t u) H(0,1,0,y) ) / ( \^2 ) - ( t (s\^2+(t-2 u) s-3 t u) H(0,1,0,z)) / ( \^2 ) +2 H(1,y) H(0,1,0,z)+6 H(2,y) H(0,1,0,z)+4 H(3,y) H(0,1,0,z)+ H(0,1,1,z)+2 H(0,z) H(0,2,0,y)+2 H(1,z) H(0,2,0,y)+7 H(0,2,0,y)-2 H(0,z) H(0,2,2,y)- H(0,2,2,y)+4 H(1,z) H(0,3,0,y)- ( ((5 t+9 u) s\^4+(9 t\^2+28 u t+19 u\^2) s\^3+2 (2 t\^3+14 u t\^2+21 u\^2 t+5 u\^3) s\^2+2 t u (5 t\^2+14 u t+9 u\^2) s+7 t\^2 u\^2) H(0,3,2,y) ) / ( \^2 \^2 ) - 6 H(0,z) H(0,3,2,y)+4 H(1,z) H(0,3,3,y)+2 H(2,y) H(1,0,0,z)
[=]{} +7 H(1,0,0,z) +( (9 s\^5+32 s\^4+(34 t\^2+80 u t+34 u\^2) s\^3+(11 t\^3+61 u t\^2+61 u\^2 t+11 u\^3) s\^2+t u (16 t\^2+23 u t+16 u\^2) s+2 t\^2 u\^2) H(1,0,1,z) ) / ( 3 \^2 \^2 ) -2 H(1,y) H(1,0,1,z)+6 H(2,y) H(1,0,1,z) - ( 3 (s+t+u) H(1,0,2,y) ) / ( ) -2 H(0,z) H(1,0,2,y)+2 H(1,z) H(1,0,3,y)+ ( 6 H(1,1,0,y) ) / ( ) + ( (2 (t+3 u) s\^2+t (5 t+14 u) s+3 t\^2 (t+3 u)) H(1,1,0,z) ) / ( \^2 ) +2 H(1,y) H(1,1,0,z)+6 H(2,y) H(1,1,0,z)--2 H(1,z) H(1,2,3,y)+2 H(0,z) H(2,0,0,y)+2 H(1,z) H(2,0,0,y)+7 H(2,0,0,y)- H(2,0,2,y)+2 H(1,z) H(2,0,3,y)+ ( (-9 s\^3+(11 t-u) s\^2+28 u s+20 u\^2) H(2,1,0,y) ) / ( 3 \^2 ) +2 H(1,z) H(2,1,0,y)+2 H(0,z) H(2,2,0,y)- H(2,2,0,y)+4 H(1,z) H(2,2,3,y)+2 H(1,z) H(2,3,0,y)+ ( (18 s\^5+61 s\^4+5 (13 t\^2+32 u t+13 u\^2) s\^3+2 (11 t\^3+67 u t\^2+67 u\^2 t+11 u\^3) s\^2+38 t \^2 u s+13 t\^2 u\^2) H(2,3,2,y) ) / ( 3 \^2 \^2 ) +2 H(0,z) H(2,3,2,y)-8 H(1,z) H(2,3,3,y)-2 H(0,z) H(3,0,2,y)-7 H(3,0,2,y)+4 H(1,z) H(3,0,3,y)-2 H(0,z) H(3,2,0,y)-7 H(3,2,0,y)+ H(3,2,2,y)+4 H(1,z) H(3,3,0,y)+4 H(0,z) H(3,3,2,y)+ H(3,3,2,y)-16 H(1,z) H(3,3,3,y)+8 H(0,0,1,0,z)+2 H(0,0,1,1,z)+2 H(0,0,2,2,y)+2 H(0,1,0,1,z)+6 H(0,1,1,0,z)+2 H(0,2,0,2,y)+4 H(0,2,1,0,y)+2 H(0,2,2,0,y)+4 H(0,3,0,2,y)+4 H(0,3,2,0,y)+4 H(0,3,3,2,y)+6 H(1,0,1,0,z)+2 H(1,0,3,2,y)+2 H(1,1,0,0,z)+4 H(1,1,0,1,z)+6 H(1,1,1,0,z)-2 H(1,2,3,2,y)+2 H(2,0,0,2,y)+2 H(2,0,1,0,y)+2 H(2,0,2,0,y)+2 H(2,0,3,2,y)+2 H(2,1,0,2,y)+2 H(2,1,2,0,y)+2 H(2,2,0,0,y)+4 H(2,2,1,0,y)+4 H(2,2,3,2,y)+2 H(2,3,0,2,y)+2 H(2,3,2,0,y)-8 H(2,3,3,2,y)+4 H(3,0,3,2,y)+4 H(3,3,0,2,y)+4 H(3,3,2,0,y)-16 H(3,3,3,2,y)+ ) }
[=]{} + C\_A C\_F { \_4 ( ) + \_3 ( 22 H(1,y)-18 H(2,y)+4 H(1,z)+ ( 803 s\^5 +4 s\^4 (424 t\^2+965 t u+424 u\^2)+s\^3 (821 t\^3+5329 t\^2 u+5329 t u\^2+821 u\^3)+s\^2 (-72 t\^4+1966 t\^3 u+5336 t\^2 u\^2+1966 t u\^3-72 u\^4)+s t u (-144 t\^3+1307 t\^2 u+1307 t u\^2-144 u\^3)-72 t\^2 u\^2 (t\^2+u\^2) ) / ( 18 s \^2 \^2 ) ) + \_2 ( -( 2 H(1,y) (18 s\^2+23 s t+11 s u-6 t\^2) ) / ( 3 s ) +( t H(0,z) (-10 s\^3-3 s\^2 (7 t+2 u)-9 s t +2 t\^3) ) / ( s \^2 ) +( u H(0,y) (-10 s\^3-3 s\^2 (2 t+7 u)-9 s u+2 u\^3) ) / ( s \^2 ) +( H(2,y) (144 s\^5 +s\^4 (289 t\^2+554 t u+289 u\^2)+4 s\^3 (32 t\^3+147 t\^2 u+147 t u\^2+32 u\^3)+s\^2 (-17 t\^4+90 t\^3 u+166 t\^2 u\^2+90 t u\^3-17 u\^4)-2 s t u (35 t\^3+111 t\^2 u+111 t u\^2+35 u\^3)-t\^2 u\^2 (71 t\^2+166 t u+71 u\^2)) ) / ( 6 \^2 \^2 \^2 ) -( H(1,z) (-72 s\^6 \^2-s\^5 (101 t\^3+231 t\^2 u+183 t u\^2+53 u\^3)+4 s\^4 \^2 (8 t\^2+25 t u+26 u\^2)+s\^3 (61 t\^5+427 t\^4 u+1120 t\^3 u\^2+1216 t\^2 u\^3+523 t u\^4+61 u\^5)+2 s\^2 \^2 u (79 t\^3+236 t\^2 u+79 t u\^2-12 u\^3)+s t u\^2 (115 t\^4+369 t\^3 u+321 t\^2 u\^2+19 t u\^3-48 u\^4)-24 t\^2 \^2 u\^4) ) / ( 6 s \^2 \^2 \^3 ) -2 H(1,y) H(0,z)-6 H(0,y) H(1,z)-2 H(1,y) H(1,z)-4 H(2,y) H(0,z)+2 H(0,2,y)+6 H(1,0,y)-2 H(1,2,y)-4 H(2,0,y)-8 H(2,1,y)+8 H(2,2,y)+2 H(0,1,z)+2 H(1,0,z)-2 H(1,1,z) +( 19 s\^3 +s\^2 (19 t\^2+182 t u+19 u\^2)+145 s t u+108 t\^2 u\^2 ) / ( 6 s ) )
[=]{} + ( ( (26 t\^2+26 s t+33 u t+15 s u) H(0,y) ) / ( 3 ) + (( 18 t u ) / ( s ) +3) H(0,z) H(0,y)+ ( (242 u\^2+215 s u+134 t u+134 s t) H(1,z) H(0,y) ) / ( 9 ) + ( ((13 t+31 u) s\^2+u (44 t+71 u) s+40 u\^2) H(0,1,z) H(0,y) ) / ( 3 \^2 ) - ( (19 s t\^2+62 s u t+31 s u\^2-6 u\^2) H(1,0,z) H(0,y) ) / ( 3 s \^2 ) - H(1,1,z) H(0,y)+4 H(0,0,1,z) H(0,y)+2 H(0,1,0,z) H(0,y)+8 H(0,1,1,z) H(0,y)+2 H(1,0,1,z) H(0,y)-4 H(1,1,0,z) H(0,y)+ ( (26 u\^2+26 s u+33 t u+15 s t) H(0,z) ) / ( 3 ) + H(1,z) + ( (242 t\^2+215 s t+134 u t+134 s u) H(0,z) H(2,y) ) / ( 9 ) +2 H(1,z) H(2,y)+ H(2,y) - ( (466 s\^2+493 s+520 t u) H(1,z) H(3,y) ) / ( 9 ) - 14 H(1,z) H(0,0,y)-14 H(2,y) H(0,0,z)- ( ((251 t+332 u) s\^2+(251 t\^2+637 u t+359 u\^2) s+2 t u (139 t+193 u)) H(0,1,z) ) / ( 9 ) - ( (36 s\^5+112 s\^4+(107 t\^2+268 u t+107 u\^2) s\^3+(31 t\^3+191 u t\^2+191 u\^2 t+31 u\^3) s\^2+2 t u (22 t\^2+35 u t+22 u\^2) s+4 t\^2 u\^2) H(2,y) H(0,1,z) ) / ( 3 \^2 \^2 ) - H(3,y) H(0,1,z)-4 H(0,0,y) H(0,1,z) + ( (242 u\^2+215 s u+134 t u+134 s t) H(0,2,y) ) / ( 9 ) - ( ((29 t+47 u) s\^2+t (49 t+76 u) s+20 t\^2 ) H(0,z) H(0,2,y) ) / ( 3 \^2 ) - H(1,z) H(0,2,y)-4 H(0,0,z) H(0,2,y)+6 H(0,1,z) H(0,2,y)+ ( (2 (7 t+13 u) s\^4+5 (5 t\^2+16 u t+11 u\^2) s\^3+(11 t\^3+79 u t\^2+121 u\^2 t+29 u\^3) s\^2+4 t u (7 t\^2+20 u t+13 u\^2) s+20 t\^2 u\^2) H(1,z) H(0,3,y) ) / ( \^2 \^2 )
[=]{} + 4 H(0,1,z) H(0,3,y)+ ( (-2 (76 t+103 u) s\^2+(37 t-233 u) u s+162 t u\^2) H(1,0,y) ) / ( 9 s ) + ( 2 (3 t\^2+8 s t+14 s u) H(0,z) H(1,0,y) ) / ( 3 s ) + ( + 9) H(1,z) H(1,0,y)-2 H(0,1,z) H(1,0,y)+ ( ((t-2 u) s\^2+t (t+19 u) s+18 t\^2 u) H(1,0,z) ) / ( s ) + ( (18 s\^3+(31 t\^2+32 u t+13 u\^2) s\^2-t (t\^2+17 u t-8 u\^2) s-2 t\^2 (7 t\^2+20 u t+7 u\^2)) H(2,y) H(1,0,z) ) / ( 3 \^2 \^2 ) +20 H(3,y) H(1,0,z)-4 H(0,0,y) H(1,0,z)+8 H(0,2,y) H(1,0,z)+16 H(0,3,y) H(1,0,z)+2 H(1,0,y) H(1,0,z)+ H(3,y) H(1,1,z)-8 H(0,0,y) H(1,1,z)+4 H(0,3,y) H(1,1,z)+2 H(1,1,z)+6 H(0,1,z) H(1,2,y)-2 H(1,0,z) H(1,2,y)+ ( (242 u\^2+215 s u+134 t u+134 s t) H(2,0,y) ) / ( 9 ) - ( (47 t\^3+153 u t\^2+153 u\^2 t+47 u\^3) H(0,z) H(2,0,y) ) / ( 3 \^3 ) - H(1,z) H(2,0,y)-4 H(0,0,z) H(2,0,y)-2 H(0,1,z) H(2,0,y)-6 H(1,0,z) H(2,0,y)- H(0,z) H(2,2,y)-8 H(0,0,z) H(2,2,y)-16 H(0,1,z) H(2,2,y)-8 H(1,0,z) H(2,2,y)+2 H(2,2,y)-( (36 s\^5+98 s\^4+(79 t\^2+212 u t+79 u\^2) s\^3+(17 t\^3+121 u t\^2+121 u\^2 t+17 u\^3) s\^2+2 t u (8 t\^2+7 u t+8 u\^2) s-10 t\^2 u\^2) H(1,z) H(2,3,y) ) / ( 3 \^2 \^2 ) +12 H(0,1,z) H(2,3,y)-4 H(1,0,z) H(2,3,y)+20 H(1,z) H(3,0,y)-4 H(0,1,z) H(3,0,y)+4 H(1,1,z) H(3,0,y)- ( (466 s\^2+493 s+520 t u) H(3,2,y) ) / ( 9 ) +20 H(0,z) H(3,2,y)+ H(1,z) H(3,2,y)-4 H(0,1,z) H(3,2,y)-4 H(1,0,z) H(3,2,y)- H(1,z) H(3,3,y)+28 H(0,1,z) H(3,3,y)-12 H(1,0,z) H(3,3,y)-8 H(1,1,z) H(3,3,y)+ ( (4 (4 t-5 u) s\^4+(41 t\^2-8 u t-49 u\^2) s\^3+(25 t\^3+53 u t\^2-73 u\^2 t-29 u\^3) s\^2+8 t u (4 t\^2-u t-5 u\^2) s-2 t\^2 u\^2) H(0,0,1,z) ) / ( 3 \^2 \^2 )
[=]{} +6 H(1,y) H(0,0,1,z)+16 H(3,y) H(0,0,1,z)-8 H(0,z) H(0,0,2,y)-8 H(1,z) H(0,0,2,y)-14 H(0,0,2,y)+4 H(1,z) H(0,0,3,y)+ ( (-6 u\^4-s (17 t\^2+22 u t+17 u\^2) u\^2+s\^2 (2 t\^2+19 u t-7 u\^2) u+2 s\^3 (5 t\^2+13 u t+2 u\^2)) H(0,1,0,y) ) / ( 3 s \^2 \^2 ) +2 H(0,z) H(0,1,0,y)+2 H(1,z) H(0,1,0,y)+ ( (-6 t\^4+s (t\^2+14 u t+u\^2) t\^2+s\^2 (29 t\^2+91 u t+38 u\^2) t+s\^3 (22 t\^2+62 u t+28 u\^2)) H(0,1,0,z) ) / ( 3 s \^2 \^2 ) -2 H(1,y) H(0,1,0,z)-12 H(2,y) H(0,1,0,z)-4 H(3,y) H(0,1,0,z)-4 H(3,y) H(0,1,1,z)+ H(0,1,1,z)-8 H(1,z) H(0,2,0,y)-14 H(0,2,0,y)+4 H(0,z) H(0,2,2,y)- H(0,2,2,y)+2 H(1,z) H(0,2,3,y)-8 H(1,z) H(0,3,0,y)+ ( (2 (7 t+13 u) s\^4+5 (5 t\^2+16 u t+11 u\^2) s\^3+(11 t\^3+79 u t\^2+121 u\^2 t+29 u\^3) s\^2+4 t u (7 t\^2+20 u t+13 u\^2) s+20 t\^2 u\^2) H(0,3,2,y) ) / ( \^2 \^2 ) +16 H(0,z) H(0,3,2,y)+4 H(1,z) H(0,3,2,y)-12 H(1,z) H(0,3,3,y)+4 H(0,z) H(1,0,0,y)+4 H(1,z) H(1,0,0,y)+14 H(1,0,0,y)-4 H(2,y) H(1,0,0,z)- ( (18 s\^5+49 s\^4+(35 t\^2+88 u t+35 u\^2) s\^3+4 (t\^3+5 u t\^2+5 u\^2 t+u\^3) s\^2-2 t u (5 t\^2+28 u t+5 u\^2) s-23 t\^2 u\^2) H(1,0,1,z) ) / ( 3 \^2 \^2 ) +6 H(1,y) H(1,0,1,z)-18 H(2,y) H(1,0,1,z)-4 H(3,y) H(1,0,1,z) +( + 9) H(1,0,2,y)+4 H(0,z) H(1,0,2,y)-6 H(1,z) H(1,0,3,y)- ( 2 (18 s\^2+23 t s+11 u s-6 t\^2) H(1,1,0,y) ) / ( 3 s ) - ( (2 u (3 t+u) s\^3+(3 t\^3+21 u t\^2+6 u\^2 t-4 u\^3) s\^2+t (3 t\^3+18 u t\^2+3 u\^2 t-8 u\^3) s-4 t\^2 u\^2) H(1,1,0,z) ) / ( s \^2 \^2 ) -2 H(1,y) H(1,1,0,z)-18 H(2,y) H(1,1,0,z)-4 H(3,y) H(1,1,0,z) + (+9) H(1,2,0,y)+2 H(0,z) H(1,2,0,y)
[=]{} +6 H(1,z) H(1,2,3,y)-4 H(0,z) H(2,0,0,y)-8 H(1,z) H(2,0,0,y)-14 H(2,0,0,y)+2 H(0,z) H(2,0,2,y)- H(2,0,2,y)-4 H(1,z) H(2,0,3,y)+ ( (6 s\^3+(9 t\^2+20 u t+15 u\^2) s\^2+u (12 t\^2+13 u t+9 u\^2) s-4 t u\^3) H(2,1,0,y) ) / ( \^2 \^2 ) -2 H(0,z) H(2,1,0,y)-2 H(1,z) H(2,1,0,y)-4 H(0,z) H(2,2,0,y)- H(2,2,0,y)-16 H(1,z) H(2,2,3,y)-4 H(1,z) H(2,3,0,y)- ( (36 s\^5+98 s\^4+(79 t\^2+212 u t+79 u\^2) s\^3+(17 t\^3+121 u t\^2+121 u\^2 t+17 u\^3) s\^2+2 t u (8 t\^2+7 u t+8 u\^2) s-10 t\^2 u\^2) H(2,3,2,y) ) / ( 3 \^2 \^2 ) -4 H(0,z) H(2,3,2,y)+16 H(1,z) H(2,3,3,y)+8 H(0,z) H(3,0,2,y)+4 H(1,z) H(3,0,2,y)+20 H(3,0,2,y)-12 H(1,z) H(3,0,3,y)+4 H(1,z) H(3,2,0,y)+20 H(3,2,0,y)+4 H(0,z) H(3,2,2,y)+ H(3,2,2,y)-8 H(1,z) H(3,2,3,y)-12 H(1,z) H(3,3,0,y)-12 H(0,z) H(3,3,2,y)-8 H(1,z) H(3,3,2,y)- H(3,3,2,y)+40 H(1,z) H(3,3,3,y)+4 H(0,0,1,0,y)-8 H(0,0,1,0,z)-8 H(0,0,1,1,z)-8 H(0,0,2,2,y)+4 H(0,0,3,2,y)-8 H(0,1,0,1,z)+2 H(0,1,0,2,y)-12 H(0,1,1,0,z)+2 H(0,1,2,0,y)-8 H(0,2,0,2,y)-2 H(0,2,1,0,y)-8 H(0,2,2,0,y)+2 H(0,2,3,2,y)-8 H(0,3,0,2,y)-8 H(0,3,2,0,y)+4 H(0,3,2,2,y)-12 H(0,3,3,2,y)-2 H(1,0,0,1,z)+4 H(1,0,0,2,y)+4 H(1,0,1,0,y)-4 H(1,0,1,0,z)+4 H(1,0,2,0,y)-6 H(1,0,3,2,y)-12 H(1,1,0,1,z)-14 H(1,1,1,0,z)+4 H(1,2,0,0,y)+4 H(1,2,1,0,y)+6 H(1,2,3,2,y)-8 H(2,0,0,2,y)-8 H(2,0,2,0,y)-4 H(2,0,3,2,y)+4 H(2,1,0,0,y)-2 H(2,1,0,2,y)-8 H(2,1,1,0,y)-2 H(2,1,2,0,y)-8 H(2,2,0,0,y)-8 H(2,2,1,0,y)-16 H(2,2,3,2,y)-4 H(2,3,0,2,y)-4 H(2,3,2,0,y)+16 H(2,3,3,2,y)+8 H(3,0,1,0,y)+4 H(3,0,2,2,y)-12 H(3,0,3,2,y)+4 H(3,2,0,2,y)-8 H(3,2,1,0,y)+4 H(3,2,2,0,y)-8 H(3,2,3,2,y)-12 H(3,3,0,2,y)-12 H(3,3,2,0,y)-8 H(3,3,2,2,y)+40 H(3,3,3,2,y)- ) }
[=]{} + C\_F\^2 { \_4 ( -22 ) + \_3 ( -16 H(1,y)+8 H(2,y)-8 H(1,z)- ( 2 (25 s\^5 +s\^4 (51 t\^2+112 t u+51 u\^2)+4 s\^3 (6 t\^3+37 t\^2 u+37 t u\^2+6 u\^3)-2 s\^2 (t\^4-27 t\^3 u-69 t\^2 u\^2-27 t u\^3+u\^4)+s t u (-4 t\^3+33 t\^2 u+33 t u\^2-4 u\^3)-2 t\^2 u\^2 (t\^2+u\^2)) ) / ( s \^2 \^2 ) ) + \_2 ( ( 2 t H(0,z) (4 s\^3+2 s\^2 (4 t+u)+3 s t -t\^3) ) / ( s \^2 ) + ( 2 u H(0,y) (4 s\^3+2 s\^2 (t+4 u)+3 s u-u\^3) ) / ( s \^2 ) + ( 2 H(2,y) (s\^4 (5 t\^2+12 t u+5 u\^2)+s\^3 (11 t\^3+39 t\^2 u+39 t u\^2+11 u\^3)+s\^2 (6 t\^4+42 t\^3 u+76 t\^2 u\^2+42 t u\^3+6 u\^4)+2 s t u (7 t\^3+26 t\^2 u+26 t u\^2+7 u\^3)+t\^2 u\^2 (9 t\^2+20 t u+9 u\^2)) ) / ( \^2 \^2 \^2 ) + ( 2 H(1,z) (s\^5 (5 t\^2+16 t u+9 u\^2)+s\^4 (11 t\^3+47 t\^2 u+53 t u\^2+17 u\^3)+s\^3 (6 t\^4+46 t\^3 u+92 t\^2 u\^2+54 t u\^3+6 u\^4)+2 s\^2 u (7 t\^4+29 t\^3 u+29 t\^2 u\^2+6 t u\^3-u\^4)+s t u\^2 (9 t\^3+20 t\^2 u+5 t u\^2-4 u\^3)-2 t\^2 u\^4) ) / ( s \^2 \^2 \^2 ) + ( 4 t (2 s-t) H(1,y) ) / ( s ) -8 H(2,y) H(1,z)+8 H(0,1,y)-8 H(0,2,y)+8 H(1,1,y)+8 H(2,1,y)-16 H(2,2,y)+ ( 2 (6 s\^3 +s\^2 (6 t\^2+4 t u+6 u\^2)-s t u-6 t\^2 u\^2) ) / ( s ) )
[=]{} + ( - ( 12 t u H(0,y) H(0,z) ) / ( s ) - ( 2 (3 s (2 t+u)+t (7 t+3 u)) H(2,y) H(0,z) ) / ( ) + ( 2 ((4 t+6 u) s\^2+t (7 t+10 u) s+3 t\^2 ) H(0,2,y) H(0,z) ) / ( \^2 ) - ( 2 (t\^2+s (t+3 u)) H(1,0,y) H(0,z) ) / ( s ) + ( (6 t\^3+22 u t\^2+22 u\^2 t+6 u\^3) H(2,0,y) H(0,z) ) / ( \^3 ) +12 H(2,2,y) H(0,z)-12 H(3,2,y) H(0,z)+8 H(0,0,2,y) H(0,z)-8 H(0,3,2,y) H(0,z)-4 H(2,0,2,y) H(0,z)-8 H(3,0,2,y) H(0,z)-8 H(3,2,2,y) H(0,z)+8 H(3,3,2,y) H(0,z)- ( 2 (3 s (t+2 u)+u (3 t+7 u)) H(0,y) H(1,z) ) / ( ) -9 H(1,z)+9 H(1,z) H(2,y)-9 H(2,y)+ ( (20 s\^2+22 s+24 t u) H(1,z) H(3,y) ) / ( ) + ( 2 ((4 t+7 u) s\^2+(4 t\^2+13 u t+8 u\^2) s+t u (5 t+9 u)) H(0,1,z) ) / ( ) - ( 2 u (+u) (2 s+3 u) H(0,y) H(0,1,z) ) / ( \^2 ) - ( 2 (2 s\^4+(5 t\^2+12 u t+5 u\^2) s\^3+(3 t\^3+19 u t\^2+19 u\^2 t+3 u\^3) s\^2+2 t u (4 t\^2+11 u t+4 u\^2) s+6 t\^2 u\^2) H(2,y) H(0,1,z) ) / ( \^2 \^2 ) +12 H(3,y) H(0,1,z) - ( 2 (3 s (t+2 u)+u (3 t+7 u)) H(0,2,y) ) / ( ) + 12 H(1,z) H(0,2,y)-4 H(0,1,z) H(0,2,y)- ( 2 (4 (t+2 u) s\^4+(7 t\^2+24 u t+17 u\^2) s\^3+(3 t\^3+23 u t\^2+37 u\^2 t+9 u\^3) s\^2+8 t u (t\^2+3 u t+2 u\^2) s+6 t\^2 u\^2) H(1,z) H(0,3,y) ) / ( \^2 \^2 ) + ( 2 ((3 t+5 u) s\^2+(6 u\^2-4 t u) s-6 t u\^2) H(1,0,y) ) / ( s ) -6 H(1,z) H(1,0,y)+4 H(0,1,z) H(1,0,y)
[=]{} - ( 2 t (s\^2+(t+7 u) s+6 t u) H(1,0,z) ) / ( s ) + ( 2 u (2 s (2 t+u)- u) H(0,y) H(1,0,z) ) / ( s \^2 ) + ( 2 t (2 (t+2 u) s\^2+(5 t\^2+11 u t+2 u\^2) s+t (3 t\^2+8 u t+3 u\^2)) H(2,y) H(1,0,z) ) / ( \^2 \^2 ) -12 H(3,y) H(1,0,z)-8 H(0,2,y) H(1,0,z)-8 H(0,3,y) H(1,0,z)+12 H(0,y) H(1,1,z)-24 H(3,y) H(1,1,z)+8 H(0,0,y) H(1,1,z)-8 H(0,3,y) H(1,1,z)+9 H(1,1,z)-4 H(0,1,z) H(1,2,y)- ( 2 (3 s (t+2 u)+u (3 t+7 u)) H(2,0,y) ) / ( ) +12 H(1,z) H(2,0,y)-4 H(0,1,z) H(2,0,y)+8 H(0,0,z) H(2,2,y)+16 H(0,1,z) H(2,2,y)+9 H(2,2,y)
[=]{} - ( 2 (8 s\^4+(17 t\^2+36 u t+17 u\^2) s\^3+(9 t\^3+49 u t\^2+49 u\^2 t+9 u\^3) s\^2+2 t u (10 t\^2+23 u t+10 u\^2) s+12 t\^2 u\^2) H(1,z) H(2,3,y) ) / ( \^2 \^2 ) -12 H(1,z) H(3,0,y)-8 H(1,1,z) H(3,0,y)+ ( (20 s\^2+22 s+24 t u) H(3,2,y) ) / ( ) -24 H(1,z) H(3,2,y)+8 H(0,1,z) H(3,2,y)+24 H(1,z) H(3,3,y)-8 H(0,1,z) H(3,3,y)+8 H(1,0,z) H(3,3,y)+16 H(1,1,z) H(3,3,y)- ( 2 s (t-u) (2 s\^3+5 s\^2+(3 t\^2+10 u t+3 u\^2) s+4 t u) H(0,0,1,z) ) / ( \^2 \^2 ) - 4 H(1,y) H(0,0,1,z)+8 H(2,y) H(0,0,1,z)+8 H(1,z) H(0,0,2,y)-8 H(1,z) H(0,0,3,y)+ ( 2 ( u\^4+s (9 t\^2+16 u t+9 u\^2) u\^2+2 s\^2 (7 t\^2+12 u t+7 u\^2) u+2 s\^3 (3 t\^2+5 u t+3 u\^2)) H(0,1,0,y) ) / ( s \^2 \^2 ) -4 H(1,z) H(0,1,0,y)+ ( 2 t (-2 u s\^3+2 (t\^2+u\^2) s\^2+t (3 t\^2+4 u t+3 u\^2) s+t\^3 ) H(0,1,0,z) ) / ( s \^2 \^2 ) +4 H(2,y) H(0,1,0,z)-8 H(0,y) H(0,1,1,z)+8 H(3,y) H(0,1,1,z)-12 H(0,1,1,z)+8 H(1,z) H(0,2,0,y)+12 H(0,2,2,y)-4 H(1,z) H(0,2,3,y)- ( 2 (4 (t+2 u) s\^4+(7 t\^2+24 u t+17 u\^2) s\^3+(3 t\^3+23 u t\^2+37 u\^2 t+9 u\^3) s\^2+8 t u (t\^2+3 u t+2 u\^2) s+6 t\^2 u\^2) H(0,3,2,y) ) / ( \^2 \^2 ) -8 H(1,z) H(0,3,2,y)+8 H(1,z) H(0,3,3,y)-8 H(1,z) H(1,0,0,y)- ( 2 (5 s\^4+(11 t\^2+24 u t+11 u\^2) s\^3+(6 t\^3+34 u t\^2+34 u\^2 t+6 u\^3) s\^2+2 t u (7 t\^2+17 u t+7 u\^2) s+9 t\^2 u\^2) H(1,0,1,z) ) / ( \^2 \^2 ) -4 H(0,y) H(1,0,1,z)-4 H(1,y) H(1,0,1,z)
[=]{} +12 H(2,y) H(1,0,1,z)+8 H(3,y) H(1,0,1,z)-6 H(1,0,2,y)+4 H(1,z) H(1,0,3,y)+ ( 4 (2 s-t) t H(1,1,0,y) ) / ( s ) - ( 2 ((t\^2-2 u t-u\^2) s\^3+(t\^3-7 u t\^2-2 u\^2 t+2 u\^3) s\^2+(4 t u\^3-6 t\^3 u) s+2 t\^2 u\^2) H(1,1,0,z) ) / ( s \^2 \^2 ) +4 H(2,y) H(1,1,0,z)-6 H(1,2,0,y)-4 H(1,z) H(1,2,3,y)+8 H(1,z) H(2,0,0,y)+12 H(2,0,2,y)- ( 2 ((6 t\^2+8 u t+4 u\^2) s\^2+u (10 t\^2+13 u t+7 u\^2) s+u\^2 (3 t\^2+4 u t+3 u\^2)) H(2,1,0,y) ) / ( \^2 \^2 ) -4 H(1,z) H(2,1,0,y)+12 H(2,2,0,y)+16 H(1,z) H(2,2,3,y)- ( 2 (8 s\^4+(17 t\^2+36 u t+17 u\^2) s\^3+(9 t\^3+49 u t\^2+49 u\^2 t+9 u\^3) s\^2+2 t u (10 t\^2+23 u t+10 u\^2) s+12 t\^2 u\^2) H(2,3,2,y) ) / ( \^2 \^2 ) -8 H(1,z) H(3,0,2,y)-12 H(3,0,2,y)+8 H(1,z) H(3,0,3,y)-8 H(1,z) H(3,2,0,y)-12 H(3,2,0,y)-24 H(3,2,2,y)+16 H(1,z) H(3,2,3,y)+8 H(1,z) H(3,3,0,y)+16 H(1,z) H(3,3,2,y)+24 H(3,3,2,y)-16 H(1,z) H(3,3,3,y)+8 H(0,0,1,1,z)+8 H(0,0,2,2,y)-8 H(0,0,3,2,y)+8 H(0,1,0,1,z)-4 H(0,1,0,2,y)+8 H(0,1,1,0,y)+8 H(0,1,1,0,z)-4 H(0,1,2,0,y)+8 H(0,2,0,2,y)-4 H(0,2,1,0,y)+8 H(0,2,2,0,y)-4 H(0,2,3,2,y)-8 H(0,3,2,2,y)+8 H(0,3,3,2,y)+4 H(1,0,0,1,z)-8 H(1,0,0,2,y)+4 H(1,0,1,0,y)-8 H(1,0,2,0,y)+4 H(1,0,3,2,y)+8 H(1,1,0,0,y)+8 H(1,1,0,1,z)+8 H(1,1,1,0,y)+8 H(1,1,1,0,z)-8 H(1,2,0,0,y)-4 H(1,2,1,0,y)-4 H(1,2,3,2,y)+8 H(2,0,0,2,y)-4 H(2,0,1,0,y)+8 H(2,0,2,0,y)-8 H(2,1,0,0,y)-4 H(2,1,0,2,y)+8 H(2,1,1,0,y)-4 H(2,1,2,0,y)+8 H(2,2,0,0,y)+16 H(2,2,3,2,y)-8 H(3,0,1,0,y)-8 H(3,0,2,2,y)+8 H(3,0,3,2,y)-8 H(3,2,0,2,y)+8 H(3,2,1,0,y)-8 H(3,2,2,0,y)+16 H(3,2,3,2,y)+8 H(3,3,0,2,y)+8 H(3,3,2,0,y)+16 H(3,3,2,2,y)-16 H(3,3,3,2,y)+ ) }
[=]{} + C\_A n\_f { \_3 ( - ) + \_2 ( - H(0,y)+ H(2,y)- H(0,z)+ H(1,z)-- ) + ( H(0,y) H(0,z) (-)-+H(1,0,z) (-)+++ H(0,y) H(1,z)-H(0,y) H(0,0,z)+ H(0,y) H(0,1,z)-H(0,y) H(1,0,z)+ H(0,y) H(1,1,z)+ H(2,y) H(0,z)+H(2,y) H(1,z)- H(3,y) H(1,z)-H(0,0,y) H(0,z)-H(0,0,y) H(1,z)-H(2,y) H(0,0,z)- H(2,y) H(0,1,z)- H(3,y) H(0,1,z)- H(0,2,y) H(0,z)+ H(0,2,y) H(1,z)+H(0,3,y) H(1,z)+H(3,y) H(1,0,z)- H(3,y) H(1,1,z)-H(2,0,y) H(0,z)+ H(2,0,y) H(1,z)+ H(2,2,y) H(0,z)- H(2,3,y) H(1,z)+H(3,0,y) H(1,z)+H(3,2,y) H(0,z)- H(3,2,y) H(1,z)- H(3,3,y) H(1,z)+ H(2,y)- H(0,0,y)+ H(0,2,y)+ H(2,0,y)+H(2,2,y)- H(3,2,y)-H(0,0,2,y)-H(0,2,0,y)+ H(0,2,2,y)+H(0,3,2,y)-H(2,0,0,y)+ H(2,0,2,y)- H(2,1,0,y)+ H(2,2,0,y)- H(2,3,2,y)+H(3,0,2,y)+H(3,2,0,y)- H(3,2,2,y)- H(3,3,2,y)+ H(1,z)- H(0,0,z)- H(0,1,z)+H(1,1,z)- H(0,0,1,z)- H(0,1,1,z)-H(1,0,0,z)- H(1,0,1,z)- ) }
[=]{} + C\_F n\_f { \_3 ( - ) + \_2 ( H(1,y)- H(2,y)- H(1,z)+ ) + ( --- H(0,y) H(1,z)- H(0,y) H(0,1,z)+ H(0,y) H(1,0,z)- H(0,y) H(1,1,z)- H(2,y) H(0,z)-2 H(2,y) H(1,z)+ H(3,y) H(1,z)+2 H(0,0,y) H(1,z)+2 H(2,y) H(0,0,z)+ H(2,y) H(0,1,z)+ H(3,y) H(0,1,z)+ H(0,2,y) H(0,z)- H(0,2,y) H(1,z)-2 H(0,3,y) H(1,z)- H(1,0,y) H(0,z)- H(2,y) H(1,0,z)-2 H(3,y) H(1,0,z)+ H(3,y) H(1,1,z)+ H(2,0,y) H(0,z)- H(2,0,y) H(1,z)- H(2,2,y) H(0,z)+ H(2,3,y) H(1,z)-2 H(3,0,y) H(1,z)-2 H(3,2,y) H(0,z)+ H(3,2,y) H(1,z)+ H(3,3,y) H(1,z)- H(2,y)- H(0,2,y)+ H(1,0,y)- H(2,0,y)-2 H(2,2,y)+ H(3,2,y)+2 H(0,0,2,y)- H(0,1,0,y)+2 H(0,2,0,y)- H(0,2,2,y)-2 H(0,3,2,y)-2 H(1,0,0,y)+ H(1,1,0,y)+2 H(2,0,0,y)- H(2,0,2,y)- H(2,2,0,y)+ H(2,3,2,y)-2 H(3,0,2,y)-2 H(3,2,0,y)+ H(3,2,2,y)+ H(3,3,2,y)- H(1,z)+ H(0,1,z)+ H(1,0,z)-2 H(1,1,z)+ H(0,0,1,z)- H(0,1,0,z)+ H(0,1,1,z)+ H(1,0,1,z)+ ) }
[=]{} + n\_f\^2 { H(0,y) H(0,z)- H(0,y)+ H(0,0,y)- H(0,z)+ H(0,0,z)+ }
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[^1]:
[^2]: We thank T. Gehrmann for providing relevant analytically continued HPLs and 2d HPLs.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We compute the class of a divisor on ${\overline{\mathcal{M}}_{g,n}}$ given as the closure of the locus of smooth pointed curves $\left[ C;\, x_1,\, \dots,\, x_n \right]$ for which $\sum d_j x_j$ has an effective representative, where $d_j$ are integers summing up to $g-1$, not all positive. The techniques used are a vector bundle computation, a pushdown argument reducing the number of marked points, and the method of test curves.'
address: 'Humboldt-Universität zu Berlin, Institut für Mathematik, 10099 Berlin'
author:
- Fabian Müller
bibliography:
- 'paper.bib'
title: 'The pullback of a Theta divisor to ${\overline{\mathcal{M}}_{g,n}}$'
---
Introduction {#sec:introduction}
============
It has long been known classically that if $C$ is a smooth curve of genus $g \geq 2$ and $C_{g-1}$ denotes its $(g-1)$-fold symmetric product, the Abelian sum map $C_{g-1} \to \operatorname{Pic}^{g-1}(C)$, which to $g-1$ unordered points $x_1,\, \dots,\, x_{g-1}$ associates the line bundle ${\mathcal{O}}_C(x_1 + \dots +
x_{g-1})$, has as image a divisor, which becomes a theta divisor under an identification of $\operatorname{Pic}^{g-1}(C)$ with the Jacobian of $C$. This result can be globalized to a map ${\mathcal{C}}_{g,g-1} \to \operatorname{Pic}_g^{g-1}$, where $${\mathcal{C}}_{g,g-1} = \left( {\mathcal{M}_{g,1}} \times_{\mathcal{M}_g} \dots \times_{\mathcal{M}_g}
{\mathcal{M}_{g,1}} \right) \big/ S_{g-1}$$ is the $(g-1)$-fold symmetric product of the universal curve, and $\operatorname{Pic}_g^{g-1}$ is the universal Picard variety of degree $g-1$. The image of this map is again a divisor, which we denote by $\Theta_g$. Given an integer vector ${\underline{d}}= (d_1,\, \dots,\, d_n) \in \mathbb{Z}^n$ satisfying $\sum_{j=1}^n
d_j = g - 1$, we can define a map ${\varphi}_{{\underline{d}}} \negthinspace:\, {\mathcal{M}_{g,n}}\to
\operatorname{Pic}_g^{g-1}$ by associating to a pointed curve $\left[ C;\, x_1,\, \dots,\,
x_n \right]$ the line bundle ${\mathcal{O}}_C(d_1 x_1 + \dots + d_n x_n)$ on $C$. If at least one of the $d_j$ is negative the image of ${\varphi}_{{\underline{d}}}$ is not contained in $\Theta_g$, and we can ask what is the class of the pullback $D_{{\underline{d}}} :=
{\varphi}_{{\underline{d}}}^* \Theta_g$ and its closure on ${\overline{\mathcal{M}}_{g,n}}$. Unraveling the concepts involved, we arrive at the following equivalent definition:
Let ${\underline{d}}= (d_1,\, \dots,\, d_n)$ be an $n$-tuple of integers satisfying $\sum_{j=1}^n d_j = g - 1$, with at least one $d_j$ negative. Denote by $$D_{{\underline{d}}} := \left\{ \left[ C;\, x_1,\, \dots,\, x_n \right] \in {\mathcal{M}_{g,n}}\,\Big|\,
h^0\left(C,\, d_1 x_1 + \dots + d_n x_n \right) \geq 1 \right\},$$ which is a divisor on ${\mathcal{M}_{g,n}}$, and let ${{\overline{D}}_{{\underline{d}}}}$ be its closure in ${\overline{\mathcal{M}}_{g,n}}$.
Note that since the $x_j$ are distinct, the condition $h^0\left(C,\, d_1 x_1 +
\dots + d_n x_n \right) \geq 1$ is equivalent to postulating that there is a pencil of degree $d_{S_+} := \sum_{j:d_j>0} d_j$ on $C$ that contains the divisor $\sum_{j: d_j > 0} d_j x_j$ and has a section that vanishes to order $-d_j$ at $x_j$ for all $j \in S_- := \big\{ j \,\big|\, d_j < 0 \big\}$. As it ties in nicely with the limit linear series characterization on reducible curves, we will always use this reformulation from now on.
The main result of this paper, which is proven in Theorem \[thm:class\_Dd\], is the computation of the class of this divisor in $\operatorname{Pic}({\overline{\mathcal{M}}_{g,n}})$. It is given by $$\label{eq:class_Dd}
\begin{split}
\left[ {{\overline{D}}_{{\underline{d}}}}\right] =& -\lambda + \sum_{j=1}^n \binom{d_j + 1}{2} \psi_j -
0 \cdot \delta_0 \\
& - \sum_{\substack{i,\, S\\ S \subseteq S_+}}
\binom{{\left| d_S - i \right|} + 1}{2} \delta_{i:S} - \sum_{\substack{i,\, S\\ S
\not\subseteq S_+}} \binom{d_S - i + 1}{2} \delta_{i:S},
\end{split}$$ where $S_+ := \big\{ j \,\big|\, d_j > 0 \big\}$ and $d_S := \sum_{j \in S}
d_j$. Thus the next to last summand corresponds to boundary classes that parameterize reducible curves where the points indexed by $S_-$ lie on a single component, while the last one corresponds to classes parameterizing curves which have points from $S_-$ on both components.
In the special case ${\underline{d}}= (d_1,\, \dots,\, d_{n-1};\, -1)$ with $d_1,\,
\dots,\, d_{n-1} > 0$, the divisor ${{\overline{D}}_{{\underline{d}}}}$ is just the pullback to ${\overline{\mathcal{M}}_{g,n}}$ of the divisor of pointed curves $\left[ C;\, x_1,\, \dots,\, x_{n-1} \right] \in
{\overline{\mathcal{M}}_{g,n-1}}$ having a $g^1_g$ containing $d_1 x_1 + \dots + d_{n-1} x_{n-1}$, which was considered by A. Logan [@bib:logan]. For $n = 2$, it is the pullback of the Weierstraß divisor on ${\overline{\mathcal{M}}_{g,1}}$, whose class has been computed by F. Cukierman [@bib:cukierman] to be $$\label{eq:weierstrass_divisor_class}
\left[ {\overline{\mathcal{W}}_g}\right] = -\lambda + \binom{g + 1}{2} \psi_1 - \sum_{i=1}^{g-1}
\binom{g - i + 1}{2} \delta_{i:1}.$$ For more details on this, see Remarks \[rmk:weierstrass\_divisor\] and \[rmk:logan\_divisor\].
A divisor similar to ${{\overline{D}}_{{\underline{d}}}}$ was studied by R. Hain [@bib:hain]: On an open subset $U$ of ${\overline{\mathcal{M}}_{g,n}}$ (or a covering of such) where there is a globally defined theta characteristic $\alpha$, one can define a morphism ${\varphi}_{{\underline{d}}}'
\negthinspace:\, U \to \operatorname{Pic}_g^0$ mapping a pointed curve $\left[ C;\, x_1,\,
\dots,\, x_n \right]$ to the line bundle ${\mathcal{O}}_C(d_1 x_1 + \dots + d_n x_n -
\alpha) \in \operatorname{Pic}^0(C)$. The class of the closure in ${\overline{\mathcal{M}}_{g,n}}$ of the pullback $D_{{\underline{d}}}' := \big({\varphi}_{{\underline{d}}}' \big)^* \Theta_\alpha$ is computed in [@bib:hain Theorem 11.7]; expressed in our notation it is $$\left[ {{\overline{D}}_{{\underline{d}}}}' \right] = -\lambda + \sum_{j=1}^n \binom{d_j + 1}{2} \psi_j +
\delta_0/8 - \sum_{i,S} \binom{d_S - i + 1}{2} \delta_{i:S} \in \operatorname{Pic}({\overline{\mathcal{M}}_{g,n}})
\otimes {\mathbb{Q}}.$$ Both this result and our Theorem \[thm:class\_Dd\] are reproven in a recent preprint by S. Grushevsky and D. Zakharov [@bib:grushevsky-zakharov Theorem 6], where it is also shown that the divisor considered by Hain is reducible and decomposes as ${{\overline{D}}_{{\underline{d}}}}$ together with some boundary components, with multiplicities according to the generic vanishing order of the theta function.
This paper is organized as follows: In Section \[sec:preliminaries\] we will collect some results on pullbacks and pushforwards of divisors on ${\overline{\mathcal{M}}_{g,n}}$ that we will need during the course of the paper. In Section \[sec:main\_coefficients\] the coefficients of the $\lambda$ and $\psi_j$ classes in the expression for $\left[ {{\overline{D}}_{{\underline{d}}}}\right]$ are computed by a vector bundle technique. The rest of the coefficients are computed via test curves. The actual test curve computations are done in Section \[sec:test\_curves\], and the results are applied in Section \[sec:boundary\] together with a pushdown technique to finish the proof of the main result.
Notation {#notation .unnumbered}
--------
By a *nodal curve*, we shall mean a reduced connected 1-dimensional scheme of finite type over a field $k$ whose only singularities are ordinary nodes. A nodal curve is said to be of *compact type* if its dual graph is a tree, or equivalently if its Jacobian is compact.
We use the shorthand $[n] := \{ 1,\, \dots,\, n \}$. If $a$ is any expression, we write $(a)_+ := \max(a,\, 0)$. Occasionally we will write down a binomial coefficient $\binom{a}{2}$ with $a < 0$, by which we just mean $a
(a - 1) / 2$.
If ${\underline{d}}= (d_1,\, \dots,\, d_n)$ is an $n$-tuple of integers, we write $S_+$ (resp. $S_-$) for the set of indices $j \in [n]$ with $d_j > 0$ (resp. $d_j <
0$). Moreover, if $S \subseteq [n]$ is an arbitrary set of indices, we write $d_S := \sum_{j \in S} d_j$. When convenient, we will assume that the positive $d_j$ come first and in the notation ${{\overline{D}}_{{\underline{d}}}}$ separate them with a semicolon from the negative ones.
When summing over boundary classes $\delta_{i:S}$ in $\operatorname{Pic}({\overline{\mathcal{M}}_{g,n}})$, the summation range $\sum_{i,S}$ (and obvious analogues) will be implicitly taken to involve only admissible combinations (e. g. ${\left| S \right|} \geq 2$ for $i = 0$) and to contain every divisor only once (e. g. by postulating $1 \in S$ or $i
\leq g/2$). By $\pi_n\negthickspace: {\overline{\mathcal{M}}_{g,n}}\to {\overline{\mathcal{M}}_{g,n-1}}$ we will denote the forgetful map which forgets the $n$-th point, while by $\pi_{(jk \mapsto
\bullet)}$ we mean the map which identifies the divisor $\Delta_{0:jk}
\subseteq {\overline{\mathcal{M}}_{g,n}}$ with ${\overline{\mathcal{M}}_{g,n-1}}$ by removing the rational component and introducing the new marking $\bullet$ for the former point of attachment. By $\pi\negthickspace: \mathcal{M}_{g,1} \times_{\mathcal{M}_g} {\mathcal{M}_{g,n}}=: {\mathcal{U}}\to {\mathcal{M}_{g,n}}$ we denote the universal family over ${\mathcal{M}_{g,n}}$ with sections $\sigma_1,\, \dots,\, \sigma_n \negthinspace: {\mathcal{M}_{g,n}}\to {\mathcal{U}}$, and by $\omega_\pi
\in \operatorname{Pic}({\mathcal{U}}/{\mathcal{M}_{g,n}})$ the relative dualizing sheaf of the map $\pi$. Picard groups are always understood in the functorial sense, i. e. as groups of divisor classes on the moduli stack.
Acknowledgements {#acknowledgements .unnumbered}
----------------
This work is part of my PhD thesis. I am very grateful to my advisor Gavril Farkas for many helpful discussions and comments. My thanks also go to the referee for a detailed reading and numerous suggestions for improvement. I am supported by the DFG Priority Project SPP 1489.
Preliminaries {#sec:preliminaries}
=============
The Picard group of ${\overline{\mathcal{M}}_{g,n}}$ {#ssec:picard_group}
----------------------------------------------------
We quickly recall the well-known description of $\operatorname{Pic}({\overline{\mathcal{M}}_{g,n}})$. The pushforward $\mathbb{E} := \pi_* \omega_\pi$ of $\omega_\pi$ to ${\mathcal{M}_{g,n}}$ is called the *Hodge bundle*. It is a vector bundle of rank $g$ whose determinant line bundle is denoted by $\lambda := \bigwedge^g {\mathbb{E}}$. For $j =
1,\, \dots,\, n$, the pullback of $\omega_\pi$ via the section $\sigma_j$ is denoted by $\psi_j := \sigma_j^* \omega_\pi$. Moreover, we denote by $\delta_0$ the line bundle corresponding to irreducible nodal pointed stable curves, and by $\delta_{i:S}$ the one corresponding to pointed stable curves consisting of two components of genera $i$ and $g-i$ that meet at a node, with the marked points indexed by $S$ lying on the former. In [@bib:arbarello-cornalba-picard-groups] it is proven that $\operatorname{Pic}({\overline{\mathcal{M}}_{g,n}})$ is freely generated by $\lambda$, the $\psi_j$, $\delta_0$ and the $\delta_{i:S}$.
Limit linear series
-------------------
Throughout this paper, we will make extensive use of the theory of limit linear series, as first developed by Eisenbud and Harris [@bib:eisenbud-harris-lls]. Here we briefly recall the most important concepts and results. Recall that a *linear series* of degree $d$ and dimension $r$ on a smooth curve $C$ (in short, a $g^r_d$) is given by a pair $\ell = ({\mathscr{L}},\, V)$, where ${\mathscr{L}}$ is a line bundle of degree $d$ on $C$ and $V
\subseteq H^0(C,\, {\mathscr{L}})$ is a subspace of projective dimension $r$. The *vanishing sequence* $a^\ell(p) = (0 \leq a^\ell_0(p) < \dots <
a^\ell_r(p) \leq d)$ of $\ell$ at a point $p \in C$ is the set $\left\{ \operatorname{ord}_p(\sigma) \,\big|\, \sigma \in V \right\}$ of vanishing orders of sections of $\ell$, ordered ascendingly.
Let $C$ be a nodal curve of compact type with irreducible components $C_1,\,
\dots,\, C_s$ and $r$, $d$ natural numbers. A *limit $g^r_d$* on $C$ is a collection $\ell$ of linear series $\ell_i = ({\mathscr{L}}_i,\, V_i)$ of degree $d$ and dimension $r$ on each component $C_i$, satisfying the compatibility conditions $$a^{\ell_i}_m(\nu) + a^{\ell_j}_{r-m}(\nu) \geq d,\qquad m = 0,\,
\dots,\, r$$ for each node $\nu$ at which the components $C_i$ and $C_j$ meet. The $\ell_i$ are called the *aspects* of $\ell$. A *section* of $\ell$ is a collection $\sigma = (\sigma_1,\, \dots,\, \sigma_s)$ of sections $\sigma_i \in V_i$ satisfying the compatibility conditions $$\operatorname{ord}_\nu(\sigma_i) + \operatorname{ord}_\nu(\sigma_j) \geq d,\qquad m = 0,\, \dots,\, r$$ for each node $\nu$ at which $C_i$ and $C_j$ meet. If $p \in C$ is a smooth point, the *vanishing sequence* of $\ell$ at $p$ and the *vanishing order* of a section $\sigma$ of $\ell$ at $p$ are respectively defined to be $a^\ell(p) := a^{\ell_i}(p)$ and $\operatorname{ord}_p(\sigma) :=
\operatorname{ord}_p(\sigma_i)$, where $C_i$ is the component of $C$ on which $p$ lies.
The usefulness of the concept of limit linear series lies in the fact that they are indeed limits of linear series: By [@bib:eisenbud-harris-lls Section 2], if a nodal curve of compact type lies in the closure of the locus of curves admitting a $g^r_d$, then it admits a limit $g^r_d$, and for $r = 1$ the converse is also true (see [@bib:eisenbud-harris-lls Proposition 3.1]). This result remains true even if one prescribes fixed vanishing sequences at points specializing to smooth points on the nodal curve.
We finally recall two well-known facts about linear series on curves: a generic curve $C$ of genus $g$ has a $g^r_d$ if and only if the *Brill-Noether-number* $$\rho(g,\, r,\, d) = g - (r + 1) (g - d + r)$$ is non-negative, and postulating a vanishing sequence $a = (a_0,\, \dots,\,
a_r)$ at a generic point of $C$ imposes $\sum_{i=0}^r (a_i - i)$ conditions on the space of $g^r_d$’s on $C$.
Pushforward and pullback formulas
---------------------------------
For computing pullbacks of divisor classes, we need the following formulas, which can be found in [@bib:arbarello-cornalba-picard-groups p. 161]:
\[lem:pullback\_forgetful\] If $\pi_n: {\overline{\mathcal{M}}_{g,n}}\to {\overline{\mathcal{M}}_{g,n-1}}$ is the forgetful map forgetting the last point, then we have the following formulas for pullbacks of divisor classes:
1. $\pi_n^* \lambda = \lambda,$
2. $\pi_n^* \psi_j = \psi_j - \delta_{0:jn},$
3. $\pi_n^* \delta_0 = \delta_0,$
4. $\pi_n^* \delta_{i:S} = \delta_{i:S} + \delta_{i:S \cup \{ n \}},$ except that $\pi_1^* \delta_{g/2:\emptyset} = \delta_{g/2:\emptyset}$.
To apply the Grothendieck-Riemann-Roch formula in Section \[sec:main\_coefficients\], we need certain formulas for pushforwards of intersections of cycles on the universal family, which can be found for example in [@bib:farkas-mustata-popa Lemma 3.13]. We reproduce the ones that concern us here:
\[lem:pushforward\_universal\] With notation as given in Section \[sec:introduction\],
1. $\pi_* \big( c_1(\omega_\pi)^2 \big) = 12 \lambda$,
2. $\pi_* \big( c_1(\omega_\pi) c_1(\sigma_j) \big) = \psi_j$, and
3. $\pi_* \big( c_1(\sigma_j)^2 \big) = -\psi_j$.
In order to be able to apply a pushdown technique in Section \[sec:boundary\], we also need various formulas for pushforwards of intersections of basis divisor classes via the map $\pi_{(jk \mapsto
\bullet)}$ which identifies the divisor $\Delta_{0:jk}$ with ${\overline{\mathcal{M}}_{g,n-1}}$. They can be found in a table in [@bib:logan Theorem 2.8]; we list the relevant ones here:
\[lem:pushforward\_forgetful\] The following formulas for pushforwards of intersection cycles hold:
1. $\mathrlap{\pi_{(1n \mapsto \bullet)*}(\lambda \cdot \delta_{0:1n})}
\phantom{\pi_{(1n \mapsto \bullet)*}(\delta_{i:S} \cdot \delta_{0:1n})} =
\lambda,$
2. $\mathrlap{\pi_{(1n \mapsto \bullet)*}(\psi_j \cdot \delta_{0:1n})}
\phantom{\pi_{(1n \mapsto \bullet)*}(\delta_{i:S} \cdot \delta_{0:1n})} =
\begin{cases}
0 & \text{for $j = 1,\, n$,}\\ \psi_j & \text{for $j = 2,\, \dots,\, n - 1$,}
\end{cases}$
3. $\mathrlap{\pi_{(1n \mapsto \bullet)*}(\delta_0 \cdot \delta_{0:1n})}
\phantom{\pi_{(1n \mapsto \bullet)*}(\delta_{i:S} \cdot \delta_{0:1n})} =
\delta_0,$
4. $\mathrlap{\pi_{(1n \mapsto \bullet)*}(\delta_{0:1n}^2)}
\phantom{\pi_{(1n \mapsto \bullet)*}(\delta_{i:S} \cdot \delta_{0:1n})} =
-\psi_\bullet,$
5. $\pi_{(1n \mapsto \bullet)*}(\delta_{i:S} \cdot \delta_{0:1n}) =
\begin{cases} \delta_{i:S} & \text{if $1,\, n \not\in S$,}\\
\delta_{i:S'} & \text{if $1,\, n \in S$,}\\
0 & \text{if $1 \in S,\, n \notin S$ or $1 \notin S,\, n \in S$,}
\end{cases}$\
where $S' := \big( S \setminus \{ 1,\, n \} \big) \cup \{
\bullet \}$.
The corresponding formulas for the pushforwards of intersections of divisors with other boundary divisor classes of the form $\delta_{0:jk}$ can easily be obtained from Lemma \[lem:pushforward\_forgetful\] by applying the $S_n$-action permuting the points on ${\overline{\mathcal{M}}_{g,n}}$. Note that when we take out the basis elements of $\operatorname{Pic}({\overline{\mathcal{M}}_{g,n}})$ that get mapped to $0$ in the above formulas, the map $\alpha \mapsto \pi_{(1n \mapsto \bullet)*}(\alpha \cdot
\delta_{0:1n})$ is injective on the span of the remaining basis elements, a fact we will make use of in Section \[sec:boundary\] (see Remark \[rmk:pushdown\]).
Finally, for applying the pushdown technique we also need to know how the divisor ${{\overline{D}}_{{\underline{d}}}}$ behaves under intersection and pushforward:
\[lem:pushforward\_Dd\] If $j,\, k \in [n]$ are two indices such that $d_j$ and $d_k$ have the same sign, then $$\pi_{(jk \mapsto \bullet)*}({{\overline{D}}_{{\underline{d}}}}\cdot \delta_{0:jk}) = {\overline{D}}_{{\underline{d}}'},$$ where ${\underline{d}}' = (d_1,\, \dots,\, \widehat{d_j},\, \dots,\, \widehat{d_k},\,
\dots,\, d_n,\, d_\bullet = d_j + d_k)$.
This is an easy generalization of the proof of [@bib:logan Proposition 5.3].
Computation of the main coefficients {#sec:main_coefficients}
====================================
We write the class of the divisor ${{\overline{D}}_{{\underline{d}}}}$ as $$\label{eq:D_coefficients}
\left[ {{\overline{D}}_{{\underline{d}}}}\right] = a \lambda + \sum_{j=1}^n c_j \psi_j + b_0 \delta_0 +
\sum_{i,S} b_{i:S} \delta_{i:S}.$$ In this section we determine the coefficients $a$ and $c_j$ by expressing $D_{{\underline{d}}}$ as the degeneracy locus of a map of vector bundles of the same rank and applying Porteous’ formula. These calculations will also be instrumental in computing some of the boundary coefficients $b_0$ and $b_{i:S}$ in Section \[sec:boundary\], while the remaining ones will be obtained by intersecting the closure ${{\overline{D}}_{{\underline{d}}}}$ with suitably chosen test curves.
The top Chern class $\lambda_g := c_g({\mathbb{E}})$ of the Hodge bundle is known to have class $0$ in $A^g({\mathcal{M}_{g,n}})$ (see [@bib:looijenga]). Therefore we can find a nowhere vanishing section of ${\mathbb{E}}$, or equivalently, a relative section of $\omega_\pi$ over ${\mathcal{M}_{g,n}}$, whose zero locus cuts out a canonical divisor on every fiber of $\pi$. We denote that zero locus by ${\mathscr{K}}$. Furthermore, we denote by ${\mathscr{D}}:= \sum_{j=1}^n d_j \sigma_j \in \operatorname{Pic}({\mathcal{U}}/{\mathcal{M}_{g,n}})$ the relative divisor which on every fiber cuts out the divisor given by the linear combination of the marked points.
We now consider the restriction map $\rho \negthinspace:\, \omega_\pi({\mathscr{D}})
\to \omega_\pi({\mathscr{D}})\big|_{\mathscr{K}}$ and its direct image $$\label{eq:vector_bundle_map}
{\varphi}:= R^0\pi_*\rho \negthinspace:\,
R^0\pi_*\big(\omega_\pi({\mathscr{D}})\big) \to
R^0\pi_*\big(\omega_\pi({\mathscr{D}})\big|_{\mathscr{K}}\big).$$ Since ${\mathscr{D}}$ has relative degree $g-1$, we find that $R^1\pi_*(\omega_\pi({\mathscr{D}})) = 0$. Similarly, $\omega_\pi({\mathscr{D}})|_{\mathscr{K}}$ is torsion on fibers, so we also have $R^1\pi_*(\omega_\pi({\mathscr{D}})|_{\mathscr{K}}) = 0$. Thus by Grauert’s theorem, both sheaves in are in fact locally free, and by Riemann-Roch they are easily seen to both have rank $2g-2$.
We are now in a position to compute the main coefficients of ${{\overline{D}}_{{\underline{d}}}}$.
\[prop:main\_coefficients\] In the expression $\eqref{eq:D_coefficients}$ for $\left[ {{\overline{D}}_{{\underline{d}}}}\right]$, we have $a = -1$ and $c_j = \binom{d_j + 1}{2}$.
The short exact sequence $$\label{eq:degeneracy_SES}
0 \to {\mathcal{O}}_{\mathcal{U}}({\mathscr{D}}) \to \omega_\pi({\mathscr{D}}) \stackrel{\rho}{\to}
\omega_\pi({\mathscr{D}})\big|_{\mathscr{K}}\to 0,$$ yields after pushing down the long exact sequence $$\label{eq:degeneracy_LES}
\begin{split}
0 &\to R^0\pi_*\big({\mathcal{O}}_{\mathcal{U}}({\mathscr{D}})\big) \to R^0\pi_*\big(\omega_\pi({\mathscr{D}})\big)
\stackrel{{\varphi}}{\to} R^0\pi_*\big(\omega_\pi({\mathscr{D}})\big|_{\mathscr{K}}\big) \\
&\to R^1\pi_*\big({\mathcal{O}}_{\mathcal{U}}({\mathscr{D}})\big) \to 0.
\end{split}$$ Since $\sum_{j=1}^n d_j = g - 1$ implies $h^0(C,\, \sum_{j=1}^n d_j x_j) =
h^1(C,\, \sum_{j=1}^n d_j x_j)$ for every point $[C;\, x_1,\, \dots,\, x_n]
\in {\mathcal{M}_{g,n}}$, the sequence stays exact after passing to a fiber. Thus, the divisor $D_{{\underline{d}}}$ is exactly the degeneracy locus of the map ${\varphi}$, and by Porteous’ formula it follows that $$\label{eq:porteous_expression}
\left[ D_{{\underline{d}}} \right] =
c_1\big(R^0\pi_*\big(\omega_\pi({\mathscr{D}})\big|_{\mathscr{K}}\big)\big) -
c_1\big(R^0\pi_*\big(\omega_\pi({\mathscr{D}})\big)\big).$$ We can calculate the two terms in by a Grothendieck-Riemann-Roch computation. For the first one, we obtain $$\begin{split}
\operatorname{ch}\big(\pi_!\big(\omega_\pi({\mathscr{D}})\big|_{\mathscr{K}}\big)\big) &=
\operatorname{ch}\big(\pi_*\big(\omega_\pi({\mathscr{D}})\big|_{\mathscr{K}}\big)\big)\\
&= \pi_* \Big[ \operatorname{ch}\big(\omega_\pi({\mathscr{D}})\big|_{\mathscr{K}}\big) \cdot
\operatorname{td}\big(\omega_\pi^\vee\big) \Big]\\
&= \pi_* \Big[ \big(\operatorname{ch}\big(\omega_\pi({\mathscr{D}})\big) -
\operatorname{ch}\big({\mathcal{O}}_{\mathcal{U}}({\mathscr{D}})\big)\big) \cdot
\operatorname{td}\big(\omega_\pi^\vee\big) \Big] \qquad \text{(by
\eqref{eq:degeneracy_SES})}\\
&= \pi_* \Big[ \big(\operatorname{ch}(\omega_\pi) - 1\big) \cdot \operatorname{ch}({\mathscr{D}}) \cdot
\operatorname{td}\big(\omega_\pi^\vee\big) \Big]\\
&= \pi_* \Big[ \big( c_1(\omega_\pi) + \frac{1}{2} c_1^2(\omega_\pi) +
\dots \big) \cdot \big( 1 + c_1({\mathscr{D}}) + \frac{1}{2} c_1^2({\mathscr{D}}) + \dots
\big) \cdot \\
& \phantom{=\pi_*} \cdot \big( 1 - \frac{1}{2} c_1(\omega_\pi) + \frac{1}{12}
c_1^2(\omega_\pi) + \dots \big) \Big] \\
&= (2g - 2) + \pi_* \Big[ c_1(\omega_\pi) c_1({\mathscr{D}}) + \dots \Big] \\
&= (2g-2) + \sum_{j=1}^n d_j \psi_j + \dots \qquad \text{(by Lemma
\ref{lem:pushforward_universal})},
\end{split}$$ while for the second one we compute $$\begin{split}
\operatorname{ch}\big(\pi_!\big(\omega_\pi({\mathscr{D}})\big)\big) &=
\operatorname{ch}\big(\pi_*\big(\omega_\pi({\mathscr{D}})\big)\big)\\
&= \pi_* \Big[ \operatorname{ch}(\omega_\pi) \cdot \operatorname{ch}({\mathscr{D}}) \cdot
\operatorname{td}\big(\omega_\pi^\vee\big) \Big]\\
&= \pi_* \Big[ \big( 1 + c_1(\omega_\pi) + \frac{1}{2} c_1^2(\omega_\pi) +
\dots \big) \cdot \big( 1 + c_1({\mathscr{D}}) + \frac{1}{2} c_1^2({\mathscr{D}}) + \dots
\big) \cdot \\
& \phantom{=\pi_*} \cdot \big( 1 - \frac{1}{2} c_1(\omega_\pi) + \frac{1}{12}
c_1^2(\omega_\pi) + \dots \big) \Big] \\
&= (2g-2) + \lambda + \frac{1}{2} \sum_{j=1}^n (d_j - d_j^2) \psi_j + \dots
\qquad \text{(by Lemma \ref{lem:pushforward_universal})}.
\end{split}$$ Putting these together into yields the result.
Intersections with test curves {#sec:test_curves}
==============================
For later use in Section \[sec:boundary\], we will gather here several computations of intersections of ${{\overline{D}}_{{\underline{d}}}}$ with families of pointed curves which are wholly contained in the boundary of ${\overline{\mathcal{M}}_{g,n}}$. This constitutes the main work in computing the class of ${{\overline{D}}_{{\underline{d}}}}$, the remaining part being mainly a properly engineered application of the results presented here.
\[rmk:Schubert\_fully\_ramified\] In proving the results of this section, we will often come across questions of the following form: Given a curve $C$ of genus $g$ and a positive integer $d$, how many $g^1_d$’s $\ell$ are there on $C$ satisfying some ramification conditions whose codimensions add up to $\rho(g,\, 1,\, d)$?
In our cases, among the conditions there will always be one of *full ramification*, where we require $\ell$ to contain some fixed effective divisor $D$ of degree $d$. This reduces the problem to a Schubert calculus computation in the Grassmannian $\mathbb{G}(1,\, r)$, where $r := r(D) =
h^0(C,\, D) - 1$. Postulating the vanishing sequence $(a,\, b)$ at a generic point of $C$ corresponds to the Schubert cycle $\sigma_{a,b-1}$, and requiring $\ell$ to contain $D$ amounts to intersecting with $\sigma_{r-1} :=
\sigma_{0,r-1}$. Since $$\sigma_{\alpha_1,\beta_1} \cdot \ldots \cdot \sigma_{\alpha_k,\beta_k} \cdot
\sigma_{r-1} = 1 \qquad \text{for } \sum_{i=1}^k (\alpha_i + \beta_i) = r - 1,$$ in such cases $\ell$ is always unique.
We first consider the case $n = 2$, where we write ${\underline{d}}= (g + b - 1;\, -b)$ with $b > 0$. Here and in the following, the intersection numbers of the families in question with generators of $\operatorname{Pic}({\overline{\mathcal{M}}_{g,n}})$ that are not explicitly mentioned in the Lemmas are implied (and easily seen) to be 0.
\[lem:family\_elliptic\_tail\] Let $\left( C;\, x_1,\, x_2,\, y \right)$ be a generic 3-pointed curve of genus $g - 1$, and let $F$ be the family in ${\overline{\mathcal{M}}_{g,2}}$ obtained by gluing the marked point $y$ to a base point of a generic plane cubic pencil. Then we have $$\begin{aligned}
& \mathrlap{F \cdot {{\overline{D}}_{{\underline{d}}}}= 0,}\\
& F \cdot \lambda = 1, && F \cdot \delta_0 = 12, && F \cdot \delta_{g-1:12} =
-1.\end{aligned}$$
A member of $F$ lying in ${{\overline{D}}_{{\underline{d}}}}$ has a limit $g^1_{d_1}$ whose $C$-aspect $\ell_C$ is spanned by $d_1 x_1$ and $b x_2 + \sigma$ for some $\sigma
\in {\big| d_1 x_1 - b x_2 \big|}$. By Riemann-Roch, $h^0(C,\, (g + b - 1) x_1 - b
x_2) = 1$ for $x_1,\, x_2$ generic, so $\ell_C$ is unique, and since $y$ is also generic, it has vanishing sequence $a^{\ell_C}(y) = (0,\, 1)$. Thus the aspect on the elliptic tail would have to have vanishing sequence $(d_1 - 1,\,
d_1)$ at the base point, which is impossible.
The remaining intersection numbers are well known and can be found e. g. in [@bib:harris-morrison p. 173f.].
\[lem:family\_2\_pos\] Let $\left( C;\, x_2 \right)$ be a generic $1$-pointed curve of genus $g$, and let $F$ be the family in ${\overline{\mathcal{M}}_{g,2}}$ obtained by letting a point $x_1$ move along $C$. Then we have $$\begin{aligned}
& \mathrlap{F \cdot {{\overline{D}}_{{\underline{d}}}}= g (d_1^2 - 1),}\\
& F \cdot \psi_1 = 2g - 1, && F \cdot \psi_2 = 1, && F \cdot \delta_{0:12} =
1.\end{aligned}$$
We compute the intersection number $F \cdot {{\overline{D}}_{{\underline{d}}}}$ by degenerating $C$ to a *comb curve* $R \cup_{y_1} E_1 \cup \dots \cup_{y_g} E_g$ consisting of a rational spine $R$ to which are attached $g$ elliptic tails at generic points $y_1,\, \dots,\, y_g$, with the point $x_2$ lying on $R$. As shown in [@bib:eisenbud-harris-cuspidal-rational-curves Section 9], the variety of limit $g^r_d$’s is reduced on a generic such curve, so all we have to do is count the number of limit linear series $\ell = (\ell_R,\, \ell_{E_1},\,
\dots,\, \ell_{E_g})$ of type $g^1_{d_1}$ satisfying the given vanishing conditions at $x_1$ and $x_2$.
By [@bib:eisenbud-harris-lls Proposition 1.1], we must have $x_1 \in E_i$ for some $i$. The $E_j$-aspect of each elliptic tail $E_j$ with $j \neq i$ must satisfy $a^{\ell_{E_j}}(y_j) \leq (d_1 - 2,\, d_1)$, giving $a^{\ell_R}(y_j) \geq (0,\, 2)$ for these $j$. Thus the $R$-aspect of $\ell$ is a $g^1_{d_1}$ that contains the divisor $d_1 y_i$, vanishes to order $b$ at $x_2$ and is simply ramified at $(g-1)$ further points, corresponding to the Schubert cycle $$\sigma_{a_0^{\ell_R}(y_i), d_1 - 1} \cdot \sigma_{b-1} \cdot \sigma_1^{g-1}
\qquad \text{in } \mathbb{G}(1,\, d_1).$$ Counting dimensions, this is non-empty only if $a_0^{\ell_R}(y_i) = 0$, and then $\ell_R$ is unique by Remark \[rmk:Schubert\_fully\_ramified\]. We thus get the upper bound $a^{\ell_R}(y_i) \leq (0,\, d_1)$, which by the compatibility conditions is equivalent to $a^{\ell_{E_i}}(y_i) \geq (0,\,
d_1)$. Since also $a^{\ell_{E_i}}(x_1) \geq (0,\, d_1)$, this is possible only if equality holds everywhere and $x_1 - y_i$ is a non-trivial $d_1$-torsion point in $\operatorname{Pic}^0(E_i)$. Thus each of the $g$ elliptic tails gives exactly $(d_1^2 - 1)$ possibilities for $x_1$.
The remaining intersection numbers can be found by standard techniques.
\[lem:family\_2\_neg\] Let $\left( C;\, x_1 \right)$ be a generic $1$-pointed curve of genus $g$, and let $F$ be the family in ${\overline{\mathcal{M}}_{g,2}}$ obtained by letting a point $x_2$ move along $C$. Then we have $$\begin{aligned}
& \mathrlap{F \cdot {{\overline{D}}_{{\underline{d}}}}= g (b^2 - 1),}\\
& F \cdot \psi_1 = 1, && F \cdot \psi_2 = 2g - 1, && F \cdot \delta_{0:12} =
1.\end{aligned}$$
We proceed as in the proof of Lemma \[lem:family\_2\_pos\], degenerating $C$ to a comb curve where now $x_1 \in R$. Reasoning as before, we find that $x_2 \in E_j$ for some $j$ and $a^{\ell_R}(y_j) \leq (0,\, b)$ for dimension reasons, so $a^{\ell_{E_j}}(y_j) \geq (g - 1,\, g + b - 1)$. Together with $a^{\ell_{E_j}}(y_j) \geq (0,\, b)$ this implies that $x_2 - y_j$ is a non-trivial $b$-torsion point in $\operatorname{Pic}^0(E_j)$, so each of the $g$ elliptic tails contributes $(b^2 - 1)$ possibilities for $x_2$.
\[lem:family\_G\_i:12\_\*\] Let $\left(C_1;\, x_1,\, x_2,\, y \right)$ be a generic $3$-pointed curve of genus $g-i$, $C_2$ a generic curve of genus $i \geq 2$, and let $F$ denote the family in ${\overline{\mathcal{M}}_{g,2}}$ obtained by gluing $y$ to a moving point of $C_2$. Then we have $$\begin{aligned}
& F \cdot {{\overline{D}}_{{\underline{d}}}}= i (i^2 - 1),\\
& F \cdot \delta_{g-i:12} = 2 - 2i.\end{aligned}$$
Let $\ell = (\ell_{C_1},\, \ell_{C_2})$ be a limit $g^1_{d_1}$ on $C$. By genericity, the family of $g^1_{d_1}$’s on $C_1$ with the required vanishing at $x_1$ and $x_2$ has dimension $\rho(g - i,\, 1,\, d_1) - (d_1 -
1) - (b - 1) = i - 1$, so for $y \in C_1$ also generic we must have $a_1^{\ell_{C_1}}(y) \leq i$. The compatibility relations then force $a_0^{\ell_{C_2}}(y) \geq d_1 - i$. Since $\ell_{C_2}$ contains the divisor $d_1 y$, this means that ${\big| i y \big|}$ is a $g^1_i$ on $C_2$, i. e. $y$ is one of the $i (i^2 - 1)$ Weierstraß point of $C_2$. Since $C_2$ is generic, it has only ordinary Weierstraß points, so we must have equality, and $\ell$ is unique by Remark \[rmk:Schubert\_fully\_ramified\].
We now turn to cases where $n = 3$. We will first suppose that $d_1,\, d_2 >
0$, while $d_3 < 0$, and we write $b := -d_3$ and $d := d_1 + d_2 = g + b -
1$.
For $b = 1$, the following result was already proven in [@bib:logan Proposition 3.3] and [@bib:diaz-thesis Lemma 6.2].
\[lem:family\_F2\_2pos\] Let $\left( C;\, x_2,\, x_3 \right)$ be a generic $2$-pointed curve of genus $g$, and let $F$ be the family in ${\overline{\mathcal{M}}_{g,3}}$ obtained by letting a point $x_1$ vary on $C$. Then we have $$\begin{aligned}
& \mathrlap{F \cdot {{\overline{D}}_{{\underline{d}}}}= g d_1^2 - (g - d_2)_+,}\\
& F \cdot \psi_1 = 2g, && F \cdot \psi_2 = 1, && F \cdot \psi_3 = 1,\\
& F \cdot \delta_{0:12} = 1, && F \cdot \delta_{0:13} = 1.\end{aligned}$$
Suppose first that $g = 1$, i. e. $b = d$. Then a $g^1_d$ containing the divisors $d_1 x_1 + d_2 x_2$ and $d x_3$ exists if and only if these are linearly equivalent, and since $d_2 > 0$ this gives $d_1^2$ possibilities for $x_1$ as claimed.
If $g > 1$, we degenerate $C$ to a transverse union $C = E \cup_y C'$ such that $\left( E;\, x_2,\, y \right)$ is a generic $2$-pointed elliptic curve and $\left( C';\, y,\, x_3 \right)$ is a generic $2$-pointed curve of genus $g
- 1$. Then there is a decomposition $F = F_E + F_{C'}$ of $1$-cycles on ${\overline{\mathcal{M}}_{g,3}}$, where $F_E$ and $F_{C'}$ correspond to the cases $x_1 \in E$ and $x_1 \in C'$. These are in a natural way pushforwards via gluing morphisms of $1$-cycles $F_E'$ and $F_{C'}'$ on ${\overline{\mathcal{M}}_{1,3}}$ and ${\overline{\mathcal{M}}_{g-1,3}}$, respectively. We will show that $$\begin{aligned}
\label{eq:FEdotDd}
F_E \cdot {{\overline{D}}_{{\underline{d}}}}&= F_E' \cdot {\overline{D}}_{(d_1, d_2; -d)} \text{\quad(}= d_1^2 \text{
by the above)}, \\
\label{eq:FCdotDd}
F_{C'} \cdot {{\overline{D}}_{{\underline{d}}}}&=
\begin{cases}
F_{C'}' \cdot {\overline{D}}_{(d_1, d_2 - 1; d_3)} & \text{if } d_2 > 1, \\
(g - 1)(d_1^2 - 1) & \text{if } d_2 = 1,
\end{cases}\end{aligned}$$ and by induction we conclude that $$F \cdot {{\overline{D}}_{{\underline{d}}}}= \sum_{i=1}^{d_2} d_1^2 + \sum_{i=d_2+1}^g (d_1^2 - 1) = g d_1^2
- (g - d_2)_+.$$
For showing , let $\ell = (\ell_E,\, \ell_{C'})$ be a $g^1_d$ having the required vanishing. Then $\ell_E$ has a section not vanishing at $y$, so by the compatibility conditions $\ell_{C'}$ must be totally ramified there. Counting dimensions as in the proof of Lemma \[lem:family\_G\_i:12\_\*\], we find that the latter cannot have a base point at $y$, so again by compatibility $\ell_E$ needs to have a section vanishing to order $d$ at $y$. This is equivalent to requiring $(E;\, x_1,\, x_2,\, y)$ to lie in ${\overline{D}}_{(d_1,d_2;-d)}$.
Now consider . Since $\ell_E$ contains $d_1 y + d_2 x_2$, and by genericity $d_2 x_2 \not\equiv d_2 y$, it cannot also contain the divisor $d y$, so we must have $a_1^{\ell_{E}}(y) \leq d - 1$. By the compatibility condition then $a_0^{\ell_{C'}}(y) \geq 1$, and after removing the base point we obtain a $g^1_{d-1}$ on $C'$ containing the divisor $d_1 x_1
+ (d_2 - 1) y$ and having a section vanishing to order $b$ at $x_3$. For $d_2
> 1$ this is equivalent to $(C';\, x_1,\, y,\, x_3) \in {\overline{D}}_{(d_1, d_2-1;
d_3)}$, while for $d_2 = 1$ the answer is given in Lemma \[lem:family\_2\_pos\].
\[lem:family\_F2\_i\_2pos\] Let $\left( C_1;\, y \right)$ be a generic $1$-pointed curve of genus $i \geq
1$ , $\left( C_2;\, x_2,\, x_3,\, y \right)$ a generic $3$-pointed curve of genus $g - i$, $\left( C = C_1 \cup_y C_2;\, x_2,\, x_3 \right)$ the $2$-pointed curve obtained by gluing $C_1$ and $C_2$ at $y$, and $F$ the family in ${\overline{\mathcal{M}}_{g,3}}$ obtained by letting a point $x_1$ move along $C_1$. Then we have $$\begin{aligned}
& \mathrlap{F \cdot {{\overline{D}}_{{\underline{d}}}}= i (d_1^2 - 1) + (i - d_1)_+,}\\
& F \cdot \psi_1 = 2i - 1, && F \cdot \delta_{i:1} = -1, && F \cdot
\delta_{i:\emptyset} = 1.\end{aligned}$$ These formulas also hold for $d_2 = 0$.
Let $\ell = (\ell_{C_1},\, \ell_{C_2})$ be a limit $g^1_d$ on $C$ satisfying the given vanishing conditions and write $\ell_{C_2} = a_0 y + \ell_{C_2}'$, where $a_0 := a_0^{\ell_{C_2}}(y)$. Then $\ell_{C_2}'$ contains the divisor $(d_1-a_0)y + d_2 x_2$ and has a section vanishing to order $b$ at $x_3$, so it corresponds to the Schubert cycle $\sigma_{r-1} \cdot \sigma_{b-1}$ in $\mathbb{G}(1,\, r)$, where $$r := h^0(C_2,\, (d_1-a_0)y + d_2 x_2) - 1 = d - a_0 - g + i$$ by Riemann-Roch. This is non-empty only if $b \leq r$, or equivalently if $a_0
\leq i - 1$.
In case $a_0 < d_1$, we have $a_1^{\ell_{C_2}}(y) = d_1$, so $a_0^{\ell_{C_1}}(y) \geq d_2$. Then $\ell_{C_1}' := \ell_{C_1} - d_2 y$ is a $g^1_{d_1}$ fully ramified at $x_1$. Since $C_1$ is generic and therefore has only ordinary Weierstraß points, this is possible only if $d_1 \geq i$. Since $a_1^{\ell_{C_1}}(y) \geq d - a_0 \geq d - i + 1$, $\ell_{C_1}'$ vanishes to order $d_1 - i + 1$ at $y$, so by Lemma \[lem:family\_2\_pos\] the number of possibilities is $$F \cdot {{\overline{D}}_{{\underline{d}}}}= i (d_1^2 - 1).$$
If on the other hand $a_0 = d_1$, we have $d_1 \leq i - 1$ by the above. By another Schubert cycle computation for $\ell_{C_2}'$ we find that we need to have $a_1^{\ell_{C_2}}(y) \leq i$, so $a_0^{\ell_{C_1}}(y) \geq d - i$. Thus $\ell_{C_1} - (d-i) y$ is now a $g^1_i$ having $d_1 x_1 + (i - d_1) y$ as a section. Applying Lemma \[lem:family\_F2\_2pos\] with ${\underline{d}}= (d_1,\, i - d_1;\,
-1)$, we find that $$F \cdot {{\overline{D}}_{{\underline{d}}}}= i d_1^2 - d_1.$$
Both arguments also go through when $d_2 = 0$.
Still considering cases where $n = 3$, we now suppose that $d_1 > 0$, while $d_2,\, d_3 < 0$, and we write $b_j := -d_j$ for $j = 2,\, 3$ and $b :=
-d_2 - d_3$, so that $d_1 = g + b - 1$.
\[lem:family\_F2\_2neg\] Let $\left( C;\, x_1,\, x_3 \right)$ be a generic $2$-pointed curve of genus $g$, and let $F$ be the family in ${\overline{\mathcal{M}}_{g,3}}$ obtained by letting a point $x_2$ vary on $C$. Then we have $$\begin{aligned}
& \mathrlap{F \cdot {{\overline{D}}_{{\underline{d}}}}= g d_2^2,}\\
& F \cdot \psi_1 = 1, && F \cdot \psi_2 = 2g, && F \cdot \psi_3 = 1,\\
& F \cdot \delta_{0:12} = 1, && F \cdot \delta_{0:23} = 1.\end{aligned}$$
This is similar to the proof of Lemma \[lem:family\_F2\_2pos\]: Let $C = E
\cup_y C'$ again with now $x_1 \in E$ and $x_3 \in C'$. Then $F = F_E +
F_{C'}$ with $$\begin{aligned}
F_E \cdot {{\overline{D}}_{{\underline{d}}}}&= F_E' \cdot {\overline{D}}_{(d_1;d_2,d_3-g+1)} = d_2^2 \quad \text{and}\\
F_{C'} \cdot {{\overline{D}}_{{\underline{d}}}}&= F_{C'}' \cdot {\overline{D}}_{(d_1-1;d_2,d_3)}.\end{aligned}$$ The only difference to before is that now $\ell_{C'}$ has a $b_2$-fold base point at $y$ in case $x_2 \in E$. The result follows by induction.
\[lem:family\_F2\_i\_2neg\] Let $\left( C_1;\, y \right)$ be a generic $1$-pointed curve of genus $i \geq
1$, $\left( C_2;\, x_1,\, x_3,\, y \right)$ a generic 3-pointed curve of genus $g - i$, $\left( C = C_1 \cup_y C_2;\, x_1,\, x_3 \right)$ the $2$-pointed curve obtained by gluing $C_1$ and $C_2$ at $y$, and $F$ the family in ${\overline{\mathcal{M}}_{g,3}}$ obtained by letting a point $x_2$ move along $C_1$. Then we have $$\begin{aligned}
& \mathrlap{F \cdot {{\overline{D}}_{{\underline{d}}}}= i (d_2^2 - 1),}\\
& F \cdot \psi_2 = 2i - 1, && F \cdot \delta_{i:2} = -1, && F \cdot
\delta_{i:\emptyset} = 1.\end{aligned}$$ These formulas also hold for $d_3 = 0$.
Let $\ell = (\ell_{C_1},\, \ell_{C_2})$ be a limit $g^1_{d_1}$ on $C$ satisfying the given vanishing conditions. Then $\ell_{C_2}$ must include the divisor $d_1 x_1$, so $a_0^{\ell_{C_2}}(y) = 0$. The section of $\ell_{C_2}$ vanishing to order $b_3$ at $x_3$ must also vanish to order $a_1 :=
a_1^{\ell_{C_2}}(y)$ at $x_1$: otherwise the corresponding section of $\ell_{C_1}$ would have to be fully ramified at $y$ while at the same time vanishing to order $b_2$ at $x_2$, which is absurd. We thus need $$h^0( C_2,\, d_1 x_1 - a_1 y - b_3 x_3 ) = b_2 + i - a_1 \geq 1 \iff a_1 \leq
b_2 + i - 1,$$ where we used Riemann-Roch and the genericity of the points on $C_2$. By compatibility, $a_0^{\ell_{C_1}}(y) \geq g - i + b_3$, and thus $\ell_{C_1} -
(g - i + b_3) y$ is a $g^1_{i+b_2-1}$ which is fully ramified at $y$ and vanishes to order $b_2$ at $x_2$. By Lemma \[lem:family\_2\_neg\] there are $i
(d_2^2 - 1)$ possibilities for $x_2$.
We now finally consider the situation $n = 4$ with $d_1,\, d_2 > 0$ and $d_3,\, d_4 < 0$. We write $b_j := -d_j$ for $j = 3,\, 4$ and $b := b_3 +
b_4$, so that $d := d_1 + d_2 = g + b - 1$.
\[lem:family\_F\_13\_on\_i\] Let $\left( C_1;\, x_1,\, x_3 \right)$ be a generic $2$-pointed curve of genus $i$ with $1 \leq i \leq g$, $\left( C_2;\, x_2,\, x_4,\, y \right)$ a generic $3$-pointed curve of genus $g-i$, and let $F$ be the family in ${\overline{\mathcal{M}}_{g,4}}$ obtained by gluing $y$ to a moving point of $C_1$. Then we have $$\begin{aligned}
& \mathrlap{F \cdot {{\overline{D}}_{{\underline{d}}}}= i (d_1 + d_3 - i + 1)^2 - (i - d_1)_+,}\\
& F \cdot \psi_1 = 1, && F \cdot \psi_3 = 1,\\
& F \cdot \delta_{i:13} = -2i, && F \cdot \delta_{i:1} = 1, && F \cdot
\delta_{i:3} = 1.\end{aligned}$$
Let $\ell = (\ell_{C_1},\, \ell_{C_2})$ be a limit $g^1_d$ on $C$ satisfying the given vanishing conditions. Then $\ell$ contains the divisor $(d_1 x_1 + d_2 y,\, d_1 y + d_2 x_2)$.
Suppose first that $d_1 + d_3 \geq i$, or $d_2 + d_4 \leq g - i - 1$. Then the base locus of $\ell_{C_2}$ cannot contain $d_1 y$, since $h^0(C_2,\, d_2
x_2 - b_4 x_4) = 0$ by Riemann-Roch and genericity. Hence $a_1^{\ell_{C_2}}(y) = d_1$, and by a dimension count $a_0^{\ell_{C_2}}(y) \leq i - d_3 - 1$, with equality attained for a unique $g^1_d$. Thus $a^{\ell_{C_1}}(y) \geq (d_2,\, g - i - d_4)$, and we can apply Lemma \[lem:family\_F2\_2neg\] with ${\underline{d}}= (d_1;\, d_2 + d_4 - g +
i,\, d_3)$ to find $$F \cdot {{\overline{D}}_{{\underline{d}}}}= i (d_2 + d_4 - g + i)^2.$$
If $d_1 + d_3 < i - 1$, then $h^0(C_1,\, d_1 x_1 - b_3 x_3) = 0$, so $d_2
y$ cannot be in the base locus of $\ell_{C_1}$, forcing $a_1^{\ell_{C_1}}(y) =
d_2$ and thus $a_0^{\ell_{C_2}}(y) = d_1$. As in the proof of Lemma \[lem:family\_F2\_i\_2neg\], we find that $a^{\ell_{C_2}}(y) \leq (d_1,\, i - d_3 - 1)$, so $a^{\ell_{C_1}}(y) \geq (g -
i - d_4,\, d_2)$. Applying Lemma \[lem:family\_F2\_2pos\] with ${\underline{d}}= (d_2 +
d_4 - g + i,\, d_1;\, d_3)$ then gives $$F \cdot {{\overline{D}}_{{\underline{d}}}}= i (d_2 + d_4 - g + i)^2 - (i - d_1)_+.$$
Finally, if $d_1 + d_3 = i - 1$ we obtain $a^{\ell_{C_2}}(y) = (d_1,\, d_1 +
1)$ and $\ell_{C_2} - d_1 y$ must have a section vanishing to order $1$ at $y$ and $b_4$ at $x_4$. Since $h^0(C_2,\, d_2 x_2 - y - b_4 x_4) = 0$, this is impossible, so in this case $F \cdot {{\overline{D}}_{{\underline{d}}}}= 0$, which is consistent with the other two formulas.
Computation of the boundary coefficients {#sec:boundary}
========================================
For computing the boundary coefficients of ${{\overline{D}}_{{\underline{d}}}}$ we will use a bootstrapping approach, considering first the easiest non-trivial case $n = 2$, then generalizing to the case $n > 2$ with exactly one $d_j < 0$, and finally tackling the most general situation.
The case $n = 2$
----------------
For ease of notation, we will write ${\underline{d}}= (d_1,\, d_2) = (g + b - 1,\, -b)$ with $b \geq 1$ and denote the corresponding divisor by ${\overline{D}}_{(g+b-1,b)} =:
{\overline{D}}_b$.
\[prop:case\_n=2\] The class of ${\overline{D}}_b$ is given by $$\begin{split}
\left[ {\overline{D}}_b \right] =& -\lambda + \binom{g + b}{2} \psi_1 + \binom{b}{2}
\psi_2 - 0 \cdot \delta_0 - \binom{g + 1}{2} \delta_{0:12}\\
& -\sum_{i=1}^{g-1} \left[ \binom{g - i + b}{2} \delta_{i:1} + \binom{g - i +
1}{2} \delta_{i:12} \right].
\end{split}$$
From Section \[sec:main\_coefficients\] we know that $a = -1$, $c_1 =
\binom{g + b}{2}$ and $c_2 = \binom{-b + 1}{2} = \binom{b}{2}$. Intersecting ${\overline{D}}_b$ with the family from Lemma \[lem:family\_G\_i:12\_\*\], we find that $b_{g-i:12} = -\binom{i + 1}{2}$, or dually $b_{i:12} = -\binom{g - i + 1}{2}$ for $i = 0,\, \dots,\, g-2$. From the family in Lemma \[lem:family\_F2\_i\_2pos\] (taking $d_2 = 0$), we get $$b_{i:1} = (2i - 1) c_1 + b_{g-i:12} - i ((g + b - 1)^2 - 1) = -\binom{g - i
+ b}{2} \quad \text{for } i = 2,\, \dots,\, g-1,$$ and Lemma \[lem:family\_F2\_i\_2neg\] with $d_3 = 0$ gives $b_{g-1:12}
= -1$. Using Lemma \[lem:family\_F2\_i\_2pos\] once more, we get the value for $b_{1:1}$, while finally Lemma \[lem:family\_elliptic\_tail\] leads to $b_0 =
(b_{g-1:12} - a)/12 = 0$.
\[rmk:weierstrass\_divisor\] Note that when we pull back from ${\overline{\mathcal{M}}_{g,1}}$ the Weierstraß divisor ${\overline{\mathcal{W}}_g}$, whose class is given in , we get by Lemma \[lem:pullback\_forgetful\] that $$\begin{split}
\left[ \pi_2^* {\overline{\mathcal{W}}_g}\right] &= -\lambda + \binom{g + 1}{2} \psi_1 - \binom{g +
1}{2} \delta_{0:12} - \sum_{i=1}^{g-1} \binom{g - i + 1}{2} \big( \delta_{i:1}
+ \delta_{i:12} \big) \\
&= \left[ {\overline{D}}_1 \right]
\end{split}$$ as expected. Furthermore it is easy to see that a 2-pointed curve $(C = C'
\cup_y {\mathbb{P}}^1;\, x_1,\, x_2)$ with $x_1,\, x_2 \in {\mathbb{P}}^1$ is in ${\overline{D}}_b$ exactly when it has a limit $g^1_{g+b-1}$ whose $C'$-aspect satisfies $a^{\ell_{C'}}(y) = (b - 1,\, g + b - 1)$, which is the case if and only if $y$ is a Weierstraß point of $C'$. From Lemma \[lem:pushforward\_forgetful\] we obtain accordingly $$\pi_{(12 \mapsto \bullet)*} (\left[ {\overline{D}}_b \right] \cdot \delta_{0:12}) =
-\lambda + \binom{g + 1}{2} \psi_\bullet - \sum_{i=1}^{g-1} \binom{g - i +
1}{2} \delta_{i:\bullet} = \left[ {\overline{\mathcal{W}}_g}\right].$$
The case of exactly one negative $d_j$
--------------------------------------
We now consider the next simplest case where exactly one of the $d_j$ is negative (for definiteness, and without loss of generality, we take $d_n <
0$).
\[rmk:pushdown\] Here and in the next section we will several times apply a “pushdown” argument which runs as follows: Let $j,\, k \in [n]$ be two indices such that $d_j$ and $d_k$ have the same sign, and suppose that $\alpha \in
\operatorname{Pic}({\overline{\mathcal{M}}_{g,n}})$ is one of the basic divisor classes described in Section \[ssec:picard\_group\] satisfying $\beta := \pi_{(jk \mapsto
\bullet)*}(\alpha \cdot \delta_{0:jk}) \neq 0$. Since then no other basis element is mapped to $\beta$ and $\pi_{(jk \mapsto \bullet)*}(\left[ {{\overline{D}}_{{\underline{d}}}}\right] \cdot \delta_{0:jk}) = \left[ {\overline{D}}_{{\underline{d}}'} \right]$ with ${\underline{d}}'$ as in Lemma \[lem:pushforward\_Dd\], the coefficient of $\alpha$ in the expression for ${{\overline{D}}_{{\underline{d}}}}$ is the same as the coefficient of $\beta$ in the class of ${\overline{D}}_{{\underline{d}}'}$.
\[prop:case\_one\_negative\_dj\] If $d_j > 0$ for $j = 1,\, \dots,\, n-1$, then the class of ${{\overline{D}}_{{\underline{d}}}}$ is given by $$\left[ {{\overline{D}}_{{\underline{d}}}}\right] = -\lambda + \sum_{j=1}^n \binom{d_j + 1}{2} \psi_j - 0
\cdot \delta_0 - \sum_{i, S \subseteq [n-1]} \binom{{\left| d_S - i \right|} + 1}{2}
\delta_{i:S}.$$
We already know from Section \[sec:main\_coefficients\] that $a = -1$ and $c_j = \binom{d_j + 1}{2}$.
For the $b_{0:jk}$ with $j,\, k \in [n-1]$, we can apply the pushdown argument explained in Remark \[rmk:pushdown\] to the divisor class $\delta_{0:jk}$ itself, which gets mapped to $-\psi_\bullet$. Thus we have $b_{0:jk} = -c_\bullet$, where $c_\bullet$ is the coefficient of $\psi_\bullet$ in the expression for $\pi_{(jk \mapsto \bullet)*}\big({{\overline{D}}_{{\underline{d}}}}\,\cdot\, \delta_{0:jk}\big)$. Since $j,\, k \leq n-1$, we have $d_j,\, d_k >
0$, so we can apply Lemma \[lem:pushforward\_Dd\] to find that $b_{0:jk} =
-\binom{d_j + d_k + 1}{2}$. Similarly, in order to compute $b_{0:S}$ for $S
\subseteq [n-1]$, we can intersect with one divisor $\delta_{0:jk}$ with $j,\, k \in S$ at a time and push down via the appropriate forgetful maps; by inductively reasoning as before we find $$b_{0:S} = -\binom{d_S + 1}{2} \qquad \text{for } S \subseteq [n - 1].$$
Looking at Lemma \[lem:pushforward\_forgetful\] and using a simple induction again, we see that when we successively let all of the points $x_1,\, \dots,\,
x_{n-1}$ come together and push down via the appropriate forgetful maps, the divisor $\delta_{i:\emptyset}$ is mapped to $\delta_{i:\emptyset} =
\delta_{g-i:12}$ on ${\overline{\mathcal{M}}_{g,2}}$, so by Lemma \[lem:pushforward\_Dd\] and Proposition \[prop:case\_n=2\] again we see that $$b_{i:\emptyset} = -\binom{i + 1}{2} \qquad \text{for } i \geq 1.$$
Next, using the test family from Lemma \[lem:family\_F2\_i\_2pos\] we get that $$b_{i:j} = (2i - 1) c_j + b_{i:\emptyset} - i (d_j^2 - 1) - (i - d_j)_+ =
-\binom{{\left| d_j - i \right|} + 1}{2},$$ for $j \in [n - 1]$, and using a pushdown argument once again we arrive at $$b_{i:S} = -\binom{{\left| d_S - i \right|} + 1}{2} \qquad \text{for } S \subseteq [n - 1]
\text{ and } i \geq 1.$$
Finally, the fact $b_0 = 0$ follows again from letting all of the points $x_1,\, \dots,\, x_{n-1}$ coalesce, pushing down to ${\overline{\mathcal{M}}_{g,2}}$ and recurring to Proposition \[prop:case\_n=2\].
\[rmk:logan\_divisor\] If $b = 1$, we expect ${{\overline{D}}_{{\underline{d}}}}$ to be the pullback to ${\overline{\mathcal{M}}_{g,n}}$ of the divisor $$D = \left\{ \left[ C; x_1,\, \dots,\, x_{n-1} \right] \,\Big|\, h^0 \left(
C,\, d_1 x_1 + \dots + d_{n-1} x_{n-1} \right) \geq 2 \right\}$$ which was considered by Logan [@bib:logan]. Indeed, we have for $S
\subseteq [n-1]$ that $$b_{i:S \cup \{ n \}} = b_{g-i:[n-1] \setminus S} = -\binom{{\left| g - d_S -
g + i \right|} + 1}{2} = b_{i:S},$$ and moreover $c_n = 0$ and $b_{0:jn} = -c_j$ for $j \in [n-1]$. Lemma \[lem:pullback\_forgetful\] thus shows that $$\left[ D \right] = -\lambda + \sum_{j=1}^{n-1} \psi_j - 0 \cdot \delta_0 -
\binom{{\left| d_S - i \right|} + 1}{2} \delta_{i:S},$$ which is consistent with the computations in [@bib:logan].
The general case
----------------
We will now finally deal with the most general case where there are at least two $d_j$ of either sign, thereby proving formula . We exclude the degenerate case where some $d_j$ equals $0$, since in this case the divisor ${{\overline{D}}_{{\underline{d}}}}$ is just a pullback of some ${\overline{D}}_{{\underline{d}}'}$ from some moduli space with fewer marked points, so its class can easily be computed from Theorem \[thm:class\_Dd\] with the help of the formulas in Lemma \[lem:pullback\_forgetful\].
\[thm:class\_Dd\] The class of ${{\overline{D}}_{{\underline{d}}}}$ in $\operatorname{Pic}({\overline{\mathcal{M}}_{g,n}})$ is given by $$\begin{split}
\left[ {{\overline{D}}_{{\underline{d}}}}\right] =& -\lambda + \sum_{j=1}^n \binom{d_j + 1}{2} \psi_j -
0 \cdot \delta_0 \\
& - \sum_{\substack{i,\, S\\ S \subseteq S_+}}
\binom{{\left| d_S - i \right|} + 1}{2} \delta_{i:S} - \sum_{\substack{i,\, S\\ S
\not\subseteq S_+}} \binom{d_S - i + 1}{2} \delta_{i:S}.
\end{split}$$
From Section \[sec:main\_coefficients\] we know that $a = -1$ and $c_j =
\binom{d_j + 1}{2}$. Using the by now familiar pushdown technique, we get from Proposition \[prop:case\_one\_negative\_dj\] that $b_0 = 0$ and $$b_{i:S} = -\binom{{\left| d_S - i \right|} + 1}{2} \qquad \text{for } S \subseteq S_+.$$ Thus we are left with computing the $b_{i:S}$ where the points indexed by $S_-$ do not all lie on the same component.
Suppose first that $\emptyset \neq S \subsetneq S_-$. By letting the points from $S_+$, $S$ and $S_- \setminus S$ respectively come together, we can reduce to the case $n = 3$ with $d_1 = d_{S_+} > 0$, $d_2 = d_S < 0$ and $d_3
= d_{S_- \setminus S} < 0$. The divisor $\delta_{i:S}$ is mapped to $-\psi_2$ for $i = 0$ and to $\delta_{i:2}$ for $i > 0$. We know that $c_2 =
\binom{d_2 + 1}{2}$, while for $i > 0$ we get from Lemma \[lem:family\_F2\_i\_2neg\] that $$b_{i:2} = (2i - 1) \binom{d_2 + 1}{2} + b_{i:\emptyset} - i(d_2^2 - 1) =
-\binom{d_2 - i + 1}{2}.$$ Thus in total we deduce by Lemma \[lem:pushforward\_Dd\] that $$b_{i:S} = -\binom{d_S - i + 1}{2} \qquad \text{for } \emptyset \neq S
\subsetneq S_-.$$
Finally, let $S = S_1 \cup S_2$ with $\emptyset \neq S_1 \subsetneq S_+$ and $\emptyset \neq S_2 \subsetneq S_-$. Letting the points from $S_1$, $S_+
\setminus S_1$, $S_2$ and $S_- \setminus S_2$ respectively come together, we reduce to the computation of $b_{i:13}$ in the case $n = 4$. Taking the family from Lemma \[lem:family\_F\_13\_on\_i\] we find $$\begin{split}
b_{i:13} &= \frac{1}{2i} \left( c_1 + c_3 + b_{i:1} + b_{i:3} - i (d_1 + d_3 -
i + 1)^2 + (i - d_1)_+ \right)\\
&= -\binom{d_1 + d_3 - i + 1}{2}.
\end{split}$$ Note that although in Lemma \[lem:family\_F\_13\_on\_i\] we require $i \geq 1$, the above formula is invariant under the substitution $(i,\, d_1,\, d_3)
\mapsto (g - i,\, d_2,\, d_4)$, so it holds also for $i = 0$. Thus in total we get $$b_{i:S} = -\binom{d_S - i + 1}{2} \qquad \text{for } S = S_+ \cup S_- \text{
with } \emptyset \neq S_1 \subsetneq S_+ \text{ and } \emptyset \neq S_2
\subsetneq S_-,$$ which finishes the computation of $\left[ {{\overline{D}}_{{\underline{d}}}}\right]$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We study index theory for manifolds with Baas-Sullivan singularities using geometric $K$-homology with coefficients in a unital $C^*$-algebra. In particular, we define a natural analog of the Baum-Connes assembly map for a torsion-free discrete group in the context of these singular spaces. The cases of singularities modelled on $k$-points (i.e., ${{\mathbb{Z}}/k{\mathbb{Z}}}$-manifolds) and the circle are discussed in detail. In the case of the former, the associated index theorem is related to the Freed-Melrose index theorem; in the case of latter, the index theorem is related to work of Rosenberg.'
author:
- 'Robin J. Deeley'
title: 'Index theory for manifolds with Baas-Sullivan singularities'
---
Introduction
============
We consider index theory for manifolds with Baas-Sullivan singularities using the framework of geometric K-homology. We also discuss the Baum-Connes conjecture in this context. Manifolds with Baas-Sullivan singularities were introduced in [@Baas] following work in [@MS]. They are a generalization of the concept of a ${{\mathbb{Z}}/k{\mathbb{Z}}}$-manifold and are well studied objects in differential topology. The reader can find introductions to these objects in [@Bot] or [@Rud Chapter VIII].
Let $P$ be a smooth, compact, ${\rm spin^c}$-manifold. Informally (see Definition \[Pmfld\] for the precise definition), a smooth, compact, ${\rm spin^c}$ manifold with Baas-Sullivan singularities modelled on $P$ is the following data: a compact, ${\rm spin^c}$ manifold with boundary, $Q$, whose boundary is diffeomorphic, in a ${\rm spin^c}$-preserving way, to $P \times \beta Q$ for some closed smooth, compact, ${\rm spin^c}$ manifold, $\beta Q$. We denote such an object by $(Q,\beta Q)$ and refered to such an object simply as a $P$-manifold. A ${{\mathbb{Z}}/k{\mathbb{Z}}}$-manifold is a $P$-manifold in the case when $P=k$-points. Our goal is the construction of index theorems for such objects; prototypical examples are the Freed-Melrose index theorem for ${{\mathbb{Z}}/k{\mathbb{Z}}}$-manifolds [@Fre; @FM] and the index theorem for $S^1$-manifolds in [@Ros]. In addition, the results of this paper are a generalization of our previous work in [@Dee1; @Dee2]; in those papers, only the case of ${{\mathbb{Z}}/k{\mathbb{Z}}}$-manifolds was considered.
In order to construct the index map and Baum-Connes assembly map for these objects, we define an abelian group via Baum-Douglas type cycles; the reader is directed to any of [@BD; @BD2; @BHS; @Rav; @Wal] for further details on these cycles and the resulting model. Before discussing our construction, we recall the basic definitions of the Baum-Douglas model of K-homology of a finite CW-complex, $X$, with coefficients in a unital $C^*$-algebra, $A$ (see for example [@Wal]). A cycle in the theory is a triple, $(M,E,f)$, where $M$ is a smooth, compact, ${\rm spin^c}$-manifold, $E$ is a locally trivial bundle over $M$ with finitely generated, projective Hilbert $A$-modules as fibers (we refer to such an object as an $A$-vector bundle), and $f:M \rightarrow X$ is a continuous function. An abelian group denoted by $K_*(X;A)$ is defined using these cycles modulo an equivalence relation which is geometrically defined. The group, $K_*(X;A)$, is via an explicit map isomorphic to $KK^*(C(X),A)$. When $X=pt$, the isomorphism is given by the map $$\label{isoInCaseOfPt}
(M,E) \in K_*(pt;A) \mapsto {\rm ind_{AS}}(D_{(M,E)}) \in K_*(A) \cong KK^*({\mathbb{C}}, A)$$ where ${\rm ind_{AS}}(D_{(M,E)})$ denotes the higher index of the Dirac operator on $M$ twisted by the bundle $E$. Based on this idenification, we can view $K_*(pt;A)$ as a natural “geometric" home for our indices.
The basic idea of our construction is to replace the manifold $M$ in the Baum-Douglas model with a $P$-manifold. In more detail, the construction is as follows. As input, we take a finite CW-complex, $X$, a smooth, compact, ${\rm spin^c}$ manifold, $P$, and unital $C^*$-algebra, $A$. A cycle, with respect to this input, is a triple, $((Q, \beta Q), (E_Q, E_{\beta Q}), f)$ where
1. $(Q, \beta Q)$ is a smooth, compact, ${\rm spin^c}$ $P$-manifold;
2. $(E_Q, E_{\beta Q})$ and $f: (Q,\beta Q) \rightarrow X$ are respectively the natural “$P$-version" of an $A$-vector bundle and continuous function; the precise definitions of these objects are given in Definitions \[PCtsMapX\], \[PBundle\], and \[cycWithBun\].
From these cycles, we construct an abelian group (see Definition \[GeoGroup\]); it is denoted by $K_*(X;P;A)$. In the case of $P=k$-points and $A={\mathbb{C}}$, we obtain the geometric realization of $K$-homology with coefficients in ${{\mathbb{Z}}/k{\mathbb{Z}}}$ constructed in [@Dee1; @Dee2]. Here, we consider the case of more general “singularities" (i.e., choices of $P$). However, we do not reach full generality since for the proofs of the key properties of $K_*(X;P;A)$, we require $P$ to have a trivial stable normal bundle. This condition implies that there is a “good" notion of normal bundle for $P$-manifolds.
The main results in regards to fundamental properties of the geometric group are the generalized Bockstein sequence, Theorem \[BockTypeSeq\], and a “uniqueness" theorem, Theorem \[uniquenessThm\]. These theorems should be compared respectively with Theorem 1.6 and Proposition 1.14 in [@Rud Chapter VIII]. Next, in Section 4, we define the assembly map. When the group is torsion-free, this map is the natural analog of the Baum-Connes assembly map. In particular, for a torsion-free group and non-singular manifolds, the equivalence of the definition of assembly considered here and the standard definition is well-known; a detailed proof is given in [@land]. It follows from the generalized Bockstein sequence and Five Lemma that the assembly map for $P$-manifolds is an isomorphism whenever the Baum-Connes assembly map is an isomorphism (see Theorem \[BCForPMfld\]).
As we mentioned above, the indices we construct are elements in $K_*(pt;P;A)$, which is defined via Baum-Douglas type cycles. Based on Equation \[isoInCaseOfPt\], one would like to identify this group with a more well-known group – ideally, via an explicit map which is analogous to the higher index map discussed in Equation \[isoInCaseOfPt\]. We give the full details of this construction in two examples. These are the case when $P$ is $k$-points and the circle. In these cases, we compute $K_*(pt;P;A)$ and relate the geometrically defined cycles with an analytic index. For $k$-points, the resulting index theorem (see Theorem \[higFMThm\]) is a higher version of the Freed-Melrose index theorem; for the circle, the resulting index theorem is related to work of Rosenberg, [@Ros]. Based on the “uniqueness" theorem (i.e., Theorem \[uniquenessThm\]) mentioned above, the examples of $k$-points, the circle, and the $2$-sphere lead to the determination of $K_*(pt;P;A)$ for any choice of $P$ with trivial stable normal bundle and well-defined dimension. The final section of the paper contains a brief discussion of some generalizations.
A summary of the notation introduced to this point is as follows: $X$ is a finite CW-complex, $K^*(X)$ is the K-theory of $X$, $K_*(X)$ is the geometric K-homology of $X$, $A$ is a unital $C^*$-algebra, $K_*(A)$ is the K-theory of $A$, $K_*(X;A)$ is the geometric K-homology of $X$ with coefficients in $A$, $K^*(X;A):=K_*(C(X)\otimes A)$, and $P$ is a smooth, compact, ${\rm spin^c}$-manifold with trivial stable normal bundle. It is convenient to assume that $P$ has a well-defined dimension mod two; that is, we assume the dimensions of the connected components of $P$ are all the same dimension modulo two. An $A$-vector bundle over $X$ is a locally trivial bundle over $X$ with finitely generated, projective Hilbert $A$-modules as fibers. Also, $\Gamma$ is a finitely generated discrete group, $B\Gamma$ is the classifying space of $\Gamma$ (for simplicity, we assume it is a finite CW-complex), $C^*(\Gamma)$ is the reduced group $C^*$-algebra of $\Gamma$. Finally, if $M$ is a smooth, compact, ${\rm spin^c}$ manifold and $E$ is a smooth $A$-vector bundle over it, then $D_{(M,E)}$ denotes the Dirac operator associated to the ${\rm spin^c}$-structure of $M$ twisted by the $A$-vector bundle $E$. This operator has a well-defined higher index, which we denote by ${\rm ind_{AS}}(D_{(M,E)})$ or ${\rm ind}(D_{(M,E)})$; this index is an element of $ K_{{\rm dim}(M)}(A)$.
Preliminaries
=============
Manifolds with Baas-Sullivan singularities
------------------------------------------
(see [@Baas])\
A closed, smooth, ${\rm spin^c}$-manifold with Baas-Sullivan singularities modeled on $P$ (i.e., a closed, smooth, ${\rm spin^c}$ $P$-manifold) is $(Q,\beta Q, \varphi)$ where
1. $Q$, a smooth, compact, ${\rm spin^c}$manifold with boundary;
2. $\beta Q$ a smooth, compact, ${\rm spin^c}$-manifold;
3. $\varphi: \beta Q \times P \rightarrow \partial Q$ a ${\rm spin^c}$-preserving diffeomorphism.
We denote such an object as $(Q,\beta Q, \varphi)$ or just as $(Q,\beta Q)$. \[Pmfld\]
There is a similar definition in the case of spin-manifolds and the constructions we give in this paper in the ${\rm spin^c}$-manifold case generalize in a natural way to the case of spin-manifolds and $KO$-theory. The choice to surpress the diffeomorphism, $\varphi$, from the notation causes an abuse of notation: often but not always, we will refer to the projection map $\beta Q \times P \rightarrow \beta Q$ when we should refer to the map $\partial Q \cong \beta Q \times P \rightarrow \beta Q$.
Two prototypical examples are $P= k$-points, in which case a $P$-manifold is exactly a ${{\mathbb{Z}}/k{\mathbb{Z}}}$-manifold, see any of [@Dee1; @Fre; @MS]) and $P=S^1$, see [@Ros]. In $KO$-theory another low-dimensional example is $P=S^1$ with its non-bounding spin-structure, again see [@Ros].
\[PmfldWithBound\] A ${\rm spin^c}$ $P$-manifold with boundary is a pair of smooth, compact, ${\rm spin^c}$-manifolds with boundary $\bar{Q}$ and, $\beta \bar{Q}$ and a smooth embedding which respects the ${\rm spin^c}$-structures $\bar{\varphi}:\beta \bar{Q} \times P \hookrightarrow \partial \bar{Q}$. The boundary of $(\bar{Q}, \beta \bar{Q}, \bar{\varphi})$ is given by $$(\partial \bar{Q} - {\rm int}(\varphi(\beta \bar{Q})), \partial \beta \bar{Q}, \bar{\varphi}|_{\partial \beta Q \times P}).$$ Suppressing the embedding from the notation, we have that $(Q, \beta Q)$ is the boundary of $(\bar{Q}, \beta \bar{Q})$ if $$\partial \beta \bar{Q} = \beta Q \hbox{ and }\partial \bar{Q} = Q \cup_{\partial Q} \beta \bar{Q} \times P.$$ As in the definition of a closed $P$-manifold, we will usually suppress the embedding (i.e., $\bar{\varphi}$) from the notation.
\[PCtsMapX\] A continuous map from a $P$-manifold, $(Q, \beta Q, \varphi)$, to a locally compact Hausdorff space (e.g., a finite CW-complex), $X$, is a pair of continuous maps, $f: Q \rightarrow X$ and $f_{\beta Q}: \beta Q \rightarrow X$, such that $$f|_{\partial Q} = f_{\beta Q} \circ \pi \circ \varphi$$ where $\pi: \beta Q \times P \rightarrow \beta Q$ is the projection map. We will often denote $(f, f_{\beta Q})$ simply as $f$.
A continuous map from one $P$-manifold, $(Q, \beta Q, \varphi)$, to another, $(\tilde{Q}, \beta \tilde{Q}, \tilde{\varphi})$, is a pair of continuous maps, $f: Q \rightarrow \tilde{Q}$ and $f_{\beta Q}: \beta Q \rightarrow \beta \tilde{Q}$, such that $$\tilde{\pi} \circ \tilde{\varphi} \circ f|_{\partial Q} = f_{\beta Q} \circ \pi \circ \varphi$$ where $\pi: \beta Q \times P \rightarrow \beta Q$ and $\tilde{\pi}: \beta \tilde{Q} \times P \rightarrow \beta \tilde{Q}$ are the natural projection maps.
Bundles and $K$-theory
----------------------
\[PBundle\] A $P$-vector bundle, $(V_Q, V_{\beta Q}, \varphi)$ over a $P$-manifold, $(Q, \beta Q, \varphi)$, is a triple, $(V_{Q}, V_{\beta Q}, \alpha)$, where
1. $V_Q$ is a vector bundle over $Q$;
2. $V_{\beta Q}$ is a vector bundle over $\beta Q$;
3. $\alpha: V_Q|_{\partial Q} \rightarrow \pi^*(V_{\beta Q})$ is a bundle isomorphism which is a lift of $\varphi$; we note that $\pi$ is the projection map $\beta Q \times P \rightarrow \beta Q$.
We will usually denote such a triple simply as $(V_Q, V_{\beta Q})$ and refer to such objects as $P$-bundles.
Many definitons from the theory of vector bundles generalize to $P$-vector bundle: in particular, two $P$-vector bundles, $(V_Q, V_{\beta Q}, \alpha)$ and $(V^{\prime}_Q, V^{\prime}_{\beta Q}, \alpha^{\prime})$, over $(Q,\beta Q)$ are isomorphic if there exists vector bundle isomorphisms $\psi_Q : V_Q \rightarrow V^{\prime}_Q$ and $\psi_{\beta Q}: V_{\beta Q} \rightarrow V^{\prime}_{\beta Q}$ such the following diagram commutes:
$\begin{CD}
(V_Q)|_{\partial Q} @>(\psi_Q)|_{\partial V}>> (V^{\prime}_Q)|_{\partial Q} \\
@V\alpha VV @VV\alpha^{\prime} V \\
\pi^*(V_{\beta Q}) @>\tilde{\psi}_{\beta Q}>> \pi^*(V^{\prime}_{\beta Q})
\end{CD}$
where $\tilde{\psi}_{\beta Q}$ is the lift of the isomorphism (from $V_{\beta Q}$ to $V^{\prime}_{\beta Q}$) to an isomorhism from $\pi^*(V_{\beta Q})$ to $\pi^*(V^{\prime}_{\beta Q})$. There are similar definitions in the context of $P$-bundles for other standard definitions in vector bundle theory (e.g., pullback bundle, complementary bundle, stably isomorphic, etc).
Every $P$-bundle has a complementary $P$-bundle. Moreover, any two complementary $P$-bundles are stably isomorphic. \[compBundle\]
The result is certainly known; we give a proof in the case of complex $P$-bundles. Let $(V_Q, V_{\beta Q})$ be a $P$-bundle and $\pi: \beta Q \times P \rightarrow \beta Q$ be the projection map. Then $V_Q$ has a complementary vector bundle (say $W$); that is, for some $n\in {\mathbb{Z}}$, we have $V_Q \oplus W \cong Q \times {\mathbb{C}}^n$. In particular, $V_Q |_{\partial Q} \oplus W|_{\partial Q} \cong \partial Q \times {\mathbb{C}}^n$. The definition of $P$-bundle implies that $$\pi^*(V_{\beta Q}) \oplus W|_{\partial Q} \cong \pi^*(\beta Q \times {\mathbb{C}}^n).$$ Let $\tilde{W}$ be a vector bundle over $\beta Q$ such that $V_{\beta Q} \oplus \tilde{W} \cong \beta Q \times {\mathbb{C}}^n$. Then $$W|_{\partial Q} \oplus \partial Q \times {\mathbb{C}}^n \cong \pi^*(\tilde{W} \oplus \beta Q \times {\mathbb{C}}^n).$$ In other words, $V^c=W \oplus Q \times {\mathbb{C}}^n$ can be given the structure of a $P$-vector bundle; it is also clear that $(V^c,\tilde{W} \oplus \beta Q \times {\mathbb{C}}^n)$ is a complementary $P$-bundle for $(V_Q, V_{\beta Q})$.
Finally, let $(W_Q, W_{\beta Q})$ and $(\tilde{W}_Q, \tilde{W}_{\beta Q})$ be two complementary bundle for $(V_Q, V_{\beta Q})$. Then, for some integers $n_1$ and $n_2$, we have $$(W_Q, W_{\beta Q}) \oplus Q \times {\mathbb{C}}^{n_1} \cong (W_Q, W_{\beta Q}) \oplus (V_Q, V_{\beta Q}) \oplus (\tilde{W}_Q, \tilde{W}_{\beta Q}) \cong Q\times {\mathbb{C}}^{n_2}\oplus (\tilde{W}_Q, \tilde{W}_{\beta Q}).$$
If $(Q, \beta Q)$ is a $P$-manifold, then the tangent bundle of $Q$ as a manifold with boundary is [*not*]{} a $P$-bundle over $(Q, \beta Q)$. In certain cases, there is a way of working around this issue. In increasing generality, we have the following examples:
1. If $(Q,\beta Q)$ is a ${{\mathbb{Z}}/k{\mathbb{Z}}}$-manifold, then $(TQ, T(\beta Q \times (0,1])$ can be given the structure of a ${{\mathbb{Z}}/k{\mathbb{Z}}}$-bundle.
2. If $P$ is a manifold with trivial tangent bundle, then there exists $n\in {\mathbb{N}}$ such that $(TQ, T(\beta Q) \oplus \beta Q \times {\mathbb{R}}^n)$ can be given the structure of a $P$-bundle.
3. If $P$ is a manifold with trivial stable normal bundle, then there exists $n_1$ and $n_2$ such that $(TQ \oplus Q \times {\mathbb{R}}^{n_1}, T(\beta Q) \oplus \beta Q\times {\mathbb{R}}^{n_2})$ can be given the structure of a $P$-bundle.
Let $P$ be a manifold with trivial stable normal bundle and $(Q, \beta Q)$ be a $P$-manifold. Then, a $P$-normal bundle for $(Q, \beta Q)$ is a complementary $P$-bundle to $(TQ \oplus Q \times {\mathbb{R}}^{n_1}, T(\beta Q) \oplus \beta Q\times {\mathbb{R}}^{n_2})$ where $n_1$ and $n_2$ are as in the previous example. In particular, a normal bundle for a manifold $M$ is a complementary vector bundle of its tangent bundle.
\[norBunCom\] Proposition \[compBundle\] implies if $P$ has trivial stable normal bundle, then that every $P$-manifold has a $P$-normal bundle and any two $P$-normal bundles are stably isomorphic.
Let $A$ be a unital $C^*$-algebra. An $A$-vector bundle over $X$ is a locally trivial bundle with finitely generated, projective Hilbert $A$-modules as fibers. An $A$-vector bundle (or more correctly an “($A$, $P$)-vector bundle") over a $P$-manifold is defined in the same way as in a $P$-vector bundle only one replaces the vector bundles in that definition with $A$-vector bundles.
\[MishBunEx\] Let $\Gamma$ be a discrete group, $B \Gamma$ be its classifying space, $(Q, \beta Q)$ be a $P$-manifold, and $f: (Q, \beta Q) \rightarrow B\Gamma$ be a continuous map. The Mishchenko bundle is given by $\mathcal{L}_{B\Gamma} := E\Gamma \times_{\Gamma} C^*(\Gamma)$. Then, $$V_{Q} = f^*(\mathcal{L}_{B \Gamma}) \hbox{ and } V_{\beta Q} = (f_{\beta Q})^*(\mathcal{L}_{B\Gamma})$$ can be given the structure of a $C^*(\Gamma)$-bundle over $(Q,\beta Q)$. In fact, the pullback along any continuous function $f: (Q,\beta Q) \rightarrow X$ of any $A$-vector bundle over $X$ is an $A$-vector bundle over $(Q, \beta Q)$.
Let $K^0(Q,\beta Q; P;A)$ be the Grothendieck group of the semi-group of isomorphism classes of ($P$,$A$)-vector bundles over $(Q,\beta Q)$.
The group $K^0(Q,\beta Q; P;A)$ shares many properties with standard K-theory. For example, it has similar functorial properties and there is a Thom isomorphism.
\[KthProp\] The following sequence is exact in the middle: $$K^0(Q,\beta Q; P; A) \rightarrow K^0(Q;A) \oplus K^0(\beta Q;A) \rightarrow K^0(\beta Q\times P; A)$$ where the maps are defined at the level of bundle data to $K$-classes as follows:
1. $(V_Q,V_{\beta Q}) \in K^0(Q,\beta Q;P;A) \mapsto ([V_Q],[V_{\beta Q}])\in K^0(Q;A) \oplus K^0(\beta Q;A)$;
2. $ (E_1,E_2)\in K^0(Q;A) \oplus K^0(\beta Q;A) \mapsto [E_1 |_{\partial Q}] - [\pi^*(E_2)]$.
where $\pi: \beta Q \times P \rightarrow \beta Q$ is the projection map.
The proof is standard; we give the proof for $P$-vector bundles, but the details generalize to the case of $A$-vector bundles. The isomorphism in the definition of a $P$-vector bundle implies that the composition of the two maps is zero. Thus, to show exactness, we need to show that if $$([E]-[E^{\prime}], [F] - [F^{\prime}]) \in K^0(Q) \oplus K^0(\beta Q) \mapsto 0 \in K^0(\beta Q \times P),$$ then there exists element in $K^0(Q,\beta Q;P)$ which maps to $([E]-[E^{\prime}], [F] - [F^{\prime}])$.
We can assume that $E^{\prime}$ and $F^{\prime}$ are trivial. Using this fact, the assumption that $([E]-[E^{\prime}], [F] - [F^{\prime}]) \mapsto 0$, and basic bundle theory, there exists trivial bundles over $Q$, $\varepsilon_Q$ and $\varepsilon^{\prime}_Q$, and bundle isomorphism $$\alpha: (E \oplus \varepsilon_Q)|_{\partial Q} \cong \pi^*(F)\oplus (\varepsilon^{\prime}_Q)|_{\partial Q}.$$ Let $\varepsilon_{\beta Q}$ respectively, $\varepsilon^{\prime}_{\beta Q}$ be the trivial vector bundle over $\beta Q$ of the same rank as $\varepsilon_Q$ respectively, $\varepsilon^{\prime}_Q$. One then checks that the desired class (i.e., the required preimage) is given by $$[( E \oplus \varepsilon_Q, F\oplus \varepsilon^{\prime}_{\beta Q}, \alpha)] - [(\varepsilon_Q \oplus \varepsilon^{\prime}_Q, \varepsilon_{\beta Q} \oplus \varepsilon^{\prime}_{\beta Q}, id)].$$
Given $\xi \in K^0(Q, \beta Q;P, A)$, we denote by $\xi_{Q}$ or $\xi|_{Q}$ (resp. $\xi_{\beta Q}$ or $\xi|_{\beta Q}$) the image of $\xi$ under the map to $K^0(Q;A)$ (resp. $K^0(\beta Q;A)$); we use similar notation for $K$-theory classes of $P$-manifolds with boundary. A number of proofs require a $K$-theory group that is rather similar to $K^0(Q, \beta Q;P, A)$:
Let $(Q, \beta Q)$ be $P$-manifold and $W$ be a manifold with boundary such that $\partial W = \beta Q$. Furthermore, let $$K^0(Q \cup_{\partial Q} W \times P, W;A)$$ be the Grothendieck group of isomorphism classes of triples of the form $$(E_{Q\cup_{\partial Q} W\times P}, E_{W}, \varphi)$$ where
1. $E_{Q\cup_{\partial Q} W\times P}$ is an $A$-vector bundle over $Q\cup_{\partial Q} W\times P$;
2. $E_{W}$ is an $A$-vector bundle over $W$;
3. $\varphi: E_{Q\cup_{\partial Q} W\times P}|_{W\times P} \rightarrow \pi^*(E_W)$ is a bundle isomorphism. We note that $\pi$ is the projection map $W\times P \rightarrow P$.
The proof of the next proposition is similar to the proof of Proposition \[KthProp\] and is omitted.
\[gluePmfldWithbound\] Using the notation of the previous definition, we have that the following sequence is exact in the middle: $$K^0(Q \cup_{\partial Q} W \times P, W;A) \rightarrow K^0(Q, \beta Q;A)\oplus K^0(W;A) \rightarrow K^0(\beta Q;A),$$ where the maps are the natural maps on K-theory induced from $$\begin{aligned}
(E_{Q\cup_{\partial Q}W \times P}, E_W) & \mapsto & ( (E_{Q\cup_{\partial Q}W \times P})|_Q, E_W|_{\partial W=\beta Q}), E_W) \\
((F_Q, F_{\beta Q}), V_W) & \mapsto & [V_{W|_{\partial W=\beta Q}}]-[F_{\beta Q}]\end{aligned}$$
Geometric models {#geoModSec}
================
\[cycWithBun\] A geometric cycle with bundle data, over $X$ with respect to $P$ and $A$ is a triple $((Q,\beta Q), (E_Q, E_{\beta Q}), f)$ where
1. $(Q,\beta Q)$ is a compact, smooth, ${\rm spin^c}$ $P$-manifold;
2. $(E_Q, E_{\beta Q})$ is a smooth $A$-vector bundle over $(Q, \beta Q)$;
3. $f$ is a continuous map from $(Q,\beta Q)$ to $X$.
\[cycWithKth\] A geometric cycle with $K$-theory data, over $X$ with respect to $P$ and $A$ is a triple $((Q,\beta Q), \xi , f)$ where
1. $(Q,\beta Q)$ is a compact, smooth, ${\rm spin^c}$ $P$-manifold;
2. $\xi \in K^0(Q,\beta Q; P; A)$;
3. $f$ is a continuous map from $(Q,\beta Q)$ to $X$.
As in the case of the Baum-Douglas model for $K$-homology, $(Q,\beta Q)$ need not be connected and there is a natural ${\mathbb{Z}}/2$-grading on cycles defined using the dimensions of the connected components of $(Q,\beta Q)$ modulo two. Furthermore, there is a natural notion of isomorphism for cycles; when we refer to a “cycle", we mean “an isomorphism class of a cycle". Addition of cycles is defined using disjoint union; we denote this operation by $\dot{\cup}$.
\[borForKthGrp\] A bordism or a cycle with boundary with respect to $X$, $P$, and $A$ is $((\bar{Q}, \beta \bar{Q}), \bar{\xi}, \bar{f})$ where
1. $(\bar{Q},\beta \bar{Q})$ is a compact, smooth, ${\rm spin^c}$ $P$-manifold with boundary;
2. $\bar{\xi} \in K^0(\bar{Q}, \beta \bar{Q}; P;A)$;
3. $\bar{f}: (\bar{Q},\beta \bar{Q}) \rightarrow X$ is a continuous map.
The boundary of a bordism, $((\bar{Q}, \beta \bar{Q}), \bar{\xi}, \bar{f})$, is given by $$((\partial \bar{Q} - {\rm int}(\beta \bar{Q}), \partial \beta \bar{Q}), \bar{\xi}|_{\partial \bar{Q} - {\rm int}(\beta \bar{Q})}, \bar{f}|_{\partial \bar{Q} - {\rm int}(\beta \bar{Q})}).$$ By construction, it is a cycle in the sense of Definition \[cycWithKth\].
Let $((Q,\beta Q), \xi , f)$ be a cycle and $(V_Q,V_{\beta Q})$ a ${\rm spin^c}$ $P$-vector bundle with even dimensional fibers over $(Q,\beta Q)$. We define the vector bundle modification of $((Q,\beta Q), \xi , f)$ by $(V_Q,V_{\beta Q})$ to be $$( (S(V_Q\oplus {\bf 1}),S(V_{\beta Q}\oplus {\bf 1})), \pi^*(\xi) \otimes \beta, f\circ \pi)$$ where
1. ${\bf 1}$ denotes the trivial real line bundle;
2. $\pi$ the vector bundle projection;
3. $\beta$ is the Bott class (see for example [@Rav Section 2.5]).
We denote the cycle so obtained by $((Q,\beta Q), \xi , f)^{(V_Q,V_{\beta Q})}$; the reader should verify that the resulting triple is a cycle (in the sense of Definiton \[cycWithKth\]). We will also find it useful to denote the bundle we are modifying by simply as $V_Q$; however, we emphasize that one can only vector bundle modify by $P$-vector bundles, which are ${\rm spin^c}$ and have even dimensional fibers.
\[GeoGroup\] Let $K_*(X;P;A):=\{(Q,\beta Q), \xi , f)\}/\sim$ where $\sim$ is the equivalence relation generated by bordism and vector bundle modification. We denote the bordism relation by $\sim_{bor}$.
The disjoint union operation gives $K_*(X;P;A)$ the structure of an abelian group. In a similar way, one can define $K_*(X;P;A)$ using cycles with “bundle data" (see Definition \[cycWithBun\]). To do so, one needs to use slightly different definitions of bordism and vector bundle modification and add a direct sum/disjoint union relation. The process is completely analogous to the difference between the original Baum-Douglas model [@BD] and the model discussed in [@Rav]. We will use the “bundle data" model for the definition of the assembly map and the detailed discussion of the case $P=k$-points and $P=S^1$.
We have the following maps:
1. $ \Phi_P : K_*(X;A) \rightarrow K_{*+{\rm dim}(P)}(X;A) $ defined at the level of cycles via $$\Phi(M,\xi,f)=(M\times P, \pi^*(\xi) , f\circ \pi)$$ where $\pi$ denotes the projection map from $M\times P$ to $M$.
2. $r_P : K_*(X;A) \rightarrow K_*(X;P;A) $ defined at the level of cycles via $$r(M, \xi, f) = ((M,\emptyset), \xi, f).$$
3. $\delta_P : K_*(X;P;A) \rightarrow K_{*-{\rm dim}(P)-1} (X;A)$ defined at the level of cycles via $$\delta((Q,\beta Q), \xi, f) = (\beta Q, \xi|_{\beta Q}, f|_{\beta Q} ).$$
\[norBordism\] Let $((Q,\beta Q), \xi, f)$ and $((Q^{\prime}, \beta Q^{\prime}), \xi^{\prime}, f^{\prime})$ be two cycles in $K_*(X;P;A)$. Then, these two cycles are normally bordant if there exists normal $P$-bundles $N$ and $N^{\prime}$ such that $((Q,\beta Q), \xi, f)^N \sim_{bor} ((Q^{\prime}, \beta Q^{\prime}), \xi^{\prime}, f^{\prime})^{N^{\prime}}$. If this is the case, then we write $$((Q,\beta Q), \xi, f) \sim_{nor} ((Q^{\prime}, \beta Q^{\prime}), \xi^{\prime}, f^{\prime}).$$ We call this relation normal bordism.
The next two lemmas are similar, in both statement and proof, to lemmas in [@Rav Sections 4.4 and 4.5] (also see [@Dee1 Section 2.2.1], [@DeeGeoRelKhom Section 4.2], and [@DeeGofSur Section 3.1]). The proofs are omitted.
\[directSumVBM\] Let $((Q,\beta Q), \xi, f)$ be a cycle and $V_1$ and $V_2$ be even rank ${\rm spin^c}$ $P$-bundles over $(Q, \beta Q)$. Then $$((Q,\beta Q), \xi, f)^{(V_1\oplus V_2)} \sim_{bor} (((Q,\beta Q), \xi, f)^{V_1})^{(p^*(V_2))}$$ where $p$ is the projection map from $S(V_1\oplus {\bf 1})$ to $Q$.
Normal bordism is equal to the equivalence relation generated from bordism and vector bundle modification on cycles with $K$-theory data.
A cycle $((Q,\beta Q), \xi, f)$ is trivial in $K_*(X;P;A)$ if and only if there exists normal $P$-bundle, $N$, such $((Q,\beta Q), \xi, f)^N$ is a boundary.
\[BockTypeSeq\] Let $X$ be a finite CW-complex, $A$ be a unital $C^*$-algebra, and $P$ be a smooth compact ${\rm spin^c}$-manifold that has trivial stable normal bundle and well-defined dimension mod two. Then, the following sequence is exact
$\begin{CD}
K_0(X;A) @>\Phi>> K_{{\rm dim}(P)}(X;A) @>r>> K_{{\rm dim}(P)}(X;P;A) \\
@A\delta AA @. @V\delta VV \\
K_{{\rm dim}(P)+1}(X;P;A) @<r<< K_{{\rm dim}(P)+1}(X;A) @<\Phi<< K_1(X;A)
\end{CD}$
where the maps were defined above, just before Definition \[norBordism\].
Before beginning the proof, we note that if $P$ is $k$-points and $A={\mathbb{C}}$, then this theorem is exactly [@Dee1 Theorem 2.20]. The proof is similar to the proof of that theorem; in particular, the notion of normal bordism plays a key role.
We begin by noting that the proof that the maps are well-defined follows from the compatiblity of the relations used to defined the various groups. In addition, the notion of bordism in the various groups implies that the composition of successive maps is zero.
These observations reduce the proof to showing $${\rm ker}(\delta)\subseteq {\rm im}(r), {\rm ker}(r)\subseteq {\rm im}(\Phi), {\rm ker}(\Phi)\subseteq {\rm im}(\delta).$$ Since the argument is very close to the proof of Theorem 2.20 in [@Dee1], we only give a detailed proof that ${\rm ker}(\delta)\subseteq {\rm im}(r)$.
Suppose that $((Q, \beta Q), \xi, f)$ is in ${\rm ker}(\delta)$. We begin by proving that $((Q, \beta Q), \xi, f)$ is equivalent to a cycle in $K_*(X;P;A)$ whose image under $\delta$ is a boundary rather than just trivial. Remark \[norBunCom\] and [@Rav Corollary 4.5.16] imply that there exists normal $P$-bundle $(N_Q, N_{\beta Q})$ (for $(Q,\beta Q)$) such that $(\beta Q, \xi|_{\beta Q}, f|_{\beta Q})^{N_{\beta Q}}$ is a boundary and $((Q, \beta Q), \xi, f)^{(N_Q, N_{\beta Q})}$ is equivalent via the vector bundle modification relation to the original cycle.
Hence, without loss of generality, we can assume there exists bordism with respect to $K_{*-1}(X;A)$, $(W, \eta, g)$, such that $(\beta Q, \xi|_{\beta Q}, f|_{\beta Q}) = \partial (W, \eta, g)$. We form the cycle in $K_*(X;A)$: $$(Q \cup_{\partial Q} (W \times P), \xi_{Q} \cup \pi^*(\eta), f \cup g)$$ where $\pi: \beta Q \times P \rightarrow \beta Q$ is the projection map and the $K$-theory class is obtained as follows. Let $\nu$ to be any preimage of $(\xi_Q, \eta)$ in the sequence which, by Proposition \[gluePmfldWithbound\], is exact in the middle: $$K^0(Q \cup_{\partial Q} W \times P, W;A) \rightarrow K^0(Q, \beta Q;A)\oplus K^0(W;A) \rightarrow K^0(\beta Q;A).$$ Then $ \xi_{Q} \cup \pi^*(\eta)$ is defined to be the image of $\nu$ under the natural map $K^0(Q \cup_{\partial Q} W, W;A) \rightarrow K^0(Q\cup_{\partial Q} W;A)$; this class in not unique, but any choice will satisfy the properties required in the rest of the proof.
The proof will be complete upon showing that $r(Q \cup_{\partial Q} (W \times P), \xi_{Q} \cup \pi^*(\eta), f \cup g)$ is equivalent to $((Q, \beta Q), \xi, f)$. This follows from the following bordism with respect to the group $K_*(X;P;A)$: $$((Q \cup_{\partial Q} (W \times P) \times [0,1], W\times \{0\}), \bar{\xi}, (f \cup g)\circ \tilde{\pi})$$ where $\tilde{\pi}$ is the projection map $(Q \cup_{\partial Q} (W \times P)) \times [0,1] \rightarrow (Q \cup_{\partial Q} (W \times P))$ and $\bar{\xi}$ is a $K$-theory class such that $$\bar{\xi}|_{Q \cup_{\partial Q} (W \times P)\times \{1\}} = \xi_{Q} \cup \pi^*(\eta) \hbox{ and } \bar{\xi}|_{Q \cup_{\partial Q} (W \times P)\times \{0\}}= \nu.$$ Such a class can be constructed by more or less pulling back the class $\nu$ discussed above.
\[uniquenessThm\] Let $X$ be a finite CW-complex, $A$ be a unital $C^*$-algebra, and $P$ and $P^{\prime}$ be smooth, compact, ${\rm spin^c}$-manifolds of dimensions $n$ and $n'$ respectively. Assume also that both $P$ and $P^{\prime}$ have trivial stable normal bundle. If $(P, [P\times {\mathbb{C}}])\sim (P^{\prime}, [P^{\prime}\times {\mathbb{C}}])$ as elements in $K_*(pt)$, then $K_*(X;P;A) \cong K_*(X;P^{\prime};A)$. Moreover, the isomorphism is natural with respect to both $X$ and $A$.
A number of projection maps are required in the proof; we use the following notation: let $M_1$ and $M_2$ be two manifolds possibly with boundary. Then, for $i=1$ and $2$, we let $\pi^{M_1\times M_2}_{M_i} : M_1 \times M_2 \rightarrow M_i$ denote the projection map.
The equivalence relation on cycles in $K_*(pt)$ is equivalent to normal bordism [@Rav Corollary 4.5.16]. Hence, there exists normal bundles, $N$ and $N^{\prime}$, over respectively $P$ and $P^{\prime}$, such that $$\label{norBorInUniProof}
(P, [P\times {\mathbb{C}}])^{N} \sim_{bor} (P^{\prime}, [P^{\prime}\times {\mathbb{C}}])^{N^{\prime}}.$$ Moreover, since both $P$ and $P^{\prime}$ have trivial stable normal bundle and normal bundles are stably isomorphic, we can and will assume that both $N$ and $N^{\prime}$ are trivial bundles. However, since $P$ and $P^{\prime}$ are not necessarily connected, the ranks of these trivial bundles may vary over the connected components of $P$ and $P^{\prime}$.
Our first goal is to show that we can take $N$ and $N^{\prime}$ each with constant rank. Decompose $P$ and $P^{\prime}$ as follows: $$P= P_1 \dot{\cup} \ldots \dot{\cup} P_k \hbox{ and }P^{\prime} = P^{\prime}_1 \dot{\cup} \ldots \dot{\cup} P^{\prime}_{k^{\prime}}$$ where
1. ${\rm rank}(N|_{P_i})$ is constant for each $1 \le i \le k$;
2. ${\rm rank}(N^{\prime}|_{P^{\prime}_j})$ is constant for each $1 \le j \le k^{\prime}$;
3. ${\rm rank}(N_1) > {\rm rank}(N_2) > \ldots >{\rm rank}(N_k)$ where $N_i:=N|_{P_i}$;
4. ${\rm rank}(N^{\prime}_1) > {\rm rank}(N^{\prime}_2) > \ldots >{\rm rank}(N^{\prime}_{k^{\prime}})$ where $N^{\prime}_j:=N^{\prime}|_{P^{\prime}_j}$.
The existence of the bordism in Equation and dimensional reasons imply that $k=k^{\prime}$ and that, for each $i=1, \ldots k$, $${\rm rank}(N_1) - {\rm rank}(N_i) = {\rm rank}(N^{\prime}_1) - {\rm rank}(N_i).$$ Furthermore, if $W$ is the manifold with boundary in a bordism from Equation , then $$W= W_1 \dot{\cup} \ldots \dot{\cup} W_k$$ where, for each $i=1, \ldots k$, $W_i$ is a bordism between $P_i^{N_i}$ and $(P^{\prime}_i)^{N^{\prime}_i}$.
Let $V$ be the vector bundle over $P$ which is the trivial bundle of rank equal to ${\rm rank}(N_1)-{\rm rank}(N_i)$ on each $P_i$ where $i=1, \ldots, k$. Likewise, let $V^{\prime}$ be the vector bundle over $P^{\prime}$ which is the trivial bundle of rank equal to $${\rm rank}(N_1)-{\rm rank}(N_i)={\rm rank}(N^{\prime}_1)-{\rm rank}(N^{\prime}_i)$$ on each $P^{\prime}_j$ where $j=1, \ldots, k$. Using the fact that trivial bundles of constant rank extend across the bordisms $W_1, \ldots W_k$ and [@Rav Lemma 4.4.3], we obtain $$\begin{aligned}
(P, [P\times {\mathbb{C}}])^{N \oplus V} & \sim_{bor} & ((P, [P\times {\mathbb{C}}])^N)^{\pi(V)} \\
& \sim_{bor} & ((P^{\prime}, [P^{\prime}\times {\mathbb{C}})^N)^{(\pi^{\prime})^*(V^{\prime})} \\
& \sim_{bor} & (P^{\prime}, [P^{\prime}\times {\mathbb{C}}])^{N^{\prime} \oplus V^{\prime}} \end{aligned}$$ where $\pi$ and $\pi^{\prime}$ are the projection maps in the vector bundle modifications of $P$ by $N$ and $P^{\prime}$ by $N^{\prime}$ respectively. Finally, by construction, the ranks of $N \oplus V$ and $N^{\prime} \oplus V^{\prime}$ are each constant. Thus, with loss of generality, we can assume that $$(P, [P\times {\mathbb{C}}])^{N} \sim_{bor} (P^{\prime}, [P^{\prime}\times {\mathbb{C}}])^{N^{\prime}}$$ where $N$ and $N^{\prime}$ are trivial bundles with constant rank.
Let $2n$ (resp. $2n^{\prime}$) be the rank of $N$ (resp. $N^{\prime}$) and $\beta_{2n}$ (resp. $\beta_{2n^{\prime}}$) be the Bott class on $S^{2n}$ (resp. $S^{2n^{\prime}})$; the construction of the Bott class can be found in [@Rav Section 2.5]. Using this notation and the definition of vector bundle modification, we have that $$(P \times S^{2n}, (\pi^{P\times S^{2n}}_{S^{2n}})^*(\beta_{2n})) \sim_{bor} (P^{\prime}\times S^{2n^{\prime}}, (\pi^{P^{\prime}\times S^{2n^{\prime}}}_{S^{2n^{\prime}}})^*(\beta_{2n^{\prime}}))$$ where, following the notation introduced at the start of the proof, $$\pi^{P\times S^{2n}}_{S^{2n}}: P \times S^{2n} \rightarrow S^{2n} \hbox{ and } \pi^{P^{\prime}\times S^{2n^{\prime}}}_{S^{2n^{\prime}}}: P^{\prime}\times S^{2n^{\prime}} \rightarrow S^{2n^{\prime}}$$ are the projection maps. Let $(W, \nu)$ denote a fixed choice of bordism between these two cycles; we note that $W$ is a smooth, compact ${\rm spin^c}$-manifold with boundary and $\nu \in K^0(W)$. Let $\partial_0 W$ (resp. $\partial_1 W$) denote the component of $\partial W$ diffeomorphic to $P\times S^{2n}$ (resp. $P^{\prime}\times S^{2n^{\prime}}$). Moreover, we fix vector bundles $F$ and $\tilde{F}$ over $W$, $F_{S^{2n}}$ and $\tilde{F}_{S^{2n}}$ over $S^{2n}$, and $F^{\prime}_{S^{2n^{\prime}}}$ and $\tilde{F}^{\prime}_{S^{2n^{\prime}}}$ over $S^{2n^{\prime}}$ such that $$\begin{aligned}
\nu & = & [F]-[\tilde{F}], \label{FixBunIsoFirst} \\
F|_{\partial_0 W} \cong (\pi^{P\times S^{2n}}_{S^{2n}})^*(F_{S^{2n}}) & \hbox{ and } & \tilde{F}|_{\partial_0 W} \cong (\pi^{P\times S^{2n}}_{S^{2n}})^*(\tilde{F}_{S^{2n}}), \label{FixBunIsoSecond} \\
F^{\prime}|_{\partial_1 W} \cong (\pi^{P^{\prime}\times S^{2n^{\prime}}}_{S^{2n^{\prime}}})^*(F^{\prime}_{S^{2n^{\prime}}}) & \hbox{ and } &\tilde{F}^{\prime}|_{\partial_1 W} \cong (\pi^{P^{\prime}\times S^{2n^{\prime}}}_{S^{2n^{\prime}}})^*(\tilde{F}^{\prime}_{S^{2n^{\prime}}}). \label{FixBunIsoLast}\end{aligned}$$
We use this data including the explicit choices of vector bundle isomorphisms in the previous equations to define the isomorphism; its definition is somewhat involved.
Let $((Q, \beta Q), \xi ,f)$ be a cycle in $K_*(X;P;A)$ and $(E_Q, E_{\beta Q})$ and $(\tilde{E}_Q, \tilde{E}_{\beta Q})$ be $A$-vector bundles over $(Q, \beta Q)$ such that $\xi=[(E_Q, E_{\beta Q})]-[(\tilde{E}_Q, \tilde{E}_{\beta Q})]$.
Using the decomposition $$\partial (\beta Q \times W)= (\beta Q \times \partial_0 W) \dot{\cup} (\beta Q \times \partial_1 W) \cong (\beta Q \times S^{2n}\times P) \dot{\cup} (\beta Q \times S^{2n^{\prime}} \times P^{\prime}).$$ form the $P^{\prime}$-manifold $$(Q\times S^{2n} \cup_{\partial Q \times S^{2n}} \beta Q \times W, \beta Q \times S^{2n^{\prime}})$$
The class in $K^0((Q\times S^{2n} \cup_{\partial Q \times S^{2n}} \beta Q \times W, \beta Q \times S^{2n^{\prime}}))$ is constructed as follows. For a cocycle $(E_Q, E_{\beta Q})$, we let $$\gamma(E_Q, E_{\beta Q}):=[(E^{\prime}, E^{\prime}_{\beta Q})] - [(\hat{E}^{\prime}, \hat{E}^{\prime}_{\beta Q})]$$ where $$\begin{aligned}
E^{\prime} & = & (\pi^{Q\times S^{2n}}_Q)^*(E_Q) \otimes (\pi^{Q\times S^{2n}}_{S^{2n}})^*(F_{S^n}) \cup_{P\times \beta Q \times S^{2n}} (\pi^{\beta Q \times W}_{\beta Q})^*(E_{\beta Q})\otimes (\pi^{\beta Q \times W}_W)^*(F) \\
E^{\prime}_{\beta Q} & = & (\pi^{\beta Q \times S^{2n^{\prime}}}_{\beta Q})^*(E_{\beta Q}) \otimes (\pi^{\beta Q \times S^{2n^{\prime}}}_{S^{2n^{\prime}}})^*(F^{\prime}_{S^{2n^{\prime}}}) \\
\hat{E}^{\prime} & = & (\pi^{Q\times S^{2n}}_Q)^*(E_Q) \otimes (\pi^{Q\times S^{2n}}_{S^{2n}})^*(\tilde{F}_{S^n}) \cup_{P\times \beta Q \times S^{2n}} (\pi^{\beta Q \times W}_{\beta Q})^*(E_{\beta Q})\otimes (\pi^{\beta Q \times W}_W)^*(\tilde{F}) \\
\hat{E}^{\prime}_{\beta Q} & = & (\pi^{\beta Q \times S^{2n^{\prime}}}_{\beta Q})^*(E_{\beta Q}) \otimes (\pi^{\beta Q \times S^{2n^{\prime}}}_{S^{2n^{\prime}}})^*(\tilde{F}^{\prime}_{S^{2n^{\prime}}})\end{aligned}$$ We make note of the use of the identification $\partial Q \times S^{2n} \cong P\times \beta Q \times S^{2n}\cong \partial_0 W \times \beta Q$ and that the required clutching isomorphisms are also included in the data (via the definition of $P$-bundle and the fixed vector bundle isomorphisms in Equations \[FixBunIsoSecond\] and \[FixBunIsoLast\]). Then, for $\xi=[(E_Q, E_{\beta Q})]-[(\tilde{E}_Q, \tilde{E}_{\beta Q})]$, we let $$\xi^{\prime}:=\gamma(E_Q, E_{\beta Q})- \gamma(\tilde{E}_Q, \tilde{E}_{\beta Q})$$ Using standard methods in topological $K$-theory, one checks that $\xi^{\prime} \in K^0 (Q\times S^{2n} \cup_{\partial Q \times S^{2n}} \beta Q \times W, \beta Q \times S^{2n^{\prime}})$ and $\xi^{\prime}$ do not depend on the choice of cocycles $(E_Q, E_{\beta Q})$ and $(\tilde{E}_Q, \tilde{E}_{\beta Q})$, but only on the class $\xi$ and, of course, on the choice of $W$, $F$, $\tilde{F}$, etc.
Finally, the function, denoted by $g$, is defined to be $f$ on $Q$ and $f\circ \pi_W$ on $\beta Q \times W$.
Let $\Psi_{(W,\nu)} : K_*(X;P;A) \rightarrow K_*(X;P^{\prime};A)$ be the map defined via the process just described; that is, $$\Psi_{(W,\nu)}((Q, \beta Q), \xi, f):=((Q\times S^{2n} \cup_{\partial Q \times S^{2n}} \beta Q \times W, \beta Q \times S^{2n^{\prime}}), \xi^{\prime}, g).$$ We must show that this map is well-defined.
Suppose that $((Q,\beta Q), \xi, f)$ is the boundary in the sense of Definition \[borForKthGrp\] of the bordism $((\bar{Q}, \beta \bar{Q}), \bar{\xi}, \bar{f})$. Definitions \[PmfldWithBound\] and \[borForKthGrp\] imply that $\partial \beta \bar{Q}= \beta Q$; hence, we can form the closed, smooth, ${\rm spin^c}$ manifold, $$Q\times S^{2n} \cup_{\partial Q \times S^{2n}} \beta Q \times W \cup_{\beta Q \times S^{2n^{\prime}} \times P^{\prime}} \beta \bar{Q} \times S^{2n^{\prime}}\times P^{\prime}$$ There are also an associated K-theory class and function: For the K-theory class, we take $$\xi^{\prime}|_{(Q\times S^{2n} \cup_{\partial Q \times S^{2n}} \beta Q \times W} \cup (\pi^{\beta \bar{Q} \times S^{2n^{\prime}}}_{\beta \bar{Q}})^*(\bar{\xi})|_{\beta \bar{Q}\times S^{2n^{\prime}}}$$ and, for the function, we take $$g \cup (\bar{f}|_{\beta \bar{Q}} \circ \pi^{\beta \bar{Q} \times S^{2n^{\prime}}}_{\beta \bar{Q}}).$$ Our goal is to show that this manifold is a boundary; we must also check that the bordism respects the K-theory class and continuous function. Definition \[PmfldWithBound\] implies that $(Q \cup \beta \bar{Q}\times P)\times S^{2n}$ is a boundary. Moreover, the K-theory and function also respect this construction. Hence, it is a bordism in the sense of the Baum-Douglas model of K-homology.
Furthermore, we will show that $$Q\times S^{2n} \cup_{\partial Q \times S^{2n}} \beta Q \times W \cup_{\beta Q \times S^{2n^{\prime}} \times P^{\prime}} \beta \bar{Q} \times S^{2n^{\prime}}\times P^{\prime} \dot{\cup} -((Q \cup \beta \bar{Q}\times P)\times S^{2n})$$ is bordant to $$(\beta \bar{Q}\times S^{2n}\times P) \cup_{\beta Q \times S^{2n} \times P} (\beta Q \times W) \cup_{\beta Q \times S^{2n^{\prime}} \times P^{\prime}} (\beta \bar{Q} \times S^{2n^{\prime}}\times P^{\prime})$$ The bordism we have in mind is formed in a similar manner to the construction of the “pair of pants" bordism by gluing the manifolds $[0,1]\times [0,1]$ and $S^1\times [0,1]$ together; the details are as follows: the manifold with boundary in the bordims is constructed by applying the “straightening the angle" technique (see [@CFPerMap] or [@Rav Appendix]) to the manifold with corners formed by gluing $ Q \times S^{2n} \times [0,1] $ to $$\left( (\beta \bar{Q}\times S^{2n}\times P) \cup_{\beta Q \times S^{2n} \times P} (\beta Q \times W) \cup_{\beta Q \times S^{2n^{\prime}} \times P^{\prime}} (\beta \bar{Q} \times S^{2n^{\prime}}\times P^{\prime}) \right) \times [0,1]$$ along $(\partial Q \times S^{2n}) \times [0,1] \cong \left ( (\beta Q \times \partial_0 W) \times [0,1] \right) \times \{1\}$. We note that we consider $\left( (\beta Q \times \partial_0 W) \times [0,1] \right)$ as a subspace of $$\left( (\beta \bar{Q}\times S^{2n}\times P) \cup_{\beta Q \times S^{2n} \times P} (\beta Q \times W) \cup_{\beta Q \times S^{2n^{\prime}} \times P^{\prime}} (\beta \bar{Q} \times S^{2n^{\prime}}\times P^{\prime}) \right)$$ Finally, by “straightening the angle" (see [@CFPerMap] or [@Rav Appendix]) of $\beta \bar{Q} \times W$, we obtain a smooth compact ${\rm spin}^c$ manifold with boundary; it has boundary $$\partial \beta \bar{Q} \times W \cup \beta Q \times \partial W=(\beta \bar{Q}\times S^{2n}\times P) \cup_{\beta Q \times S^{2n} \times P} (\beta Q \times W) \cup_{\beta Q \times S^{2n^{\prime}} \times P^{\prime}} (\beta \bar{Q} \times S^{2n^{\prime}}\times P^{\prime})$$ Combining these three observations completes the construction of the required bordism at least at the level of the “manifold part" of the cycle. That this bordism respects the continuous function data also follows from standard results in bordism theory. In particular, by [@CFPerMap], the “straightening the angle" process respects the continuous functions involved in the bordism. That the bordism respects the K-theory data is a bit move involved, but it also follows from the fact that the “straightening the angle" process respects K-theory data (see [@Rav Appendix]). This completes the proof that $\Psi_{(W,\nu)}$ respects the bordism relation.
For vector bundle modification, let $((Q, \beta Q), \xi, f)$ be a cycle and $((Q^{\prime}, \beta Q^{\prime}), \xi^{\prime}, f^{\prime})$ be its image under $\Psi_{(W,\nu)}$. Also, let $(V_Q, V_{\beta Q})$ be a ${\rm spin^c}$ $P$-bundle of even rank over $((Q, \beta Q)$. Form the vector bundle $ (\pi^{Q\times S^{2n}}_Q)^*(V_Q) \cup_{\partial Q} (\pi^{W\times \beta Q}_{\beta Q})^*(V_{\beta Q})$ over $Q^{\prime}$; it is also ${\rm spin^c}$, of even rank, and can be given the structure of a $P^{\prime}$-bundle. As such, we can consider the vector bundle modification of $((Q^{\prime}, \beta Q^{\prime}), \xi^{\prime}, f^{\prime})$ by this bundle. Moreover, the definition of vector bundle modification implies that $$\left( (Q^{\prime}, \beta Q^{\prime}), \xi^{\prime}, f^{\prime} \right)^{ \left( (\pi^{Q\times S^{2n}}_Q )^*(V_Q) \cup_{\partial Q} (\pi^{W\times \beta Q}_{\beta Q})^*(V_{\beta Q}) \right) } = \Psi_{(W,\nu)} \left( ((Q, \beta Q), \xi, f \right)^{(V_Q, V_{\beta Q})}$$ This completes the proof that $\Psi_{(W,\nu)}$ is well-defined.
To see that this map is an isomorphism, we can use the generalized Bockstein sequence and Five Lemma. However, there is a more direct approach as follows. We recall that $(-W, \nu)$ denotes the opposite of $(W, \nu)$ and that $(-W,\nu)$ is therefore a bordism from $(P^{\prime}, [P^{\prime}\times {\mathbb{C}}])^{N^{\prime}}$ to $(P,[P\times {\mathbb{C}}])^N$. Using the same choice for vector bundles representing $\nu$ (see Equations \[FixBunIsoFirst\] to \[FixBunIsoLast\]), we have the map $$\Psi_{(-W,\nu)}: K_*(X;P^{\prime};A) \rightarrow K_*(X;P;A).$$ Moreover, it follows from standard results in bordism theory (essentially the observation that the double of a manifold is a boundary) imply that $\Psi_{(-W,\nu)}$ is the inverse of $\Psi_{(W,\nu)}$.
Finally, that isomorphism constructed in this proof is natural with respect to $X$ and $A$ follows from the explicit nature of the map.
\[remThmDepKhomClass\] The previous theorem implies that if we can compute $K_*(X;P;A)$ for $P$ equaling $k$-points, $S^1$, and $S^2$, then we have determined $K_*(X;P;A)$ for any $P$, which satisfy the conditions in the statement of the previous theorem and such that $(P, [P\times {\mathbb{C}}])$ is equivalent in $K_*(pt)$ to $(k$-points,$[k$-points$\times {\mathbb{C}}])$, $(S^1, [S^1\times {\mathbb{C}}])$, or $(S^2, [S^2\times {\mathbb{C}}])$. We note that this list of classes contains all elements in $K_*(pt)$ and that these three examples will be discussed in detail in Section \[exSec\].
The assembly map for $P$-manifolds
==================================
Let $\Gamma$ be a finitely generated discrete group, $B\Gamma$ be the classifying space of $\Gamma$, $C^*(\Gamma)$ be the reduced group $C^*$-algebra of $\Gamma$, although similar results hold for the full group $C^*$-algebra. For simpility, we assume $B\Gamma$ is a finite CW-complex. Recall (see [@Wal]) that the Baum-Connes assembly map, $\mu: K_*(B\Gamma) \rightarrow K_*(pt;C^*(\Gamma))$, can be defined at the level of geometric cycles as follows: $$(M,E,f) \mapsto (M,E\otimes_{{\mathbb{C}}} f^*(\mathcal{L}_{B\Gamma}))$$ where $\mathcal{L}_{B\Gamma}$ is the Mishchenko line bundle; it was defined in Example \[MishBunEx\]. This definition of assembly generalizes to the $P$-manifold setting as follows.
The assembly map, $\mu_{P}:K_*(B\Gamma;P) \rightarrow K_*(pt;P;C^*(\Gamma))$, is defined at the level of cycles via $$((Q,\beta Q), (E_Q,E_{\beta Q}), f) \mapsto ((Q,\beta Q), (E_Q \otimes_{{\mathbb{C}}} f^*(\mathcal{L}_{B\Gamma}), E_{\beta Q} \otimes_{{\mathbb{C}}} (f_{\beta Q})^*(\mathcal{L}_{B\Gamma}))).$$
\[BCForPMfld\] The map $\mu_P$ is well-defined. Moreover, the following diagram is commutative:
$\begin{CD}
@>>> K_0(B\Gamma) @>\Phi_P>> K_{{\rm dim}(P)}(B\Gamma) @>r_P>> K_0(B\Gamma;P) @>\delta_P>> K_{{\rm dim}(P)-1}(B\Gamma) @>>> \\
@. @V\mu VV @V\mu VV @V\mu_P VV @V\mu VV @. \\
@>>> K_0(pt;C^*(\Gamma)) @>\Phi_P >> K_{{\rm dim}(P)}(pt; C^*(\Gamma)) @>r_P >> K_0(pt;P;C^*(\Gamma)) @>\delta_P >> K_{{\rm dim}(P)-1}(pt;C^*(\Gamma)) @>>>
\end{CD}$
In particular, if $\mu: K_*(B\Gamma) \rightarrow K_*(pt;C^*(\Gamma))$ is an isomorphism, then $\mu_P$ is an isomorphism.
The first two statements follow by observing that all the groups are defined using geometric cycles and the maps are defined at the level of these cycle. As such, the proof is a matter of showing the relations on the various cycles are compatible, which can be checked directly. The last statement in the theorem follows from the first two and the Five Lemma.
Examples {#exSec}
========
In this section, we work with the bundle model of $K_*(X;P;A)$. That is, we use cycles as in Definition \[cycWithBun\]. The definitions of the operations (e.g., addition, opposite, etc) and the relation (e.g., bordism, vector bundle modification, etc) are the natural “$P$-manifold version" of those given in any of [@BD; @BHS; @Wal]. We include the definition of bordism in this context as a prototypical example.
\[borForBunGrp\] A bordism or a cycle with boundary with respect to $X$, $P$, and $A$ in the bundle model is $((\bar{Q}, \beta \bar{Q}), (\bar{E}_Q, \bar{E}_{\beta Q}), \bar{f})$ where
1. $(\bar{Q},\beta \bar{Q})$ is a compact, smooth, ${\rm spin^c}$ $P$-manifold with boundary;
2. $(\bar{E}_Q, \bar{E}_{\beta Q})$ is a smooth $A$-vector bundle over $(\bar{Q}, \beta \bar{Q})$;
3. $\bar{f}: (\bar{Q},\beta \bar{Q}) \rightarrow X$ is a continuous map.
The boundary of a bordism, $((\bar{Q}, \beta \bar{Q}), (\bar{E}_Q, \bar{E}_{\beta Q}), \bar{f})$, is given by $$((\partial \bar{Q} - {\rm int}(\beta \bar{Q}), \partial \beta \bar{Q}), ((\bar{E}_Q) |_{\partial \bar{Q} - {\rm int}(\beta \bar{Q})}, \bar{E}_{\beta Q}|_{\partial \beta \bar{Q}}), \bar{f}|_{\partial \bar{Q} - {\rm int}(\beta \bar{Q})}).$$ By construction, it is a cycle in the sense of Definition \[cycWithBun\].
In the first example below we make use of K-homology with coefficients in ${{\mathbb{Z}}/k{\mathbb{Z}}}$. The reader can find details of the analytic (i.e., KK-theory) construction of this group in [@Schochet]. In particular, results in [@Schochet] imply that we can define the K-homology of $X$ with cofficients in ${{\mathbb{Z}}/k{\mathbb{Z}}}$ as follows: $$K_*(X;{{\mathbb{Z}}/k{\mathbb{Z}}}):= KK^*(C(X),C_{\phi}).$$ where $C_{\phi}$ is the mapping cone of the unital inclusion of the complex numbers in the $k$ by $k$ matrices. In similar way, $$K_*(X;A;{{\mathbb{Z}}/k{\mathbb{Z}}}):=KK^*(C(X), A\otimes C_{\phi}) \hbox{ and } K_*(A;{{\mathbb{Z}}/k{\mathbb{Z}}}):=K_*(A\otimes C_{\phi})$$
In this example, we work with the bundle model of $K_*(X;P;A)$ in the case when $P=k$-points. Results in [@Dee1; @Dee2] imply that $K_*(X;P;{\mathbb{C}}) \cong K_*(X;{\mathbb{C}};{{\mathbb{Z}}/k{\mathbb{Z}}})$. The methods used in those papers can be generalized to the case of general $A$; that is, $$K_*(X;P;A) \cong K_*(X;A;{{\mathbb{Z}}/k{\mathbb{Z}}})$$ Moreover, in the case when $X=pt$ and $A={\mathbb{C}}$, the map $$K_0(pt;\{ k\hbox{-points}\};{\mathbb{C}}) \cong K_0(pt;{{\mathbb{Z}}/k{\mathbb{Z}}}) \cong {{\mathbb{Z}}/k{\mathbb{Z}}}$$ can be taken to be the map defined at the level of cycle $$((Q, \beta Q), (E_{Q}, E_{\beta Q})) \mapsto {\rm ind_{APS}}(D_{Q, E_Q}) \hbox{ mod }k.$$ where ${\rm ind}_{APS}(\: \cdot \:)$ denotes the Atiyah-Patodi-Singer index (see [@APS1]).
In the case of general $A$, using results in [@Schochet], one has the six-term exact sequence:
$\begin{CD}
K_0(X;A) @>k>> K_0(X;A) @>r>> K_0(X;A;{{\mathbb{Z}}/k{\mathbb{Z}}}) \\
@A\delta AA @. @V\delta VV \\
K_1(X;A;{{\mathbb{Z}}/k{\mathbb{Z}}}) @<r<< K_1(X;A) @<k<< K_1(X;A)
\end{CD}$
If $K_*(A)$ contains no torsion of order $k$, then this six-term exact sequence reduces to the following two short exact sequences: $$0 \rightarrow K_0(pt;A) \rightarrow K_0(pt;A) \rightarrow K_0(pt;A;{{\mathbb{Z}}/k{\mathbb{Z}}}) \rightarrow 0$$ $$0 \rightarrow K_1(pt;A) \rightarrow K_1(pt;A) \rightarrow K_1(pt;A;{{\mathbb{Z}}/k{\mathbb{Z}}}) \rightarrow 0.$$ In this case, an index map $$K_*(pt;\{k\hbox{-points}\};A) \rightarrow K_*(pt;A;{{\mathbb{Z}}/k{\mathbb{Z}}}) \cong K_*(A;{{\mathbb{Z}}/k{\mathbb{Z}}})$$ can be defined using higher APS-index theory; the reader can find more details on this theory in [@LP] and the references therein.
A key object in higher APS-index theory is the notion of a spectral section (see [@LP]). There exists a spectral section with respect to a ${\rm spin^c}$-manifold $M$ and an $A$-bundle over it if and only if the index of the assoicated twisted Dirac operator vanishes. Hence, the existence of a spectral section for the boundary of a manifold with boundary $W$ and the restriction of an $A$-vector bundle over $W$ to $\partial W$ follows from the cobordism invariance of the higher index. In the case of a ${{\mathbb{Z}}/k{\mathbb{Z}}}$-manifold $(Q, \beta Q)$ and $A$-vector bundle $(E_{Q}, E_{\beta Q})$, we need to choose a spectral section with respect to $\beta Q$ and $E_{\beta Q}$; such a spectral section does not in general exist. In general, we have $$0={\rm ind}(D_{\partial Q, E_{Q}|_{\partial Q}})= k \cdot {\rm ind}(D_{\beta Q, E_{\beta Q}}).$$ However, if we assume that $K_*(A)$ contains no torsion of order $k$, then this equality implies that ${\rm ind}(D_{\beta Q, E_{\beta Q}})=0$ and hence that a spectral section for $\beta Q$ and $E_{\beta Q}$ exists (see [@LP Section 2]).
As such, under the assumption that $K_*(A)$ contains no torison of order $k$, we define the index map at the level of cycles via $$((Q, \beta Q), (E_Q, E_{\beta Q})) \mapsto r({\rm ind_{APS}}(D_{(Q, E_Q)}(\mathcal{P}_{\beta Q}))$$ where
1. $\mathcal{P}_{\beta Q}$ denotes the pullback from $\beta Q$ to $\beta Q \times P \cong \partial Q$ of a spectral section for the manifold $\beta Q$ and bundle $E_{\beta Q}$;
2. $D_{(Q, E_Q)}(\mathcal{P}_{\beta Q})$ denotes the Dirac operator on $Q$ twisted by $E_Q$ with the higher APS boundary condition associated with the spectral section $\mathcal{P}_{\beta Q}$;
3. ${\rm ind_{APS}}(\: \cdot \:)$ denotes the higher APS index;
4. $r: K_*(A) \rightarrow K_*(A;{{\mathbb{Z}}/k{\mathbb{Z}}})$ denotes the map from the exact sequence above.
We must show that this map is well-defined; it is not even clear that the map is well-defined at the level of cycles. The analytic data required to define the higher APS index is a metric, compatible connections on both the spinor and vector bundles, and a choice of a spectral section. To show that the above map is independent of these choices, we suppose that we have continuous families of this data. That is, we let
1. $\{g_t\}_{t\in [0,1]}$ be a one parameter family of Riemannian metrics on $(Q, \beta Q)$;
2. $\nabla_{E_{\beta Q},t}$ be a one parameter family of connections on $E_{\beta Q}$ which is compatible with $g_t|_{\beta Q}$;
3. $\nabla_{E_Q,t}$ be a one parameter family of connections on $E_Q$ which is compatible with $g_t$ and with the family of connections $\nabla_{E_{\beta Q},t}$;
4. $\hat{\mathcal{P}}_0$ and $\hat{\mathcal{P}}_1$ are the pullback of two choices of spectral section for $D_{\beta Q, E_{\beta Q}}$.
Let $D_{(Q, E_Q, 0)}(\hat{\mathcal{P}}_0)$ and $D_{(Q, E_Q, 1)}(\hat{\mathcal{P}}_1)$ the Dirac operators associated to the end points of the family of data fixed above. Also, let $\tilde{D}_t$ denote the family of boundary operators associated to the above data restricted to the boundary; the reader is directed to [@LP] for more details on this family of operators.
With all of the above data fixed, we can apply [@LP Proposition 8 and Theorems 6 and 7] to obtain $${\rm ind_{APS}}(D_{(Q, E_Q, 0)}(\hat{\mathcal{P}}_0))-{\rm ind_{APS}}(D_{(Q, E_Q, 1)}(\hat{\mathcal{P}}_1))= {\rm sf}(\tilde{D}_t, \hat{\mathcal{P}}_1, \hat{\mathcal{P}}_0)$$ where ${\rm sf}(\tilde{D}_t, \hat{\mathcal{P}}_1, \hat{\mathcal{P}}_0)$ is the spectral flow of the family of operators, $\tilde{D}_t$ (see [@LP] for details).
Applying the map $r$, observing that right hand side of the equation (i.e., the spectral flow term) is in the image of $k$, and exactness imply the required result: $$r({\rm ind_{APS}}(D_{(Q, E_Q, 0)}(\hat{\mathcal{P}}_0)))=r({\rm ind_{APS}}(D_{(Q, E_Q, 1)}(\hat{\mathcal{P}}_1))).$$
Next, we must show that the map respects the three relations used to define $K_*(pt;\{k-\hbox{points}\};A)$. For the direct sum/disjoint union relation the proof follows from basic properties of the higher APS-index.
For the bordism relation, suppose $((Q, \beta Q), (E_Q, E_{\beta Q}), f)$ is the boundary of the bordism $((\bar{Q}, \beta \bar{Q}), (\bar{E}_{\bar{Q}}, \bar{E}_{\beta \bar{Q}}), g)$. After fixing the required analytic data to define the higher APS index, we have that [@LP Theorems 8 and 9] imply that $${\rm ind}_{AS}(D_{\partial \bar{Q}, \bar{E}_{\bar{Q}}|_{\partial \bar{Q}}}) = {\rm ind_{APS}}(D_{(Q, E_Q)}(\mathcal{P})) + k \cdot {\rm ind_{APS}}(D_{(\beta \bar{Q}, \beta \bar{E}_{\bar{Q}})}(\tilde{\mathcal{P}}))$$ where
1. $\pi: \partial Q \cong \beta Q \times P \rightarrow \beta Q$ is the projection map;
2. $\pi^*(\tilde{\mathcal{P}})=\mathcal{P}$; recall that, by assumption, $\mathcal{P}$ is the pullback of a spectral section associated to $\beta Q$ and $\beta E$.
Applying the map $r$ to this equation and using exactness, we obtain $$r({\rm ind}_{AS}(D_{\partial \bar{Q}, \bar{E}_{\bar{Q}}|_{\partial \bar{Q}}})) = r({\rm ind_{APS}}(D_{(Q, E_Q)}(\mathcal{P}))).$$ Finally, the bordism invariance of the higher AS-index implies that ${\rm ind}_{AS}(D_{\partial \bar{Q}, \bar{E}_{\bar{Q}}|_{\partial \bar{Q}}})$ vanishes. Hence the proof of the required invariance under the bordism relation is complete. Finally, for vector bundle modification the result follows from [@DeeAnaTopIndMaps Proposition 4.2].
The discussion in the previous example gives a proof of the next theorem. We use the notation introduced in that example and note that the assumption that ${\rm ind_{AS}}(D_{\beta Q, E_{\beta Q}})=0$ implies that there exists a spectral section with respect to $\partial Q$ and $E_Q|_{\partial Q}$ that is the pullback of a spectral section with respect to $\beta Q$ and $E_{\beta Q}$ (see [@LP Section 2]).
\[higFMThm\] Let $((Q, \beta Q), (E_Q, E_{\beta Q}))$ be a cycle in $K_*(pt;k$-points$;A)$ such that ${\rm ind_{AS}}(D_{\beta Q, E_{\beta Q}})=0$. Then $$r({\rm ind_{APS}}(D_{Q, E_Q}(\mathcal{P})))$$ is a topological invariant (i.e., independent of the metric, connection, etc). Moreover, it is also, in the sense of Definition \[borForKthGrp\], a cobordism invariant.
In fact, there is a topological formula for $r({\rm ind_{APS}}(D_{Q, E_Q}(\mathcal{P}))$. It is obtained in the same way as in the Freed-Melrose index theorem; that is, one considers an embedding of the ${{\mathbb{Z}}/k{\mathbb{Z}}}$-manifold into a suitable half-space (see [@Fre] or [@Dee1 Section 1.4]) and defines a wrong-way map in $K$-theory. However, we will not discuss this in detail.
In this example, we work with the bundle model of $K_*(X;P;A)$ in the case when $P$ is the circle, $S^1$. We will show that $$K_0(pt; S^1; A) \cong K_0(A) \oplus K_0(A) \hbox{ and } K_1(pt;S^1;A) \cong K_1(A)\oplus K_1(A).$$ In particular, if $A = {\mathbb{C}}$, then $K_0(pt;S^1) \cong {\mathbb{Z}} \oplus {\mathbb{Z}}$ and $K_1(pt;S^1)\cong \{0\}$.
This result can be obtained indirectly using the exact sequence in Theorem \[BockTypeSeq\]. However, there is an explicit isomorphism defined as follows: $${\rm ind}_{S^1}: ((Q,\beta Q), (E_{Q}, E_{\beta Q})) \mapsto \left( {\rm ind_{AS}}(D_{(\beta Q, \beta E)}), {\rm ind_{AS}}(D_{Q\cup_{\partial Q} \beta Q \times {\mathbb{D}}, E_{Q} \cup \pi^*(E_{\beta Q})}) \right)$$ where
1. ${\mathbb{D}}$ denotes the unit disk;
2. $E_{Q} \cup \pi^*(E_{\beta Q})$ denotes the vector bundle obtained by clutching via the isomorphism assoicated to the $P$-bundle $(E_Q, E_{\beta Q})$;
3. $D_{M,E}$ denotes the Dirac operator of the manifold $M$ twisted by the bundle $E$;
4. $\pi: \beta Q \times P \rightarrow \beta Q$ denotes the projection map.
We must prove that this map is well-defined. That the map respects the direct sum/disjoint union relation follows using definitions and basic properties of the index map. For the bordism relation, suppose that $((Q,\beta Q), (E_{Q}, E_{\beta Q}))$ is the boundary of the bordism $((\bar{Q},\beta \bar{Q}), (\bar{E}_{Q}, \bar{E}_{\beta Q}))$. We are required to show that $${\rm ind}(D_{(\beta Q, \beta E)})=0 \hbox{ and } {\rm ind}(D_{Q\cup_{\partial Q} \beta Q \times{\mathbb{D}}, E_{Q} \cup \pi^*(E_{\beta Q})})=0.$$ The first of the two holds by the cobordism invariance of the index and the fact that $(\beta Q, \beta E)= \partial (\beta \bar{Q}, \bar{E}_{\beta Q})$.
The proof of the second equality is more involved. Let $\tilde{\pi}: \beta \bar{Q} \times S^1 \rightarrow \beta \bar{Q}$ be the projection map. The definition of bordism as in Definition \[borForBunGrp\] implies that the cycle $(Q\cup \beta \bar{Q}\times S^1,E\cup\tilde{\pi}^*(\bar{E}_{\beta \bar{Q}}))$ is a boundary; it is the boundary of $(\bar{Q}, \bar{E})$. Moreover, standard results in bordism theory imply that $$(Q\cup_{\partial Q} \beta \bar{Q}\times S^1, E\cup \tilde{\pi}^*(E_{\beta \bar{Q}})) \dot{\cup} -(Q \cup_{\partial Q} \beta Q \times {\mathbb{D}}, E_{Q} \cup \pi^*(E_{\beta Q}))$$ is bordant to $$(\beta Q \times {\mathbb{D}} \cup_{\partial Q} \beta \bar{Q}\times S^1, \pi^*(E_{\beta Q}) \cup \tilde{\pi}^*(E_{\beta \bar{Q}}) ).$$ By “straightening the angle" (see [@CFPerMap] or [@Rav Appendix]) of $\beta \bar{Q} \times {\mathbb{D}}$, one can show that $ (\beta Q \times {\mathbb{D}} \cup_{\partial Q} \beta \bar{Q}\times S^1, \pi^*(E_{\beta Q}) \cup \tilde{\pi}^*(E_{\beta \bar{Q}}))$ is a boundary. The combination of the three bordisms discussed in this paragraph imply that $(Q \cup_{\partial Q} \beta Q \times {\mathbb{D}}, E_{Q} \cup \pi^*(E_{\beta Q}))$ is a boundary. The cobordism invariance of the index, then implies the required vanishing result.
For the vector bundle modification relation, if $(V, V_{\beta Q})$ is an even rank ${\rm spin^c}$ $P$-vector bundle over $(Q, \beta Q)$, then one can form the vector bundle $V \cup \pi^*(V_{\beta Q})$; it is an even rank ${\rm spin^c}$ vector bundle over $Q \cup \beta Q \times {\mathbb{D}}$. This fact and the invariance of the index under vector bundle modification in the standard Baum-Douglas model give the required result.
Next, we prove that the following diagram is commutative:
$\begin{CD}
@>>> K_1(pt;A) @>\Phi_{S^1}>> K_0(pt;A) @>r_{S^1}>> K_0(pt;S^1;A) @>\delta_{S^1}>> K_0(pt;A) @>\Phi_{S^1}>> K_1(pt;A) @>>> \\
@. @V{\rm ind} VV @V{\rm ind} VV @V{\rm ind}_{S^1} VV @V{\rm ind} VV @V{\rm ind}VV @. \\
@>>> K_1(A) @>\Phi >> K_0(A) @>r >> K_0(A) \oplus K_0(A) @>\delta >> K_0(A) @>\Phi>> K_1(A) @>>>
\end{CD}$
To begin, we note that the $\Phi_{S_1}$ is the zero map because $S^1$ is a boundary. Secondly, if $(M,E)\in K_*(pt;A)$, then $$({\rm ind}_{S^1}\circ r_{S^1})(M,E) = ({\rm ind}_{AS}(D_E), 0)= r({\rm ind}(M,E))$$ as required. Finally, if $((Q,\beta Q), (E_Q, E_{\beta Q})) \in K_*(pt;S^1;A)$, then $$({\rm ind}\circ \delta_{S^1})((Q,\beta Q), (E_Q, E_{\beta Q}))={\rm ind}_{AS}(D_{(\beta Q, E_{\beta Q})})=(\delta \circ {\rm ind}_{S^1})((Q, \beta Q), (E_Q, E_{\beta Q}))$$ The Five Lemma and the previous commutative diagram imply the main result of this example (i.e., that the index map $K_*(pt;S^1;A)$ to $K_*(A)\oplus K_*(A)$ is an isomorphism).
For other examples of $P$, one can also compute $K_*(pt;P;A)$. For example, if $P$ is the two-sphere, $S^2$, then an isomorphism, $K_*(pt;S^2;A) \rightarrow K_{*+1}(A)\oplus K_*(A)$, can defined via $$((Q, \beta Q), (E_{Q}, E_{\beta Q})) \mapsto \left( {\rm ind}(D_{(\beta Q, \beta E)}), {\rm ind}(D_{Q\cup_{\partial Q} \beta Q \times {\mathbb{D}}^2, E_{Q} \cup \pi^*(E_{\beta Q})}) \right)$$ where
1. ${\mathbb{D}}^2$ is the two-dimensional ball;
2. $\pi:\beta Q \times {\mathbb{D}}^2 \rightarrow \beta Q$ is the projection map;
3. $D_{\beta Q, E_{\beta Q}}$ and ${\rm ind_{AS}}$ are defined as in the previous example.
The proof that this map is an isomorphism is similar to the proof that ${\rm ind}_{S^1}$ is an isomorphism given in the previous example. Furthermore, these methods can be used to show that $K_*(pt;S^n;A) \cong K_*(A) \oplus K_{*-n-1}(A)$ for any $n$.
Generalizations
===============
The constructions in the previous sections of the paper have a number of generalizations. The most obvious to the reader familiar with the Baum-Douglas model for K-homology are to various equivariant settings. For the corresponding equivarent theories one should replace our cycles (i.e., Definition \[cycWithKth\]) with the natural generalization of the cycles in [@BOOSW] in the case of a compact Lie group or [@BHSProGrpAct] in the case of a discrete group which acts properly. Both these two generalization can be obtained using standard techniques and tools developed to this point. For example, in the case of proper discrete group actions, the reader should see [@Rav Section 4.5] for more on the correct notion of “normal bordism". We will not give a detailed developement. On the other hand, in the case of the action of a groupoid the correct generalization is less clear, but would be an interesting project. The reader might find [@EM] a useful starting point. This completes the discussion of the equivariant generalizations.
Another generalization is as follows. Based on the statement of Theorem \[uniquenessThm\], one is lead to consider the map on K-homology induced by more general cycles in $K_*(pt)$; the precise definition of the relevant cycles is given below in Definition \[defGenCyc\]. Let $(P,F)$ be a cycle in $K_*(pt)$, then we can consider the map $\Phi_{(P,F)}: K_*(X;A) \rightarrow K_{*+{\rm dim}(P)}(X;A)$ defined at the level of cycles via $$(M,E,f) \mapsto (M \times P, (\pi_M)^*(E) \otimes (\pi_P)^*(F), f \circ \pi_M)$$ where $\pi_M$ (resp. $\pi_P$) denote the projection map from $M\times P$ to $M$ (resp. $P$). One is naturally led to ask if the constructions in the previous sections generalize to this setting; this is the case so long as we continue to assume $P$ has trivial stable normal bundle. To be precise, we have the following definition of a cycle with respect to $(P,F)$:
\[defGenCyc\] A geometric cycle with bundle data, over $X$ with respect to $(P,F)$ and $A$ is a triple $((Q,\beta Q), (E_Q, E_{\beta Q}), f)$ where
1. $(Q,\beta Q)$ is a compact, smooth, ${\rm spin^c}$ $P$-manifold;
2. $E_Q$ is a smooth $A$-vector bundle over $Q$;
3. $E_{\beta Q}$ is a smooth $A$-vector bundle over $\beta Q$;
4. $\alpha: E_Q|_{\partial Q} \rightarrow (\pi_{\beta Q})^*(E_{\beta Q}) \otimes (\pi_{P})^*(F)$ is an isomorphism of $A$-vector bundles; we note that $\pi_{\beta Q}$ (respectively $\pi_P$) denotes the projection map from $\partial Q \cong \beta Q\times P$ to $\beta Q$ (respectively $P$);
5. $f$ is a continuous map from $(Q,\beta Q)$ to $X$.
The reader should note that when $F=P\times {\mathbb{C}}$, Definition \[defGenCyc\] is exactly the same as Definition \[cycWithBun\]. Let $K_*(X;(P,F);A)$ denote the set of isomorphism classes of cycles modulo the natural generalization of the equivalence discussed in Definition \[GeoGroup\]. The generalizations of the two main theorems in Section \[geoModSec\] are as follows. We will not give the proofs since they can be obtained using our previous proofs with only additional notational complexity.
Let $X$ be a finite CW-complex, $A$ be a unital $C^*$-algebra, $P$ be a smooth compact $spin^c$-manifold that has trivial stable normal bundle and well-defined dimension modulo two, and $F$ be a vector bundle over $P$. Then, the following sequence is exact
$\begin{CD}
K_0(X;A) @>\Phi>> K_{{\rm dim}(P)}(X;A) @>r>> K_{{\rm dim}(P)}(X;(P,F);A) \\
@A\delta AA @. @V\delta VV \\
K_{{\rm dim}(P)+1}(X;(P,F);A) @<r<< K_{{\rm dim}(P)+1}(X;A) @<\Phi<< K_1(X;A)
\end{CD}$
where the maps are defined as follows
1. $\Phi$ is defined at the level of cycles via $$(M,E,f) \mapsto (M \times P, (\pi_M)^*(E) \otimes (\pi_P)^*(F), f \circ \pi_M)$$ where $\pi_M$ (respectively $\pi_P$) denotes the projection map from $M\times P$ to $M$ (respectively $P$);
2. $r$ is defined at the level of cycles via $$(M, E, f) \mapsto ((M,\emptyset), E, f);$$
3. $\delta$ is defined at the level of cycles via $$((Q,\beta Q), (E_Q, E_{\beta Q}), f) \mapsto (\beta Q, E_{\beta Q}, f|_{\beta Q}).$$
Let $X$ be a finite CW-complex, $A$ be a unital $C^*$-algebra, $P$ and $P^{\prime}$ be smooth, compact, ${\rm spin^c}$-manifolds, each of which has a trivial stable normal bundle and well-defined dimension, and $F$ and $F^{\prime}$ be vector bundles over $P$ and $P^{\prime}$ respectively. If $(P, [F])\sim (P^{\prime}, [F^{\prime}])$ as elements in $K_*(pt)$, then $K_*(X;(P,F);A) \cong K_*(X;(P^{\prime},F^{\prime});A)$; moreover, the isomorphism is natural (with respect to both $X$ and $A$).
Let $(P,F)=(S^2, F_{{\rm Bott}})$ where $F_{{\rm Bott}}$ is the Bott bundle (see for example [@BD]). Then the map $\Phi_{(S^2,F_{{\rm Bott}})}$ is the Bott periodicity isomorphism; hence $K_*(X;(P,F);A)\cong \{0\}$.
\
[**Acknowledgments**]{}\
The author thanks Magnus Goffeng and Thomas Schick for discussions. This work began while the auther held an NSERC Postdoctoral Fellowship at Georg-August Universit${\rm \ddot{a}}$t, G${\rm \ddot{o}}$ttingen.
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Email address: [email protected]\
[ Universit${\rm \acute{e}}$ Blaise Pascal, Clermont-Ferrand II, Laboratoire de Math${\rm \acute{e}}$matiques, Campus des C${\rm \acute{e}}$zeaux B.P. 80026 63177 Aubi${\rm \grave{e}}$re cedex, France]{}
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abstract: 'The Kontsevich-Soibelman solution of the cyclic version of Deligne’s conjecture and the formality of the operad of little discs on a cylinder provide us with a natural homotopy calculus structure on the pair $({C^{\bullet}}(A), {C_{\bullet}}(A))$ “Hochschild cochains $+$ Hochschild chains” of an associative algebra $A$. We show that for an arbitrary smooth algebraic variety $X$ over a field ${{\mathbb K}}$ of characteristic zero the sheaf $({C^{\bullet}}({{\cal O}}_X), {C_{\bullet}}({{\cal O}}_X))$ of homotopy calculi is formal. This result was announced in paper [@TT1] by the second and the third author.'
author:
- 'Vasiliy Dolgushev, Dmitry Tamarkin, and Boris Tsygan'
title: 'Formality of the homotopy calculus algebra of Hochschild (co)chains'
---
[*To Mikhail Olshanetsky on the occasion of his 70th birthday.*]{}
Introduction
============
The standard Cartan calculus on polyvector fields and exterior forms can be naturally extended to the Hochschild cohomology $HH^{{{\bullet}}}(A,A)$ and the Hochschild homology $HH_{{{\bullet}}}(A,A)$ of an arbitrary associative algebra $A$ [@DGT], [@Rinehart]. This calculus is induced by simple operations on Hochschild (co)chains, and the identities of this algebraic structure hold for these operations up to homotopy.
The Kontsevich-Soibelman proof of the cyclic version of Deligne’s conjecture [@K-Soi1] and the formality of the operad of little discs on a cylinder[^1] imply that this nice collection of the operations on the pair $({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A))$ “(normalized) Hochschild cochains $+$ (normalized) Hochschild chains” can be extended to an $\infty$- or homotopy calculus structure.
This homotopy calculus structure on the pair $({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A))$ is a natural generalization of the homotopy Gerstenhaber algebra structure on the cochains ${C^{\bullet}_{{\rm norm}}}(A)$. In paper [@BLT] we proved the formality of this homotopy Gerstenhaber algebra on ${C^{\bullet}_{{\rm norm}}}(A)$ for an arbitrary regular commutative algebra $A$ over a field ${{\mathbb K}}$ of characteristic zero. In this paper we extend this result to the homotopy calculus algebra on the pair $({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A))$.
As well as in [@BLT] we also consider the situation when the algebra $A$ is replaced by the structure sheaf ${{\cal O}}_X$ of a smooth algebraic variety $X$ over the field ${{\mathbb K}}$. More precisely, we consider the homotopy calculus algebra on the pair $({C^{\bullet}_{{\rm norm}}}({{\cal O}}_X), {C_{\bullet}^{{\rm norm}}}({{\cal O}}_X))$ where ${C^{\bullet}_{{\rm norm}}}({{\cal O}}_X)$ and ${C_{\bullet}^{{\rm norm}}}({{\cal O}}_X)$ is, respectively, the sheaf of (normalized) Hochschild cochains and the sheaf of (normalized) Hochschild chains of ${{\cal O}}_X$. In this paper we show that the sheaf of homotopy calculi $({C^{\bullet}_{{\rm norm}}}({{\cal O}}_X), {C_{\bullet}^{{\rm norm}}}({{\cal O}}_X))$ is formal.
If $A$ is an associative algebra (with unit), the pair $({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A))$ is also equipped with an algebraic structure defined by a degree $-1$ Lie bracket on ${C^{\bullet}_{{\rm norm}}}(A)$, a degree $-1$ Lie module structure on ${C_{\bullet}^{{\rm norm}}}(A)$ over ${C^{\bullet}_{{\rm norm}}}(A)$, and Connes’ operator on ${C_{\bullet}^{{\rm norm}}}(A)$ which is compatible with the Lie module structure. In the paper we refer to such algebra structures as ${{\Lambda}}{{\bf Lie}}^+_{{{\delta}}}$-algebra. (See Definition \[lie+de\].)
In paper [@Tsygan] the third author conjectured that if $A$ is a regular commutative algebra then this ${{\Lambda}}{{\bf Lie}}^+_{{{\delta}}}$-algebra structure on $({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A))$ is formal. This conjecture was proved in [@W] (at least in the case ${{\mathbb R}}\subset {{\mathbb K}}$) by Willwacher who used the constructions of B. Shoikhet [@Sh] and the first author [@FTHC].
In general $Ho({{\Lambda}}{{\bf Lie}}^+_{{{\delta}}})$-part of the homotopy calculus structure on $({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A))$ derived from [@K-Soi1] may not coincide with the ${{\Lambda}}{{\bf Lie}}^+_{{{\delta}}}$-algebra on the pair $({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A))$. However, we show that this homotopy calculus algebra on $({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A))$ is quasi-isomorphic to another homotopy calculus algebra on $({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A))$ whose $Ho({{\Lambda}}{{\bf Lie}}^+_{{{\delta}}})$-part is the ordinary ${{\Lambda}}{{\bf Lie}}^+_{{{\delta}}}$-algebra given by the above Lie bracket on ${C^{\bullet}_{{\rm norm}}}(A)$, the Lie algebra module on ${C_{\bullet}^{{\rm norm}}}(A)$ over ${C^{\bullet}_{{\rm norm}}}(A)$ and Connes’ operator on ${C_{\bullet}^{{\rm norm}}}(A)$. In this sense, the formality of the homotopy calculus algebra on $({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A))$ is a generalization of Willwacher’s cyclic formality theorem [@W].
The organization of the paper is as follows. In Section 2 we fix the notation and recall required results about (co)operads and (co)algebras. Section 3 is devoted to $\infty$- or homotopy versions for the algebras over the operads ${{\bf calc}}$, ${{\bf e_2}}$, and ${{\bf Lie}}^+_{{{\delta}}}$. In Section 4 we recall the Kontsevich-Soibelman operad and the operad ${{\rm C y l}\,}$ of little discs on a cylinder. We show that the homology operad $H_{-{{\bullet}}}({{\rm C y l}\,}, {{\mathbb K}})$ of ${{\rm C y l}\,}$ with the reversed grading is the operad of calculi. Finally we recall required results from [@K-Soi1] and prove a useful property of the Kontsevich-Soibelman operad. Section 5 is devoted to properties of the homotopy calculus algebra on the pair $({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A))$. In Section 6 we formulate and prove the main result of this paper. (See Theorem \[main\] on page .) In the concluding section we discussion applications and generalizations of Theorem \[main\]. We also discuss recent articles related to our main result.
\
[**Acknowledgment.**]{} A bigger part of this work was done when V.D. was a Boas Assistant Professor of Mathematics Department at Northwestern University. During these two years V.D. benefited from working at Northwestern so much that he feels as if he finished one more graduate school. V.D. cordially thanks Mathematics Department at Northwestern University for this time. The results of this work were presented at the conference Poisson 2008 in Lausanne. We would like to thank the participants of this conference for questions and useful comments. V.D. would like to thank Pavel Snopok for showing him a very convenient drawing program “Inkscape”. D.T. and B.T. are supported by NSF grants. The work of V.D. is partially supported by the Grant for Support of Scientific Schools NSh-8065.2006.2.
Preliminaries
=============
(Co)operads and (co)algebras
----------------------------
Most of the notation and conventions for (co)operads and their (co)algebras are borrowed from [@BLT].
Depending on a context our underlying symmetric monoidal category is either the category of graded vector spaces, or the category of chain complexes, or the category of compactly generated topological spaces, or the category of sets. By suspension ${{\bf s}\,}{{\cal V}}$ of a graded vector space (or a chain complex) ${{\cal V}}$ we mean ${{\varepsilon}}\otimes {{\cal V}}$, where ${{\varepsilon}}$ is a one-dimensional vector space placed in degree $+1$. For a vector $v\in {{\cal V}}$ we denote by $|v|$ its degree. The symmetric group of permutations of $n$ letters is denoted by $S_n$. The underlying field ${{\mathbb K}}$ has characteristic zero.
For an operad ${{\cal O}}$ we denote by ${{\rm Alg}\,}_{{{\cal O}}}$ the category of algebras over the operad ${{\cal O}}$. Dually, for a cooperad ${{\cal C}}$ we denote by ${{\rm Coalg}\,}_{{{\cal C}}}$ the category of nilpotent[^2] coalgebras over the cooperad ${{\cal C}}$. By [*corestriction*]{} we mean the canonical map $$\label{corest}
\rho_{{{\cal V}}} : {{\mathbb F}}_{{{\cal C}}}({{\cal V}}) \to {{\cal V}}$$ from the free coalgebra ${{\mathbb F}}_{{{\cal C}}}({{\cal V}})$ to the vector space of its cogenerators ${{\cal V}}$. We often omit the subscript in the notation $\rho_{{{\cal V}}}$ for the corestriction.
For a polynomial functor ${{\cal P}}$ we denote by ${{\mathbb T}}({{\cal P}})$ (resp. ${{\mathbb T}}^*({{\cal P}})$) the free operad (resp. the free cooperad) (co)generated by ${{\cal P}}$. The notation $\bullet$ is reserved for the monoidal product of the polynomial functors. Thus, if ${{\cal P}}$ and ${{\cal Q}}$ are polynomial functors then $$\label{bullet}
{{\cal P}}\bullet {{\cal Q}}(n) =
\bigoplus_{k_1+ \dots + k_m = n}
{{\cal P}}(m) \otimes_{S_m} ( {{\cal Q}}(k_1) \otimes
\dots \otimes {{\cal Q}}(k_m) )\,.$$ This formula can be easily generalized to the colored polynomial functors.
By “suspension” of a (co)operad ${{\cal O}}$ of graded vector spaces (or chain complexes) we mean the (co)operad ${{\Lambda}}({{\cal O}})$ whose $m$-th vector space is $$\label{susp-op}
{{\Lambda}}({{\cal O}})(m) = {{\bf s}\,}^{1-m} {{\cal O}}(m) \otimes {{\rm s g n}}_{m}\,,$$ where ${{\rm s g n}}_{m}$ is the sign representation of the symmetric group $S_m$.
For a commutative algebra ${{\cal B}}$ and a ${{\cal B}}$-module ${{\cal V}}$ we denote by $S_{{{\cal B}}}({{\cal V}})$ the symmetric algebra of ${{\cal V}}$ over ${{\cal B}}$. $S_{{{\cal B}}}^m({{\cal V}})$ stands for the $m$-th component of this algebra. If ${{\cal B}}={{\mathbb K}}$ then ${{\cal B}}$ is omitted from the notation. The abbreviation “DGLA” stands for differential graded Lie algebra.
We denote by $\ast$ the polynomial functor $$\label{ast}
\ast(n) =
\begin{cases}
{{\mathbb K}}\,, \qquad {\rm if} ~~ n = 1 \,, \\
0 \,, \qquad {\rm otherwise}\,.
\end{cases}$$ This functor carries the unique structure of the operad (resp. the cooperad) such that $\ast$ is the initial (resp. the terminal) object in the category of operads (resp. cooperads) of graded vector spaces or chain complexes. There is an obvious generalization of $\ast$ (\[ast\]) to the category of sets and to the category of topological spaces. However, we will need $\ast$ only for linear (co)operads, i.e. the (co)operads in the category of graded vector spaces or the category of chain complexes.
All the linear operads (resp. linear cooperads), we consider, are equipped with an augmentation (resp. coaugmentation). In other words, for every operad ${{\cal O}}$ we will have a chosen morphism of operads: $$\label{aug}
\tau\, :\, {{\cal O}}\to \ast\,.$$ Dually for every cooperad ${{\cal C}}$ we will have a chosen morphism of cooperads $$\label{coaug}
{{\kappa}}\, :\, \ast \to {{\cal C}}\,.$$
We are going to deal with $2$-colored (co)operads. Throughout the paper we label the two colors of all $2$-colored (co)operads by ${{\mathfrak{c}}}$ and ${{\mathfrak{a}}}$. For example, the notation ${{\bf Lie}}^+$ is reserved for the $2$-colored operad which governs the pairs “Lie algebra ${{\cal V}}$ and a Lie algebra module ${{\cal W}}$ over ${{\cal V}}$.” Vectors of the Lie algebra ${{\cal V}}$ are colored by ${{\mathfrak{c}}}$ and vectors of the module ${{\cal W}}$ are colored by ${{\mathfrak{a}}}$.
For a linear $2$-colored operad ${{\cal O}}$ we will denote by ${{\cal O}}^{{{\mathfrak{c}}}}(n,k)$ (resp. ${{\cal O}}^{{{\mathfrak{a}}}}(n,k)$) the vector space of operations producing a vector with the color ${{\mathfrak{c}}}$ (resp. ${{\mathfrak{a}}}$) from $n$ vectors with the color ${{\mathfrak{c}}}$ and $k$ vectors with the color ${{\mathfrak{a}}}$. We use the same notation for the linear $2$-colored cooperads and for topological $2$-colored operads.
The polynomial functor $\ast$ (\[ast\]) has the obvious generalization to the category of linear $2$-colored (co)operads: $$\label{ast-color}
\begin{array}{c}
\ast^{{{\mathfrak{c}}}}(n,k) =
\begin{cases}
{{\mathbb K}}\,, \qquad {\rm if} ~~ (n,k)= (1,0) \,, \\
0 \,, \qquad {\rm otherwise}\,.
\end{cases} \\[0.5cm]
\ast^{{{\mathfrak{a}}}}(n,k) =
\begin{cases}
{{\mathbb K}}\,, \qquad {\rm if} ~~ (n,k)= (0,1)\,, \\
0 \,, \qquad {\rm otherwise}\,.
\end{cases}
\end{array}$$
For a linear operad ${{\cal O}}$ we denote by $Bar({{\cal O}})$ its bar construction. Dually, for a linear cooperad ${{\cal C}}$ we denote by $Cobar({{\cal C}})$ its cobar construction.
We recall that, as a cooperad of graded vector spaces, $Bar({{\cal O}})$ is freely generated by the polynomial functor ${{\bf s}\,}^{-1}{\overline{\cal O}}$, where ${\overline{\cal O}}$ is the kernel of the augmentation (\[aug\]). Dually, as an operad of graded vector spaces, $Cobar({{\cal C}})$ is freely generated by the polynomial functor ${{\bf s}\,}{\overline{\cal C}}$, where ${\overline{\cal C}}$ is cokernel of the coaugmentation (\[coaug\]). The differential ${{\partial}}^{Bar}$ on the operad $Bar({{\cal O}})$ is defined using the multiplication of the operad ${{\cal O}}$ and the differential ${{\partial}}^{Cobar}$ on the cooperad $Cobar({{\cal C}})$ is defined using the comultiplication of the cooperad ${{\cal C}}$. See Chapter 3 in [@Fresse] or Section 2 in [@GJ] for details.
For a quadratic operad ${{\cal O}}$ there is a natural sub-cooperad ${{\cal O}}^{\vee}$ of $Bar({{\cal O}})$ which satisfies the property: $${{\partial}}^{Bar} \Big|_{{{\cal O}}^{\vee}} = 0\,.$$ The details of the construction of ${{\cal O}}^{\vee}$ can be found in Section 5.2 in [@Fresse]. Following [@GK] we call ${{\cal O}}^{\vee}$ the Koszul dual cooperad of ${{\cal O}}$.
For a linear operad ${{\cal O}}$ (resp. linear cooperad ${{\cal C}}$) and a vector space ${{\cal V}}$ we denote by ${{\mathbb F}}_{{{\cal O}}}({{\cal V}})$ (resp. by ${{\mathbb F}}_{{{\cal C}}}({{\cal V}})$) the free algebra (resp. free coalgebra) over the operad ${{\cal O}}$ (resp. cooperad ${{\cal C}}$). For a linear $2$-colored (co)operad ${{\cal O}}$ the functor[^3] ${{\mathbb F}}_{{{\cal O}}}$ splits according to the colors ${{\mathfrak{c}}}$ and ${{\mathfrak{a}}}$ as $${{\mathbb F}}_{{{\cal O}}}({{\cal V}}, {{\cal W}}) =
{{\mathbb F}}_{{{\cal O}}}({{\cal V}}, {{\cal W}})_{{{\mathfrak{c}}}} \oplus
{{\mathbb F}}_{{{\cal O}}}({{\cal V}}, {{\cal W}})_{{{\mathfrak{a}}}}\,,$$ where $${{\mathbb F}}_{{{\cal O}}}({{\cal V}}, {{\cal W}})_{{{\mathfrak{c}}}} =
\bigoplus_{n,k}{{\cal O}}^{{{\mathfrak{c}}}}(n,k)
\otimes_{S_n \times S_k} {{\cal V}}^{\otimes \, n} \otimes {{\cal W}}^{\otimes \, k}\,,$$ and $${{\mathbb F}}_{{{\cal O}}}({{\cal V}}, {{\cal W}})_{{{\mathfrak{a}}}} =
\bigoplus_{n,k}{{\cal O}}^{{{\mathfrak{a}}}}(n,k)
\otimes_{S_n \times S_k} {{\cal V}}^{\otimes \, n} \otimes {{\cal W}}^{\otimes \, k}\,.$$
We need to recall some facts about algebras over the operad $Cobar({{\cal C}})$ for a coaugmented cooperad ${{\cal C}}$.
Since $Cobar({{\cal C}})$ is freely generated by the suspension ${{\bf s}\,}{\overline{\cal C}}$ of the cokernel ${\overline{\cal C}}$ of the coaugmentation (\[coaug\]) a $Cobar({{\cal C}})$-algebra structure on a chain complex ${{\cal V}}$ is uniquely determined by the restriction of the multiplication map $$\mu: {{\mathbb F}}_{Cobar({{\cal C}})}({{\cal V}}) \to {{\cal V}}$$ to the subspace $${{\mathbb F}}_{{{\bf s}\,}{\overline{\cal C}}}({{\cal V}}) \subset {{\mathbb F}}_{Cobar({{\cal C}})}({{\cal V}})\,.$$ In other words, a $Cobar({{\cal C}})$-algebra structure on ${{\cal V}}$ is uniquely determined by a degree $1$ map from ${{\mathbb F}}({\overline{\cal C}})({{\cal V}})$ to ${{\cal V}}$.
It turns out that the maps from ${{\mathbb F}}({\overline{\cal C}})({{\cal V}})$ to ${{\cal V}}$ have a elegant description in terms of coderivations of the free coalgebra ${{\mathbb F}}_{{{\cal C}}}({{\cal V}})$. To recall this description we introduce the Lie subalgebra $$\label{Coder-prime}
{\rm C o d e r}'({{\mathbb F}}_{{{\cal C}}}({{\cal V}})) =
\{Q \in {\rm C o d e r}({{\mathbb F}}_{{{\cal C}}}({{\cal V}})) ~|~
Q\Big|_{{{\cal V}}} = 0 \}\,,$$ where ${{\cal V}}$ is considered as a subspace of ${{\cal C}}(1)\otimes {{\cal V}}$ via the coaugmentation (\[coaug\]). In other words, the elements of ${\rm C o d e r}'({{\mathbb F}}_{{{\cal C}}}({{\cal V}}))$ are coderivations of the free coalgebra ${{\mathbb F}}_{{{\cal C}}}({{\cal V}})$ which can be factored through the projection $${{\mathbb F}}_{{{\cal C}}}({{\cal V}}) \to {{\mathbb F}}_{{\overline{\cal C}}}({{\cal V}})\,.$$ It is not hard to see that the subspace (\[Coder-prime\]) is closed under the commutator and the differentials coming from ${{\cal C}}$ and ${{\cal V}}$. Thus the graded vector space ${\rm C o d e r}'({{\mathbb F}}_{{{\cal C}}}({{\cal V}}))$ is in fact a DGLA.
Let us recall from [@GJ] the following proposition
\[coder-cofree\] For a coaugmented cooperad ${{\cal C}}$ the composition with the corestriction (\[corest\]) $\rho_{{{\cal V}}}: {{\mathbb F}}_{{{\cal C}}}({{\cal V}}) \to {{\cal V}}$ induces an isomorphism of graded vector spaces $$\label{der-free}
Coder'({{\mathbb F}}_{{{\cal C}}}({{\cal V}})) \cong
{{\rm Hom}\,}({{\mathbb F}}_{{\overline{\cal C}}}({{\cal V}}), {{\cal V}})\,,$$ where, as above, ${\overline{\cal C}}$ is the cokernel of the coaugmentation (\[coaug\]) of ${{\cal C}}$.
Due to this proposition a $Cobar({{\cal C}})$-algebra structure on a chain complex ${{\cal V}}$ is uniquely determined by a degree $1$ coderivation $$\label{deriv-Q}
Q \in Coder'({{\mathbb F}}_{{{\cal C}}}({{\cal V}}))\,.$$
According to Proposition 2.15 from [@GJ] the compatibility of the $Cobar({{\cal C}})$-algebra structure on ${{\cal V}}$ with the total differential on $Cobar({{\cal C}})$ and the differential on ${{\cal V}}$ is equivalent to the Maurer-Cartan equation for the corresponding derivation (\[deriv-Q\]): $$\label{MC-deriv-Q}
[{{\partial}}^{{{\cal C}}} + {{\partial}}^{{{\cal V}}},Q] + \frac{1}{2}[Q, Q] =0\,,$$ where ${{\partial}}^{{{\cal C}}}$ is the differential on ${{\mathbb F}}_{{{\cal C}}}({{\cal V}})$ induced by the one on the cooperad ${{\cal C}}$ and ${{\partial}}^{{{\cal V}}}$ comes from that on ${{\cal V}}$.
In other words,
\[GJ\] There is a natural bijection between the Maurer-Cartan elements of the DGLA $Coder'({{\mathbb F}}_{{{\cal C}}}({{\cal V}}))$ and and the $Cobar({{\cal C}})$-algebra structures on ${{\cal V}}$.
If we have a map $$\label{mu}
\mu : {{\cal C}}_1 \to {{\cal C}}_2$$ of coaugmented cooperads then the corresponding map between the cobar constructions $$Cobar(\mu) : Cobar({{\cal C}}_1) \to Cobar({{\cal C}}_2)$$ allows us to pull $Cobar({{\cal C}}_2)$-algebra structure on ${{\cal V}}$ to a $Cobar({{\cal C}}_1)$-algebra on ${{\cal V}}$.
We claim that
\[how-to-pull\] If $Q_1$ is a Maurer-Cartan element of the DGLA $Coder'({{\mathbb F}}_{{{\cal C}}_1}({{\cal V}}))$ corresponding to a $Cobar({{\cal C}}_1)$-algebra structure on ${{\cal V}}$ and $Q_2$ is a Maurer-Cartan element of the DGLA $Coder'({{\mathbb F}}_{{{\cal C}}_2}({{\cal V}}))$ corresponding to a $Cobar({{\cal C}}_2)$-algebra structure on ${{\cal V}}$ then $$\label{pull-MC}
\rho_{{{\cal V}}} \circ Q_1 = \rho_{{{\cal V}}} \circ Q_2 \circ {{\mathbb F}}(\mu)\,,$$ where the map $${{\mathbb F}}(\mu) : {{\mathbb F}}_{{\overline{\cal C}}_1}({{\cal V}}) \to {{\mathbb F}}_{{\overline{\cal C}}_2}({{\cal V}})$$ is induced by (\[mu\]).
[**Proof.**]{} Let $$\nu_2 :
{{\mathbb F}}_{Cobar({{\cal C}}_2)}({{\cal V}}) \to {{\cal V}}$$ be the $Cobar({{\cal C}}_2)$-algebra structure on ${{\cal V}}$. Then the $Cobar({{\cal C}}_1)$-algebra structure on ${{\cal V}}$ $$\nu_1 :
{{\mathbb F}}_{Cobar({{\cal C}}_1)}({{\cal V}}) \to {{\cal V}}$$ is obtained by composing the map $\nu_2$ with the map $${{\mathbb F}}(Cobar(\mu)) :
{{\mathbb F}}_{Cobar({{\cal C}}_1)}({{\cal V}}) \to
{{\mathbb F}}_{Cobar({{\cal C}}_2)}({{\cal V}})\,.$$ It is not hard to see that the restriction of $\nu_1$ to the subspace ${{\mathbb F}}_{{{\bf s}\,}{\overline{\cal C}}_1}({{\cal V}})$ coincides with the composition of the maps $${{\mathbb F}}_{{{\bf s}\,}{\overline{\cal C}}_1}({{\cal V}}) \stackrel{{{\mathbb F}}(\mu)}{\to}
{{\mathbb F}}_{{{\bf s}\,}{\overline{\cal C}}_2}({{\cal V}})$$ and $$\nu_2 \Big|_{{{\mathbb F}}_{{{\bf s}\,}{\overline{\cal C}}_2}({{\cal V}})} :
{{\mathbb F}}_{{{\bf s}\,}{\overline{\cal C}}_2}({{\cal V}}) \to {{\cal V}}\,.$$ Thus the proposition follows from the equation $$\nu_i \Big|_{{{\mathbb F}}_{{{\bf s}\,}{\overline{\cal C}}_i}({{\cal V}})}
= \rho_{{{\cal V}}}\circ Q_i \circ {{\sigma}}\,,$$ where $\rho_{{{\cal V}}}$ is the corestriction (\[corest\]) and ${{\sigma}}$ is the suspension isomorphism $${{\sigma}}: {{\mathbb F}}_{{{\bf s}\,}{\overline{\cal C}}_i}({{\cal V}}) \to {{\mathbb F}}_{{\overline{\cal C}}_i}({{\cal V}})\,,$$ and $i=1,2$. $\Box$
We will freely use Propositions \[coder-cofree\], \[GJ\] and \[how-to-pull\] for colored cooperads.
We remark that all $2$-colored (co)operads, we consider, satisfy the following property: [*an argument with the color ${{\mathfrak{a}}}$ can enter an operation at most once. If an argument with this color enters an operation then the resulting color is also ${{\mathfrak{a}}}$. Otherwise the resulting color is ${{\mathfrak{c}}}$.* ]{} In other words, for every $n$ $$\label{P}
\begin{array}{cc}
{{\cal O}}^{{{\mathfrak{c}}}}(n,k) = {{\cal O}}^{{{\mathfrak{a}}}}(n,k) = \{ \bf 0\}\,, &
\forall ~~ k > 1\,,\\[0.3cm]
{{\cal O}}^{{{\mathfrak{a}}}}(n,0) = \{ \bf 0\}\,, & {{\cal O}}^{{{\mathfrak{c}}}}(n,1) = \{ \bf 0\}
\end{array}$$ for the (co)operads of graded vector spaces or chain complexes and $$\label{P-top}
\begin{array}{cc}
{{\cal O}}^{{{\mathfrak{c}}}}(n,k) = {{\cal O}}^{{{\mathfrak{a}}}}(n,k) = \emptyset\,, &
\forall ~~ k > 1\,,\\[0.3cm]
{{\cal O}}^{{{\mathfrak{a}}}}(n,0) = \emptyset\,, & {{\cal O}}^{{{\mathfrak{c}}}}(n,1) = \emptyset
\end{array}$$ for the (co)operads of topological spaces or sets.
It is not hard to see that bar and cobar constructions the (co)operads of graded vector spaces or chain complexes preserve property (\[P\]).
Let us recall that
\[G-alg\] A graded vector space ${{\cal V}}$ is a Gerstenhaber algebra if it is equipped with a commutative and associative product $\wedge$ of degree $0$ and a Lie bracket $[\,,\,]$ of degree $-1$. These operations have to be compatible in the sense of the following Leibniz rule $$\label{cup-Lie}
[a, b \wedge c] =
[a, b] \wedge c +
(-1)^{(|a|+1)|b|} b \wedge [a, c]\,,$$ where $a,b,c$ are homogeneous vectors of ${{\cal V}}$.
\[precalc\] A precalculus is a pair of a Gerstenhaber algebra $({{\cal V}}, \wedge, [,])$ and a graded vector space ${{\cal W}}$ together with
- a module structure $i_{{{\bullet}}} \,:\, {{\cal V}}\otimes {{\cal W}}\mapsto {{\cal W}}$ of the graded commutative algebra ${{\cal V}}$ on ${{\cal W}}$,
- an action $l_{{{\bullet}}} \,:\, {{\bf s}\,}^{-1} {{\cal V}}\otimes {{\cal W}}\mapsto {{\cal W}}$ of the Lie algebra ${{\bf s}\,}^{-1} {{\cal V}}$ on ${{\cal W}}$ which are compatible in the sense of the following equations $$\label{l-i}
i_a l_b - (-1)^{|a|(|b|+1)}
l_b i_a = i_{[a,b]}\,,$$ and $$\label{l-cup}
l_{a\wedge b} = l_a i_{b} + (-1)^{|a|}i_a l_b \,.$$
Furthermore,
\[calc\] A calculus is a precalculus $({{\cal V}},{{\cal W}}, [,], \wedge, i_{{{\bullet}}}, l_{{{\bullet}}})$ with a degree $-1$ unary operation ${{\delta}}$ on ${{\cal W}}$ such that $$\label{l-i-delta}
\delta \, i_{a} - (-1)^{|a|} i_{a} \, \delta =
l_a\,,$$ and[^4] $$\label{de-2}
{{\delta}}^2 = 0\,.$$
We call $l$ and $i$ the Lie derivative and the contraction, respectively.
We will use the following list of (co)operads:
- ${{\bf Lie}}$ (resp. ${{\bf coLie}}$) is the operad of Lie algebras (resp. the cooperad of Lie coalgebras),
- ${{\bf comm}}$ (resp. ${{\bf cocomm}}$) is the operad of commutative (associative) algebras (resp. the operad of cocommutative coassociative coalgebras),
- ${{\bf e_2}}$ denotes the operad of Gerstenhaber algebras, (see Definition \[G-alg\]),
- ${{\bf K S}}$ denotes the operad of M. Kontsevich and Y. Soibelman. This operad[^5] is described in sections 11.1, 11.2 and 11.3 of [@K-Soi1],
- ${{\bf Lie}}^+$ (resp. ${{\bf coLie}}^+$) denotes the $2$-colored operad of pairs “Lie algebra $+$ its module” (resp. the $2$-colored cooperad of pairs “Lie coalgebra $+$ its comodule”),
- ${{\bf comm}}^+$ (resp. ${{\bf cocomm}}^+$) denotes the $2$-colored operad of pairs “commutative algebra $+$ its module” (resp. the $2$-colored cooperad of pairs “cocommutative coalgebra $+$ its comodule”),
- ${{\bf pcalc}}$ denotes the $2$-colored operad of precalculi, (see Definition \[precalc\]),
- ${{\bf calc}}$ denotes the $2$-colored operad of calculi, (see Definition \[calc\]),
- ${{\bf assoc}}$ is the non-symmetric operad of sets controlling unital monoids; each set ${{\bf assoc}}(n)$, $n\geq 0$, is a point.
It is not hard to show that for the vector space of the free calculus algebra generated by the pair $({{\cal V}}, {{\cal W}})$ we have $$\label{free-calc}
{{\mathbb F}}_{{{\bf calc}}}({{\cal V}}, {{\cal W}}) \cong {{\mathbb F}}_{{{\bf comm}}^+}({{\mathbb F}}_{{{\Lambda}}{{\bf Lie}}^+}({{\cal V}},
{{\cal W}}\oplus {{\bf s}\,}^{-1}\, {{\cal W}}))\,.$$ In other words, for the color components we have the isomorphisms of graded vector spaces: $$\label{free-calc-mc}
{{\mathbb F}}_{{{\bf calc}}}({{\cal V}}, {{\cal W}})_{{{\mathfrak{c}}}} \cong {{\mathbb F}}_{{{\bf comm}}}( {{\mathbb F}}_{{{\Lambda}}{{\bf Lie}}}({{\cal V}}) )\,,$$ and $$\label{free-calc-ma}
{{\mathbb F}}_{{{\bf calc}}}({{\cal V}}, {{\cal W}})_{{{\mathfrak{a}}}} \cong
{{\mathbb F}}_{{{\bf comm}}^+}({{\mathbb F}}_{{{\Lambda}}{{\bf Lie}}}({{\cal V}}), {{\mathbb F}}_{{{\Lambda}}{{\bf Lie}}^+}({{\cal V}},
{{\cal W}}\oplus {{\bf s}\,}^{-1}\, {{\cal W}})_{{{\mathfrak{a}}}} )_{{{\mathfrak{a}}}} \,.$$
Hochschild (co)chain complexes
------------------------------
For an associative algebra $A$ $${C^{\bullet}}(A) = {{\rm Hom}\,}(A^{\otimes {{\bullet}}}, A)$$ denotes the Hochschild cochain complex and $${C_{\bullet}}(A)= A\otimes A^{\otimes (-{{\bullet}})}$$ stands for the Hochschild chain complex of $A$ with the reversed grading.
For the normalized versions of these complexes we reserve the notation: $${C^{\bullet}_{{\rm norm}}}(A) = \{ P\in
Hom (A^{\otimes {{\bullet}}}, A)\,\, |\,\,
P(\dots ,1, \dots) = 0 \}$$ and $${C_{\bullet}^{{\rm norm}}}(A)= A\otimes (A/{{\mathbb K}}\, 1)^{\otimes (-{{\bullet}})}\,.$$
- The notation ${{\partial}}^{Hoch}$ is reserved both for the Hochschild coboundary operator on ${C^{\bullet}_{{\rm norm}}}(A)$ and Hochschild boundary operator on ${C_{\bullet}^{{\rm norm}}}(A)$ $$({{\partial}}^{Hoch} P)(a_0, a_1, \dots, a_k) = a_0 P(a_1, \dots, a_k) -
P(a_0 a_1, \dots, a_k) + P(a_0, a_1 a_2, a_3, \dots , a_k) -
\dots$$ $$+ (-1)^{k} P(a_0, \dots , a_{k-2}, a_{k-1} a_k) +
(-1)^{k+1} P(a_0, \dots , a_{k-2}, a_{k-1}) a_k$$
$${{\partial}}^{Hoch}(a_0, a_1, \dots, a_m) =
(a_0 a_1, a_2, \dots, a_m) - (a_0, a_1 a_2, a_3, \dots, a_m) +
\dots +$$ $$(-1)^{m-1}(a_0, \dots, a_{m-2}, a_{m-1} a_m) +
(-1)^m (a_m a_0, a_1, a_2, \dots, a_{m-1})\,,$$ $$a_i\in A\,, \qquad P \in C^k_{{\rm norm}}(A)\,.$$
- The notation $\cup$ is reserved for the cup-product on ${C^{\bullet}_{{\rm norm}}}(A)$ $$\label{cup}
P_1\cup P_2 (a_1, a_2, \dots, a_{k_1 + k_2}) =
P_1(a_1, \dots, a_{k_1}) P_2(a_{k_1+1}, \dots, a_{k_1+k_2})\,,$$ $$P_i \in C^{k_i}_{{\rm norm}}(A)\,.$$
- $[\, , \, ]_G$ stands for the Gerstenhaber bracket on ${C^{\bullet}_{{\rm norm}}}(A)$ $$[Q_1, Q_2]_{G} =$$ $$\label{Gerst}
\sum_{i=0}^{k_1}(-1)^{i k_2}
Q_1(a_0,\,\dots , Q_2 (a_i,\,\dots,a_{i+k_2}),\, \dots,
a_{k_1+k_2}) -
(-1)^{k_1 k_2} (1 \leftrightarrow 2)\,,$$ $$Q_i \in C^{k_i+1}_{{\rm norm}}(A)\,.$$
- $I_P(c)$ is the contraction of a Hochschild cochain $P \in C^k_{{\rm norm}}(A)$ with a Hochschild chain $c=(a_0, a_1, \dots, a_m)$ $$\label{I-P}
I_P (a_0, a_1, \dots, a_m) =
\begin{cases}
(a_0 P(a_1, \dots, a_k), a_{k+1}, \dots , a_m)\,, \qquad {\rm if} ~~ m\ge k \,, \\
0 \,, \qquad {\rm otherwise}\,.
\end{cases}$$
- $L_Q(c)$ denotes the Lie derivative of a Hochschild chain $c = (a_0, a_1, \dots, a_m)$ along a Hochschild cochain $Q\in C^{k+1}_{{\rm norm}}(A)$ $$\label{L-Q}
L_{Q}(a_0, a_1, \dots, a_m)=
\sum_{i=0}^{m-k}(-1)^{ki} (a_0 , \dots,
Q(a_i, \dots, a_{i+k}), \dots, a_m) +$$ $$\sum_{j=m-k}^{m-1}(-1)^{m(j+1)} ( Q(a_{j+1}, \dots, a_m, a_0, \dots, a_{k+j-m}),
a_{k+j+1-m}, \dots , a_j )\,.$$
- $B: {C_{\bullet}^{{\rm norm}}}(A) \to C^{{\rm norm}}_{{{\bullet}}-1}(A)$ denotes Connes’ operator $$\label{B}
B(a_0, a_1, \dots, a_m)=
\sum_{i=0}^{m}(-1)^{m i}
(1, a_i , \dots, a_m, a_0, a_1, \dots, a_{i-1} )\,.$$
The notation $HH^{{{\bullet}}}(A)$ (resp. $HH_{{{\bullet}}}(A)$) is used for the Hochschild cohomology (resp. homology groups) of $A$ with coefficients in $A$ $$HH^{{{\bullet}}}(A) = H^{{{\bullet}}}({C^{\bullet}_{{\rm norm}}}(A), {{\partial}}^{Hoch})\,,$$ $$HH_{{{\bullet}}}(A) = H^{{{\bullet}}}({C_{\bullet}^{{\rm norm}}}(A), {{\partial}}^{Hoch})\,.$$
To describe algebraic structures on pairs $({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A))$ and $(HH^{{{\bullet}}}(A), HH_{{{\bullet}}}(A))$ we use the the language of operads. Thus, the Gerstenhaber bracket $[\,,\,]_G$ equips the cochain complex ${C^{\bullet}_{{\rm norm}}}(A)$ with an algebra structure over the operad ${{\Lambda}}{{\bf Lie}}$ and the Lie derivative (\[L-Q\]) equips the pair $({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A))$ with the algebra structure over the operad ${{\Lambda}}{{\bf Lie}}^+$.
In order to add Connes’ operator (\[B\]) into this operadic picture we give one more definition
\[lie+de\] We say that the pair of graded vector spaces $({{\cal V}},{{\cal W}})$ is an algebra over the operad ${{\bf Lie}}^+_{{{\delta}}}$ if ${{\cal V}}$ is a Lie algebra, ${{\cal W}}$ is a module over ${{\cal V}}$ and ${{\cal W}}$ is equipped with a degree $-1$ unary operation ${{\delta}}$ satisfying the equations $$\label{Lie-de-2}
{{\delta}}^2 = 0\,,$$ and $$\label{l-delta}
[\delta, l_a] = 0\,, \qquad \forall ~~ a\in {{\cal V}}\,,$$ where $l$ is the action of ${{\cal V}}$ on ${{\cal W}}$.
Adding Connes’ operator $B$ into the picture we may say that the pair $({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A))$ is a ${{\Lambda}}{{\bf Lie}}^+_{{{\delta}}}$-algebra.
The operations (\[cup\]), (\[Gerst\]), (\[I-P\]), (\[L-Q\]) and (\[B\]) are closed with respect to the (co)boundary operator ${{\partial}}^{Hoch}$.
According to [@G] the operations $\cup$ (\[cup\]) and $[\,,\,]_G$ (\[Gerst\]) induce on $HH^{{{\bullet}}}(A)$ the structure of a Gerstenhaber algebra. Furthermore, it is known [@DGT] that the operations (\[cup\]), (\[Gerst\]), (\[I-P\]), (\[L-Q\]) and (\[B\]) induce on the pair $(HH^{{{\bullet}}}(A), HH_{{{\bullet}}}(A))$ the structure of the calculus algebra.
The operads ${{{\rm Ho}\,}({\bf calc})}$, ${{{\rm Ho}\,}({\bf e_2})}$, and ${{\rm Ho}\,}({{\bf Lie}}^+_{{{\delta}}})$
=====================================================================================================================
In this section we describe the homotopy versions for the algebras over the operads ${{\bf calc}}$, ${{\bf e_2}}$, and ${{\bf Lie}}^+_{{{\delta}}}$.
Description of the operads ${{{\rm Ho}\,}({\bf calc})}$ and ${{{\rm Ho}\,}({\bf e_2})}$
---------------------------------------------------------------------------------------
To describe the homotopy version of ${{\bf calc}}$-algebras we use the canonical cofibrant resolution $Cobar(Bar({{\bf calc}}))$. In other words, we set $$\label{Calc}
{{{\rm Ho}\,}({\bf calc})}= Cobar(Bar({{\bf calc}}))\,.$$
The cooperad $Bar({{\bf calc}})$ will be used throughout the paper. For this reason we reserve a short-hand notation $$\label{bB}
{{\bf B}}= Bar ({{\bf calc}})$$ for this cooperad.
Recall that, as a cooperad of graded vector spaces, ${{\bf B}}= Bar({{\bf calc}})$ is freely generated by the polynomial functor ${{\bf s}\,}^{-1}\overline{{{\bf calc}}}$, where $\overline{{{\bf calc}}}$ is the kernel of the augmentation.
We represent elements of the free coalgebra ${{\mathbb F}}_{{{\bf B}}}({{\cal V}}, {{\cal W}})$ and elements of the cooperad ${{\bf B}}$ graphically. Thus Figures \[prod\], \[brack\] represent the simplest elements of ${{\mathbb F}}_{{{\bf B}}}({{\cal V}}, {{\cal W}})_{{{\mathfrak{c}}}}$ with ${{\gamma}}_1$ and ${{\gamma}}_2$ being vectors in ${{\cal V}}$.
{width="35.00000%"} {width="35.00000%"}\
Figures \[contr\], \[Lieder\] show the simplest elements of ${{\mathbb F}}_{{{\bf B}}}({{\cal V}}, {{\cal W}})_{{{\mathfrak{a}}}}$ with ${{\gamma}}\in {{\cal V}}$ and $c\in {{\cal W}}$.
{width="45.00000%"} {width="45.00000%"}\
Figures \[d\] and \[u-m\] represent simple elements of ${{\bf B}}^{{{\mathfrak{a}}}}(0,1)$.
\[\][$\delta$]{}
{width="45.00000%"} {width="45.00000%"}\
The dashed line in figures \[contr\], \[Lieder\], \[d\], and \[u-m\] is used to label the arguments of the color ${{\mathfrak{a}}}$ and the solid line is used to label the arguments of the color ${{\mathfrak{c}}}$.
Using this graphical notation we may perform simple computations in the coalgebra ${{\mathbb F}}_{{{\bf B}}}({{\cal V}}, {{\cal W}})$. For example, using equation (\[l-i-delta\]), we present on Figure \[formula\] a simple computation with the bar differential ${{\partial}}^{Bar}$.
\[\][$\delta$]{} \[\][$\delta\,i$]{} \[\][$i\,\delta$]{} \[\][$\partial^{Bar}$]{} \[\][$+$]{} \[\][$=$]{} \[\][$\gamma$]{} \[\][$c$]{} \[\][$-(-1)^{|{{\gamma}}|}$]{} \[\][$l$]{} \[\][$i$]{}
{width="90.00000%"}\
Here ${{\gamma}}\in {{\cal V}}$ and $c\in {{\cal W}}$.
For the operad ${{\bf e_2}}$ we use a resolution which is simpler than the canonical one $Cobar(Bar({{\bf e_2}}))$. More precisely, as in [@BLT], we set $$\label{HoGer}
{{{\rm Ho}\,}({\bf e_2})}= Cobar({{\bf e_2}}^{\vee})\,.$$ Due to koszulity of the operad ${{\bf e_2}}$ the inclusions $$\label{Ger-Koszul}
{{\iota}}_{{{\bf e_2}}} : {{\bf e_2}}^{\vee} \hookrightarrow Bar({{\bf e_2}})$$ and $$\label{Ger-Koszul1}
Cobar({{\iota}}_{{{\bf e_2}}}) :
Cobar({{\bf e_2}}^{\vee}) \hookrightarrow
Cobar(Bar({{\bf e_2}}))$$ are quasi-isomorphisms of cooperads and operads, respectively. It is the second quasi-isomorphism (\[Ger-Koszul1\]) which allows us to replace the canonical resolution $Cobar(Bar({{\bf e_2}}))$ by (\[HoGer\]).
To get a more tractable description of algebras over the operads ${{{\rm Ho}\,}({\bf calc})}$ and ${{{\rm Ho}\,}({\bf e_2})}$ we introduce the following DGLAs $$\label{coder-Ger}
{\rm C o d e r}'({{\mathbb F}}_{{{\bf e_2}}^{\vee}}({{\cal V}})) =
\{Q \in {\rm C o d e r}({{\mathbb F}}_{{{\bf e_2}}^{\vee}}({{\cal V}})) ~|~
Q\Big|_{{{\cal V}}} = 0 \}\,,$$ $$\label{coder-calc}
{\rm C o d e r}'({{\mathbb F}}_{{{\bf B}}}({{\cal V}},{{\cal W}})) =
\{Q \in {\rm C o d e r}({{\mathbb F}}_{{{\bf B}}}({{\cal V}}, {{\cal W}})) ~|~
Q\Big|_{{{\cal V}}\oplus {{\cal W}}} = 0 \}\,,$$ where ${\rm C o d e r}({{\mathbb F}}_{{{\bf e_2}}^{\vee}}({{\cal V}}))$ (resp. ${\rm C o d e r}({{\mathbb F}}_{{{\bf B}}}({{\cal V}}, {{\cal W}}))$ ) is the DGLA of coderivations of the free coalgebra ${{\mathbb F}}_{{{\bf e_2}}^{\vee}}({{\cal V}})$ (resp. the free coalgebra ${{\mathbb F}}_{{{\bf B}}}({{\cal V}}, {{\cal W}})$). Furthermore, ${{\cal V}}$ (resp. ${{\cal V}}\oplus {{\cal W}}$) is considered as a subspace of ${{\mathbb F}}_{{{\bf e_2}}^{\vee}}({{\cal V}})$ (resp. ${{\mathbb F}}_{{{\bf B}}}({{\cal V}}, {{\cal W}})$) via the corresponding coaugmentation.
According to Proposition \[GJ\] the ${{{\rm Ho}\,}({\bf e_2})}$-algebra structures on ${{\cal V}}$ are in bijection with the Maurer-Cartan elements of the DGLA ${\rm C o d e r}'({{\mathbb F}}_{{{\bf e_2}}^{\vee}}({{\cal V}}))$. Similarly, the ${{{\rm Ho}\,}({\bf calc})}$-algebra structures on the pair $({{\cal V}}, {{\cal W}})$ are in bijection with the Maurer-Cartan elements of the DGLA ${\rm C o d e r}'({{\mathbb F}}_{{{\bf B}}}({{\cal V}},{{\cal W}}))$. Moreover, due to Proposition \[coder-cofree\] the Maurer-Cartan element $Q$ of the DGLA (\[coder-Ger\]) (resp. the DGLA (\[coder-calc\])) is uniquely determined by its composition $\rho_{{{\cal V}}}\circ Q$ (resp. $\rho_{{{\cal V}}, {{\cal W}}} \circ Q$) with the corestriction $\rho_{{{\cal V}}}: {{\mathbb F}}_{{{\bf e_2}}^{\vee}}({{\cal V}}) \to {{\cal V}}$ (resp. the corestriction $\rho_{{{\cal V}}, {{\cal W}}}: {{\mathbb F}}_{{{\bf B}}}({{\cal V}}, {{\cal W}})
\to {{\cal V}}\oplus {{\cal W}}$).
The vector space of the free coalgebra ${{\mathbb F}}_{{{\bf B}}}({{\cal V}}, {{\cal W}})$ splits according to the two colors $({{\mathfrak{c}}}, {{\mathfrak{a}}})$ as $$\label{bB-mc-ma}
{{\mathbb F}}_{{{\bf B}}}({{\cal V}}, {{\cal W}}) = {{\mathbb F}}_{{{\bf B}}}({{\cal V}}, {{\cal W}})_{{{\mathfrak{c}}}}
\oplus {{\mathbb F}}_{{{\bf B}}}({{\cal V}}, {{\cal W}})_{{{\mathfrak{a}}}}\,,$$ where $$\label{bB-Ger}
{{\mathbb F}}_{{{\bf B}}}({{\cal V}}, {{\cal W}})_{{{\mathfrak{c}}}} = {{\mathbb F}}_{Bar({{\bf e_2}})}({{\cal V}})\,.$$
Thus for every ${{{\rm Ho}\,}({\bf calc})}$-algebra $({{\cal V}}, {{\cal W}})$ the graded vector space ${{\cal V}}$ is an algebra over the operad $Cobar(Bar({{\bf e_2}}))$. Using this algebra structure over $Cobar(Bar({{\bf e_2}}))$ and the embedding (\[Ger-Koszul1\]) we get a ${{{\rm Ho}\,}({\bf e_2})}$-algebra structure on ${{\cal V}}$.
To describe the relationship between these algebras we denote by $Q_{{{\cal V}}, {{\cal W}}}$ the Maurer-Cartan element of the DGLA ${\rm C o d e r}'({{\mathbb F}}_{{{\bf B}}}({{\cal V}},{{\cal W}}))$ corresponding to the ${{{\rm Ho}\,}({\bf calc})}$-algebra structure on $({{\cal V}}, {{\cal W}})$. Next, we denote by $Q_{{{\cal V}}}$ the Maurer-Cartan element of the DGLA ${\rm C o d e r}'({{\mathbb F}}_{{{\bf e_2}}^{\vee}}({{\cal V}}))$ corresponding to the ${{{\rm Ho}\,}({\bf e_2})}$-algebra structure on ${{\cal V}}$.
Proposition \[how-to-pull\] implies that $$\label{calc-Ger}
\rho_{{{\cal V}}} \circ Q_{{{\cal V}}} =
\rho_{{{\cal V}}\oplus {{\cal W}}} \circ Q^{{{\mathfrak{c}}}}_{{{\cal V}}, {{\cal W}}}
\circ {{\mathbb F}}({{\iota}}_{{{\bf e_2}}})\,,$$ where ${{\iota}}_{{{\bf e_2}}}$ is the embedding (\[Ger-Koszul\]) and $$Q^{{{\mathfrak{c}}}}_{{{\cal V}}, {{\cal W}}} = Q_{{{\cal V}}, {{\cal W}}}
\Big|_{{{\mathbb F}}_{{{\bf B}}}({{\cal V}}, {{\cal W}})_{{{\mathfrak{c}}}}} \,.$$
Due to Proposition \[coder-cofree\] the coderivation $Q_{{{\cal V}}}$ (resp. the coderivation $Q_{{{\cal V}}, {{\cal W}}}$) is uniquely determined by the composition $\rho_{{{\cal V}}} \circ Q_{{{\cal V}}}$ (resp. $\rho_{{{\cal V}}, {{\cal W}}} \circ Q_{{{\cal V}}, {{\cal W}}}$). Thus equation (\[calc-Ger\]) indeed describes the relationship between the ${{{\rm Ho}\,}({\bf calc})}$-algebra structure on $({{\cal V}}, {{\cal W}})$ and the ${{{\rm Ho}\,}({\bf e_2})}$-algebra structure on ${{\cal V}}$.
\
[**Remark.**]{} The vector space of operations of the cooperad ${{\bf B}}$ with no arguments having color ${{\mathfrak{c}}}$ is $$\label{bB-01}
{{\bf B}}^{{{\mathfrak{a}}}}(0,1) = {{\mathbb K}}[u]\,,$$ where $u$ is an auxiliary variable of degree $-2$. The monomial $u^m$ corresponds to the element of ${{\bf B}}^{{{\mathfrak{a}}}}(0,1)$ which is drawn on Figure \[u-m\] (See page ).
Description of the operad ${{\rm Ho}\,}({{\bf Lie}}^+_{{{\delta}}})$
--------------------------------------------------------------------
The canonical cofibrant resolution $Cobar(Bar({{\bf Lie}}^+_{{{\delta}}}))$ can be simplified. In this subsection we construct a sub-cooperad $({{\bf Lie}}^+_{{{\delta}}})^{\vee}$ of $Bar({{\bf Lie}}^+_{{{\delta}}})$ such that the embedding of operads $$Cobar(({{\bf Lie}}^+_{{{\delta}}})^{\vee}) \hookrightarrow
Cobar(Bar({{\bf Lie}}^+_{{{\delta}}}))$$ is a quasi-isomorphism. This construction goes along the lines of [@Fresse], [@GK]. (See also Definition 3.2.1 in [@Hinich].) It allows us to set $${{\rm Ho}\,}({{\bf Lie}}^+_{{{\delta}}}) =
Cobar(({{\bf Lie}}^+_{{{\delta}}})^{\vee})\,.$$
Let us first recall that algebras over the operad ${{\bf Lie}}^+_{{{\delta}}}$ are pairs $({{\cal V}}, {{\cal W}})$ where ${{\cal V}}$ is a Lie algebra ${{\cal W}}$ is a Lie algebra module over ${{\cal V}}$ and ${{\cal W}}$ is equipped with degree $-1$ unary operation ${{\delta}}$ which satisfies the identities $$\label{delta-2}
{{\delta}}^2 = 0\,.$$ and $$\label{l-delta1}
\delta\, l_a - (-1)^{|a|} l_a \, {{\delta}}= 0\,,$$ where $l: {{\cal V}}\otimes {{\cal W}}\to {{\cal W}}$ is the action of ${{\cal V}}$ on ${{\cal W}}$.
Thus the operad ${{\bf Lie}}^+_{{{\delta}}}$ is generated by the elementary operations $[\,,\,]$, $l$ and ${{\delta}}$, where $[\,,\,]$ denotes the Lie bracket. These operations are subject to the homogeneous quadratic relations: the Jacobi identity for the Lie bracket $[\,,\,]$, and the compatibility equation between $l$ and $[\,,\,]$ $$\label{l-brack}
l_a l_b - (-1)^{|a|\, |b|} l_a l_b = l_{[a,b]}$$ and, finally, equations (\[delta-2\]) and (\[l-delta1\]).
To construct the cooperad $({{\bf Lie}}^+_{{{\delta}}})^{\vee}$ we introduce the polynomial functor $S$ spanned linearly by the elementary operations $[\,,\,]$, $l$, and ${{\delta}}$ of the operad ${{\bf Lie}}^+_{{{\delta}}}$.
We also introduce the linear span $R$ of the homogeneous quadratic relations of ${{\bf Lie}}^+_{{{\delta}}}$ between the elementary operations.
Next, we consider the free cooperad ${{\mathbb T}}^*({{\bf s}\,}^{-1} S)$ generated by the polynomial functor ${{\bf s}\,}^{-1} S$. The cooperad ${{\mathbb T}}^*({{\bf s}\,}^{-1} S)$ may be viewed as a sub-cooperad of $Bar({{\bf Lie}}^+_{{{\delta}}})$ if we forget about the differential ${{\partial}}^{Bar}$.
Let us remark that, the cooperad ${{\mathbb T}}^*({{\bf s}\,}^{-1} S)$ is equipped with the natural grading $$\label{grading}
{{\mathbb T}}^*({{\bf s}\,}^{-1} S) =
\bigoplus_{m=0}^{\infty} {{\mathbb T}}^*_m({{\bf s}\,}^{-1} S)\,,
\qquad
{{\mathbb T}}^*_0({{\bf s}\,}^{-1} S) = \ast\,,$$ where $\ast$ is the terminal object (\[ast-color\]) in the category of $2$-colored cooperads and ${{\mathbb T}}^*_m({{\bf s}\,}^{-1} S)$ consists of the elements of degree $m$ in the elementary operations. Thus, since the relations between the elementary operations are quadratic, ${{\bf s}\,}^{-2} R$ is a subspace of ${{\mathbb T}}^*_2({{\bf s}\,}^{-1} S)$.
First, we construct the cooperad $({{\bf Lie}}^+_{{{\delta}}})^{\vee}$ as a sub-cooperad of ${{\mathbb T}}^*({{\bf s}\,}^{-1} S)$ and then we will show that $({{\bf Lie}}^+_{{{\delta}}})^{\vee}$ belongs to the kernel of the bar differential ${{\partial}}^{Bar}$.
We construct $({{\bf Lie}}^+_{{{\delta}}})^{\vee}$ by induction on the degree $m$ in (\[grading\]). The base of the induction is given by the equations $$\label{base1}
({{\bf Lie}}^+_{{{\delta}}})^{\vee} \cap {{\mathbb T}}^*_0({{\bf s}\,}^{-1} S)
\oplus {{\mathbb T}}^*_1({{\bf s}\,}^{-1} S)=
{{\mathbb T}}^*_0({{\bf s}\,}^{-1} S) \oplus {{\mathbb T}}^*_1({{\bf s}\,}^{-1} S)\,,$$ $$\label{base11}
({{\bf Lie}}^+_{{{\delta}}})^{\vee} \cap {{\mathbb T}}^*_2({{\bf s}\,}^{-1} S) =
{{\bf s}\,}^{-2} R\,,$$ and the step is given by the condition: [*a vector $v\in {{\mathbb T}}^*_m({{\bf s}\,}^{-1} S)$ belongs to $({{\bf Lie}}^+_{{{\delta}}})^{\vee}$ provided*]{} $${{\widetilde{\Delta}}}(v) \in ({{\bf Lie}}^+_{{{\delta}}})^{\vee}
{{\bullet}}({{\bf Lie}}^+_{{{\delta}}})^{\vee} \,.$$ Here ${{\Delta}}$ is the coproduct: $${{\Delta}}: {{\mathbb T}}^*({{\bf s}\,}^{-1} S) \to {{\mathbb T}}^*({{\bf s}\,}^{-1} S) \bullet
{{\mathbb T}}^*({{\bf s}\,}^{-1} S)\,,$$ and $${{\widetilde{\Delta}}}(v) = {{\Delta}}(v) -
v \otimes (1 \otimes \dots \otimes 1) -
1 \otimes (v \otimes 1 \otimes \dots \otimes 1)-
1 \otimes (1 \otimes v \otimes 1 \dots \otimes 1) -
\dots$$ $$- 1 \otimes (1 \otimes \dots \otimes 1 \otimes v)\,.$$
By construction $({{\bf Lie}}^+_{{{\delta}}})^{\vee} $ is a sub-cooperad of ${{\mathbb T}}^*({{\bf s}\,}^{-1} S)$.
Equation (\[base11\]) imply immediately that $${{\partial}}^{Bar}\, v = 0\,, \qquad
\forall ~~
v \in
({{\bf Lie}}^+_{{{\delta}}})^{\vee} \cap
{{\mathbb T}}^*_2({{\bf s}\,}^{-1} S) \,.$$
Then the compatibility of ${{\partial}}^{Bar}$ with the coproduct ${{\Delta}}$: $${{\Delta}}\, {{\partial}}^{Bar} = (
{{\partial}}^{Bar} \otimes (1 \otimes \dots \otimes 1) +
1 \otimes ({{\partial}}^{Bar} \otimes 1 \otimes \dots \otimes 1)+
\dots )\, {{\Delta}}$$ and the inductive definition of $({{\bf Lie}}^+_{{{\delta}}})^{\vee} $ imply that $$\label{pa-Bar-Lie-vee}
{{\partial}}^{Bar}\, v = 0\,, \qquad
\forall ~~
v \in
({{\bf Lie}}^+_{{{\delta}}})^{\vee}\,.$$
Thus $({{\bf Lie}}^+_{{{\delta}}})^{\vee}$ belongs to the kernel of the bar differential ${{\partial}}^{Bar}$ in $Bar({{\bf Lie}}^+_{{{\delta}}})$.
The following proposition gives us a description of the coalgebras over the cooperad $({{\bf Lie}}^+_{{{\delta}}})^{\vee}$
\[lie-de-vee\] A pair $({{\cal V}}, {{\cal W}})$ of graded vector spaces forms a coalgebra over the cooperad $({{\bf Lie}}^+_{{{\delta}}})^{\vee}$ if $({{\cal V}}, {{\cal W}})$ is a coalgebra over the cooperad ${{\Lambda}}{{\bf cocomm}}^+$ and ${{\cal W}}$ is equipped with a degree $2$ endomorphism $${{\delta}}^{\vee} : {{\cal W}}\to {{\cal W}}$$ satisfying the equation $$l^{\vee} \circ {{\delta}}^{\vee} =
(1 \otimes {{\delta}}^{\vee}) l^{\vee}\,,$$ where $l^{\vee}$ is the coaction of ${{\cal V}}$ on ${{\cal W}}$ $$l^{\vee} : {{\cal W}}\to {{\bf s}\,}^{-1}( {{\cal V}}\otimes {{\cal W}})\,.$$
[**Proof.**]{} Let us consider the restricted dual vector space $$\label{dual-T-star}
[{{\mathbb T}}^*({{\bf s}\,}^{-1} S)]^{*} =
{{\rm Hom}\,}_{{{\rm restr}\,}}({{\mathbb T}}^*({{\bf s}\,}^{-1} S), {{\mathbb K}})$$ of the free cooperad ${{\mathbb T}}^*({{\bf s}\,}^{-1} S)$ with respect to the grading (\[grading\]). It is not hard to see that $$[{{\mathbb T}}^*({{\bf s}\,}^{-1} S)]^{*} = {{\mathbb T}}({{\bf s}\,}S^*)$$ is the free operad ${{\mathbb T}}({{\bf s}\,}S^*)$ generated by the suspension ${{\bf s}\,}S^*$ of the linear dual $S^*$ of the polynomial functor $S$.
From the construction of $({{\bf Lie}}^+_{{{\delta}}})^{\vee}$ it follows that the restricted dual $[({{\bf Lie}}^+_{{{\delta}}})^{\vee}]^*$ of the cooperad $({{\bf Lie}}^+_{{{\delta}}})^{\vee}$ is the quotient of the free operad ${{\mathbb T}}({{\bf s}\,}S^*)$ with respect to the ideal generated by the polynomial functor of dual relations $$\label{dual-R}
R^* = \{r \in
{{\rm Hom}\,}({{\mathbb T}}^*_2({{\bf s}\,}^{-1} S), {{\mathbb K}})\,, ~~|~~
r\Big|_{R} =0
\}\,.$$
Let $\{[\,,\,]^*, l^*, {{\delta}}^{*}\}$ be the basis of $S^*$ which is dual to the basis $\{[\,,\,], l, {{\delta}}\}$ of $S$.
Dualizing the Jacobi relation for $[\,,\,]$ and the compatibility (\[l-brack\]) of $l$ with $[\,,\,]$ we see that the operation ${{\bf s}\,}[\,,\,]^*$ satisfies the axioms of an associative commutative product and the operation ${{\bf s}\,}l^*$ satisfies the axiom of a module over an associative and commutative algebra. Dualizing the relation (\[l-delta1\]) we see that ${{\bf s}\,}l^*$ and ${{\bf s}\,}{{\delta}}^*$ are compatible in the sense of the following relation $$\label{l-delta-star}
{{\bf s}\,}{{\delta}}^* \, {{\bf s}\,}l^* = {{\bf s}\,}l^* \, (1 \otimes {{\bf s}\,}{{\delta}}^*) \,.$$ Finally the presence of the relation (\[delta-2\]) implies that we should not impose any additional condition on ${{\bf s}\,}{{\delta}}^*$ besides (\[l-delta-star\]).
Thus a pair $({\widetilde{{\cal V}}}, {\widetilde{{\cal W}}})$ is an algebra over the operad $[({{\bf Lie}}^+_{{{\delta}}})^{\vee}]^*$ if $({\widetilde{{\cal V}}}, {\widetilde{{\cal W}}})$ is a ${{\Lambda}}^{-1}{{\bf comm}}^+$-algebra and ${\widetilde{{\cal W}}}$ is equipped with a degree $2$ endomorphism ${{\bf s}\,}{{\delta}}^*$ which is compatible with the action of ${\widetilde{{\cal V}}}$ on ${\widetilde{{\cal W}}}$ in the sense of (\[l-delta-star\]).
Taking the dual partner of an algebra over the operad $[({{\bf Lie}}^+_{{{\delta}}})^{\vee}]^*$ we get the statement of the proposition. $\Box$
Proposition \[lie-de-vee\] implies that a free coalgebra over the cooperad $({{\bf Lie}}^+_{{{\delta}}})^{\vee}$ generated by a pair $({{\cal V}}, {{\cal W}})$ is $$\label{liede-vee}
{{\mathbb F}}_{({{\bf Lie}}^+_{{{\delta}}})^{\vee}}({{\cal V}},{{\cal W}})=
{{\mathbb F}}_{{{\Lambda}}{{\bf cocomm}}^+}({{\cal V}},{{\cal W}}[[u]])\,,$$ where $u$ is an auxiliary variable of degree $-2$.
We claim that
\[Lie-de-vee-Kos\] The operad ${{\bf Lie}}^+_{{{\delta}}}$ is Koszul. In other words the embedding $$Cobar(({{\bf Lie}}^+_{{{\delta}}})^{\vee}) \to
Cobar (Bar({{\bf Lie}}^+_{{{\delta}}}))$$ is a quasi-isomorphism of operads.
[**Proof.**]{} The criterion of Ginzburg and Kapranov [@GK] (theorem $4.2.5$) reduces this question to computation of the homology of a free ${{\bf Lie}}^+_{{{\delta}}}$-algebra. More precisely, we need to show that for every pair $({{\cal V}}, {{\cal W}})$ of vector spaces the complex $$\label{complex-of-free}
{{\mathbb F}}_{({{\bf Lie}}^+_{{{\delta}}})^{\vee}} \circ {{\mathbb F}}_{{{\bf Lie}}^+_{{{\delta}}}} ({{\cal V}},{{\cal W}})$$ has nontrivial cohomology only in degree $0$.
Here the differential on the complex (\[complex-of-free\]) is defined along the lines of [@GJ] using the twisting cochain of the pair $({{\bf Lie}}^+_{{{\delta}}}, ({{\bf Lie}}^+_{{{\delta}}})^{\vee})$. (See Section 2.4 in [@GJ] for more details.)
If we split the complex (\[complex-of-free\]) according to the colors ${{\mathfrak{c}}}$ and ${{\mathfrak{a}}}$ and use equation (\[liede-vee\]) then we get two complexes: $$\label{complex-mc}
{{\mathbb F}}_{({{\bf Lie}}^+_{{{\delta}}})^{\vee}} \circ {{\mathbb F}}_{{{\bf Lie}}^+_{{{\delta}}}} ({{\cal V}},{{\cal W}})_{{{\mathfrak{c}}}}
= {{\mathbb F}}_{{{\Lambda}}{{\bf cocomm}}} \circ {{\mathbb F}}_{{{\bf Lie}}}({{\cal V}})\,,$$ and $$\label{complex-ma}
{{\mathbb F}}_{({{\bf Lie}}^+_{{{\delta}}})^{\vee}} \circ {{\mathbb F}}_{{{\bf Lie}}^+_{{{\delta}}}} ({{\cal V}},{{\cal W}})_{{{\mathfrak{a}}}}
= {{\mathbb F}}_{{{\Lambda}}{{\bf cocomm}}^+} ({{\mathbb F}}_{{{\bf Lie}}}({{\cal V}}), T({{\cal V}}) \otimes ({{\cal W}}\oplus {{\delta}}{{\cal W}})
[[u]])_{{{\mathfrak{a}}}}\,,$$ where $T({{\cal V}})$ denotes the tensor algebra of ${{\cal V}}$, ${{\delta}}$ is the unary operation of ${{\bf Lie}}^+_{{{\delta}}}$ and $u$ is an auxiliary variable of degree $-2$.
The first complex is exactly the Harrison complex of the free Lie algebra generated by ${{\cal V}}$ and it is known that this complex has nontrivial cohomology only in degree $0$.
The second complex is the tensor product of the Harrison complex of the free module generated by ${{\cal W}}$ over the free Lie algebra ${{\mathbb F}}_{{{\bf Lie}}}({{\cal V}})$ and the De Rham complex $$({{\mathbb K}}[[u]] \oplus {{\delta}}\, {{\mathbb K}}[[u]], {{\delta}}\frac{{{\partial}}}{{{\partial}}u} )$$ of the algebra ${{\mathbb K}}[[u]]$. Thus the second complex also has nontrivial cohomology only in degree $0$. $\Box$
This Proposition implies immediately that the embedding $$Cobar({{\Lambda}}({{\bf Lie}}^+_{{{\delta}}})^{\vee}) \hookrightarrow
Cobar (Bar({{\Lambda}}{{\bf Lie}}^+_{{{\delta}}}))$$ is a quasi-isomorphism of operads. Thus we may set $$\label{HoLaLie}
{{\rm Ho}\,}({{\Lambda}}{{\bf Lie}}^+_{{{\delta}}}) = Cobar({{\Lambda}}({{\bf Lie}}^+_{{{\delta}}})^{\vee})\,.$$
We would also like to remark that equation (\[liede-vee\]) implies that $$\label{liede-vee1}
{{\mathbb F}}_{{{\Lambda}}({{\bf Lie}}^+_{{{\delta}}})^{\vee}}({{\cal V}},{{\cal W}})=
{{\mathbb F}}_{{{\Lambda}}^2{{\bf cocomm}}^+}({{\cal V}},{{\cal W}}[[u]])\,,$$ where $u$ is an auxiliary variable of degree $-2$.
The Kontsevich-Soibelman operad and the operad of little discs on a cylinder
============================================================================
The Kontsevich-Soibelman operad ${{\bf K S}}$
---------------------------------------------
Let us describe the auxiliary operad ${{\cal H}}$ (of sets) of “natural”[^6] operations on the pair $$({C^{\bullet}}(A), {C_{\bullet}}(A))\,.$$ This operad is going to have a countable set of colors $$\label{colors}
\Xi = {{\mathbb Z}}^+\, \sqcup\, {{\mathbb Z}}^- \,,$$ where ${{\mathbb Z}}^+$ (resp. ${{\mathbb Z}}^-$) denotes the set of nonnegative (resp. nonpositive) integers.
The numbers from the set ${{\mathbb Z}}^+$ label the degrees of the Hochschild cochains and the numbers from the set ${{\mathbb Z}}^-$ label the degrees of Hochschild chains.
Using ${{\cal H}}$ we construct the DG operad ${{\bf K S}}$ of Kontsevich and Soibelman. The latter operad[^7] is described in sections 11.1, 11.2 and 11.3 of [@K-Soi1].
For the $\Xi$-colored operad ${{\cal H}}$ we only allow the operations in which a chain may enter as an argument at most once. If a chain enters then the result of the operation is also a chain. Otherwise the result is a cochain. We denote the set of operations producing a cochain from $n$ cochains by ${{\cal H}}(n,0)$. The set of operations producing a chain from $n$ cochains and $1$ chain is denoted by ${{\cal H}}(n,1)$.
${{\cal H}}(n,0)$ is the set of equivalence classes of rooted[^8] planar trees $T$ with marked vertices. The equivalence relation is the finest one in which two such trees are equivalent if one of them can be obtained from the other by either:
- the contraction of an edge with unmarked ends or
- removing an unmarked vertex with only one edge originating from it and joining the two edges adjacent to this vertex into one edge.
If a marked vertex is internal then it is reserved for a cochain which enters as an argument of the operation. The number of the incoming edges of such vertex is the degree of the corresponding cochain. If a marked vertex is terminal then it is reserved either for a cochain of degree $0$ or for an argument of the cochain produced by the operation.
The unmarked vertices (both internal and terminal) are reserved for the operations of the non-symmetric operad ${{\bf assoc}}$ which controls unital monoids. For example, an unmarked terminal vertex is reserved for unit of $A$, an unmarked vertex of valency $2$ is reserved for the identity transformation on $A$, and an unmarked vertex of valency $3$ is reserved for the associative product on $A$.
The root vertex is special. Since our trees are rooted this vertex has always valency $1$. It is always marked and reserved for the outcome of the cochain produced by the operation corresponding to the tree.
The tree on figure \[primer11\] represents an operation which produces the $2$-cochain: $$a_1\otimes a_2 \to
Q(a_1, a_2, 1) P$$ from a degree $0$ cochain $P$ and a degree $3$ cochain $Q$.
\[\][$Q$]{} \[\][$P$]{} \[\][$a_1$]{} \[\][$a_2$]{}
{width="30.00000%"}\
Marked vertices in this figure are labeled by small circles. The unmarked terminal vertex corresponds to the insertion of the unit into $Q(a_1, a_2, 1)$. The unmarked 3-valent vertex gives the product of $P$ and $Q(a_1, a_2, 1)$.
Let us denote by ${{\cal H}}^{m_a}_{m_r}(n,1)$ the set of operations producing a chain in $C_{-m_r}(A)$ from $n$ cochains and a chain in $C_{-m_a}(A)$.
${{\cal H}}^{m_a}_{m_r}(n,1)$ is described using forests of rooted trees drawn on the standard cylinder $$\label{Si}
{{\Sigma}}= S^1 \times [0,1]$$ and subject to the following conditions:
1. every tree of the forest has its root vertex on the boundary $S^1\times \{0\}$,
2. all vertices of the forest lying on the boundary of the cylinder are marked:
- the vertices lying on the boundary $S^1\times \{1\}$ are marked by integers $0, 1, \dots, m_a$ in the counterclockwise order; these vertices are reserved for the components of the chain which enters as an argument,
- the roots are marked by integers $0, 1, \dots, m_r$ in the same counterclockwise order; they are reserved for components of the resulting chain,
3. all other marked vertices of the forest lie on the lateral surface $S^1\times (0,1)$ of the cylinder and there are exactly $n$ such marked vertices.
On the set of these forests we introduce the finest equivalence relation in which two such forests are equivalent if one of them can be obtained from the other by either:
- isotopy, or
- the contraction of an edge with unmarked ends, or
- removing an unmarked vertex with only one edge originating from it and joining the two edges adjacent to this vertex into one edge.
${{\cal H}}^{m_a}_{m_r}(n,1)$ is the set of the corresponding equivalence classes.
As we see from the conditions, all unmarked vertices lie on the lateral surface $S^1\times (0,1)$ of the cylinder. As above, these vertices are reserved for operations of ${{\bf assoc}}$. The marked vertices lying on the lateral surface $S^1\times (0,1)$ are reserved for cochains.
We allow forests with no marked vertices lying on the lateral surface $S^1\times (0,1)$. Such forests represent operations which produce a chain from a chain.
Figure \[primer1\] gives an example of an operation of ${{\cal H}}(2,1)$ which produces the chain $$\label{chain}
(b_0, b_1, b_2, b_3) =
(P a_3, Q(a_0,1, a_1), 1, a_2)$$ from a degree $0$ cochain $P$, a degree $3$ cochain $Q$ and a degree $-3$ chain $(a_0, a_1, a_2, a_3)$.
\[\][$Q$]{} \[\][$P$]{} \[\][$a_0$]{} \[\][$a_1$]{} \[\][$a_2$]{} \[\][$a_3$]{} \[\][$b_0$]{} \[\][$b_1$]{} \[\][$b_2$]{} \[\][$b_3$]{}
{width="50.00000%"}\
Marked vertices in this figure are labeled by small circles. The unmarked 3-valent vertex gives the product of $P$ and $a_3$, the two unmarked terminal vertices give units of $A$ and the unmarked 2-valent vertex gives the identity operation on $A$. The vertices lying on the boundary $S^1\times \{1\}$ are marked by the components of the chain $(a_0, a_1, a_2, a_3)$ and the roots are marked by the components of the chain (\[chain\]).
It is clear how the operad ${{\cal H}}$ acts on the pair $({C^{\bullet}}(A), {C_{\bullet}}(A))$. From this action it is also clear how to compose the operations. For example, the composition of operations from ${{\cal H}}(n_1,1)$ and ${{\cal H}}(n_2,1)$ corresponds to putting one cylinder on the top of the other matching the roots of the first cylinder with the vertices lying on the upper circle of the second cylinder, and then shrinking the resulting cylinder to the required height.
Recall that the operad ${{\cal H}}$ is colored by degrees of the cochains and degrees of the chains. It is not hard to see that ${{\cal H}}(n,0)$ is a cosimplicial set with respect to the degree of the resulting cochain and a polysimplicial set with respect to the degrees of the cochains entering as arguments.
Similarly, ${{\cal H}}(n,1)$ is a cosimplicial set with respect to the degree of the chain entering as an argument and a polysimplicial set with respect to the degrees of the cochains entering as arguments and the degree of the resulting chain.
These poly-simplicial/cosimplicial structure is compatible with the compositions and we get
The DG operad ${{\bf K S}}$ is the realization of the operad ${{\cal H}}$ in the category of chain complexes.
It follows from the construction that ${{\bf K S}}$ is a $2$-colored operad which acts on the pair $({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A))$.
It is not hard to see that the operations $\cup$ (\[cup\]), $[\,,\,]_G$ (\[Gerst\]), $I$ (\[I-P\]), $L$ (\[L-Q\]) and $B$ (\[B\]) come from the action of the operad ${{\bf K S}}$.
\
[**Remark 1.**]{} The operad ${{\bf K S}}$ with its action on $$({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A))$$ was introduced by Kontsevich and Soibelman in[^9] [@K-Soi1] in the case when $A$ is an $A_{\infty}$-algebra. Here we recall the construction of ${{\bf K S}}$ in the case when $A$ is simply an associative algebra. It is this assumption on $A$ which allows us to utilize the natural cosimplicial/simplicial structure on $({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A))$.
\
[**Remark 2.**]{} If we restrict ourselves to the subspace of operations of ${{\bf K S}}$ which do not involve chains then we get the minimal operad of Kontsevich and Soibelman described in [@K-Soi].
The operad of little discs on a cylinder
----------------------------------------
A “topological partner” of ${{\bf K S}}$ is the operad ${{\rm C y l}\,}$ of discs on a cylinder [@K-Soi1], [@TT1]. As well as the operad of Kontsevich and Soibelman ${{\rm C y l}\,}$ is a $2$-colored operad satisfying the property (\[P-top\]).
The spaces ${{\rm C y l}\,}^{{{\mathfrak{c}}}}(n,0)$, $n\ge 1$ are the spaces of the little disc operad.
To introduce the space ${{\rm C y l}\,}^{{{\mathfrak{a}}}}(n,1)$ for $n\ge 1$ we consider cylinders $S^1\times [a,c]$ for $a,c \in {{\mathbb R}}$, $a < c$ with the natural flat metric and define the topological space ${\widetilde{\rm C y l}\,}_n$.
A point of the space ${\widetilde{\rm C y l}\,}_n$ is a cylinder $S^1\times [a,c]$ together with a configuration of $n\ge 1$ discs on the lateral surface $S^1\times (a,c)$ and a position of two points $b$ and $t$ lying on the boundaries $S^1\times {a}$ and $S^1\times {c}$, respectively. The topology on the space ${\widetilde{\rm C y l}\,}_n$ is defined in the obvious way using the flat metric on the cylinder.
The space ${\widetilde{\rm C y l}\,}_n$ is equipped with a free action of the group $S^1 \times {{\mathbb R}}$. The subgroup $S^1\subset S^1 \times {{\mathbb R}}$ simultaneously rotates all the cylinders and the subgroup ${{\mathbb R}}\subset S^1 \times {{\mathbb R}}$ acts by parallel shifts $$S^1\times [a,c] \to S^1\times [a+l,c+l]\,, \qquad
l\in {{\mathbb R}}\,.$$
The space ${{\rm C y l}\,}^{{{\mathfrak{a}}}}(n,1)$ for $n\ge 1$ of the operad ${{\rm C y l}\,}$ is the quotient $$\label{C-col-2}
{{\rm C y l}\,}^{{{\mathfrak{a}}}}(n,1) = {\widetilde{\rm C y l}\,}_n / S^1 \times {{\mathbb R}}\,.$$
The space ${{\rm C y l}\,}^{{{\mathfrak{a}}}}(0,1)$ is the space of configurations of two (possibly coinciding) points $b$ and $t$ on the circle $S^1$ considered modulo rotations. Although it is obvious that ${{\rm C y l}\,}^{{{\mathfrak{a}}}}(0,1)$ is homeomorphic to the circle $S^1$ we still define ${{\rm C y l}\,}^{{{\mathfrak{a}}}}(0,1)$ using the configuration space in order to better visualize the operations of the operad.
The insertions of the type $${{\rm C y l}\,}^{{{\mathfrak{c}}}}(n,0)\times {{\rm C y l}\,}^{{{\mathfrak{c}}}}(m,0) \to {{\rm C y l}\,}^{{{\mathfrak{c}}}}(n+m-1,0)$$ are defined in the same as for the operad of little squares. The operations of the type $${{\rm C y l}\,}^{{{\mathfrak{c}}}}(n,0)\times {{\rm C y l}\,}^{{{\mathfrak{a}}}}(m,1) \to {{\rm C y l}\,}^{{{\mathfrak{a}}}}(n+m-1,1)$$ are insertions of the configuration of little discs of ${{\rm C y l}\,}^{{{\mathfrak{c}}}}(n,0)$ in a little disc on the lateral surface of a cylinder. Finally the operations of the type $${{\rm C y l}\,}^{{{\mathfrak{a}}}}(n,1)\times {{\rm C y l}\,}^{{{\mathfrak{a}}}}(m,1) \to {{\rm C y l}\,}^{{{\mathfrak{a}}}}(n+m,1)$$ correspond to putting the first cylinder under the second one while the second cylinder is rotated in such a way that the point $b$ of the second cylinder coincides with the point $t$ of the first cylinder. The composition involving degenerate configurations of ${{\rm C y l}\,}^{{{\mathfrak{a}}}}(0,1)$ are defined in the obvious way.
To describe the operad of homology groups of ${{\rm C y l}\,}$ we will need some results about the configuration spaces of distinct points on the punctured plane ${{\mathbb R}}^2\setminus \{{\bf 0}\}$.
Let us denote by ${{\rm Conf}\,}_n({{\mathbb R}}^2 \setminus \{ {\bf 0} \})$ the configuration space of $n$ distinct points on the punctured plane ${{\mathbb R}}^2 \setminus \{{\bf 0}\}$ and consider the following projections $$\label{p-k}
p_k : {{\rm Conf}\,}_n({{\mathbb R}}^2 \setminus \{ {\bf 0} \})
\to {{\mathbb R}}^2 \setminus \{ {\bf 0} \}\,,$$ $$p_k ({{\bf x}}_1, \dots, {{\bf x}}_n) = {{\bf x}}_k\,.$$
Due to E. Fadell and L. Neuwirth [@FN] we have
\[FN-teo\] For every $k=1, 2, \dots, n$ the map $p_k$ is a locally trivial fibration.
Using the ideas of E. Fadell and L. Neuwirth [@FN] we show that
The map $$p : {{\rm Conf}\,}_n({{\mathbb R}}^2 \setminus \{{\bf 0}\} ) \to
{{\rm Conf}\,}_{n-1}({{\mathbb R}}^2 \setminus \{{\bf 0}\} )\,,$$ $$\label{p}
p ({{\bf x}}_1, {{\bf x}}_2, \dots, {{\bf x}}_n) =
({{\bf x}}_2, \dots, {{\bf x}}_n)\,.$$ is a locally trivial fibration. Furthermore, the fiber $F_n$ of $p$ is $$\label{F-n}
F_n = {{\mathbb R}}^2 \setminus
\{{\bf 0}, {{\bf q}}_2, \dots, {{\bf q}}_{n}\} \,,$$ where ${{\bf q}}_2, \dots, {{\bf q}}_{n}$ are $n-1$ distinct points of the punctured plane ${{\mathbb R}}^2\setminus \{{\bf 0}\}$.
[**Proof.**]{} For the open unit disc $D_1$ on ${{\mathbb R}}^2$ centered at the origin there exists a continuous map $$\label{theta}
{\theta}: D_1 \times \bar{D}_1 \to \bar{D}_1$$ satisfying the following properties:
– for all ${{\bf x}}\in D_1$ the map ${\theta}({{\bf x}},\,\, ): \bar{D}_1 \to \bar{D}_1 $ is a a homeomorphism having ${{\partial}}\bar{D}_1$ fixed.
– for all ${{\bf x}}\in D_1$ we have ${\theta}({{\bf x}},{{\bf x}})={\bf 0}$.
For distinct points ${{\bf q}}_2, \dots, {{\bf q}}_{n}$ on the punctured plane ${{\mathbb R}}^2\setminus \{{\bf 0}\}$ we choose open discs $$\label{discs1}
D_{{{\bf q}}_2}, \, D_{{{\bf q}}_3}, \,
\dots, \, D_{{{\bf q}}_{n}}\,,$$ which are centered at ${{\bf q}}_2, {{\bf q}}_3, \dots, {{\bf q}}_{n}$, respectively. Each disc $\bar{D}_{{{\bf q}}_j}$ avoids the origin ${\bf 0}$ and for each $i \neq j$ $$\bar{D}_{{{\bf q}}_i} \cap
\bar{D}_{{{\bf q}}_j} = \emptyset\,.$$
Let us denote respectively by $r_2, \dots, r_{n}$ the radii of the discs (\[discs1\]) and let $U_b$ be the following neighborhood of ${{\rm Conf}\,}_{n-1}({{\mathbb R}}^2 \setminus \{\bf 0\} )$: $$\label{U-b}
U_b = \{({{\bf x}}_2, \dots, {{\bf x}}_n) \,\,|\,\,
{{\bf x}}_2 \in D_{{{\bf q}}_2}\,, \quad
{{\bf x}}_3 \in D_{{{\bf q}}_3}\,, \, \dots \,,
{{\bf x}}_n \in D_{{{\bf q}}_n} \} \,.$$
The desired homeomorphism $h$ from $p^{-1}(U_b) \to F_n\times U_b$ is given by the formula: $$\label{h}
h({{\bf x}}_1, {{\bf x}}_2, \dots, {{\bf x}}_n) =
\begin{cases}
\displaystyle
({{\bf x}}_1, {{\bf x}}_2, \dots, {{\bf x}}_n)
\,, \qquad {\rm if} ~~ {{\bf x}}_1 \notin
\bigcup_{j=2}^n D_{{{\bf q}}_j}\,, \\[0.3cm]
\displaystyle
({{\bf q}}_j + r_j\,
{\theta}\Big(\frac{{{\bf x}}_j-{{\bf q}}_j}{r_j},
\frac{{{\bf x}}_1- {{\bf q}}_j}{r_j}\Big), {{\bf x}}_2, \dots, {{\bf x}}_n)\,,
\quad {\rm if} ~~ {{\bf x}}_1
\in \bar{D}_{{{\bf q}}_j}\,,
\end{cases}$$ where $r_j$ is the radius of the $j$-th disc $D_{{{\bf q}}_j}$.
Since the open subsets of the form $U_b$ (\[U-b\]) cover ${{\rm Conf}\,}_{n-1}({{\mathbb R}}^2 \setminus \{\bf 0\} )$ the map $p$ (\[p\]) is indeed a locally trivial fibration. $\Box$
Fiber $F_n$ of $p$ (\[p\]) is homotopy equivalent to the wedge sum $\vee^n S^1$ of $n$ circles. Hence, the homology groups of $F_n$ (\[F-n\]) are $$\label{H-F-n}
H_{{{\bullet}}} (F_n, {{\mathbb K}}) =
\begin{cases}
{{\mathbb K}}\,, \qquad {\rm if} ~~ {{\bullet}}= 0\,, \\
{{\mathbb K}}^n\,, \qquad {\rm if} ~~ {{\bullet}}= 1\,, \\
0 \,, \qquad {\rm otherwise}\,.
\end{cases}$$
Let us show that
\[acts-triv\] The fundamental group $\pi_1({{\rm Conf}\,}_{n-1}({{\mathbb R}}^2 \setminus \{\bf 0\} ))$ acts trivially on the homology groups $H_{{{\bullet}}} (F_n, {{\mathbb K}})$ of $F_n$.
[**Proof.**]{} It is obvious that we only need to consider the action of $\pi_1({{\rm Conf}\,}_{n-1}({{\mathbb R}}^2 \setminus \{\bf 0\} ))$ on $H_1(F_n, {{\mathbb K}})$.
To get the cycles representing elements of a basis for $H_1(F_n, {{\mathbb K}})$ we choose closed discs $$\label{discs}
D_{\bf 0}, \, D_{{{\bf q}}_2}, \, D_{{{\bf q}}_3}, \,
\dots, \, D_{{{\bf q}}_{n}}\,,$$ which are centered at ${\bf 0}, {{\bf q}}_2, {{\bf q}}_3, \dots, {{\bf q}}_{n}$, respectively. The discs (\[discs\]) are chosen in such a way that their closures $$\bar{D}_{\bf 0}, \, \bar{D}_{{{\bf q}}_2}, \, \bar{D}_{{{\bf q}}_3}, \,
\dots, \, \bar{D}_{{{\bf q}}_{n}}$$ are pairwise disjoint.
The boundaries of the discs (\[discs\]) are cycles representing the elements of a basis for $H_1(F_n, {{\mathbb K}})$.
Let us identify ${{\mathbb R}}^2$ with the complex plane ${{\mathbb C}}$ and consider the following loop in ${{\rm Conf}\,}_{n-1}({{\mathbb R}}^2 \setminus \{\bf 0\} )$ $$\label{loop}
f(t) =
(e^{2 \pi i t}{{\bf q}}_2, e^{2 \pi i t} {{\bf q}}_3, \dots,
e^{2 \pi i t} {{\bf q}}_{n} )
\,:\, [0,1] \to {{\rm Conf}\,}_{n-1}({{\mathbb R}}^2 \setminus \{\bf 0\} ) \,,$$ where $({{\bf q}}_2, {{\bf q}}_3, \dots, {{\bf q}}_{n})$ is a fixed collection of the distinct points of ${{\mathbb R}}^2 \setminus \{\bf 0\}$.
The loop (\[loop\]) lifts to the following loop in ${{\rm Conf}\,}_{n}({{\mathbb R}}^2 \setminus \{\bf 0\} )$ $$\label{loop1}
{{\widetilde{f}}}(t) =
(e^{2 \pi i t} {{\bf x}}_1,
e^{2 \pi i t}{{\bf q}}_2, e^{2 \pi i t} {{\bf q}}_3, \dots,
e^{2 \pi i t} {{\bf q}}_{n} )
\,:\, [0,1] \to
{{\rm Conf}\,}_{n}({{\mathbb R}}^2 \setminus \{\bf 0\} ) \,.$$ As we go around the loop (\[loop1\]) the point ${{\bf x}}_1$ of the fiber $F_n$ returns to its original position. Thus the element $[f]\in \pi_1({{\rm Conf}\,}_{n-1}({{\mathbb R}}^2 \setminus \{\bf 0\} ))$ represented by the loop $f$ (\[loop\]) acts trivially on $H_{{{\bullet}}}(F_n,{{\mathbb K}})$.
Let now $g$ be an arbitrary loop in ${{\rm Conf}\,}_{n-1}({{\mathbb R}}^2 \setminus \{\bf 0\} )$.
To find the action of the homotopy class $[g]$ on $H_{{{\bullet}}}(F_n,{{\mathbb K}})$ we need to lift the map $$\label{gamma}
{{\gamma}}(y, t) = g(t): F_n \times [0,1] \to
{{\rm Conf}\,}_{n-1}({{\mathbb R}}^2 \setminus \{\bf 0\} )$$ to a map $$\label{gamma-t}
{{\widetilde{\gamma}}}(y, t): F_n \times [0,1] \to
{{\rm Conf}\,}_{n}({{\mathbb R}}^2 \setminus \{\bf 0\} )$$ which makes the diagram $$\label{gamma-diag}
\begin{array}{ccc}
F_n \times \{0\} & \hookrightarrow & {{\rm Conf}\,}_{n}({{\mathbb R}}^2 \setminus \{ {\bf 0} \} ) \\[0.3cm]
\downarrow & ~^{{{\widetilde{\gamma}}}}\nearrow & \downarrow^{p} \\[0.3cm]
F_n \times [0,1] & \stackrel{{{\gamma}}}{\rightarrow} &
{{\rm Conf}\,}_{n-1}({{\mathbb R}}^2 \setminus \{{\bf 0}\} )
\end{array}$$ commutative.
To construct the lift ${{\widetilde{\gamma}}}$ we divide the segment $[0,1]$ into small enough subsegments $[t_i, t_{i+1}]$ satisfying the property $$\label{subseg-prop}
g([t_i, t_{i+1}]) \subset V\,,$$ where $V$ is an open subset of ${{\rm Conf}\,}_{n-1}({{\mathbb R}}^2 \setminus \{\bf 0\} ) $ of the form (\[U-b\]). Then each individual lift $$\label{gamma-t-i}
{{\widetilde{\gamma}}}\Big|_{F_n \times [t_i, t_{i+1}]} :
F_n \times [t_i, t_{i+1}] \to {{\rm Conf}\,}_{n}({{\mathbb R}}^2 \setminus \{ {\bf 0} \} )$$ can be constructed using the trivialization (\[h\]).
With this construction in mind we consider the compositions $p_k \circ g$ of $g$ with the projections $p_k$ (\[p-k\]) for $k \in \{2,\, \dots,\, n\}$. Since the image $p_k \circ g([0,1])$ of the segment $[0,1]$ is a compact subset in ${{\mathbb R}}^2 \setminus \{ {\bf 0} \}$ we may choose the disc $D_{\bf 0}$ in such a way that the closure $\bar{D}_{\bf 0}$ avoids the points of the images $p_k \circ g([0,1])$ for all $k \in \{2,\, \dots,\, n\}$. Therefore the lift ${{\widetilde{\gamma}}}$ (\[gamma-t\]) can be chosen in such a way that $${{\widetilde{\gamma}}}(y,t) = y\,, \qquad \forall ~~ y\in \bar{D}_{\bf 0}\subset F_n\,.$$ Hence the action of the homotopy class $[g]$ on the homology class represented by the boundary of the disc $D_{\bf 0}$ is trivial.
Let us examine the loop $p_k\circ g$ in ${{\mathbb R}}^2\setminus \{\bf 0\}$ more closely.
Since the fundamental group of ${{\mathbb R}}^2\setminus \{\bf 0\}$ is generated by the homotopy class of the loop $$l(t) = e^{2 \pi i t} {{\bf q}}_k : [0,1] \to
{{\mathbb R}}^2\setminus \{\bf 0\}$$ around the origin there exists an integer $N \in {{\mathbb Z}}$ such that the loop $p_k \circ (g * f^N)$ is null-homotopic. Here $f$ is the loop (\[loop\]) in ${{\rm Conf}\,}_{n-1}({{\mathbb R}}^2 \setminus \{\bf 0\} )$.
But the class of $f$ acts trivially on the homology of the fiber $F_n$. Therefore, without loss of generality, we may assume that the composition $p_k \circ g$ is null-homotopic.
Thus, due to Theorem \[FN-teo\] of Fadell and Neuwirth we may further assume that $p_k \circ g$ is a constant map $$\label{p-k-g}
p_k \circ g (t) \equiv {{\bf q}}_k\,.$$ In other words, the $k$-th point ${{\bf x}}_k$ does not move as we go along the loop $g$.
Since for each $j\neq k$ the image $p_j \circ g([0,1])$ of the segment $[0,1]$ is a compact subset in ${{\mathbb R}}^2 \setminus \{ {\bf 0} \}$ we may choose the disc $D_{{{\bf q}}_k}$ in such a way that the closure $\bar{D}_{{{\bf q}}_k}$ avoids the points of the images $p_j \circ g([0,1])$ for all $j\neq k$.
Thus, using the partition of the segment $[0,1]$ satisfying the property (\[subseg-prop\]) and constructing the lift (\[gamma-t\]) using the trivializations of the form (\[h\]) we see that the lift ${{\widetilde{\gamma}}}$ can be chosen in such a way that $${{\widetilde{\gamma}}}(y,t) = y\,, \qquad \forall~~ y\in \bar{D}_{{{\bf q}}_k}\subset F_n\,.$$ Therefore the action of the homotopy class $[g]$ on the homology class represented by the boundary of the disc $D_{{{\bf q}}_k}$ is trivial.
The proposition is proved. $\Box$
Generalizing the result of F. R. Cohen [@Cohen] we get the following
\[H-Cyl\] The homology operad $H_{-{{\bullet}}}({{\rm C y l}\,}, {{\mathbb K}})$ of ${{\rm C y l}\,}$ with the reversed grading is the operad ${{\bf calc}}$ of calculi.
[**Proof.**]{} Since the operad ${{\rm C y l}\,}$ has two colors algebras over $H_{-{{\bullet}}}({{\rm C y l}\,}, {{\mathbb K}})$ are pairs of graded vector spaces $({{\cal V}}, {{\cal W}})$.
The components ${{\rm C y l}\,}^{{{\mathfrak{c}}}}(n,0)$ form the operad of little discs. Thus, due to Theorem 1.2 in[^10] [@Cohen] ${{\cal V}}$ is a Gerstenhaber algebra.
The space ${{\rm C y l}\,}^{{{\mathfrak{c}}}}(0,0)$ is empty, the space ${{\rm C y l}\,}^{{{\mathfrak{c}}}}(1,0)$ is a point and the space ${{\rm C y l}\,}^{{{\mathfrak{c}}}}(n,0)$ for $n > 1$ is homotopy equivalent to the space ${{\rm Conf}\,}_n({{\mathbb R}}^2)$ of configurations of $n$ distinct points on ${{\mathbb R}}^2$. The map $$\label{h-equiv}
{{\rm C y l}\,}^{{{\mathfrak{c}}}}(n,0) \stackrel{\sim}{\to} {{\rm Conf}\,}_n({{\mathbb R}}^2)$$ which establishes the equivalence associates with a configuration of disjoint discs the configuration of their centers.
The space ${{\rm Conf}\,}_2({{\mathbb R}}^2)$ is, in turn, homotopy equivalent to $S^1$. Thus the generator of $H_0(S^1)$ represents the commutative product on ${{\cal V}}$ and the generator of $H_1(S^1)$ represents the bracket on ${{\cal V}}$. This bracket has degree $-1$ because we use the reversed grading on the homology groups.
The operations without inputs of color ${{\mathfrak{c}}}$ correspond to homology classes of the space ${{\rm C y l}\,}^{{{\mathfrak{a}}}}(0,1)$. Since this space is homeomorphic to the circle $S^1$ we have $$H_{{{\bullet}}}({{\rm C y l}\,}^{{{\mathfrak{a}}}}(0,1), {{\mathbb K}}) =
\begin{cases}
{{\mathbb K}}\,, \qquad {\rm if} ~~ {{\bullet}}= 0,1 \,, \\
0 \,, \qquad {\rm otherwise}\,.
\end{cases}$$ The generator of $H_{0}({{\rm C y l}\,}^{{{\mathfrak{a}}}}(0,1))$ corresponds to the identity transformation of ${{\cal W}}$ and the generator ${{\delta}}$ of $H_{1}({{\rm C y l}\,}^{{{\mathfrak{a}}}}(0,1))$ corresponds to a unary operation on ${{\cal W}}$. The operation ${{\delta}}$ has degree $-1$ because we use the reversed grading on the homology groups.
The identity $${{\delta}}^2 = 0$$ follows immediately from the fact that $$H_{2}({{\rm C y l}\,}^{{{\mathfrak{a}}}}(0,1), {{\mathbb K}}) = 0\,.$$
Let us construct a homotopy equivalence between the space ${{\rm C y l}\,}^{{{\mathfrak{a}}}}(n,1)$ for $n\ge 1$ and the space $$\label{Conf-col-n}
{{\rm Conf}\,}_n({{\mathbb R}}^2 \setminus {\bf 0}) \times S^1\,,$$ where ${{\rm Conf}\,}_n({{\mathbb R}}^2 \setminus {\bf 0})$ is the configuration space of $n$ distinct points on the punctured plane ${{\mathbb R}}^2 \setminus {\bf 0}$.
First, we kill the rotation symmetry by fixing the position of the point $t$ on the upper boundary $S^1\times \{c\}$ of the cylinder $S^1 \times [a,c]$. Second, we kill translation symmetry by setting $c=0$.
Next we assign to each configuration of discs on the lateral surface $S^1\times (a,0)$ the configuration of centers of the discs. In this way we get a homotopy equivalence between the space ${{\rm C y l}\,}^{{{\mathfrak{a}}}}(n,1)$ and the space $$\label{Conf-Cyl}
{{\rm Conf}\,}_n(S^1 \times (a,0)) \times S^1\,,$$ where the points on the factor $S^1$ correspond to positions of the point $b$ on $S^1\times \{a\}$.
Finally, using the map $$\chi: S^1 \times (a,0) \to {{\mathbb R}}^2 \setminus {\bf 0}\,,$$ $$\chi({{\varphi}}, y) = \Big( \frac{y}{a-y}\cos({{\varphi}})\,,\, \frac{y}{a-y}\sin({{\varphi}})
\Big)$$ we get the desired homotopy equivalence $$\label{homot-equiv}
{{\rm C y l}\,}^{{{\mathfrak{a}}}}(n,1) \simeq
{{\rm Conf}\,}_n({{\mathbb R}}^2 \setminus {\bf 0}) \times S^1\,.$$
Let us consider the homology groups of ${{\rm C y l}\,}^{{{\mathfrak{a}}}}(1,1)$ in more details.
The space ${{\rm C y l}\,}^{{{\mathfrak{a}}}}(1,1)$ is homotopy equivalent to $${{\mathbb R}}^2\setminus {\bf 0}\, \times \, S^1$$ and the latter space is, in turn, homotopy equivalent to $S^1 \times S^1$. Thus $$\label{H-C-col-1}
{{\rm C y l}\,}^{{{\mathfrak{a}}}}(1,1) \simeq S^1 \times S^1\,.$$ Therefore $$H_{-{{\bullet}}}({{\rm C y l}\,}^{{{\mathfrak{a}}}}(1,1), {{\mathbb K}}) =
{{\mathbb K}}\oplus {{\bf s}\,}^{-1}{{\mathbb K}}^2 \oplus
{{\bf s}\,}^{-2}{{\mathbb K}}\,.$$ To identify the cycles representing the homology classes we parameterize the circle $S^1$ by the angle variable ${{\varphi}}\in [0, 2\pi]$ and the torus $S^1\times S^1$ by the pair of angle variables ${{\varphi}}_1, {{\varphi}}_2 \in [0, 2\pi]$.
The zeroth homology space $H_{0}({{\rm C y l}\,}^{{{\mathfrak{a}}}}(1,1), {{\mathbb K}})$ is one-dimensional and its generator corresponds to the contraction $i$ of elements of ${{\cal V}}$ with elements of ${{\cal W}}$. The second homology space $H_{2}({{\rm C y l}\,}^{{{\mathfrak{a}}}}(1,1), {{\mathbb K}})$ is also one-dimensional. Its generator corresponds to the composition ${{\delta}}\, i\, {{\delta}}$.
The cycle $$\label{cyc-id}
{{\varphi}}\to (0, {{\varphi}}) : S^1 \hookrightarrow S^1\times S^1$$ represents the homology class corresponding to the composition $i \, {{\delta}}$. In order to get this cycle in ${{\rm C y l}\,}^{{{\mathfrak{a}}}}(1,1)$ we need to revolve the point $b$ on the lower boundary about the vertical axis as it is shown on Figure \[id\].
\[\][$b$]{} \[\][$t$]{}
{width="40.00000%"}\
The composition ${{\delta}}i$ is, in turn, represented by the diagonal $$\label{cyc-di}
{{\varphi}}\to ({{\varphi}}, {{\varphi}}) : S^1 \to S^1 \times S^1$$ of the torus. To get this cycle we need to revolve simultaneously the disc and the point $b$ about the vertical axis as it is shown on Figure \[di\].
\[\][$b$]{} \[\][$t$]{}
{width="40.00000%"}\
The homology classes ${{\delta}}i$ and $i {{\delta}}$ form a basis of $H_{1}({{\rm C y l}\,}^{{{\mathfrak{a}}}}(1,1), {{\mathbb K}})$.
We would like to remark that the homology class represented by the cycle $$\label{cyc-l}
{{\varphi}}\to ({{\varphi}}, 0) : S^1 \hookrightarrow S^1\times S^1$$ equals to the combination $${{\delta}}\, i - i \, {{\delta}}\,.$$ Indeed it is easy to see that the cycles (\[cyc-id\]), (\[cyc-di\]), and (\[cyc-l\]) form the boundary of the following $2$-simplex in $S^1\times S^1$ $$\{({{\varphi}}_1, {{\varphi}}_2)\,\,|\,\, {{\varphi}}_1 \le {{\varphi}}_2 \} \subset S^1\times S^1\,.$$ Thus the cycle (\[cyc-l\]) represents the homology class corresponding to the Lie derivative $l$. To get this cycle in ${{\rm C y l}\,}^{{{\mathfrak{a}}}}(1,1)$ we need to revolve the disc about the vertical axis as it is shown on Figure \[l\].
\[\][$b$]{} \[\][$t$]{}
{width="40.00000%"}\
In general, for $n\ge 1$ the homology of the space ${{\rm C y l}\,}^{{{\mathfrak{a}}}}(n,1)$ can be computed with the help of the homological version of Lemma 6.2 from [@Cohen]. Due to this lemma we have $$\label{H-Conf}
H_{{{\bullet}}}({{\rm Conf}\,}_n({{\mathbb R}}^2 \setminus {\bf 0}), {{\mathbb K}}) =
\bigotimes_{j=1}^{n}
H_{{{\bullet}}} (\vee^j \, S^1, {{\mathbb K}})\,.$$ Using the homotopy equivalence (\[homot-equiv\]) and the Künneth formula, we deduce that $$\label{H-Cyl-ma}
H_{{{\bullet}}}({{\rm C y l}\,}^{{{\mathfrak{a}}}}(n,1), {{\mathbb K}}) =
\bigotimes_{j=1}^{n}
H_{{{\bullet}}} (\vee^j \, S^1, {{\mathbb K}}) \,\, \otimes \,\,
H_{{{\bullet}}}(S^1, {{\mathbb K}})\,.$$
Let us show that the operad $H_{-{{\bullet}}}({{\rm C y l}\,}, {{\mathbb K}})$ is generated by operations of $H_{-{{\bullet}}}({{\rm C y l}\,}^{{{\mathfrak{c}}}}(2,0), {{\mathbb K}})$, $H_{-{{\bullet}}}({{\rm C y l}\,}^{{{\mathfrak{a}}}}(0,1)$, and $H_{-{{\bullet}}}({{\rm C y l}\,}^{{{\mathfrak{a}}}}(1,1), {{\mathbb K}})$.
Since the case of the operad of little discs was already considered by F. Cohen [@Cohen] we should only consider the operations of $H_{-{{\bullet}}}({{\rm C y l}\,}^{{{\mathfrak{a}}}}(n,1), {{\mathbb K}})$.
Due to homotopy equivalence (\[homot-equiv\]) the homology classes of ${{\rm C y l}\,}^{{{\mathfrak{a}}}}(n,1)$ are of the forms $$\label{form0}
{{\alpha}}\otimes 1\,,$$ and $$\label{form1}
{{\alpha}}\otimes \phi\,,$$ where ${{\alpha}}\in H_{-{{\bullet}}}( {{\rm Conf}\,}_n({{\mathbb R}}^2 \setminus {\bf 0}), {{\mathbb K}}) $, $1$ is the generator of $H_0(S^1, {{\mathbb K}})$ and $\phi$ is the generator of $H_0(S^1, {{\mathbb K}})$.
It is obvious that homology classes of the form (\[form1\]) are obtained by composing the homology classes of the form (\[form0\]) with the generator ${{\delta}}$ of $ H_{1}({{\rm C y l}\,}^{{{\mathfrak{a}}}}(0,1), {{\mathbb K}})$.
To analyze the homology classes (\[form0\]) we consider the Serre spectral sequence corresponding to the fibration (\[p\]). Due to Proposition \[acts-triv\] the $E^2$ term of the sequence is $$\label{E-2}
E^2_{p,q} = H_p({{\rm Conf}\,}_{n-1}({{\mathbb R}}^2 \setminus {\bf 0}), {{\mathbb K}})
\otimes H_q(F_n, {{\mathbb K}})\,,$$ where $F_n$ is the fiber (\[F-n\]) of $p$.
Since $F_n$ is homotopy equivalent to the wedge $\vee^n S^1$ of $n$ circles equation (\[H-Conf\]) implies that the vector spaces $$\bigoplus_{p}
E^2_{p, {{\bullet}}-p}$$ and $H_{{{\bullet}}}({{\rm Conf}\,}_{n}({{\mathbb R}}^2 \setminus {\bf 0}), {{\mathbb K}})$ have the same dimension. Thus, using the fact that spectral sequence corresponding to the fibration (\[p\]) converges to $H_{{{\bullet}}}({{\rm Conf}\,}_{n}({{\mathbb R}}^2 \setminus {\bf 0}), {{\mathbb K}})$, we deduce that this spectral sequence degenerates at $E_2$ and $$\label{H-Conf1}
H_{{{\bullet}}}({{\rm Conf}\,}_{n}({{\mathbb R}}^2 \setminus {\bf 0}), {{\mathbb K}})
=\bigoplus_{p+q= {{\bullet}}}
H_p({{\rm Conf}\,}_{n-1}({{\mathbb R}}^2 \setminus {\bf 0}), {{\mathbb K}}) \otimes H_q(F_n, {{\mathbb K}})$$ Using equation (\[H-F-n\]) we reduce this expression further to $$H_{{{\bullet}}}({{\rm Conf}\,}_{n}({{\mathbb R}}^2 \setminus {\bf 0}), {{\mathbb K}}) =$$ $$\label{H-Conf11}
H_{{{\bullet}}}({{\rm Conf}\,}_{n-1}({{\mathbb R}}^2 \setminus {\bf 0}), {{\mathbb K}})\otimes H_0(F_n, {{\mathbb K}}) \,\,
\oplus \,\,
H_{{{\bullet}}-1}({{\rm Conf}\,}_{n-1}({{\mathbb R}}^2 \setminus {\bf 0}), {{\mathbb K}})
\otimes H_1(F_n, {{\mathbb K}})\,.$$ Let $v_1, v_2, \dots, v_n\in {{\cal V}}$ and $w\in {{\cal W}}$ be the arguments of an operation corresponding to a homology class in (\[H-Conf11\]). It is not hard to see that the generator of $H_0(F_n, {{\mathbb K}})= {{\mathbb K}}$ corresponds to the contraction $i$ with $v_1$ and the generators of $H_1(F_n, {{\mathbb K}}) = {{\mathbb K}}^n$ correspond[^11] to the brackets $[v_1, v_j]$ for $j \in \{2, 3, \dots, n\}$ and the Lie derivative $l_{v_1}$ along $v_1$.
Thus the homology classes of $H_{-{{\bullet}}}({{\rm C y l}\,}^{{{\mathfrak{a}}}}(n,1), {{\mathbb K}})$ are all produced by the operadic compositions of the classes in $H_{-{{\bullet}}}({{\rm C y l}\,}^{{{\mathfrak{a}}}}(n-1,1), {{\mathbb K}})$, $H_{-{{\bullet}}}({{\rm C y l}\,}^{{{\mathfrak{a}}}}(1,1), {{\mathbb K}})$, and $H_{-{{\bullet}}}({{\rm C y l}\,}^{{{\mathfrak{c}}}}(2,0), {{\mathbb K}})$.
This inductive argument allows us to conclude that the operad $H_{-{{\bullet}}}({{\rm C y l}\,}, {{\mathbb K}})$ is generated by the classes $$\label{gener-H-Cyl}
\begin{array}{c}
\begin{array}{ccc}
i\in H_{0}({{\rm C y l}\,}^{{{\mathfrak{a}}}}(1,1), {{\mathbb K}})\,, &
l \in H_{1}({{\rm C y l}\,}^{{{\mathfrak{a}}}}(1,1), {{\mathbb K}})\,, &
{{\delta}}\in H_{1}({{\rm C y l}\,}^{{{\mathfrak{a}}}}(0,1), {{\mathbb K}})\,,
\end{array}\\[0.5cm]
\begin{array}{cc}
\wedge \in H_0({{\rm C y l}\,}^{{{\mathfrak{c}}}}(2,0), {{\mathbb K}})\,, &
[\,,\,] \in H_1({{\rm C y l}\,}^{{{\mathfrak{c}}}}(2,0), {{\mathbb K}})\,.
\end{array}
\end{array}$$
Using equation (\[free-calc-ma\]) and the fact [@Klyach] that $\dim {{\bf Lie}}(n) = (n-1)!$ it is not hard to show that $${{\bf calc}}^{{{\mathfrak{a}}}}(n,1) \cong
{{\bf calc}}^{{{\mathfrak{a}}}}(n-1,1) \otimes ({{\mathbb K}}\oplus {{\bf s}\,}^{-1} {{\mathbb K}}^n)$$ as graded vector spaces.
On the other hand, equation (\[H-Cyl-ma\]) gives us the same isomorphism $$H_{-{{\bullet}}}( {{\rm C y l}\,}^{{{\mathfrak{a}}}}(n,1), {{\mathbb K}}) \cong
H_{-{{\bullet}}}( {{\rm C y l}\,}^{{{\mathfrak{a}}}}(n-1,1), {{\mathbb K}})
\otimes ({{\mathbb K}}\oplus {{\bf s}\,}^{-1} {{\mathbb K}}^n)$$ of graded vector spaces for $H_{-{{\bullet}}}({{\rm C y l}\,}, {{\mathbb K}})$.
Therefore, the dimensions of graded components of $H_{-{{\bullet}}}({{\rm C y l}\,}^{{{\mathfrak{a}}}}(n,1), {{\mathbb K}})$ and ${{\bf calc}}^{{{\mathfrak{a}}}}(n,1)$ are equal.
Thus, in order the complete the proof of the theorem, we need to show the operations (\[gener-H-Cyl\]) satisfy the identities of calculus algebra (see Definition \[calc\]).
The identities of the Gerstenhaber algebra were already checked in [@Cohen]. The identity (\[l-i-delta\]) was checked above. On Figure \[iden\]
\[\][$b$]{} \[\][$t$]{} \[\][$+$]{} \[\][$\sim$]{}
{width="80.00000%"}\
we show how to check the identity $$\label{i-l-brack}
[i, l] = i_{[\,,\,]}\,.$$ The remaining identities can be checked in the similar way.
Theorem \[H-Cyl\] is proved. $\Box$
Required results from [@K-Soi1]
-------------------------------
We will need
\[KS-teo\] The operad ${{\bf K S}}$ is quasi-isomorphic to the operad of singular chains of the topological operad ${{\rm C y l}\,}$. The homology operad $H_{-{{\bullet}}}({{\bf K S}}, {{\mathbb K}})$ is generated by the classes of the operations $\cup$ (\[cup\]), $[\,,\,]_G$ (\[Gerst\]), $I$ (\[I-P\]), $L$ (\[L-Q\]) and $B$ (\[B\]).
Furthermore, (See Proposition 11.3.3 on page 50 in [@K-Soi1])
\[discs-formal\] The operad of singular chains of the topological operad ${{\rm C y l}\,}$ is formal.
Combining these two statements with Theorem \[H-Cyl\] we get the following corollary.
\[vot-ono\] The pair $({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A))$ is a homotopy calculus algebra. The operations of this algebra structure are expressed in terms of operations of ${{\cal H}}$. The induced calculus structure on the pair $(HH^{{{\bullet}}}(A), HH_{{{\bullet}}}(A))$ coincides with the one in [@DGT].
A useful property of the operad ${{\bf K S}}$
---------------------------------------------
For the $2$-colored operad ${{\bf K S}}$ of chain complexes we have the following
\[low-degrees\] The elements of ${{\bf K S}}^{{{\mathfrak{c}}}}(k,0)$ have degrees $$\label{deg1}
\deg \ge 1 - k$$ and the elements of ${{\bf K S}}^{{{\mathfrak{a}}}}(k,1)$ have degrees $$\label{deg11}
\deg \ge -1-k\,.$$
[**Proof.**]{} Let us start with ${{\bf K S}}^{{{\mathfrak{c}}}}(k,0)$ for $k=1$. Operations of ${{\bf K S}}^{{{\mathfrak{c}}}}(1,0)$ produce a Hochschild cochain from a Hochschild cochain. In order to prove that these operations have nonnegative degrees we need to show that the operations from ${{\cal H}}(1,0)$ which lower the number the number of arguments of the cochain do not contribute to the realization of ${{\cal H}}$.
Let $T$ be a tree representing an operation from ${{\cal H}}(1,0)$ and let $v_a$ be the vertex adjacent to the root. If $v_a$ is marked then it is marked by the only cochain $P$ which enters as an argument. All other marked vertices are necessarily terminal and they are reserved for the arguments of the cochain produced by the operation.
In order to lower the number of arguments, we need to insert the unit into the cochain $P$. The insertion of the unit is a degeneracy of the simplicial structure on ${{\cal H}}(1,0)$. Hence all operations from ${{\cal H}}(1,0)$ which lower the degree of the cochain do not contribute to the realization.
If $v_a$ is unmarked then, starting with the marked vertex $v_P$ reserved for the cochain $P$, we can form the proper maximal subtree with $v_P$ being the vertex adjacent to the root. In order to contribute to the realization the operation corresponding to this subtree has to have a nonnegative degree. Hence, so does the operation corresponding to the whole tree.
We proved (\[deg1\]) for $k=1$.
Let us take it as a base of the induction and assume that (\[deg1\]) is proved for all $m < k $.
We consider a tree $T$ which represents an operation from ${{\cal H}}(k,0)$ and denote the vertex adjacent to the root of $T$ by $v_a$. Let us consider the case when the vertex $v_a$ is marked. Say, $v_a$ is reserved for a cochain $P_1$ of degree $q_1$.
Then the tree $T$ has exactly $q_1$ maximal proper subtrees whose root vertex is $v_a$. We denote these subtrees by $T_1, T_2, \dots, T_{q_1}$. The number $q_1$ splits into the sum $$\label{q1}
q_1 = p_{n} + p_{y}\,,$$ where $p_{n}$ is the number of the subtrees with no vertices reserved for cochains and $p_{y}$ is the number of the subtrees in which at least one vertex is reserved for a cochain.
Let $P$ denote the cochain produced by the operation in question and let $r$ by the number of arguments of this cochain. We will find an estimate for $r$ using the inductive assumption.
Every subtree with no vertices reserved for cochains has to give at least one argument for the cochain $P$. Otherwise, we have to insert the unit as an argument of the cochain $P_1$. In this case the operation in question is a obtained from another operation by degeneracy. Therefore this operation would not contribute to the realization of ${{\cal H}}$.
If $T_j$ is a subtree with exactly $k_j$ vertices reserved for cochains then, obviously, $k_j < k$. Hence, applying the assumption of the induction, we get that the number of arguments of $P$ coming from $T_j$ is greater or equal $$(1- k_j) + q_j\,,$$ where $q_j$ is the total degree of all cochains entering as arguments of the operation corresponding to the subtree $T_j$.
Thus the number $r$ of the arguments of the cochain $P$ produced by the operation in question can be estimated by $$\label{r-ineq}
r \ge p_{n} + \sum_{j=1}^{p_y}(1- k_j + q_j)\,.$$ This inequality can be rewritten as $$r - (\, p_n + p_y + \sum_{j=1}^{p_y} q_j \,) \ge
- \sum_{j=1}^{p_y} k_j\,.$$ Due to equation (\[q1\]) the sum $$p_n + p_y + \sum_{j=1}^{p_y} q_j$$ is the total degree of all cochains entering as arguments of the operation. Furthermore, since the vertex $v_a$ is reserved for one of the cochains $$\sum_{j=1}^{p_y} k_j = k-1$$ and the desired inequality (\[deg1\]) is proved in this case.
Let us now consider the case when the vertex $v_a$ is unmarked.
The valency of the vertex $v_a$ has to be at least $2$. Otherwise the operation will have the empty set of arguments. If the valency of this vertex is $2$ then we remove it using the equivalence transformation.
Thus, without loss of generality, we may assume that the vertex $v_a$ has at least $2$ incoming edges. Let us denote by $s$ the number of these incoming edges and let $T_1, \dots, T_s$ be the maximal proper subtrees of $T$ whose root vertex is $v_a$.
If the vertices reserved for cochains belong to only one of these subtrees $T_i$ then excising the subtrees $T_j$ for $j\neq i$ we get another operation whose degree is less or equal the degree of the original operation. Since in the new tree the unmarked vertex $v_a$ has the valency $2$ we may remove this vertex by the corresponding equivalence transformation.
If in this modified tree the vertex adjacent to the root is marked then we deduce the desired inequality to the case considered above.
Otherwise, we should only consider the case when the vertex $v_a$ is unmarked, its valency is at least $3$ and each maximal proper subtree $T_j$ of $T$ with root vertex $v_a$ has at least one vertex reserved for a cochain.
Let $s$ be the number of the maximal proper subtrees of $T$ whose root vertex is $v_a$. Since the number of these subtrees is greater or equal than $2$ therefore every subtree $T_j$ represents an operation with the number of arguments $k_j < k$.
Let $\deg(T_j)$ be the degree of the operation corresponding to the $j$-th subtree $T_j$. Applying the assumption of the induction we get the inequality $$\label{m-j}
\deg(T_j) \ge 1 - k_j\,.$$ It is clear that the degree $\deg(T)$ of the operation corresponding to the tree $T$ is the sum of degrees of the operations corresponding to the subtrees $T_1, \dots, T_s$. Therefore $$\deg(T) \ge \sum_{i=1}^s (1- k_i)\,.$$ On the other hand $\displaystyle \sum_{i=1}^s k_i = k$. Therefore, $$\deg(T) \ge s - k$$ and inequality (\[deg1\]) follows from the fact that $s \ge 2$.
To prove the second inequality (\[deg11\]) we denote by ${{\cal H}}^{m_a}_{m_r}(k,1)$ the set of operations producing a chain $$\label{chain-r}
(b_0, b_1, \dots, b_{m_r})$$ in $C_{-m_r}(A)$ from $k$ cochains and a chain $$\label{chain-a}
(c_0, c_1, \dots, c_{m_a})$$ in $C_{-m_a}(A)$.
Let $F$ be a forest on the cylinder (\[Si\]) representing an operation from ${{\cal H}}^{m_a}_{m_r}(k,1)$ which contribute to the realization of ${{\cal H}}(k,1)$. Our purpose is to prove the inequality $$\label{ineq-ch}
-m_r \ge -m_a + q -1 - k \,,$$ where $q$ is the total degree of all $k$ cochains of the operation.
This inequality is equivalent to $$\label{ineq-ch1}
m_a \ge m_r + q -1 - k\,.$$
By construction the forest $F$ has exactly $m_r$ trees. Let us denote these trees by $S_1, \dots, S_{m^n_r}$, $T_1, \dots, T_{m^y_r}$ where the trees $S_1, \dots, S_{m^n_r}$ have no vertices reserved for cochains and each tree $T_i$ has at least one vertex reserved for a cochain. Obviously, $$\label{m-r}
m_r = m^n_r + m^y_r\,.$$
The roots of the trees $S_1, \dots, S_{m^n_r}$, $T_1, \dots, T_{m^y_r}$ are marked by components of the chain (\[chain-r\]). If the root of the tree $S_i$ is marked by the component $b_j$ of the for $j \neq 0$ then $S_i$ has to have at least one terminal vertex marked by a component of the chain (\[chain-a\]). Otherwise we have to insert the unit as the $j$-th component of (\[chain-r\]) for $j\neq 0$. In this case the operation in question is a composition of the another operation and a degeneracy. Hence, this operation would not contribute to the realization of ${{\cal H}}(k,1)$.
Let us denote by $m_i$ the number of the terminal vertices of the tree $T_i$ marked by components of the chain (\[chain-a\]). If the tree $T_i$ has exactly $k_i$ vertices reserved for the cochains then $T_i$ represents an operation from ${{\cal H}}(k_i,0)$. Furthermore, if the operation corresponding to the forest $F$ contributes to the realization then so does the operation corresponding to the tree $T_i$. Hence, the number $m_i$ can be estimated using the inequality (\[deg1\]) $$m_i \ge q_i +1 - k_i\,,$$ where $q_i$ is the total degree of all cochains of the operation corresponding to the tree $T_i$.
Thus we get the following inequality for $m_a$ $$\label{ineq-m-a}
m_a \ge (m^n_r -1) + \sum_{i=1}^{m^y_r} (q_i +1 - k_i)\,,$$ where the first term $(m^n_r -1)$ in the right hand side comes from estimate of the number of the marked terminal vertices of the trees $S_1, \dots, S_{m^n_r}$.
Inequality (\[ineq-m-a\]) can be rewritten as $$\label{ineq-m-a1}
m_a \ge m^n_r + m^y_r + q - 1 - k\,,$$ where $q$ is the total degree of all $k$ cochains of the operation in question.
Due to equation (\[m-r\]) inequality (\[ineq-m-a1\]) coincides exactly with the desired inequality (\[ineq-ch1\]).
The proposition is proved. $\Box$
The homotopy calculus on the pair $({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A))$.
===================================================================================================
Since the operad ${{\Lambda}}{{\bf Lie}}^+_{{{\delta}}}$ is a suboperad of ${{\bf calc}}$, Corollary \[vot-ono\] implies that the pair $$({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A))$$ carries a natural $Ho({{\Lambda}}{{\bf Lie}}^+_{{{\delta}}})$-algebra structure. In this section we show that the homotopy calculus on $({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A))$ can be modified in such a way that its $Ho({{\Lambda}}{{\bf Lie}}^+_{{{\delta}}})$-algebra part becomes the ${{\Lambda}}{{\bf Lie}}^+_{{{\delta}}}$-algebra structure given by the operations $[\,,\,]_G$ (\[Gerst\]), $L$ (\[L-Q\]), and $B$ (\[B\]).
Due to Proposition \[GJ\] a homotopy calculus structure on the pair $({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A))$ is a Maurer-Cartan element $$\label{Q}
Q \in {\rm C o d e r}'( {{\mathbb F}}_{{{\bf B}}}({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A)) )\,.$$
In other words, $Q$ is a degree $1$ coderivation of the coalgebra ${{\mathbb F}}_{{{\bf B}}}({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A))$ satisfying the condition $$Q\Big|_{{C^{\bullet}_{{\rm norm}}}(A) \oplus {C_{\bullet}^{{\rm norm}}}(A)} = 0\,,$$ and the equation $$\label{MC-Q}
[{{\partial}}^{Bar} + {{\partial}}^{Hoch},Q] + Q \circ Q =0\,,$$ where ${C^{\bullet}_{{\rm norm}}}(A) \oplus {C_{\bullet}^{{\rm norm}}}(A)$ is considered as subspace of ${{\mathbb F}}_{{{\bf B}}}({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A))$ via coaugmentation and ${{\partial}}^{Hoch}$ is the differential coming from the Hochschild (co)boundary operator on $({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A))$.
It is convenient to reserve a notation for the Lie algebra ${\rm C o d e r}'({{\mathbb F}}_{{{\bf B}}}({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A)))$ $$\label{bbL}
{{\mathbb L}}= {\rm C o d e r}'({{\mathbb F}}_{{{\bf B}}}({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A)))\,.$$
Proposition \[coder-cofree\] implies that the coderivation $Q$ is uniquely determined by its composition with the corestriction $\rho : {{\mathbb F}}_{{{\bf B}}}({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A)) \to {C^{\bullet}_{{\rm norm}}}(A)\oplus {C_{\bullet}^{{\rm norm}}}(A) $ $$\label{q}
q = \rho \circ Q : {{\mathbb F}}_{{{\bf B}}}({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A)) \to
{C^{\bullet}_{{\rm norm}}}(A)\oplus {C_{\bullet}^{{\rm norm}}}(A) \,,$$ while equation (\[MC-Q\]) is equivalent to $$\label{MC-Qq}
[{{\partial}}^{Hoch}, q] + q \circ {{\partial}}^{Bar}
+ q \circ Q = 0\,.$$
The coalgebra ${{\mathbb F}}_{{{\bf B}}}({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A))$ is equipped with a natural increasing filtration[^12] $${C^{\bullet}_{{\rm norm}}}(A) \oplus {C_{\bullet}^{{\rm norm}}}(A)[[u]] =
{{\cal F}}^1\, {{\mathbb F}}_{{{\bf B}}}({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A))
\subset
{{\cal F}}^2\, {{\mathbb F}}_{{{\bf B}}}({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A))
\subset \dots$$ $$\label{filtr-calc}
\begin{array}{c}
\displaystyle
{{\cal F}}^m\, {{\mathbb F}}_{{{\bf B}}}({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A))_{{{\mathfrak{c}}}}
=
\bigoplus_{k\le m}
{{\bf B}}^{{{\mathfrak{c}}}}(k,0)\,\otimes_{S_{k}}\,({C^{\bullet}_{{\rm norm}}}(A))^{\otimes\, k}\,, \\[0.3cm]
\displaystyle
{{\cal F}}^m\, {{\mathbb F}}_{{{\bf B}}}({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A))_{{{\mathfrak{a}}}}
=
\bigoplus_{k+1\le m}
{{\bf B}}^{{{\mathfrak{a}}}}(k,1)\,\otimes_{S_{k}}\,({C^{\bullet}_{{\rm norm}}}(A))^{\otimes\, k} \otimes
{C_{\bullet}}(A)\,,
\end{array}$$ where $u$ is an auxiliary variable of degree $-2$.
Using (\[filtr-calc\]) we endow the Lie algebra (\[bbL\]) with the following decreasing filtration $${{\mathbb L}}={{\cal F}}^0\, {{\mathbb L}}\supset {{\cal F}}^1\, {{\mathbb L}}\supset
{{\cal F}}^2\, {{\mathbb L}}\supset \dots$$ $$\label{filtr-coder}
{{\cal F}}^m\, {{\mathbb L}}= \{\, Y\in {{\mathbb L}}~~|~~
Y\Big|_{{{\cal F}}^m\, {{\mathbb F}}_{{{\bf B}}}({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A))} = 0
\, \}\,.$$
Since $${{\mathbb L}}=
\lim_{m} {{\mathbb L}}/
{{\cal F}}^m\, {{\mathbb L}}$$ the Lie subalgebra ${{\cal F}}^1 {{\mathbb L}}^0$ is pronilpotent.
Therefore ${{\cal F}}^1 {{\mathbb L}}^0$ integrates to a prounipotent group $$\label{bbG}
{{\mathbb G}}= \exp({{\cal F}}^1 {{\mathbb L}}^0)$$ which acts on the Maurer-Cartan elements of ${{\mathbb L}}$. This action is defined by the formula: $$\label{action}
\exp(Y)\, Q =
\exp([\,\,,Y])\, Q + f([\,\,,Y])\, [{{\partial}}^{Bar}
+ {{\partial}}^{Hoch}, Y]\,,$$ where $f$ is the power series of the function $$f(x) = \frac{e^x - 1}{x}$$ at the point $x=0$.
Let $Q$ be a Maurer-Cartan element of the DGLA ${{\mathbb L}}$ which corresponds to the homotopy calculus structure on the pair $({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A))$ which comes from the action of the operad ${{\bf K S}}$.
For every $Y \in {{\cal F}}^1 {{\mathbb L}}$ the homotopy calculus on the pair $({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A))$ corresponding to the Maurer-Cartan element $\exp(Y)\, Q$ is quasi-isomorphic to the homotopy calculus corresponding the original Maurer-Cartan element $Q$. Indeed, the desired ${{{\rm Ho}\,}({\bf calc})}$-quasi-isomorphism is expressed in terms of $Y$ as $$\label{q-iso-Y}
\exp([\,\,,Y]) : ({{\mathbb F}}_{{{\bf B}}}({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A)), Q)
\to ({{\mathbb F}}_{{{\bf B}}}({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A)), \exp(Y)\, Q)\,.$$
Thus we get a family of mutually quasi-isomorphic homotopy calculus structures on $({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A))$. Let us denote this family by ${{\bf S}}_{{{\bf calc}}}$.
We claim that
\[valid-beer\] The family ${{\bf S}}_{{{\bf calc}}}$ contains a homotopy calculus structure whose $Ho({{\Lambda}}{{\bf Lie}}^+_{{{\delta}}})$-algebra part is the ${{\Lambda}}{{\bf Lie}}^+_{{{\delta}}}$-algebra structure given by the operations $[\,,\,]_G$ (\[Gerst\]), $L$ (\[L-Q\]), and $B$ (\[B\]).
[**Proof.**]{} According to Proposition \[GJ\] and equation (\[HoLaLie\]) the ${{\rm Ho}\,}({{\Lambda}}{{\bf Lie}}^+_{{{\delta}}})$ is given by a Maurer-Cartan element $M$ of the DGLA $$\label{esche-odna}
{\rm C o d e r}'(\,{{\mathbb F}}_{{{\Lambda}}({{\bf Lie}}^+_{{{\delta}}})^{\vee}}({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A))\, )\,.$$ Due to Proposition \[coder-cofree\] this Maurer-Cartan element is, in turn, uniquely determined by the composition with the corestriction $\rho$ onto ${C^{\bullet}_{{\rm norm}}}(A) \oplus {C_{\bullet}^{{\rm norm}}}(A)$ $$\label{m}
m =
\rho \circ M : {{\mathbb F}}_{{{\Lambda}}({{\bf Lie}}^+_{{{\delta}}})^{\vee}}({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A))
\to {C^{\bullet}_{{\rm norm}}}(A) \oplus {C_{\bullet}^{{\rm norm}}}(A)\,.$$
Finally, Proposition \[how-to-pull\] shows that the map $m$ is related to the map $q$ (\[q\]) by the equation $$\label{m-q}
m = q \Big|_{{{\mathbb F}}_{{{\Lambda}}({{\bf Lie}}^+_{{{\delta}}})^{\vee}}({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A))}\,,$$ where ${{\Lambda}}({{\bf Lie}}^+_{{{\delta}}})^{\vee}$ is considered as a sub-cooperad of ${{\bf B}}= Bar({{\bf calc}})$ via the chain of embeddings: $${{\Lambda}}({{\bf Lie}}^+_{{{\delta}}})^{\vee} \hookrightarrow Bar({{\Lambda}}{{\bf Lie}}^+_{{{\delta}}})
\hookrightarrow Bar({{\bf calc}})\,.$$
The ${{\rm Ho}\,}({{\Lambda}}{{\bf Lie}})$-algebra structure on ${C^{\bullet}_{{\rm norm}}}(A)$ is induced by the ${{{\rm Ho}\,}({\bf e_2})}$-algebra structure which, in turn, comes from the action of the minimal operad of Kontsevich and Soibelman [@K-Soi] on ${C^{\bullet}_{{\rm norm}}}(A)$. It was proved in [@BLT] (see Theorem 2) that this ${{\rm Ho}\,}({{\bf Lie}})$-algebra is in fact a genuine Lie algebra given by $[\,,\,]_G$.
Thus it remains to take care about the operations involving a chain.
Due to (\[liede-vee1\]) all the operations of the ${{\rm Ho}\,}({{\Lambda}}{{\bf Lie}}^+_{{{\delta}}})$-algebra involving a chain are combined into a single degree $1$ map $$m^{{{\mathfrak{a}}}} =$$ $$\label{m-ma}
m \Big|_{{{\mathbb F}}_{{{\Lambda}}^2{{\bf cocomm}}^+}({C^{\bullet}_{{\rm norm}}}(A),{C_{\bullet}^{{\rm norm}}}(A)[[u]])_{{{\mathfrak{a}}}}} :
{{\mathbb F}}_{{{\Lambda}}^2{{\bf cocomm}}^+}({C^{\bullet}_{{\rm norm}}}(A),{C_{\bullet}^{{\rm norm}}}(A)[[u]])$$ $$\to {C_{\bullet}^{{\rm norm}}}(A)\,.$$ such that $$\label{m-00}
m^{{{\mathfrak{a}}}}\Big|_{{C_{\bullet}^{{\rm norm}}}(A)} = 0\,.$$ The latter equation follows from the fact that the coderivation $M$ belongs to the DGLA (\[esche-odna\]).
In other words, we have the infinite collection of operations: $$\label{m-kn}
m^{{{\mathfrak{a}}}}_{k,n} : S^k({C^{\bullet}_{{\rm norm}}}(A))\otimes {C_{\bullet}^{{\rm norm}}}(A) \to {C_{\bullet}^{{\rm norm}}}(A)$$ of degrees $1- 2 k- 2n$, where $S^k({C^{\bullet}_{{\rm norm}}}(A))$ is the $k$-th component of the symmetric algebra $S({C^{\bullet}_{{\rm norm}}}(A))$ of ${C^{\bullet}_{{\rm norm}}}(A)$.
Due to Proposition \[low-degrees\] the operation $m^{{{\mathfrak{a}}}}_{k,n}$ vanish if $k+2n > 2$. Furthermore, due to equation (\[m-00\]) we have $m^{{{\mathfrak{a}}}}_{0,0}=0$. Thus we need to analyze only there operations: one unary operation $$\label{m-01}
m^{{{\mathfrak{a}}}}_{0,1} : {C_{\bullet}^{{\rm norm}}}(A) \to {C_{\bullet}^{{\rm norm}}}(A)$$ of degree $-1$, one binary operation $$\label{m-10}
m^{{{\mathfrak{a}}}}_{1,0} : {C^{\bullet}_{{\rm norm}}}(A) \otimes {C_{\bullet}^{{\rm norm}}}(A) \to {C_{\bullet}^{{\rm norm}}}(A)$$ of degree $-1$ and one ternary operation $$\label{m-20}
m^{{{\mathfrak{a}}}}_{2,0} : S^2({C^{\bullet}_{{\rm norm}}}(A))\otimes {C_{\bullet}^{{\rm norm}}}(A) \to {C_{\bullet}^{{\rm norm}}}(A)$$ of degree $-3$.
Theorems \[H-Cyl\] and \[KS-teo\] imply that the operation (\[m-01\]) differs from Connes’ operator $B$ by an exact operation. Namely, $$m^{{{\mathfrak{a}}}}_{0,1}(c) = B(c) + {{\partial}}^{Hoch}\, \beta(c) - \beta({{\partial}}^{Hoch}\, c) \,,$$ where $\beta$ is an operation in ${{\bf K S}}$ $$\beta : {C_{\bullet}^{{\rm norm}}}(A) \to {C_{\bullet}^{{\rm norm}}}(A)$$ of degree $-2$.
Using Proposition \[low-degrees\] we deduce that $\beta$ is zero. Hence $m^{{{\mathfrak{a}}}}_{0,1}= B$.
Due to Proposition \[how-to-pull\] the operation (\[m-10\]) is expressed in terms of $q$ (\[q\]) as $$\label{L-1}
m^{{{\mathfrak{a}}}}_{1,0}(P, c) = q(b_1)\,,$$ where $P\in {C^{\bullet}_{{\rm norm}}}(A)$, $c\in {C_{\bullet}^{{\rm norm}}}(A)$, and the element $b_1 \in {{\mathbb F}}_{{{\bf B}}}({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A))$ is depicted on Figure \[b1b2\].
Theorems \[H-Cyl\] and \[KS-teo\] imply that $m^{{{\mathfrak{a}}}}_{1,0}$ differs from the action $L$ of cochains on chains by an exact operation. In other words, $$\label{L-11}
m^{{{\mathfrak{a}}}}_{1,0} (P, c) = -(-1)^{|P|}L_P \,c + {{\partial}}^{Hoch} \psi(P, c)
- \psi({{\partial}}^{Hoch}P, c) - (-1)^{|P|} \psi(P, {{\partial}}^{Hoch} c)\,,$$ where $|P|$ is the degree of $P$ and $$\psi: {C^{\bullet}_{{\rm norm}}}(A) \otimes {C_{\bullet}^{{\rm norm}}}(A) \to {C_{\bullet}^{{\rm norm}}}(A)$$ is an operation in ${{\bf K S}}^{{{\mathfrak{a}}}}(1,1)$ of degree $-2$.
We remark that $\psi$ may be considered as a map $$\label{psi}
\psi: {{\Lambda}}({{\bf Lie}}^+)^{\vee}(1,1)\otimes {C^{\bullet}_{{\rm norm}}}(A) \otimes {C_{\bullet}^{{\rm norm}}}(A)
\to {C_{\bullet}^{{\rm norm}}}(A)$$ of degree $0$. Our purpose is to extend $\psi$ “by zero” to the whole vector space of the coalgebra ${{\mathbb F}}_{{{\bf B}}}({C^{\bullet}_{{\rm norm}}}(A) , {C_{\bullet}^{{\rm norm}}}(A))$. This extension depends on the choice of basis in ${{\bf calc}}^{{{\mathfrak{a}}}}(1,1)$.
We choose the basis $$\label{basis}
\{ l, ~ i, ~ i\,{{\delta}}, ~ l \, {{\delta}}\}$$ and extend $\psi$ to $ {{\bf B}}^{{{\mathfrak{a}}}}(1,1)\otimes {C^{\bullet}_{{\rm norm}}}(A) \otimes {C_{\bullet}^{{\rm norm}}}(A)$ $$\label{ono}
\psi: {{\bf B}}^{{{\mathfrak{a}}}}(1,1)\otimes {C^{\bullet}_{{\rm norm}}}(A) \otimes {C_{\bullet}^{{\rm norm}}}(A)
\to {C_{\bullet}^{{\rm norm}}}(A)\,,$$ as $$\psi(b_1) = \psi(P, c)\,, \qquad \psi(b_2) = \psi(b_3)= \psi(b_4) = 0\,,$$ $$\psi(b_{{{\lambda}}})=0\,,$$ where the elements $$b_1, b_2, b_3, b_4, b_{{{\lambda}}}\in {{\bf B}}^{{{\mathfrak{a}}}}(1,1)\otimes {C^{\bullet}_{{\rm norm}}}(A) \otimes {C_{\bullet}^{{\rm norm}}}(A)$$ are depicted
\[\][$b_1$]{} \[\][$b_2$]{} \[\][$P$]{} \[\][$c$]{} \[\][$=$]{} \[\][$l$]{} \[\][$i$]{}
{width="70.00000%"}\
on Figures \[b1b2\], \[b3b4\], \[bl\],
\[\][$b_3$]{} \[\][$b_4$]{} \[\][$P$]{} \[\][$c$]{} \[\][$=$]{} \[\][$l\,\delta$]{} \[\][$i\,\delta$]{}
{width="70.00000%"}\
${{\lambda}}$ is an arbitrary element of the basis (\[basis\]),
\[\][$b_{\lambda}$]{} \[\][$\delta$]{} \[\][$P$]{} \[\][$c$]{} \[\][$=$]{} \[\][$\lambda$]{}
{height="70.00000%"}\
and $P \in {C^{\bullet}_{{\rm norm}}}(A)$, $c\in {C_{\bullet}^{{\rm norm}}}(A)$.
Next we extend $\psi$ by zero to the whole vector space of the coalgebra $$\label{ona}
\begin{array}{c}
{{\mathbb F}}_{{{\bf B}}}({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A)) = \\[0.4cm]
\displaystyle
\bigoplus_n {{\bf B}}^{{{\mathfrak{c}}}}(n,0)\otimes_{S_n} ({C^{\bullet}_{{\rm norm}}}(A))^{\otimes\, n}
\,\, \oplus \,\,
\bigoplus_n {{\bf B}}^{{{\mathfrak{a}}}}(n,1)\otimes_{S_n} ({C^{\bullet}_{{\rm norm}}}(A))^{\otimes\, n}
\otimes {C_{\bullet}^{{\rm norm}}}(A)\,.
\end{array}$$
Then according to Proposition \[coder-cofree\] the equation $$\rho \circ \Psi = \psi$$ defines a derivation $\Psi$ of the coalgebra (\[ona\]). The derivation $\Psi$ has degree $0$ since so does the map $\psi$. Furthermore, it is obvious that $$\Psi \in {{\cal F}}^1\, {{\mathbb L}}\,.$$
Applying the element $\exp(-\Psi)$ of the group ${{\mathbb G}}$ (\[bbG\]) to the Maurer-Cartan element $Q$ (\[Q\]) we adjust the component $m^{{{\mathfrak{a}}}}_{1,0}$ by killing this additional exact term ${{\partial}}^{Hoch} \psi(P, c)
- \psi({{\partial}}^{Hoch}P, c) -
(-1)^{|P|} \psi(P, {{\partial}}^{Hoch} c) $ in (\[L-11\]). In doing this we do not change the unary operations because $\Psi\in {{\cal F}}^1\, {{\mathbb L}}$.
Thus we are left with only one non-vanishing operation (\[m-20\]).
The Maurer-Cartan equation (\[MC-Qq\]) implies that $m^{{{\mathfrak{a}}}}_{2,0}$ should be closed with respect to the differential ${{\partial}}^{Hoch}$.
Since the degree of $m^{{{\mathfrak{a}}}}_{2,0}$ is $-3$, using Theorems \[H-Cyl\] and \[KS-teo\], we deduce that up to ${{\partial}}^{Hoch}$-exact terms the operation $m^{{{\mathfrak{a}}}}_{2,0}$ is made of the following “building blocks”: $$L_{[P_1, P_2]_G}\, B\, c\,, \qquad L_{P_1} L_{P_2}\, B\, c\,,
\qquad L_{P_2} L_{P_1}\, B\, c\,,$$ where $P_1, P_2 \in {C^{\bullet}_{{\rm norm}}}(A)$ and $c\in {C_{\bullet}^{{\rm norm}}}(A)$.
Using the symmetry in the arguments $P_1$, $P_2$ and the compatibility with ${{\partial}}^{Hoch}$ it is not hard to show that (up to ${{\partial}}^{Hoch}$-exact terms) the most general expression for $m^{{{\mathfrak{a}}}}_{2,0}$ is $$\label{L-2}
m^{{{\mathfrak{a}}}}_{2,0} (P_1, P_2, c) = (-1)^{|P_1|}\mu L_{[P_1, P_2]_G}\, B\, c\,,$$ where $\mu\in {{\mathbb K}}$, $P_1, P_2\in {C^{\bullet}_{{\rm norm}}}(A)$ and $c\in {C_{\bullet}^{{\rm norm}}}(A)$.
If necessary, we apply the above trick with the action (\[action\]) of the group (\[bbG\]) to modify $Q$ (\[Q\]) so that equation (\[L-2\]) indeed holds.
To kill $m^{{{\mathfrak{a}}}}_{2,0}$ we will introduce the map $$\label{y}
y: {{\bf B}}^{{{\mathfrak{a}}}}(1,1)\otimes {C^{\bullet}_{{\rm norm}}}(A) \otimes {C_{\bullet}^{{\rm norm}}}(A) \to
{C_{\bullet}^{{\rm norm}}}(A)\,.$$ This map is defined by the equations $$y(b_1) = -\mu L_P\, B\, c\, \,, \qquad y(b_2) = y(b_3)= y(b_4) =0\,,$$ $$y(b_{{{\lambda}}})=0\,,$$ where the elements $$b_1, b_2, b_3, b_4, b_{{{\lambda}}}\in {{\bf B}}^{{{\mathfrak{a}}}}(1,1)\otimes {C^{\bullet}_{{\rm norm}}}(A) \otimes {C_{\bullet}^{{\rm norm}}}(A)$$ are depicted on Figures \[b1b2\], \[b3b4\], \[bl\], ${{\lambda}}$ is an arbitrary element of the basis (\[basis\]) in ${{\bf calc}}^{{{\mathfrak{a}}}}(1,1)$ and $$P \in {C^{\bullet}_{{\rm norm}}}(A)\,, \qquad c\in {C_{\bullet}^{{\rm norm}}}(A)\,.$$ Then we extend $y$ by zero to the whole vector space of the coalgebra (\[ona\]). It is not hard to see that $y$ is of degree $0$.
According to Proposition \[coder-cofree\] the equation $$\rho \circ Y = y$$ defines a degree $0$ coderivation $Y$ of the coalgebra (\[ona\]). Furthermore, $$\label{Hoch-Y}
[{{\partial}}^{Hoch}, Y] = 0\,,$$ and $$\label{YF1}
Y \in {{\cal F}}^1\, {{\mathbb L}}\,.$$
Applying the element $\exp(Y)$ of the group ${{\mathbb G}}$ (\[bbG\]) to the Maurer-Cartan element $Q$ (\[Q\]) we get another homotopy calculus structure on $({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A))$. This homotopy calculus structure is determined by the Maurer-Cartan element $\exp(Y)\, Q$.
Let us denote by ${{\tilde{m}}}^{{{\mathfrak{a}}}}_{0,1}$, ${{\tilde{m}}}^{{{\mathfrak{a}}}}_{1,0}$, and ${{\tilde{m}}}^{{{\mathfrak{a}}}}_{2,0}$ the operations (\[m-01\]), (\[m-10\]), (\[m-20\]) of the $Ho({{\Lambda}}{{\bf Lie}}^+_{{{\delta}}})$-algebra corresponding to the new homotopy calculus $\exp(Y)\, Q$. Since $Y\in {{\cal F}}^1 {{\mathbb L}}$ the unary operation cannot change. Thus $${{\tilde{m}}}^{{{\mathfrak{a}}}}_{0,1} = m^{{{\mathfrak{a}}}}_{0,1} = B\,.$$ For the binary operation we have $$\label{tm-10}
{{\tilde{m}}}^{{{\mathfrak{a}}}}_{1,0} (P,c) = \rho \circ \exp(Y)\, Q (b_1)\,,$$ where $\rho$ is the projection from ${{\mathbb F}}_{{{\bf B}}}({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A))$ onto $ {C^{\bullet}_{{\rm norm}}}(A) \oplus {C_{\bullet}^{{\rm norm}}}(A) $ and the element $b_1 \in {{\mathbb F}}_{{{\bf B}}}({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A))_{{{\mathfrak{a}}}}$ is depicted on Figure \[b1b2\].
Using (\[action\]), (\[Hoch-Y\]), and (\[YF1\]) we simplify equation (\[tm-10\]) as follows $${{\tilde{m}}}^{{{\mathfrak{a}}}}_{1,0} (P,c)=
m^{{{\mathfrak{a}}}}_{1,0} (P,c) + [(q\circ Y - y \circ Q) + y \circ {{\partial}}^{Bar}]\, b_1\,.$$ It is obvious that ${{\partial}}^{Bar} b_1 = 0$. Furthermore, it is not hard to see that $q \circ Y(b_1)=0$ and $y \circ Q (b_1)=0$. Thus the binary operation (\[m-10\]) is also unchanged.
For the ternary operation ${{\tilde{m}}}^{{{\mathfrak{a}}}}_{2,0}$ we have $${{\tilde{m}}}^{{{\mathfrak{a}}}}_{2,0} (P_1,P_2,c) = \rho \circ \exp(Y)\, Q (b)=$$ $$= m^{{{\mathfrak{a}}}}_{2,0} (P_1,P_2,c) + (q\circ Y - y \circ Q)(b)$$ $$+ y \circ {{\partial}}^{Bar} (b) - y \circ {{\partial}}^{Bar} \circ Y(b) +
\frac{1}{2} y \circ Y \circ {{\partial}}^{Bar}(b)\,,$$ where the element $b \in {{\mathbb F}}_{{{\bf B}}}({C^{\bullet}_{{\rm norm}}}(A), {C_{\bullet}^{{\rm norm}}}(A))_{{{\mathfrak{a}}}}$ is depicted on Figure \[b\]
\[\][$[\,,\,]$]{} \[\][$-$]{} \[\][$P_1$]{} \[\][$P_2$]{} \[\][$c$]{} \[\][$l$]{} \[\][$(-1)^{|P_1|}$]{} \[\][$-(-1)^{|P_1||P_2|}$]{}
{height="70.00000%"}\
and, as above, we used (\[Hoch-Y\]) and (\[YF1\]).
It is obvious that ${{\partial}}^{Bar}(b)=0$ and it is not hard to check that ${{\partial}}^{Bar}\circ Y(b)=0$. Furthermore a direct computation shows that $$(q\circ Y - y \circ Q)(b) = - (-1)^{|P_1|}\mu L_{[P_1, P_2]_G}\, B\, c\,.$$ Thus $$\label{tm-20}
{{\tilde{m}}}^{{{\mathfrak{a}}}}_{2,0} = 0$$ and Theorem \[valid-beer\] is proved. $\Box$
Formality theorem
=================
Enveloping algebra of a Gerstenhaber algebra
--------------------------------------------
Let $({{\cal V}}, {{\cal W}})$ be a pair of graded vector spaces. For our purposes we will need to consider ${{\bf calc}}$-algebras on $({{\cal V}}, {{\cal W}})$ with a fixed Gerstenhaber algebra structure $({{\cal V}}, \wedge, [\,,\,])$ on ${{\cal V}}$. For such ${{\bf calc}}$-algebras we call ${{\cal W}}$ [*a ${{\bf calc}}$-module over $({{\cal V}}, \wedge, [\,,\,])$*]{}.
The category of ${{\bf calc}}$-modules over ${{\cal V}}$ is equivalent to a category of ordinary modules over the enveloping algebra [@TT] of the Gerstenhaber algebra ${{\cal V}}$. In this section we recall the construction of this enveloping algebra and describe its properties.
Let us start with the following definition:
For a Gerstenhaber algebra $({{\cal V}}, \wedge, [\,,\,])$ we define an associative algebra ${{\cal Y}}_0({{\cal V}})$ which is generated by two sets of symbols $$\label{generators}
l_{v},\,\, i_{v} \qquad v \in {{\cal V}}$$ of degrees $$\label{degrees}
|i_{v}| = |v|\,, \qquad |l_{v}|= |v|-1\,.$$ These symbols are ${{\mathbb K}}$-linear in $v$ and they are subject to the following relations $$\label{Y-0relations}
\begin{array}{c}
\begin{array}{cc}
i_{v_1 \cdot v_2} = i_{v_1} i_{v_2}\,, &
[i_{v_1}, l_{v_2}] = i_{[v_1, v_2]}\,,
\end{array} \\[0.7cm]
l_{v_1 \cdot v_2} = l_{v_1} i_{v_2} +
(-1)^{|v_1|} i_{v_1} l_{v_2}\,,
\qquad
[l_{v_1}, l_{v_2}] = l_{[v_1, v_2]}\,.
\end{array}$$
Furthermore,
If $({{\cal V}}, \cdot, [\,,\,])$ is a Gerstenhaber algebra then the associative algebra ${{\cal Y}}({{\cal V}})$ is generated by symbols (\[generators\]) and an element ${{\delta}}$ of degree $-1$. The symbols (\[generators\]) are ${{\mathbb K}}$-linear in $v$ and they are subject to the following relations $$\label{Y-relations}
\begin{array}{cc}
{{\delta}}^2 = 0\,, & [{{\delta}}, i_v] = l_v\,, \\[0.3cm]
i_{v_1 \cdot v_2} = i_{v_1} i_{v_2}\,, &
[i_{v_1}, l_{v_2}] = i_{[v_1, v_2]}\,.
\end{array}$$
It is obvious that the category of ${{\bf calc}}$-modules over ${{\cal V}}$ is equivalent to the category of ordinary modules over the associative algebra ${{\cal Y}}({{\cal V}})$.
Let us give the following natural definition
\[regular\] A DG commutative algebra ${{\cal V}}$ is called regular if the module ${{\Omega}}^1({{\cal V}})$ of its Kähler differentials is cofibrant.
[**Remark.**]{} Notice that if ${{\cal V}}$ is a commutative algebra concentrated in degree $0$ then the above condition of regularity means that the ${{\cal V}}$-module ${{\Omega}}^1({{\cal V}})$ is projective.
We claim that
\[Y-alm-exact\] Let ${{\cal V}}$ be a DG Gerstenhaber algebra and ${{\cal R}}\to {{\cal V}}$ be its cofibrant resolution. If the corresponding DG commutative algebra ${{\cal V}}$ is regular then the induced map $$\label{Y-map}
{{\cal Y}}({{\cal R}}) \to {{\cal Y}}({{\cal V}})$$ is a quasi-isomorphism of DG associative algebras.
[**Proof.**]{} Due to the obvious equality of chain complexes $$\label{cY-cY-0}
{{\cal Y}}({{\cal V}}) = {{\cal Y}}_0({{\cal V}}) \oplus
{{\cal Y}}_0({{\cal V}}) \, {{\delta}}$$ it suffices to show that the map $$\label{Y-0-map}
{{\cal Y}}_0({{\cal R}}) \to {{\cal Y}}_0({{\cal V}})$$ is a quasi-isomorphism.
For this purpose we introduce the following Lie-Rinehart algebra structure [@Rinehart] on the pair $({{\cal V}},{{\Omega}}^1({{\cal V}}))$, where ${{\Omega}}^1({{\cal V}})$ is the module of Kähler differentials of the DG commutative algebra ${{\cal V}}$. The Lie bracket $\{\,,\,\}$ on ${{\Omega}}^1({{\cal V}})$ and the action ${{\mathfrak{l}}}$ of ${{\Omega}}^1({{\cal V}})$ on ${{\cal V}}$ are defined in terms of the Lie bracket on ${{\cal V}}$ as follows $$\label{brack-OmV}
\{ a_1\, d a_2, b_1 \, d b_2 \} =
(-1)^{|a_2|+1} a_1 [a_2, b_1]\, d b_2
+(-1)^{(|a_2|+1) |b_1|} a_1 b_1\, d([a_2, b_2]) +$$ $$(-1)^{(|a_1|+|a_2|+1)(|b_1|+|b_2|+1) + |b_2|}
\, b_1 [b_2, a_1] \, d a_2 \,,$$ $$\label{action-OmV}
{{\mathfrak{l}}}_{a_1\, d a_2}(b) = (-1)^{|a_2|+1}a_1 [a_2, b]\,.$$ The identities of a Gerstenhaber algebra imply that equations (\[brack-OmV\]) and (\[action-OmV\]) indeed define a Lie-Rinehart algebra on the pair $({{\cal V}}, {{\Omega}}^1({{\cal V}}))$.
Next we remark that for every (DG) Gerstenhaber algebra ${{\cal V}}$ the associative algebra ${{\cal Y}}_0({{\cal V}})$ is nothing but the enveloping algebra of the Lie-Rinehart algebra $({{\cal V}}, {{\Omega}}^1({{\cal V}}))$. Indeed, the required isomorphism is defined on generators as $$a \mapsto i_a\,,
\qquad
d b \mapsto l_b\,, \qquad
a,b \in {{\cal V}}\,.$$
Then the PBW-filtration on ${{\cal Y}}_0({{\cal V}})$ is $$\label{Y-0-filtr}
{{\cal V}}\cong {{\cal F}}^0 {{\cal Y}}_0({{\cal V}})
\subset {{\cal F}}^1 {{\cal Y}}_0({{\cal V}}) \subset
{{\cal F}}^2 {{\cal Y}}_0({{\cal V}}) \subset \dots$$ where $${{\cal F}}^{k}{{\cal Y}}_0({{\cal V}})$$ is spanned by monomials in which the number of symbols $l_{v}\,,$ $v \in {{\cal V}}$ is less or equal to $k$.
Since the DG commutative algebra ${{\cal V}}$ is regular we can apply the PBW-theorem [@Rinehart] to the Lie-Rinehart algebra $({{\cal V}}, {{\Omega}}^1({{\cal V}}))$. Using this theorem (see Theorem 3.1 in [@Rinehart]) we conclude that the associated graded algebra is isomorphic to the symmetric algebra $S_{{{\cal V}}}({{\Omega}}^1({{\cal V}}))$ $$\label{Assoc-Gr}
\bigoplus_k {{\cal F}}^{k} {{\cal Y}}_0({{\cal V}})/
{{\cal F}}^{k-1} {{\cal Y}}_0({{\cal V}})
\cong S_{{{\cal V}}}({{\Omega}}^1({{\cal V}}))\,.$$
Since ${{\cal R}}$ is a cofibrant resolution of ${{\cal V}}$ the same argument with PBW theorem from [@Rinehart] works for ${{\cal Y}}_0({{\cal R}})$. The map (\[Y-0-map\]) is obviously compatible with the filtrations (\[Y-0-filtr\]) on ${{\cal Y}}_0({{\cal R}})$ and ${{\cal Y}}_0({{\cal V}})$. Furthermore, these filtrations are cocomplete $${{\cal Y}}_0({{\cal V}}) = {{\rm c o l i m}}_{k} \, {{\cal F}}^k {{\cal Y}}_0({{\cal V}})\,,
\qquad
{{\cal Y}}_0({{\cal R}}) = {{\rm c o l i m}}_{k} \, {{\cal F}}^k {{\cal Y}}_0({{\cal R}})\,.$$ Hence, in order to prove that the map (\[Y-0-map\]) is a quasi-isomorphism, we need to show that so is the map $$\label{Om-map}
S_{{{\cal R}}}({{\Omega}}^1({{\cal R}}))
\to S_{{{\cal V}}}({{\Omega}}^1({{\cal V}}))\,.$$ This statement follows from the regularity of ${{\cal V}}$. Thus the proposition is proved. $\Box$
Sheaves of Hochschild (co)chains on an algebraic variety
--------------------------------------------------------
Let $X$ be a smooth algebraic variety over ${{\mathbb K}}$ with the structure sheaf ${{\cal O}}_X$. We denote by ${V^{\bullet}}_X$ the sheaf of polyvector fields and by ${{\Omega_X^{-\bullet}}}$ be the sheaf of exterior forms with reversed grading. ${{\cal D}}_X$ denotes the sheaf of differential operators on $X$ and ${{\cal D}}_{{{\Omega_X^{-\bullet}}}}$ denotes the sheaf of differential operators on the sheaf of (graded) commutative algebras ${{\Omega_X^{-\bullet}}}$.
In the affine case $X = {\rm S p e c}(A)$ we will use the short-hand notation for the corresponding modules of global sections ${V^{\bullet}}(A) = {{\Gamma}}(X,{V^{\bullet}}_{X})$, ${{\Omega^{-\bullet}}}(A)={{\Gamma}}(X, {{\Omega_X^{-\bullet}}})$, ${{\cal D}}(A) = {{\Gamma}}(X, {{\cal D}}_{X})$, and finally ${{\cal D}}({{\Omega^{-\bullet}}}(A))= {{\Gamma}}(X, {{\cal D}}_{{{\Omega_X^{-\bullet}}}})$.
The pair $({V^{\bullet}}_X, {{\Omega_X^{-\bullet}}})$ is a calculus algebra with respect to the operations: the exterior product $\wedge$ on ${V^{\bullet}}_X$, the Schouten-Nijenhuis bracket $[\,,\,]_{SN}$ on ${V^{\bullet}}_X$, the contraction ${{\cal I}}$ of a form with a polyvector, the Lie derivative ${{\cal L}}$ of a form along a polyvector, and finally the de Rham differential $d$ on the exterior forms.
Using the contraction ${{\cal I}}$ we define the natural ${{\cal O}}_X$-linear pairing ${\langle}\,,\,{\rangle}$ between the sheaves ${V^{\bullet}}_X$ and ${{\Omega_X^{-\bullet}}}$ $${\langle}\,,\, {\rangle}: {V^{\bullet}}_X \otimes_{{{\cal O}}_X} {{\Omega_X^{-\bullet}}}\to {{\cal O}}_X$$ $$\label{pairing}
{\langle}{{\gamma}}, \eta {\rangle}=
\begin{cases}
{{\cal I}}_{{{\gamma}}} \eta\,, \qquad {\rm if} ~~ |\eta|= - |{{\gamma}}| \,, \\
0 \,, \qquad {\rm otherwise}\,.
\end{cases}$$ Here ${{\gamma}}$ and $\eta$ are local sections of ${V^{\bullet}}_X$ and ${{\Omega_X^{-\bullet}}}$, respectively.
An appropriate version of Hochschild cochain complex for the structure sheaf ${{\cal O}}_X$ is the sheaf of polydifferential operators [@Swan], [@Y]. We will denote this sheaf by ${C^{\bullet}}({{\cal O}}_X)$. For example $C^0({{\cal O}}_X)$ is the structure sheaf ${{\cal O}}_X$ and $C^1({{\cal O}}_X)$ is the sheaf ${{\cal D}}_X$ of differential operators on $X$.
Let us also denote by ${C^{\bullet}_{{\rm norm}}}({{\cal O}}_X)$ the sheaf of normalized polydifferential operators. These are the polydifferential operators satisfying the property $$P(\dots, 1, \dots) = 0\,.$$
Similarly an appropriate version of Hochschild chain complex for the structure sheaf ${{\cal O}}_X$ is the sheaf of polyjets [@Tsygan]: $$\label{polyjets}
{C_{\bullet}}({{\cal O}}_X) = {\mathcal Hom\,}_{{{\cal O}}_X}(C^{-{{\bullet}}}({{\cal O}}_X), {{\cal O}}_X)\,,$$ where ${\mathcal Hom\,}$ denotes the sheaf-Hom and ${C^{\bullet}}({{\cal O}}_X)$ is considered with its natural left ${{\cal O}}_X$-module structure. For example $C_{0}({{\cal O}}_X)$ is the structure sheaf ${{\cal O}}_X$ and $C_{-1}({{\cal O}}_X)$ is the sheaf of $\infty$-jets.
There are natural analogs of the degenerate Hochschild chains $$c = (c_0, \dots, 1, \dots)$$ and these degenerate chains form a subsheaf $C^{{\rm degen}}_{\bullet}({{\cal O}}_X)$ of ${C_{\bullet}}({{\cal O}}_X)$. Furthermore the subsheaf $C^{{\rm degen}}_{\bullet}({{\cal O}}_X)$ is closed with respect to the Hochschild boundary operator ${{\partial}}^{Hoch}$.
We define the sheaf ${C_{\bullet}^{{\rm norm}}}({{\cal O}}_X)$ of normalized Hochschild chains as the quotient sheaf ${C_{\bullet}}({{\cal O}}_X)/C^{{\rm degen}}_{\bullet}({{\cal O}}_X) $. It is not hard to show that $$\label{n-polyjets}
{C_{\bullet}^{{\rm norm}}}({{\cal O}}_X) = {\mathcal Hom\,}_{{{\cal O}}_X}(C^{-\bullet}_{norm}({{\cal O}}_X), {{\cal O}}_X)\,.$$
As well as for Hochschild complexes of an associative algebra the inclusion $${C^{\bullet}_{{\rm norm}}}({{\cal O}}_X) \hookrightarrow {C^{\bullet}}({{\cal O}}_X)$$ and the projection $${C_{\bullet}}({{\cal O}}_X) \to {C_{\bullet}^{{\rm norm}}}({{\cal O}}_X)$$ are quasi-isomorphisms of complexes of sheaves. Furthermore, the action of the Kontsevich-Soibelman operad ${{\bf K S}}$ on the pair of sheaves $({C^{\bullet}_{{\rm norm}}}({{\cal O}}_X), {C_{\bullet}^{{\rm norm}}}({{\cal O}}_X))$ is well-defined[^13].
Thus the pair $({C^{\bullet}_{{\rm norm}}}({{\cal O}}_X), {C_{\bullet}^{{\rm norm}}}({{\cal O}}_X))$ is a sheaf of homotopy calculi.
Let us recall that the embedding $$\label{HKR1}
{{\varrho}}_{{{\mathfrak{c}}}}: {V^{\bullet}}_X \hookrightarrow {C^{\bullet}_{{\rm norm}}}({{\cal O}}_X)$$ is called the Hochschild-Kostant-Rosenberg map. It is known [@HKR] that ${{\varrho}}_{V}$ is a quasi-isomorphism of complexes of sheaves where the sheaf ${V^{\bullet}}_X$ is considered with the zero differential.
The corresponding quasi-isomorphism for Hochschild chains $$\label{HKR11}
{{\varrho}}_{{{\mathfrak{a}}}}: {C_{\bullet}^{{\rm norm}}}({{\cal O}}_X) \to {{\Omega_X^{-\bullet}}}$$ is called the Connes-Hochschild-Kostant-Rosenberg map. This map is defined by the equation $$\label{HKR111}
{\langle}{{\gamma}},
{{\varrho}}_{{{\mathfrak{a}}}}(c) {\rangle}= c({{\varrho}}_{{{\mathfrak{c}}}}({{\gamma}}))\,,$$ where $c$ is a local section of $C_{-m}^{{\rm norm}}({{\cal O}}_X)$, ${{\gamma}}$ is a local section of $V^m_X$ and the pairing ${\langle}\, , \, {\rangle}$ is defined in (\[pairing\]).
It is known [@Boris-book] that the maps ${{\varrho}}_{{{\mathfrak{c}}}}$ and ${{\varrho}}_{{{\mathfrak{a}}}}$ are compatible with the operations of the Cartan calculus on the pair $({V^{\bullet}}_X, {{\Omega_X^{-\bullet}}})$ and the operations $\cup$ (\[cup\]), $[\,,\,]_G$ (\[Gerst\]), $I$ (\[I-P\]), $L$ (\[L-Q\]) and $B$ (\[B\]) on the pair $$({C^{\bullet}_{{\rm norm}}}({{\cal O}}_X), {C_{\bullet}^{{\rm norm}}}({{\cal O}}_X))$$ [*up to homotopy*]{}. We upgrade this observation to the following theorem.
\[main\] If $X$ is a smooth algebraic variety over a field ${{\mathbb K}}$ of characteristic zero then the sheaf $({C^{\bullet}_{{\rm norm}}}({{\cal O}}_X), {C_{\bullet}^{{\rm norm}}}({{\cal O}}_X))$ of homotopy calculi is quasi-isomorphic to the sheaf $({V^{\bullet}}_X, {{\Omega}}^{-{{\bullet}}}_X)$ of calculi.
The proof of this theorem is given in Subsection \[proof\].
Morita equivalence
------------------
In this subsection we will show that the sheaf of algebras ${{\cal Y}}({V^{\bullet}}_X)$ is Morita equivalent to the sheaf ${{\cal D}}_X[{{\mathfrak{d}}}]/({{\mathfrak{d}}}^2)$ where ${{\mathfrak{d}}}$ is an auxiliary variable of degree $-1$ commuting with all the differential operators.
First we remark that ${{\cal Y}}_0({V^{\bullet}}_X)$-module structure on the sheaf ${{\Omega_X^{-\bullet}}}$ gives us a natural map $$\label{Y-D-Om-map}
{{\cal Y}}_0({V^{\bullet}}_X) \to {{\cal D}}_{{{\Omega_X^{-\bullet}}}}$$ between the sheaves of associative algebras. We claim that
\[Y-D-Om\] The map (\[Y-D-Om-map\]) is an isomorphism of sheaves of associative algebras.
[**Proof.**]{} We need to show that (\[Y-D-Om-map\]) gives us an isomorphism on stalks $${{\cal Y}}_0({V^{\bullet}}_X)_x \to ({{\cal D}}_{{{\Omega_X^{-\bullet}}}})_x$$ for every point $x\in X$.
Thus, since $X$ is smooth, it suffices to show that the map $$\label{Y-D-A}
{{\cal Y}}_0({V^{\bullet}}(A)) \to {{\cal D}}_{{{\Omega^{-\bullet}}}(A)}$$ is an isomorphism for every local regular (commutative) algebra $A$ over ${{\mathbb K}}$.
It is easy to see that the associative algebra ${{\cal Y}}_0({V^{\bullet}}(A))$ is generated by symbols: $$\label{symbols}
i_a,\,\, i_v, \,\, l_a,\,\, l_v\,,$$ where $a\in A$ and $v\in V^1(A)$.
Under the map (\[Y-D-A\]) the symbols go to $$i_a \to {{\cal I}}_a\,, \qquad i_v \to {{\cal I}}_v\,,
\qquad l_a \to {{\cal L}}_a \,, \qquad l_v \to {{\cal L}}_v\,.$$ Since the images of the symbols (\[symbols\]) satisfy the same relations therefore the map (\[Y-D-A\]) is injective.
To show that (\[Y-D-A\]) is surjective we remark that the algebra ${{\cal D}}_{{{\Omega^{-\bullet}}}(A)}$ is generated by differential operators of the form $$a\, \cdot\,, \qquad d\, b\, \cdot\,, \qquad a,b\in A$$ and derivations ${{\rm Der}\,}_{{{\mathbb K}}}({{\Omega^{-\bullet}}}(A))$ of ${{\Omega^{-\bullet}}}(A)$.
Since $a\, \cdot\, = {{\cal I}}_a$ and $d\,b \, \cdot = {{\cal L}}_b$ it remains to show that every derivation $W\in {{\rm Der}\,}_{{{\mathbb K}}}({{\Omega^{-\bullet}}}(A))$ belongs to the image of the map (\[Y-D-A\]).
The regularity of $A$ implies that the $A$-modules ${{\Omega}}^{1}(A)$ and ${{\Omega^{-\bullet}}}(A)$ are free. More precisely, if $x_1, \dots, x_n$ is a regular system of parameters in $A$ then the $A$-module ${{\Omega}}^1(A)$ is freely generated by the $1$-forms $$\label{1forms}
d\,x^i\,, \qquad i =1, 2, \dots, n\,,$$ and the $A$-module ${{\Omega^{-\bullet}}}(A)$ is freely generated by the forms $$\label{forms}
d x^{i_1}\, d x^{i_2}\, \dots \, d x^{i_k}\,,
\qquad 1 \le i_1 < i_2 < \dots < i_k \le n\,.$$
Dually the $A$-module $V^1(A) = {{\rm Der}\,}_{{{\mathbb K}}}(A)$ is freely generated by $$\label{vectors}
e_1, e_2, \dots, e_n\,,$$ where $e_i$ is a derivation of $A$ defined by the equation $${{\cal I}}_{e_i}(d x^j) = {{\delta}}^j_i\,.$$
Since the $A$-module ${{\Omega^{-\bullet}}}(A)$ is freely generated by the forms (\[forms\]) therefore every derivation $W\in {{\rm Der}\,}_{{{\mathbb K}}}({{\Omega^{-\bullet}}}(A))$ is uniquely determined by its values on the elements of $A$ and the $1$-forms (\[1forms\]). In general we have $$W(a) = \sum_{1 \le i_1 < i_2 < \dots < i_k \le n}
W_{i_1 \dots i_k}(a)\, d x^{i_1} d x^{i_2} \dots
d x^{i_k}$$ and $$W(d x^i) = \sum_{1 \le i_1 < i_2 < \dots < i_k \le n} W^i_{i_1 \dots i_k}
d x^{i_1} d x^{i_2} \dots d x^{i_k}\,,$$ where $ W_{i_1 \dots i_k}$ are derivations of $A$ over ${{\mathbb K}}$ and $W^i_{i_1 \dots i_k}\in A$.
Let $W_1$ be the following derivation of ${{\Omega^{-\bullet}}}(A)$: $$W_1 = \sum_{1 \le i_1 < i_2 < \dots < i_k \le n}
d x^{i_1} d x^{i_2} \dots d x^{i_k} \, {{\cal L}}_{W_{i_1 \dots i_k}}\,.$$ It is obvious that the difference $W-W_1$ is $A$-linear. Hence $W_2 = W - W_1$ is uniquely determined by its values on $1$-forms (\[1forms\]): $$W_2(d x^i) = \sum_{1 \le i_1 < i_2 < \dots < i_k \le n} {{\widetilde{W}}}^i_{i_1 \dots i_k}
d x^{i_1} d x^{i_2} \dots d x^{i_k}\,,$$ where ${{\widetilde{W}}}^i_{i_1 \dots i_k}\in A$. Thus the derivation $W$ can be rewritten as $$W= \sum_{1 \le i_1 < i_2 < \dots < i_k \le n}
d x^{i_1} d x^{i_2} \dots d x^{i_k} \, \cdot\, {{\cal L}}_{W_{i_1 \dots i_k}} +
\sum_{1 \le i_1 < i_2 < \dots < i_k \le n}
d x^{i_1} d x^{i_2} \dots d x^{i_k} \, \cdot\, {{\cal I}}_{{{\widetilde{W}}}_{i_1 \dots i_k}}\,,$$ where ${{\widetilde{W}}}_{i_1 \dots i_k}$ is the derivation of $A$ defined by $${{\widetilde{W}}}_{i_1 \dots i_k} = \sum_i {{\widetilde{W}}}^i_{i_1 \dots i_k} e_i\,.$$ This shows that the map (\[Y-D-A\]) is surjective. Hence so is the map (\[Y-D-Om-map\]). $\Box$
Using the pairing (\[pairing\]) we introduce the following map of sheaves of ${{\cal O}}_X$-bimodules: $$\label{map-r}
r\,:\, {{\Omega_X^{-\bullet}}}{\otimes_{{\cal O}_X}}{{\cal D}}_X {\otimes_{{\cal O}_X}}{V^{\bullet}}_X \, \to \, {{\cal D}}_{{{\Omega_X^{-\bullet}}}}\,,$$ $$r(\eta, D, {{\gamma}})({{\varphi}}) = \eta \, D {\langle}{{\gamma}},{{\varphi}}{\rangle}\,,$$ where $\eta$ and ${{\varphi}}$ are local sections of ${{\Omega_X^{-\bullet}}}$, $D$ is a local section of ${{\cal D}}_X$, ${{\gamma}}$ is a local section of ${V^{\bullet}}_X$, and ${{\cal D}}_X$ is considered as sheaf of bimodules over ${{\cal O}}_X$.
We claim that
\[map-r-pred\] The map $r$ in (\[map-r\]) is an isomorphism of sheaves of ${{\cal O}}_X$-bimodules.
[**Proof.**]{} Again, it suffices to prove that the map $r$ gives an isomorphism on stalks $$({{\Omega_X^{-\bullet}}}{\otimes_{{\cal O}_X}}{{\cal D}}_X {\otimes_{{\cal O}_X}}{V^{\bullet}}_X)_x \to ({{\cal D}}_{{{\Omega_X^{-\bullet}}}})_x$$ for every point $x\in X$.
Hence, due to smoothness of $X$, we need to show that the map $$\label{map-r-A}
r_A: {{\Omega^{-\bullet}}}(A) \otimes_A {{\cal D}}(A) \otimes_A {V^{\bullet}}(A) \to {{\cal D}}_{{{\Omega^{-\bullet}}}(A)}$$ is an isomorphism for every local regular (commutative) algebra over ${{\mathbb K}}$.
Since ${{\Omega^{-\bullet}}}(A)$ and ${V^{\bullet}}(A)$ are free modules over $A$ the injectivity of $r_A$ follows easily from the injectivity of the restriction $$r_A \Big|_{{{\cal D}}(A)} : {{\cal D}}(A) \to {{\cal D}}_{{{\Omega^{-\bullet}}}(A)}\,.$$
Due to Proposition \[Y-D-Om\] the algebra ${{\cal D}}_{{{\Omega^{-\bullet}}}(A)}$ is generated by differential operators of the form $$\label{ops}
{{\cal I}}_a = a\, \cdot\,, \qquad {{\cal L}}_a = d a\, \cdot\,, \qquad a\in A\,,$$ $$\label{ops1}
{{\cal I}}_v\,, \qquad {{\cal L}}_v\,, \qquad v\in {{\rm Der}\,}_{{{\mathbb K}}}(A)\,.$$ Thus, to show that $r_A$ is surjective it suffices to prove that the operators (\[ops\]) and (\[ops1\]) belong to the image of $r_A$.
If we choose a regular system of parameters $x_1, \dots, x_n$ in $A$ then using the generators (\[forms\]) and (\[vectors\]) we may rewrite the operators ${{\cal I}}_a = a \,\cdot$, ${{\cal L}}_a = d a\, \cdot$ as follows: $$a \cdot {{\varphi}}= \sum_{k; \, 1 \le i_1 < i_2 < \dots < i_k \le n}
d x^{i_1} d x^{i_2} \dots d x^{i_k}\, a {\langle}e_{i_k}\wedge \dots
\wedge e_{i_2} \wedge e_{i_1}, {{\varphi}}{\rangle}\,,$$ and $$d a \cdot {{\varphi}}= \sum_{k; \, 1 \le i_1 < i_2 < \dots < i_k \le n}
d a \, d x^{i_1} d x^{i_2} \dots d x^{i_k} \, \,
{\langle}e_{i_k} \wedge \dots \wedge e_{i_2} \wedge e_{i_1}, {{\varphi}}{\rangle}\,.$$ Thus the operators (\[ops\]) belong to the image of $r_A$.
Every derivation $v\in {{\rm Der}\,}_{{{\mathbb K}}}(A)$ can be uniquely written as $$v = \sum_i v^i e_i\,,$$ where $v^i \in A$.
Using this decomposition we rewrite the operators ${{\cal I}}_v$ and ${{\cal L}}_v$ as $${{\cal I}}_v {{\varphi}}= \sum_{k; \, 1 \le i_1 < i_2 < \dots < i_k \le n}
\sum_{s=1}^k (-1)^{s-1}
v^{i_s} \, d x^{i_1} \dots \widehat{d x^{i_s}} \dots d x^{i_k} \, \,
{\langle}e_{i_k} \wedge \dots \wedge e_{i_2} \wedge e_{i_1}, {{\varphi}}{\rangle}\,,$$ $$\label{L-v}
{{\cal L}}_v {{\varphi}}= \sum_{k; \, 1 \le i_1 < i_2 < \dots < i_k \le n}
d x^{i_1} d x^{i_2} \dots d x^{i_k}\, v \,
{\langle}e_{i_k} \wedge \dots \wedge e_{i_2} \wedge e_{i_1}, {{\varphi}}{\rangle}+$$ $$\sum_{k; \, 1 \le i_1 < i_2 < \dots < i_{k+1} \le n}
\sum_{t,s=1}^{k+1} (-1)^{s-t} e_{i_t}(v^{i_s})\,
d x^{i_1} \dots \widehat{d x^{i_s}} \dots
d x^{i_{k+1}} \,\,
{\langle}e_{i_{k+1}} \wedge \dots \wedge
\widehat{e_{i_t}}\wedge \dots \wedge e_{i_1}, {{\varphi}}{\rangle}\,,$$ where the symbol $\widehat{~}$ over $d x^{i}$ (resp. $e_i$) means that the $1$-form $d x^{i}$ (resp. the vector $e_i$) is omitted. The vector $v$ in the right hand side of (\[L-v\]) is considered as a differential operator on $A$.
Thus the operators (\[ops1\]) also belong to the image of $r_A$. This concludes the proof of the proposition. $\Box$
We remark that the sheaf $$\label{cP}
{{\cal P}}= {{\Omega_X^{-\bullet}}}{\otimes_{{\cal O}_X}}{{\cal D}}_X$$ has the natural left ${{\cal D}}_{{{\Omega_X^{-\bullet}}}}$-module structure and the natural right ${{\cal D}}_X$-module structure. Similarly, the sheaf $$\label{cQ}
{{\cal Q}}= {{\cal D}}_X {\otimes_{{\cal O}_X}}{V^{\bullet}}_X$$ has the natural left ${{\cal D}}_X$-module structure and a natural right ${{\cal D}}_{{{\Omega_X^{-\bullet}}}}$-module structure.
Proposition \[map-r-pred\] implies that $$\label{Morita-PQ}
{{\cal P}}\otimes_{{{\cal D}}_X} {{\cal Q}}\cong {{\cal D}}_{{{\Omega_X^{-\bullet}}}}\,.$$ Furthermore, it is obvious that $$\label{Morita-QP}
{{\cal Q}}\otimes_{{{\cal D}}_{{{\Omega_X^{-\bullet}}}}} {{\cal P}}\cong {{\cal D}}_{X}\,.$$
Thus we arrive at the following statement
\[Morita0\] The sheaves ${{\cal P}}$ (\[cP\]) and ${{\cal Q}}$ (\[cQ\]) establish a Morita equivalence between the sheaves of associative algebras ${{\cal D}}_{{{\Omega_X^{-\bullet}}}}$ and ${{\cal D}}_X$. $\Box$
Combining this statement with Proposition \[Y-D-Om\] we conclude that the sheaves ${{\cal Y}}_0({V^{\bullet}}_X)$ and ${{\cal D}}_{X}$ are Morita equivalent.
In order to get the sheaf of algebras ${{\cal Y}}({V^{\bullet}}_X)$ from ${{\cal Y}}_0({V^{\bullet}}_X)$ we need to tensor ${{\cal Y}}_0({V^{\bullet}}_X)$ with the constant sheaf $${{\mathbb K}}[{{\delta}}]/({{\delta}}^2)\,, \qquad \deg({{\delta}}) = -1$$ and impose the equation $$\label{de-hde}
[{{\delta}}, P] = {\widehat{\delta}}(P)\,,$$ where ${\widehat{\delta}}$ is the derivation of ${{\cal Y}}_0({V^{\bullet}}_X)$ defined by $$\label{hat-delta}
{\widehat{\delta}}(i_{{{\gamma}}}) = l_{{{\gamma}}}\,, \qquad {\widehat{\delta}}(l_{{{\gamma}}}) = 0\,,$$ $P$ is a local section of ${{\cal Y}}_0({V^{\bullet}}_X)$ and ${{\gamma}}$ is a local section of ${V^{\bullet}}_X$.
On the other hand, for every local section $P$ of ${{\cal Y}}_0({V^{\bullet}}_X)$ we have $${\widehat{\delta}}(P) = [d, P]\,,$$ where $d$ is the de Rham differential.
The de Rham differential $d$ is a global section of the sheaf ${{\cal D}}_{{{\Omega_X^{-\bullet}}}}$. Hence, due to Proposition \[Y-D-Om\], $d$ is a global section of ${{\cal Y}}_0({V^{\bullet}}_X)$.
Thus, switching from ${{\delta}}$ to $$\label{md}
{{\mathfrak{d}}}= {{\delta}}- d$$ we get the following isomorphism of the sheaves of algebras $$\label{Y-Y-0}
{{\cal Y}}({V^{\bullet}}_X) \cong {{\cal Y}}_0({V^{\bullet}}_X)[{{\mathfrak{d}}}]/({{\mathfrak{d}}}^2)\,,$$ where ${{\mathfrak{d}}}$ has degree $-1$ and $$[{{\mathfrak{d}}}, P] =0$$ for every local section $P$ of ${{\cal Y}}_0({V^{\bullet}}_X)$.
Combining this observation with Proposition \[Morita0\] we arrive at the following statement
\[Morita\] Let ${{\cal P}}$ and ${{\cal Q}}$ be the sheaves defined in (\[cP\]) and (\[cQ\]), respectively. The sheaves $${{\cal P}}[{{\mathfrak{d}}}]/({{\mathfrak{d}}}^2)\,, \qquad {{\cal Q}}[{{\mathfrak{d}}}]/({{\mathfrak{d}}}^2)$$ establish a Morita equivalence between the sheaf of associative algebras ${{\cal Y}}({V^{\bullet}}_X)$ and the sheaf $${{\cal D}}_X[{{\mathfrak{d}}}]/({{\mathfrak{d}}}^2)\,,$$ where ${{\mathfrak{d}}}$ has degree $-1$ and $$[{{\mathfrak{d}}}, D] =0$$ for every local section $D$ of ${{\cal D}}_X$. $\Box$
Proof of Theorem \[main\] {#proof}
-------------------------
Let us recall that, due to Proposition \[GJ\], a homotopy calculus structure on the pair $$({C^{\bullet}_{{\rm norm}}}({{\cal O}}_X), {C_{\bullet}^{{\rm norm}}}({{\cal O}}_X))$$ is a Maurer-Cartan element $Q$ of the DGLA $$\label{coder-calc1}
{\rm C o d e r}'( {{\mathbb F}}_{{{\bf B}}}({C^{\bullet}_{{\rm norm}}}({{\cal O}}_X), {C_{\bullet}^{{\rm norm}}}({{\cal O}}_X)) )\,,$$ where ${{\bf B}}$ is as above the cooperad $Bar({{\bf calc}})$ and the DGLA (\[coder-calc1\]) consists of the coderivations of ${{\mathbb F}}_{{{\bf B}}}({C^{\bullet}_{{\rm norm}}}({{\cal O}}_X), {C_{\bullet}^{{\rm norm}}}({{\cal O}}_X))$ satisfying the condition $$Q \Big|_{{C^{\bullet}_{{\rm norm}}}({{\cal O}}_X) \oplus {C_{\bullet}^{{\rm norm}}}({{\cal O}}_X)} =0\,.$$
The codifferential on the sheaf of coalgebras ${{\mathbb F}}_{{{\bf B}}}({C^{\bullet}_{{\rm norm}}}({{\cal O}}_X), {C_{\bullet}^{{\rm norm}}}({{\cal O}}_X))$ is the sum $$\label{pa}
{{\partial}}= {{\partial}}^{Bar} + {{\partial}}^{Hoch}\,,$$ where ${{\partial}}^{Bar}$ comes from the bar differential on ${{\bf B}}= Bar({{\bf calc}})$ and ${{\partial}}^{Hoch}$ comes from the Hochschild (co)boundary operators on ${C^{\bullet}_{{\rm norm}}}({{\cal O}}_X)$ and ${C_{\bullet}^{{\rm norm}}}({{\cal O}}_X)$.
Using the Maurer-Cartan element $Q$ we shift the codifferential ${{\partial}}$ by $[Q, \,]$ and get the new codifferential on ${{\mathbb F}}_{{{\bf B}}}({C^{\bullet}_{{\rm norm}}}({{\cal O}}_X), {C_{\bullet}^{{\rm norm}}}({{\cal O}}_X))$: $$\label{pa-Q}
{{\partial}}^Q = {{\partial}}^{Bar} + {{\partial}}^{Hoch} + [Q,\,]\,.$$ Let us denote the resulting sheaf of DG ${{\bf B}}$-coalgebras by $C_Q$: $$\label{C-Q}
C_Q = \Big( {{\mathbb F}}_{{{\bf B}}}({C^{\bullet}_{{\rm norm}}}({{\cal O}}_X), {C_{\bullet}^{{\rm norm}}}({{\cal O}}_X)),
{{\partial}}^Q \Big)\,.$$
We use $C_Q$ to get the canonical free resolution[^14] ${{\cal R}}$ of the sheaf $({C^{\bullet}_{{\rm norm}}}({{\cal O}}_X), {C_{\bullet}^{{\rm norm}}}({{\cal O}}_X))$ of homotopy calculi. As a sheaf of calculi, $$\label{cR}
{{\cal R}}= {{\mathbb F}}_{{{\bf calc}}}(C_Q)$$ and the differential on ${{\cal R}}$ is $$\label{d-cR}
{{\partial}}^{{{\cal R}}} = {{\partial}}^{{\rm t w}} + {{\partial}}^Q\,,$$ where ${{\partial}}^{Q}$ comes from the differential on $C_Q$ and ${{\partial}}^{{\rm t w}}$ is defined using the twisting cochain between operad ${{\bf calc}}$ and cooperad ${{\bf B}}=Bar({{\bf calc}})$. (See Section 2.3 in [@GJ] on twisting cochain and the construction of the differential ${{\partial}}^{{\rm t w}}$ for algebras over an abstract operad.)
The sheaf of DG ${{\bf calc}}$-algebras ${{\cal R}}$ splits according to the colors $({{\mathfrak{c}}}, {{\mathfrak{a}}})$ as $${{\cal R}}= {{\cal R}}_{{{\mathfrak{c}}}} \oplus {{\cal R}}_{{{\mathfrak{a}}}}\,,$$ where $$\label{cR-ma}
{{\cal R}}_{{{\mathfrak{a}}}} =
{{\mathbb F}}_{{{\bf calc}}}\Big({{\mathbb F}}_{Bar({{\bf calc}})}({C^{\bullet}_{{\rm norm}}}({{\cal O}}_X), {C_{\bullet}^{{\rm norm}}}({{\cal O}}_X)) \Big)_{{{\mathfrak{a}}}}\,,$$ and $$\label{cR-mc}
{{\cal R}}_{{{\mathfrak{c}}}} =
{{\mathbb F}}_{{{\bf e_2}}}\circ {{\mathbb F}}_{Bar({{\bf e_2}})}({C^{\bullet}_{{\rm norm}}}({{\cal O}}_X))\,.$$ Thus the sheaf ${{\cal R}}_{{{\mathfrak{c}}}}$ with the differential $${{\partial}}^{{{\cal R}}} \Big|_{{{\cal R}}_{{{\mathfrak{c}}}}}$$ is a free resolution of the sheaf ${C^{\bullet}_{{\rm norm}}}({{\cal O}}_X)$ of homotopy Gerstenhaber algebras. This resolution can be simplified. More precisely, we may consider the subsheaf $$\label{Ger-simple0}
{{\mathbb F}}_{{{\bf e_2}}}\circ {{\mathbb F}}_{{{\bf e_2}}^{\vee}}({C^{\bullet}_{{\rm norm}}}({{\cal O}}_X))
\subset {{\cal R}}_{{{\mathfrak{c}}}}$$ with the differential obtained by restricting the one on (\[cR-mc\]). Then, using the fact that the inclusion ${{\iota}}_{{{\bf e_2}}}$ (\[Ger-Koszul\]) is a quasi-isomorphism of cooperads one can show that the inclusion $$\label{Ger-simple-map}
{{\mathbb F}}_{{{\bf e_2}}}({{\iota}}_{{{\bf e_2}}})\, : \,
{{\mathbb F}}_{{{\bf e_2}}}\circ {{\mathbb F}}_{{{\bf e_2}}^{\vee}}({C^{\bullet}_{{\rm norm}}}({{\cal O}}_X))\,
\stackrel{\sim}{\hookrightarrow} \,
{{\mathbb F}}_{{{\bf e_2}}}\circ {{\mathbb F}}_{Bar({{\bf e_2}})}({C^{\bullet}_{{\rm norm}}}({{\cal O}}_X))$$ is a quasi-isomorphism of sheaves of DG Gerstenhaber algebras.
Thus the sheaf (\[Ger-simple0\]) is also a free resolution of the sheaf ${C^{\bullet}_{{\rm norm}}}({{\cal O}}_X)$ of homotopy Gerstenhaber algebras.
We denote the differential on the sheaf (\[Ger-simple0\]) by ${{\partial}}^{{{\cal R}}}_{{{\mathfrak{c}}}}$ and reserve the notation ${{\cal R}}({C^{\bullet}_{{\rm norm}}}({{\cal O}}_X))$ for this resolution $$\label{Ger-simple}
{{\cal R}}({C^{\bullet}_{{\rm norm}}}({{\cal O}}_X)) =
(\, {{\mathbb F}}_{{{\bf e_2}}}\circ {{\mathbb F}}_{{{\bf e_2}}^{\vee}}({C^{\bullet}_{{\rm norm}}}({{\cal O}}_X))\, , \,
{{\partial}}^{{{\cal R}}}_{{{\mathfrak{c}}}} \,)\,.$$
The quasi-isomorphism (\[Ger-simple-map\]) provides the sheaf ${{\cal R}}_{{{\mathfrak{a}}}}$ (\[cR-ma\]) with a (DG) ${{\bf calc}}$-module structure over the sheaf ${{\cal R}}({C^{\bullet}_{{\rm norm}}}({{\cal O}}_X))$ (\[Ger-simple\]). Thus, in order to prove Theorem \[main\], we need to show that the sheaf $({{\cal R}}({C^{\bullet}_{{\rm norm}}}({{\cal O}}_X)), {{\cal R}}_{{{\mathfrak{a}}}})$ of calculi is quasi-isomorphic to the sheaf $({V^{\bullet}}_X, {{\Omega_X^{-\bullet}}})$.
In paper [@BLT] we constructed a chain of quasi-isomorphisms of sheaves of DG Gerstenhaber algebras which connects the sheaf ${{\cal R}}({C^{\bullet}_{{\rm norm}}}({{\cal O}}_X))$ to the sheaf ${V^{\bullet}}_X$.
In this construction we use the sheaf of ${{\bf e_2}}^{\vee}$-coalgebras: $$\label{DT}
\Xi_X = {{\mathbb F}}_{{{\Lambda}}^{2}{{\bf cocomm}}} \circ
{{\mathbb F}}_{{{\Lambda}}{{\bf coLie}}^+} ({{\cal O}}_X, V^1_X)\,,$$ where ${{\bf coLie}}^+$ is the cooperad which governs pairs “a Lie coalgebra $+$ its comodule.”
We also use the canonical free resolution $$\label{R-V-X}
({{\mathbb F}}_{{{\bf e_2}}}\circ {{\mathbb F}}_{{{\bf e_2}}^{\vee}}({V^{\bullet}}_X),
{{\partial}}^{{{\cal R}}}_V)$$ of the sheaf of Gerstenhaber algebras ${V^{\bullet}}_X$. Here the differential ${{\partial}}^{{{\cal R}}}_V$ on the sheaf (\[R-V-X\]) comes from the twisting cochain [@GJ] of the pair $({{\bf e_2}}, {{\bf e_2}}^{\vee})$.
It is obvious that $\Xi_X$ is a subsheaf of $${{\mathbb F}}_{{{\bf e_2}}^{\vee}}({C^{\bullet}_{{\rm norm}}}({{\cal O}}_X))\,.$$ and a subsheaf of $${{\mathbb F}}_{{{\bf e_2}}^{\vee}}({V^{\bullet}}_X)\,.$$ Due to this observation we have two inclusions of the sheaves of free Gerstenhaber algebras $$\label{incl1}
{{\sigma}}_1 :{{\mathbb F}}_{{{\bf e_2}}}(\Xi_X) \hookrightarrow
{{\cal R}}({C^{\bullet}_{{\rm norm}}}({{\cal O}}_X))\,,$$ and $$\label{incl11}
{{\sigma}}_2 :{{\mathbb F}}_{{{\bf e_2}}}(\Xi_X) \hookrightarrow
{{\mathbb F}}_{{{\bf e_2}}}\circ {{\mathbb F}}_{{{\bf e_2}}^{\vee}}({V^{\bullet}}_X)\,.$$
It was shown in [@BLT] that the sheaf ${{\mathbb F}}_{{{\bf e_2}}}(\Xi_X)$ is closed both with respect to the differential ${{\partial}}^{{{\cal R}}}_{{{\mathfrak{c}}}}$ on ${{\cal R}}({C^{\bullet}_{{\rm norm}}}({{\cal O}}_X))$ and the differential ${{\partial}}^{{{\cal R}}}_V$ on the sheaf (\[R-V-X\]). Furthermore, the restriction of the differential ${{\partial}}^{{{\cal R}}}_{{{\mathfrak{c}}}}$ to ${{\mathbb F}}_{{{\bf e_2}}}(\Xi_X)$ coincides with the restriction of the differential ${{\partial}}^{{{\cal R}}}_{V}$. In other words, the sheaf of free Gerstenhaber algebras ${{\mathbb F}}_{{{\bf e_2}}}(\Xi_X)$ is equipped with a canonical differential.
Composing ${{\sigma}}_2$ (\[incl11\]) with the quasi-isomorphism $${{\mathbb F}}_{{{\bf e_2}}}\circ {{\mathbb F}}_{{{\bf e_2}}^{\vee}}({V^{\bullet}}_X)
\, \stackrel{\sim}{\to} \, {V^{\bullet}}_X$$ we arrive at the following pair of maps of sheaves of Gerstenhaber algebras: $$\label{chain-Ger}
\begin{array}{ccccc}
{V^{\bullet}}_X & \stackrel{{{\lambda}}}{\leftarrow} &
{{\mathbb F}}_{{{\bf e_2}}}(\Xi_X)
& \stackrel{{{\sigma}}_1}{\hookrightarrow} & {{\cal R}}({C^{\bullet}_{{\rm norm}}}({{\cal O}}_X))\,.
\end{array}$$ In [@BLT] it was shown that both ${{\lambda}}$ and ${{\sigma}}_1$ are quasi-isomorphisms of complexes of sheaves.
The quasi-isomorphism ${{\sigma}}_1$ in (\[chain-Ger\]) provides the sheaf ${{\cal R}}_{{{\mathfrak{a}}}}$ (\[cR-ma\]) with a (DG) ${{\bf calc}}$-module structure over the sheaf ${{\mathbb F}}_{{{\bf e_2}}}(\Xi_X)$. Thus, in order to prove Theorem \[main\] we need to show that the sheaf of calculi $({{\mathbb F}}_{{{\bf e_2}}}(\Xi_X), {{\cal R}}_{{{\mathfrak{a}}}})$ is quasi-isomorphic to the sheaf $({V^{\bullet}}_X, {{\Omega_X^{-\bullet}}})$.
For this purpose we introduce the bar resolution of the the sheaf ${{\cal R}}_{{{\mathfrak{a}}}}$ of ${{\cal Y}}({{\mathbb F}}_{{{\bf e_2}}}(\Xi_X))$-modules: $$\label{BR-a}
{{\cal B}}{{\cal R}}_{{{\mathfrak{a}}}} = \bigoplus_{k \ge 1} {{\bf s}\,}^{1-k} \,
{{\cal Y}}({{\mathbb F}}_{{{\bf e_2}}}(\Xi_X))^{\otimes \, k} \otimes {{\cal R}}_{{{\mathfrak{a}}}}\,.$$
The map ${{\lambda}}$ in (\[chain-Ger\]) induces the following map of sheaves of associative algebras $$\label{cY-la}
{{\cal Y}}({{\lambda}}) : {{\cal Y}}({{\mathbb F}}_{{{\bf e_2}}}(\Xi_X)) \to
{{\cal Y}}({V^{\bullet}}_X)\,.$$ Considering the map ${{\cal Y}}({{\lambda}})$ on the level of stalks at a point $x\in X$ we get the map of associative algebras $$\label{cY-la-stalk}
{{\cal Y}}({{\mathbb F}}_{{{\bf e_2}}}(\Xi(A))) \to {{\cal Y}}({V^{\bullet}}(A))\,,$$ where $A$ is the local algebra at the point $x$, $\Xi(A) = \Xi_{{\rm S p e c}(A)}$ and ${V^{\bullet}}(A)= {V^{\bullet}}_{{\rm S p e c}(A)}$.
Since the variety $X$ is smooth the local algebra $A$ and hence the graded commutative algebra ${V^{\bullet}}(A)$ is regular. Furthermore, the Gerstenhaber algebra ${{\mathbb F}}_{{{\bf e_2}}}(\Xi(A))$ is a free resolution of ${V^{\bullet}}(A)$. Thus, due to Proposition \[Y-alm-exact\], the map (\[cY-la-stalk\]) a quasi-isomorphism. Hence (\[cY-la\]) is a quasi-isomorphism of complexes of sheaves.
Recall that ${{\cal B}}{{\cal R}}_{{{\mathfrak{a}}}}$ is the free resolution (\[BR-a\]) of the sheaf ${{\cal R}}_{{{\mathfrak{a}}}}$ of ${{\cal Y}}({{\mathbb F}}_{{{\bf e_2}}}(\Xi_X))$-modules. Therefore, applying the functor $$\otimes_{{{\cal Y}}({{\mathbb F}}_{{{\bf e_2}}}(\Xi_X))}\, {{\cal B}}{{\cal R}}_{{{\mathfrak{a}}}}$$ to the quasi-isomorphism (\[cY-la\]) we get the quasi-isomorphism of sheaves of ${{\cal Y}}({{\mathbb F}}_{{{\bf e_2}}}(\Xi_X))$-modules: $$\label{cY-mod-map}
{{\cal B}}{{\cal R}}_{{{\mathfrak{a}}}} \, \stackrel{\sim}{\rightarrow} \,
{{\cal Y}}({V^{\bullet}}_X) \, \otimes_{{{\cal Y}}({{\mathbb F}}_{{{\bf e_2}}}(\Xi_X))} \, {{\cal B}}{{\cal R}}_{{{\mathfrak{a}}}}\,.$$
Thus we need to show that the sheaf of ${{\cal Y}}({V^{\bullet}}_X)$-modules ${{\cal Y}}({V^{\bullet}}_X) \, \otimes_{{{\cal Y}}({{\mathbb F}}_{{{\bf e_2}}}(\Xi_X))} \, {{\cal B}}{{\cal R}}_{{{\mathfrak{a}}}}$ is quasi-isomorphic to ${{\Omega_X^{-\bullet}}}$.
For this purpose we remark that the sheaf of (graded) commutative algebras ${{\Omega_X^{-\bullet}}}$ can be realized as a subsheaf of ${{\cal Y}}({V^{\bullet}}_X)$. Indeed, $$\label{Omb-cY}
{{\Omega_X^{-\bullet}}}= {{\cal Y}}_0({{\cal O}}_X) \subset {{\cal Y}}({V^{\bullet}}_X)\,.$$
Using the global section ${{\bf 1}}\in {{\Gamma}}(X, C^{{\rm norm}}_0({{\cal O}}_X))$ we introduce the following global section $$\label{cycle-E}
E = 1_{{{\cal Y}}({V^{\bullet}}_X)} \, \otimes \, 1_{{{\cal Y}}({{\mathbb F}}_{{{\bf e_2}}}(\Xi_X))}\,
\otimes \, {{\bf 1}}\in {{\Gamma}}(X,
{{\cal Y}}({V^{\bullet}}_X) \, \otimes_{{{\cal Y}}({{\mathbb F}}_{{{\bf e_2}}}(\Xi_X))} \, {{\cal B}}{{\cal R}}_{{{\mathfrak{a}}}} )$$ of the sheaf ${{\cal Y}}({V^{\bullet}}_X) \, \otimes_{{{\cal Y}}({{\mathbb F}}_{{{\bf e_2}}}(\Xi_X))} \, {{\cal B}}{{\cal R}}_{{{\mathfrak{a}}}}$. Here ${{\bf 1}}\in {{\Gamma}}(X, C^{\rm norm}_0({{\cal O}}_X))$ is also considered as a global section of the sheaf $${{\cal R}}_{{{\mathfrak{a}}}} =
{{\mathbb F}}_{{{\bf calc}}}\Big({{\mathbb F}}_{{{\bf B}}}({C^{\bullet}_{{\rm norm}}}({{\cal O}}_X), {C_{\bullet}^{{\rm norm}}}({{\cal O}}_X)) \Big)_{{{\mathfrak{a}}}}\,,$$ via the unit of the operad ${{\bf calc}}$ and the coaugmentation of the cooperad ${{\bf B}}$.
It is obvious that $E$ is closed with respect to the total differential on $${{\Gamma}}(X, {{\cal Y}}({V^{\bullet}}_X) \, \otimes_{{{\cal Y}}({{\mathbb F}}_{{{\bf e_2}}}(\Xi_X))} \, {{\cal B}}{{\cal R}}_{{{\mathfrak{a}}}})\,.$$ Furthermore, for every point $x\in X$ the cohomology class of the germ $E_x$ corresponds to the cohomology class of the germ ${{\bf 1}}_x$.
Using the cycle $E$ and equation (\[Omb-cY\]) we define the following map of sheaves $$\nu :{{\Omega_X^{-\bullet}}}\to
{{\cal Y}}({V^{\bullet}}_X) \, \otimes_{{{\cal Y}}({{\mathbb F}}_{{{\bf e_2}}}(\Xi_X))} \, {{\cal B}}{{\cal R}}_{{{\mathfrak{a}}}}\,,$$ $$\label{nu}
\nu (a\, d b_1 \, d b_2 \, \dots \, d b_m) =
(i_a\, l_{b_1}\, l_{b_2}\, \dots \, l_{b_m} \,,\, E)\,,$$ where $a, b_1, b_2, \dots, b_m $ are local sections of the structure sheaf ${{\cal O}}_X$.
It is obvious that $\nu$ is compatible with ${{\Omega_X^{-\bullet}}}$-module structures.
We claim that
\[nu-q-iso\] The map $\nu$ (\[nu\]) is a quasi-isomorphism of complexes of sheaves.
[**Proof.**]{} Indeed, let us consider the corresponding map of stalks $$\nu_x :({{\Omega_X^{-\bullet}}})_x \to
\Big({{\cal Y}}({V^{\bullet}}_X) \, \otimes_{{{\cal Y}}({{\mathbb F}}_{{{\bf e_2}}}(\Xi_X))} \, {{\cal B}}{{\cal R}}_{{{\mathfrak{a}}}}
\Big)_x$$ at a point $x\in X$. Let $a, b_1, b_2, \dots, b_m$ be germs of functions on $X$ at $x$.
Since the cohomology class of the germ $E_x$ corresponds to the cohomology class of the germ ${{\bf 1}}_x\in C^{\rm norm}_0({{\cal O}}_X)_x$ therefore the cohomology class of the element $$(i_a\, l_{b_1}\, l_{b_2}\, \dots \, l_{b_m} \,,\, E_x)$$ corresponds to the cohomology class of the Hochschild cycle $$\label{vot-chain}
\sum_{{{\sigma}}\in S_m} (-1)^{|{{\sigma}}|}
(a, b_{{{\sigma}}(1)}, b_{{{\sigma}}(2)}, \dots, b_{{{\sigma}}(m)})
= I_a\, L_{b_1}\, L_{b_2}, \dots\, L_{b_m} {{\bf 1}}_x
\in C^{{\rm norm}}_m(A)\,.$$ Furthermore, under Connes-Hochschild-Kostant-Rosenberg map (\[HKR11\]) the chain (\[vot-chain\]) goes to the form $a\, d b_1 \, d b_2 \, \dots \, d b_m$. Thus $\nu_x$ induces isomorphism on the level of cohomology groups. $\Box$
Recall that according to Proposition \[Y-D-Om\] and equation (\[Y-Y-0\]) the sheaf of associative algebras ${{\cal Y}}({V^{\bullet}}_X)$ is isomorphic to the sheaf $${{\cal D}}_{{{\Omega_X^{-\bullet}}}}[{{\mathfrak{d}}}]/({{\mathfrak{d}}}^2)\,,$$ where ${{\cal D}}_{{{\Omega_X^{-\bullet}}}}$ is the sheaf of differential operators on exterior forms and ${{\mathfrak{d}}}$ is an auxiliary variable of degree $-1$ which commutes with local sections of ${{\cal D}}_{{{\Omega_X^{-\bullet}}}}$. Let us also recall that due to the isomorphism (\[map-r\]) the sheaf ${{\cal D}}_X {\otimes_{{\cal O}_X}}{V^{\bullet}}_X[{{\mathfrak{d}}}]/({{\mathfrak{d}}}^2)$ has a natural structure of left ${{\cal D}}_{{{\Omega_X^{-\bullet}}}}[{{\mathfrak{d}}}]/({{\mathfrak{d}}}^2)$-module.
It is not hard to see that for every sheaf ${{\cal M}}$ of ${{\cal D}}_{{{\Omega_X^{-\bullet}}}}[{{\mathfrak{d}}}]/({{\mathfrak{d}}}^2)$-modules we have the natural isomorphism of sheaves of ${{\cal O}}_{X}$-modules $$\label{cQotimes}
{{\cal D}}_X {\otimes_{{\cal O}_X}}{V^{\bullet}}_X[{{\mathfrak{d}}}]/({{\mathfrak{d}}}^2) \,\,
\otimes_{{{\cal D}}_{{{\Omega_X^{-\bullet}}}}[{{\mathfrak{d}}}]/({{\mathfrak{d}}}^2)} \,\, {{\cal M}}\cong {{\cal O}}_X \otimes_{{{\Omega_X^{-\bullet}}}} {{\cal M}}\,,$$ where the ${{\Omega_X^{-\bullet}}}$-module structure on ${{\cal O}}_X$ is given by the equation $$f\, \eta = {\langle}\eta, f {\rangle}\,,
\qquad \eta\in {{\Gamma}}(U, {{\Omega_X^{-\bullet}}})\,,
\quad f\in {{\Gamma}}(U, {{\cal O}}_X)\,,$$ and the pairing ${\langle}\,,\,{\rangle}$ is defined in (\[pairing\]).
Having in mind Proposition \[Morita\], we apply the functor $${{\cal D}}_X {\otimes_{{\cal O}_X}}{V^{\bullet}}_X[{{\mathfrak{d}}}]/({{\mathfrak{d}}}^2) \,\,
\otimes_{{{\cal D}}_{{{\Omega_X^{-\bullet}}}}[{{\mathfrak{d}}}]/({{\mathfrak{d}}}^2)}$$ to the map $\nu$ (\[nu\]) and get the following quasi-isomorphism of sheaves of ${{\cal O}}_X$-modules $$\label{q-iso1}
{{\widetilde{\nu}}}\,\, : \,\, {{\cal O}}_X \,
\stackrel{\sim}{\rightarrow} \,
{{\cal D}}_X {\otimes_{{\cal O}_X}}{V^{\bullet}}_X[{{\mathfrak{d}}}]/({{\mathfrak{d}}}^2) \,
\otimes_{{{\cal Y}}({{\mathbb F}}_{{{\bf e_2}}}(\Xi_X))} \, {{\cal B}}{{\cal R}}_{{{\mathfrak{a}}}}\,,$$ where the right ${{\cal Y}}({{\mathbb F}}_{{{\bf e_2}}}(\Xi_X))$-module structure on ${{\cal D}}_X {\otimes_{{\cal O}_X}}{V^{\bullet}}_X[{{\mathfrak{d}}}]/({{\mathfrak{d}}}^2)$ is obtained from the right ${{\cal Y}}({V^{\bullet}}_X)$-module structure via the map ${{\lambda}}$ in (\[chain-Ger\]).
It is not hard to see that the ${{\cal D}}_X$-module structure on $$\label{sheaf}
{{\cal O}}_X \cong
{{\cal D}}_X {\otimes_{{\cal O}_X}}{V^{\bullet}}_X[{{\mathfrak{d}}}]/({{\mathfrak{d}}}^2) \,
\otimes_{{{\cal D}}_{{{\Omega_X^{-\bullet}}}}[{{\mathfrak{d}}}]/({{\mathfrak{d}}}^2)} \, {{\Omega_X^{-\bullet}}}$$ is the standard one. Furthermore, the element ${{\mathfrak{d}}}$ acts on the sections of the sheaf ${{\cal O}}_X$ in (\[sheaf\]) by zero simply because ${{\mathfrak{d}}}$ has degree $-1$ and ${{\cal O}}_X$ is concentrated in the single degree $0$.
Let us denote the target of the map ${{\widetilde{\nu}}}$ (\[q-iso1\]) by ${{\cal G}}$: $$\label{cG}
{{\cal G}}=
{{\cal D}}_X {\otimes_{{\cal O}_X}}{V^{\bullet}}_X[{{\mathfrak{d}}}]/({{\mathfrak{d}}}^2) \,
\otimes_{{{\cal Y}}({{\mathbb F}}_{{{\bf e_2}}}(\Xi_X))} \, {{\cal B}}{{\cal R}}_{{{\mathfrak{a}}}}\,.$$ We also denote by ${{\partial}}^{{{\cal G}}}$ the total differential on this sheaf.
For the next proposition we will need the Cech resolution ${\check{{\mathcal C}}^{\bullet}}({{\cal G}})$ of the sheaf of ${{\cal D}}_X[{{\mathfrak{d}}}]/({{\mathfrak{d}}}^2)$-modules ${{\cal G}}$ in the category of sheaves.
\[pravilnoe-pivo\] The map (\[q-iso1\]) extends to an $A_{\infty}$ quasi-isomorphism ${{\Upsilon}}$ from the sheaf of ${{\cal D}}_X[{{\mathfrak{d}}}]/({{\mathfrak{d}}}^2)$-modules ${{\cal O}}_X$ to the the Cech resolution ${\check{{\mathcal C}}^{\bullet}}({{\cal G}})$ of the sheaf ${{\cal G}}$ (\[cG\]).
[**Proof.**]{} First, let us prove that the map ${{\widetilde{\nu}}}$ (\[q-iso1\]) is compatible with the action of the sheaf ${{\cal D}}_X[{{\mathfrak{d}}}]/({{\mathfrak{d}}}^2)$ up to homotopy on the level of stalks.
Let $A$ be the stalk $({{\cal O}}_X)_x$ of the structure sheaf ${{\cal O}}_X$ at a point $x$. Since $X$ is smooth $A$ is a local regular algebra.
Let, as above, $x_1, \dots, x_n$ be a regular system of parameters in $A$. The module ${{\Omega}}^1(A)$ of Kähler differentials is freely generated by the $1$-forms (\[1forms\]) and the module of derivations ${{\rm Der}\,}(A)$ is freely generated by (\[vectors\]).
Since the algebra ${{\cal D}}(A)[{{\mathfrak{d}}}]/({{\mathfrak{d}}}^2)$ is generated by $A$, derivations of $A$ and the element ${{\mathfrak{d}}}$ it suffices to show that the action of the element ${{\mathfrak{d}}}$ and a derivation $v$ of $A$ sends the element $$\label{el}
1_{{{\cal D}}(A) {\otimes_{{\cal O}_X}}{V^{\bullet}}(A)[{{\mathfrak{d}}}]/({{\mathfrak{d}}}^2)} \,\,
\otimes_{{{\cal Y}}({V^{\bullet}}(A))}\,\, E_x$$ to cohomologically trivial elements of the chain complex $$\label{cG-x}
{{\cal G}}_x =
{{\cal D}}(A) {\otimes_{{\cal O}_X}}{V^{\bullet}}(A)[{{\mathfrak{d}}}]/({{\mathfrak{d}}}^2) \,
\otimes_{{{\cal Y}}({V^{\bullet}}(A))}\, {{\cal Y}}({V^{\bullet}}(A))
\otimes_{{{\cal Y}}({{\mathbb F}}_{{{\bf e_2}}}(\Xi(A)))} \, ({{\cal B}}{{\cal R}}_{{{\mathfrak{a}}}})_x\,.$$
The isomorphism $${{\cal Y}}({V^{\bullet}}(A)) \cong {{\Omega^{-\bullet}}}(A) \otimes_A {{\cal D}}(A) \otimes_A {V^{\bullet}}(A)[{{\mathfrak{d}}}]/({{\mathfrak{d}}}^2)$$ allows us to consider ${{\mathfrak{d}}}$ and the derivation $v\in {{\rm Der}\,}(A)$ as elements of the algebra ${{\cal Y}}({V^{\bullet}}(A))$. Thus we need to show that the cocycles $$v\, E_x$$ and $${{\mathfrak{d}}}\, E_x$$ are cohomologically trivial.
Since the map ${{\widetilde{\nu}}}$ (\[q-iso1\]) is a quasi-isomorphism of complexes of sheaves, the cohomology of the complex (\[cG-x\]) is concentrated in the degree $0$. Hence, the cocycle ${{\mathfrak{d}}}\, E_x$ is a coboundary because it has degree $-1$.
Next, using the generators (\[1forms\]) and (\[vectors\]) we rewrite the cocycle $v\, E_x$ as $$\label{v-Y}
v\, E_x = \frac{1}{n!}\, l_v \prod_{k=1}^n \Big( k -
\sum_{j=1}^n l_{x^j} i_{e_j} \Big) \, E_x\,,$$ where the element $$\frac{1}{n!}\prod_{k=1}^n \Big( k - \sum_{j=1}^n l_{x^j} i_{e_j}
\Big)\in
{{\cal D}}({{\Omega^{-\bullet}}}(A))$$ operates as a projection on the degree $0$ forms.
Since the cohomology class of $E_x$ corresponds to the cohomology class of ${{\bf 1}}_x = 1 \in C^{\rm norm}_0(A)$ the cohomology class of the element (\[v-Y\]) corresponds to the class of $$\frac{1}{n!}
L_v \prod_{k=1}^n \Big( k - \sum_{j=1}^n L_{x^j} I_{e_j} \Big)\,\, {{\bf 1}}_x
\in C^{{\rm norm}}_0(A)\,.$$
It is easy to see that $$\frac{1}{n!}
L_v \prod_{k=1}^n \Big( k - \sum_{j=1}^n L_{x^j} I_{e_j} \Big)\,\, {{\bf 1}}_x =0\,.$$
Thus the map ${{\widetilde{\nu}}}$ (\[q-iso1\]) is indeed compatible with the action of the sheaf ${{\cal D}}_X[{{\mathfrak{d}}}]/({{\mathfrak{d}}}^2)$ up to homotopy on the level of stalks.
An $A_{\infty}$ morphism from the sheaf of ${{\cal D}}_X[{{\mathfrak{d}}}]/({{\mathfrak{d}}}^2)$-modules ${{\cal O}}_X$ to the sheaf ${\check{{\mathcal C}}^{\bullet}}({{\cal G}})$ is the degree $0$ element $$\label{Ups}
{{\Upsilon}}\in \bigoplus_{k \ge 0}\,
{{\bf s}\,}^k \, {{\rm Hom}\,}\Big( ({{\cal D}}_X[{{\mathfrak{d}}}]/({{\mathfrak{d}}}^2)) ^{\otimes \, k} \otimes
{{\cal O}}_X \,\,, \,\, {\check{{\mathcal C}}^{\bullet}}({{\cal G}}) \Big)$$ satisfying the cocycle condition $$\label{Ups-eq}
({{\partial}}^{{{\cal G}}} + {\check{\partial}}+ {{\cal D}}^{Hoch}) {{\Upsilon}}= 0\,,$$ where ${{\partial}}^{{{\cal G}}}$ is the differential on the sheaf (\[cG\]), ${\check{\partial}}$ is the Cech differential and ${{\cal D}}^{Hoch}$ is the Hochschild coboundary operator of the sheaf ${{\cal D}}_X[{{\mathfrak{d}}}]/({{\mathfrak{d}}}^2)$ with values in the sheaf of bimodules ${\mathcal Hom\,}({{\cal O}}_x, {\check{{\mathcal C}}^{\bullet}}({{\cal G}}))$.
In other words, ${{\Upsilon}}$ can be defined by the infinite collection of maps $$\label{Ups-k}
{{\Upsilon}}_k \in
{{\rm Hom}\,}\Big( ({{\cal D}}_X[{{\mathfrak{d}}}]/({{\mathfrak{d}}}^2)) ^{\otimes \, k} \otimes
{{\cal O}}_X \,\,, \,\, {\check{{\mathcal C}}^{\bullet}}({{\cal G}}) \Big)\,, \qquad
k =0, 1, 2 \dots$$ such that ${{\Upsilon}}_k$ has degree $-k$ and all the maps satisfy the equations $$\label{Ups-k-eq}
({{\partial}}^{{{\cal G}}} + {\check{\partial}}){{\Upsilon}}_{k+1} + {{\cal D}}^{Hoch}\, {{\Upsilon}}_k = 0\,.$$ Our purpose is to show that there is exists an $A_{\infty}$-morphism ${{\Upsilon}}$ with $${{\Upsilon}}_0 = {{\widetilde{\nu}}}\,.$$
Let us show that there exists ${{\Upsilon}}_1$ satisfying the equation $$\label{Ups-0-eq}
({{\partial}}^{{{\cal G}}} + {\check{\partial}}){{\Upsilon}}_{1} + {{\cal D}}^{Hoch}\, {{\Upsilon}}_0 = 0\,.$$ We will find ${{\Upsilon}}_1$ by induction in degrees of the Cech complex. In general, $$\label{Ups-1}
{{\Upsilon}}_1 = \sum_{q=0}^{\infty} {{\Upsilon}}^q_1\,,$$ where $$\label{Ups-q-1}
{{\Upsilon}}^q_1 \in
{{\rm Hom}\,}\Big( {{\cal D}}_X[{{\mathfrak{d}}}]/({{\mathfrak{d}}}^2) \otimes
{{\cal O}}_X \,\,, \,\, {\check{{\mathcal C}}}^q({{\cal G}}^{-1-q}) \Big)$$ and the equation (\[Ups-0-eq\]) is equivalent to $$\label{Ups-0-0-eq}
{{\partial}}^{{{\cal G}}} \, {{\Upsilon}}^0_1 + {{\cal D}}^{Hoch}\, {{\Upsilon}}_0 = 0\,.$$ $$\label{Ups-0-q-eq}
{{\partial}}^{{{\cal G}}}\, {{\Upsilon}}^{q+1}_1 + {\check{\partial}}\,{{\Upsilon}}^q_1 = 0\,,
\qquad
q= 0,1,2, \dots\,.$$ If we set ${{\Upsilon}}_0= {{\widetilde{\nu}}}$ then there exists a map ${{\Upsilon}}^0_1$ satisfying equation (\[Ups-0-0-eq\]) because ${{\widetilde{\nu}}}$ is compatible with the action of ${{\cal D}}_X[{{\mathfrak{d}}}]/({{\mathfrak{d}}}^2)$ up to homotopy on the level of stalks.
Due to equation (\[Ups-0-0-eq\]) the element ${\check{\partial}}\,{{\Upsilon}}^0_1$ is closed with respect to ${{\partial}}^{{{\cal G}}}$ $${{\partial}}^{{{\cal G}}}\, ({\check{\partial}}\,{{\Upsilon}}^0_1) = 0\,.$$ But $${\check{\partial}}\,{{\Upsilon}}^0_1 \in
{{\rm Hom}\,}\Big( {{\cal D}}_X[{{\mathfrak{d}}}]/({{\mathfrak{d}}}^2) \otimes
{{\cal O}}_X \,\,, \,\, {\check{{\mathcal C}}}^1({{\cal G}}^{-1}) \Big)\,.$$ Thus, using the fact that the cohomology of the stalk ${{\cal G}}_x$ is concentrated in degree $0$ we conclude that there exists the next map ${{\Upsilon}}^1_1$ in (\[Ups-q-1\]) satisfying the equation $${{\partial}}^{{{\cal G}}}\, {{\Upsilon}}^{1}_1 + {\check{\partial}}\,{{\Upsilon}}^0_1 = 0\,.$$ This is the base of the induction.
Let us now assume that for $m > 0$ there exists the collection of maps ${{\Upsilon}}^q_1$ (\[Ups-q-1\]) for $q < m$ satisfying equation (\[Ups-0-q-eq\]) for $q<m-1$. Then, due to equation (\[Ups-0-q-eq\]) for $q = m-2$ the map ${\check{\partial}}\, {{\Upsilon}}^{m-1}_1$ is closed with respect to the differential ${{\partial}}^{{{\cal G}}}$: $${{\partial}}^{{{\cal G}}}\,({\check{\partial}}\, {{\Upsilon}}^{m-1}_1) = 0\,.$$ But $${\check{\partial}}\,{{\Upsilon}}^{m-1}_1 \in
{{\rm Hom}\,}\Big( {{\cal D}}_X[{{\mathfrak{d}}}]/({{\mathfrak{d}}}^2) \otimes
{{\cal O}}_X \,\,, \,\, {\check{{\mathcal C}}}^m({{\cal G}}^{-m}) \Big)\,.$$ Thus, using the fact that the cohomology of the stalk ${{\cal G}}_x$ is concentrated in degree $0$ we conclude that there exists the next map ${{\Upsilon}}^m_1$ in (\[Ups-q-1\]) satisfying equation (\[Ups-0-q-eq\]) for $q = m-1$.
We proved the existence of the map ${{\Upsilon}}_1$ in (\[Ups-k\]) satisfying equation (\[Ups-k-eq\]) for $k=0$.
Now we proceed by induction on $k$ in (\[Ups-k\]) and (\[Ups-k-eq\]).
Let us assume that ${{\Upsilon}}_k$ (\[Ups-k\]) are constructed for $k < m$ and equation (\[Ups-k-eq\]) holds for $k < m-1$. Then equation (\[Ups-k-eq\]) for $k = m-2$ implies that the element $$\label{on-samyj}
{{\cal D}}^{Hoch} {{\Upsilon}}_{m-1} \in {{\rm Hom}\,}\Big( ({{\cal D}}_X[{{\mathfrak{d}}}]/({{\mathfrak{d}}}^2)) ^{\otimes \, (m-1)} \otimes
{{\cal O}}_X \,\,, \,\, {\check{{\mathcal C}}^{\bullet}}({{\cal G}}) \Big)$$ is closed with respect to the differential ${{\partial}}^{{{\cal G}}} + {\check{\partial}}$.
Since the sheaf ${\check{{\mathcal C}}^{\bullet}}({{\cal G}})$ is acyclic with respect to the functor of global sections the map ${{\widetilde{\nu}}}$ (\[q-iso1\]) induces the quasi-isomorphism between the chain complex $$\label{Rhom-cG}
{{\rm Hom}\,}\Big( ({{\cal D}}_X[{{\mathfrak{d}}}]/({{\mathfrak{d}}}^2)) ^{\otimes \, (m-1)} \otimes {{\cal O}}_X
\,\,, \,\, {\check{{\mathcal C}}^{\bullet}}({{\cal G}}) \Big)$$ and the chain complex $$\label{Rhom-cO}
{{\rm Hom}\,}\Big( ({{\cal D}}_X[{{\mathfrak{d}}}]/({{\mathfrak{d}}}^2)) ^{\otimes \, (m-1)} \otimes {{\cal O}}_X
\,\,, \,\, {\check{{\mathcal C}}^{\bullet}}({{\cal O}}_X) \Big)\,.$$ It is obvious that the cohomology of the latter complex is concentrated only in non-negative degrees.
On the other hand the cocycle (\[on-samyj\]) has the negative degree $-m+1$. Hence there exists the next map ${{\Upsilon}}_m$ satisfying equation (\[Ups-k-eq\]) for $k = m-1$.
Proposition \[pravilnoe-pivo\] is proved. $\Box$
Thus the sheaf (\[sheaf\]) of ${{\cal D}}_X[{{\mathfrak{d}}}]/({{\mathfrak{d}}}^2)$-modules is quasi-isomorphic to the sheaf $${{\cal G}}= {{\cal D}}_X {\otimes_{{\cal O}_X}}{V^{\bullet}}_X[{{\mathfrak{d}}}]/({{\mathfrak{d}}}^2) \,
\otimes_{{{\cal Y}}({{\mathbb F}}_{{{\bf e_2}}}(\Xi_X))} \, {{\cal B}}{{\cal R}}_{{{\mathfrak{a}}}}\,.$$
Combining this observation with Proposition \[Morita\] we see that the sheaves $${{\Omega_X^{-\bullet}}}$$ and $${{\cal Y}}({V^{\bullet}}_X) \, \otimes_{{{\cal Y}}({{\mathbb F}}_{{{\bf e_2}}}(\Xi_X))} \, {{\cal B}}{{\cal R}}_{{{\mathfrak{a}}}}$$ are quasi-isomorphic as sheaves of ${{\cal Y}}({V^{\bullet}}_X)$-modules.
Theorem \[main\] is proved. $\Box$
Applications and generalizations
================================
Let, as above, $X$ be a smooth algebraic variety over a field ${{\mathbb K}}$ of characteristic zero. The homotopy calculus algebra on the pair $({C^{\bullet}_{{\rm norm}}}(X),{C_{\bullet}^{{\rm norm}}}(X))$ gives us a ${{\bf comm}}^+$-module structure on the pair $({V^{\bullet}}_X, {{\Omega_X^{-\bullet}}})$. Theorem \[KS-teo\] implies that this ${{\bf comm}}^+$-module structure on $({V^{\bullet}}_X, {{\Omega_X^{-\bullet}}})$ is given by the $\wedge$-product of polyvector fields and contraction of polyvectors with forms.
According to [@Swan] and [@Y] the Hochschild cohomology $HH^{{{\bullet}}}(X)$ of the variety $X$ is the hypercohomology of the sheaf ${C^{\bullet}_{{\rm norm}}}({{\cal O}}_X)$: $$HH^{{{\bullet}}}(X) = {{\mathbb H}}^{{{\bullet}}}({C^{\bullet}_{{\rm norm}}}({{\cal O}}_X))\,.$$ Furthermore, according to [@Cald], the Hochschild homology $HH_{{{\bullet}}}(X)$ of the variety $X$ is the hypercohomology of the sheaf ${C_{\bullet}^{{\rm norm}}}({{\cal O}}_X)$ $$HH_{{{\bullet}}}(X) = {{\mathbb H}}^{{{\bullet}}}({C_{\bullet}^{{\rm norm}}}({{\cal O}}_X))\,.$$
Thus, using Theorem \[main\] we get the following generalization of Corollary 2 from [@BLT]
For every smooth algebraic variety $X$ over a field ${{\mathbb K}}$ of characteristic zero the ${{\bf comm}}^+$-algebras $$( H^{{{\bullet}}}(X, {V^{\bullet}}_X), H^{{{\bullet}}}(X, {{\Omega_X^{-\bullet}}}))$$ and $$(\,HH^{{{\bullet}}}(X), HH_{{{\bullet}}}(X) \,)$$ are isomorphic. $\Box$
This statement is the existence part of Caldararu’s conjecture [@Cald1] on the Hochschild structure of a smooth algebraic variety. The cohomological part of this conjecture was proved in [@CV]. As far as we know, D. Calaque, C. Rossi, and M. Van den Bergh are currently writing an article [@CRV] with a proof of homological part of Caldararu’s conjecture.
Combining Theorem \[valid-beer\] with Theorem \[main\] we deduce the statement of cyclic formality conjecture (see Conjecture 3.3.2 in [@Tsygan]) from [@Tsygan] for an arbitrary smooth algebraic variety over a field ${{\mathbb K}}$ of characteristic zero:
\[Will\] If $X$ is a smooth algebraic variety a field ${{\mathbb K}}$ of characteristic zero then the sheaf of ${{\Lambda}}{{\bf Lie}}^+_{{{\delta}}}$-algebras $({C^{\bullet}}({{\cal O}}_X), {C_{\bullet}}({{\cal O}}_X))$ is formal. $\Box$
[**Remark.**]{} Strictly speaking the methods used by T. Willwacher in [@W] require an additional assumption ${{\mathbb R}}\subset {{\mathbb K}}$. Theorems \[valid-beer\] and \[main\] allow us to remove the assumption ${{\mathbb R}}\subset {{\mathbb K}}$ from the statement of Corollary \[Will\].
The proof of Theorem \[main\] can be easily modified for the following two cases:
- $X$ is complex manifold with ${{\cal O}}_X$ being the sheaf of holomorphic functions,
- $X$ is a real manifold with ${{\cal O}}_X$ being the sheaf of $C^{\infty}$ functions.
Thus we get the following obvious modification of Theorem \[main\]
\[main1\] If $X$ is a complex manifold (resp. real manifold) with ${{\cal O}}_X$ being the sheaf of holomorphic functions (resp. the sheaf of $C^{\infty}$ real functions) then the sheaf $$({C^{\bullet}_{{\rm norm}}}({{\cal O}}_X), {C_{\bullet}^{{\rm norm}}}({{\cal O}}_X))$$ of homotopy calculi is quasi-isomorphic to the sheaf $({V^{\bullet}}_X, {{\Omega}}^{-{{\bullet}}}_X)$ of calculi.
For $C^{\infty}$ real case we also get the following statement
\[glob-sect\] If $X$ is a real manifold with ${{\cal O}}_X$ being the sheaf of $C^{\infty}$ functions then the homotopy calculus algebra $$\Big({{\Gamma}}(X, {C^{\bullet}_{{\rm norm}}}({{\cal O}}_X)), {{\Gamma}}(X, {C_{\bullet}^{{\rm norm}}}({{\cal O}}_X)) \Big)$$ is quasi-isomorphic to the calculus algebra $$\Big({{\Gamma}}(X, {V^{\bullet}}_X), {{\Gamma}}(X, {{\Omega_X^{-\bullet}}}) \Big)\,.$$
[**Proof.**]{} In the $C^{\infty}$ real case the chain of quasi-isomorphisms connecting the sheaves $$({C^{\bullet}_{{\rm norm}}}({{\cal O}}_X), {C_{\bullet}^{{\rm norm}}}({{\cal O}}_X))$$ and $$({V^{\bullet}}_X, {{\Omega}}^{-{{\bullet}}}_X)$$ consists of soft sheaves. Hence, applying the functor ${{\Gamma}}(X, \,\,)$ of global sections we get the desired result. $\Box$
We would like to mention recent papers [@CW] and [@CF]. In paper [@CF] A. Cattaneo and G. Felder consider the DG Lie algebra module $CC^-_{-{{\bullet}}}({{\cal O}}_X)$ of negative cyclic chains over the DGLA ${C^{\bullet}}({{\cal O}}_X)$ of Hochschild cochains on a $C^{\infty}$ real manifold equipped with a volume form. Using an interesting modification of the Poisson sigma model A. Cattaneo and G. Felder construct a curious $L_{\infty}$ morphism (not a quasi-isomorphism!) from this DG Lie algebra module to a DG Lie algebra module modeled on polyvector fields using the volume form. A. Cattaneo and G. Felder also apply this result to a construction of a specific trace on the deformation quantization algebra of a unimodular Poisson manifold. Although this trace can be constructed using the formality quasi-isomorphism for Hochschild chains [@Sh], [@W] the relation of the $L_{\infty}$ morphism of A. Cattaneo and G. Felder to the formality quasi-isomorphism is a mystery.
Paper [@CW] is devoted to the proof of Kontsevich’s cyclic formality conjecture for cochains formulated in paper [@Sh1]. We suspect that the statement of this conjecture may be related to Theorem \[main\] and Corollary \[Will\] via the Van den Bergh duality [@VB] between Hochschild cohomology and Hochschild homology.
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\
<span style="font-variant:small-caps;">Department of Mathematics, University of California at Riverside,\
900 Big Springs Drive,\
Riverside, CA 92521, USA\
*E-mail address:* [**[email protected]**]{}</span>
\
<span style="font-variant:small-caps;">Mathematics Department, Northwestern University,\
2033 Sheridan Rd.,\
Evanston, IL 60208, USA\
*E-mail addresses:* [**[email protected]**]{}, [**[email protected]**]{}</span>
[^1]: See Proposition 11.3.3 on page 50 in [@K-Soi1].
[^2]: For the definition of [*nilpotent*]{} coalgebra see section $2.4.1$ in [@Hinich].
[^3]: ${{\mathbb F}}_{{{\cal O}}}$ is called the Schur functor.
[^4]: Although ${{\delta}}^2=0$, the operation ${{\delta}}$ is never considered as a part of the differential on ${{\cal W}}$.
[^5]: In [@K-Soi1] this operad is denoted by $P$.
[^6]: We are not sure if these operations are natural in the sense of category theory.
[^7]: In [@K-Soi1] this operad is denoted by $P$.
[^8]: Recall that a tree called [*rooted*]{} if if its root vertex has valency $1$.
[^9]: See sections 11.1, 11.2, and 11.3 in [@K-Soi1].
[^10]: We only need the operations which survive in characteristic zero.
[^11]: This picture is very reminiscent of the consideration of Hochschild-Serre spectral sequence in the proof of Proposition 4.1 in [@Dima-Disc].
[^12]: See Equation (\[bB-01\]).
[^13]: Obvious extensions of the operations on Hochschild chains to the operations on polyjets is discussed in details in [@CDH].
[^14]: The construction of this free resolution is known in topology as [*the rectification*]{} [@BVogt]. We describe this construction in more details in [@BLT] (See Proposition 3 therein).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'It is a recurrent issue in astronomical data analysis that observations are unevenly sampled or incomplete maps with missing patches or intentionaly masked parts. In addition, many astrophysical emissions are non stationary processes over the sky. Hence spectral estimation using standard Fourier transforms is no longer reliable. Spectral matching ICA (SMICA) is a source separation method based on covariance matching in Fourier space which is successfully used for the separation of diffuse astrophysical emissions in Cosmic Microwave Background observations. We show here that wavelets, which are standard tools in processing non stationary data, can profitably be used to extend SMICA. Among possible applications, it is shown that gaps in data are dealt with more conveniently and with better results using this extension, wSMICA, in place of the original SMICA. The performances of these two methods are compared on simulated CMB data sets, demonstrating the advantageous use of wavelets.'
address: |
$^1$ DAPNIA/SEDI-SAP, CEA/Saclay, F-91191 Gif-sur-Yvette, France\
$^2$ CNRS/ENST, 46 rue Barrault, F-75634 Paris, France\
$^3$ CNRS/PCC, Collège de France, 11 place Marcelin Berthelot, F-75231 Paris, France\
\
[email protected], [email protected], [email protected], [email protected]\
title: |
Blind component separation in wavelet space.\
Application to CMB analysis.
---
[ ***Keywords** : blind source separation, cosmic microwave background, wavelets, data analysis*]{}
Introduction
============
The detection of Cosmic Microwave Background (CMB) anisotropies on the sky has been over the past three decades subject of intense activity in the cosmology community.
The CMB, discovered in 1965 by Penzias and Wilson, is a relic radiation emitted some 13 billion years ago, when the Universe was about 370.000 years old. Small fluctuations of this emission, tracing the seeds of the primordial homogeneities which gave rise to present large scale structures as galaxies and clusters of galaxies, have been observed by a number of experiments such as Archeops [@2002APh....17..101B], Boomerang [@2000Natur.404..955D], Maxima [@2000ApJ...545L...5H] and WMAP [@2003ApJS..148....1B] .
The precise measurement of these fluctuations is of utmost importance for Cosmology. Their statistical properties (spatial power spectrum, Gaussianity) strongly depend upon the cosmological scenarios describing the properties and evolution of our Universe as a whole, and thus permit to constrain these models as well as to measure the cosmological parameters describing the matter content, the geometry, and the evolution of our Universe [@jungman96].
Accessing this information, however, requires disentangling in the data the contribution of several distinct astrophysical sources, all of which emit radiation in the frequency range used for CMB observations [@fb-rg99]. This problem of component separation, in the field of CMB studies, has thus been the object of many dedicated studies in the past.\
To first order, the total sky emission is modelled as a linear mixture of a few independent processes. The observation of the sky with detector $d$ is then a noisy linear mixture of $N_c$ components: $$y_{d}(\theta, \phi)
=
\sum_{j=1}^{N_c} A_{d j} s_{j}(\theta, \phi) +
n_{d}(\theta, \phi)
\label{eq:mixture}$$ where $s_{j}$ is the emission template for the $j$th astrophysical process, herein referred to as a *source* or a *component*. The coefficients $A_{dj}$ reflect emission laws while $n_d$ accounts for noise. When $N_d$ detectors provide independent observations, this equation can be put in vector-matrix form: $$X(\theta, \phi) =AS(\theta, \phi)+N(\theta, \phi)
\label{model1}$$ where $X$ and $N$ are vectors of length $N_d$, $S$ is a vector of length $N_c$, and A is the $N_d\times N_c$ mixing matrix.
Given the observations of such a set of independent detectors, component separation consists in recovering estimates of the maps of the sources $s_{j}(\theta, \phi)$. Explicit component separation has been investigated first in CMB applications by [@1996MNRAS.281.1297T], [@fb-rg99], and [@1998MNRAS.300....1H]. In these applications, recovering component maps is the primary target, and all the parameters of the model (mixing matrix $A_{dj}$, noise levels, statistics of the components, including the spatial power spectra) are assumed to be known and used as priors to invert the linear system.
Recent research has addressed the case of an imperfectly known mixing matrix. It is then necessary, to estimate it (or at least some of its entries) directly from the data. For instance, Tegmark *et al.* assume power law emission spectra for all components except CMB and SZ, and fit spectral indices to the observations [@2000ApJ...530..133T]. More recently, blind source separation or independent component analysis (ICA) methods have been implemented specifically for CMB studies. The work of [@Bac2000], further extended by [@Maino2002] implements a blind source separation method exploiting the non–Gaussianity of the sources for their separation, which permits to recover the mixing matrix $A$ and the maps of the sources.
Delabrouille et al. [@Del2003] propose an approach exploiting the spectral diversity of components, with the new point of view that spatial power spectra are actually the main unknown parameters of interest for CMB observations. The estimation of a set of parameters of the model, among which the spatial power spectra of the components, is made using a set of band-averaged spectral covariance matrices in Fourier space.
While working in the Fourier domain has a number of advantages, it also has a number of drawbacks. When components or noise are strongly non-stationary, one may wish to avoid the averaging induced by Fourier transforms. In addition, when dealing with real-life observations, quite often the coverage is incomplete for a reason or another. Either the instrument observes only a fraction of the sky, or some regions of the sky have to be rejected due to localised strong astrophysical sources of contamination : compact radiosources or galaxies, strong emitting regions in the galactic plane.
Blind component separation (and in particular estimation of the mixing matrix), as discussed by Cardoso [@3easy], can be achieved in several different ways. The first of these exploits non-Gaussianity of all but possibly one components. However, this is not recommended for mixtures where one component is close to Gaussian and all observations suffer from additive Gaussian noise. The component separation method of Baccigalupi [@Bac2000] and Maino [@Maino2002] is based on this method. The second, which exploits spectral diversity (or non-stationarity in Fourier domain), has the advantage that detector–dependent beams can be handled easily, since the convolution with a point spread function in direct space becomes a simple product in Fourier space. SMICA is an extension of this approach to noisy observations. Finally, the third method exploits non-stationarity in real space. It is adapted to situations where components are strongly non-stationary in real space.
As an extension of these last two methods, it is natural to investigate the possible benefits of exploiting both non-stationarity and spectral diversity for blind component separation using wavelets. Indeed wavelets are powerful tools in revealing the spectral content of non-stationary data. In what follows, we first recall in section \[smica\] the fundamental principles of Spectral Matching ICA. Then, after a brief reminder of the *à trous* wavelet transform, we discuss in section \[wavelets\] the extension of SMICA for component separation in wavelet space in order to deal with non-stationary data. Considering the problem of incomplete data as a model case of practical significance for the comparison of SMICA and its extension wSMICA, numerical experiments and results are reported in section \[simulations\] . From these, conclusions are drawn in section \[conclusion\].
SMICA
=====
This paragraph recalls the main hypotheses and equations of the SMICA algorithm which we actually extended to deal with gapped data. For ease of presentation, we concentrate on the 1D case since the extension to two dimensional data is straightforward. Detailled descriptions and discussions of this method can be found in [@Pha2; @Car] and results of previous applications to CMB analysis can be read in [@Del2003; @Pat].
Model and cost function
-----------------------
Spectral matching ICA is a blind source separation technique that overcomes the inseparability of Gaussian sources using standard ICA methods by relying on their assumed spectral diversity : SMICA allows us to recover independent Gaussian colored sources from observed noisy mixtures provided their spectra are substantially not proportional [@Car2].\
Considering the linear instantaneous mixing model with additive noise defined by (\[model1\]), with the assumption that noise and source processes are centered, stationary and independent, and denoting $R_{X}(\nu)$, $R_{S}(\nu)$ and $R_{N}(\nu)$ the spectral covariances of $X$, $S$ and $N$ respectively, it follows from (\[model1\]) that for any value of the reduced frequency $\nu \in [-0.5, 0.5]$, $$R_{X}(\nu) = A R_{S}(\nu) A ^\dagger + R_{N}(\nu)
\label{model2}$$ when we further assume independence between source and noise processes. Clearly, independence also implies that $R_{S}(\nu)$ and $R_{N}(\nu)$ are diagonal matrices.\
Given a batch of $T$ regularly spaced experimental data samples $X_{t=1\rightarrow T}$ and a set $\{ \nu_{q, q=1\rightarrow Q} \}$ of Q different reduced frequencies chosen *a priori* , estimates $\widehat{R}_{X}(\nu_q)$ of $R_{X}(\nu_q)$ of the spectral covariance at these frequencies can be computed easily in a number of ways. The basic idea of spectral matching is to fit the model covariances of equation (\[model2\]) to these experimental covariances by minimizing, over all or a subset of the model parameters $\theta = \{ R_{S}(\nu_q), R_{N}(\nu_q), A \}$, the functional $$\phi (\theta) = \sum _{q=1}^{Q} \alpha_q \mathcal{D} \Big( \widehat{R}_{X}(\nu_q), A
R_{S}(\nu_q) A^{\dagger} + R_{N}(\nu_q) \Big)
\label{Cost1}$$ where $\mathcal{D} (.,.) $ is a measure of the divergence between two covariance matrices, and $\alpha_q$ are weights which depend on $q$. This adjustment results in estimates $\widehat{\theta} = \{ \widehat{R}_{S}(\nu_q), \widehat{R}_{N}(\nu_q), \widehat{A} \}$ of the model parameters and hence enables us to achieve the desired source separation. It is worth highlighting that resorting to covariances highly reduces data dimension, which is of great interest to astrophysical applications where data sets tend to become very large. Moreover, it may be argued in the stationary Gaussian case that this reduction is without significant loss of information [@Del2003].\
Although any reasonable set of weights $\alpha_q$ and divergence $\mathcal{D} (.,.) $ can be used in (\[Cost1\]) to assess spectral mismatch, this will affect the statistical properties of the estimated model parameters $\widehat{\theta} = \{ \widehat{R}_{S}(\nu_q), \widehat{R}_{N}(\nu_q), \widehat{A} \}$. Deriving a mismatch criterion from higher statistical principles such as maximum likelihood should lead to better such estimates.\
In the SMICA method, the divergence $\mathcal{D}$ used is given by $$\mathcal{D}_{KL} (R_1, R_2 ) = \frac{1}{2} \Big( \textrm{Tr}(R_1R_2^{-1}) - \textrm{log}\textrm{det}(R_1R_2^{-1}) - m \Big)$$ which actually derives from the Kullback-Leibler divergence between two centered Gaussian distributions with size $m\times m$ covariance matrices $R_1$ and $R_2$. Moreover, assuming constant source $R_{S,q}^f$ and noise $R_{N,q}^f$ power spectra, over frequency domains $\{F_q\}_{q\in [1, Q]}$ , SMICA uses refined unbiased estimates $\widehat{R}_{X,q}^f$ of the mixture covariance matrices $R_{X,q}$ defined by : $$\hat{R}_{X,q}^f = \frac{1}{n_q}\sum_{p=0,\frac{p}{T}\in F_q}^{T-1} \widetilde{X}(\frac{p}{T})\widetilde{X}(\frac{p}{T})^\dagger
\label{Esti1}$$ where $\tilde{X}$ is the discrete Fourier transform of $X$, $$\widetilde{X}(\nu)= \frac{1}{\sqrt{T}} \sum_{t=0}^{T-1} X(t)e^{-2\pi j \nu t},$$ the $F_q$ are non-overlapping domains in $[-1/2, 1/2]$, symmetric with respect to zero, with their positive parts centered on $\nu_q$, and $n_q$ is the number of $\frac{p}{T}$ that fall in $F_q$. It follows from this definition that the entries of $\widehat{R}_{X,q}^f$ are in fact all real. The statistical grounds and implications of these choices are explored in [@Pha2; @Car2] where it is shown that SMICA can be derived asymptotically from the maximum likelihood principle in the particular case of stationary processes in the Whittle approximation. This latter approximation asserts that the Fourier coefficients $\widetilde{X}(\frac{p}{T})$ of a stationary process $X(t)$ are asymptotically Gaussian, uncorrelated, centered with spectral covariance equal to $R_X(\frac{p}{T})$.
As a result, the model covariance (\[model2\]) is finally rewritten as: $$R_{X,q}^f = A R_{S,q}^f A^{\dagger} + R_{N,q}^f$$ and the derived spectral matching criterion is given by $$\phi(\theta) = \sum _{q=1}^{Q} n_q \mathcal{D}_{KL} \Big( \widehat{R}_{X,q}^f,
A R_{S,q}^f A^{\dagger} + R_{N,q}^f \Big)
\label{Cost2}$$ to be minimized with respect to the new set of parameters $\theta = (A,R_{S,q}^f, R_{N,q}^f )$.\
The previous definitions are easily extended for the method to be applied to real images. The above $F_q$ are naturally replaced by 2D domains in the frequency plane [@Car]. These are best chosen, based on available prior information relative to source spectra, to enhance spectral diversity [@Car2]. Regarding our application to CMB analysis, the supposed spatial stationarity and isotropy of the sources strongly suggests taking rings centered on the null frequency which are finally simply described as 1D frequency bands.\
An especially important limiting case, for simulation purposes, is when the mixing matrix is square and invertible, and when the mixtures are assumed without noise. Then, as shown in [@Pha2], the likelihood can be directly related to a joint diagonalization criterion of spectral covariance matrices for which an efficient optimization algorithm is actually available.\
Parameter optimization {#OPTIM}
----------------------
Finding the model closest to the data in the sense of SMICA’s objective function benefits from the latter’s connection to the maximum likelihood principle and indeed the EM algorithm is shown to be a fruitful search method in [@Del2003] where it is fully described. Actually, this latter algorithm was slightly modified in order to deal with the case of colored noise $N$ in (\[model1\]). Another useful enhancement was to allow for constraints to be set on the model parameters so that prior information such as bounds on some entries of the mixing matrix $A$ could be included. The details of this constrained EM algorithm are given in appendix \[annexe1\].\
Eventually, using the EM algorithm in simulation, it appeared that after a quick start, convergence slowed down dramatically in a second stage possibly owing to poor signal to noise ratio in some frequency bands. In order to speed convergence back up, it was found profitable to alternately use fixed numbers of EM steps and BFGS steps [@Del2003; @Pat] in a heuristic procedure.\
An unavoidable issue in optimization is that of initiating the search method and this, obviously, is most critical when the function to be optimized is strongly suspected to be multimodal. Such may very well be the case with (\[Cost1\]). This point though is left aside in what follows since our prime interest is in the study of the statistical performances of different estimators of the model parameters $\theta$. In the simulations discussed further down, the optimal values of the parameters are sought starting from the true mixing matrix and the spectral covarinces estimated from the initial separate source and noise maps.\
Component map estimation
------------------------
As by-products of the SMICA method, estimates $\widehat{R}_{S,q}^f$ and $\widehat{R}_{N,q}^f$ of the different signal and noise covariances are obtained in the model fitting step and can be used for reconstructing the source maps *via* Wiener filtering the data maps in Fourier space, in each frequency band $\nu \in F_q$ : $$\widehat{S}(\nu) = (\widehat{A}\adj \widehat{R}_{N,q}^{f-1} \widehat{A} + \widehat{R}_{S,q}^{f-1})\inv \widehat{A}\adj \widehat{R}_{N,q}^{f-1} X(\nu)
\label{wiener}$$ In the limiting case where noise is small compared to signal components, $\widehat{R}_{S,q}^{f-1}$ is negligible and the above filter reduces to $$\widehat{S}(\nu) = (\widehat{A}\adj \widehat{R}_{N,q}^{f-1} \widehat{A})\inv \widehat{A}\adj \widehat{R}_{N,q}^{f-1} X(\nu)
\label{wiener}$$ which is also the generalized least square solution under Gaussian statistics.
Note however that the Wiener filter is only one possibility among others for inverting (\[model1\]). Its optimality is true in the restricted case of Gaussian noise and signal processes. In real case applications, other inverting schemes should also be experimented [@Del2003].
Wavelets and SMICA {#wavelets}
===================
The SMICA method for spectral matching in Fourier space has proven to be a very powerful tool for CMB spectral estimation in multidetector experiments. It is particularly useful to identify and remove residuals of poorly known correlated systematics and astrophysical foreground emissions contaminating CMB maps. However, SMICA suffers from several practical difficulties when dealing with real data.
Indeed, actual components are known to depart slightly from the ideal linear mixture of equation (\[model1\]). The mixing matrix (in particular those columns of $A$ which correspond to galactic emissions) is known to depend somewhat on the direction of observation or on spatial frequency. Measuring the dependence $A(\theta,\phi)$ is of interest for future experiments as Planck, and can not be achieved directly with SMICA. Further, the components are known to be both correlated and non stationary. For instance, galactic dust emissions are strongly peaked towards the galactic plane. A Fourier (or spherical harmonics) transform inevitably mixes contributions from high galactic sky, nearly free of foreground contamination, and contributions from within the galactic plane. Noise levels themselves may be quite non stationary, with high SNR regions observed for a long time and low SNR regions poorly observed.
When there are sharp edges on the maps or gaps in the data, corresponding to unobserved or masked regions, spectral estimation using the periodogram or the Daniell-like smoothed periodogram as in (\[Esti1\]) is also not the most satisfactory procedure. Although apodizing windows may help cope with edge effects in Fourier analysis, they are not very straightforward to use in the case of arbitrarily shaped 2D maps with arbitrarily shaped 2D gaps, such as provided by the Archeops experiment [@2002APh....17..101B]. Clearly, the spectral analysis of gapped data requires tools different from those used to process full data sets, if only because the hypothesized stationarity of the data is greatly disturbed by the missing samples.\
Common such methods often amount to first trying to fill the gaps with estimates of the missing samples and then using standard spectral estimators. However, the data interpolation stage is critical and cannot be completed without prior assumptions on the data [@Sto]. We prefered to rely on methods intrinsically dedicated to the analysis of non-stationnary data such as the wavelet transform, widely used to reveal variations in the spectral content of time series or images, as they permit to single out regions in direct space while retaining localization in the frequency domain. We see next how to reformulate (\[Cost1\]) so to take advantage of wavelet transforms when dealing with non-stationary data. A particular case in which wavelets are shown to be an especially powerful tool is that of incomplete data. Note that in what follows, the locations of the missing samples are always known.\
Wavelet transform: the *à trous* algorithm
------------------------------------------
We give here the necessary background on the *à trous* algorithm which, among the several possible wavelet transform implementations, is the one we retained in our simulations. With the compact supported cubic $B_3$ spline as scaling function $\phi(k)$, or its 2D quasi-isotropic extension $\phi(k)\phi(l)$, the *à trous* algorithm has been shown to be well suited to the analysis of atrophysical data where translation invariance is desirable and the accent is seldom set on data compression [@Sta]. For this choice of scaling function, the scaling equation (\[ScalingEquation\]) is satisfied and therefore fast implementations of the decomposition and reconstruction steps of the *à trous* tranform are available [@Sta].\
Consider for instance a sampled 1D signal $c_0(k)$ of length $T$. The *à trous* algorithm recursively produces smoother approximations $c_i$ to $c_0$ on a dyadic resolution scale using a low-pass filter $h$ according to : $$c_i(k) = \sum_{u}h(u)c_{i-1}(k + 2^{i-1} u) = \sum_{u} \frac{1}{2^i}\phi(\frac{k-u}{2^i}) c_0(u)$$ where $h = \{ 1/16, 1/4, 3/8, 1/4, 1/16 \}$ is actually the set of coefficients in the scaling equation for the cubic spline : $$\phi(k) = \sum_u h(u) \phi(2k - u)
\label{ScalingEquation}$$ We note that each $c_i$ is the same size as the original data $c_0$ and that the lowest resolution $J_{max}$ is obviously limited by data size $T$. Then, taking the difference between two consecutive approximations gives the details at that scale or the wavelet coefficients $$w_i(k) = c_{i-1}(k) - c_{i}(k) = \sum_{u} \frac{1}{2^{i-1}}\psi(\frac{k-u}{2^{i-1}}) c_0(u)$$ where the wavelet function $\psi(k)$ is defined by: $$\psi(k) = \phi(k) - \frac{1}{2} \phi(\frac{k}{2})$$ The $w_i$’s and $c_i$’s given using the *à trous* algorithm actually are obtained by passing the original signal $c_0$ through a set of finite impulse response (FIR) filters $\psi_1, \psi_2, \ldots, \psi_J, \phi_J$. An essential property of these filters is that an inverse transform exists. In fact, reconstruction results simply from adding all the wavelet scales together with the last smooth approximation: $$\forall k, c_0(k) = c_J(k) + w_J(k) +w_{J-1}(k) + \ldots + w_2(k) + w_1(k)
\label{IWT}$$
The above [*à trous*]{} algorithm is easily extendable to two-dimensional images: $$\begin{aligned}
c_{i}(k,l) \!\!\!& = &\!\!\! \sum_u \sum_v h(u,v) c_{i-1}(k+2^{i-1}u,l+2^{i-1}v) \\
w_{i}(k,l) \!\!\!& = &\!\!\! c_{i-1}(k,l) - c_{i}(k,l)\end{aligned}$$ and the reconstruction is still a simple co-addition of the wavelet scales and the smooth array: $$\begin{aligned}
c_{0}(k,l) = c_{J}(k,l) + \sum_{i=1}^{J} w_{i}(k,l)
\label{eqn_rec}\end{aligned}$$
The use of the $B_3$ spline leads to a convolution with the $5 \times 5$ mask $h$: $$\frac{1}{256}\left( \begin{array}{ccccc}
1 & 4 & 6 & 4 & 1 \\
4 & 16 & 24 & 16 & 4 \\
6 & 24 & 36 & 24 & 6 \\
4 & 16 & 24 & 16 & 4 \\
1 & 4 & 6 & 4 & 1
\end{array} \right)$$ but it is faster to compute the convolution in a separable way (first on rows, and then on the resulting columns).
Spectral matching in wavelet space : wSMICA
-------------------------------------------
Consider the set of ideal band pass filters $\mathcal F_q$ associated with non-overlapping frequency domains $F_q$ as used by the Fourier space implementation of SMICA. Let $Y_q$ denote the stationary Gaussian random processes obtained by passing the observations $X$ of size $m$ through filter $\mathcal F_q$. Let $\widetilde{Y}_q$ be their Fourier coefficients. Because of the unitary property of the Fourier transform, considering a batch of $T$ samples $X_{t=1,T}$, the following equality between joint probabilities holds : $$P(Y_{1;t=1,T},...,Y_{Q;t=1,T} ) = P(\widetilde{Y}_{1;k=1,T},...,\widetilde{Y}_{Q;k=1,T} )$$ Assuming uncorrelated Fourier coefficients as in the above mentioned maximum likelihood derivation of SMICA based on the the Whittle approximation, and because of the non-overlapping filters, it follows that the $Y_{q;t}$ for different $q$’s are also decorrelated so that: $$-\mathrm{log} P(Y_{1;t=1,T},...,Y_{Q;t=1,T} ) = -\sum _{q=1}^Q \mathrm{log} P(\widetilde{Y}_{q;k=1,T})$$ and that $\forall q$: $$\label{weights}
\begin{split}
-\mathrm{log} P(Y_{q;k=1,T}) & = -\mathrm{log} P(\widetilde{Y}_{q;k=1,T}) \\
& = n_q \mathcal{D}_{KL} \Big( \widehat{R}_{X,q}^f, A R_{S,q}^f A^{\dagger} + R_{N,q}^f \Big)
\end{split}$$
Now define mixture, source and noise covariances $R_{X,q}^t$, $R_{S,q}^t$ and $R_{N,q}^t$ in the time domain at the output of the above filters. The former matrices can be estimated from the available data using: $$\widehat{R}_{X,q}^t = \frac{1}{T}\sum_{t=0}^{T-1} Y_{q;t}Y_{q;t}^\dagger
\label{cov_temps}$$ and nothing opposes attempting component separation by spectral matching in the time domain using these latter covariances by minimizing $$\phi (\theta) = \sum _{q=1}^{Q} \alpha_q \mathcal{D} \Big( \widehat{R}_{X,q}^t, A
R_{S,q}^t A^{\dagger} + R_{N,q}^t\Big)
\label{Cost_time}$$ with respect to $\theta = (A,R_{S,q}^t, R_{N,q}^t )$, provided the estimated covariances are full rank matrices. However, deriving adequate weights $\alpha_q$ in order to get a good approximation of the likelihood is not straightforward because of the correlations between the $Y_{q;t}$’s at different $t$’s. In fact, owing to these correlations, the convergence of $\widehat{R}_{X,q}^t$ to $R_{X,q}^t$ can be very slow. The helpful point equation (\[weights\]) actually makes is that taking $\alpha_q = n_q$ will correctly reflect our confidence in the estimated covariances $\widehat{R}_{X,q}^t$.\
The next step is obviously to use another set of filters in place of the ideal band pass filters used by SMICA. In fact, in dealing with non stationary data or, as a special case, with gapped data, it is especially attractive to consider finite impulse response filters. Indeed, provided the response of such a filter is short enough compared to data size $T$ and gap widths, not all the samples in the filtered signal will be affected by the gaps. Therefore, using these latter samples exclusively, one may expect better estimation of the statistical properties of the original data *i.e.* without the gaps. We choose in what follows to use filters $\psi_1, \psi_2, \ldots, \psi_J, \phi_J$ (see figure \[Filtres2\]) and the wavelet *à trous* algorithm described previously. An immediate consequence of this choice is that the decorrelation between the different filter outputs no longer holds, due to their overlapping responses in Fourier space. However, we do benefit from the fast filtering algorithms and, which is quite significant, from the possibility of reconstructing estimated source templates.\
Let us consider again a batch of $T$ regularly spaced data samples $X_{t=1\rightarrow T}$. Possible gaps in the data are simply described with a mask $\mu$ *i.e.* a vector of zeroes and ones the same length as $X$ with ones corresponding to samples outside the gaps. Denoting $W_1, W_2, \ldots, W_J$ and $C_J$ the wavelet scales and smooth approximation of $X$, obtained with the *à trous* transform and $\mu_1, \ldots, \mu_{J+1}$ the masks for the different scales determined from the original mask $\mu(t)$ knowing the different filter lengths, wavelet covariances are estimated as follows: $$\begin{split}
\widehat{R}_{X,1\leq i\leq J}^w = & \frac{1}{l_i } \sum_{t = 1}^{T}\mu_i(t) W_i(t) W_i(t) ^\dagger \\
\widehat{R}_{X,J+1}^w = & \frac{1}{l_{J+1}} \sum_{t = 1}^{T}\mu_{J+1}(t) C_J(t) C_J(t) ^\dagger
\end{split}$$ where $l_i$ is the number of non zero samples in $\mu_i$. With source and noise covariances $R_{S,i}^w, R_{N,i}^w$ defined in a similar way, the covariance model in wavelet space becomes $$R_{X,i}^w = A R_{S,i}^w A^{\dagger} + R_{N,i}^w$$ and minimizing $$\phi (\theta) = \sum _{q=1}^{Q} \alpha_q \mathcal{D} \Big( \widehat{R}_{X,q}^w, A
R_{S,q}^w A^{\dagger} + R_{N,q}^w\Big)
\label{Cost_wavelet}$$ with respect to the model parameters $\theta_w = (A,R_{S,i}^w, R_{N,i}^w )$ achieves the desired component separation.\
However, in order for $\phi (\theta)$ to be a good approximation to the likelihood, the weights $\alpha_q$ again have to be determined with care. These weights should account for the correlations between wavelet coefficients from different or the same scales, especially in the lower frequencies. Actually, exagerating the so-called decorrelating property of the wavelet transform, we assume coefficients from different scales are uncorrelated. Nevertheless, coefficients from one same scale are strongly correlated, especially with the adopted *à trous* redundant transform. Then, in the case of complete data sets *i.e.* without gaps, and because the 1D wavelet filter length in the time domain doubles from scale to scale, the transposition of equation (\[weights\]) leads to taking: $$\{ \alpha_1, \alpha_2, ..., \alpha_J, \alpha_{J+1} \} = \{ \frac{1}{2}, \frac{1}{4}, ..., \frac{1}{2^J}, \frac{1}{2^J} \}$$ In the 2D case, this becomes: $$\{ \alpha_1, \alpha_2, ..., \alpha_J, \alpha_{J+1} \} = \{ \frac{3}{4}, \frac{3}{16}, ..., \frac{3}{4^J}, \frac{1}{4^J} \}$$ However, when there are gaps in the data, the Fourier modes can be strongly correlated and the Whittle approximation is no longer appropriate. In order to derive an approximate likelihood function, consider the orthogonal discrete wavelet transform. In the 1D case, this is a non-redundant transform in which the number of coefficients is halved from scale to scale. It is common and quite convenient to assume these coefficients are uncorrelated. Denoting $l_i^{DWT}$ the number of DWT coefficients unaffected by the gaps in scale $i$, these have the same statistical significance or information content as the $l_i \approx 2^i\times l_i^{DWT}$ coefficients in scale $i$ determined with the *à trous* wavelet transform. Finally, a good approximation to the likelihood is obtained taking $$\{ \alpha_1, \alpha_2, ..., \alpha_J, \alpha_{J+1} \} = \{ \frac{l_1}{2}, \frac{l_2}{4}, ..., \frac{l_J}{2^J}, \frac{l_{J+1}}{2^J} \}$$ or, in the 2D case,: $$\{ \alpha_1, \alpha_2, ..., \alpha_J, \alpha_{J+1} \} = \{ \frac{3l_1}{4}, \frac{3l_2}{16}, ..., \frac{3l_J}{4^J}, \frac{l_{J+1}}{4^J} \}$$ in equation (\[Cost\_wavelet\]). We will refer to this combination of principles from SMICA and wavelet transforms as wSMICA.\
A point to be stressed here is that the number of bands in the case of wSMICA is very much limited by the original data size, which is not as strongly the case with SMICA. But this limitation is mostly a requirement for reconstruction using (\[wiener\]) and (\[IWT\]) to make sense. If the mixing matrix $A$ is a parameter of greater interest and if there is no real need to estimate source maps $S$, then there is no objection in principle to using more redundant transforms such as the continuous wavelet transform, or in fact any set of linear filters (of finite impulse response to cope easily with edges and gaps). This in turn raises the question of optimally choosing this set of filters as in [@Sto].
![Magnitudes of the cubic spline wavelet filters $\psi_1, \psi_2, \ldots, \psi_5$ used in the simulations described further down. The vertical dotted lines for $\nu = \{ 0.013, 0.025, 0.045, 0.09, 0.2, 0.5\}$ delimit the five frequency bands used with SMICA in these simulations. []{data-label="Filtres2"}](filters2.eps){width="7cm"}
NUMERICAL EXPERIMENTS {#simulations}
=====================
Simulated data
--------------
The methods described above were applied to synthetic observations consisting of $m = 6$ mixtures of $n = 3$ components namely CMB, galactic dust and SZ emissions for which typical templates, shown on figure \[Templates\], were obtained as described in [@Del2003].\
![Simulated component templates for CMB *(top)*, DUST *(middle)*, SZ *(bottom)*.[]{data-label="Templates"}](CMB.eps "fig:"){width="5cm"}\
![Simulated component templates for CMB *(top)*, DUST *(middle)*, SZ *(bottom)*.[]{data-label="Templates"}](DUST.eps "fig:"){width="5cm"}\
![Simulated component templates for CMB *(top)*, DUST *(middle)*, SZ *(bottom)*.[]{data-label="Templates"}](SZ.eps "fig:"){width="5cm"}\
The templates, and thus the mixtures in each simulated data set, consist of $300 \times 300$ pixel maps corresponding to a $12.5^\circ \times 12.5 ^\circ$ field located at high galactic latitude. The six mixtures in each set mimic observations that will eventually be acquired in the six frequency channels of the Planck-HFI on part-sky, local maps. The entries of the mixing matrix $A$ used in these simulations actually are estimated values of the electromagnetic emission laws of the original components at $100$, $143$, $217$, $353$, $545$ and $857~\textrm{GHz}$. These values are grouped in table \[MatrixA\].\
White Gaussian noise was added to the mixtures according to equation (\[model1\]) in order to simulate instrumental noise. While the relative noise standard deviations between channels were set according to the nominal values of the Planck HFI, we experimented five *global* noise levels at $-20$, $-6$, $-3$, $0$ and $+3$ dB from nominal values. Table \[Energie1\] gives the typical energy fractions that are contributed by each of the $n=3$ original sources and noise, to the total energy of each of the $m=6$ mixtures, considering Planck nominal noise variance. In fact, because SMICA and wSMICA actually work on spectral bands, a much better indication of signal to noise ratio in these simulations is given by figure \[mixtures1\] where it is shown how noise and source energy contributions distribute with respect to frequency in the six mixtures.\
![Energy contributed by each source and noise to each of the six mixtures *(mixture 1 : top left, mixture 6 : bottom right)* as a function of frequency, for the nominal noise variance on the Planck HFI channels. Note how SZ is expected to always be below nominal noise, that CMB and dust strongly dominate in different channels and that CMB and dust spectra, without being proportional, display the same general behaviour dominated by low modes.[]{data-label="mixtures1"}](channel1.eps "fig:"){width="4.2cm"} ![Energy contributed by each source and noise to each of the six mixtures *(mixture 1 : top left, mixture 6 : bottom right)* as a function of frequency, for the nominal noise variance on the Planck HFI channels. Note how SZ is expected to always be below nominal noise, that CMB and dust strongly dominate in different channels and that CMB and dust spectra, without being proportional, display the same general behaviour dominated by low modes.[]{data-label="mixtures1"}](channel2.eps "fig:"){width="4.2cm"} ![Energy contributed by each source and noise to each of the six mixtures *(mixture 1 : top left, mixture 6 : bottom right)* as a function of frequency, for the nominal noise variance on the Planck HFI channels. Note how SZ is expected to always be below nominal noise, that CMB and dust strongly dominate in different channels and that CMB and dust spectra, without being proportional, display the same general behaviour dominated by low modes.[]{data-label="mixtures1"}](channel3.eps "fig:"){width="4.2cm"} ![Energy contributed by each source and noise to each of the six mixtures *(mixture 1 : top left, mixture 6 : bottom right)* as a function of frequency, for the nominal noise variance on the Planck HFI channels. Note how SZ is expected to always be below nominal noise, that CMB and dust strongly dominate in different channels and that CMB and dust spectra, without being proportional, display the same general behaviour dominated by low modes.[]{data-label="mixtures1"}](channel4.eps "fig:"){width="4.2cm"} ![Energy contributed by each source and noise to each of the six mixtures *(mixture 1 : top left, mixture 6 : bottom right)* as a function of frequency, for the nominal noise variance on the Planck HFI channels. Note how SZ is expected to always be below nominal noise, that CMB and dust strongly dominate in different channels and that CMB and dust spectra, without being proportional, display the same general behaviour dominated by low modes.[]{data-label="mixtures1"}](channel5.eps "fig:"){width="4.2cm"} ![Energy contributed by each source and noise to each of the six mixtures *(mixture 1 : top left, mixture 6 : bottom right)* as a function of frequency, for the nominal noise variance on the Planck HFI channels. Note how SZ is expected to always be below nominal noise, that CMB and dust strongly dominate in different channels and that CMB and dust spectra, without being proportional, display the same general behaviour dominated by low modes.[]{data-label="mixtures1"}](channel6.eps "fig:"){width="4.2cm"}
Finally, in order to investigate the benefits of using wSMICA in place of SMICA when gaps are inserted in the data, the mask shown on figure \[Masks\] was applied onto the mixture maps. The case where no data is missing was also considered for the sake of comparison. In each of these two particular configurations, spectral matching was assessed and optimized both at the output of the five wavelet filters $\psi_1,\ldots, \psi_5$ associated to higher frequency details, and on the corresponding five bands in Fourier space, as shown on figure \[Filtres2\]. This latter choice of frequency bands is simply made to ease comparison between SMICA and wSMICA. It may be argued that this choice is probably not optimal to run SMICA. But, in fact, the optimal selection of filters is clearly a meaningful question both for SMICA and wSMICA. This will require further investigation.\
![Mask used to simulate a gap in the data *(top left)*, and the modified masks at scales 1 *(top right)* through 5 *(bottom left)*. The discarded pixels are in black. []{data-label="Masks"}](Masque.eps "fig:"){width="3cm"} ![Mask used to simulate a gap in the data *(top left)*, and the modified masks at scales 1 *(top right)* through 5 *(bottom left)*. The discarded pixels are in black. []{data-label="Masks"}](Masque1.eps "fig:"){width="3cm"} ![Mask used to simulate a gap in the data *(top left)*, and the modified masks at scales 1 *(top right)* through 5 *(bottom left)*. The discarded pixels are in black. []{data-label="Masks"}](Masque2.eps "fig:"){width="3cm"} ![Mask used to simulate a gap in the data *(top left)*, and the modified masks at scales 1 *(top right)* through 5 *(bottom left)*. The discarded pixels are in black. []{data-label="Masks"}](Masque3.eps "fig:"){width="3cm"} ![Mask used to simulate a gap in the data *(top left)*, and the modified masks at scales 1 *(top right)* through 5 *(bottom left)*. The discarded pixels are in black. []{data-label="Masks"}](Masque4.eps "fig:"){width="3cm"} ![Mask used to simulate a gap in the data *(top left)*, and the modified masks at scales 1 *(top right)* through 5 *(bottom left)*. The discarded pixels are in black. []{data-label="Masks"}](Masque5.eps "fig:"){width="3cm"}
Preliminary results
-------------------
Preliminary experiments were conducted in the case of vanishing instrumental noise variance, with a square $3 \times 3$ mixing matrix. It was mentioned before that in this limit, the spectral matching objective boils down to the joint diagonalization of covariance matrices. Further, taking the mixing matrix to be the identity matrix (*i.e.* try to separate sources which are not actually mixed ), it is possible to gain some insight on the spectral diversity of the independent sources, for a given choice of bands or filters. Indeed, the performance of the independent component separation methods based on spectral matching depend highly on spectral diversity.\
\
The following steps were repeated 1000 times:\
- [ randomly pick one of each component maps out of the available 200 CMB maps, 30 dust maps and 1500 SZ maps. ]{}
- [ calculate covariance matrices in the five wavelet or Fourier bands, both with and without masking part of the maps, as is all described above.]{}
- [ normalize each source so that its total energy over the five bands is equal to one.]{}
- [use the algorithm in [@Pha2] to jointly diagonalize the covariances in each configuration, and keep the resulting separating matrices.]{}\
If the sources have satisfactory spectral properties, the obtained separating matrices should not depart drastically from the identity matrix. Moreover, denoting $\mathcal{A}$ any invertible $3 \times 3$ mixing matrix, and $\widehat{\mathcal{A}}^{-1} $ the resulting separating matrix, it is shown in [@Car2] that the variances of the off-diagonal terms in $\widehat{ \mathcal{A}}^{-1} \mathcal{A} $ depend only on spectral diversity, in the case of Gaussian sources. In fact, to assess the effect of any non-Gaussianity or non-stationarity in the source templates, the same experiment was repeated on Gaussian maps generated with the same spectra as the CMB, Dust and SZ components. In any case, the independent source components are separated using: $$\widehat{S} = \widehat{ \mathcal{A}}^{-1} \mathcal{A} S = \mathcal{I}S
\label{SEP1}$$ so that with the above normalization, the square of any off-diagonal term $\mathcal{I}_{ij}$ is directly related to the residual level of component $j$ in the recovered component $i$.\
![Histograms of the off diagonal term corresponding to the residual corruption of “CMB” by “Dust” while separating Gaussian maps generated with the same power spectra as the astrophysical components, by joint diagonalization of covariance matrices in Fourier *(left)* and wavelet *(right)* space, with *(black, which appears grey when seen through white )* and without *(white)* masking part of the the data. The dark widest histogram on the left highlights the impact of masking on source separation based on Fourier covariances. []{data-label="histo_cmb"}](histogramme.eps){width="8cm"}
The histograms on figure \[histo\_cmb\] are for the off diagonal term corresponding to the residual corruption of CMB by *Gaussian* Dust in the second set of experiments. In tables \[RES2\] and \[RES3\], the results obtained with the synthetic component maps are given as well as those obtained with the Gaussian maps, in terms of the standard deviations of the off-diagonal entries $\mathcal{I}_{ij}$ defined by (\[SEP1\]).\
Interestingly, when working on Gaussian maps without masks, using covariances in Fourier space or in wavelet space gives similar performances. It is also satisfactory, when covariances in wavelet space are used with Gaussian maps, that each computed standard deviation only slightly increases when a mask is applied on the data. Indeed, as a consequence of incomplete coverage, there are less samples from which to estimate the covariances. This increase is also observed when covariances in Fourier space are used with the Gaussian maps but it can be as high as five-fold and it does not affect all coefficients the same way. Although this can again be attributed to the reduced data size, the lowered spectral diversity between components, because of the correlations and smoothing induced in Fourier space by the mask, is also part of the explanation. In fact, as shown on figure \[mixtures1\], CMB and dust spatial power spectra are somewhat similar, *i.e.* show low spectral diversity, and further smoothing can only degrade the performance of the source separation algorithm based on Fourier covariances.\
In the case of realistic component maps, we note first that the comparison of the performance of component separation using wavelet covariances with and without mask again agrees with the different data sizes, which is not the case with covariances in Fourier space. Next, whether covariances in Fourier or wavelet space are used, we note that the terms coupling CMB and Dust are again much higher in magnitude, even on complete maps. It seems that the actual non-stationarity and non-Gaussianity of the realistic component maps are relevant issues. Another point is that the CMB and Dust templates as in figure \[Templates\] exhibit sharp edges compared to SZ and this inevitably disturbs spectral estimation using a simple DFT. To assess this effect, simulations were also conducted where the covariances in Fourier space were computed after an apodizing Hanning window was applied on the complete data maps. The results reported in table \[RES2\], to be compared to table \[RES3\], do indicate a slightly positive effect of windowing, but still the separation using wavelet covariances appears better.\
\
Realistic experiments
---------------------
The above preliminary results clearly point out in the noiseless case the advantageous use of wavelets to easily escape the very bad impact that gaps and sharp edges actually have on the performance of the source separation using covariances in Fourier space. Hence this is strong encouragement to move on to investigating the effect of additive noise on the mixture maps according to (\[model1\]), using SMICA and its extension wSMICA. We note that although in the case of wSMICA the link with maximum likelihood is not as strongly asserted as with SMICA, the optimization algorithm used in the simulations hereafter consists in both cases of the same heuristic succession of EM and BFGS steps and initialization is done as discussed in paragraph \[OPTIM\].\
Picking at random one of each component maps out of the available 200 CMB maps, 30 dust maps and 1500 SZ maps, $1000$ synthetic mixture maps were generated as previously described, for each of the 5 noise levels chosen. Then, component separation was conducted using the spectral matching algorithms SMICA and wSMICA both with and without part of the maps being masked. Now, each run of SMICA and wSMICA on the data returns estimates $\widehat{A}_f$ and $\widehat{A}_w$ of the mixing matrix. Clearly, these estimates are subject to the indeterminacies inherent to the instantaneous linear mixture model (\[model1\]). Indeed, in the case where optimization is over all parameters $\theta$, it is obvious that any simultaneous permutation of the columns of $A$ and of the lines of $S$ leaves the model unchanged. The same occurs when exchanging a scalar possibly negative factor between any column in $A$ and the corresponding line in $S$. Therefore, columnwise comparison of $\widehat{A}_f$ and $\widehat{A}_w$ to the original mixing matrix $A$ requires first fixing these indeterminacies. This is done by hand after $\widehat{A}_f$ and $\widehat{A}_w$ have been normalized columnwise.\
![Comparison of the mean squared errors on the estimation of the emissivity of **CMB** as a function of noise in five different configurations namely : wSMICA without mask, wSMICA with mask, fSMICA without mask, fSMICA with mask, fSMICA with Hanning apodizing window.[]{data-label="cmb_emis"}](cmb_emis.eps){width="7cm"}
![Comparison of the mean squared errors on the estimation of the emissivity of **DUST** as a function of noise in five different configurations namely : wSMICA without mask, wSMICA with mask, fSMICA without mask, fSMICA with mask, fSMICA with Hanning apodizing window.[]{data-label="dust_emis"}](dust_emis.eps){width="7cm"}
![Comparison of the mean squared errors on the estimation of the emissivity of **SZ** as a function of noise in five different configurations namely : wSMICA without mask, wSMICA with mask, fSMICA without mask, fSMICA with mask, fSMICA with Hanning apodizing window.[]{data-label="sz_emis"}](sz_emis.eps){width="7cm"}
The results we report next concentrate on the statistical properties of $\widehat{A}_f$ and $\widehat{A}_w$ as estimated from the 1000 runs of the two competing methods in the several configurations retained. In fact, the correct estimation of the mixing matrix in model (\[model1\]) is a relevant issue for instance when it comes to dealing with the cross calibration of the different detectors. Figures \[cmb\_emis\], \[dust\_emis\] and \[sz\_emis\] show the results obtained, using the quadratic norm $$QE_j =\left( \sum_{i = 1}^m \left( A_{ij} - \widehat{A}_{ij} \right)^2 \right)^\frac{1}{2}$$ with $\widehat{A} = \widehat{A}_f\textrm{ or } \widehat{A}_w$ and $j = \textrm{CMB, DUST or SZ}$, to assess the residual errors on the estimated emissivities of each component. The plotted curves show how the mean of the above positive error measure varies with increasing noise variance. For the particular case of CMB, table \[RES4\] gives the estimated standard deviations of the relative errors $$\frac{A_{ij} - \widehat{A}_{ij} }{ A_{ij} }$$ on the estimated CMB emissivity in the six channels of Planck’s HFI in the different configurations retained.\
Closer to our source separation objective, a more significant way of assessing the quality of $\widehat{A}_f$ and $\widehat{A}_w$ as estimators of the mixing matrix $A$, would be to use the following signal to interference ratio: $$ISR_j = \frac{ \mathcal{I}_{j,j} ^2 \sigma_j^2 }{ \sum_{i \neq j} \mathcal{I}_{j,i} ^2 \sigma_i^2 }$$ where the $\sigma_j$ are the source variances and $$\mathcal{I} = (\widehat{A}\adj \widehat{R}_{N}^{-1} \widehat{A})\inv \widehat{A}\adj \widehat{R}_{N}^{-1} A$$ with $R_N$ the noise covariance. The plots on figures \[CMB\_res\], \[Dust\_res\] and \[SZ\_res\] show how the mean ISR from the 1000 runs of SMICA and wSMICA in different configurations, varies with increasing noise.
![Comparison of the mean ISR for **CMB** as a function of noise in five different configurations namely : wSMICA without mask, wSMICA with mask, fSMICA without mask, fSMICA with mask, fSMICA with Hanning apodizing window.[]{data-label="CMB_res"}](residuals_cmb.eps){width="7cm"}
![Comparison of the mean ISR for **DUST** as a function of noise in five different configurations namely : wSMICA without mask, wSMICA with mask, fSMICA without mask, fSMICA with mask, fSMICA with Hanning apodizing window.[]{data-label="Dust_res"}](residuals_dust.eps){width="7cm"}
![Comparison of the mean ISR for **SZ** as a function of noise in five different configurations namely : wSMICA without mask, wSMICA with mask, fSMICA without mask, fSMICA with mask, fSMICA with Hanning apodizing window.[]{data-label="SZ_res"}](residuals_sz.eps){width="7cm"}
We note again that the performance of wSMICA behaves as expected when noise increases and if part of the data is missing. However this is not always the case with SMICA. Finally this set of simulations, conducted in a more realistic setting with respect to ESA’s Planck mission, again confirms the higher performance, over Fourier analysis, that we indeed expected from the use of wavelets. The latter are able to correctly grab the spectral content of partly masked data maps and from there allow for better component separation.
Conclusion
==========
This paper has presented an extension of the Spectral Matching ICA algorithm to the case where the collected data is both correlated and non stationary, considering maps with gaps as a particular instance of practical significance. It was shown that simply substituting covariance matching in Fourier space by covariance matching in wavelet space enables to cope in the most general and straightforward way with gaps of possibly any shape. Mainly, it is the FIR nature of the wavelet filters used that allows the impact of edges and gaps on the estimated covariances and hence on component separation to be lowered. Optimally choosing the FIR filter-bank regarding a particular application is a possible further enhancement.\
Results obtained with simulated astrophysical data as expected from the Planck mission were given and these confirm the benefits of correctly processing existing gaps. Clearly, other possible types of non-stationarities in the collected data such as spatially varying noise or component variance, etc. can be dealt with very simply in a similar fashion using the wavelet extension of SMICA.\
In the CMB application, the mixed components have quite different statistical properties : some are expected to be very close to Gaussian whereas others are strongly non Gaussian. Standard ICA methods exploit the non Gaussianity of the mixed components. However, it is not clear yet how best to combine non Gaussianity and spectral diversity in order to perform better source separation. Other features of wavelets which are known to be powerful tools for the analysis and sparse representation of structured data might reveal useful here.
Appendix : EM algorithm with constraints on the mixing matrix {#annexe1}
=============================================================
Considering $Q$ separate frequency bands of size $n_q$ with $\sum n_q = 1$, the EM functional derived for the instantaneous mixing model (\[model1\]) with independent Gaussian stationary sources $S$ and noise $N$ is: $$\Phi(\underline \theta, \theta) = \mathcal{E} \left\{ \log p(X,S|\underline \theta) | \theta \right\}$$ with $\theta=(A, R_{S,1}, \ldots, R_{S,Q}, R_{N,1},\ldots , R_{N,Q}) $ and $\underline \theta=(\underline A, \underline R_{S,1} , \ldots, \underline
R_{S,Q}, \underline R_{N,1},\ldots ,\underline R_{N,Q}) $. The maximization step of the EM algorithm seeks then to maximize $\Phi(\underline \theta, \theta)$ with respect to $\underline \theta$ and the optimal $\underline \theta$ is used as the value for $\theta$ at the next EM step, and so on until satisfactory convergence is reached. Explicit expressions are easily derived for the optimal $\underline \theta$ in the white noise case where an interesting decoupling occurs between the re-estimating equations for noise variances, source variances and the mixing matrix [@Car].\
Linear equality constraints {#linear-equality-constraints .unnumbered}
----------------------------
When $A$ is subject to linear constraints, the joint maximization of the EM functional with respect to all model parameters is no longer easily achieved in general. In fact, one cannot simply decouple the re-estimating rules for the noise parameters and the mixing matrix and these have to be optimized separately. We give next the modified re-estimating equations for the mixing matrix and the source variances in the case of constant noise (*i.e.* $\theta=(A, R_{S,1}, \ldots, R_{S,Q})$ ).\
First, let us exhibit the quadratic dependence of the EM functional $\Phi(\underline\theta , \theta)$ on $\underline A$ : $$\begin{gathered}
\label{EM1}
\Phi(\underline\theta , \theta) = -\frac{1}{2} \sum_q n_q \trace \Big( \underline{A} R_q^{ss} \underline{A}\adj R_{N,q}\inv \\
-\underline{A} R_q^{xs\dagger} R_{N,q}\inv - R_q^{xs} \underline{A}\adj R_{N,q}\inv \Big) + const_{\underline A}\end{gathered}$$ where $$\begin{aligned}
C_q &=& (A\adj R_{N,q} \inv A + R_{S,q}\inv )\inv \\
W_q &=& (A\adj R_{N,q} \inv A + R_{S,q}\inv )\inv A\adj R_{N,q}\inv \\
R_q^{xs} &=& \widehat R_{X,q} W_q\adj \\
R_q^{ss} &=& W_q \widehat R_{X,q} W_q\adj + C_q \end{aligned}$$
**In the white noise case**, $R_{N,q}=R_N$, equation (\[EM1\]) becomes: $$\begin{gathered}
\Phi(\underline\theta , \theta) = -\frac{1}{2} \trace \Big( (\underline{A} - R^{xs} R^{ss-1} )R^{ss}\\
(\underline{A} - R^{xs}R^{ss-1} )^\dagger R_N\inv\Big) + const_{\underline A}\end{gathered}$$ where : $$R^{xs}=\sum_q n_q R_q^{xs} \quad \textrm{and} \quad R^{ss} = \sum_q n_q R_q^{ss}$$ Again, this can be re-written as : $$\Phi(\underline\theta , \theta) = -\frac{1}{2} ( \underline{\mathcal{A}} - \mathcal{M} )
\mathcal{Q}(\underline{\mathcal{A}} - \mathcal{M} )\adj + const_{\underline A}$$ where: $$\underline{\mathcal{A}} = \vect \underline{A} \quad \textrm{,} \quad \mathcal{Q} = \underline{R_N}\inv \otimes \sum_q n_q R_q^{ss}$$ $$\mathcal{M} = \vect \left(\left( \sum_q n_q R_q^{ys}\right) \left(\sum_q n_q R_q^{ss} \right)\inv \right)$$ With “$\vect$”, we build a column vector with the entries of a matrix taken along its lines. Now let us consider linear constraints on the mixing matrix, specified as follows : $$\mathcal{C}\adj( \underline{\mathcal{A}} -\mathcal{A}_0 ) =0$$ where $\mathcal{C}$ is a matrix with as many columns as constraints, and the columns of $\mathcal{C}$ are the same size as $\mathcal{A}$. The maximum of the EM functional with respect to $\underline \theta$ subject to the specified linear constraints is then reached for: $$\underline{\mathcal{A}} = \mathcal{M} - \mathcal{Q} \mathcal{C} \left( \mathcal{C}\adj \mathcal{Q} \mathcal{C} \right)\inv \mathcal{C}\adj( \mathcal{M} -\mathcal{A}_0 )$$ and $$\underline R_{S,q} = \textrm{diag} ( R_q^{ss})$$ where “$\textrm{diag}$” returns a matrix with the same diagonal entries as its input argument.\
**In the free noise case**, things are quite similar except that the noise covariance matrices $R_{N,q}$ do not factorize out as nicely. The EM functional is again expressed as : $$\Phi(\underline\theta , \theta) = -\frac{1}{2} ( \underline{\mathcal{A}} - \mathcal{M} )
\mathcal{Q}(\underline{\mathcal{A}} - \mathcal{M} )\adj + const_{\underline A}$$ where in this case: $$\mathcal{Q} = \sum_q n_q R_{N,q}\inv \otimes R_q^{ss}$$ and $$\mathcal{M} = \mathcal{Q} \inv \vect \left( \sum_q n_q R_{N,q}\inv R_q^{xs} \right)$$ Then, the maximum of the EM functional with respect to $\underline \theta$ subject to the specified linear constraints is again reached for: $$\underline{\mathcal{A}} = \mathcal{M} - \mathcal{Q} \mathcal{C} \left( \mathcal{C}\adj \mathcal{Q} \mathcal{C} \right)\inv \mathcal{C}\adj( \mathcal{M} -\mathcal{A}_0 )$$ and $$\underline R_{S,q} = \textrm{diag} ( R_q^{ss})$$
These expressions of the re-estimates of the mixing matrix can become algorithmically very simple when for instance the linear constraints to be dealt with affect separate lines of $A$, or even simpler when the constraints are such that the entries of $A$ are affected separately.
Positivity constraints on the entries of $A$ {#positivity-constraints-on-the-entries-of-a .unnumbered}
--------------------------------------------
Suppose a subset of entries of $A$ are constrained to be positive. The maximization step of the EM algorithm on $A$ alone, again has to be modified. We suggest dealing with such constraints in a combinatorial way rephrasing the problem in terms of equality constraints. If the unconstrained maximum of the EM functional is not in the specified domain, then one has to look for a maximum on the borders of that domain: on a hyperplane, on the intersection of two, or three, or more hyperplanes. One important point is that the maximum of the EM functional with respect to $A$ subject to a set of equality constraints will necessarily be lower than the maximum of the same functional considering any subset of these equality constraints. Hence, not all combinations need be explored, and a Branch and Bound type algorithm is well suited [@Nar]. A straightforward extension allows to deal with the case where a set of entries of the mixing matrix are constrained by upper and lower bounds.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In recent work, Renaud, Venaille, and Bouchet (RVB) [@RVB2016] revisit the equilibrium statistical mechanics theory of the shallow water equations, within a microcanonical approach, focusing on a more careful treatment of the energy partition between inertial gravity wave and eddy motions in the equilibrium state, and deriving joint probability distributions for the corresponding dynamical degrees of freedom. The authors derive a Liouville theorem that determines the underlying phase space statistical measure, but then, through some physical arguments, actually compute the equilibrium statistics using a measure that *violates* this theorem. Here, using a more convenient, but essentially equivalent, grand canonical approach, the full statistical theory consistent with the Liouville theorem is derived. The results reveal several significant differences from the previous results: (1) The microscale wave motions lead to a strongly fluctuating thermodynamics, including long-ranged correlations, in contrast to the mean-field-like behavior found by RVB. The final effective model is equivalent to that of an elastic membrane with a nonlinear wave-renormalized surface tension. (2) Even when a mean field approximation is made, a rather more complex joint probability distribution is revealed. Alternative physical arguments fully support the consistency of the results. Of course, the true fluid final steady state relies on dissipative processes not included in the shallow water equations, such as wave breaking and viscous effects, but it is argued that the current theory provides a more mathematically consistent starting point for future work aimed at assessing their impacts.'
author:
- 'Peter B. Weichman'
title: 'Competing turbulent cascades and eddy–wave interactions in shallow water equilibria'
---
Introduction {#sec:intro}
============
The modern era of exact statistical treatments of the late-time steady states of 2D fluid flows, properly accounting for the infinite number of conserved integrals of the motion, began with the Miller–Robert–Sommeria (MRS) theory of the 2D Euler equation [@M1990; @RS1991; @MWC1992; @LB1967], generalizing earlier approximate treatments going all the way back to the seminal work of Onsager [@O1949], and progressing through the Kraichnan Energy–Enstrophy theory [@K1975], and various formulations of the point vortex problem (see, e.g., [@MJ1974; @LP1977]). Since then, the theory has been applied to significantly more complex systems, containing multiple interacting fields (in contrast to the Euler equation, which reduces to a single scalar equation for the vorticity), but still possessing an infinite number of conserved integrals [@HMRW1985]. These include, for example, magnetohydrodynamic equilibria [@JT1997; @W2012], 3D axisymmetric flow [@TDB2013], and the shallow water equations [@WP2001; @CS2002], as well as numerous other geophysical applications [@BV2012].
The theory of the shallow water system was recently revisited in Ref. [@RVB2016] (hereinafter referred to as RVB). The work highlighted simplifying approximations made in previous work on this system [@WP2001; @CS2002], and aimed to move beyond them in order to generate more quantitative predictions. Previous simplifications mainly involved the problem of dissipation of microscale gravity wave fluctuations. Such physical effects are certainly physically present, in the form of nonlinear phenomena such as wave breaking or shock wave dissipation, but lie beyond the shallow water approximation (which, in particular, assumes the length scale of horizontal motions to be much larger than the fluid depth). In previous work the small scale free surface fluctuations were simply set to zero at a convenient point in the calculation (citing untreated dissipation processes), and mean field variational equations describing the remaining large scale eddy motion were then derived [@WP2001]. In RVB, the shallow water system, though idealized, is taken at face value, and an attempt is made to treat the wave fluctuations in a more consistent manner, but also within a mean field approximation. The result is a very interesting equilibrium state that includes both steady large scale eddy motions and finite microscale wave fluctuations.
The key underlying physics here, also motivating earlier studies, is that the two nonlinearly interacting fields, surface height and eddy vorticity, when viewed in isolation, have very different turbulent dynamics. Two-dimensional eddy systems governed by Navier–Stokes turbulence tend to self-organize into long-lived, large-scale coherent structures such as cyclones (exemplified by Jupiter’s Great Red Spot) and jets, a consequence of the famous 2D inverse energy cascade [@K1967; @KM1980]. However, weak turbulence theory [@ZFL1998] predicts that interacting acoustic waves, similar to 3D Navier–Stokes turbulence, possesses a forward cascade of energy, transporting it from larger to smaller scales where it is ultimately acted upon by viscosity or other microscopic dissipation mechanisms. When both motions are present, the question arises as to what the final disposition of the energy is. The RVB results propose a quantitative answer, predicting the equilibrium distribution of energy (and other quantities of interest) between the large-scale eddy and microscale wave motions, depending of course on all of the conserved integral values set, for example, by a flow initial condition.
The purpose of the present paper is to revisit deeper simplifying mathematical assumptions made in RVB that strongly impact the derived statistical equilibrium state. Two key features are highlighted. First, the variational mean field results are at odds with other recent results for systems with multiple interacting fields which are only partially constrained by conservation laws (in contrast, e.g., to the Euler equation, in which the vorticity field completely specifies the dynamics, while at the same time its fluctuations are strongly limited by the conservation laws). For example, for magnetohydrodynamic equilibria, the unconstrained degrees possess finite microscale fluctuations that lead to a non-mean field thermodynamic description of the large scale flow [@W2012]. An analogous result is derived here: the surface height fluctuations are not controlled by the vorticity conservation laws, and lead to a strongly fluctuating equilibrium thermodynamics. Physically, the microscale surface height fluctuations lead to a fluctuating effective Coulomb-like interaction between vortices that does not self-average even on large length scales. A mean field description emerges only in an approximation where this effect is ignored.
Second, the formalism of statistical mechanics relies on identification of the correct phase space measure used to compute the thermodynamic free energy and perform statistical averages. This measure is determined by a Liouville theorem that characterizes the geometry of phase space flows. In particular, when expressed in terms of the correct combination of fields, these flows are incompressible, and this constrains the phase space measure to be a function only of the conserved integrals of the motion (expressed in terms of these particular field combinations). An issue addressed in this paper is that the correct Liouville theorem is indeed derived in RVB, but is not actually implemented correctly to define the phase space measure. The authors recognize this, but propose various physical arguments why their chosen implementation, which simplifies the mathematics (in particular, it makes the fields statistically independent), also makes more physical sense.
If the motivation of the study was to follow the full consequences of the shallow water equations, prior to speculating on the effects neglected physics, there appears to be a basic inconsistency here. In the following, the full statistical theory is derived using the correct equilibrium phase space measure. The resulting theory leads to much more complex behavior, and indeed has some unusual physical consequences—for example, the microscale fluctuations lead to an equilibrium-averaged flow that does *not* satisfy the time-independent shallow water equations. Of course, which theory more closely reflects physical reality remains an interesting question, but the point of view taken here is that one should at least start by adhering as rigorously as possible to the mathematically consistent predictions of the model. Only following this should one attempt to insert physical considerations at various points to see what their affect might be. For example, a key consequence of the shallow water model is that the surface height fluctuations cascade to arbitrarily small wavelengths while at the same time maintaining a finite amplitude, thereby generating a kind of finite-thickness surface “foam”. It is the dynamics of this foam that leads to both the strongly fluctuating equilibrium and to the violation of the time-independent equations, and was suppressed at the outset in previous work [@WP2001; @CS2002]. These are consistent predictions of the model, but is obviously inconsistent with any physical final state, which must emerge by inserting a dissipation step to obtain a “true” equilibrium. How to best accomplish this lies beyond the scope of this paper, and would be an interesting topic for future work.
Outline {#sec:outline}
-------
The aim of this paper is to formulate general statistical models of shallow water equilibrium states, and then explore some of their key, high-level features. More detailed, physically motivated, investigations of model predictions are left for future work.
The outline of the remainder of this paper is as follows. In Sec. \[sec:bkgnd\] the shallow water equations are summarized, and the infinite number of conserved potential vorticity integrals are identified. In Sec. \[sec:canonfields\] all quantities of interest are expressed in terms of the basic vorticity (velocity curl), compressional (velocity divergence), and fluid height fields. The free slip boundary conditions play a key role here, especially in multiply connected domains where a set of circulation integrals about each connected component of the boundary is separately conserved. The latter lead to an additional set of “potential flow” contributions to the energy, and also to the expressions for the linear or angular momentum (in the case of translation or rotation invariant domains, respectively, where they are conserved). These have not been previously considered in the context of the shallow water system. In Sec. \[sec:statmech\] the equilibrium statistical mechanics formalism is introduced, with the conservation laws handled by introducing conjugate “chemical potentials” within the grand canonical approach. Application of a Kac–Hubbard–Stratanovich transformation allows one to exactly integrate out the fluid fields, and reduce the problem to that of a single effective field whose equilibrium average determines the large scale flow. The resulting statistical model is equivalent that of a fluctuating, scalar nonlinear elastic membrane problem [@W2012]. The model also has a dual description in terms of the vortex degrees of freedom interacting through a fluctuating Coulomb-like interaction. In Sec. \[sec:furtherprops\], we consider simplifying limits in which fluctuations are neglected. An approximate saddle point variational approach (analogous to, but quantitatively different from, that derived by RVB) is then used to illustrate further properties of the model. Equivalent forms of this theory are derived from both the elastic membrane and Coulomb models. The latter is closer in spirit to the RVB microcanonical approach. In Sec. \[sec:eulercomp\] an interesting order-of-limits paradox (infinite gravity $g$ vs. perfect rigid lid Euler equation boundary condition) is examined. The two limits produce very different forms of the Liouville theorem, and the paradox is resolved in terms of the finite contribution of microscale gravity waves to the free energy due to the simultaneous divergence of the wave speed $c \approx \sqrt{g h}$. The paper is concluded in Sec. \[sec:conclude\]. Two Appendices \[app:liouville\] and \[app:liouvilleinequiv\] prove a very general form of the Liouville theorem and review its relation to the statistical phase space integration measure. Some formal energy and momentum calculational details are relegated to App. \[app:KEPi\].
![Shallow water geometry and fields.[]{data-label="fig:swcartoon"}](ShallowWaterCartoon.png){width="3.0in"}
Background {#sec:bkgnd}
==========
The (2D) shallow water equations take the form [@foot:2Dcompress] $$\begin{aligned}
\partial_t {\bf v} + ({\bf v} \cdot \nabla) {\bf v}
+ f {\bf \hat z} \times {\bf v} &=& -g \nabla \eta
\nonumber \\
\partial_t h + \nabla \cdot (h {\bf v}) &=& 0
\label{2.1}\end{aligned}$$ where ${\bf v}$ is the (horizontal) velocity field, $h({\bf r})$ is the fluid layer thickness, $f({\bf r})$ is the Coriolis parameter, $h_b({\bf r})$ is the bottom height, and $$\eta({\bf r}) = h({\bf r}) + h_b({\bf r}) - H_0
\label{2.2}$$ is the surface height deviation from its average value $$H_0 = \int_D \frac{d{\bf r}}{A_D} h({\bf r})
\label{2.3}$$ (see Fig. \[fig:swcartoon\]). Here $A_D$ is the area of the domain $D$, and we normalize the average bottom height to vanish, $$\int_D \frac{d{\bf r}}{A_D} h_b({\bf r}) = 0.
\label{2.4}$$ The second equation in (\[2.1\]) expresses conservation of 3D fluid density through the mass current, or momentum (areal) density, $${\bf j} = \rho_0 h {\bf v}.
\label{2.5}$$ The (fixed, uniform) fluid 3D mass density $\rho_0$ is included here for convenience in order to maintain a consistent set of physical units ($\rho_0$ drops out of the equations of motion). One may simply set $\rho_0 = 1$ if one wishes.
Conservation laws {#sec:conslaws}
-----------------
### Potential vorticity {#subsec:potvort}
The potential vorticity, $$\Omega = \frac{\omega + f}{h},\ \ \omega = \nabla \times {\bf v}
\label{2.6}$$ which includes the combined effect of Earth and fluid rotation, is advectively conserved: $$\frac{D\Omega}{Dt} \equiv \partial_t \Omega
+ ({\bf v} \cdot \nabla) \Omega = 0.
\label{2.7}$$ It follows that, for any function $w(\Omega)$, $hw(\Omega)$ is a conserved density, $$\partial_t [h w(\Omega)] + \nabla \cdot [h w(\Omega) {\bf v}] = 0,
\label{2.8}$$ and hence that any integral of the form $$I_w = \int_D d{\bf r} h({\bf r}) w[\Omega({\bf r})]
\label{2.9}$$ is conserved, $\partial_t I_w = 0$. All such conservation laws may be conveniently summarized by the function $$g(\sigma) = \int_D d{\bf r} h({\bf r}) \delta[\sigma - \Omega({\bf r})],
\label{2.10}$$ which is then conserved for each value of $-\infty < \sigma < \infty$. One may recover any $I_w$ from $g(\sigma)$ in the form $$I_w = \int d\sigma g(\sigma) w(\sigma).
\label{2.11}$$
An important consequence of (\[2.9\]) is that, choosing $w(\Omega) = \Omega$, one obtains $$I_1 = \int_D d{\bf r} h\Omega = \int_D d{\bf r} (\omega+f)
= \int_{\partial D} {\bf v} \cdot d{\bf l} + \int_D d{\bf r} f.
\label{2.12}$$ It follows that the total circulation is conserved. In a multiply connected domain, it can be shown that the individual circulations $$\Gamma_l = \int_{\partial D_l} {\bf v} \cdot d{\bf l}
\label{2.13}$$ about any connected component $\partial D_l$, $l = 1,2,\ldots,N_D$, of the boundary are conserved as well. These generate an additional $N_D-1$ independent conserved integrals that are not expressible in terms of $g(\sigma)$. We adopt the sign convention here that $\partial D_1$ is the outermost boundary, so the circulation integral direction on all other $\partial D_l$, $l \geq 2$, is opposite. In particular, the total circulation appearing in (\[2.12\]) is given by $$\Gamma = \int_{\partial D} {\bf v} \cdot d{\bf l} = \Gamma_1 - \sum_{l=2}^{N_D} \Gamma_l.
\label{2.14}$$
![**Top:** Translation invariant domain relevant to linear momentum conservation. Periodic boundary conditions are applied along $x$. **Bottom:** Rotation invariant domain relevant to angular momentum conservation.[]{data-label="fig:symdomains"}](PeriodicStrip.png "fig:"){width="2.8in"} ![**Top:** Translation invariant domain relevant to linear momentum conservation. Periodic boundary conditions are applied along $x$. **Bottom:** Rotation invariant domain relevant to angular momentum conservation.[]{data-label="fig:symdomains"}](Annulus.PNG "fig:"){width="2.5in"}
### Energy and momentum {#subsec:energy}
The conserved energy is a sum of kinetic and potential contributions: $$E = \frac{\rho_0}{2} \int_D d{\bf r}
\left[h({\bf r}) |{\bf v}({\bf r})|^2
+ g \eta({\bf r})^2 \right].
\label{2.15}$$
The canonical linear momentum is given by $${\bf P} = \rho_0 \int_D d{\bf r} \, h[{\bf v} + {\bf A}],
\label{2.16}$$ where the vector potential is defined by $f = \nabla \times {\bf A}$ [@foot:vecpot]. If the system is translation invariant along a direction which we call ${\bf \hat x}$ (including the case of periodic boundary conditions along this direction, illustrated in the upper panel of Fig. \[fig:symdomains\]), the momentum component $P_x = {\bf P} \cdot {\bf \hat x} $ is conserved. More explicitly, if $f = f(y)$ and $h_b = h_b(y)$ depend only on the orthogonal coordinate $y$, one may choose ${\bf A} = -F(y) {\bf \hat x}$ where $\partial_y F = f$, and one obtains the conserved integral $$P_x = \rho_0 \int_D d{\bf r} h({\bf r}) [v_x({\bf r}) - F(y)].
\label{2.17}$$ Using (\[2.1\]), and judicious application of integration by parts and the boundary conditions, it is straightforward to verify directly that $\partial_t P_x = 0$.
The translation symmetry corresponds to the following Galilean transformation of the fields themselves: $$\begin{aligned}
{\bf \bar v}({\bf r},t) &=& {\bf v}({\bf r} - {\bf \hat x} v_0 t,t)
+ v_0 {\bf \hat x}
\nonumber \\
\bar h({\bf r},t) &=& h({\bf r} - {\bf \hat x} v_0 t,t)
\nonumber \\
\bar \omega({\bf r},t) &=& \omega({\bf r} - {\bf \hat x} v_0 t,t)
\nonumber \\
\bar h_b(y) &=& h_b(y) - \frac{v_0}{g} F(y),
\label{2.18}\end{aligned}$$ Thus, the same flow pattern boosted by an arbitrary velocity $v_0$ is a solution to (\[2.2\]) if one imposes an additional bottom tilt proportional to $F(y)$. In the magnetic analogy [@foot:vecpot], the latter corresponds to a “Hall voltage” that compensates for the change in Coriolis force induced by the change in the mean flow.
Similarly, in the presence of a rotational symmetry (circular or annular domain, illustrated in the lower panel of Fig. \[fig:symdomains\])), the canonical angular momentum $$L = \rho_0 \int_D d{\bf r} \, h {\bf r} \times ({\bf v} + {\bf A})
\label{2.19}$$ is conserved. Here, the 2D vector cross product produces the scalar quantity ${\bf r} \times {\bf j} = x j_y - y j_x$. In this case $f = f(r)$ and $h_b = h_b(r)$ depend only on the radial coordinate, and one may choose azimuthal ${\bf A} = \hat {\bm \theta} F(r)$, with $f = r^{-1} \partial_r(rF)$ to obtain the explicit form $$L = \rho_0 \int_D d{\bf r} h({\bf r}) [{\bf r} \times {\bf v}({\bf r}) + rF(r)].
\label{2.20}$$ It is again straightforward to verify directly that $\partial_t L = 0$.
The field symmetry corresponding to (\[2.20\]) is the rotational Galilean transformation $$\begin{aligned}
{\bf \bar v}({\bf r},t) &=& {\bf \hat R}_{\omega_0 t}
{\bf v}({\bf \hat R}_{-\omega_0 t}{\bf r},t)
+ \omega_0 r \hat {\bm \theta}
\nonumber \\
\bar h({\bf r},t) &=& h({\bf \hat R}_{-\omega_0 t}{\bf r},t)
\nonumber \\
\bar \omega({\bf r},t) &=& \omega({\bf \hat R}_{-\omega_0 t}{\bf r},t) + 2\omega_0
\nonumber \\
\bar h_b(r) &=& h_b(r) + \frac{\omega_0}{g} rF(r) - \frac{\omega_0^2}{2g} r^2
\nonumber \\
\bar f(r) &=& f(r) - 2\omega_0,
\label{2.21}\end{aligned}$$ where $\hat {\bm \theta} = {\bf \hat z} \times {\bf \hat r}$ is the azimuthal unit vector, and ${\bf \hat R}_\alpha {\bf r} = r [\cos(\alpha){\bf \hat r} + \sin(\alpha) \hat {\bm \theta}]$ applies the 2D rotation by angle $\alpha$. In this case, the transformation preserves the identical flow pattern, but it now undergoes a net rotation at arbitrary angular rate $\omega_0$, and is maintained by both a bottom tilt correction \[this time including also a centrifugal potential $\propto (\omega_0 r)^2$\] and a change in the Coriolis parameter itself.
Expressions in terms of canonical fields $\Omega,Q,h$ {#sec:canonfields}
=====================================================
Given the fundamental role of the conservation laws in the statistical mechanical treatment, it is useful to express quantities in terms of $\Omega$ and the compressional part of the velocity field $$Q = \frac{q}{h},\ \ q \equiv \nabla \cdot {\bf v}.
\label{3.1}$$ To this end, one decomposes ${\bf v}$ into rotational and compressional components: $${\bf v} = \nabla \times \psi - \nabla \phi,
\label{3.2}$$ where the 2D curl of a scalar is defined by $\nabla \times \psi = (\partial_y \psi, -\partial_x \psi)$. Both terms are chosen transverse to any free-slip boundary: ${\bf \hat l} \cdot \nabla \psi = 0$, ${\bf \hat n} \cdot \nabla \phi = 0$, where ${\bf \hat l}$ and ${\bf \hat n}$ are the boundary tangent and normal unit vectors, respectively. Both obey any periodic boundary condition that might present as well. Substituting the form (\[3.2\]) into (\[2.6\]) and (\[3.1\]), one obtains $$\left[\begin{array}{c}
\omega \\ q
\end{array} \right]
= \left[\begin{array}{c}
h\Omega - f \\ hQ
\end{array} \right]
= -\nabla^2
\left[\begin{array}{c}
\psi \\ \phi
\end{array} \right].
\label{3.3}$$
![**Top:** Simply connected domain. **Bottom:** Multiply connected domain. Independent circulation integrals $\Gamma_l$ and stream function values $\psi^0_l$ are associated with each internal boundary $\partial D_l$, and are related via (\[3.13\]) through the potential flow circulations (\[3.11\]).[]{data-label="fig:simmultdomains"}](SimplyConnected.PNG){width="2.5in"}
![**Top:** Simply connected domain. **Bottom:** Multiply connected domain. Independent circulation integrals $\Gamma_l$ and stream function values $\psi^0_l$ are associated with each internal boundary $\partial D_l$, and are related via (\[3.13\]) through the potential flow circulations (\[3.11\]).[]{data-label="fig:simmultdomains"}](MultiplyConnected.PNG){width="2.5in"}
Potential and non-potential flow decomposition {#sec:vortpot}
----------------------------------------------
The free-slip condition on $\psi$ implies that it is constant on each connected component of the boundary. For a simply connected domain (top panel of Fig. \[fig:simmultdomains\]), one may specify the Dirichlet condition $\psi|_{\partial D} \equiv 0$. However, for a multiply connected domain (bottom panel of Fig. \[fig:simmultdomains\]), one has the interesting consequence that the boundary value differences of $\psi$ can fluctuate. Multiply connected domains are emphasized here because they play a key role in the presence of conserved momenta. To account for these conservation laws in the statistical mechanics treatment, one must separate out the corresponding *potential flow* contributions.
To account for this dynamical degree of freedom we write $\psi$ in the form of a superposition: $$\psi({\bf r}) = \psi^V({\bf r}) + \psi^P({\bf r}),
\label{3.4}$$ in which the vortical component $\psi^V$ *vanishes* on every free slip boundary component, and contains all contributions to $\omega$, $$-\nabla^2 \psi^V = \omega
\label{3.5}$$ while the “potential flow” field $\psi^P$ matches the boundary values of $\psi$, $$\psi^P|_{\partial D} = \psi|_{\partial D},
\label{3.6}$$ while producing zero circulation and compression: $$-\nabla^2 \psi^P = 0 \ \ \Leftrightarrow \ \
\nabla \times {\bf v}^P
= 0 = \nabla \cdot {\bf v}^P,
\label{3.7}$$ where we define $${\bf v}^P = \nabla \times \psi^P,\ \
{\bf v}^V = \nabla \times \psi^V,\ \
{\bf v}^C = -\nabla \phi.
\label{3.8}$$ The orthogonality conditions $$\int_D d{\bf r} {\bf v}^I \cdot {\bf v}^J = 0,
\label{3.9}$$ for $I \neq J = P,V,C$ follows through integration by parts, and the use of the boundary conditions.
Since both $\psi^V$ and $\phi$ satisfy homogeneous boundary conditions, one obtains the inverse relations $$\begin{aligned}
\psi^V({\bf r}) &=& \int_D d{\bf r}' G_D({\bf r},{\bf r}') \omega({\bf r}')
\nonumber \\
\phi({\bf r}) &=& \int_D d{\bf r}' G_N({\bf r},{\bf r}') q({\bf r}'),
\label{3.10}\end{aligned}$$ in which $G_D$ and $G_N$ are, respectively, the Dirichlet and Neumann Green functions of the Laplacian for the domain $D$.
The aim in what follows is to show that both $\psi^V$ and $\psi^P$ are fully determined by $\omega$ and the conserved circulations (\[2.13\]). To this end, we further decompose $$\psi^P({\bf r}) = \psi^0_1 + \sum_{l=2}^{N_D} (\psi^0_l - \psi^0_1) \psi^P_l({\bf r}),
\label{3.11}$$ in which $\psi^0_l = \psi|_{\partial D_l}$ is the value of $\psi$ on connected boundary component $\partial D_l$, $l=1,2,\ldots,N_D$, and the “potential flow eigenfunctions” are independent solutions to the Laplace equation on $D$ obeying $$\psi^P_l({\bf r})|_{\partial D_m} = \delta_{lm},\
l=2,3,\ldots,N_D,
\label{3.12}$$ i.e., the boundary value is nonzero only on the matching boundary component. We define as well the symmetric, positive definite array of inner products $$\Gamma^P_{lm} = \int_D d{\bf r} {\bf v}_l^P \cdot {\bf v}_m^P
= \int_{\partial D_l} {\bf v}_m^P \cdot d{\bf l}
= \int_{\partial D_m} {\bf v}_l^P \cdot d{\bf l},
\label{3.13}$$ in which the boundary integrals follow by substituting ${\bf v}^P_l = \nabla \times \psi^P_l$, integrating by parts, and using (\[3.9\]).
The potential eigenfunctions may also be used to decompose the circulation integrals (\[2.13\]) into potential and vortex contributions (the contribution from the compressional component $\phi$ trivially vanishes). Through integration by parts, and recalling the sign convention (\[2.14\]), it is easy to check that $$\begin{aligned}
\int_D d{\bf r} \psi^P_l({\bf r}) \omega({\bf r})
&=& \int_{\partial D} \psi^P_l {\bf v} \cdot d{\bf l}
+ \int_D d{\bf r} {\bf v} \cdot {\bf v}^P_l
\nonumber \\
&=& -\Gamma_l + \sum_{m=2}^{N_D} \Gamma^P_{lm}
(\psi^0_m - \psi^0_1).\ \ \ \ \ \
\label{3.14}\end{aligned}$$ It follows that the conserved circulation integrals (\[2.13\]) may be decomposed in the form $$\begin{aligned}
\Gamma_l &=& \Gamma^V_l + \Gamma^P_l
\nonumber \\
\Gamma^V_l &=& \int_{\partial D_l} {\bf v}^V \cdot d{\bf l}
= -\int_D d{\bf r} \psi^P_l({\bf r}) \omega({\bf r})
\nonumber \\
\Gamma^P_l &=& \int_{\partial D_l} {\bf v}^P \cdot d{\bf l}
= \sum_{m=2}^{N_D} \Gamma^P_{lm} (\psi^0_m - \psi^0_1).
\label{3.15}\end{aligned}$$ This leads to the interpretation of $\psi^P({\bf r})$ as the circulation about boundary component $l$ due to a unit point vortex at ${\bf r}$. One obtains, in particular, $$\psi^0_l - \psi^0_1 = \sum_{l=2}^{N_D}
[\Gamma^P]^{-1}_{lm} (\Gamma_m - \Gamma^V_m),
\label{3.16}$$ demonstrating, as required, that the inhomogeneous boundary values, though fluctuating with the flow, are in fact fully specified by the vorticity field and the conserved integrals.
### Periodic strip geometry {#subsec:periodicstrip}
Relevant to systems with linear momentum conservation (\[2.17\]), the two connected boundary components are the lower and upper boundaries, $y_1 < y_2$, of the periodic strip of length $L_x$ (top panel of Fig. \[fig:symdomains\]). There is a single potential flow eigenfunction, representing uniform flow along the channel: $$\psi^P_2({\bf r}) = \frac{y-y_1}{L_y},\ \
{\bf v}^P_2 = \frac{1}{L_y} {\bf \hat x},
\label{3.17}$$ where $L_y = y_2 = y_1$. The circulation integral follows in the form $$\Gamma^P \equiv \Gamma^P_{22} = \frac{L_x}{L_y}.
\label{3.18}$$
Well known analytic series forms for the Green functions $G_N,G_D$ entering (\[3.10\]) may be derived using the method of images.
### Annular geometry {#subsec:annulus}
Relevant to systems with angular momentum conservation, for an annular geometry, with inner and outer radii $0 \leq R_2 < R_1$ (lower panel of Fig. \[fig:symdomains\]), the single eigenfunction corresponds to the axial flow $$\psi^P_2({\bf r}) = \frac{\ln(r/R_1)}{\ln(R_2/R_1)},\ \
{\bf v}^P_2 = \frac{1}{\ln(R_2/R_1) r} \hat {\bm \theta}.
\label{3.19}$$ The circulation integral takes the form $$\Gamma^P = \frac{2\pi}{\ln(R_2/R_1)}.
\label{3.20}$$
Once again, well known analytic series forms for the Green functions in (\[3.10\]) may be derived in polar coordinates.
Kinetic energy {#sec:ke}
--------------
The substitution of the decomposition ${\bf v} = {\bf v}^V + {\bf v}^C + {\bf v}^P$, along with the representations (\[3.8\]), (\[3.10\]) and (\[3.11\]), allows one to express into the kinetic part of the energy (\[2.15\]), $$E_K = \frac{\rho_0}{2} \int_D d{\bf r} h({\bf r}) |{\bf v}({\bf r})|^2,
\label{3.21}$$ as a nonlocal quadratic functional of $\omega,q,\psi_l^0$, including also $h$. Note that in the periodic strip or the annulus, there is only a single term in the sum (\[3.11\]), $l = m = N_D = 2$. Only $\Gamma^P_{22}$ enters (\[3.15\]), given by the explicit forms (\[3.18\]) or (\[3.20\]), respectively. Substituting $\omega = h\Omega - f$ and $q = hQ$, as well as (\[3.16\]), provides the explicit representation in terms of the basic fields $\Omega,Q,h$. The result is quite messy, including nonvanishing cross-terms, despite (\[3.9\]), due to presence of $h$. This expression is not actually needed in the analysis below, but for completeness is written out in App. \[app:KEPi\].
Conserved momenta {#sec:consmomenta}
-----------------
The kinetic parts of the linear and angular momenta, (\[2.17\]) and (\[2.20\]), may similarly be decomposed into vortical, compressional, and potential components. It is useful to write these in the form $$\begin{aligned}
\Pi &=& \Pi_K + \Pi_h
\nonumber \\
\Pi_K &=& \rho_0 \int_D d{\bf r} h({\bf r})
{\bf v}_\Pi({\bf r}) \cdot {\bf v}({\bf r})
\nonumber \\
\Pi_h &=& \rho_0 \int_D d{\bf r} h({\bf r}) F_\Pi({\bf r})
\label{3.22}\end{aligned}$$ where $$\begin{aligned}
{\bf v}_\Pi({\bf r}) &=& \left\{\begin{array}{ll}
\hat {\bf x}, & \Pi = P_x \\
r \hat {\bm \theta}, & \Pi = L
\end{array} \right.
\nonumber \\
F_\Pi({\bf r}) &=& \left\{\begin{array}{ll}
-F(y), & \Pi = P_x \\
r F(r), & \Pi = L.
\end{array} \right.
\label{3.23}\end{aligned}$$ Substituting the decomposition (\[3.8\]) of ${\bf v}$, $\Pi_K$ may be written out as a linear functional of $\omega,q,\psi^0_l$, depending also nonlocally on $h$. These expressions, given in App. \[app:KEPi\], will again not actually be needed below.
Example: flat-bottom Euler equation {#sec:eulereg}
-----------------------------------
The Euler equation on a flat bottom is obtained by setting $h = H_0$, $\eta = h_b = 0$, $\nabla \cdot {\bf v} = 0$, hence ${\bf v} = \nabla \times \psi$, and $\phi = 0$ (no compressional component). The potential flow eigenfunction expansion (\[3.11\]) remains exactly as before.
The vortex contribution to the stream function is still given by first line of (\[3.10\]), and the vortex contribution to the kinetic energy $E_K = E_K^V + E_K^P$ follows in the familiar Coulomb-like form $$E_K^V = \frac{1}{2} \rho_0 H_0 \int_D d{\bf r} \int_D d{\bf r}'
\omega({\bf r}) G_D({\bf r},{\bf r}') \omega({\bf r}').
\label{3.24}$$ Since $\Gamma_{lm}[h] = H_0 \Gamma_{lm}^P$, the potential flow contribution is given by $$E_K^P = \frac{1}{2} \rho_0 H_0 \sum_{l,m = 2}^{N_D}
\Gamma_{lm}^P (\psi^0_l - \psi^0_1)(\psi^0_m - \psi^0_1)
\label{3.25}$$ The cross term vanishes by orthogonality (\[3.9\]).
With linear momentum conservation on a periodic strip, one obtains from (\[2.16\]) the form $$P_x = \rho_0 (v_0 - v_f) V_D,
\label{3.26}$$ where $V_D = H_0 A_D$ is the system volume, $$v_0 = \frac{\psi^0_2 - \psi^0_1}{L_y}
\label{3.27}$$ is the (conserved) mean flow speed along the periodic dimension, $L_y = y_2-y_1$ is the strip width, and $$v_f = \frac{1}{L_y} \int_{y_1}^{y_2} F(y) dy
\label{3.28}$$ is a speed defined by the Coriolis effect. The momentum resides entirely in the potential component of the flow in this case, and the boundary values $\psi_{1,2}^0$ are both conserved. In particular, the value of $P_x$ fully specifies the boundary conditions and the energy in the potential flow. It fully specifies the potential contribution to the kinetic energy as well: $$E_K^P = \frac{1}{2} \rho_0 H_0 \Gamma^P (\psi^0_2 - \psi^0_1)^2
= \frac{1}{2} \rho_0 V_D v_0^2.
\label{3.29}$$
The circulation integral $\Gamma_2 = \Gamma_2^V + \Gamma^P (\psi^0_2 - \psi^0_1)$ follows directly from (\[3.15\]). Inserting (\[3.17\]), one sees that the vorticity contribution $$\Gamma^V_2 = - \frac{1}{L_y} \int_D d{\bf r} (y-y_1) \omega({\bf r})
\label{3.30}$$ is separately conserved, and also equivalent to momentum conservation.
For the annular geometry, one may express $$\begin{aligned}
\int_D d{\bf r} {\bf r} \times {\bf v}
&=& -\frac{1}{2} \int_D d{\bf r}
{\bf v}({\bf r}) \cdot \nabla \times (r^2-R_1^2)
\label{3.31} \\
&=& \frac{1}{2} (R_1^2 - R_2^2) \Gamma_2
- \frac{1}{2} \int_D d{\bf r} (r^2 - R_1^2) \omega.
\nonumber\end{aligned}$$ The angular momentum may therefore be written in the form $$\begin{aligned}
L &=& \rho_0 H_0 \left[L_2 + \frac{1}{2}(R_1^2 - R_2^2) \Gamma_2
+ F_2 \right]
\nonumber \\
L_2 &=& \frac{1}{2} \int_D d{\bf r} (R_1^2 - r^2) \omega({\bf r})
\nonumber \\
F_2 &=& 2\pi \int_{R_2}^{R_1} r^2 F(r) dr,
\label{3.32}\end{aligned}$$ which expresses it entirely in terms of the vorticity field and the conserved boundary circulations. Conservation of $L$ therefore produces the new conserved vorticity second moment $L_2$, analogous to the first moment (\[3.30\]).
The potential flow is equivalent to a point vortex at the origin, and one obtains $$E_K^P = \frac{\pi (\psi^0_2 - \psi^0_1)^2}{\rho_0 H_0 \ln(R_1/R_2)}.
\label{3.33}$$ Using (\[3.15\]), (\[3.19\]) and (\[3.20\]), the vorticity contribution to the circulation integral is $$\Gamma_2^V = \int_D d{\bf r} \frac{\ln(r/R_1)}{\ln(R_1/R_2)} \omega({\bf r})
\label{3.34}$$
Unlike for the linear momentum case, $\Gamma_2^V$, along with the boundary value $\psi_2^0 - \psi_1^0$, is not conserved, hence fluctuates with the flow. The reason for the difference is that in the linear momentum case ${\bf v}_{P_x}({\bf r}) = \hat {\bf x} = L_y {\bf v}^P_2({\bf r})$ happens to coincide with the potential eigenfunction, whereas ${\bf v}_L({\bf r}) = r \hat {\bm \theta}$ is distinct from ${\bf v}^P_2 \propto \hat {\bm \theta}/r$. In particular, the former has nonzero vorticity $\omega_L = 2$.
Fluid system statistical mechanics {#sec:statmech}
==================================
We seek a description of the equilibrium flows of the shallow water system, with conserved integrals defined by the energy (\[2.15\]), advection constraints (Casimirs) (\[2.10\]), the circulation integrals (\[2.13\]), and momentum (\[2.17\]) or (\[2.20\]), if present. The equilibrium phase space measure $d\nu(\Gamma) = \rho(\Gamma) d\Gamma$, and the Liouville theorem from which it follows, are described in detail in App. \[app:liouville\]. We work in the grand canonical ensemble with phase space probability density $$\rho = \frac{1}{Z} e^{-\beta {\cal K}},\
Z \equiv \int d\Gamma e^{-\beta {\cal K}}
\label{4.1}$$ and generalized Hamiltonian $${\cal K}[h,{\bf v}] = E - \alpha \Pi
- \sum_{l=2}^{N_D} \gamma_l \Gamma_l
- \int_D d{\bf r} h({\bf r}) \mu[\Omega({\bf r})].
\label{4.2}$$ The function $\mu(\sigma)$ is the Lagrange multiplier function conjugate to $g(\sigma)$, and $\Pi$ denotes the conserved momentum ($P_x$ or $L$), if present—see (\[3.22\]). The objective is to use this form to compute the free energy density $${\cal F}[\beta,\alpha,{\bm \gamma},\mu]
= -\frac{1}{\beta A_D} \ln(Z),
\label{4.3}$$ which characterizes the equilibrium state.
The phase space integral (\[4.1\]) is a formal infinite-dimensional functional integral over all possible fluid field configurations, weighted by the density $\rho(\Gamma)$. In order to perform computations, a finite-dimensional approximation is first constructed by discretizing the domain $D$ using a finite mesh (for simplicity, here taken as a uniform square mesh), replacing ${\bf r} \to {\bf r}_i$ by a discrete index $i$. To make physical sense, the continuum limit, taken at the end, must produce a finite, well defined form for ${\cal F}$, and this requirement will enforce nontrivial scaling of some parameters, especially the temperature $T = 1/\beta$.
Given the prominent role played by the potential vorticity, we use the statistical measure (\[A25\]), defined in terms of unrestricted integrals over each grid value of $(\Omega,Q,h)$, as well as the Dirichlet boundary values $\psi_l^0$. The partition function takes the form $$\begin{aligned}
Z &=& \prod_{l=2}^{N_D} \int d\psi^0_l \frac{\rho_0}{P_0}
\prod_i \int h_i^4 dh_i
\frac{\rho_0^2 \Delta x^2}{H_0 P_0^2}
\nonumber \\
&&\times\ \int dQ_i d\Omega_i
e^{-\beta {\cal K}[\Omega,Q,h,{\bm \psi}^0]}
\label{4.4}\end{aligned}$$ where $\Delta x \to 0$ is the mesh size, and, as discussed in App. \[app:liouville\], the constant factors $\rho_0/P_0$ and $\rho_0^2/H_0 P_0^2$ are introduced for convenience to make the partition function dimensionless ($P_0$ has dimensions of momentum or mass current density ${\bf j}$).
Form of generalized Hamiltonian {#sec:formgenham}
-------------------------------
In a symmetric domain, the $\alpha \Pi$ term is present, and some manipulations are required to put the combination $E - \alpha \Pi$ into a convenient form. By completing the square in various terms one obtains $$\begin{aligned}
F &\equiv& E - \alpha \Pi = F^v + F^h + F^0
\nonumber \\
F^v &=& \frac{1}{2} \int d{\bf r} h \left|{\bf v} - \alpha {\bf v}_\Pi \right|^2
\nonumber \\
F^h &=& \frac{1}{2} \rho_0 g \int_D d{\bf r} \bar \eta^2
\nonumber \\
F^0 &=& -\rho_0 g \int_D d{\bf r}
\left[(h_b-H_0) \delta h_b + \frac{1}{2} \delta h_b^2 \right]
\label{4.5}\end{aligned}$$ in which we define $$\begin{aligned}
\bar \eta &=& h + \bar h_b - H_0
\nonumber \\
\bar h_b &=& h_b + \delta h_b
\nonumber \\
\delta h_b &=& -\frac{\alpha}{g}
\left(F_\Pi + \frac{1}{2} \alpha |{\bf v}_\Pi|^2 \right).
\label{4.6}\end{aligned}$$ The term $F^0(\alpha)$ is constant, but does depend on the Lagrange multiplier $\alpha$. Note that only the full velocity ${\bf v}$ appears: the decompositions (\[3.2\]) and (\[3.8\]) will be exploited at a later step.
One may write the vorticity combination $$\begin{aligned}
\omega - \alpha \omega_\Pi &=& hQ - \bar f
\nonumber \\
\bar f({\bf r}) &=& f({\bf r}) + \alpha \omega_\Pi({\bf r})
\nonumber \\
&=& \left\{\begin{array}{ll}
f(y), & \Pi = P_x \\
f(r) + 2 \alpha, & \Pi = L.
\end{array} \right.
\label{4.7}\end{aligned}$$ The factor of $\omega_L = 2$ is obtained from (\[3.23\]). The transformations (\[4.6\]) and (\[4.7\]) correspond precisely to the symmetry transformations (\[2.18\]) and (\[2.21\]) with $v_0 = -\alpha$ and $\omega_0 = -\alpha$, respectively. In this way, the $\alpha \Pi$ term effectively identifies the frame of reference in which the translation or rotation velocity vanishes.
KHS transformation {#sec:khs}
------------------
In order to simplify the calculation, we perform a Kac–Hubbard–Stratanovich (KHS) transformation by introducing an auxilliary Laplace transform 2D current density field ${\bf J}$. Its equilibrium average will eventually be related to the large scale flow. This field is used to convert the kinetic energy term into a term linear in the velocity via the Gaussian identity $$e^{-\frac{1}{2} \beta \rho_0 \Delta x^2 h_i |{\bf V}_i|^2}
= \int_C d{\bf J}_i
\frac{e^{\beta \Delta x^2 |{\bf J}_i|^2/2 \rho_0 h_i}}
{2\pi \rho_0 h_i/\beta \Delta x^2}
e^{-\beta \Delta x^2 {\bf J}_i \cdot {\bf V}_i}
\label{4.8}$$ applied independently to each site $i$, and used with ${\bf V} = {\bf v} - \alpha {\bf v}_\Pi$. The subscript $C$ is a complex integration contour, for each component of ${\bf J}_i$, that runs parallel to the imaginary axis. Here and below, for any 2D vector, we adopt the notation $|{\bf J}|^2 = {\bf J} \cdot {\bf J}$, which does *not* include a complex magnitude.
The result (\[4.8\]) holds for arbitrary real axis intersection point, but saddle point and other considerations will determine a convenient choice below. This identity is sensible in the limit $\Delta x \to 0$ only if the combination $$\bar \beta = \beta \Delta x^2
\label{4.9}$$ remains finite. Thus, the fluid hydrodynamic temperature $T = 1/\beta = \Delta x^2 \bar T$ (in contrast to the physical thermodynamic temperature) must *vanish* in the continuum limit in order to obtain nontrivial macroscopic flows. The physical motivation for this scaling, which recognizes that large scale hydrodynamic flows cannot be in equilibrium with microscopic thermal fluctuations, has been discussed extensively in the literature, see, e.g., Ref. [@MWC1992]. The Gaussian integral also converges only if $\beta > 0$: as observed in [@RVB2016], the inclusion of height fluctuations precludes the negative temperature states observed for the Euler equation [@M1990; @RS1991; @MWC1992].
Inserting this identity for each $i$, one obtains $$\begin{aligned}
Z &=& \prod_{l=2}^{N_D} d\psi^0_l \frac{\rho_0}{P_0}
\prod_i \int h_i^3 dh_i
\frac{\bar \beta \rho_0 \Delta x^2}{2\pi H_0 P_0^2}
\nonumber \\
&&\times\ \int_C d{\bf J}_i \int dQ_i d\Omega_i
e^{-\beta \tilde {\cal F}[{\bf J},h,{\bf v}]},
\label{4.10}\end{aligned}$$ in which the free energy functional takes the continuum form $$\begin{aligned}
\tilde {\cal F}[{\bf J},h,{\bf v}] &=& \int_D d{\bf r}
\left\{{\bf J}({\bf r}) \cdot
[{\bf v}({\bf r}) - \alpha {\bf v}_\Pi({\bf r})]
- \frac{|{\bf J}({\bf r})|^2}{2 \rho_0 h({\bf r})} \right\}
\nonumber \\
&&+\ F^h + F^0 - \sum_{l=2}^{N_D} \gamma_l \Gamma_l
- \int_D d{\bf r} h({\bf r}) \mu[\Omega({\bf r})]
\nonumber \\
\label{4.11}\end{aligned}$$
We now reexpress the ${\bf J} \cdot {\bf V}$ term in terms of the canonical fields. Given that there is a component of ${\bf J}$ associated with each nonzero component of ${\bf v}$, it makes sense to enforce the free slip boundary condition ${\bf J} \cdot \hat {\bf n} = 0$ on ${\bf J}$ as well. It follows that one may apply the same decomposition (\[3.8\]) to obtain $${\bf J} = \nabla \times \Psi - \nabla \Phi,\ \ \Psi = \Psi^V + \Psi^P.
\label{4.12}$$ Substituting (\[3.8\]), (\[3.9\]), and (\[3.10\]), one obtains $$\begin{aligned}
&&\int_D d{\bf r} {\bf J} \cdot ({\bf v} - \alpha {\bf v}_\Pi)
= \int_D d{\bf r} [(h\Omega-\bar f) \Psi^V + h Q \Phi]
\nonumber \\
&&+\ \sum_{l=2}^{N_D} (\Psi_l^0 - \Psi_1^0)
\left[\sum_{m=2}^{N_D} \Gamma_{lm}^P (\psi_m^0 - \psi_1^0)
- \alpha \Gamma_{\Pi,l} \right] \ \ \ \ \ \
\label{4.13}\end{aligned}$$ in which the momentum circulations (defined only for the case $N_D = 2$) are obtained from (\[3.23\]) in the form $$\Gamma_{\Pi,2} = \int_D d{\bf r} {\bf v}_\Pi \cdot {\bf v}^P_2
= \left\{\begin{array}{ll}
L_x, & \Pi = P_x \\
\frac{2\pi (R_1^2 - R_2^2)}{\ln(R_2/R_1)}, & \Pi = L
\end{array} \right.
\label{4.14}$$ and one identifies the explicit forms $$\begin{aligned}
\Psi^V({\bf r}) &=& \int d{\bf r}' G_D({\bf r},{\bf r}')
\nabla' \times {\bf J}({\bf r}')
\nonumber \\
\Phi({\bf r}) &=& \int d{\bf r}' G_N({\bf r},{\bf r}')
\nabla' \cdot {\bf J}({\bf r}').
\label{4.15}\end{aligned}$$ The potential flow component is similarly decomposed in the form $$\Psi^P({\bf r}) = \Psi^0_1
+ \sum_{l=2}^{N_D} (\Psi^0_l - \Psi^0_1) \psi^P_l({\bf r}).
\label{4.16}$$
With these substitutions, and using the circulation representation (\[3.15\]), the free energy functional takes the explicit form
$$\begin{aligned}
\tilde {\cal F}[{\bf J},h,{\bf v}] &=& \int_D d{\bf r}
\left\{\frac{1}{2} \rho_0 g \bar \eta({\bf r})^2 - \bar f({\bf r})
\left[\Psi^V({\bf r}) + \Psi^{\bm \gamma}({\bf r}) \right]
- \frac{|{\bf J}({\bf r})|^2}{2 \rho_0 h({\bf r})}
\right\}
\nonumber \\
&&+\ \int_D d{\bf r} h({\bf r}) \left\{
\left[\Psi^V({\bf r}) + \Psi^{\bm \gamma}({\bf r}) \right] \Omega({\bf r})
+ \Phi({\bf r}) Q({\bf r}) - \mu[\Omega({\bf r})]\right\}
\nonumber \\
&&+\ \sum_{l,m=2}^{N_D} \Gamma_{lm}^P
(\Psi_l^0 - \Psi_1^0 - \gamma_l) (\psi_m^0 - \psi_1^0)
- \alpha \sum_{l=2}^{N_D} \Gamma_{\Pi,l} (\Psi_l^0 - \Psi_1^0 - \gamma_l)
+ \bar F^0(\alpha,{\bm \gamma}),
\label{4.17}\end{aligned}$$
where we define $$\begin{aligned}
\bar F^0(\alpha,{\bm \gamma}) &=& F^0(\alpha)
- \alpha \sum_{l=2}^{N_D} \gamma_l \Gamma^\Pi_l
\nonumber \\
\Gamma^\Pi_2 &\equiv& \int_{\partial D_l} {\bf v}_\Pi \cdot d{\bf l}
= \Gamma_{\Pi,2} - \int_D d{\bf r} \omega_\Pi \psi^P_2
\nonumber \\
&=& \left\{\begin{array}{ll}
L_x, & \Pi = P_x \\
2\pi R_2^2, & \Pi = L
\end{array} \right.
\nonumber \\
\Psi^{\bm \gamma}({\bf r}) &=& \sum_{l=2}^{N_D} \gamma_l \psi^P_l({\bf r}).
\label{4.18}\end{aligned}$$
The form (\[4.17\]) achieves the goal of being entirely local in $h,\Omega,Q$: for given ${\bf J}$, the statistical factor $e^{-\beta \tilde {\cal F}}$ can be expressed as an independent product over sites $i$, allowing the integration over $h_i,Q_i,\Omega_i,\psi^0_m$ to be carried out explicitly. To proceed, we note first that the only dependence on $Q$ is in the second line of (\[4.17\]). Choosing $\Phi_i$ to be pure imaginary, the former may be integrated out to produce a factor $$\prod_i 2\pi \delta(i \bar \beta h_i \Phi_i)
= \prod_i \frac{2\pi}{\bar \beta h_i} \delta(i\Phi_i).
\label{4.19}$$ Directly analogous to the change of variable from ${\bf v}$ in (\[A23\]) to $(\Omega,Q,{\bm \psi}^0)$ in (\[A24\]), one may change variables ${\bf J} \to (\Psi,\Phi,{\bm \Psi}^0)$, with constant Jacobian $\Delta x^{-2N_E}$: $$\prod_i \int_C d{\bf J}_i
= \prod_{l=2}^{N_D} \int_C d\Psi^0_l
\prod_i \int_C \frac{d\Phi_i d\Psi^V_i}{\Delta x^2}.
\label{4.20}$$ In each case, $C$ is again a contour parallel to the imaginary axis. The result of the $\Phi$ integral is therefore to simply set $$\Phi_i \equiv 0 \ \ \forall i
\label{4.21}$$ in $\tilde {\cal F}$ [@foot:Qint]. The factor $\prod_i (\bar \beta h_i)^{-1}$ produced by (\[4.19\]) encompasses the contribution to the free energy from the fluctuations in $Q$ that have now been fully integrated out.
Similarly, ${\bm \psi}^0$ appears only in the last term in (\[4.17\]). Choosing the ${\bm \Psi}^0$ contours so that $\Psi^0_l - \Psi^0_1 - \gamma_l$ are all pure imaginary, the ${\bm \psi}^0$ integrals produce the factor $$\frac{1}{\det(\Gamma^P)} \prod_{l=2}^{N_D}
\delta[i(\Psi^0_l - \Psi^0_1 - \gamma_l)].
\label{4.22}$$ The result of the ${\bm \Psi}^0$ integrals is therefore to simply replace $$\Psi^0_l - \Psi^0_1 = \gamma_l,\ l=2,3,\ldots,N_D.
\label{4.23}$$ Using (\[4.12\]), (\[4.16\]) and (\[4.21\]), one identifies $$\begin{aligned}
\Psi({\bf r}) &=& \Psi^V({\bf r}) + \Psi^{\bm \gamma}({\bf r})
\nonumber \\
|{\bf J}({\bf r})|^2 &=& |\nabla \times \Psi({\bf r})|^2
= |\nabla \Psi({\bf r})|^2
\label{4.24}\end{aligned}$$ The first line implies that the circulation Lagrange multipliers simply enforce the boundary conditions $\Psi|_{\partial D_l} = \gamma_l$. We reiterate, here and below, that $|\nabla \Psi|^2 = \nabla \Psi \cdot \nabla \Psi$ does *not* include a complex magnitude.
The end result of eliminating $Q,\Phi,{\bm \Psi}^0,{\bm \psi}^0$ is the partially reduced free energy functional $$\begin{aligned}
\hat {\cal F}[\Psi,h,\Omega] &=& \bar F^0 + \int_D d{\bf r}
\bigg\{\frac{1}{2} \rho_0 g \bar \eta({\bf r})^2
- \frac{|\nabla \Psi({\bf r})|^2}{2 \rho_0 h({\bf r})}
\nonumber \\
&&+\ [\omega({\bf r}) - \alpha \omega_\Pi({\bf r})] \Psi({\bf r})
- \mu[\Omega({\bf r})] \bigg \}.
\nonumber \\
\label{4.25}\end{aligned}$$
Final effective models {#sec:modelfinal}
----------------------
There are two ways to proceed in order to further reduce (\[4.25\]), each providing a rather different (but obviously equivalent) view of the underlying physics. The first is to integrate out $h,\Omega$ to obtain an effective theory in terms of the stream function $\Psi$ alone. This yields an effective nonlinear elastic membrane interpretation. The second is to integrate out $\Psi$ to obtain a dual effective theory in terms of $\Omega,h$. This yields the generalized Coulomb system interpretation. The latter, which is now completely independent of the KHS field ${\bf J}$, could also have been obtained by integrating out $Q,{\bm \psi}$ from ${\cal K}$ in (\[4.4\]). However, the intermediate KHS route actually provides the more transparent derivation. We derive both models in sequence.
### Nonlinear elastic membrane model {#subsec:nonlinmembrane}
In order to handle the $h_i,\Omega_i$ integrals, we define a function $W$ of three scalar arguments by $$\begin{aligned}
e^{\bar \beta W(\tau,h_0,\xi)}
&=& \frac{\rho_0}{P_0}
\int_0^\infty \lambda^2 d\lambda \int d\sigma
e^{\bar \beta \lambda[\mu(\sigma) - \sigma \tau]}
\nonumber \\
&&\ \ \ \ \ \ \times\
e^{-\frac{1}{2} \bar \beta
[\xi/\rho_0 \lambda + \rho_0 g(\lambda + h_0)^2]}.\ \ \ \ \
\label{4.26}\end{aligned}$$ The factor $\lambda^2$ originates from the factor $\bar \beta h_i^3$ in (\[4.10\]), divided the factor $\bar \beta h_i$ in (\[4.19\]). The remaining factor $\rho_0/P_0$ makes the result dimensionless. We observe here again that (1) this function makes sense only if $\bar \beta$ (not $\beta$) is finite, and (2) that the $\lambda$-integral converges only if $\bar \beta, \xi > 0$.
Combining (\[4.20\]), (\[4.23\]), and (\[4.24\]), the partition function may be put in the form $$Z = \prod_i \int_C \frac{d\Psi^V_i}{P_0 H_0}
e^{-\beta {\cal F}[{\bm \Psi}]}
\equiv \int D[\Psi^V] e^{-\beta {\cal F}[{\bm \Psi}]}
\label{4.27}$$ with fully reduced (continuum limit) free energy functional
$$\begin{aligned}
{\cal F}[\Psi] = \bar F^0(\alpha,{\bm \gamma})
- \int_D d{\bf r} \left\{\bar f({\bf r})\Psi({\bf r})
+ W\left[\Psi({\bf r}), \bar h_b({\bf r}) - H_0,\
-|\nabla \Psi({\bf r})|^2 \right] \right\}.
\label{4.28}\end{aligned}$$
The physical interpretation of this model is that of an inhomogeneous, nonlinear elastic fluctuating membrane (see Ref. [@W2012] for a similar analogy in the context of the theory of magnetohydrodynamic equilibria). In the limit $\beta = \bar \beta/\Delta x^2 \to \infty$, the dependence on $|\nabla \Psi({\bf r})|^2$ ensures that $\Psi$ is continuous, with $\delta \Psi = \Psi - \Psi^\mathrm{eq} = O(\Delta x/\sqrt{\bar \beta})$ differing only microscopically from its (smooth) equilibrium average $\Psi^\mathrm{eq}({\bf r}) = \langle \Psi({\bf r}) \rangle$ [@foot:C]. However, it follows that $|\nabla \Psi({\bf r})|^2 = O(1/\bar \beta)$ is a finite random variable, varying on the microscale $\Delta x$. For non-gradient terms inside ${\cal F}$, one is therefore free to replace $\Psi \to \Psi^\mathrm{eq}$ (whose form must eventually be determined self-consistently), and the first two arguments of $W$ may then be viewed as smooth, deterministic functions of ${\bf r}$. However, the third argument remains a fluctuating field, contributing nontrivially to the functional integral.
If $W(\xi)$ were a slowly varying function of its third argument, on the scale $\bar T = 1/\bar \beta$, then $W(\xi) \simeq W(\xi_0) + \partial_\xi W(\xi_0) (\xi - \xi_0)$, where $\xi_0 = -|\nabla \Psi^\mathrm{eq}|^2$, and the membrane becomes linear (though still inhomogeneous), with effective local surface tension defined by $\partial_\xi W(\xi_0)$. However, with increasing $\bar T$ the linear approximation fails, the surface tension depends on $\xi$ itself, and the model becomes intrinsically nonlinear. One may understand this effect from the point of view of the original shallow water system. With increasing $\bar T$ the microscopic height fluctuations, correlated with the current density fluctuations $\nabla \times \Psi$, increase to the point where the height field excursions become comparable to $H_0$, and one exits the regime of linear surface waves. In this sense, the behavior here is significantly more complex than that found in the magnetohydrodynamic problem, where terms equivalent to $|\nabla \Psi|^2$ always enter the free energy functional linearly [@W2012].
Since $|\nabla \Psi|^2$ varies by $O(1)$ on the lattice scale $\Delta x$, one might hope that it possesses only short range correlations. If this were true, one could independently integrate it out at each point ${\bf r}$ according to its single site-statistics, as we did the fields $\Omega,Q,h,\Phi$ in obtaining ${\cal F}[\Psi]$ from $\tilde {\cal F}[{\bf J},\Omega,Q,h]$. Unfortunately precisely the opposite is the case: the curl-free condition on $\nabla \Psi$, makes it highly correlated from site to site. For example, for the simplest, linear, homogenous model one obtains logarithmic correlations $\langle [\Psi({\bf r}) - \Psi({\bf r}')]^2 \rangle \sim \ln(|{\bf r}-{\bf r}'|/\Delta x)$. Correspondingly, one obtains macroscopic-scale dipole-like correlations of the current $\nabla \times \Psi$ [@W2012]. Thus, ${\cal F}$ generates a highly nontrivial, strongly correlated statistical model, with no simple analytic form for the free energy. In Sec. \[sec:furtherprops\] we will consider limits in which the fluctuations are small, and in which more explicit analytic progress can be made.
If, for convenience one separates [@foot:C]
$$\begin{aligned}
{\cal F}[\Psi] &=& {\cal F}[\Psi^\mathrm{eq}]
+ {\cal F}^\mathrm{fluct}[\Psi^\mathrm{eq},\delta \Psi]
\nonumber \\
{\cal F}[\Psi^\mathrm{eq},\delta \Psi]
&=& -\int_D d{\bf r} \left\{W\left[\Psi^\mathrm{eq},\bar h_b - H_0,
-|\nabla (\Psi^\mathrm{eq} + \delta \Psi)|^2 \right]
- W\left[\Psi^\mathrm{eq},\bar h_b - H_0,
-|\nabla \Psi^\mathrm{eq}|^2 \right] \right\}
\label{4.29}\end{aligned}$$
into static and fluctuating parts, then the equilibrium free energy takes the form $$\begin{aligned}
F[\Psi^\mathrm{eq}] &=& {\cal F}[\Psi^\mathrm{eq}]
+ F^\mathrm{fluct}[\Psi^\mathrm{eq}]
\label{4.30}\\
F^\mathrm{fluct}[\Psi^\mathrm{eq}] &=& -\frac{1}{\beta}
\ln\left\{\int D[\delta \Psi] e^{-\beta {\cal F}^\mathrm{fluct}
[\Psi^\mathrm{eq},\delta \Psi]} \right\},
\nonumber\end{aligned}$$ which explicitly exposes the “mean field” and fluctuating components. The self-consistent equation for the large-scale equilibrium flow follows by minimizing $F$: $$\frac{\delta F}{\delta \Psi^\mathrm{eq}({\bf r})} = 0.
\label{4.31}$$
The functional derivative (\[4.31\]) may be conveniently evaluated by first defining an intermediate average over the fields $\Omega,h$ using the functional $W$: $$\begin{aligned}
n_{\Omega,h}({\bf r},\sigma,\lambda)
&\equiv& \langle \delta[\Omega({\bf r}) - \sigma]
\delta[h({\bf r}) - \lambda] \rangle_W
\nonumber \\
&=& \frac{\frac{\rho_0}{P_0} \lambda^2 e^{\bar \beta \lambda
\left[\mu(\sigma) - \sigma \Psi({\bf r}) \right]}
e^{\frac{1}{2} \bar \beta
\left\{|\nabla \Psi({\bf r})|^2/\rho_0\lambda
- \rho_0 g[\lambda + \bar h_b({\bf r}) - H_0]^2 \right\}}}
{e^{\bar \beta W\left[\Psi({\bf r}), \, \bar h_b({\bf r})-H_0, \,
-|\nabla \Psi({\bf r})|^2 \right]}},
\label{4.32}\end{aligned}$$ which may be interpreted as the probability density for potential vorticity and fluid height at the point ${\bf r}$, for a given fixed realization of the field $\Psi$. The $e^{\bar \beta W}$ denominator ensures that the distribution is normalized. This interpretation is most easily derived by following the identical sequence of integration steps to obtain the results (\[4.21\]) and (\[4.23\]), but in the integration over the $h$ and $\Omega$ fields (in advance of the $\Psi$ integration), the delta functions then produce (\[4.32\]) in place of free integration result (\[4.26\]). With this definition one obtains $$\begin{aligned}
-\partial_\tau W
&=& \int_0^\infty \lambda d\lambda
\int d\sigma \sigma n_{\Omega,h}({\bf r},\sigma,\lambda)
= \langle h({\bf r}) \Omega({\bf r}) \rangle_W
= \langle \omega({\bf r}) \rangle_W + f({\bf r})
\nonumber \\
2\rho_0 \partial_\xi W
&=& \int_0^\infty \frac{d\lambda}{\lambda}
\int d\sigma n_{\Omega,h}({\bf r},\sigma,\lambda)
= \left\langle \frac{1}{h({\bf r})} \right\rangle_W
\nonumber \\
-\partial_{h_0} W &=& \rho_0 g \int_0^\infty d\lambda
(\lambda + \bar h_b - H_0)
\int d\sigma n_{\Omega,h}({\bf r},\sigma,\lambda)
= \rho_0 g \langle \bar \eta({\bf r}) \rangle_W,
\label{4.33}\end{aligned}$$
and one may express (\[4.31\]) in the form $$\begin{aligned}
\nabla \times {\bf V}^\mathrm{eq}
&=& \langle h({\bf r}) \Omega({\bf r}) \rangle + f({\bf r})
= \langle \omega({\bf r}) \rangle
\nonumber \\
{\bf V}^\mathrm{eq} &\equiv& \left\langle
\frac{\nabla \times \Psi({\bf r})}{\rho_0 h({\bf r})} \right\rangle
+ \alpha {\bf v}_\Pi({\bf r}),
\label{4.34}\end{aligned}$$ in which the averages now include $\Psi$: $\langle \cdot \rangle \equiv \langle \langle \cdot \rangle_W \rangle_{\cal F}$. In the presence of momentum conservation, ${\bf V}^\mathrm{eq}$ is the instantaneous mean flow velocity seen in the laboratory frame, while ${\bf J} = \rho_0 \langle h ({\bf v} - \alpha {\bf v}_\Pi) \rangle$ is current density in the translating or rotating frame of reference (hence, generated by the net vorticity $\Delta \omega^\mathrm{eq} = \langle \omega \rangle - \alpha \omega_\Pi$). In the latter frame, the equilibrium flow is time-independent, obtained from the transformation (\[2.18\]) or (\[2.21\]), with $v_0 = -\alpha$ or $\omega_0 = -\alpha$, respectively. The incompressibility condition on ${\bf J}^\mathrm{eq}$ still allows, in general, a nonzero compressible velocity field component $\langle q \rangle = \nabla \cdot {\bf V}^\mathrm{eq}$.
The equilibrium form $\Psi$ depends on the Lagrange multipliers $\beta,\mu,\alpha,{\bm \gamma}$, which must then be tuned to obtain prescribed values of the conserved integrals. The latter may be derived as equilibrium averages in the form
$$\begin{aligned}
g(\sigma) &=& - \frac{\delta {\cal F}}{\delta \mu(\sigma)}
= \langle h({\bf r}) \delta[\Omega({\bf r}) - \sigma] \rangle
= \int_D d{\bf r} \int_0^\infty \lambda d\lambda \,
\langle n_{\Omega,h}({\bf r},\sigma,\lambda) \rangle_{\cal F}
\nonumber \\
\Gamma_l &=& -\frac{\partial {\cal F}}{\partial \gamma_l}
= -\frac{\partial \bar F^0}{\partial \gamma_l}
+ 2 \int_{\partial D_l} \langle \partial_\xi W
(\nabla \times \Psi) \rangle \cdot d{\bf l}
= \int_{\partial D_l} \left\langle
\frac{{\bf J} + \rho_0 h \alpha {\bf v}_\Pi}{\rho_0 h}
\right \rangle \cdot d{\bf l}
\nonumber \\
&=& \int_{\partial D_l} {\bf V} \cdot d{\bf l}
= \sum_{m=2}^{N_D} \Gamma^P_{lm}
\langle \psi^0_l - \psi^0_1 \rangle
- \int_D d{\bf r} \psi^P_l({\bf r})
\langle \omega({\bf r}) \rangle
\nonumber \\
\Pi &=& -\frac{\partial {\cal F}}{\partial \alpha}
= -\frac{\partial \bar F^0}{\partial \alpha}
+ \int_D d{\bf r} \left\{\omega_\Pi \langle \Psi \rangle
- \frac{1}{g} \langle \partial_{h_0}W \rangle
[F_\Pi + \alpha |{\bf v}_\Pi|^2] \right\}
\nonumber \\
&=& \int_D d{\bf r} \left({\bf v}_\Pi \cdot
\langle {\bf J} + \rho_0 h \alpha {\bf v}_\Pi \rangle
+ \rho_0 \langle h \rangle F_\Pi \right)
= \rho_0 \int_D d{\bf r} \langle h({\bf r}) \rangle
\left[{\bf v}_\Pi({\bf r}) \cdot \tilde {\bf V}({\bf r})
+ F_\Pi({\bf r}) \right]
\nonumber \\
E &=& \left[\frac{\partial (\bar \beta {\cal F})}
{\partial \bar \beta}\right]_{\bar \beta \alpha,
\bar \beta \mu, \bar \beta {\bm \gamma}}
= \frac{1}{2} \int_D d{\bf r} \left\{
\left\langle \frac{|{\bf J}
+ \rho_0 h \alpha {\bf v}_\Pi|^2}{\rho_0 h} \right\rangle
+ \rho_0 g \langle \eta^2 \rangle \right\},
\label{4.35}\end{aligned}$$
which correspond to averages of (\[2.10\]), (\[3.15\]), (\[3.22\]), and (\[2.15\]). In the last expression for $\Pi$, we define a somewhat different measure of the mean velocity field $\tilde {\bf V}$ by $$\tilde {\bf V}({\bf r})
= \frac{\langle {\bf J} + \rho_0 h \alpha {\bf v}_\Pi \rangle}
{\rho_0 \langle h \rangle}.
\label{4.36}$$ Clearly, $\tilde {\bf V}$ and ${\bf V}$ become equivalent if the fluctuations in $h$ are small. In the computation of $\Gamma_l$, only the surface term survives in the first line by virtue of (\[4.34\]). The last expression for $\Gamma_l$ uses (\[3.15\]) to alternatively express the average potential flow circulation $\langle \Gamma^P_l \rangle = \int_D d{\bf r} {\bf v}^P_l \cdot {\bf V}$ in terms of an average of the boundary values.
### Generalized Coulomb model {#subsec:coulomb}
Alternatively, one may integrate out $\Psi$ from (\[4.25\]). For fixed height field $h$ the integral is Gaussian, and one obtains
$$\begin{aligned}
Z &=& \prod_i \int_0^\infty h_i^2 dh_i \frac{\rho_0}{P_0 H_0^{1/2}}
\int d\Omega_i \sqrt{\det(G_h)} e^{-\beta \hat K[\Omega,h]}
\nonumber \\
\hat {\cal K}[\Omega,h] &=& \frac{1}{2} \int_D d{\bf r} \int_D d{\bf r}'
[\omega({\bf r}) - \alpha \omega_\Pi({\bf r})] G_h({\bf r},{\bf r}')
[\omega({\bf r}) - \alpha \omega_\Pi({\bf r})]
+ \int_D d{\bf r} \left\{\frac{1}{2} \rho_0 g_0 \bar \eta({\bf r})^2
- h({\bf r}) \mu[\Omega({\bf r})] \right\}
\nonumber \\
&&-\ \sum_{l=2}^{N_D} \gamma_l \int_{\partial D_l}
\tilde {\bf v} \cdot d{\bf l}
+ \frac{A_D}{2 \bar \beta}
\ln\left(\frac{2\pi P_0^2}{\rho_0 H_0} \bar \beta \right) + F^0(\alpha),
\label{4.37}\end{aligned}$$
in which, as before, $\omega = h\Omega - f$, and the Green function $G_h$ is defined by $$-\nabla \cdot \frac{1}{\rho_0 h({\bf r})}
\nabla G_h({\bf r},{\bf r}') = \delta({\bf r}-{\bf r}'),
\label{4.38}$$ with Dirichlet boundary conditions on $\partial_D$. The form (\[4.37\]) differs significantly from the form of the tensor Green function (\[C3\]) before $Q$ is integrated out (compare, especially, the $\omega$-$\omega$ block of ${\cal \hat G}_h$). The $\det(G_h) \approx \prod_i h_i$ term comes from the normalization of the Gaussian integral and adjusts the phase space measure defining $\int D[h]$. The $\ln(\bar \beta)$ term generates the equipartition contribution $1/2\bar \beta = \bar T/2$ to the energy density coming the fluctuating $Q$-field that has been integrated out. The quantity $$\tilde \Psi({\bf r}) = \int_D d{\bf r}' G_h({\bf r},{\bf r}')
[\omega({\bf r}') - \alpha \omega_\Pi({\bf r}')]
\label{4.39}$$ obeys $\nabla \times (h^{-1} \nabla \times \tilde \Psi) = \omega - \alpha \omega_\Pi$, and $$\tilde {\bf j}({\bf r}) \equiv
h({\bf r})[\tilde {\bf v}({\bf r})
- \alpha {\bf v}_\Pi({\bf r})]
= \nabla \times \tilde \Psi({\bf r})
\label{4.40}$$ therefore represents the divergence free component of the current density. This also defines the quantity $\tilde {\bf v}$ appearing in the circulation term in (\[4.37\]).
For constant $h \equiv H_0$, $G_h$ becomes the Dirichlet Coulomb potential, and the corresponding term in the free energy coincides with that for the Euler equation. The smoothness of this potential, together with the constrained fluctuations in $\omega$, produce an energy that is completely dominated by the large scale flow [@M1990; @RS1991; @MWC1992]. The energy may therefore be obtained by substituting $\langle \omega \rangle$ for $\omega$, and this in turn produces an exact variational form for the free energy. On the other hand, the presence of $1/h$ here, with $O(1)$ fluctuations on the scale $\Delta x$, produces a finite microscale fluctuation energy contribution: $G_h$ (as well as $\tilde \Psi$) is continuous, but its gradient fluctuates on scale $\Delta x$, and is highly correlated with $h$. It follows that one *cannot* simply substitute $\langle \omega \rangle$ for $\omega$ and $\langle G_h \rangle$ for $G_h$. The reasons for this failure are equivalent to the correlated site-to-site fluctuations of $|\nabla \Psi|^2$ found in the membrane formulation (\[4.27\]). One concludes again that the model free energy does not reduce to a variational mean field form.
Simplifying limits and further properties of the models {#sec:furtherprops}
=======================================================
In this section we consider simplifying limits in which more explicit computations can be carried out, and used these to explore further properties of the models. The critical assumption will be that the fluctuations are small, so that $\Psi \simeq \Psi^\mathrm{eq}$ may be treated as a fixed, nonfluctuating field.
Variational limit {#sec:varlimit}
-----------------
The variational or mean field limit is defined by neglecting $F^\mathrm{fluct}$. In particular one sets $\Psi = \Psi^\mathrm{eq}$ inside the functional $W$ (in both the first and last arguments), and in the distribution (\[4.32\]) as well. The condition (\[4.31\]) applied to ${\cal F}[\Psi]$ then leads to the Euler–Lagrange equations $$2 \nabla \cdot (\partial_\xi W \, \nabla \Psi)
= \partial_\tau W + \bar f.
\label{5.1}$$ It is important here that the variation is with respect to $\Psi^V$, which ensures that there are no boundary terms. Using (\[4.33\]), equation (\[4.34\]) reduces to $$\begin{aligned}
\nabla \times {\bf V}^\mathrm{eq} &=& \langle \omega({\bf r}) \rangle_W
\nonumber \\
{\bf V}^\mathrm{eq} &=& \left \langle \frac{1}{\rho_0 h({\bf r})}
\right \rangle_W \nabla \times \Psi^\mathrm{eq} + \alpha {\bf v}_\Pi.
\label{5.2}\end{aligned}$$ Equation (\[5.2\]) is the basic result of this section. Its solution allows one to derive the large scale mean flow encoded in $\Psi^\mathrm{eq}$ in the presence of the microscopic height and compressional fluctuations encoded in $n_{\Omega,h}({\bf r},\sigma,\lambda)$. The result therefore represents a mean field self-consistency condition, in the form of a highly nonlinear PDE, whose solution $\Psi^\mathrm{eq}({\bf r})$ also fully determines $n_{\Omega,h}$.
The solution $\Psi^\mathrm{eq}$ again depends on the Lagrange multipliers $\beta,\mu,\alpha,{\bm \gamma}$, which must be tuned to obtain prescribed values of the conserved integrals. The latter are given by (\[4.35\]), but with all averages now with respect to $W$ at fixed $\Psi = \Psi^\mathrm{eq}$. There is one subtlety here, however. The conserved energy takes the form $$\begin{aligned}
E &=& \frac{1}{2} \int_D d{\bf r} \left\{\left\langle
\frac{|\nabla \times \Psi^\mathrm{eq}
+ \rho_0 h \alpha {\bf v}_\Pi|^2}{\rho_0 h} \right\rangle_W \right.
\nonumber \\
&&\hskip 0.75in
\left. +\ \rho_0 g \langle \eta^2 \rangle_W
+ \frac{1}{\bar \beta} \right\}
\nonumber \\
&=& \frac{1}{2} \int_D d{\bf r} \left\{
\left\langle \frac{1}{\rho_0 h}
\right\rangle_W |\nabla \times \Psi^\mathrm{eq}|^2
+ 2 \alpha {\bf v}_\Pi \cdot \nabla \times \Psi^\mathrm{eq} \right.
\nonumber \\
&&+\ \left. \rho_0 \langle h \rangle_W \alpha^2 |{\bf v}_\Pi|^2
+ \rho_0 g \langle \eta^2 \rangle_W
+ \frac{1}{\bar \beta} \right\}.
\label{5.3}\end{aligned}$$ The $\bar T = 1/\bar \beta$ constant term in the energy is the equipartition energy due to the quadratic fluctuations of $\Psi$ about the equilibrium value, and is produced by the Gaussian integral about saddle point in the steepest descent calculation. This term remains finite even when fluctuations are small, and represents precisely the contribution of the compressional degree of freedom $Q$ that gave rise to the $\ln(\bar \beta)$ term in (\[4.37\]).
Variational equations derived from the generalized Coulomb representation {#sec:vareqcoulomb}
-------------------------------------------------------------------------
Variational equations equivalent to (\[5.1\]) and (\[5.2\]) can also be derived from the generalized Coulomb representation (\[4.37\]). Since the latter is expressed entirely in terms of the original $h,\Omega$ fields, the derivation is much closer in spirit to the microcanonical approach used by RVB. Central to this approach is the local distribution function $n_{\Omega,h}({\bf r},\sigma,\lambda) = \langle \delta[\Omega({\bf r}) - \sigma] \delta[h({\bf r}) - \lambda] \rangle$ characterizing the local microscopic vorticity and height fluctuations. The grand canonical form is given in (\[4.32\]), and the corresponding microcanonical form will be rederived here by a different route. One could in principle consider as a starting point a more fundamental three-field correlation function that includes $Q$ (see Sec. \[sec:Qdistr\] below), and attempt to work directly with the original generalized Hamiltonian ${\cal K}$ defined in (\[4.2\]). However, the divergent fluctuations in $Q \sim 1/\Delta x$ lead to the failure of the key self-averaging property used below, and hence make ${\cal K}$ a less convenient starting point. We work then with the representation (\[4.37\]) in which $Q$ has already been integrated out.
The derivation proceeds by considering, in addition to the microscale $\Delta x$, a mesoscale $\Delta X$, both vanishing in the continuum limit, but with $\Delta X/\Delta x \to \infty$. On the scale $\Delta X$, one may define the joint probability density whose limiting form is obtained by counting the number of joint occurrences of the field levels across the $\Delta x$-cells in the given $\Delta X$-cell centered on point ${\bf r}$: $$n_{\Omega,h}({\bf r},\sigma,\lambda)
= \lim_{\Delta V_g \to 0} \lim_{\Delta x \to 0}
\frac{\Delta x^2}{\Delta X^2}
\frac{\nu_{ik}}{\Delta V_g},
\label{5.4}$$ where $i$ labels $\Delta X$-cell centers ${\bf r}_i$, $\{\sigma_k, \lambda_k \}_{k=1}^{N_g}$ is a 2D gridding of $(\Omega, h)$-space, with 2D cell volume $\Delta V_g = \Delta \Omega \Delta h$, and $\nu_{ik}$ \[normalized so that $\sum_k \nu_{ik} = (\Delta X/\Delta x)^2$\] counts the number of $\Delta x$-cells in $\Delta X$-cell $i$ (in the $\Delta X^2$ neighborhood of the point ${\bf r}$) with parameter value $\sigma_k,\lambda_k$ (in the $\Delta V_g$ neighborhood of $\sigma,\lambda$). The form (\[5.4\]) ensures the normalization $$\int d\sigma \int_0^\infty d\lambda \,
n_{\Omega,h}({\bf r},\sigma,\lambda) = 1
\label{5.5}$$ and the conserved quantities are expressed in the same form (\[4.35\]).
The phase space integral is now performed using a separation of scales: First one assigns field values for *fixed* $n_{\Omega,h}$, then one integrates over all possible $n_{\Omega,h}$. The former includes all permutations of $\Delta x$-cells within a given $\Delta X$-cell (which clearly leaves $n_{\Omega,h}$ fixed, as well as all Casimirs). This sum, via the usual permutation count familiar from the lattice hard core ideal gas, produces an entropic contribution to the partition function of the form [@MWC1992] $$\begin{aligned}
e^{S[n_{\Omega,h}]/\Delta x^2} &=& e^{\beta \bar T S[n_{\Omega,h}]}
\nonumber \\
S[n_{\Omega,h}] &\equiv& -\int_D d{\bf r}
\int d\sigma \int_0^\infty d\lambda
\, n_{\Omega,h}({\bf r},\sigma,\lambda)
\nonumber \\
&& \times\ \ln[R_0 n_{\Omega,h}({\bf r},\sigma,\lambda)],
\label{5.6}\end{aligned}$$ where $R_0 = P_0/\rho_0 H_0^2$ has dimensions $[\sigma \lambda] = [\Omega h]$, and is required to make the argument of the logarithm dimensionless. This information theoretic form for $S$ is equivalent to the Sanov theorem result used by RVB.
The key assumption underlying (\[5.6\]) is that microscale fluctuations are uncorrelated across $\Delta X$-cells: In addition to the Casimirs (which are clearly unchanged, for arbitrary shuffling of $\Omega$ values around the domain $D$), the energy and momentum should also be unchanged. Arbitrarily shuffling $h$ values, even over the entire $D$, obviously does not change the potential energy term. However, the singular fashion in which $h$ enters the Green function $G_h$ defined in (\[4.38\]) (as well as the tensor Green function $\hat {\cal G}_h$ defined in App. \[app:KEPi\]), in the form of a gradient acting on a field with $O(1)$ variations on the scale $\Delta x$, *does* in fact lead to strong correlations across $\Delta X$ cells, invalidating (\[5.6\]).
Recognizing that the result is at best approximate, we proceed now in a manner equivalent to the variational approach, by neglecting such correlations. We define the microscale averaged Green function $\overline{G_h}$ by $$-\nabla \cdot \left\langle \frac{1}{\rho_0 h({\bf r})} \right\rangle_0
\nabla \overline{G_h}({\bf r},{\bf r}').
\label{5.7}$$ Using this in place of $G_h$ one may express all quantities in terms of $n_{\Omega,h}$: $$\begin{aligned}
E[n_{\Omega,h}] &=& \frac{1}{2} \int_D d{\bf r}
\bigg\{\frac{1}{\rho_0}
\left\langle \frac{1}{h({\bf r})} \right\rangle_0
|\langle {\bf j}({\bf r}) \rangle_0|^2
\nonumber \\
&&+\ g \rho_0 \langle [h({\bf r}) + h_b({\bf r}) - H_0]^2
\rangle_0 + \frac{1}{\bar \beta} \bigg\}
\nonumber \\
\Pi[n_{\Omega,h}] &=& \rho_0 \int d{\bf r}
[{\bf v}_\Pi({\bf r})
\cdot \langle {\bf j}({\bf r}) \rangle_0
+ \langle h({\bf r}) \rangle_0 F_\Pi({\bf r})]
\nonumber \\
\Gamma_l[n_{\Omega,h}] &=& \int_{\partial D_l}
\left\langle \frac{1}{h({\bf r})} \right\rangle_0
\langle {\bf j}({\bf r}) \rangle_0 \cdot d{\bf l}
\nonumber \\
g_\sigma[n_{\Omega,h}] &=& \int_D d{\bf r}
\int_0^\infty \lambda d\lambda \,
n_{\Omega,h}({\bf r},\sigma,\lambda).
\label{5.8}\end{aligned}$$ Here local averages $\langle \cdot \rangle_0$ are defined in the obvious way: $$\langle F[\Omega({\bf r}),h({\bf r})] \rangle_0
= \int d\sigma \int_0^\infty d\lambda F(\sigma,\lambda)
n_{\Omega,h}({\bf r},\sigma,\lambda),
\label{5.9}$$ while the mean current density $\langle {\bf j}({\bf r}) \rangle_0$, obeying $\nabla \cdot \langle {\bf j}({\bf r}) \rangle_0$, is defined by the analog of (\[5.2\]): $$\nabla \times \left[\left\langle \frac{1}{h({\bf r})} \right\rangle_0
\langle {\bf j}({\bf r}) \rangle_0 \right]
= \langle h({\bf r}) \Omega({\bf r}) \rangle_0 + f({\bf r}),
\label{5.10}$$ which, in addition to the circulation constraint in (\[5.8\]), fully specifies its form. Along the same lines as (\[4.39\]) and (\[4.40\]), the formal solution may be expressed in terms of $\overline{G_h}$.
The total microcanonical entropy ${\cal S}$ is now given by a functional integral over all $n_{\Omega,h}$, constrained by particular values of all of the conserved quantities: $$\begin{aligned}
e^{{\cal S}(\varepsilon,p,g)/\Delta x^2}
&=& \int D[n_{\Omega,h}] e^{S[n_{\Omega,h}]/\Delta x^2}
\delta(\varepsilon - E[n_{\Omega,h}])
\nonumber \\
&\times& \delta(p - \Pi[n_{\Omega,h}])
\prod_{l=2}^{N_D} \delta(c_l - \Gamma_l[n_{\Omega,h}])
\nonumber \\
&\times& \prod_\sigma \delta(g(\sigma) - g_\sigma[n_{\Omega,h}]),
\label{5.11}\end{aligned}$$ The computation of ${\cal S}$ proceeds now by noting the appearance of the divergent factors $1/\Delta x^2$ in the exponentials, which produces a saddle point solution: ${\cal S}$ is the maximum of $S[n_{\Omega,h}]$ over all $n_{\Omega,h}$ obeying the constraint conditions (and, for this reason, the precise definition of the measure $\int D[n_{\Omega,h}]$ is not important here). We handle these constraints via the *ordinary* use of Lagrange multipliers: rather than invoking them, via the grand canonical ensemble, at the level of the phase space integration, which introduces more stringent conditions on valid free energy minima, we use them here only to perform the constrained minimization of the functional ${\cal S}[n_{\Omega,h}]$. Thus, we introduce Lagrange multipliers $\beta = \bar \beta/\Delta x^2, \alpha, \gamma_l$, respectively, for the energy, momentum, circulation constraints, a function $\mu(\sigma)$ defining a functional $${\cal C}_\mu[n_0] = \int_D d{\bf r}
\int \mu(\sigma) d\sigma \int_0^\infty \lambda d\lambda \,
n_{\Omega,h}({\bf r},\sigma,\lambda)
\label{5.12}$$ that is used to enforce the Casimir constraints, and an additional function $\zeta({\bf r})$ to enforce for the normalization constraint (\[5.5\]): $${\cal N}_\zeta[n_{\Omega,h}]
= \int_D \zeta({\bf r}) d{\bf r}
\int_0^\infty d\lambda \int d\sigma \,
n_{\Omega,h}({\bf r},\sigma,\lambda).
\label{5.13}$$ We therefore seek the minimum with respect to $n_{\Omega,h}$ of the microcanonical variational free energy $$\begin{aligned}
{\cal F}_\mathrm{micro} &=& E[n_{\Omega,h}] - \bar T S[n_{\Omega,h}]
- \alpha \Pi[n_{\Omega,h}]
\nonumber \\
&&-\ \sum_{l=2}^{N_D} \gamma_l \Gamma_l[n_{\Omega,h}]
- {\cal C}_\mu[n_{\Omega,h}] - {\cal N}_\zeta[n_{\Omega,h}]
\nonumber \\
&&-\ 2 \bar T \int d{\bf r} \langle \ln(H_0/h({\bf r})) \rangle_0.
\label{5.14}\end{aligned}$$ The last term accounts for the net $h^2$ factor in the phase space measure that also appears in (\[4.26\]). The Euler-Lagrange equation, $\delta {\cal F}_\mathrm{micro}/\delta n_{\Omega,h}({\bf r},\sigma,\lambda) = 0$, produces $$\begin{aligned}
&&\bar T \ln[(P_0/\rho_0) n_{\Omega,h}({\bf r},\sigma,\lambda)/\lambda^2]
= - \frac{\rho_0 g}{2} [\lambda + \bar h_b({\bf r}) - H_0]^2
\nonumber \\
&&\ \ \ \ \ +\ \frac{|{\bf J}({\bf r})|^2}{2\rho_0 \lambda}
+ \lambda [\mu(\sigma) - \sigma \Psi({\bf r})] - {\cal N}({\bf r}).
\label{5.15}\end{aligned}$$ Here, $${\bf J}({\bf r}) \equiv \nabla \times \Psi({\bf r})
= \langle {\bf j}({\bf r}) \rangle_0
- \alpha \rho_0 \langle h({\bf r}) \rangle_0 {\bf v}_\Pi({\bf r})
\label{5.16}$$ includes the momentum term, as does the shift (\[4.6\]) to $\bar h_b$, and the circulation constraint enforces the boundary value $\Psi|_{\partial D_l} = \gamma_l$—equivalent to the first line of (\[4.24\]). The normalization ${\cal N}({\bf r})$ combines various other constant terms with $\zeta({\bf r})$. Exponentiating this result precisely reproduces (\[4.32\]). Inserting this result into (\[5.10\]) produces the self-consistent variational equation for $\Psi$, equivalent to (\[5.2\]). Inserting it into (\[5.8\]) produces equations for the Lagrange multipliers.
Equilibrium properties of the field $Q$ {#sec:Qdistr}
---------------------------------------
All of the previous results were derived by freely integrating out the compressional field $q = hQ$, resulting, via (\[4.19\]), in $\Phi \equiv 0$ and confirming that the large scale mean flow is completely determined by the remaining field $\Psi$. The distribution $n_{\Omega,h}$, defined by (\[4.32\]), is fundamental and allows one to compute all inputs to the variational equation (\[5.2\]). However, it may be of interest to compute equilibrium properties of $q$ as well. As observed above, its mean $\langle q({\bf r}) \rangle = \nabla \cdot {\bf V}({\bf r})$ is trivially determined from the previously computed mean flow. More interesting are its statistical fluctuations about the mean.
To illustrate such a computation (but still within the variational approximation), we extend the two-field distribution (\[4.32\]) to the three-field distribution function $$n_0({\bf r},\sigma,\kappa,\lambda) = \langle \delta[\Omega({\bf r}) - \sigma]
\delta[\bar Q({\bf r}) - \kappa] \delta[h({\bf r}) - \lambda] \rangle,
\label{5.17}$$ whose integral over $\kappa$ must reduce to (\[4.32\]). Here $\bar Q = \Delta x Q$ will be seen to be the correct continuum limit scaling [@foot:QRVB]: the fluctuations in $Q$ are $O(1/\Delta x)$, leading to order unity fluctuations in the compressional part of the velocity ${\bf v}^C = -\nabla \phi$, and a continuous velocity potential $\phi$.
The computation again begins with the KHS-transformed free energy functional (\[4.17\]). Integration over the field $Q$ now replaces (\[4.19\]) by $$e^{-\bar \beta \lambda \Phi({\bf r}) \kappa/\Delta x}
\prod_{{\bf r}_i \neq {\bf r}} 2\pi \delta(i\bar \beta h_i \Phi_i).
\label{5.18}$$ where we have substituted $h({\bf r}) = \lambda$. The result of the $\Phi$ integral is again to set $\Phi_i = 0$ for all ${\bf r}_i \neq {\bf r}$, but now leaving a single nontrivial integral over $\Phi({\bf r})$. The dependence on $\Phi({\bf r})$, via the $|{\bf J}|^2/2\rho_0 h$ term, is quadratic, with $\nabla \Phi(\bar {\bf r}) = \Delta x^{-1} [\Phi(\bar {\bf r} + \Delta x {\bf \hat x}) - \Phi(\bar {\bf r}),\ \Phi(\bar {\bf r} + \Delta x {\bf \hat y}) - \Phi(\bar {\bf r})]$ nonzero only on the neighboring sites ${\bf r}$, ${\bf r}_x = {\bf r} - \Delta x {\bf \hat x}$, and ${\bf r}_y = {\bf r} - \Delta x {\bf \hat y}$. The result is the (normalized) Gaussian integral $$\begin{aligned}
n_G({\bf r},\kappa) &=& \frac{\bar \beta \lambda}{\Delta x}
\int_C d\Phi({\bf r})
e^{-\bar \beta \lambda \Phi({\bf r})
(\kappa - \bar \kappa)/\Delta x}
\nonumber \\
&&\ \ \ \ \ \ \times\ e^{[\bar \beta \lambda \Phi({\bf r})/\Delta x]^2 \Delta \kappa^2/2}
\nonumber \\
&=& \frac{e^{-(\kappa - \bar \kappa)^2/2 \Delta \kappa^2}}
{\sqrt{2\pi \Delta \kappa^2}},
\label{5.19}\end{aligned}$$ where we define the mean and variance $$\begin{aligned}
\bar \kappa({\bf r},\lambda,\lambda_x,\lambda_y)
&=& \frac{1}{\rho_0 \lambda}
\left[\frac{\partial_x \Psi({\bf r})
- \partial_y \Psi({\bf r})}{\lambda} \right.
\nonumber \\
&&- \frac{\partial_x \Psi({\bf r}_y)}{\lambda_y}
+\ \left. \frac{\partial_y \Psi({\bf r}_x)}{\lambda_x} \right]
\nonumber \\
\Delta \kappa(\lambda,\lambda_x,\lambda_y)^2
&=& \frac{1}{\bar \beta \lambda^2}
\left[\frac{2}{\lambda} + \frac{1}{\lambda_x}
+ \frac{1}{\lambda_y} \right]
\label{5.20}\end{aligned}$$ in which $\lambda_x = h({\bf r}_x)$, $\lambda_y = h({\bf r}_y)$, and, for future reference, $\sigma_x = \Omega({\bf r}_x)$, $\sigma_y = \Omega({\bf r}_y)$ [@foot:dualpsi]. The result for $n_G$ is independent of $\Delta x$, as claimed.
The integral over the fields $\Omega,h$ now produce a factor $e^{\bar \beta W(\bar {\bf r})}$ \[defined by (\[4.26\]), with the same argument substitutions as in (\[4.28\])\], for every $\bar {\bf r} \notin \{{\bf r},{\bf r}_x,{\bf r}_y \}$, while the remaining integrals produce a factor $$\begin{aligned}
n_{\bar Q}({\bf r},\kappa|\lambda)
&=& \int_0^\infty d\lambda_x \int d\sigma_x \,
n_{\Omega,h}({\bf r}_x,\sigma_x,\lambda_x)
\nonumber \\
&&\times\ \int_0^\infty d\lambda_y \int d\sigma_y \,
n_{\Omega,h}({\bf r}_y,\sigma_y,\lambda_y)
\nonumber \\
&&\ \ \ \ \ \ \times\ n_G({\bf r},\kappa|\lambda,\lambda_x,\lambda_y),
\label{5.21}\end{aligned}$$ which differs from unity by the presence of the Gaussian factor (\[5.19\]). One may view the result as a superposition of Gaussian densities in which the mean $\bar \kappa$ and variance $\Delta \kappa^2$ range over values weighted by the probability distribution $n_{\Omega,h}$. The normalization (\[5.19\]) ensures that $n_{\bar Q}(\kappa)$ is a probability density for any fixed values of the other parameters.
With these inputs, the final result for $n_0$ is given by $$n_0({\bf r},\sigma,\kappa,\lambda)
= n^\mathrm{eq}_{\Omega,h}({\bf r},\sigma,\lambda)
n^\mathrm{eq}_{\bar Q}({\bf r},\kappa|\lambda,\sigma)
\label{5.22}$$ in which “eq” superscript indicates that in the continuum limit one simply substitutes the variational solution $\Psi = \Psi^\mathrm{eq}$. In this same limit one may replace all appearances of $\Psi$ and its derivatives on neighboring lattice sites by their values at ${\bf r}$ wherever they appear in (\[5.20\]) and (\[5.21\]).
A key observation is that the fluctuation statistics predicted by $n_{\Omega,h}$ and $n_0$ are not independent. In particular, independent products of various terms for fixed height field $h$ become strongly mixed after the functional integral over $h$. This is in strong contrast to the results of RVB, in which the different choice of phase space measure does produce independent statistics [@foot:QRVB]. Nevertheless, despite this statistical entanglement, we have seen in Sec. \[sec:khs\] that $Q$ can still be straightforwardly integrated out to produce a relatively transparent effective free energy (\[4.25\]) for $\Omega,h$.
Note as well that the microscale Gaussian form (\[5.19\]), and hence the precise form of $n_{\bar Q}$, is sensitive to the definition of the discrete derivative used here. One could imagine using a non-square lattice, and/or further-neighbor discrete difference forms. Given that the microscale fluctuations on the grid scale $\Delta x$ contain a finite fraction of the system energy, this sensitivity to the precise form of the grid is physically consistent. However, this sensitivity disappears upon integrating out $\kappa$. Thus, $n_{\Omega,h}$ depends only on the macroscopic flow $\Psi^\mathrm{eq}$, and produces continuum limit equilibrium forms that are insensitive to grid details. This is entirely consistent with the trivial equipartition contribution to the energy observed in (\[5.3\]) arising from fluctuations in $Q$.
Small height fluctuation limit {#sec:smallhfluct}
------------------------------
In order to begin to make contact with equilibria, previously treated in the literature [@WP2001; @CS2002], in which surface height fluctuations were neglected, we consider here the limit in which small scale fluctuations in $h$ are assumed very small. This allows one to further reduce the problem to a simultaneous extremum problem for $\Psi$ and $h$. We continue to work within the variation approximation, though one may expect that for parameter ranges that do indeed produce small fluctuations, this approximation may often become exact (though quantifying this is beyond the scope of this paper).
Assuming that thermodynamic parameters are chosen in such a way that (\[4.26\]) produces a very narrow distribution for $h$ about a (yet to be determined) mean, one may simplify $W$ to the form $$\begin{aligned}
e^{\bar \beta W(\tau,h_0,\xi)}
&=& \frac{h^2}{H_0^2} e^{V(\tau,\bar \beta h)}
e^{\frac{1}{2} \bar \beta[\xi/\rho_0 h - \rho_0 g(h+h_0)^2]}
\nonumber \\
e^{V(\tau,\gamma)} &\equiv& \frac{\rho_0 H_0^3}{P_0}
\int d\sigma e^{\gamma [\mu(\sigma) - \sigma \tau]},
\label{5.23}\end{aligned}$$ and the free energy functional (\[4.28\]) now reduces to the form
$${\cal F}[\Psi,h] = \bar F^0(\alpha,{\bm \gamma})
- \int_D d{\bf r} \left[\frac{|\nabla \Psi|^2}{2 \rho_0 h}
- \frac{1}{2} \rho_0 g (h + \bar h_b - H_0)^2 + \bar f \Psi
+ \frac{1}{\bar \beta} V(\Psi,\bar \beta h) \right],
\label{5.24}$$
whose minimum describes the large scale equilibrium flow, and simultaneously self-consistently determines the value of the mean surface height $h$. The Euler–Lagrange equations now produce the forms $$\begin{aligned}
&&-\nabla \cdot \left(\frac{1}{\rho_0 h} \nabla \Psi \right) + \bar f
= -\frac{1}{\bar\beta} \partial_\tau V(\Psi,\bar \beta h)
\label{5.25} \\
&&\frac{|\nabla \Psi|^2}{2\rho_0 h^2} + \rho_0 g(h + \bar h_b - H_0)
= \partial_\gamma V(\Psi,\bar \beta h)
- \frac{2}{\bar \beta h}.
\nonumber\end{aligned}$$ Analogous to (\[4.32\]), the potential vorticity distribution function for given $\Psi,h$ is $$\begin{aligned}
n_\Omega({\bf r},\sigma)
&=& \langle \delta[\Omega({\bf r}) - \sigma] \rangle
\label{5.26} \\
&=& \frac{\rho_0 H_0^3}{P_0} e^{V(\Psi,\bar\beta h)}
e^{\bar \beta h [\mu(\sigma) - \sigma \Psi)]},
\nonumber\end{aligned}$$ from which one identifies $$\begin{aligned}
-\frac{1}{\bar \beta} \partial_\tau V
&=& h({\bf r}) \int \sigma d\sigma \, n_\Omega({\bf r},\sigma)
\nonumber \\
&=& h({\bf r}) \langle \Omega({\bf r}) \rangle
= \langle \omega({\bf r}) \rangle + f({\bf r}).
\label{5.27}\end{aligned}$$ Analogous to (\[5.2\]), if we define the equilibrium flow velocity ${\bf V}$ by $${\bf V} - \alpha {\bf v}_\Pi = (\rho_0 h)^{-1} \nabla \times \Psi,
\label{5.28}$$ the first line of (\[5.25\]) reproduces (\[5.1\]), while the second line produces the generalized Bernouilli equation $$\frac{1}{2} \rho_0 |{\bf V} - \alpha {\bf v}_\Pi|^2 + \rho_0 g \bar \eta
= \partial_\gamma V(\Psi,\bar \beta h)
- \frac{2}{\bar \beta h}.
\label{5.29}$$
As discussed in [@RVB2016], by perturbatively treating small, but finite, fluctuations in $h$ around the mean value defined by (\[5.25\]), the resulting theory is that of a weakly coupled system consisting of large-scale eddy motions with superimposed small scale fluctuations.
### Vlasov and Bernoulli conditions {#subsec:vlasovbernoulli}
The forms (\[5.25\]) appear to violate the Vlasov and Bernoulli conditions, namely that the right hand sides should depend only on the stream function $\Psi$. These conditions follow from the observations that the first line of (\[2.1\]) and (\[2.7\]), respectively, require that steady state flows (i.e., time-independent, in the appropriate frame of reference if momentum is conserved) obey, $$\begin{aligned}
({\bf v}-\alpha{\bf v}_\Pi) \cdot \nabla
\left(\frac{1}{2} |{\bf v} - \alpha {\bf v}_\Pi|^2 + g \bar \eta \right) &=& 0
\nonumber \\
({\bf v}-\alpha {\bf v}_\Pi) \cdot \nabla \Omega &=& 0.
\label{5.30}\end{aligned}$$ The steady state condition $\nabla \cdot {\bf J} = 0$ implied by the second line of (\[2.1\]) allows one to express ${\bf J} \equiv h({\bf v}-\alpha{\bf v}_\Pi) = \nabla \times \Psi$ in terms of a current density stream function $\Psi$. Equations (\[5.30\]) then imply that the level curves of $\Psi$, $\Omega$, and $B \equiv \frac{1}{2} |{\bf v} - \alpha {\bf v}_\Pi|^2 + g \bar \eta$ all coincide, and hence one may formally write $\Omega = f_V(\Psi)$ and $B = f_B(\Psi)$ for some fixed pair of 1D functions $f_\Omega,f_B$.
To resolve the paradox implied by the failure of the equilibrium equations to produce this functional dependence, one must understand the limits under which surface height fluctuations are small, and show that these indeed restore the Vlasov and Bernoulli conditions. We consider the cases of (1) strong gravity, $g \to \infty$, (2) low temperature $\bar \beta \to \infty$, and (3) the effects of physical processes that dissipate small scale fluctuations and hence lead to a quiescent surface.
#### Case (1):
The limit $g \to \infty$ turns out to be surprisingly subtle, and is discussed in detail in Sec. \[sec:eulercomp\]. This limit indeed produces a fluctuation-free surface, $\eta \to 0$, hence $h = H_0 - \bar h_b$ independent of $\Psi$. However, due to the increased surface wave speed $c \approx \sqrt{gH_0}$, even as $g \to \infty$ one finds finite amplitude fluctuations in the compressional part of ${\bf v}$. Even though these fluctuations can still be integrated out freely \[see equation (\[4.19\])\], this still leads to violations of the Vlasov condition because the advective term ${\bf v} \cdot \nabla \Omega$ in (\[2.7\]) contains finite amplitude, correlated fluctuations in both ${\bf v}$ and $\Omega$, with the result that $\langle {\bf v} \cdot \nabla \Omega \rangle \neq {\bf V} \cdot \nabla \langle \Omega \rangle$. The same considerations apply to the Bernoulli condition, which then fails because $\nabla \cdot \langle h{\bf v} \rangle \neq \nabla \cdot (\langle h \rangle {\bf V})$.
As shown in Sec. \[sec:eulercomp\], if one imposes a strict “rigid lid” condition on the surface, corresponding to the Euler equation limit, the Vlasov condition is restored, but the absence of microscopic fluctuations in ${\bf v}$ and $\eta$ leads to a quantitatively different equilibrium theory. Only in the additional $\bar \beta \to \infty$ limit, discussed next, do the two theories match.
#### Case (2):
In the limit $\bar \beta \to \infty$ the term $2/\bar \beta h$ may be neglected, while a steepest descent evaluation of $V(\tau,\gamma)$ is appropriate. The latter produces $$V(\tau,\gamma) \approx \gamma
\{\mu[\sigma_0(\tau)] - \tau \sigma_0(\tau) \}
\label{5.31}$$ where $\sigma_0(\tau)$ is the solution to the stationary condition $$\tau = \mu'(\sigma).
\label{5.32}$$ This leads to $$\begin{aligned}
&&\frac{1}{\bar \beta} V(\Psi,\bar \beta h)
\approx h \{\mu[\sigma_0(\Psi)] - \Psi \sigma_0(\Psi) \}
\nonumber \\
&&\Rightarrow\ \left\{\begin{array}{l}
\partial_\tau V/\bar \beta h = -\sigma_0(\Psi) \\
\partial_\gamma V = \mu[\sigma_0(\Psi)]
- \Psi \sigma_0(\Psi),
\end{array} \right.
\label{5.33}\end{aligned}$$ which are indeed both functions of $\Psi$ alone. Thus, zero temperature, non-fluctuating flows indeed satisfy the requisite stream line conditions. Equation (\[5.32\]), taking the form $\Psi = \mu'(\Omega)$, directly exhibits the Lagrange multiplier function.
#### Case (3):
This case is the most speculative, and was in fact the basis for the treatment of shallow water equilibria in Ref. [@WP2001]. There, at a critical step in the analysis, fluctuations in $h$ and $Q$ were simply assumed to have been suppressed by some set of dissipative mechanisms (e.g., viscosity, wave breaking). The resulting variational equation for $\Psi,h$ was then developed in a form similar to (\[5.24\]) and (\[5.25\]).
By appealing to dissipative mechanisms lying outside of the shallow water system, the theory is removed, at least temporarily, from the purely equilibrium statistical mechanics arena. The supporting notion is that in a number of physically relevant cases, a strong separation develops between the large scale eddies and the small scale wave motions, and the latter are preferentially dissipated with negligible effect on the large scale flow. The result is to remove a certain fraction of the total energy from the system, while the remainder would be proposed to lie entirely in a “renormalized” equilibrium flow with vanishing height fluctuations. The appropriate effective theoretical description could then be a version of case (2), in which corresponding renormalized values of the Lagrange multipliers are sought that reproduce the observed values of the conserved integrals.
An interesting consequence is that negative temperatures are no longer precluded [@WP2001]. Thus, $V(\tau,\gamma)$, unlike $W(\tau,h_0,\xi)$, is perfectly well defined for $\gamma < 0$, and so extrema of (\[5.24\]) may be sought for both positive and negative $\bar \beta$ (in particular, for both $\bar \beta \to \pm \infty$). In principle, negative temperature equilibria are unstable to leakage of energy into (positive temperature) wave motions, but the physical coupling of large scale flows to small scale wave generation is extremely small, and it makes sense to develop a theory along these lines that neglects such effects. The key observation here is that compact eddy structures, such as Jupiter’s Great Red Spot, having vorticity maxima confined away from the system boundaries, can only be interpreted as negative temperature states [@MWC1992]. Such structures therefore lie outside the strict shallow water theory presented here, and nonequilibrium dissipation arguments *must* therefore be invoked in order to make contact with the effective equilibrium descriptions ubiquitous in the literature [@BV2012].
We note finally that there is no reason for the more general result (\[5.1\])—or, for that matter, the fully fluctuating result (\[4.34\])—to satisfy these conditions because the microscale flows are not steady state. The conditions need only be restored when such fluctuations are assumed to be absent. An interesting point is that RVB found that, even in their general theory, both conditions to be satisfied, and cite this as a supporting feature [@RVB2016]. Their result occurs because, in contrast to (\[5.2\]), their version of the phase space measure produces independent microscale fluctuations of $h,Q,\Omega$. This leads in particular to $\langle {\bf v} \cdot \nabla Q \rangle = {\bf V} \cdot \nabla \langle Q \rangle$ and $\langle \nabla \cdot (h{\bf v}) = \nabla \cdot (\langle h \rangle {\bf V})$, and this is then reflected in the desired $\Psi$-dependence of the equilibrium equations. The feature therefore is a direct consequence of the inconsistency of their measure choice with the Liouville theorem (see Apps. \[app:liouville\] and \[app:liouvilleinequiv\]), and we argue therefore that it should not (in absence of much deeper arguments) be considered as supporting the validity of the approach.
Comparison with Euler equilibria {#sec:eulercomp}
================================
In this section, shallow water equilibria will be compared to those of the Euler equation, including variable bottom topography $h_b({\bf r})$, but now with a fixed rigid-lid surface. It will be shown that the latter leads to an equilibrium phase space measure with non-uniform gridding of the domain $D$, determined by $h_b$—consistent in this case with the choice made by RVB. This is significantly different from the limit $g \to \infty$ in the shallow water results of Sec. \[sec:statmech\], which continues to require uniform gridding. The paradox is resolved by showing that the equilibria are in fact expected to be physically *different*, with microscale height fluctuation effects present even in the limit $g \to \infty$. These results serve again to highlight the inconsistency of the RVB nonuniform grid choice with that implied by the shallow water Liouville equation.
The Euler equation, including variable bottom topography, is described by $$\begin{aligned}
\partial_t {\bf v} + ({\bf v} \cdot \nabla) {\bf v}
+ f {\bf \hat z} \times {\bf v} &=& -\frac{1}{\rho_0} \nabla p
\nonumber \\
\nabla \cdot (h {\bf v}) &=& 0,
\label{6.1}\end{aligned}$$ and is equivalent to the shallow water equations (\[2.1\]) but with $h({\bf r}) = H_0 - h_b({\bf r})$ now a fixed function, and the pressure $p$ enforcing the incompressibility condition (and an equation for which is obtained by multiplying both sides of the first line by $h$ and taking the divergence). The potential vorticity is still given by (\[2.6\]), and continues to be advectively conserved \[equation (\[2.7\])\].
The incompressibility condition implies that $${\bf j} = \rho_0 h {\bf v} = \nabla \times \psi
\label{6.2}$$ is purely transverse. The velocity and potential vorticity $$\begin{aligned}
{\bf v} &=& \frac{1}{\rho_0 h} \nabla \times \psi
\nonumber \\
h\Omega - f &=& -\nabla \cdot \left(\frac{1}{\rho_0 h} \nabla \psi \right)
\label{6.3}\end{aligned}$$ are completely determined in terms of the single scalar function $\psi$. The equation of motion (\[2.7\]) therefore fully describes the Euler dynamics. Statistical equilibria, obeying the Vlasov condition $${\bf v} \cdot \nabla \Omega = 0,
\label{6.4}$$ are then formulated entirely in terms of $\Omega$ as well. For simplicity, we will consider only a simply connected domain $D$ with Dirichlet boundary conditions on $\psi$. Defining the (symmetric) scalar Green function $G_h$ by $$-\nabla \cdot \frac{1}{\rho_0 h} \nabla G_h({\bf r},{\bf r}')
= \delta({\bf r}-{\bf r}')
\label{6.5}$$ with Dirichlet boundary conditions \[identical to (\[4.38\]), but now with deterministic $h$\], one obtains the relation $$\psi({\bf r}) = \int_D d{\bf r} G_h({\bf r},{\bf r}')
(h\Omega - f)({\bf r}').
\label{6.6}$$
Liouville theorem and equilibrium measures {#sec:eulerliouville}
------------------------------------------
The Liouville theorem follows from the equation of motion written in the conserved form $$h \dot \Omega = -\nabla \cdot [h \Omega {\bf v}],
\label{6.7}$$ which leads to [@foot:liouville] $$h({\bf r}) \frac{\delta \dot \Omega({\bf r})}{\delta \Omega({\bf r})}
= - \nabla \cdot \left\{h({\bf r})
\frac{\delta [\Omega({\bf r}) {\bf v}({\bf r})]}{\delta \Omega({\bf r})} \right\}.
\label{6.8}$$ From the boundary conditions on $\partial D$, it follows that $$\int_D d{\bf r} h({\bf r})
\frac{\delta \dot \Omega({\bf r})}{\delta \Omega({\bf r})} = 0.
\label{6.9}$$ In order to express this in the standard form of a divergence free condition on the phase space flows, we define a mapping ${\bf r}({\bf a}): D \to D$ (clearly not unique, but still fixed by the bottom topography) with Jacobian $$J({\bf a}) \equiv \frac{\partial {\bf r}}{\partial {\bf a}}
= \frac{H_0}{h[{\bf r}({\bf a})]}.
\label{6.10}$$ Thus, ${\bf r}({\bf a})$ maps a fluid with uniform height $H_0$ to one with variable, but time-independent, height $h$. Relabeling $\Omega({\bf a}) \equiv \Omega[{\bf r}({\bf a})]$, (\[6.9\]) may be written in the form $$\int_D d{\bf a}
\frac{\delta \dot \Omega({\bf a})}{\delta \Omega({\bf a})} = 0.
\label{6.11}$$
It follows immediately from (\[6.11\]) that equilibrium statistical measures $\rho(E,P,{\cal C})$ are, as usual, functions only of the conserved integrals, and phase space averages may be defined through the continuum limit $$\rho[\Omega] D[\Omega]
= \lim_{\Delta V \to 0} \rho[\Omega] \prod_i d\Omega_i,
\label{6.12}$$ in which $\Omega_i = \Omega({\bf a}_i)$ and $\{{\bf a}_i \}$ represents a *uniform* gridding of $D$, with fixed physical fluid element volume $\Delta V = H_0 \Delta A = h_i \Delta A_i$, where $\Delta A_i$ is the image of cell $i$ under the mapping ${\bf r}({\bf a})$.
Statistical mechanics {#sec:eulerstatmech}
---------------------
The grand canonical statistical measure is given by $$\begin{aligned}
\rho &=& \frac{1}{Z} e^{-\beta {\cal K}[\Omega]}
\nonumber \\
{\cal K}[\Omega] &=& E[\Omega] - C_\mu[\Omega]
\label{6.13}\end{aligned}$$ with energy and Casimir functionals $$\begin{aligned}
E[\Omega] &=& \frac{\rho_0}{2} \int_D d{\bf r} \int_D d{\bf r}'
(h \Omega - f)({\bf r})
\nonumber \\
&&\ \ \ \ \ \ \ \ \times G_h({\bf r},{\bf r}') (h \Omega - f)({\bf r}')
\nonumber \\
C_\mu[\Omega] &=& \int_D d{\bf r} h({\bf r}) \mu[\Omega({\bf r})].
\label{6.14}\end{aligned}$$ Given the simply connected domain, momentum conservation is not considered here.
In discrete form, one obtains $$\begin{aligned}
E &=& \int_D d{\bf a} \int_D d{\bf a}' (\Omega - f/h)({\bf a})
\nonumber \\
&&\ \ \ \ \ \times G_h[{\bf r}({\bf a}),{\bf r}({\bf a}')]
(\Omega - f/h)({\bf a}')
\nonumber \\
&=& \lim_{\Delta V \to 0} \frac{\rho_0}{2} \Delta V^2 \sum_{i,j}
(\Omega - f/h)_i G_{h,ij} (\Omega - f/h)_j
\nonumber \\
C_\mu &=& H_0 \int_D d{\bf a} f[\Omega({\bf a})]
= \lim_{\Delta V \to 0} \Delta V \sum_i \mu[\Omega({\bf a}_i)].
\nonumber \\
\label{6.15}\end{aligned}$$ The KHS transformation, acting to decouple the energy, produces the partition function $$\begin{aligned}
Z &=& \prod_i \frac{P_0}{\rho_0 H_0^3} \int d\Omega_i e^{-\beta {\cal K}[\Omega]}
\nonumber \\
&=& \frac{1}{{\cal N}_h} \prod_i P_0 H_0 \int d\Psi_i e^{-\beta {\cal F}[\Psi]}
\label{6.16}\end{aligned}$$ with (continuum limit) free energy functional $${\cal F}[\Psi] = -\int d{\bf r}
\left[\frac{|\nabla \Psi|^2}{2 \rho_0 h}
+ f \Psi + h W(\Psi) \right].
\label{6.17}$$ The normalization ${\cal N}_h$ is the determinant of the quadratic form that defines $E$ in (\[6.15\]), and is a nontrivial functional of $h$. This is contrast to shallow water KHS result (\[4.10\]) where the normalization takes the form of trivial product factors. However, it is a fixed constant, and hence does not contribute to equilibrium averages.
The Lagrange multiplier function $\mu$ is now subsumed into the function $W(\tau)$ defined by $$e^{\bar \beta_E W(\tau)}
= \int d\sigma e^{\bar \beta_E [\mu(\sigma) - \sigma \tau]}
\label{6.18}$$ with renormalized temperature variable $$\bar \beta_E = \beta \Delta V,\ \bar T_E = T/\Delta V
\label{6.19}$$ remaining finite in the continuum limit $\Delta V \to 0$. Both positive and negative temperatures are allowed here since convergence of the integral is in general controlled by $\mu(\sigma)$. In this limit, one has $|\beta| \to \infty$ and the variational condition where one seeks the minimum of ${\cal F}$ emerges. In this case, since the compressional degree of freedom has been suppressed at the outset and, correspondingly, the height field is fixed, $\Psi$ is continuously differentiable, and both $\Psi$ and $\nabla \Psi$ are non-fluctuating in the continuum limit (while, of course, $\nabla^2 \Psi$ has finite fluctuations). The variational approximation is therefore exact in this case, and one obtains the Euler-Lagrange equation $$\omega_0 \equiv -\nabla \cdot \left(\frac{1}{\rho_0 h} \nabla \Psi_0 \right)
= -h W'(\Psi_0) - f.
\label{6.20}$$ The equilibrium potential vorticity obeys $$\Omega_0 = \frac{\omega_0 + f}{h} = -W'(\Psi_0),
\label{6.21}$$ which ensures that the Vlasov condition (\[6.4\]) is satisfied (i.e., $\nabla \Psi_0$ and $\nabla Q_0$ are everywhere colinear, and hence $\Psi_0$ and $\Omega_0$ share stream lines).
Comments {#sec:comments}
--------
Unlike for the shallow water equations, in which the microscale (especially surface height) fluctuations contribute in a highly nontrivial way, even in the $g \to \infty$ limit, to the variational free energy (\[4.28\]) for the large-scale vortical stream function, the rigid lid boundary conditions here suppress these entirely, and the Vlasov condition is satisfied explicitly. The key enabling result is that the velocity ${\bf v} = {\bf v}_0$ is purely large scale.
From a mathematical point of view, the Vlasov result *requires* that the function $W$ have spatial dependence through $\Psi$ alone—all dependence on $h$ escapes only to the overall multiplier of $W(\Psi)$ in (\[6.17\]). This happens only because the Liouville equation that determines the phase space measure produces, in contrast to the shallow water case, a *nonuniform* real space mesh. Intuitively, the rigid lid condition places corresponding rigid conditions on the Eulerian phase space fluid parcel distribution $\nu({\bf r},{\bf p})$ defined in App. \[app:liouville\]. Specifically, the moments defined in (\[A18\]) are restricted by $h = H_0 - h_b$ and $\nabla \cdot {\bf j} = 0$. The resulting reformulation (\[6.12\]) entirely in terms of $\Omega$, which enforces these conditions automatically, then also induces the nonuniform mesh.
Comparing to the $g \to \infty$ limit of the shallow water equations, one observes that the function $W$ in (\[4.26\]) or $V$ in (\[5.23\]) continues to depend nontrivially on $h$, even though the condition $\eta = h + h_b - H_0 \to 0$ is indeed enforced, in the small fluctuation limit, through the $g (h+h_0)^2$ term in (\[4.26\]) or $g \bar \eta^2$ term in (\[5.24\]). The resolution of this paradox is that although $\eta/H_0 = O(1/\sqrt{\bar \beta \rho_0 g H_0^2})$ is very small, the compressional part of the velocity ${\bf v}_L = O(c \eta/H_0) = O(1/\sqrt{\bar \beta \rho_0 H_0})$ remains finite because the wave speed $c = \sqrt{g H_0}$ (along with the microscopic frequency $\partial_t h/H_0$) diverges. Thus, the Vlasov combination ${\bf v} \cdot \nabla \Omega$ has finite amplitude (correlated) microscopic fluctuations in *both* ${\bf v}$ and $\Omega$, and the equilibrium average $\langle {\bf v} \cdot \nabla \Omega \rangle \neq {\bf v}_0 \cdot \nabla \Omega_0$ fails to factorize \[except in the additional low temperature limit $\bar \beta \to \infty$ described by (\[5.31\])–(\[5.33\])\]. This explains the violation of the Vlasov condition implied by the dependence on $h$ of the right hand side of the first line of (\[5.25\]).
As a final comment, we note that the heuristic suppression of small-scale wave fluctuations considered in Ref. [@WP2001] also led to a theory with a nonuniform $h$-dependent mesh. However, in this case $h$ was not fixed *a priori*, but determined, along with $\Psi$, through the free energy minimization (which, as we have seen, implicitly assumes that all of the energy is in the large scale flow, and hence provides the mathematical mechanism for suppressing small-scale waves). Self-consistently, the resulting hydrostatically balanced flows satisfied both the Vlasov and Bernoulli conditions.
Concluding remarks {#sec:conclude}
==================
An important distinction between RVB and the present approach is the use here of the grand canonical ensemble, and of the KHS transformation (Sec. \[sec:khs\]), as key tools for deriving useful reduced forms for the effective free energy functional from the generalized Hamiltonian (\[4.2\]). By expressing the generalized Hamiltonian in the purely local form (\[4.17\]), the method has the advantage of providing a mathematically complete and efficient procedure for deriving the intermediate reduced form (\[4.25\]) (integrating out $Q,\Phi$), following with either the fully reduced elastic membrane form (\[4.28\]) (integrating out $h,\Omega$, leaving only $\Psi$), or the dual generalized Coulomb form (\[4.37\]) (integrating out $\Psi$, leaving $\Omega,h$). Most importantly, it transparently exhibits the strong fluctuations and long-range correlations that survive the continuum limit. The formulation adopted by RVB misses both of these effects because the mean field approximation is implicit in their approach to separating the fields into large scale and small scale components.
The discussion in App. \[app:liouvilleinequiv\] on the connection between the Liouville theorem and equilibrium measures is based on a very general formulation (\[B2\]) or (\[B10\]) of the Liouville theorem, and does not require an appeal to an underlying Hamiltonian structure. The latter is used as part of the specific derivation in App. \[app:liouville\], but the conclusions follow much more generally. In particular, the theory leads quite generally to the construction of a phase space measure through a limiting procedure completely consistent with standard uniform area gridding of the field index ${\bf r}$, which is also fully consistent with many previous statistical mechanics applications in quantum and classical field theory.
RVB instead replace the uniform grid by a highly nonuniform “Lagrangian” grid, that is moreover *dynamically adjusted* according to the fluid height field, which is itself one of the phase space variables being integrated over. Given that $h$ has strong variations on the grid scale, this is a rather singular adjustment, and is very unlike, for example, the smooth change of variable adopted in Ref. [@WP2001] after the microscale height fluctuations were assumed to have been dissipated, or the time-independent change of variable(\[6.10\]) in the Euler case (with degree of smoothness governed by the bottom topography $h_b$). We have seen that the RVB choice corresponds to a very different form of the Liouville theorem—equivalent to a nontrivial density $w({\bf r})$ in (\[B17\]) that also includes strong variations on the grid scale.
The equilibrium theory resulting from the two choices are quantitatively different, so this is not an instance of mathematical convenience to obtain an equivalent continuum limit. In particular, we have emphasized that the shallow water equilibrium states *are not expected to be stationary, time independent solutions to the fluid equations.* Unlike the pure 2D Euler case (discussed in detail in Sec. \[sec:eulercomp\]), we have seen that the macroscopic flows are strongly dynamic, with finite energy, finite amplitude, high frequency height fluctuations (resulting from the undissipated forward cascade of wave energy). They are found to be stationary in Ref. [@RVB2016] only because the Lagrangian gridding leads to a product measure in which the basic fields have independent statistics, leading to exact factorization of key averages. In the present theory, the height field $h$ is not independent of ${\bf v}$, and this leads to the expected nonstationary equilibrium averages. As discussed in Sec. \[sec:eulercomp\], this is also what leads to the inequivalence of the rigid lid Euler flow and shallow water $g \to \infty$ limit.
This paper has concentrated on deriving general statistical models and exploring some of their key general features. Detailed studies of equilibrium solutions for specific, physically motivated choices of model parameters remains to be addressed in future work. The effects of fluctuations, and predicting the effects of various dissipation mechanisms in producing the ultimate *quiescent* equilibria [@WP2001], deserve special focus. RVB have already made some explorations along these lines within the variational theory. Significant insight can be gained by restricting the problem to a finite number of degrees of freedom. For example, the choice $\mu(\sigma) = -\frac{1}{2} \mu_0 \sigma^2$ reduces (\[4.26\]) to a Gaussian integral in the variable $\sigma$, and corresponds to a version of the Energy–Enstrophy theory [@K1975]. Perhaps more interesting are the finite-level systems $e^{\bar \beta \mu(\sigma)} = \sum_{n=1}^{N_\sigma} e^{\bar \beta \mu_n} \delta(\sigma - \sigma_n)$ [@MWC1992; @BV2012] in which the potential vorticity is permitted to take only a discrete set of values, with relative populations controlled by the corresponding discrete set of chemical potentials $\mu_n$. Even the cases $N_\sigma = 2,3$ generate an interesting variety of equilibria as the temperature and other parameters are varied.
Most previous investigations have focused on mean field equilibria, especially those of the Euler and quasigeostrophic equations for which they are exact. A very interesting feature is the set of transitions between equilibrium states that can occur as a function of the thermodynamic parameters. An important example is when a translation or rotational symmetry is broken: with increasing energy, an instability can occur in which an annular or linear jet transitions to a more compact vortex structure. Within the variational approximation, such transitions are simple bifurcations. In the presence of strong fluctuations the character of the transition remains an open question. A possibility is that it elevates to a true critical phenomenon with nontrivial critical exponents [@S1971]. Phase transitions in the context of elastic membranes include roughening of crystalline solid facets [@CL1995]. Here there is competition between a periodic confining potential which prefers a flat interface, and entropic fluctuations which prefer a rough surface with the logarithmic correlations alluded to in Sec. \[subsec:nonlinmembrane\]. In the present case the analogue of a periodic crystalline potential is absent, and the membrane is always in the rough regime. Instead, there is a large-scale conformational change of the membrane, more akin perhaps to shape changes in biological membrane systems [@PKTH2013].
Fluid system Liouville theorem and phase space measure {#app:liouville}
======================================================
In this Appendix a very general 2D fluid system Liouville theorem is derived, applicable to a much more general class of equations than just the shallow water system. The derivation is based on a Lagrangian coordinate description, in which standard Hamiltonian position and conjugate momentum coordinates may be transparently derived and applied [@WP2001]. A transformation to Eulerian coordinates is made at the end to demonstrate equivalence for the special case, specific to the shallow water equations, derived in Ref. [@RVB2016]. The equivalence lends insight to the the nature of the microscale fluctuations being considered.
Lagrangian coordinate Hamiltonian formulation {#app:lagham}
---------------------------------------------
In the presence of both a Coriolis force $f({\bf r})$ and bottom topography $h_b({\bf r})$, the Lagrangian coordinate Hamiltonian takes the form $$\begin{aligned}
{\cal H} &=& \int_D d^2a \left(\frac{|{\bf p}({\bf a})
- {\bf A}[{\bf r}({\bf a})]|^2}{2\rho_0 H_0} \right.
\nonumber \\
&&+\ \left. \frac{1}{2} \rho_0 H_0 g
\{h[{\bf r}({\bf a})] + 2 h_b[{\bf r}(a)] \}
\right)
\label{A1}\end{aligned}$$ where, incorporating a minor rescaling from that appearing in (\[2.16\]) and (\[2.19\]), the vector potential represents the Coriolis parameter via $$\rho_0 H_0 f({\bf r}) = \nabla \times {\bf A}({\bf r})
= \partial_x A_y - \partial_y A_x,
\label{A2}$$ and has the physical interpretation of the steady velocity field that produces the background coordinate system rotation. Units have been chosen so that ${\bf p}$ is an areal momentum density, i.e., it has the same dimensions as ${\bf j}$.
The Lagrangian coordinate ${\bf r}({\bf a},t)$ and conjugate momentum ${\bf p}({\bf a},t)$ represents a fluid parcel of fixed volume $H_0 d^2a$. For each $t$, ${\bf a} \to {\bf r}({\bf a},t)$ represents a mapping of the domain $D$ into itself. Unlike for the case of the Euler equation, this mapping is not in general area preserving. In fact, the height field is defined by $$\frac{H_0}{h[{\bf r}({\bf a})]} = J({\bf a})
\label{A3}$$ where $$J({\bf a}) = \det\left(\frac{\partial {\bf r}}{\partial {\bf a}}\right)
= (\partial_1 r_1)(\partial_2 r_2) - (\partial_2 r_1)(\partial_1 r_2)
\label{A4}$$ is the Jacobian of the transformation. Thus, in a slight abuse of notation, $h[{\bf r}({\bf a})]$ is actually a nontrivial functional of ${\bf r}({\bf a})$. The corresponding term in ${\cal H}$ represents the potential energy of a particle at height $h/2$ above the bottom, equivalent to that of a fluid parcel of thickness $h$.
Hamilton’s equations of motion then yield $$\begin{aligned}
\dot {\bf r}({\bf a}) &=& \frac{\delta {\cal H}}{\delta {\bf p}({\bf a})}
= \frac{{\bf p}({\bf a}) - {\bf A}[{\bf r}({\bf a})]}{\rho_0 H_0}
\nonumber \\
\dot {\bf p}({\bf a}) &=& -\frac{\delta {\cal H}}{\delta {\bf r}({\bf a})}
\nonumber \\
&=& \frac{1}{\rho_0 H_0} (\nabla {\bf A})[{\bf r}({\bf a})]
\cdot \{{\bf p}({\bf a}) - {\bf A}[{\bf r}({\bf a})]\}
\nonumber \\
&&-\ \rho_0 H_0 g \nabla \{h[{\bf r}({\bf a})] + h_b[{\bf r}({\bf a})] \},
\nonumber \\
\label{A5}\end{aligned}$$ where the gradient of $h$ is defined through the change of variable $$\nabla h[{\bf r}({\bf a})]
= \left(\frac{\partial {\bf r}}{\partial {\bf a}}\right)^{-1}
\nabla_a h[{\bf r}({\bf a})].
\label{A6}$$ Newton’s equation of motion are then obtained in the form $$\begin{aligned}
\ddot {\bf r}({\bf a}) &=& \frac{1}{\rho_0 H_0}
\{\dot {\bf p}({\bf a})
- [\dot {\bf r}({\bf a}) \cdot \nabla] {\bf A}[{\bf r}({\bf a})]\}
\label{A7} \\
&=& -f[{\bf r}({\bf a})] {\bf \hat z} \times \dot {\bf r}({\bf a})
- g \nabla \{h[{\bf r}({\bf a})] + h_b[{\bf r}({\bf a})] \},
\nonumber\end{aligned}$$ where the Coriolis term has been produced by the antisymmetric combination $$\begin{aligned}
&&(\nabla A) \cdot \dot {\bf r}
- \dot {\bf r} \cdot(\nabla A)
= [\nabla A - (\nabla A)^T] \cdot \dot {\bf r}
\nonumber \\
&&\ \ \ \ \ =\ \left(\begin{array}{cc}
0 & \partial_y A_x - \partial_x A_y \\
\partial_x A_y - \partial_y A_x & 0
\end{array} \right) \dot {\bf r}
\nonumber \\
&&\ \ \ \ \ =\ -\rho_0 H_0 f {\bf \hat z} \times \dot {\bf r}.
\label{A8}\end{aligned}$$
Equation (\[A7\]) is the Lagrangian equivalent of the first line of (\[2.1\]), where one identifies ${\bf v}[{\bf r}({\bf a},t),t] = \dot {\bf r}({\bf a},t)$ and $\frac{d}{dt} = \partial_t + {\bf v} \cdot \nabla$. The second line of (\[2.1\]) follows from the equation of motion for $h[{\bf r}({\bf a},t)]$: $$\begin{aligned}
\frac{d}{dt} h[{\bf r}({\bf a},t)]
&=& - h[{\bf r}({\bf a},t)] \mathrm{tr}
\left[\left(\frac{\partial {\bf r}}{\partial {\bf a}} \right)^{-1}
\nabla_a \dot {\bf r}({\bf a},t) \right]
\nonumber \\
&=& - h[{\bf r}({\bf a},t)] \nabla \cdot \dot {\bf r}({\bf a},t).
\label{A9}\end{aligned}$$
Lagrangian coordinate Liouville theorem {#app:liouvillelagrange}
---------------------------------------
For any Hamiltonian system, the phase space invariant measures take the form $$d\Gamma = \rho({\cal H},\{{\cal C}_i\}) D[{\bf p}] D[{\bf r}]
\label{A10}$$ where the phase space density $\rho$ is an arbitrary function of all of the conserved integrals of the motion (the proof is outlined in a more general context in App. \[app:liouvilleinequiv\]). Here $\{{\cal C}_i\}$ represents the collection of all conserved quantities, besides ${\cal H}$, including total momentum or angular momentum, depending on the domain symmetries, as well as the Casimirs (\[2.10\]), and the circulations (\[2.13\]). The choice of $\rho$ determines the ensemble. The functional integration measure, consisting of an independent product over all coordinates and momenta, may be defined by the continuum limit $$D[{\bf r}] D[{\bf p}] = \lim_{b \to 0} \frac{1}{N_b!} \prod_{j=1}^{N_b}
\frac{d{\bf p}({\bf a}_j) \, d{\bf r}({\bf a}_j)}{(P_0 b)^2},
\label{A11}$$ in which the domain $D$ is approximated by a square mesh of $N_b = A_D/b^2$ fluid parcels of *equal area* $b^2$ (and equal volume $H_0 b^2$). The $1/N_b!$ factor accounts for the arbitrary parcel relabeling symmetry. The factor $1/(P_0 b)^2$ is included to obtain a dimensionless partition function, and a properly normalized free energy, with $P_0$ an arbitrary constant with the same units as ${\bf p}$ (e.g., $P_0 = \rho_0 H_0 V_0$, where $V_0$ is a characteristic fluid velocity).
Conversion to Eulerian coordinates {#app:eulerconv}
----------------------------------
The alternative Eulerian formulation is obtained by recognizing that the coordinate measure $D[{\bf r}]D[{\bf p}]$ corresponds to a physical coordinate–momentum space *Poisson process*, in which each fluid parcel is placed independently, with uniform probability, in the domain ${\cal D} = D \times \mathbb{R}^2$, i.e., at a particular point ${\bf r}({\bf a})$ in $D$ with a particular momentum ${\bf p}({\bf a})$. In order to obtain a sensible limit, we can restrict ${\bf p}$ to a compact domain $D_P \subset \mathbb{R}^2$, with finite area $A_P$, and take the limit $D_P \uparrow \mathbb{R}^2$ in the end.
An equivalent statistical description divides ${\cal D}$ into an arbitrary fixed mesh, and counts the number of parcels $n_{lm} = n({\bf r}_l,{\bf p}_m)$ in each cell of phase space volume $\Delta V = \Delta p^2 \Delta x^2$. The limit $b \to 0$ will be taken first first, at fixed $\Delta x, \Delta p$, so that $n_{lm} \to \infty$. To obtain a sensible limit, one defines the continuous variables, $$\nu_{lm} = \frac{P_0^2 b^2}{\Delta x^2 \Delta p^2} n_{lm}
\label{A12}$$ The partition function is then obtained by freely and independently integrating over each $\nu_{lm}$. The constraint $\sum_{l,m} n_{lm} = N_b$ leads to $$\sum_{l,m} \nu_{lm} = \frac{P_0^2 A_D}{\Delta p^2 \Delta x^2}.
\label{A13}$$ With this normalization, the continuum limit $\Delta x, \Delta p \to 0$ produces $\nu_{lm} \to \nu({\bf p},{\bf r})$, with $$\sum_{l,m} \nu_{lm} \Delta x^2 \Delta p^2
\to \int d{\bf p} \int_D d{\bf r} \,
\nu({\bf r},{\bf p}) = P_0^2 A_D.
\label{A14}$$ This continuum notation is intended here only as a heuristic, because $\nu_{lm}$ fluctuates wildly from cell to cell. However, the notational intent is clear, and the more rigorous mathematical statement resides only in the underlying finite dimensional calculations that are then used to derive well defined, smoothly varying, limiting forms for thermodynamic averages.
The corresponding Eulerian phase space measure is now given by an integration over the phase space defined by all functions $\nu$: $$d\Gamma = \rho({\cal H},\{{\cal C}_i\}) D[\nu]
\label{A15}$$ defined by the continuum limit $$D[\nu] = \lim_{\Delta V \to 0} \prod_{l,m} d\nu_{lm}.
\label{A16}$$ The usual Eulerian hydrodynamic fields are obtained from the moments $$\begin{aligned}
h_l &=& \frac{H_0 b^2}{\Delta x^2} \sum_m n_{lm}
= \frac{H_0 \Delta p^2}{P_0^2} \sum_m \nu_{lm}
\nonumber \\
{\bf j}_l &=& {b^2}{\Delta x^2} \sum_m n_{lm} {\bf p}_m
= \frac{\Delta p^2}{P_0^2} \sum_m \nu_{lm} {\bf p}_m.
\label{A17}\end{aligned}$$ In continuum notation, $$\begin{aligned}
h({\bf r}) &=& H_0 \int \frac{d{\bf p}}{P_0^2} \nu({\bf r},{\bf p})
\nonumber \\
{\bf j}({\bf r}) &=& \int \frac{d{\bf p}}{P_0^2} {\bf p}\,
\nu({\bf r},{\bf p})
\label{A18}\end{aligned}$$ The velocity is defined as usual by ${\bf j} = \rho_0 h {\bf v}$.
Reduced moment description {#app:reducedmoment}
--------------------------
In principle, the phase space measure (\[A15\]) is applicable to any Hamiltonian and conserved integrals constructed as arbitrary functionals of $\nu$. However, if the phase space integrand depends only on the two moments (\[A18\]), one may reduce (\[A15\]) by integrating out all other degrees of freedom. This is accomplished formally by representing $$\begin{aligned}
\nu({\bf r},{\bf p}) &=& \sum_m M_m({\bf p}) \nu_m({\bf r})
\nonumber \\
\nu_m({\bf r}) &\equiv & \int d{\bf p} M_m({\bf p}) \nu({\bf r},{\bf p})
\label{A19}\end{aligned}$$ as an expansion in a complete set of moment functions $M_n({\bf p})$ (e.g., Legendre polynomials covering the domain $D_P$), constrained by the choice $M_0 = 1$, $M_1 = p_x$, $M_2 = p_y$. One may then write $$d\Gamma = \rho({\cal H},\{{\cal C}_i\}) \prod_n D[\nu_n].
\label{A20}$$ Since $\rho$ depends only on $\nu_0 = h$ and $(\nu_1,\nu_2) = {\bf j}$, one may freely integrate out all higher $\nu_n$, $n > 2$, to obtain the “reduced” Eulerian phase space measure $$d\Gamma_E = \rho({\cal H},\{{\cal C}_i\}) D[h] D[{\bf j}]
\label{A21}$$ defined by the continuum limit $$D[h] D[{\bf j}] = \lim_{\Delta x \to 0}
\prod_l \frac{dh_l \, d{\bf j}_l}{H_0 P_0^2}.
\label{A22}$$ The factor $1/H_0 P_0^2$ is again included for dimensional purposes. Unlike (\[A11\]), which includes the parcel area $b^2 \to 0$, the microgrid area $\Delta x^2$ does not appear in (\[A22\]) because there is no $1/N_E!$ combinatorial factor in this representation. Note that the constraint of fixed fluid volume $H_0 A_D = \sum_l h_l \Delta x^2$ is already accounted for in $\rho$ through the Casimir (\[2.9\]) with $w \equiv 1$.
The form (\[A22\]) may also be derived directly from the Eulerian fluid equations (\[2.1\]) [@RVB2016]. However, the derivation here exhibits the possibility of a much more general class of Hamiltonians and conserved integrals that could depend on the full $\nu({\bf r},{\bf p})$, not just a few of its moments.
Potential vorticity description {#app:potvort}
-------------------------------
Significant work was done in Sec. \[sec:canonfields\] to show that (for given $h$) the fluid current ${\bf j}$ is fully represented by the vorticity and compression fields $\Omega,Q$. To substitute the latter as the fundamental phase space variables, one first changes variables via $$\prod_l dh_l \, d{\bf j}_l = \prod_l \rho_0^2 h_l^2 dh_l \, d{\bf v}_l.
\label{A23}$$ With $\omega = \nabla \times {\bf v} = -\nabla^2 \psi^V$ and $q = \nabla \cdot {\bf v} = -\nabla^2 \phi$, one may further change variables ${\bf v} \to (\omega,q,{\bm \psi}^0)$, in which ${\bm \psi}^0 = \{\psi^0_m \}$ represents the potential flow component ${\bf v}^P = \nabla \times \psi^P$, with an overall constant Jacobian $J_0 = \Delta x^{2N_E}$ (where $N_E = A_D/\Delta x^2$ is the number of spatial grid cells): $$\begin{aligned}
\prod_l h_l^2 dh_l \, d{\bf v}_l
&=& \prod_{m=2}^{N_D} d\psi^0_m \,
\prod_l \Delta x^2 h_l^2 dh_l \, dq_l \, d\omega_l
\nonumber \\
&=& \prod_{m=2}^{N_D} d\psi^0_m \,
\prod_l \Delta x^2 h_l^4 dh_l \, dQ_l \, d\Omega_l,
\nonumber \\
\label{A24}\end{aligned}$$ where in the last expression we have introduced $\Omega_l = (\omega_l + f_l)/h_l$, $Q_l = q_l/h_l$. One obtains finally: $$\begin{aligned}
d\Gamma_E &=& \rho({\cal H},\{{\cal C}_i\})
\prod_{m=2}^{N_D} \frac{d\psi^0_m}{P_0 H_0}
\nonumber \\
&&\times\ \lim_{\Delta x \to 0} \prod_l \frac{\rho_0^2 \Delta x^2}{H_0 P_0^2}
h_l^4 dh_l \, dQ_l \, d\Omega_l.
\label{A25}\end{aligned}$$ This is the basic form that is used in Sec. \[sec:statmech\] to derive the shallow water equilibrium equations.
Liouville theorem and inequivalent phase space measures {#app:liouvilleinequiv}
=======================================================
Given its centrality to the differences between the present work and that of RVB [@RVB2016], for completeness we carefully summarize here, at a more general level, the constraints enforced by the Liouville theorem on the form of the infinite-dimensional invariant phase space measure. It is shown that the continuum limit obtained from the fluid parcel *area* discretization consistent with the theorem proven in App. \[app:liouville\] is explicitly inequivalent to the fluid parcel *volume* discretization adopted by RVB.
To focus the discussion, consider a phase space $\Gamma$ defined by a single continuous scalar field ${\bm \varphi} \equiv \{\varphi({\bf r}) \}_{{\bf r} \in D}$, with $D$ a spatial domain (2D in this case). The generalization to multiple fields simply adds more indices. The field is assumed to obey a first order equation of motion of the form $$\partial_t \varphi({\bf r}) = V({\bf r};{\bm \varphi}),
\label{B1}$$ where ${\bf V}[{\bm \varphi}] = \{V({\bf r};{\bm \varphi})\}_{{\bf r} \in D}$ is the phase space flow field, each of whose “components” ${\bf r}$ is a functional of ${\bf \varphi}$. A probability density $P[{\bm \varphi}]$ on the phase space obeys an equation of motion $$\partial_t P + \nabla_{\bm \varphi} \cdot (P {\bf V}) = 0,
\label{B2}$$ in which the explicit form of the phase space divergence of any vector ${\bf F}$ is defined by the functional derivative $$\nabla_{\bm \varphi} \cdot {\bf F}[{\bm \varphi}] = \int_D d{\bf r}
\frac{\delta F({\bf r};{\bm \varphi})}{\delta \varphi({\bf r})}.
\label{B3}$$ Conservation of probability follows by integrating (\[B2\]) over $\Gamma$ and applying the infinite dimensional Gauss law to the second term: $$\partial_t \int_\Gamma {\cal D}[{\bm \varphi}] P[{\bm \varphi}]
= -\int_{\partial \Gamma} P[{\bm \varphi}] {\bf V}[{\bm \varphi}]
\cdot d{\bm \Sigma}[{\bm \varphi}] \to 0,
\label{B4}$$ where $d{\bm \Sigma}$ is the outward pointing area element. The vanishing of the right hand side is based on the assumption that $P$ vanishes as the boundary $\partial \Gamma$ is pushed to infinity (or, in some models with bounded fields, that there is a well defined finite boundary through which the flows do not pass, ${\bf V}[{\bm \varphi}] \cdot d\Sigma[{\bm \varphi}] = 0)$.
Consistent with the form of (\[B3\]), and the subsequent application of Gauss’s law, the phase space functional integral underlying averages with respect to $P$ must be defined by free integration over each field component: $$\int {\cal D}[{\bm \varphi}] = \prod_{\bf r} \int d\varphi({\bf r})
\label{B5}$$ The right hand side is most conveniently defined by first approximating $D$ by a regular, uniform grid $\{{\bf r}_i\}_{i=1}^N$ of mesh size $b$ and then taking the continuum limit, $$\int {\cal D}[{\bm \varphi}] = \lim_{b \to 0}
\frac{1}{{\cal N}(b)} \prod_{i=1}^N \int d\varphi_i,
\label{B6}$$ where ${\cal N}(b)$ is an overall normalization. Of course, other, non-uniform grids may be chosen, but they are constrained by the requirement that they give the same continuum limit, e.g., for physically well defined statistical averages. Rigorous examples of this construction include Feynman path integrals, and higher dimensional random surface integrals, that may defined through Brownian motion Wiener measures. It should be emphasized here that there is nothing in principle that forbids a grid choice that gives a physically different continuum limit from generating mathematically consistent probabilities, but it must correspond to a different dynamical model than (\[B1\]).
This association between the form of the divergence and the integration measure is critical to proper application of the Liouville theorem. Specifically, the Liouville theorem holds if the phase space flows can be shown to be divergence-free: $$\nabla_{\bm \varphi} \cdot {\bf V}[{\bm \varphi}] = 0.
\label{B7}$$ This follows trivially for Hamiltonian flows, where ${\bm \varphi} = ({\bf q},{\bf p})$ is composed of conjugate pairs of coordinate and momentum variables. Then $${\bf V}({\bf q},{\bf p}) = (\nabla_{\bf p} H, -\nabla_{\bf q} H),
\label{B8}$$ is derived from a Hamiltonian $H({\bf q},{\bf p})$, and $$\nabla_{\bm \varphi} \cdot {\bf V}
= \nabla_{\bf q} \cdot \nabla_{\bf p} H
- \nabla_{\bf p} \cdot \nabla_{\bf q} H = 0.
\label{B9}$$
When (\[B7\]) is satisfied, it follows from (\[B3\]) that the probability density is freely advected by the flow, $$\partial_t P + {\bf V} \cdot \nabla_{\bm \varphi} P = 0,
\label{B10}$$ and in particular, an invariant measure $P_0$, for which $\partial_t P_0 = 0$, must obey $${\bf V} \cdot \nabla_{\bm \varphi} P_0[{\bm \varphi}] = 0.
\label{B11}$$ However, one then notes that $$\frac{d}{dt} P_0[{\bm \varphi}]
= \nabla_{\bm \varphi} P_0[{\bm \varphi}] \cdot \frac{d}{dt} {\bm \varphi}
= {\bf V}[{\bm \varphi}] \cdot \nabla_{\bm \varphi} P_0[{\bm \varphi}] = 0,
\label{B12}$$ showing that $P_0[{\bm \varphi}]$ is conserved by the flow, and hence one may write $P_0[{\bm \varphi}] = \rho(\{C_l[{\bm \varphi}]\})$, an arbitrary (normalized) function of the set of all integrals of the motion $\{C_l\}$. The microcanonical ensemble adopted in [@RVB2016] corresponds to the choice $$\rho = \prod_l \delta(c_l - C_l[{\bm \varphi}]).
\label{B13}$$
One may compare the above formulation to one in which, instead, the Liouville theorem takes the form $$\int_D d{\bf r} w({\bf r})
\frac{\delta V({\bf r};{\bm \varphi})}{\delta \varphi({\bf r})} = 0,
\label{B14}$$ where $w$ is a positive function that is in general a functional of ${\bm \varphi}$—it corresponds to the fluid height field $h$ in the shallow water application. One may construct a domain (but not area) preserving map ${\bf r}({\bf a}): D \to D$ with Jacobian $$J({\bf a}) \equiv \left|\frac{\partial {\bf r}}{\partial {\bf a}} \right|
= \frac{w_0}{w[{\bf r}({\bf a})]},\ \
w_0 \equiv \frac{1}{A_D} \int_D d{\bf r} w({\bf r}),
\label{B15}$$ where $A_D$ is the area of $D$, and with corresponding fields $\tilde \varphi({\bf a}) \equiv \varphi[{\bf r}({\bf a})]$ and $\tilde {\bf V}({\bf a};\tilde {\bm \varphi}) = {\bf V}[{\bf r}({\bf a}); {\bm \varphi}|_{{\bf r} \to {\bf r}({\bf a})}]$. One then obtains $$\int_D d{\bf r} w({\bf r})
\frac{\delta V({\bf r};{\bm \varphi})}{\delta \varphi({\bf r})}
= \int_D d{\bf a}
\frac{\delta \tilde V({\bf a};\tilde {\bm \varphi})}
{\delta \tilde \varphi({\bf a})} = 0,
\label{B16}$$ corresponding to a standard Liouville theorem in the new variables. It follows, according to (\[B5\]) and (\[B6\]), that the continuum limit should be obtained using a uniform grid in the ${\bf a}$ coordinate, hence to a nonuniform grid in the ${\bf r}$ coordinate, with elements of area $b^2 w_0/w[{\bf r}({\bf a}_i)]$. In the context of the shallow water equations, this corresponds precisely, as stated, to the equal fluid volume element choice.
Because the height field fluctuates so strongly in the shallow water model, this constitutes a huge effect that demonstrably changes the equilibrium flow equations. As an illustrative example, consider a Gaussian free energy of the form $$\begin{aligned}
F(\beta) &=& -\ln\left[\int {\cal D}[{\bm \varphi}]
e^{-\beta H[{\bm \varphi}]} \right]
\nonumber \\
H[{\bm \varphi}] &\equiv& \frac{1}{2} \int_D
d{\bf r} w({\bf r}) \varphi({\bf r})^2
\label{B17}\end{aligned}$$ The equal area discretization produces (up to a normalization constant) $$F(\beta) = \frac{1}{2} \int_D d{\bf r} \ln[\beta w({\bf r})/2\pi]
\label{B18}$$ while the equal volume discretization produces the manifestly different result $$F(\beta) = \frac{1}{2} A_D \ln[\beta w_0/2\pi].
\label{B19}$$
Kinetic energy and momenta in terms of basic fields {#app:KEPi}
===================================================
Although not needed for the statistical mechanics treatment, for completeness we provide here explicit expressions for the kinetic energy (\[3.21\]) and kinetic part of the momentum (\[3.22\]) in terms of the basic fields $\Omega,Q,h$.
Substituting the velocity decomposition (\[3.8\]), along with the expressions (\[3.10\]) and (\[3.11\]) for the potentials, into (\[3.21\]) one obtains for the kinetic energy $$\begin{aligned}
E_K = E_K^{VC} + E_K^{VCP} + E_K^P,
\label{C1}\end{aligned}$$ where the vortical and compressional components are encompassed by the term $$\begin{aligned}
E_K^{VC} &=& \frac{\rho_0}{2} \int_D d{\bf r} h({\bf r})
\left|\nabla \times \psi^V({\bf r}) - \nabla \phi({\bf r}) \right|^2
\label{C2} \\
&=& \frac{\rho_0}{2} \int_D d{\bf r}' \int_D d{\bf r}'
\left[\begin{array}{c}
\omega({\bf r}) \\ q({\bf r})
\end{array} \right]^T
\hat {\cal G}_h({\bf r},{\bf r}')
\left[\begin{array}{c}
\omega({\bf r}') \\ q({\bf r}')
\end{array} \right],
\nonumber\end{aligned}$$ in which the tensor Green function, which depends on the height field, is defined by
$$\hat {\cal G}_h({\bf r},{\bf r}') = \int_D h(\bar {\bf r}) d\bar {\bf r}
\left[\begin{array}{cc}
\bar \nabla G_D(\bar {\bf r},{\bf r}) \cdot \bar \nabla G_D(\bar {\bf r},{\bf r}') &
\bar \nabla G_D(\bar {\bf r},{\bf r}) \times \bar \nabla G_N(\bar {\bf r},{\bf r}') \\
-\bar \nabla G_N(\bar {\bf r},{\bf r}) \times \bar \nabla G_D(\bar {\bf r},{\bf r}') &
\bar \nabla G_N(\bar {\bf r},{\bf r}) \cdot \bar \nabla G_N(\bar {\bf r},{\bf r}')
\end{array} \right].
\label{C3}$$
The potential term is given by $$\begin{aligned}
E_K^P &=& \frac{\rho_0}{2} \int d{\bf r}
h({\bf r}) |{\bf v}^P({\bf r})|^2
\nonumber \\
&=& \frac{\rho_0}{2} \sum_{l,m=2}^{N_D}
\Gamma_{lm}[h] (\psi_l^0 - \psi_1^0)(\psi_l^0 - \psi_1^0)
\label{C4}\end{aligned}$$ where $$\Gamma_{lm}[h] = \Gamma_{ml}[h] = \int d{\bf r} h({\bf r})
{\bf v}^P_l({\bf r}) \cdot {\bf v}^P_m({\bf r})
\label{C5}$$ is also a (linear) functional of the height field, and does not simplify to a pure circulation integral. Finally, the cross term may be written in the form $$\begin{aligned}
E^{VCP}_K &=& \rho_0 \int_D d{\bf r} {\bf v}^P({\bf r})
\cdot [\nabla \times \psi^V({\bf r}) - \nabla \phi({\bf r})]
\nonumber \\
&=& - \rho_0 \sum_{l=2}^{N_D} (\psi_l^0 - \psi_1^0)
\nonumber \\
&&\times\ \int d{\bf r} [H^V_l({\bf r}) \omega({\bf r})
+ H^C_l({\bf r}) q({\bf r})],
\label{C6}\end{aligned}$$ where $$\begin{aligned}
H^V_l({\bf r}) &=& -\int_D h(\bar {\bf r}) d\bar {\bf r}
{\bf v}^P_l(\bar {\bf r}) \times \bar \nabla G_D(\bar {\bf r},{\bf r})
\nonumber \\
H^C_l({\bf r}) &=& \int_D h(\bar {\bf r}) d\bar {\bf r}
{\bf v}^P_l(\bar {\bf r}) \cdot \bar \nabla G_N(\bar {\bf r},{\bf r})
\label{C7}\end{aligned}$$ are inner products between the potential velocity field eigenfunction and velocity fields generated by either unit point vortex or unit compression at ${\bf r}$. Both are linear functionals of the height field, and therefore, unlike (\[3.9\]), do not vanish.
Substituting the velocity decomposition into (\[3.22\]), one obtains $$\begin{aligned}
\Pi_K &=& \Pi_K^{VC} + \Pi_K^P
\nonumber \\
\Pi_k^{VC} &=& -\rho_0 \int_D d{\bf r}
\left[H^V_\Pi({\bf r}) \omega({\bf r})
+ H^C_\Pi({\bf r}) q({\bf r}) \right]
\nonumber \\
\Pi_K^P &=& \rho_0 \sum_{l=2}^{N_D}
\Gamma^\Pi_l (\psi_l^0 - \psi_1^0)
\label{C8}\end{aligned}$$ where $$\begin{aligned}
H^V_\Pi({\bf r}) &=& -\int_D h(\bar {\bf r}) d\bar {\bf r}
{\bf v}_\Pi(\bar {\bf r}) \times \nabla G_D(\bar {\bf r},{\bf r})
\nonumber \\
H^C_\Pi({\bf r}) &=& \int_D h(\bar {\bf r}) d\bar {\bf r}
{\bf v}_\Pi(\bar {\bf r}) \cdot \nabla G_N(\bar {\bf r},{\bf r})
\nonumber \\
\Gamma^\Pi_l &=& \int_D h({\bf r}) d{\bf r}
{\bf v}_\Pi({\bf r}) \cdot {\bf v}^P_l({\bf r}).
\label{C9}\end{aligned}$$ Note that all of these are also linear functionals of the fluctuating height field $h$.
Substituting $\omega = h\Omega+f$, $q = hQ$ provides the desired representation in terms of $\Omega,Q,h$.
A. Renaud, A. Venaille, and F. Bouchet, “Equilibrium statistical mechanics and energy partition for the shallow water model,” [J. Stat. Phys. **163**, 784–-843 (2016)](http://dx.doi.org/10.1007/s10955-016-1496-x) J. Miller, [Phys. Rev. Lett. **65**, 2137 (1990)](http://dx.doi.org/10.1103/PhysRevLett.65.2137).
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Notably, a mathematically equivalent theory of galaxy dynamics (based on the Boltzmann equation) was derived much earlier:D. Lynden-Bell, [Mon. Not. R. Astron. Soc. **136**, 101 (1967)](http://dx.doi.org/10.1093/mnras/136.1.101). However, these results failed to transition to the fluid dynamics community.
L. Onsager, [Nuovo Cimento Suppl. **6**, 279 (1949)](http://dx.doi.org/10.1007/BF02780991).
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P. B. Weichman, [Phys. Rev. Lett. **109**, 235002 (2012)](http://dx.doi.org/10.1103/PhysRevLett.109.235002).
S. Thalabard, B. Dubrulle, and F. Bouchet, [J. Stat. Mech.: Theory and Experiment **1**, P01005 (2014)](http://dx.doi.org/10.1088/1742-5468/2014/01/P01005).
P. B. Weichman and D. M. Petrich, [Phys. Rev. Lett. **86**, 1761–64 (2001)](http://dx.doi.org/10.1103/PhysRevLett.86.1761).
P. H. Chavanis and J. Sommeria, [Phys. Rev. E **65**, 026302 (2002).](http://dx.doi.org/10.1103/PhysRevE.65.026302)
For a recent review, see: F. Bouchet and A. Venaille, [Phys. Rep. **515**, 227 (2012)](http://dx.doi.org/10.1016/j.physrep.2012.02.001).
R. H. Kraichnan, [Phys. Fluids **10**, 1417 (1967)](http://dx.doi.org/10.1063/1.1762301).
R. Kraichnan and D. Montgomery, [Rep. Prog. Phys. **43**, 547 (1980)](http://dx.doi.org/10.1088/0034-4885/43/5/001).
V. E. Zakharov, G. Falkovich, and V. Lvov, [*Kolmogorov Spectra of Turbulence I, Wave Turbulence* (Springer, Berlin, 1992)](http://dx.doi.org/10.1007/978-3-642-50052-7).
An alternative, entirely equivalent, interpretation of the shallow water equations is that of a purely 2D fluid with density $h$, compressibility $g$, and background potential $h_b$. Surface waves are now acoustic compressional waves.
The velocity-dependent potential in (\[2.1\]) in in the form of a Lorentz force induces by a vertical magnetic field $f {\bf \hat z}$. The relation ${\bf v} = {\bf p} - {\bf A}$ between canonical momentum and physical fluid parcel velocity (that accounts for physical momentum redirection generated by the Coriolis force) then follows in the standard way. The same combination appears in the potential vorticity $\Omega = h^{-1} \nabla \times ({\bf v} + {\bf A})$. See also the Lagrangian coordinate representation discussed in App. \[app:lagham\], where a slightly different normalization of ${\bf A}$ is used.
Note that the result equivalent to $\Phi \equiv 0$ could have been obtained in principle by integrating $Q$ out directly in (\[4.4\]), and then performing a KHS transformation on $\Omega$ alone. However, it is useful to exhibit the symmetric appearance of $\Omega$ and $Q$, with the major difference being only that there is no Lagrange multiplier function analogous to $\mu$ controlling the behavior of $Q$. The above model containing $\Omega,h$ alone, is derived in Sec. \[subsec:nonlinmembrane\] via the reverse procedure, undoing the KHS transformation.
Recall here that $\Psi^\mathrm{eq}$ is real, but the contour $C$ dictates that $\delta \Psi$ be pure imaginary, ensuring that $e^{\bar \beta W} \sim e^{\bar \beta |\nabla (i \delta \Psi)|^2/2\rho_0 h}$ is rapidly decreasing for large $|\delta \Psi|$. Note that the $h$-integral contained in the definition of $W$ converges only if the real part of $\xi = -|\nabla \Psi|^2$ is positive, and this is guaranteed for all $\delta \Psi$ only if $C$ coincides with the imaginary axis. The shifted coutour therefore, as is standard in steepest descent calculations, relies on analytic continuation of $W$ to $\mathrm{Re} \, \xi < 0$. As an example, the result $\int_0^\infty d\lambda e^{-(\xi/\lambda + \lambda)} = \sqrt{\xi} K_1(\sqrt{\xi})$ extends the definition of the integral to any complex value of $\xi$ [@GR1980].
Alternatively, RVB replace $\bar Q$ by the nonlocal quantity $\mu = (-\nabla^2)^{-1/2} q = (-\nabla^2)^{1/2} \phi$ which also produces a $\Delta x$ scaling. Their formulation also produces a well defined Gaussian distribution in the continuum limit, which is also statistically independent of the other fields. In the present formulation one could introduce the analogous form $\mu = (-\nabla \cdot h \nabla)^{1/2} \phi$, which may be useful for some purposes \[e.g., formally integrating out $Q$ in (\[4.4\]) without using the KHS transformation\], but does not produce statistical independence.
Technically, the discrete version of the field $\Psi$ actually lives on the dual lattice, while the components of the finite difference $\nabla \times \Psi(\bar {\bf r}) = \Delta x^{-1} [\Psi(\bar {\bf r} + \Delta x({\bf \hat x} + {\bf \hat y})/2) - \Psi(\bar {\bf r} + \Delta x({\bf \hat x} - {\bf \hat y})/2),\ \Psi(\bar {\bf r} + \Delta x({\bf \hat y} - {\bf \hat x})/2) - \Psi(\bar {\bf r} + \Delta x({\bf \hat x} + {\bf \hat y})/2)]$ live on the direct lattice links.
The functional derivative in (\[6.8\]) is quite singular, including divergent terms $\delta(0)$ and $\nabla \times G_h({\bf r},{\bf r}')|_{{\bf r}'={\bf r}}$. However, this form may be regularized to produce the desired limiting result. The underlying physical reasoning is that the self-advection of a vortex, which is measured by the (\[6.8\]), does not come from the singular vortex core, but only from “image flows” generated by the boundary conditions and by the bottom topography. This allows one, for example, to subtract off the logarithmic singularity in $G_h$.
I. S. Gradshteyn and I. M. Rhyzhik, [*Table of Integrals, Series, Products* (Academic Press, New York, 1980)](http://www.sciencedirect.com/science/book/9780122947605).
See, e.g., H. E. Stanley, [*Introduction to phase transitions and critical phenomena* (Oxford University Press, 1971)](http://www.oupcanada.com/catalog/9780195053166.html).
P. M. Chaikin and T. C. Lubensky, [*Principles of condensed matter physics* (Cambridge University Press, 1995)](http://www.cambridge.org/us/academic/subjects/physics/condensed-matter-physics-nanoscience-and-mesoscopic-physics/principles-condensed-matter-physics?format=PB&isbn=9780521794503).
R. Phillips, J. Kondev, J. Theriot, and H. Garcia, [*Physical Biology of the Cell, Second Edition* (Garland Science, 2013)](http://www.garlandscience.com/product/isbn/9780815344506).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The fleet management of mobile working machines with the help of connectivity can increase not only safety but also productivity. However, rare mobile working machines have taken advantage of V2X. Moreover, no one published the simulation results that are suitable for evaluating the performance of the ad-hoc network at a working site on the highway where is congested, with low mobility, and without building. In this paper, we suggested that IEEE 802.11p should be implemented for fleet management, at least for the first version. Furthermore, we proposed an analytical model for machines to estimate the ad-hoc network performance about the expected delay and the probability of packet loss in real-time based on the simulation results we made in $ns3$. The model of this paper can be used for determining when shall ad-hoc or cellular network be used in the corresponding scenario.'
author:
-
-
-
bibliography:
- 'ConnectedVehicle.bib'
title: 'Realtime Estimation of IEEE 802.11p for Mobile working Machines Communication respecting Delay and Packet Loss\'
---
V2X, IEEE 802.11p, CSMA, Mobile machines
Introduction
============
Besides artificial intelligence [@Xiang.2020], the fleet management of mobile machines is the principal research direction of the internet of things in the fields of mobile working machinery. Currently, the mobile machines are distributed sparsely in the working site and working at low transport speed to avoid a collision. With the vehicle-to-everything(V2X), we envision that the mobile machines can work more densely with each other and transport the material much quicker since the collision is impossible with sensible communication. The most challenging and research-worthy use case can be described as the task of repairing the highway. During the road is repaired, congestion of traffic is usually expected. According to the study from Triantis, traffic congestion causes significant economic losses [@Triantis.2011]. Apparently, by investing more machines with the help of V2X technology in a particular site can surely improve the working productivity, so that the economy lost due to the congestion can be diminished. However, the challenging thing is the congested passenger cars also occupy the channel load; thus, the possibility of the packet collision as well as the packet loss will increase. In this paper, we first evaluate the performance of the IEEE 802.11p standard for varying node density rates by means of simulations using $ns-3$ [@Henderson.2008]. After that, we propose an analytical model based on the simulation results for mobile machines to predict the mean delay and probability of package loss of the transmit since the simulation model is computationally expensive.
Why we use the IEEE 802.11p?
============================
Despite the fact that LTE has a series of advantages, we would like to adopt the IEEE 802.11p as our first version for connected mobile machines due to the following reasons. First of all, to fully make the advantages of C-V2X, mobile machines need a base station nearby, which varies from 10m until 10km[@Siomina.2006]. However, for the fleet of mobile machines that are working far away from urban, they might fail to find a base station nearby. Moreover, the usage of 802.11p is free of charge. Different from the cellular network which the users must pay for the service from the network operators, the 5.9 GHz band is a free but licensed spectrum [@Jiang.2008]. In addition, IEEE 802.11p is well designed for the vehicle industry so that no additional modification is needed for vehicle onboard ECU [@Bazzi.2017]. Thus, the compatibility of IEEE 802.11p is better for the mobile machines designed without the consideration of V2X. Usually, mobile machines drive at a relatively lower speed, and the communication between cars and mobile machines is not essential; thus, the lack of ability to deal with vehicle mobility by IEEE 802.11p can be ignored, based on the analysis of Alasmary’s study [@Alasmary.2012]. Although there have no consensus about which wireless technology is the more promising technology, scientists from both sides agree that the combination of LTE and 802.11p have a certain improvement in performance compared to if only one technology is used [@Bazzi.2017; @Chen.2016; @Mir.2014; @Viriyasitavat.2015]. Thus, we would like to use IEEE 802.11p as the communication technology for our initial version fleet management. Even though the passenger car industry adopts cellular technology in the future, the idea of using IEEE 802.11p for mobile machines is still sensible, because the congestion of the channel is consequently alleviated.
Modelling
=========
Mecklenbräuker has shown the common scenarios in their paper [@Mecklenbrauker.2011]. Unfortunately, for mobile machines that have the task to repair the highway, the scenario does not belong to these common scenarios. Firstly, there has usually no buildings around the working site, but the traffic is congested. Secondly, instead evaluate the communication among all the participants in the ad-hoc network, only communication among mobile machines is essential.
Propagation model
-----------------
In [@Stoffers.201287201289], a comparative analysis between different propagation models is performed. Based on Stoffer’s study, there is no best model for all cases, and the users should select the model depending on the concrete environment. Because we are mainly interested in MAC performance and the highway is more similar to an urban scenario, we used a log-distance path loss model proposed by [@Erceg.1999]. It is denoted as
$$PL(dB)= PL(d_0) + 10nlog(\frac{d}{d_0})\label{eq}$$
where PL (d0) is defined as the path loss at the reference distance(d0), and PL(d0) = 46.6777dB. n refers to the path loss distance exponent varying from the propagation environment, and n = 3.
Since the single factor that influences receive power is the distance from the transmitter, in the following simulations, the dynamic mobility model is not applied to vehicles. Still, the relative positions of the vehicles are randomly initialized.
Cooperative Awareness Message(CAM)’s generation model
-----------------------------------------------------
Venel presented that CAMs are generated at a rate in a range of 2 to 20 packets /second corresponding to multiple factors such as driver’s reaction time and vehicle speed [@Vinel.2009]. Thereby, we apply a mean value from them, namely 10 packets/ second(10 Hz). In addition, the length of a packet varies from different applications in real-world vehicular communications. In the following simulations, packet length is set to be 450 bytes, which ensures the necessary information for the safety-related application. Since the generation rate and CAM length are constant throughout the simulation, the channel load is only depended on the number of nodes in the scenario.
CDMA/CA {#AA}
-------
CSMA/CA algorithm is specified in IEEE 802.11 is to schedule transmissions over a single channel by differing the access attempt with a random back-off time. Because Physical Layer Convergence Protocol(PLCP) header is modulated with Binary Phase Shift Keying(BPSK) [@Tse.2004] and the payload is transmitted in the form of Quadrature Phase Shift Keying(QPSK) modulation, two range are expected: transmission range and sensing range.
Since the primary emphasis of this paper is on the congestion control algorithms at MAC layer and CAM length is constant, the term delay in the following part will always refer to the back-off time between the time point that a node request for channel access and the packet is forwarded from the MAC layer to the PHY layer, neglecting the transmission time depending on packet length and propagation time depending on distance. Table \[ft\_tab\_ex\] contains the vital parameters setting that we use.
Parameters Value Unit
------------------------ ------- -------
TxPower 17 dBm
Packet length 450 Bytes
Packet generation rate 10 Hz
Channel width 10 MHz
Data rate (BPSK) 3 Mbps
Data rate (QPSK) 6 Mbps
CWmin 15 -
AIFSN 7 -
Time Slot 13 µs
SIFS 32 µs
EIFS 120 µs
: Simulation parameters
\[ft\_tab\_ex\]
In short, the scenario we analyzed is a working site on the highway where the communication performance among mobile machines under the interference from cars nearby.
Empirical model for fast estimation of ad-hoc network performance
=================================================================
Although $ns-3$ can simulate the V2X performance regarding the delay and the probability of lost packet, we still need a quick estimation method, so that onboard ECU can obtain V2X performance in real-time and evaluate the plausibility of V2X data. Therefore, we build an empirical model to fast estimate the network performance based on the results from $ns-3$. Since the contention behavior due to CSMA/CA in corresponding ranges should follow the same roles, which highly depend on the number of neighbors, we introduce the analytical model as follows.
LuT generation
--------------
For each Cluster, e.g., the area within transmission range and the area between transmission and sensing range, we generate a Lookup-Table(LuT) in advance, which contains a set of crucial performance indicators in relationship with varying number of neighbors. To reduce the effect of randomness, we average the indicators from a large number of CAM transmissions.
To generate LuT for 1 cluster, we execute the following simulations. The neighbors are located at the same position with 60 meters away from the transmitter. The number of neighbors varies from 5 to 200, with a step of 5 in each scenario. Furthermore, for each of the 40 scenarios, 5 simulations are conducted, in which every single node schedules 1000 transmissions. The same simulations are executed for the 2. LuT, only the neighbors are 140m away from the transmitter.
![Packet delay probability, packet collision probability, packet loss probability and mean delay measured with varying number of neighbors in 2 cluster are included in the LuT []{data-label="fig:LUT_4"}](Figures/LUT_4.eps){width="3.95in"}
Four metrics of the transmitter are measured, as shown in Fig.\[fig:LUT\_4\], e.g. collisions probability($P_{c}$), packet delay probability($P_{d}$), packet loss probability($P_{l}$), and mean delay($t_{md}$). The term collision indicates the access attempt occurs during the duration, in which another node is transmitting. Moreover, the access attempt can also be differed due to the on-going AIFS, which follows the previous transmission, even though the channel is idle. Therefore, the percentage of delayed packets is slightly higher than the percentage of collisions. The metrics packet delay probability and mean delay indicate how probable the packet would be delayed due to an access contention, and once delay occurs, what would be the average duration.
Performance estimation
----------------------
For each on broad unit in the scenario, the number of neighbors located in each of the two Clusters are measured. The analytical result is derived from the sum of two values that are interpolated and extracted from LuTs. Furthermore, the upper limit for an analytical percentage is equal to 1. Eq. and eq. demonstrates this idea,
$$\hat{{\Phi}}_{A,t} = LoU_{t,1}(n_T) + LoU_{t,2}(n_S)\label{eq:NaiveEstimation,t}$$
$$\hat{\tilde{\Phi}}_{A,p} = min(1, LoU_{p,1}(n_T) + LoU_{p,2}(n_S)\label{eq:NaiveEstimation,p}$$
where $\hat{\tilde{\Phi}}_A $ is the naive estimation of the performance of the ad hoc using the analytical model, the footnote $t$ and $p$ denote the estimation in terms of time and probability, respectively. $n_T$ is the node numbers inside of transmission range, $n_S$ is the node numbers inside of sensing range.
Validation and calibration
==========================
In this section, we first validate the viability of the analytical model and then introduce the correction factor to eliminate the error between the naive LuT and the realistic simulation results.
In the validation simulation, the traffic scenario is set to be a 1500m long highway with 3 lanes in each direction. 500 onboard units equipped with 802.11p devices are located static. Congested traffic due to a highway worksite is assumed. The simulation is set up with a total simulation time of 100s, in which the vehicles are randomly distributed on the road.
The delay relevant metrics are simulated and estimated among all onboard units. This is because each transmission has a different channel access time, which is independent of reception. For each onboard unit, the packet loss probability is measured on a random receiver, which is located within its’ 15m range, corresponding to two cooperating mobile machines.
Fig.\[fig:fig3\] represents the correlation coefficients for each performance metric, which evaluate the strength of the association between simulated and analytical results. For an optimum fitting, the blue dots are supposed to be correctly distributed along the diagonal line, with a correlation coefficient equal to 1. The correlation coefficients for the mean delay, packet delay probability, and packet loss probability are 0.9417, 0.9277 and 0.9167, which manifest a strong correlation and satisfied estimation ability of the analytical model.
![Correlation coefficients of 3 metrics are close to 1, which indicate a good feasibility of analytical estimation. To increase estimation accuracy, we introduce $\tilde{f_{c}}$ []{data-label="fig:fig3"}](Figures/fig3.eps){width="3.7in"}
To optimize the estimation performance of the proposed analytical model, the term correction factor ($f_c$) is introduced, $$\tilde{f_c} = \frac{\tilde{\Phi}_S}{\hat{\tilde{\Phi}}_A}\label{eq:CorrectionFactor}$$ where $\tilde{\Phi}_S, \hat{\tilde{\Phi}}_A$ are the performance matrix from the simulation and the analytical model regarding the $t_{md}$, $P_{d}$,$P_{l}$, separately.
Obviously, our goal can be demonstrated as eq.
$$min(J) =\sum_{i}^{n=N} (\tilde{f_c} \cdot \hat{\tilde{\Phi}}_A - \tilde{\Phi}_S)^2 \label{eq:LossFunction}$$
where N denotes the total number of vehicles.
The $ \tilde{\Phi}_S / \hat{\tilde{\Phi}}_A $ is shown in the bottom right sub-figure in Fig. \[fig:fig3\]. The three curves from top to bottom indicate the $\tilde{f}_{c}$ for mean delay, packet delay probability and packet loss probability. The uniform color in the center area indicates that the naive analytical estimation method has stable performance and thus can be adjusted by multiplying appropriate correction factor $f_c$. Among 3 metrics, packet loss probability is dramatically underestimated and needs a larger $f_{c}$. This is because, in the Lut generation scenario, a reception is failed only due to multiple differed access attempts to access the channel simultaneously. However, in the real-time simulation, the transmissions from the hidden nodes cause interference at the receiver. Consequently, the reception is more like to corrupt due to lower SINR.
The correction factor differ in the discontinuous edge of the scenario, where hidden node problem is not obvious. In this case, we introduce another correction matrix. Tab. \[tab: Correction factors\] records the correction factor in the middle($f_{c,m}$) and the correction factor at the edge($f_{c,e}$), where the results are calculated based on eq. .
$f_c$ middle edge
-------------------------- -------- --------
Mean delay 1.0857 1.3048
Packet delay probability 0.7516 0.9671
Packet loss probability 2.2617 2.9121
: Correction factors
\[tab: Correction factors\]
After using the correction factors, the analytical model outputs a very similar result to the simulation model.
Conclusion
==========
In this paper, we suggest that the IEEE 802.11p is a better solution for the first version of the fleet management of mobile working machines based on the analysis of the ad-hoc network and the cellular network. Moreover, we propose an analytical model to let mobile working machines have a real-time sense of the packet delay probabiliy, mean delay and the probability of packet loss in the ad-hoc network. That is, the machine can estimate how probable its transmission can be delayed, how long its transmission can be delayed and how many packets can be lost in real-time. Thanks to V2X technology, mobile machines can work closer and be driven faster so that the productivity of the working site can be increased dramatically. However, our results also show the applicable conditions of IEEE 802.11p on mobile machines. As the nodes increase, the ad-hoc network may overload. Therefore, in our second version, we are going to publish a V2X solution that combines the IEEE 802.11p and 5G. In that version, machines use the analytical model proposed in this paper to decide when the 5G should be applied. Due to the limit of the pages, we just introduce the core ideas and the results. To find the full implementation, you can find our code on our Github.
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- Carlo Abate
- 'Richard J. Stancliffe'
- 'Zheng-Wei Liu'
date: 'Received ...; accepted ...'
title: |
How plausible are the proposed\
formation scenarios of CEMP-r/s stars?
---
Introduction {#intro}
============
Elements heavier than the iron group are mostly produced by neutron-capture processes. The canonical picture distinguishes two types of neutron-capture process: the *slow* or $s$-process, in which the neutron densities are relatively low, $N_{\mathrm{n}}\approx10^7-10^8\,{\mathrm{cm}}^{-3}$; and the *rapid* or $r$-process, in which the neutron densities are much higher, $N_{\mathrm{n}}\geq 10^{23}\,{\mathrm{cm}}^{-3}$ [@Pagel2009]. Three parameters essentially determine what isotopes are produced by either neutron-capture process: the neutron exposure, that is the time-integrated neutron flux, the neutron-capture cross-sections of the isotopes, and the $\beta$-decay timescales of unstable isotopes. At low neutron exposures, $\tau_0 \lesssim 1\,{\mathrm{mbarn}}^{-1}$ [e.g. @Kappeler1989; @Busso1999; @Pagel2009], the neutron-capture timescale is comparable or longer to that of most $\beta$-decays, hence nucleosynthesis by the $s$-process progresses along the $\beta$-stability valley up to bismuth ($^{209}{\mathrm{Bi}}$). In contrast, at high neutron exposures the rate of neutron captures exceeds that of the $\beta$-decays. Consequently, very neutron-rich unstable nuclei are synthetised and, when the neutron flux is interrupted, they subsequently decay into stable isotopes on the neutron-rich side of the valley of stability [@Seeger1965; @Arnould2007; @Pagel2009].
The $r$- and $s$-processes are active in completely different physical conditions. The latter is believed to take place in the intershell region of low-mass stars along the asymptotic giant branch [AGB, e.g. @Truran1981-1; @Gallino1998; @Busso1999; @Herwig2005] or in the helium-burning cores of massive stars [@Peters1968; @Truran1981-1; @Pignatari2008; @Pignatari2010; @Frischknecht2012]. The site of $r$-nucleosynthesis is as yet uncertain, and it is normally associated with the collapse of massive-star cores into neutron stars, for example in supernova explosions and accretion-induced collapses [@Burbidge1957; @Cameron1957; @Woosley1994; @Qian2003; @Cowan2004; @Sneden2008], or to the mergers of compact binary systems [@RamirezRuiz2015].
Abundances of $r$- and $s$-elements are determined in stars of all metallicities and evolutionary stages to study the relative importance of the $r$- and $s$-processes at different epochs and investigate galactic chemical evolution [e.g. @Spite1978; @Truran1981-1; @Burris2000; @Travaglio2001; @Mashonkina2003; @Simmerer2004; @Travaglio2004; @Cescutti2006; @Pagel2009; @Roederer2013]. In the past three decades the abundances of neutron-capture elements have been determined in many studies which analyse high-resolution spectra of hundreds of stars (e.g. [@Burris2000], [@Simmerer2004], [@Aoki2007; @Aoki2013-1], [@Roederer2014-2], and references therein). These studies show that high abundances of neutron-capture elements are very frequently found in carbon-enhanced metal-poor (CEMP) stars: i.e. in stars that are metal-poor (MP, i.e. with $[{\mathrm{Fe}}/{\mathrm{H}}]<-1$)[^1] stars with high carbon abundances, $[{\mathrm{C}}/{\mathrm{Fe}}]>1$.
CEMP stars are observed as a substantial fraction of the MP stars in the halo, and the cumulative fraction of CEMP stars raises with decreasing metallicity, from three percent at $[{\mathrm{Fe}}/{\mathrm{H}}]<-1$ up to about $80\%$ at $[{\mathrm{Fe}}/{\mathrm{H}}]<-4$ [e.g. @Cohen2005; @Marsteller2005; @Frebel2006; @Lucatello2006; @Lee2013; @Yong2013III; @Placco2014]. [@Aoki2007] show that at least $80\%$ of all CEMP stars in our Galaxy are also enriched in barium, whose solar-system abundance is mostly produced by the $s$-process [$\approx 89\%$, @Bisterzo2011], and are therefore called CEMP-$s$ stars. To explain the simultaneous enrichments in carbon, barium, and other $s$-elements it has been suggested that CEMP-$s$ stars underwent a mass-transfer episode from a more massive binary companion in the AGB phase of evolution. This scenario is supported by the evidence that most CEMP-$s$ stars show radial-velocity variations consistent with orbital motions [@Lucatello2006; @Starkenburg2014; @Hansen2015-4].
Many CEMP stars are enriched in elements whose solar-system abundances are dominated by the $r$-process, such as europium [$94\%$ of which comes from the $r$-process, @Bisterzo2011], as well as in $s$-elements. These stars are called CEMP-$r/s$ stars. The definitions adopted for the various categories of CEMP stars vary for different authors [e.g. @BeersChristlieb2005; @Jonsell2006; @Lugaro2009; @Masseron2010; @Lee2013]. In our work we adopt the following classification scheme.
- *CEMP stars* are defined by $[{\mathrm{C}}/{\mathrm{Fe}}]>1$. We call “carbon normal” those stars with $[{\mathrm{C}}/{\mathrm{Fe}}]\leq 1$.
- *CEMP-$s$ stars* are CEMP stars with $[{\mathrm{Ba}}/{\mathrm{Fe}}]>1$ and $[{\mathrm{Ba}}/{\mathrm{Eu}}]>0$. The lower limit on \[Ba/Eu\] is required because high abundances of barium can be produced by the $r$-process, but in that case \[Ba/Eu\] is negative. [@BeersChristlieb2005] propose to use $[{\mathrm{Ba}}/{\mathrm{Eu}}]>0.5$, however AGB-nucleosynthesis models, which only include the $s$-process, predict $[{\mathrm{Ba}}/{\mathrm{Eu}}]$ close to zero in some circumstances [@Lugaro2012; @Abate2015-1], hence our choice of a lower threshold.
- *CEMP-$r/s$ stars* are CEMP stars with $[{\mathrm{Eu}}/{\mathrm{Fe}}]>1$, $[{\mathrm{Ba}}/{\mathrm{Fe}}]>1$ and $[{\mathrm{Ba}}/{\mathrm{Eu}}]>0$. With this definition, CEMP-$r/s$ stars are a subclass of CEMP-$s$ stars. However, we note that CEMP-$s$ and CEMP-$r/s$ stars possibly have different origins.
- *CEMP-$r$ stars* are CEMP stars that exhibit $[{\mathrm{Eu}}/{\mathrm{Fe}}]>1$ and $[{\mathrm{Ba}}/{\mathrm{Eu}}]\leq0$. The upper limit on \[Ba/Eu\] is to exclude systems that are europium-rich because of the $s$-process, which would show positive \[Ba/Eu\] [e.g. @Abate2015-2; @Abate2015-1].
- *CEMP-no stars* are CEMP stars with $[{\mathrm{Ba}}/{\mathrm{Fe}}]\leq 1$ and $[{\mathrm{Eu}}/{\mathrm{Fe}}]\leq 1$.
The origin of CEMP-$r/s$ stars is debated. Although currently no study has specifically focused on the orbital properties of CEMP-$r/s$ stars, there are indications that the majority of them belongs to binary systems, as for CEMP-$s$ stars [@Lucatello2005a; @Hansen2015-4]. However, it is commonly believed that the interiors of AGB stars do not reach sufficiently high neutron densities to activate the $r$-process, and consequently the binary scenario invoked for the formation of CEMP-$s$ stars does not explain the abundances observed in CEMP-$r/s$ stars. Also, CEMP-$r/s$ stars exhibit on average higher abundances of carbon and heavy-$s$ elements (barium, lanthanum and cerium) compared to CEMP-$s$ stars with $[{\mathrm{Eu}}/{\mathrm{Fe}}]\leq 1$, whereas the two groups are observed to have approximately the same range of abundances of light-$s$ elements [strontium, yttrium and zirconium, @Abate2015-2].
[@Jonsell2006] and [@Lugaro2009], suggest and discuss a variety of formation scenarios. In many of these scenarios the origin of the $s$-elements is pollution from AGB primary stars in binary systems, as for CEMP-$s$ stars, whereas to explain the abundant $r$-elements it has been proposed that: ($i$) the gas clouds, in which the binary systems were born, were $r$-rich because of supernova explosions of previous-generation stars; ($ii$) $r$-process nucleosynthesis can be activated under some circumstances in low-metallicity AGB stars; ($iii$) $r$-elements are ejected by a third, massive star in a hierarchical triple system, or alternatively, in a binary system, by the primary star that, after the AGB evolution, ($iv$) explodes as Type-1.5 supernovae or ($v$) undergoes accretion-induced collapse into a neutron star. Alternative formation channels involve radiative levitation of the neutron-capture elements in the stellar atmospheres or self-enrichment during the AGB phase of evolution. A quantitative study of the likelihood to form CEMP-$r/s$ stars in these scenarios is as yet missing.
The aim of our paper is to answer the following question: what is the most likely scenario to explain the formation of CEMP-$r/s$ stars? For this purpose we calculate the frequency of CEMP-$r/s$ stars in each of the above formation scenarios, and we compare it with that determined in an observed sample of MP stars.
Observed sample {#data}
===============
Our observational sample is based on the SAGA database of stellar abundances [@Suda2008; @Suda2011 last updated August 2015], which is arguably the richest collection of heavy-element abundances currently available and includes $1944$ MP stars with $[{\mathrm{Fe}}/{\mathrm{H}}]<-1$. Stars with higher iron abundances in the SAGA database do not show large enrichments in either barium or europium, and therefore we exclude them from our sample. We select the MP stars with observed abundances of barium and europium, and we ignore systems in which only upper or lower limits are available. In case the chemical composition of a star is reported by multiple sources, we adopt the arithmetic mean of the observed logarithmic abundances. Table \[tab:obs\] summarises the number of MP stars of different classes included in our sample.
Figure \[fig:BaH-vs-EuH\] shows the $451$ MP stars in our sample with observed abundances of europium and barium, without any assumption about the carbon abundance. Almost all these stars exhibit abundances between the two dotted lines that indicate the barium-to-europium ratios ascribed to the $r$- and the $s$-process in the solar system, $[{\mathrm{Ba}}/{\mathrm{Eu}}]_{\odot,r}=-0.84$ and $[{\mathrm{Ba}}/{\mathrm{Eu}}]_{\odot,s}=1.2$ respectively [@Goriely1999; @Masseron2006]. The dashed line separates the majority of stars in our sample, with $[{\mathrm{Ba}}/{\mathrm{Eu}}]\leq 0$, from the 77 stars with $[{\mathrm{Ba}}/{\mathrm{Eu}}]> 0$. Grey dots indicate the $399$ stars with $[{\mathrm{Ba}}/{\mathrm{Fe}}]\leq1$ and $[{\mathrm{Eu}}/{\mathrm{Fe}}]\leq1$. The blue open squares are the $50$ “$s$-enhanced MP stars”, or MP-$s$ stars, i.e. stars with $[{\mathrm{Ba}}/{\mathrm{Fe}}]>1$ and $[{\mathrm{Ba}}/{\mathrm{Eu}}]>0$. Stars with $[{\mathrm{Eu}}/{\mathrm{Fe}}]>1$ are shown as red circles. Half of these $52$ europium-rich stars have $[{\mathrm{Ba}}/{\mathrm{Eu}}]>0$ and are classified as MP-$r/s$ stars in analogy with our definition of CEMP-$r/s$ stars.
The stars in our sample are selected according to their observed abundances of barium and europium, regardless of the carbon abundances. However, $48$ out of $50$ MP-$s$ stars and all MP-$r/s$ stars exhibit $[{\mathrm{C}}/{\mathrm{Fe}}]>1$, whereas only five out of $26$ MP-$r$ stars are also carbon-enriched. One of the two MP-$s$ stars that are not classified as CEMP-$s$ has $[{\mathrm{C}}/{\mathrm{Fe}}]\approx0.8$ and is a red giant [BD –01 2582, @Simmerer2004; @Roederer2014-1]. Hence, in the hypothesis that carbon-rich material was transferred from a companion star, the carbon enhancement could have been reduced by the first dredge-up [e.g. @Stancliffe2007; @Placco2014]. There currently is no determination of carbon in the literature for the other MP-$s$ star (G 18–24) . Hence, in our sample the proportion of CEMP stars among $s$- and $r/s$-rich MP stars is almost $100\%$, whereas among MP-$r$ stars it is consistent with the overall CEMP/MP fraction (approximately $19\%$).
The frequency of CEMP-$r/s$ stars among CEMP-$s$ stars in our sample is $26/48\approx54\%$. The SAGA database includes $55$ CEMP stars with $[{\mathrm{Ba}}/{\mathrm{Fe}}]>1$ and without the europium abundance, that is stars in which an upper limit is determined with $[{\mathrm{Ba}}/{\mathrm{Eu}}_{\mathrm{upper}}]>0$ or europium lines are not detected. If we relax our definition of CEMP-$s$ stars to also include these systems, that is we implicitly assume that if europium is not observed its abundance is sufficiently low that $[{\mathrm{Ba}}/{\mathrm{Eu}}]>0$, the ratio of CEMP-$r/s$ to CEMP-$s$ stars decreases to approximately $25\%$. The frequency of CEMP-$r/s$ stars provide useful constraints on the theoretical models proposed for the formation of these systems. However, the SAGA database is a collection of stellar abundances published in the literature, and consequently our sample is incomplete and inhomogeneous because of the different properties and selection effects of the original sources.
----------------------------------------------------------------------------------------------------------------------------------- ----------
definition of stellar group number
of stars
$[{\mathrm{Fe}}/{\mathrm{H}}]<-1$ (MP) 1944
MP with detected barium and europium (including limits) 725
$-$ MP with barium and europium abundances 451
$-$ MP with $[{\mathrm{Ba}}/{\mathrm{Fe}}]>1.0$ 60
$-$ MP-$s$ ($[{\mathrm{Ba}}/{\mathrm{Fe}}]>1.0$ and $[{\mathrm{Ba}}/{\mathrm{Eu}}]>0.0$) 50
$-$ MP-$r/s$ ($[{\mathrm{Ba}}/{\mathrm{Fe}}]>1.0$, $[{\mathrm{Ba}}/{\mathrm{Eu}}]>0.0$, and $[{\mathrm{Eu}}/{\mathrm{Fe}}]>1.0$) 26
$-$ MP-$r$ ($[{\mathrm{Eu}}/{\mathrm{Fe}}]>1.0$, $[{\mathrm{Ba}}/{\mathrm{Eu}}]\leq0.0$) 26
MP with carbon abundances 947
$-$ MP with $[{\mathrm{C}}/{\mathrm{Fe}}]\leq 1.0$ 763
$-$ MP with $[{\mathrm{C}}/{\mathrm{Fe}}]>1.0$ (CEMP) 184
$-$ CEMP-no$^\dag$ 75
$-$ CEMP with barium abundances 152
$-$ CEMP with $[{\mathrm{Ba}}/{\mathrm{Fe}}]>1.0$ without europium abundances 55
$-$ CEMP with barium and europium abundances 56
$-$ CEMP-$r$ 5
$-$ CEMP with $[{\mathrm{Ba}}/{\mathrm{Fe}}]>1.0$ (with europium abundances) 51
$-$ CEMP-$s$ 48
$-$ CEMP-$r/s$ 26
----------------------------------------------------------------------------------------------------------------------------------- ----------
: Classification of stars in our observational sample based on the SAGA database.[]{data-label="tab:obs"}
![\[Ba/H\] and \[Eu/H\] of metal-poor stars with $[{\mathrm{Fe}}/{\mathrm{H}}]<-1$ from the SAGA observational database. Grey and red dots indicate stars with $[{\mathrm{Eu}}/{\mathrm{Fe}}]\leq1$ and $[{\mathrm{Eu}}/{\mathrm{Fe}}]>1$, respectively. Stars with $[{\mathrm{Ba}}/{\mathrm{Fe}}]>1$ and $[{\mathrm{Ba}}/{\mathrm{Eu}}]>0$ are shown as blue open squares. The $+$ symbol in the bottom right corner shows the average observational uncertainty. The solar \[Ba/Eu\] ascribed to the $r$- and $s$-processes are shown as dotted lines [@Goriely1999; @Masseron2006]. The dashed line represents $[{\mathrm{Ba}}/{\mathrm{Eu}}]=0$.[]{data-label="fig:BaH-vs-EuH"}](Fig-1.pdf){width="48.00000%"}
Results
=======
Several scenarios have been proposed in the literature to explain the formation of CEMP-$r/s$ stars [e.g. @Hill2000; @Cohen2003; @Zijlstra2004; @Ivans2005; @Jonsell2006; @Lugaro2009; @Herwig2011; @Dardelet2015]. To determine the likelihood of these hypotheses, we calculate the frequency of CEMP-$r/s$ stars predicted in each scenario and we compare it with the observed proportion of CEMP-$r/s$ stars among CEMP-$s$ stars.
Pre-enrichment in $r$-rich material and pollution from an AGB star in a binary system {#pre-enrich}
-------------------------------------------------------------------------------------
In this scenario, the enrichment in $r$- and $s$-elements in CEMP-$r/s$ stars originate from two independent sources. The CEMP-$r/s$ star would be the secondary star in a binary system born in a gas cloud which was pre-enriched in $r$-elements, for example because of the explosion of a nearby supernova. The observed carbon and $s$-elements are instead produced by the primary star during the AGB phase, and subsequently transferred by stellar winds onto the low-mass companion, which is the star observed today (e.g. [@Jonsell2006], [@Bisterzo2011], and references therein).
To investigate this scenario, we use the population synthesis model developed by [@Izzard2004; @Izzard2006; @Izzard2009; @Izzard2010] to simulate a population of binary stars in which the abundances of $r$-process elements are initially enhanced by a random factor. Our model population is constructed according to the default model set A of [@Abate2015-3]. Our simulated grid consists of approximately $1.8$ million binary systems with primary and secondary masses in the intervals $[0.5,\,8]\,{M_{\odot}}$ and $[0.1,\,0.9]\,{M_{\odot}}$, respectively, orbital separations $a_i$ between $50$ and $5\times10^6\,{R_{\odot}}$, and seven different values of the mass of the partial-mixing zone, a parameter that determines the abundance distribution of neutron-capture elements synthetised in the AGB phase [@Karakas2010; @Lugaro2012; @Abate2015-1]. The distribution of primary masses in our population follows the solar-neighbourhood initial mass function (IMF) proposed by [@Kroupa1993], whereas all secondary masses are equally likely and the separation distribution is flat in $\log a_i$. We compute the wind mass-transfer process according to a wind Roche-lobe overflow model described by @Abate2013 [Eq. 9], and adopting the spherically-symmetric-wind approximation. We assume that the transferred material is mixed throughout the accreting star to mimic the effect of thermohaline mixing, which is expected to be efficient in low-mass stars [@Stancliffe2007; @Stancliffe2008].
As initial composition we adopt the solar abundance distribution of [@Asplund2009] scaled down to metallicity $10^{-4}$ (that is $[{\mathrm{Fe}}/{\mathrm{H}}]\approx-2.2$). Our method to simulate a random pre-enrichment in barium and europium produced by the $r$-process is summarised as follows. For each stable isotope $i$ of barium and europium, we take from [@Bisterzo2011] the fraction of its solar abundance which is produced by the $r$-process, $f_{i,r}$. We multiply our initial, solar-scaled abundances, $X_i$, of each isotope times their corresponding fraction $f_{i,r}$. We sum the abundances of the isotopes of the same element to calculate the total abundances of barium and europium ascribed to the $r$-process, and we calculate the decimal logarithm of these two values. Subsequently, for each synthetic binary system we select one value from a randomly generated sequence of $1.8$ million numbers uniformly distributed between $-1$ and $3$, and we add this number to the logarithmic abundances of barium and europium to simulate an initial enhancement (or depletion) of the $r$-component of these two elements. The interval in which the random numbers are generated reproduces the range of variation of $[{\mathrm{Eu}}/{\mathrm{Fe}}]$ in our observed sample. As a consequence of this artificial pre-pollution, the initial model distributions of $[{\mathrm{Eu}}/{\mathrm{Fe}}]$ and $[{\mathrm{Ba}}/{\mathrm{Fe}}]$ are flat over the abundance ranges $[-1,\,3]$ and $[-0.2,\,2.2]$, respectively.
We evolve our binary systems with these initial conditions and we determine the number of stars that are visible as a function of their luminosities according to the selection criteria of @Abate2015-3 [Sect. 2.3] with $V$-magnitude limits at $6$ and $16.5$. In Fig. \[fig:EuFe-BaFe\] we show the distributions of barium and europium relative to iron for our synthetic population. Carbon-rich stars with $[{\mathrm{C}}/{\mathrm{Fe}}]>1$ are shown in red, with darker colours representing regions of higher probability, while carbon-normal stars are shown in blue. Figure \[fig:EuFe-BaFe\] also shows the abundances of observed CEMP and carbon-normal metal-poor stars (red crosses and blue dots, respectively). Because our model is computed for $[{\mathrm{Fe}}/{\mathrm{H}}]=-2.2$, we restrict our observed sample to stars with iron abundance within a factor of five from the model value, $-2.9\leq [{\mathrm{Fe}}/{\mathrm{H}}] \leq-1.5$. This selection leaves us with $49$ CEMP, $43$ CEMP-$s$, and $22$ CEMP-$r/s$ stars, that is more than $80\%$ of our entire carbon-rich sample.
The theoretical and observed abundances differ in many aspects.
1. Observed carbon-rich and carbon-normal stars appear as two distinct and clearly separated populations. We compute the linear regression of the observed europium and barium abundances weighted with the observational errors, and we determine the following equations for the ratio of barium to europium in carbon-normal and CEMP-$s$ stars, respectively, $$\begin{aligned}
[{\mathrm{Ba}}/{\mathrm{Fe}}] &= (0.8 \pm 0.1)\times [{\mathrm{Eu}}/{\mathrm{Fe}}] - (0.4 \pm 0.1)~~, \label{eq:fitVMP}\\
[{\mathrm{Ba}}/{\mathrm{Fe}}] &= (0.7 \pm 0.1)\times [{\mathrm{Eu}}/{\mathrm{Fe}}] + (1.1 \pm 0.1)~~, \label{eq:fitCEMP-s}\end{aligned}$$ represented in Fig. \[fig:EuFe-BaFe\] with blue-dashed and magenta-solid lines. If we compute the linear fit for CEMP-$s$ stars with $[{\mathrm{Eu}}/{\mathrm{Fe}}]\le1$ and CEMP-$r/s$ stars separately we find almost identical results as in Eq. \[eq:fitCEMP-s\]. Equations \[eq:fitVMP\] and \[eq:fitCEMP-s\] show that the empirical relation between \[Eu/Fe\] and \[Ba/Fe\] has almost the same slope in the two classes of stars, and CEMP-$s$ stars have \[Ba/Fe\] on average $1.5$ dex higher than carbon-normal stars at the same \[Eu/Fe\]. The relations described by Eqs. \[eq:fitVMP\] and \[eq:fitCEMP-s\] are likely consequences of the barium-to-europium ratios produced in the $r$- and in the $s$-process, respectively. In contrast, in our simulation there is not a clear separation between the two groups (colour distributions in Fig. \[fig:EuFe-BaFe\]). Because $r$- and $s$-enrichments are independent, the initial \[Eu/Fe\] and \[Ba/Fe\] follow approximately the relation described by Eq. \[eq:fitVMP\], whereas the final abundances depend on the amount of material transferred from the AGB primary stars onto their secondary companions. Higher accreted masses cause stronger abundance enhancements of carbon and barium. Because AGB stars generally do not produce high europium abundances, \[Eu/Fe\] is essentially determined by its initial value, except for stars with initial \[Eu/Fe\] less than about $0.5$. Consequently, for each \[Eu/Fe\], which is given by the initial $r$-enrichment, we find stars with all barium abundances up to two dex higher than predicted by a pure $r$-process, and the correlation between \[Eu/Fe\] and \[Ba/Fe\] described by Eq. \[eq:fitCEMP-s\] is completely washed out. [@Lugaro2009; @Lugaro2012] and [@Abate2015-3] criticise this scenario of independent $r$- and $s$-process contributions with analogous arguments.
![\[Ba/Fe\] vs \[Eu/Fe\] in carbon-normal and carbon-rich metal-poor stars (blue dots and red crosses, respectively). Stars enriched in $s$-elements occupy the region above the horizontal and diagonal solid black lines, corresponding to zones A and B of the plot. According to our adopted definitions, CEMP-$r/s$ stars are in zone B. In zone C there are stars with $[{\mathrm{Ba}}/{\mathrm{Fe}}]>1$, $[{\mathrm{Eu}}/{\mathrm{Fe}}]>1$, and $[{\mathrm{Ba}}/{\mathrm{Eu}}]<0$. CEMP-$r$ stars and carbon-normal stars with $[{\mathrm{Eu}}/{\mathrm{Fe}}]>1$ and $[{\mathrm{Ba}}/{\mathrm{Fe}}]\leq1$ are in zone D. The dashed-blue and solid-magenta lines are linear fits of the carbon-normal stars and CEMP-$s$ stars, respectively. The red and blue distributions represent our the results of our theoretical model for CEMP and carbon-normal metal-poor stars.[]{data-label="fig:EuFe-BaFe"}](Fig-2.pdf){width="50.00000%"}
2. Only two of the $205$ carbon-normal stars in our metal-poor sample have abundances of europium and barium relative to iron higher than one (CS $29497$–$004$ and CS $31082$–$001$, in section C of Fig. \[fig:EuFe-BaFe\], have $[{\mathrm{Eu}}/{\mathrm{Fe}}] = 1.7$ and $1.6$, respectively, and $[{\mathrm{Ba}}/{\mathrm{Fe}}]=1.1$). In contrast, $12$ CEMP-$r/s$ stars have $[{\mathrm{Eu}}/{\mathrm{Fe}}]>1.7$ up to $[{\mathrm{Eu}}/{\mathrm{Fe}}] = 2.5$. If the $r$- and $s$-enrichments are independent, we expect to observe carbon-normal MP stars and CEMP-$r/s$ stars with europium abundances approximately in the same range.
3. The observed fraction of carbon-normal stars with $[{\mathrm{Eu}}/{\mathrm{Fe}}]>1$, $[{\mathrm{Ba}}/{\mathrm{Fe}}]>1$, and $[{\mathrm{Ba}}/{\mathrm{Eu}}]\leq 1$ (in zone C of Fig. \[fig:EuFe-BaFe\]) is $2/205\approx1\%$. In contrast, approximately $28\%$ of our simulated carbon-normal stars are in zone C of the plot. This discrepancy is unlikely to be caused by an observational bias against carbon-normal, $r$- and $s$-rich metal-poor stars. The spectra of metal-poor stars are selected evaluating the strength of the calcium and iron lines, and these metallicity indicators are not perturbed by the presence of barium or europium.
4. Similarly, the fraction of synthetic CEMP stars in zone C of the plot is $42\%$. In contrast, only two of the $49$ observed CEMP stars are in this region, that is approximately $4\%$. The high fraction of synthetic carbon-rich and carbon normal stars in zone C of Fig. \[fig:EuFe-BaFe\] is a consequence of our assumptions on the initial abundances of barium and europium. To reduce this fraction, it is necessary to constrain the initial barium abundances within $[{\mathrm{Ba}}/{\mathrm{Fe}}]<1$, that corresponds to europium abundances in $[{\mathrm{Eu}}/{\mathrm{Fe}}]\lesssim1.8$ [@Goriely1999; @Pagel2009]. Consequently, if we adopt these initial assumptions in our model, the europium abundances of eight observed CEMP-$r/s$ stars with $[{\mathrm{Eu}}/{\mathrm{Fe}}]>1.8$ are not reproduced. Alternatively it would be necessary to assume that the barium-to-europium ratio of a pure $r$-process at low metallicity is $[{\mathrm{Ba}}/{\mathrm{Eu}}]\approx-2$, but this hypothesis is not borne out by the observations (cf. Fig. \[fig:BaH-vs-EuH\]).
5. In our simulation the proportion of CEMP-$r/s$ among CEMP-$s$ stars is approximately $22\%$, and it underestimates the observed fraction of about $51\%$ by more than a factor of two. If we also include CEMP stars with $[{\mathrm{Ba}}/{\mathrm{Fe}}]>1$ and an upper limit on europium such that $[{\mathrm{Ba}}/{\mathrm{Eu}}]>0$, and we assume that all barium-enhanced CEMP stars with no europium detection are CEMP-$s$ stars, the number of observed CEMP-$s$ stars increases to $86$ and the observed ratio of CEMP-$r/s$ to CEMP-$s$ stars is $26\%$, that is approximately consistent with the simulations. However, even in this case, there is a large discrepancy in the barium-abundance distributions of observed and modelled CEMP-$r/s$ stars. The majority of the observed CEMP-$r/s$ stars ($16/22\approx73\%$) are strongly barium-enhanced and show $[{\mathrm{Ba}}/{\mathrm{Fe}}]>2$. In contrast, the cumulative fraction of synthetic CEMP-$r/s$ stars with barium abundance $[{\mathrm{Ba}}/{\mathrm{Fe}}]> 2$ is $4\%$, almost a factor of twenty less than the observations. Hence, this model does not adequately reproduce the observed distribution of barium abundances in CEMP-$r/s$ stars.
These discrepancies also arise if we modify our model assumptions. The abundance distributions of barium and europium are essentially independent from the assumptions about the efficiencies of the wind-accretion process and the angular-momentum loss, and the initial distributions of secondary masses and orbital separations [@Abate2015-3]. An IMF weighted towards intermediate masses favours stars that typically produce higher barium abundances than low-mass stars with $M<1.5{M_{\odot}}$ [@Abate2015-3], but the proportion of CEMP-$r/s$ stars with $[{\mathrm{Ba}}/{\mathrm{Fe}}]>2$ is low (less than $\approx5\%$). If we assume that the accreted material remains at the surface of the secondary stars until is mixed by the first dredge-up, as in model set D of [@Abate2015-3], we find the maximum frequency of CEMP-$r/s$ stars with $[{\mathrm{Ba}}/{\mathrm{Fe}}]>2$, that is about $84\%$, higher than the observations. However, this model does not reproduce the abundance distributions of carbon and $s$-elements in CEMP stars [cf. model D of @Abate2015-3] nor the observed correlation between barium and europium abundances, analogously to our default model. The discrepancy between the small number of stars observed in zone C of Fig. \[fig:EuFe-BaFe\] and the much higher frequencies of synthetic carbon-normal and carbon-rich MP stars is a consequence of the adopted initial abundances of barium and europium, and therefore does not depend on the model set.
Binary mass-transfer scenario with *i*-process nucleosynthesis in AGB stars {#i-proc}
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In this scenario, the AGB star invoked to explain the enrichment in carbon and $s$-elements is also responsible for the production of $r$-elements. [@Cohen2002] and [@Jonsell2006] dismiss this channel with the argument that in AGB stars it is not possible to create the high neutron flux required for $r$-process nucleosynthesis. However, recent simulations show that relatively high neutron densities, between $10^{12}$ and $10^{15}\,{\mathrm{cm}}^{-3}$, can be reached in very low-metallicity AGB stars [@Campbell2008; @Cristallo2009-2; @Campbell2010; @Stancliffe2011], in the very-late thermal pulse of post-AGB stars [@Herwig2011; @Bertolli2013; @Herwig2014; @Woodward2015] or during the thermal-pulse evolution of low-metallicity super-AGB stars [@Jones2015]. At these densities, which are in-between $10^8\,{\mathrm{cm}}^{-3}$ and $10^{23}\,{\mathrm{cm}}^{-3}$ (values normally adopted in $s$- and $r$-process models, respectively, [@Pagel2009]), the *intermediate* neutron-capture process ($i$-process, [@Cowan1977]) is triggered.
[@Hampel2015] use a one-zone model to simulate the properties of a gas with temperature, density and chemical composition as in the intershell region of an AGB star and calculate the nucleosynthetic products in the presence of neutron densities between $10^{12}$ and $10^{16}\,{\mathrm{cm}}^{-3}$. The author shows that the $i$-process predicts high abundances of elements traditionally associated with the main $s$-process component (e.g. barium, lanthanum, cerium, and lead) and with the $r$-process (e.g. iodine and europium), while it does not significantly modify the abundances of elements normally associated to the weak $s$-process (strontium, yttrium and zirconium). These abundance patterns are typical of CEMP-$r/s$ stars, as discussed for example by [@Abate2015-2] and [@Hollek2015]. With her model, [@Hampel2015] is able to reproduce the surface abundances of $20$ observed CEMP-$r/s$ stars, significantly improving the results obtained by [@Abate2015-2] with a standard AGB-nucleosynthesis model.
To investigate the likelihood of this scenario it is necessary to know under what circumstances AGB stars experience thermal pulses characterised by high neutron fluxes. This information is very uncertain. Proton-ingestion episodes with large neutron production are found in models of very low-metallicity ($Z\leq 10^{-4}$), low-mass ($M\leq 2{M_{\odot}}$) stars [e.g. @Campbell2008; @Cristallo2009-2; @Stancliffe2011]. At present, it has not been investigated whether these episodes occur in the whole metallicity range of CEMP-$r/s$ stars, i.e. up to $Z\approx10^{-3}$. In addition, proton ingestion is triggered in the early thermal pulses of the AGB stars. It is currently unclear whether the subsequent nucleosynthesis dominates the surface abundances at the end of AGB evolution, when most of the envelope mass is ejected, or whether traces of the $i$-process remain detectable [@Campbell2008; @Cristallo2009-2].
Super-AGB stars are normally considered to form at masses higher than $6\,{M_{\odot}}$. Because AGB stars of these masses experience hot-bottom burning [e.g. @Boothroyd1993; @Izzard2007], the material accreted from these stars is expected to be in most cases nitrogen-rich, rather than carbon-rich [@Pols2012]. Also, such massive stars are disfavoured by the IMF, and therefore the theoretical ratio of CEMP-$r/s$ to CEMP-$s$ stars is lower than the observations: about $0.04\%$ with a solar-neighbourhood IMF. This fraction increases if we adopt an IMF weighted towards intermediate-mass stars, for example that proposed by [@Lucatello2005a], which is a gaussian function with $\mu_{\log_{10} M} = 0.79$ and $\sigma_{\log_{10} M} = 1.18$. However, the probability of making a CEMP star in a binary system with a $M\ge6{M_{\odot}}$ donor star is about $2\%$, more than an order of magnitude lower than the observations. A similar argument constrains the conditions to trigger the very-late thermal pulse in post-AGB stars. If these conditions are mass-dependent, a relatively low threshold mass is necessary. To produce a proportion of CEMP-$r/s$ stars of about $30\%$ in this scenario with a solar-neighbourhood IMF, the high neutron exposures should be reached in all stars of initial mass down to $1.2\,{M_{\odot}}$. Further detailed calculations are necessary to investigate if the $i$-process can be activated at such low masses.
Triple systems with SN explosion and AGB pollution {#triple}
--------------------------------------------------
In this scenario CEMP-$r/s$ stars are formed in triple systems in which the tertiary, least massive star accretes $r$-elements from the ejecta of the supernova explosion of the primary star, and carbon and $s$-elements from the wind of the secondary star in the AGB phase of evolution. To maximise the ratio of CEMP-$r/s$ to CEMP-$s$ stars produced in this scenario, we make the following, rather unrealistic, assumptions: ($i$) all stars form in hierarchical triple systems, ($ii$) intermediate-mass, low-metallicity AGB stars have negligible mass loss, as argued by [@Wood2011], and consequently all stars of mass $M \geq 3{M_{\odot}}$ explode as supernovae, ($iii$) the accretion of material onto the low-mass component of the triple system is sufficiently efficient to enhance europium above the threshold $[{\mathrm{Eu}}/{\mathrm{Fe}}]=1$, regardless of the orbital separation of the primary star.
Under these assumptions, the necessary condition to form a CEMP-$r/s$ star is that the primary star in the triple system is more massive than $3\,{M_{\odot}}$. If we adopt the solar-neighbourhood IMF proposed by [@Kroupa1993], the probability of forming such a primary star is $P(M\geq 3{M_{\odot}})=0.014$, that is, more than a factor of ten lower than the observations. If we adopt the IMF weighted towards intermediate-mass stars proposed by [@Lucatello2005a], the probability of a star more massive than $3{M_{\odot}}$ is $P(M\geq 3{M_{\odot}})=0.31$, that is within the range of uncertainty of the observations. However, we note that [@Lucatello2005a] proposed their IMF for stars of extremely low metallicity, $[{\mathrm{Fe}}/{\mathrm{H}}]<-2.5$, and similarly the suppression of the mass loss proposed by [@Wood2011] was only found in models at $[{\mathrm{Fe}}/{\mathrm{H}}]=-4.2$, whereas the most iron-deficient CEMP-$r/s$ star in our sample has $[{\mathrm{Fe}}/{\mathrm{H}}]=-3.1$ and $15$ of our $26$ CEMP-$r/s$ stars have $[{\mathrm{Fe}}/{\mathrm{H}}]>-2.5$.
These results are obtained assuming that all stars are formed in triple systems. The multiples-to-binaries ratio estimated by [@Rastegaev2010] for MP stars is $10/64\approx0.16$. Taking into account this factor, with the IMF of [@Lucatello2005a] the CEMP-$r/s$ frequency is reduced to approximately $5\%$, that is a factor of five up to ten times lower than in our observed sample, depending on whether we count all CEMP stars without the europium abundance and $[{\mathrm{Ba}}/{\mathrm{Fe}}]>1$ (cf. Tab. \[tab:obs\]). This proportion decreases even further if we consider that mass accretion from supernova ejecta is typically inefficient, unless the orbital separation is short [@Liu2015]. In contrast, the initial period of the inner binary system has to be at least $1,\!000$ days, because the tertiary star has to transfer carbon and $s$-elements from the secondary AGB star, and binary systems in closer orbits enter in a common envelope without mass accretion. In stable triple systems the orbit of the primary star is typically much wider than that of the inner binary system [@Kiseleva1994]. The calculations of [@Liu2015] predict that at orbital periods longer than $1,\!000$ days the transferred mass is less than $10^{-5}\,{M_{\odot}}$. Consequently, if the transferred material is diluted in a layer as thin as $0.01\,{M_{\odot}}$ in the envelope of the accreting star, the amount of transferred material is not sufficient to enhance the europium abundances above $[{\mathrm{Eu}}/{\mathrm{Fe}}]=1$, even if the supernova ejecta exhibit $[{\mathrm{Eu}}/{\mathrm{Fe}}]=3$. Hence, this formation scenario of CEMP-$r/s$ stars is considered implausible.
Binary star with $1.5$-supernova pollution {#1.5SN}
------------------------------------------
In this formation scenario the CEMP-$r/s$ star would be the secondary star of a binary system in which the primary star produced the $s$-elements during the AGB phase and subsequently exploded as a Type 1.5 supernova producing the $r$-elements. It is therefore necessary to assume that the mass loss along the AGB is sufficiently high to allow significant accretion of $s$-elements, but not too high otherwise the core of the star does not reach the Chandrasekhar mass. [@Zijlstra2004] propose that low-metallicity AGB stars with initial mass higher than about $3-4\,{M_{\odot}}$ undergo this evolution. These AGB stars should also produce carbon and $s$-elements. According to the simulations of [@Abate2015-3] the proportion of carbon-enhanced stars that are formed in binary systems with primary masses higher than $3\,{M_{\odot}}$ is less than $1\%$ if a solar-neighbourhood IMF is adopted, and increases to about $14\%$ with the IMF of [@Lucatello2005a]. Simulations of the interactions of core-collapse supernova ejecta with a main-sequence companion star show that the amount of accreted mass critically depends on the binary separation, and it is less than $10^{-4}\,{M_{\odot}}$ for a $0.9\,{M_{\odot}}$ at periods longer than about $40$ days [@Liu2015]. The cumulative fraction of CEMP stars formed in closer orbits with the IMF of [@Lucatello2005a] is less than $5\%$, and consequently the ratio of CEMP-$r/s$ to CEMP-$s$ stars is less than $0.7\%$. In addition, according to many authors the explosion of a Type 1.5 supernova destroys the donor star [@Nomoto1976; @Iben1983; @Lau2008], although [@Arnett1974] argues that a remnant may survive in some cases, and therefore the binary system is disrupted. In contrast, many CEMP-$r/s$ stars are observed in binary systems [@Lucatello2005a; @Hansen2015-4] and therefore could not have originated by this formation channel.
Binary system with AGB- and AIC-pollution {#AIC}
-----------------------------------------
This scenario involves two phases of mass transfer. In the first, a relatively massive primary AGB star [$M \approx 3-12\,{M_{\odot}}$ according to @Cohen2003] transfers carbon and $s$-elements onto its low-mass ($M_*<0.9\,{M_{\odot}}$) companion, and later ends its life as a white dwarf. Subsequently, the secondary star transfers some material onto the white dwarf which collapses into a neutron star. This scenario is supported by the study of [@Qian2003] on r-element production in neutrino winds associated to accretion-induced collapse (AIC).
We consider this scenario unlikely for the following reasons. First, CEMP stars formed in binary systems with primary masses above $3\,{M_{\odot}}$ are unlikely, as also discussed in sections \[triple\] and \[1.5SN\]. We also note that AIC models have mostly investigated the collapse of oxygen-neon white dwarfs, which descend from stars more massive than $5\,{M_{\odot}}$ [e.g. @Qian2003; @Dessart2007], for which the probability of forming a CEMP star is less than $5\%$ even with the IMF of [@Lucatello2005a].
Secondly, the $s$-element distribution predicted for relatively massive AGB stars ($M\ge3\,{M_{\odot}}$) is not consistent with the abundances observed in CEMP-$s$ stars, which are mostly reproduced by stellar models of mass $M<2\,{M_{\odot}}$ [e.g. @Bisterzo2011; @Bisterzo2012; @Abate2015-1; @Abate2015-2]. Also, the $r$-process nucleosynthesis in AIC events is largely uncertain [@Qian1996; @Qian2003], therefore it is unclear whether in this scenario the total amount of $r$-elements produced and their abundance distribution reproduce the chemical composition of observed CEMP-$r/s$ stars.
Third, this scenario works in a narrow range of orbital separations, because at the end of the first mass transfer the secondary star needs to be close enough to the white dwarf to subsequently fill its Roche-lobe and undergo the second mass-transfer phase. A star of $0.9\,{M_{\odot}}$ and a white dwarf of $1\,{M_{\odot}}$ need to be in an orbit of less than about $200$ days to undergo Roche-lobe overflow during the main-sequence or the red-giant phase. This star should be already carbon-enhanced because the AIC event is responsible of the $r$-pollution. According to the simulations of [@Abate2015-3], the cumulative probability of CEMP stars in binary systems with orbital periods less than $200$ days is about $8\%$ at maximum. Hence, the frequency of CEMP-$r/s$ stars is reduced by this factor, and consequently the ratio of CEMP-$r/s$ to CEMP-$s$ stars is much lower than observed.
Radiative levitation {#radlev}
--------------------
In stars, elements with high atomic mass are normally believed to sink more than light elements because of gravitational settling. However, partially-ionized heavy elements usually have large photon-absorption cross sections, and hence they can be accelerated outwards by radiative pressure. This process, known as radiative levitation, may cause the abundance of heavier elements to increase towards the surface. Consequently, the overabundances of neutron-capture elements detected in CEMP-$r/s$ stars could in principle be explained by radiative levitation, although this process has never been studied for elements heavier than nickel at low metallicities [@Richard2002-3].
In stars, the efficiency of radiative levitation increases with decreasing convective-envelope mass and increasing effective temperature. Consequently, this process is most efficient in main-sequence stars close to the turnoff and Hertzsprung-gap stars, whereas its effect is negligible in giants ([@Richard2002-1], Matrozis et al. in prep.). However, the observed CEMP-$r/s$ stars are not preferentially main-sequence or turnoff stars. Only $15$ out of $26$ CEMP-$r/s$ stars in our sample have surface gravities in the range $3.5<{\log_{10}(g/\mathrm{cm}\,\mathrm{s}^{-2})}<4.25$ and effective temperatures higher than ${T_{\mathrm{eff}}}=5,\!800\,$K, and consequently exhibit surface abundances that have been possibly modified by radiative levitation. In contrast, $11$ CEMP-$r/s$ stars ($42\%$ of our sample) have ${\log_{10}(g/\mathrm{cm}\,\mathrm{s}^{-2})}<3.5$, hence radiative levitation cannot be generally invoked to explain the enrichments in neutron-capture elements observed in CEMP-$r/s$ stars. In addition, this scenario does not explain the differences in abundances between CEMP-$s$ and CEMP-$r/s$ stars.
Pre-enrichment in $r$-rich material and self pollution
------------------------------------------------------
In this scenario, the CEMP-$r/s$ star formed in an environment which was enriched in $r$-elements, for example because of an early supernova, and it self-enriched its surface with carbon and $s$-elements during the AGB phase of evolution [@Hill2000; @Cohen2003]. However, as mentioned in Sect. \[radlev\], more than half of the CEMP-$r/s$ stars in our sample have ${\log_{10} g}>3.5$ and have not reached the giant phase yet. This hypothesis is also dismissed by [@Jonsell2006] with similar arguments. In addition, if the $r$- and $s$-enrichments are independent, the observed correlation between europium and barium abundances is not reproduced by the models, as discussed in Sect. \[pre-enrich\]. Also, this scenario does not explain the high frequency of binary systems detected among CEMP-$r/s$ stars [@Lucatello2005a; @Hansen2015-4].
Discussion
==========
The results presented in Sect. \[results\] show that all the scenarios proposed so far to explain the formation of CEMP-$r/s$ stars have difficulties in reproducing the properties of the observed CEMP-$r/s$ population, in particular the proportion of these systems among CEMP-$s$ stars. Scenarios that involve multiple phases of mass transfer, either in binary or in triple systems, underestimate this proportion because the $r$-elements are produced by the explosion or collapse of a relatively massive star ($M>3\,{M_{\odot}}$), which are rarely formed in a solar-neighbourhood IMF and also, in most cases, even assuming an IMF weighted towards intermediate-mass stars. In addition, impact simulations of supernova explosions on binary companions show that the mass accreted from supernova ejecta is typically lower than $10^{-4}\,{M_{\odot}}$ for orbital periods longer than about $100$ days [e.g. @Pan2012; @Liu2015]. Because the fraction of CEMP stars that are formed in such close orbits is low [@Abate2015-3], very few CEMP-$r/s$ stars are consequently formed via these channels. Also, it is currently unclear whether the $r$-element distribution produced in supernovae would reproduce the abundances of observed CEMP-$r/s$ stars [@Arcones2011].
A frequency of CEMP-$r/s$ stars that approaches the observations is predicted in the hypothesis of independent enrichments in $s$- and $r$-elements, in which the latter are the result of pre-pollution of the gas in which CEMP-$r/s$ stars were born. However, models based on this hypothesis fail to reproduce the observed correlation between the barium and europium abundances in CEMP-$s$ stars (as also discussed e.g. by [@Lugaro2009; @Lugaro2012], and [@Abate2015-3]) and they predict too many carbon-normal stars with high abundances of europium and barium. [@Lugaro2012] also note that in stellar-nucleosynthesis models the final europium abundance is essentially independent of its value at the beginning of the AGB evolution, unless the initial enrichment is higher than $[r/{\mathrm{Fe}}]\approx1.5$. Hence, lower initial enhancements are essentially washed out by the $s$-process nucleosynthesis occurring along the AGB. Also, the fact that the average abundance of heavy-$s$ elements (barium, lanthanum, cerium) is more than about $0.9$ dex higher than that of the light-$s$ elements (strontium, yttrium, zirconium), i.e. one of the characteristics of most CEMP-$r/s$ stars which is not predicted by AGB models [@Abate2015-2; @Hollek2015], can only be reproduced if high initial $r$-enhancements are assumed ($[r/{\mathrm{Fe}}]\geq 1.5$, e.g. [@Bisterzo2012]).
Recent simulations suggest that neutron densities sufficiently high to trigger the $i$-process ($10^{12}-10^{16}\,{\mathrm{cm}}^{-3}$ approximately) are reached if hydrogen-rich material is injected in region processed by helium burning. These proton-ingestion episodes are found to occur during the early pulses of very low-metallicity, low-mass AGB stars [@Campbell2008; @Cristallo2009-2; @Stancliffe2011], in the late thermal pulses of super-AGB stars [@Jones2015], and in the very-late thermal pulse of post-AGB stars [@Herwig2011; @Herwig2014]. The abundance patterns predicted by nucleosynthesis models at these neutron densities reproduce the surface abundances observed in many CEMP-$r/s$ stars ([@Dardelet2015], [@Hampel2015], Hampel et al. in prep.). If the preliminary results of these theoretical models are confirmed, AGB stars that fulfill the conditions to undergo the $i$-process would be the most promising candidates to explain the CEMP-$r/s$ abundances in the context of the binary mass-transfer scenario which is also invoked for the formation of CEMP-$s$ stars. This scenario is also consistent with the observational evidence that many CEMP-$r/s$ stars are found in binary systems [@Abate2015-1; @Hansen2015-4], although a detailed analysis (similar to the studies on CEMP-$s$ stars performed by [@Lucatello2005a], [@Starkenburg2014], and [@Hansen2015-4]) focused on the orbital properties of these objects is currently missing.
The $i$-process is also invoked to explain the abundances observed in post-AGB stars in the Milky Way (such as the Sakurai’s Object, [@Herwig2011]) and in the Magellanic Clouds [@Lugaro2015], which in some cases have very different abundance distributions than CEMP-$r/s$ stars. For example, some post-AGB stars have been observed to be lead deficient [e.g. @DeSmedt2014] and with light-$s$ elements much more enhanced than heavy-$s$ elements (e.g. [@Herwig2011], and [@Jones2015]). In contrast, all CEMP-$r/s$ stars with observed lead abundances in our sample have $[{\mathrm{Pb}}/{\mathrm{Fe}}]\geq 2.5$, and positive ratios of heavy-$s$ to ligh-$s$ elements, $[{\mathrm{hs}}/{\mathrm{ls}}]\geq 0.5$. Further work is necessary to calculate if the range of element ratios produced in $i$-process nucleosynthesis spans over the wide range observed in CEMP-$r/s$ stars and post-AGB stars.
To determine whether within this “$i$-process scenario” the properties of CEMP-$r/s$ stars are reproduced, the following aspects need to be clarified.
- Following on the work of [@Herwig2011; @Herwig2014], [@Stancliffe2011], and [@Woodward2015], more three-dimensional hydrodynamical simulations are necessary to study the conditions under which proton-ingestion events occur in the helium flash of AGB stars. These simulations should precisely determine the intervals of neutron densities and neutron exposures that are produced as a function of the temperature, density, and metallicity of the stellar layers in which the proton ingestion takes place.
- Based on the results of these simulations, detailed stellar-evolution models should be used to investigate how likely it is for AGB stars of different masses and metallicities to experience a proton-ingestion event, at what stage of the evolution and how many times this event can occur, and what are its consequences on the subsequent evolution of the star. Also, it is necessary to study whether the nucleosynthetic products of the proton-ingestion event are actually mixed to the surface, and whether this occurs when the star is undergoing substantial mass loss.
- Nucleosynthesis models are required to determine whether at the range of neutron exposures predicted by the detailed hydrodynamical simulations the production of neutron-capture elements reproduce the abundances observed in CEMP-$r/s$ stars. The preliminary results of [@Dardelet2015], @Hampel2015, and Hampel et al. (in prep.) support this hypothesis. The dependence of the abundance distribution on the temperature, density and chemical composition of the gas undergoing the neutron-capture process needs to be investigated to determine the ranges of abundances and element-to-element ratios that are produced in $i$-process nucleosythesis.
- Population-synthesis models, which incorporate the physics of the detailed models described above, are necessary to study if the properties of the observed CEMP-$r/s$ population can be reproduced within the binary mass-transfer scenario, in particular the frequency of CEMP-$r/s$ stars among CEMP-$s$ stars and their chemical abundances. For this scenario to work, low-mass AGB stars down to approximately $1.2\,{M_{\odot}}$ have to experience $i$-process nucleosynthesis. Should it be proved that the $i$-process is activated at low masses and metallicities, the binary mass-transfer scenario would be confirmed as a robust and most likely mechanism to explain the formation of CEMP stars enriched in neutron-capture elements. This would also have important consequences for our understanding of the early chemical enrichment of galaxies, because it would support the hypothesis that all CEMP stars enriched in neutron-capture elements have a common origin, which is different from CEMP-no stars, as suggested by many authors [e.g. @Spite2013; @Starkenburg2014; @Bonifacio2015].
Conclusions {#concl}
===========
None of the models as yet proposed to explain the origin of CEMP-$r/s$ stars currently reproduces the properties of the observed CEMP-$r/s$ population. In particular, in all but one formation scenarios the observed frequency of CEMP-$r/s$ stars is underestimated by at least a factor of five and up to two orders of magnitude. The only model that predicts a ratio of CEMP-$r/s$ to CEMP-$s$ stars almost consistent with the observations fails to reproduce the correlation between the abundances of europium and barium observed in CEMP stars, and overestimates the proportion of $r$-rich stars by more than a factor of ten.
It has been proposed that the *intermediate* or $i$-process may be activated in some circumstances in AGB or post-AGB stars. Preliminary results show that the theoretical abundance distributions predicted by the models are consistent with those observed in CEMP-$r/s$ stars. CEMP-$r/s$ stars could therefore be the secondary stars of binary systems that in the past accreted material from the winds of AGB primary stars, that is, the same formation scenario proposed for CEMP-$s$ stars. Further calculations are necessary to determine at what masses and metallicities the $i$-process is triggered in AGB stars, what abundance distribution is produced by the $i$-process, and how likely it is to form CEMP-$r/s$ stars. If the preliminary results were confirmed, the binary mass-transfer scenario would stand out as a robust explanation of the origin of all CEMP-$s$ stars, including CEMP-$r/s$ stars. This would support the hypothesis that metal-poor stars highly enriched in neutron-capture elements are formed in binary systems, whereas other carbon-normal or carbon-enhanced metal-poor stars have a different formation history.
The authors are grateful to Dr. R. Izzard for his untiring support with the code `binary_c`. RJS is the recipient of a Sofja Kovalevskaja Award from the Alexander von Humboldt Foundation.
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[^1]: $[{\mathrm{X}}/{\mathrm{Y}}] = \log_{10}(N_{{\mathrm{X}}}/N_{{\mathrm{Y}}}) - \log_{10}(N_{{\mathrm{X}}}/N_{{\mathrm{Y}}})_{\odot}$, where $N_{{\mathrm{X}}}$ and $N_{{\mathrm{Y}}}$ are the number densities of elements X and Y, respectively.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In an attempt to find a polynomial-time algorithm for the edge-clique cover problem on cographs we tried to prove that the edge-clique graphs of cographs have bounded rankwidth. However, this is not the case. In this note we show that the edge-clique graphs of cocktail party graphs have unbounded rankwidth.'
author:
- 'Maw-Shang Chang'
- Ton Kloks
- 'Ching-Hao Liu'
title: '[Edge-Clique Graphs of Cocktail Parties have Unbounded Rankwidth]{}'
---
Introduction
============
Let $G=(V,E)$ be an undirected graph with vertex set $V$ and edge set $E$. A clique is a complete subgraph of $G$.
A edge-clique covering of $G$ is a family of complete subgraphs such that each edge of $G$ is in at least one member of the family.
The minimal cardinality of such a family is the edge-clique covering number, and we denote it by $\theta_e(G)$.
The problem of deciding if $\theta_e(G) \leq k$, for a given natural number $k$, is NP-complete [@kn:kou; @kn:orlin; @kn:holyer]. The problem remains NP-complete when restricted to graphs with maximum degree at most six [@kn:hoover]. Hoover [@kn:hoover] gives a polynomial time algorithm for graphs with maximum degree at most five. For graphs with maximum degree less than five, this was already done by Pullman [@kn:pullman]. Also for linegraphs the problem can be solved in polynomial time [@kn:orlin; @kn:pullman].
In [@kn:kou] it is shown that approximating the clique covering number within a constant factor smaller than two remains NP-complete.
Gyárfás [@kn:gyarfas] showed the following interesting lowerbound. Two vertices $x$ and $y$ are [*equivalent*]{} if they are adjacent and have the same closed neighborhood.
\[gyarfas\] If a graph $G$ has $n$ vertices and contains neither isolated nor equivalent vertices then $\theta_e(G) \geq \log_2(n+1)$.
Gyárfás result implies that the edge-clique cover problem is fixed-parameter tractable (see also [@kn:gramm]). Cygan et al showed that, under the assumption of the exponential time hypothesis, there is no polynomial-time algorithm which reduces the parameterized problem $(\theta_e(G),k)$ to a kernel of size bounded by $2^{o(k)}$. In their proof the authors make use of the fact that $\theta_e(cp(2^{\ell}))$ is a \[sic\] “hard instance for the edge-clique cover problem, at least from a point of view of the currently known algorithms.” Note that, in contrast, the parameterized edge-clique partition problem can be reduced to a kernel with at most $k^2$ vertices [@kn:mujuni]. (Mujuni and Rosamond also mention that the edge-clique cover problem probably has no polynomial kernel.)
Rankwidth of edge-clique graphs of cocktail parties
===================================================
The cocktail party graph $cp(n)$ is the complement of a matching with $2n$ vertices.
Notice that a cocktail party graph has no equivalent vertices. Thus, by Theorem \[gyarfas\], $$\theta_e(cp(n)) \geq log_2(2n+1).$$ For the cocktail party graph an exact formula for $\theta_e(cp(n))$ is given in [@kn:gregory]. In that paper Gregory and Pullman prove that $$\lim_{n \rightarrow \infty} \; \frac{\theta_e(cp(n))}{\log_2(n)}=1.$$
Let $G=(V,E)$ be a graph. The edge-clique graph $K_e(G)$ has as its vertices the edges of $G$ and two vertices of $K_e(G)$ are adjacent when the corresponding edges in $G$ are contained in a clique.
Albertson and Collins prove that there is a 1-1 correspondence between the maximal cliques in $G$ and $K_e(G)$ [@kn:albertson]. The same holds true for the intersections of maximal cliques in $G$ and in $K_e(G)$.
For a graph $G$ we denote the vertex-clique cover number of $G$ by $\kappa(G)$. Thus $$\kappa(G) = \chi(\Bar{G}).$$ Notice that, for a graph $G$, $$\theta_e(G)=\kappa(K_e(G)).$$
Albertson and Collins mention the following result (due to Shearer) [@kn:albertson] for the graphs $K_e^r(cp(n))$, defined inductively by $K_e^r(cp(n))=K_e(K_e^{r-1}(cp(n)))$. $$\alpha(K_e^r(cp(n))) \leq 3\cdot (2^r)!$$ Thus, for $r=1$, $\alpha(K_e(cp(n))) \leq 6$. However, the following is easily checked.
\[bound alpha\] For $n \geq 2$ $$\alpha(K_e(cp(n))) =4.$$
Let $G$ be the complement of a matching $\{x_i,y_i\}$, for $i \in \{1,\dots,n\}$. Let $K=K_e(G)$. Obviously, every pair of edges in the matching induces an independent set with four vertices in $K$.
Consider an edge $e=\{x_i,x_j\}$ in $G$. The only edges in $G$ that are not adjacent to $e$ in $K$, must have endpoints in $y_i$ or in $y_j$. Consider an edge $f=\{y_i,y_k\}$ for some $k \notin \{i,j\}$. The only other edge incident with $y_i$, which is not adjacent in $K$ to $f$ nor to $e$ is $\{y_i,x_k\}$.
The only edge incident with $y_j$ which is not adjacent to $e$ nor $f$ is $\{y_j,x_i\}$. This proves the lemma.
A class of graphs $\mathcal{G}$ is $\chi$-bounded if there exists a function $f$ such that for every graph $G \in \mathcal{G}$, $$\chi(G) \leq f(\omega(G)).$$
Dvořák and Král proved that the class of graphs with rankwidth at most $k$ is $\chi$-bounded [@kn:dvorak].
We now easily obtain our result.
The class of edge-clique graphs of cocktail parties has unbounded rankwidth.
It is easy to see that the rankwidth of any graph is at most one more than the rankwidth of its complement [@kn:oum]. Assume that there is a constant $k$ such that the rankwidth of $K_e(G)$ is at most $k$ whenever $G$ is a cocktail party graph. Let $$\mathcal{K}=\{\; \overline{K_e(G)} \;|\; G \simeq cp(n),
\;n \in \mathbb{N}\;\}.$$ Then the rankwidth of graphs in $\mathcal{K}$ is uniformly bounded by $k+1$. By the result of Dvořák and Král, there exists a function $f$ such that $$\kappa(K_e(G)) \leq f(\alpha(K_e(G)))$$ for every cocktail party graph $G$. This contradicts Lemma \[bound alpha\] and Theorem \[gyarfas\].
Concluding remark
=================
As far as we know, the recognition of edge-clique graphs is an open problem.
The edge-clique cover problem is NP-complete for cographs.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The usage of a large amount of CdTe(CdZnTe) semiconductor detectors for solar neutrino spectroscopy in the low energy region is investigated. Several different coincidence signals can be formed on five different isotopes to measure the neutrino line at 862 keV in real-time. The most promising one is the usage of resulting in 89 SNU. The presence of permits even the real-time detection of pp-neutrinos. A possible antineutrino flux above 713 keV might be detected by capture on .'
address: |
Oxford University, Dept. of Physics, Denys Wilkinson Building,\
Keble Road, Oxford OX1 3RH, England
author:
- 'K. Zuber'
title: Spectroscopy of low energy solar neutrinos using CdTe detectors
---
\#1\#2\#3\#4[[\#1]{} [**\#2**]{}, \#3 (\#4)]{}
[PACS: 13.15,13.20Eb,14.60.Pq,14.60.St]{}
massive neutrinos, solar neutrinos
Introduction
============
Over the last years striking evidence arose for a non-vanishing neutrino rest mass (for reviews see [@zub98; @samoil]). They all come from neutrino oscillations experiments. Among them is the long standing evidence of a solar neutrino deficit also being of fundamental importance for stellar astrophysics. The deficit is seen in radiochemical detectors, namely GALLEX/GNO [@gallex; @kirsten] and SAGE [@gavrin] using , still the only pp-neutrino detectors available, and the Homestake experiment using [@ray]. A reduced $^8$B [ ]{}flux is measured by two water Cerenkov detectors, namely Super-Kamiokande [@smy] and SNO [@snocc]. A difference in measured fluxes among the latter resulted in evidence for an active neutrino flavour coming from the sun besides [ ]{}. This is due to the fact that Super-Kamiokande is using neutrino-electron scattering and SNO inverse [$\beta$-decay]{} for detection. Recent SNO results of neutral current reactions on deuterium dramatically confirm the existence of further active neutrinos, being the dominant solar neutrino flux [@snonc]. The solution of the solar neutrino deficit has to come from particle physics, the scenario discussed most often is neutrino oscillations. Taking all experimental results together the largest effects in the solar neutrino spectrum implied by the various oscillations solutions show up in the region below 1 MeV. Furthermore this region corresponds to 99 % of the solar neutrino flux and is still very important for understanding stellar energy generation [@bah02]. An interesting idea to measure such low energy neutrinos in real time is suggested by [@rajulens] using coincidence techniques for neutrino capture on nuclei. It is already finding its practical application in the LENS [@lens], SIREN [@siren] and MOON projects [@moon]. aiming to measure pp-neutrinos in real time. The technique relies on using either a large amount of double beta isotopes (in case of LENS, [ ]{}in case of SIREN and in case of MOON) or highly forbidden beta decay emitters like (4-fold forbidden, currently under study in LENS as an alternative to ), as target material [@rajulensin]. Clearly an interesting spin off is the investigation of double beta decay [@zub00].\
In this paper the possibility to apply the same technique for CdTe(CdZnTe) semiconductor detectors and their feasibility for solar neutrino detection is explored. CdTe semiconductor detectors have already a wide field of application in $\gamma$-ray astronomy and medical physics. The study performed here is motivated by the COBRA project [@cobra], planning to use large amounts of CdTe-detectors for double beta decay searches. The usage of large amounts of semiconductors for solar neutrino detection was also considered in the past for Ge-detectors [@laura] and GaAs [@bowles] relying largely on the detection of electrons from neutrino-electron scattering. In the case discussed here, we focus on the detection of and pp-neutrinos only. Measurements of higher energetic neutrino flux components are also possible but will not be discussed. Also a possible real-time detection of pp-neutrinos via neutrino-electron scattering as well as contributions from the CNO cycle are not considered.
Solar neutrino detection via coincidence measurements
=====================================================
The detection principle for solar neutrinos using coincidences relies on the following two reactions \[eq:gs\] [ ]{}+ (A,Z) (A,Z+1)\_[g.s.]{} + e\^- (A,Z+2) + e\^- + [ ]{}\
\[eq:es\] [ ]{}+ (A,Z) (A,Z+1)\^+ e\^- (A,Z+1)\_[g.s.]{} + Therefore either coincidence between two electrons for the ground state transitions or the coincidence of an electron with the corresponding de-excitation photon(s) is required. The first one is followed by MOON, while the second one is used for LENS. The produced electrons as the first part of the coincidence have energies of $$E_e = E_{\nu} - (E_f - E_i)$$ with [ ]{}as neutrino energy, $E_f$ and $E_i$ as the energy of the final and initial nuclear state involved in the transition. In case of [neutrinos ]{}the electrons will be monoenergetic.
Solar [neutrino ]{}tags using double beta isotopes
--------------------------------------------------
Consider detection with the help of double beta isotopes first. There are 4 (5) [ ]{}emitters in CdTe (CdZnTe) [@cobra]. Three of them have sensitivity to the line of 862 keV, namely , and . The corresponding coincidence tags are shown in Fig. \[pic:bbtags\]. and are in the form of ground state transitions (Eq. \[eq:gs\]), while for an e-$\gamma$ coincidence (Eq. \[eq:es\]) is required. In case of it is the ground state transition to with a threshold of [ ]{}= 655 keV. will decay via beta decay to the ground state of in 98.9 % of all cases and a half-life of 21.14 min. However because is only present in small amounts (Zn typically replaces 10 % of Cd in CdZnTe detectors) in the detector, the expected rate is rather small. In case of it is the transition to the first excited 1$^+$ state in , lying 43.25 keV above ground state. This corresponds to a neutrino energy threshold of [ ]{}= 494 keV. The state de-excites under the emission of a 3.3 keV X-ray. With a half-life of 8.8 mins in 86 % of the cases an IT will happen resulting in a 39.95 keV photon or in 14 % a [$\beta$-decay]{} to , dominantly inot the first excited 2$^+$ state. This is connected with the emission of a 536 keV photon. The analogous [$\beta$-decay]{} of the 5$^+$ ground state of with a half-life of 12.36 h is probably inadequate to use in the coincidence. Two more 1$^+$ states exist at 254.8 keV and 349.6 keV above ground state which can be populated by , corresponding to a neutrino energy threshold of [ ]{}= 706 keV and 801 keV respectively. Furthermore there exist several low-lying states, whose quantum numbers have not been determined yet and it might be worthwile to do so. Probably the most appropriate isotope for the search is . The coincidence signal will be the two electrons of the neutrino capture to the ground state of with an energy threshold of [ ]{}= 464 keV together with the [$\beta$-decay]{} (practically 100 %) of with a half life of 14.1 s. The Q-value of the latter is 3.275 MeV. Thus a good coincidence signal can be formed among the two electrons. Such double electron tags can also be performed for detecting the pep-line at 1.445 MeV by using . Here a low energy electron of about 5 keV is in coincidence with a [$\beta$-decay]{} electron of $^{114}$In, having a half-life of 72s and a Q-value of 1.98 MeV.
The case of
------------
One might also consider the case of as target for a neutrino capture into . is one out of only three known 4-fold forbidden beta decay isotopes, besides and $^{50}$V. Indeed, since quite some time the idea to build an indium solar neutrino detector exists [@raju1], a possible realisation is considered now within the LENS experiment. It is therefore natural to ask, whether is also interesting. The possible solar neutrino tag is shown in Fig. \[pic:otags\]. As can be seen, also with this isotope spectroscopy of the sun is possible. There are two excited 3/2$^+$ states at 1029.6 and 1063.9 keV above ground state, resulting in neutrino thresholds of [ ]{}= 709 and 743 keV respectively. The electron is most of the time accompanied by a 672 or 638 keV photon. The IT of the 1/2$^-$ state to the 9/2$^+$ ground state of is associated with an additional 391.7 keV photon and a half-life of 1.65 h. Two forbidden transitions exist which in principle would allow a real time-detection of pp-neutrinos with a threshold of [ ]{}= 70 keV and 330 keV, however the expected smallness of the involved nuclear matrix elements will result in a rather small rate.\
Possible advantages with respect to are that the beta half-life of is more than one order of magnitude higher than for , resulting in correspondingly less background from that process (typical background from the [$\beta$-decay]{} is 0.24 Bq/g In). Furthermore, the endpoint energy of is only about 320 keV compared to 496 keV from . Additionally, its implementation in a semiconductors implies a better energy resolution than scintillators. Disadvantages are the smaller natural abundance with respect to , being a factor eight less, and the much higher threshold for allowed transitions. $^{50}$V as the third 4-fold forbidden beta-decay emitter is not appropriate for low energy solar neutrino searches because of its low natural abundance and lack of interesting low lying 1$^+$ states. A comparison of the three isotopes is shown in Tab. \[tab:fourfold\].
Real time pp-detection using of
--------------------------------
A chance for real-time pp-neutrino detection exists using two allowed states in (Fig \[\]). The first excited 3/2$^+$ state in is 188 keV above the ground state allowing pp-detection with a threshold of [ ]{}= 366 keV. The coincidence will be formed by a double tag of an electron in direct coincidence to the emission of a 188 keV photon or the corresponding cascade. An additional 1/2$^+$ state exist with a threshold of [ ]{}= 420 keV resulting in the emission of a 243 keV photon. Additionally two further 3/2$^+$ states can be used for detection having thresholds of 549 keV and 631 keV. Associated with them is the emision of 453.8 or 372.1 keV photons respectively.
Measuring solar antineutrinos
-----------------------------
The existence of active flavours coming from the Sun besides [ ]{}is established by SNO. However, it is an open question what this new flavour actually is. Therefore a measurement of a possible solar antineutrino flux might be useful. In CdTe (CdZnTe) there are 3 (4) isotopes available of various forms of [ ]{}decay, allowing for a tag on possible solar antineutrinos. From a principle point of view no antineutrino below 511 keV can be detected via charged current reactions on nuclei because one has to account for the positron mass. Therefore pp-antineutrinos cannot be observed by this method. The most promising candidate in CdTe is , allowing antineutrino detection with a threshold of [ ]{}= 713 keV. The tag would be a monoenergetic 149 keV positron together with a decay of $^{106}$Ag via $\beta^+$ or electron capture and a half life of 24 min.
Experimental considerations
===========================
In the following the detection of the coincidences is discussed close to the design presented in [@cobra]. It is assuming an array of CdTe detectors each of 1cm$^3$ size. Such a design has a large advantage for coincidence searches. The signals containing two electron tags have to rely on the fact, that the same crystal has to fire twice. The individual rates of such a crystal can be small and applying corresponding energy cuts appropriate for the tag will reduce background significantly. Furthermore in case of monoenergetic electrons as expected from [neutrinos ]{}the good energy resolution allows to define a rather tight constraint on the coincidence signal. To avoid background from [$\beta$-decay]{} the second electron can be required to have at least 320 keV. This is completely reducing this background, while keeping the efficiency still high, because the interesting [$\beta$-decay]{}s have Q-values well above 2 MeV (e.g. the [$\beta$-decay]{} of has a Q-value of 3.27 MeV). None of the $\gamma$-lines to be observed from the signal are in close vicinity of the strongest lines of the natural decay chains. The most dangerous might be an effect of the 46.5 keV line of $^{214}$Bi on the 39.9 keV IT line occuring in the tag. Also the 672 keV line of the tag is in vicinity of the 662 keV line of $^{137}$Cs, but both can be significantly reduced by the coincidence techniques and the good energy resolution.\
For higher energetic gammas the coincidence of neighbouring crystals has to be used, because no delayed coincidences can be formed. The efficiency of gamma-rays leaving the crystal without further interactions is increasing with energy and is beyond 60 % already at 250 keV as obtained by a GEANT4 Monte Carlo simulation [@hk]. Another step forward in signal identification would be the usage of pixelised detectors, which would constrain the vertex for two electron tags to one pixel and additionally two tracks consistent with electrons have to start from that pixel. Also multiple interactions within a crystal like an electron together with a gamma, can be probed in that way. Even without pixel detectors such a discrimination seems possible by pulse shape analysis.\
For calibration purposes several solar neutrino experiments have used MCi $^{51}$Cr source, producing monoenergetic lines of 743 keV (90 %) and 426 keV (10 %) neutrinos [@kirst]. Two more sources of $^{75}$Se with [ ]{}= 451/461 keV and $^{37}$Ar with [ ]{}= 814 keV are also under consideration [@kom; @gav]. The $^{51}$Cr source would be appropriate here as well, because all nuclear levels discussed as signal levels except one of the excited 1$^+$ states in and one of the excited 3/2$^+$ states in can be populated by the source. The latter depends on the precise Q-value of the [$\beta$-decay]{}.
Rates
=====
Observed rates can be determined for the ground state transitions by using the known ft-values of the corresponding $\beta$-decays. The cross-section can be determined via the relation [@bv] $$\sigma = \frac{2.64 \cdot 10^{-41}}{ft} \frac{2I'+1}{2I+1} p_e E_e F(Z,E_e) \quad \mbox{(cm$^2$)}$$ where $I',I$ are the involved nuclear spins, $p_e$ and $E_e$ are the momentum and energy of the outgoing electron in units of the electron mass and F(E,Z) is the Coulomb function. The used ft-values of [$\beta$-decay]{} are taken from [@ftv; @bat98]. The Gamow-Teller transition matrix elements for were measured recently [@akimune]. The expected rates from only are 89 SNU for and 10 SNU for . For the excited states transitions the GT matrix elements have to be measured or calculated and without their knowledge, rates cannot be seriously predicted. Therefore the above mentioned neutrino sources are very important. Accelerator measurements can be done using charge exchange reaction like (p,n) or ($^3$He,t) [@ejiri]. A more sophisticated analysis wil aslo include efficiencies from Monte Carlo simulations and details on the $\gamma$-emission in the nuclear de-excitation.
Summary and conclusions
=======================
The prospects of various isotopes of Cd, Zn and Te for low energy solar neutrino spectroscopy are explored. To obtain a reasonable signal various coincidence tags can be used, as compiled in Tab. \[tab:comparison\]. It allows the detection of $^7$Be in real time for five isotopes and therefore offers redundancy in the obtained results. The most promising detection signal is the ground state transition of to resulting in 89 SNU. This has to be seen as a lower limit because CNO contributions are not taken into account. In addition $^{125}$Te allows a real time detection of pp-neutrinos with a threshold of 330 keV. The usage of semiconductors is advantageous for background reduction for $^7$Be detection is because the monoenergetic electron forming the first step of the coincidence can be measured with good precision. Rates for excited state transitions cannnot be determined reliably because a lack of knowledge in the corresponding GT matrix elements, a problem also known from other low energy solar neutrino experiments. It might be worthwile to consider an experimental program to measure these matrix elements, which would also be valueable for double beta decay. As common for solar neutrino detection detector sizes of tons have to be considered, this kind of experiment is not feasable in the very near future.
Acknowledgements
================
I would like to thank H. Ejiri, Y. Ramachers and S. Schoenert for valueable discussions and comments. This work is supported by a Heisenberg-Fellowship of the Deutsche Forschungsgemeinschaft.
[99]{} K. Zuber, S. M. Bilenky, C. Giunti, W. Grimus, W. Hampel et al. , T. Kirsten, Talk presented at Neutrino 2002, Munich V. N. Gavrin, Talk presented at Neutrino 2002, Munich B.T. Cleveland et al., M. B. Smy, Talk presented at Neutrino 2002, Munich Q. R. Ahmad et al., Q. R. Ahmad et al., nucl-ex/0204008 J.N. Bahcall, Talk presented at LowNu02, Heidelberg R. S. Raghavan , C. Cattadori, Talk presented at LowNu02, Heidelberg V. A. Kudryavtsev ,Talk presented at LowNu02, Heidelberg H. Ejiri et al., R. S. Raghavan ,hep-ex/0106054 K. Zuber, K. Zuber, L. Baudis, H.V. Klapdor-Kleingrothaus, V. N. Gavrin et al., R. S. Raghavan , H. Kiel, private communication T. Kirsten , Talk presented at LowNu02, Heidelberg V. Komouhkov, Talk presented at LowNu02, Heidelberg V. N. Gavrin, Talk presented at LowNu02, Heidelberg F. Boehm, P. Vogel, Physics of massive neutrinos, Cambridge University Press, 1992 B. Singh et al , M. Bhattacharya, H. Akimune et al. , H. Ejiri ,
tabs.tex
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{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
Given a birational normal extension ${\mathcal{O}}$ of a two-dimensional local regular ring $(R,
{\mathfrak{m}})$, we describe all the equisingularity types of the complete ${\mathfrak{m}}$-primary ideals $J$ in $R$ whose blowing-up $X=Bl_J(R)$ has some point $Q$ whose local ring ${\mathcal{O}}_{X,Q}$ is analytically isomorphic to ${\mathcal{O}}$.
author:
- |
Maria Alberich-Carramiñana and Jesús Fernández-Sánchez [^1]\
\
\
title: Equisingularity classes of birational projections of normal singularities to a plane
---
Introduction {#introduction .unnumbered}
============
A sandwiched surface singularity $(X, Q)$ is a normal surface singularity that can be projected birationally to a non-singular surface. From a more algebraic point of view, the local ring ${\mathcal{O}}$ of any sandwiched singularity is a birational normal extension of a two-dimensional local regular ring $R$. Once a sandwiched surface singularity has been fixed, in this paper we address the problem of describing the equisingularity classes of all its birational projections to a plane. The problem of classifying the germs of sandwiched surface singularities was already posed by Spivakovsky. As he claims in [@Spivak] this problem has two parts: discrete and continuous. The continuous part is to some extent equivalent to the problem of the moduli of plane curve singularities, while the main result of this paper solves completely the combinatorial part. Any birational projection from a sandwiched singularity to a plane is obtained by the morphism of blowing up a complete ${\mathfrak{m}}_O$-primary ideal in the local ring of a regular point $O$ on the plane. Our goal is to give all the equisingularity types of these ideals. Namely, fixed a birational normal extension ${\mathcal{O}}$ of a local regular ring $(R, {\mathfrak{m}}_O)$, we describe the equisingularity type of any complete ${\mathfrak{m}}_O$-primary ideal $J\subset R$ such that its blowing-up $X=Bl_J(R)$ has some point $Q$ whose local ring ${\mathcal{O}}_{X,Q}$ is analytically isomorphic to ${\mathcal{O}}$. In this case, we will say that the surface $X$ contains the singularity ${\mathcal{O}}$ for short, making a slight abuse of language. This is done by describing the Enriques diagram of the cluster of base points of any such ideal $J$: such a diagram will be called an Enriques diagram for the singularity ${\mathcal{O}}$. Recall that an Enriques diagram is a tree together with a binary relation (proximity) representing the topological equivalence classes of clusters of points in the plane (see §\[ED-dualG\]). Previous works by Spivakovsky [@Spivak] and Möhring [@Mohring] describe a type of Enriques diagram that exists for any given sandwiched surface singularity (detailed in §2) and provide other types mostly in the case of cyclic quotients (see [@Mohring] 2.7) and minimal singularities (see [@Mohring] 2.5).
The organization of the paper is as follows. Section 1 is devoted to recalling some definitions concerning the language of infinitely near points, sandwiched surface singularities and graphs. Fundamental for our purpose will be the notion of Enriques diagram, introduced in [@EC85]. In Section 2, after some technical results, we introduce the concept of *contraction* for a sandwiched surface singularity ${\mathcal{O}}$. By a contraction we mean the resolution graph ${\Gamma}_{{\mathcal{O}}}$ of ${\mathcal{O}}$ (a sandwiched graph, as introduced in [@Spivak]) enriched by some proximities between their vertices, these proximities being compatible with the weights of the graph. Fixed a sandwiched graph, the problem of finding the whole list of possibilities for such proximities is the hard part of our work. This is achieved in Section 3, by proving that any contraction for ${\Gamma}_{{\mathcal{O}}}$ may be recovered from some contraction of the graph obtained from ${\Gamma}_{{\mathcal{O}}}$ by removing one end. This fact is the key result in order to describe a procedure to obtain all the contractions for ${\mathcal{O}}$. Finally, in Section 4, we explain how to complete contractions in order to obtain any Enriques diagram for ${\mathcal{O}}$.
**Acknowledgement** The authors thank E. Casas-Alvero for drawing their attention to the problem addressed in this paper. They also thank M. Spivakovsky for the conversations held on this topic.
Preliminaries
=============
In this section, we fix notation and recall some of the facts concerning sandwiched surface singularities and base points of ideals which will be used throughout this paper, and we focus on our problem. A standard reference for most of the tools and techniques treated here is the book by Casas-Alvero [@Cas00].
Infinitely near points and complete ideals
------------------------------------------
Let $(R,{{\mathfrak{m}}_O})$ be a regular local two-dimensional $\mathbb{C}$-algebra and $S=Spec(R)$. A *cluster* of points of $S$ with origin $O$ is a finite set $K$ of points infinitely near or equal to $O$ such that, for any $p\in K$, $K$ contains all points to which $p$ is infinitely near. A subset of a cluster $K$ is a *subcluster* if it is a cluster (with origin some point of $K$). By assigning integral multiplicities $\nu=\{\nu_p\}$ to the points of $K$, we obtain a *weighted cluster* ${\mathcal
K}=(K,\nu)$; the multiplicities $\nu$ are called the *virtual multiplicities* of ${\mathcal{K}}$. A point $p$ is said to be *proximate* to another point $q$ if $p$ is infinitely near to $q$ and lies on the strict transform of the exceptional divisor of blowing up $q$. We write $p\geq q$ if $p$ is infinitely near or equal to $q$, and $p\rightarrow q$ if $p$ is proximate to $q$. The relation $\geq$ is an ordering of the infinitely near points, and it will be considered as their natural ordering. A point $p\in K$ is *free* if it is proximate to only one point, which is necessarily the immediate predecessor, and $p$ is *satellite* if it is proximate to two points; otherwise, the point is necessarily the origin of the cluster. The number $\rho^{{\mathcal{K}}}_p=\nu_p- \sum_{q \rightarrow p} {\nu _q }$ is the *excess* at $p$ of ${\mathcal K}$. *Consistent clusters* are those weighted clusters with non-negative excesses at all their points. We write ${\mathcal{K}}_+$ for the set of the *dicritical* points of ${\mathcal{K}}$, that is, the points with positive excess.
If $\mathcal{K}=(K, \nu)$ and $\mathcal{K}'=(K', \nu ')$, define the *sum* $\mathcal{K}+ \mathcal{K}'$ as the weighted cluster whose set of points is $K \cup K'$ and whose virtual multiplicities are $\nu_{p}+\nu'_{p}$ for $p \in K \cup K'$ ([@Cas00] 8.4). This operation is clearly associative and commutative, thus making the set of all weighted clusters with origin at $O$ a semigroup. Consider the set $\mathbf{W}$ of all consistent clusters with origin at $O$ with positive virtual multiplicities. Again, $\mathbf{W}$ equipped with the sum is clearly a semigroup. A weighted cluster $\mathcal{K} \in \mathbf{W}$ is called *irreducible* if it is so as element of the semigroup $\mathbf{W}$, that is, $\mathcal{K}$ is not the sum of two elements of $\mathbf{W}$. To any point $p$, $p \ge O$, we associate the irreducible cluster $\mathcal{K}(p)$ in $\mathbf{W}$ which has virtual multiplicity one at $p$, which will be called the *irreducible cluster in* $\mathbf{W}$ *ending at* $p$.
Two *clusters* $K$ and $K'$ are called *similar* if there is a bijection (*similarity*) $\varphi: K
\longrightarrow K'$ so that both $\varphi$ and $\varphi ^{-1}$ preserve ordering and proximity. Two *weighted clusters* $\mathcal{K}=(K, \nu)$ and $\mathcal{K}'=(K', \nu ')$ are called *similar* if there is a similarity between $K$ and $K'$ preserving virtual multiplicities (see [@Cas00] 8.3). An analytic isomorphism $\Phi$ defined in a neighborhood of $O$ clearly induces a similarity between each cluster $K$ with origin $O$ and its image $\Phi(K)$ ([@Cas00] 3.3). Furthermore, if $\Phi$ is only a homeomorphism, then $K$ and $\Phi(K)$ are still similar ([@Cas00] 8.3.12)
If $\pi_K:S_K\longrightarrow S$ is the composition of the blowing-ups of all points in $K$, write $E_K$ for the exceptional divisor of $\pi_K$ and $\{E_p\}_{p\in K}$ for its irreducible components. We denote by $\mathbf{A}_K=(E_p\cdot E_q)_{p,q\in K}$ the *intersection matrix* of $E_K$: if $p=q$, its coefficient is just the self-intersection of $E_p$, and equals $-r_p-1$, where $r_p$ is the number of points in $K$ proximate to $p$; if $p\neq q$, $E_p\cdot E_q=1$ in case $E_p\cap E_q\neq \emptyset$, and $E_p\cdot E_q=0$ otherwise. It can be easily seen that $E_p\cap E_q\neq \emptyset$ if and only if $p$ is maximal among the points of $K$ proximate to $q$ or vice-versa (cf. [@Cas00] 4.4.2). Notice that $\mathbf{A}_K$ is an invariant of the similarity class of $K$.
If ${\mathcal K}$ is a weighted cluster, there is a well established notion for a germ of curve to go through $\mathcal{K}$ (which is a linear condition, see [@Cas00] 4.1), and the equations of all curves going through $\mathcal{K}$ define a complete ${\mathfrak{m}}_O$-primary ideal $H_{\mathcal K}$ in $R$ (see [@Cas00] 8.3). Any complete ${\mathfrak{m}}_O$-primary ideal $J$ in $R$ has a *weighted cluster of base points*, denoted by $BP(J)$, which consists of the points shared by, and the multiplicities of, the curves defined by generic elements of $J$. Moreover, the maps $J\mapsto BP(J)$ and ${\mathcal
K}\mapsto H_{\mathcal K}$ are reciprocal isomorphisms between the semigroup $\textbf{I}_R$ of complete ${\mathfrak{m}}_O$-primary ideals in $R$ (equipped with the product of ideals) and the semigroup $\textbf{W}$ (see [@Cas00] 8.4.11 for details). If $p \ge O$, denote by $J(p)$ the ideal in $\mathbf{I}_R$ corresponding by the preceding isomorphism to the irreducible cluster $\mathcal{K}(p) \in \mathbf{W}$ ending at $p$, that is, $J(p) = H_{\mathcal{K}(p)}$.
A couple of ideals $J, J'$ in $\mathbf{I}_R$ are *equisingular* if $BP(J)$ and $BP(J')$ are similar ([@Cas00] 8.3). Notice that two equisingular complete ideals in $\mathbf{I}_R$ have equisingular (that is, topologically equivalent) generic germs and equal codimensions ([@Cas00] 8.3.9).
Sandwiched surface singularities {#section 1.1}
--------------------------------
The main references here are [@Spivak] and [@FS1]. If $I\in {\mathbf{I}_R}$, we denote by $\pi_I:~X=Bl_I(R)\longrightarrow S$ the blowing-up of $I$. The surface $X$ is not regular in general, and its singularities are sandwiched singularities. Moreover, if $K$ is the set of base points of $I$, we have a commutative diagram $$\label{CommDiagr}
\xymatrix{ {S_K} \ar[r]^{f}\ar[rd]_{\pi_K} & {X} \ar[d]^{\pi_I} \\ & {S}}$$ where the morphism $f$, given by the universal property of the blowing-up, is the minimal resolution of the singularities of $X$ ([@Spivak] Remark 1.4). Let ${\mathcal{O}}$ be any singularity of $X$; then we say that $I$ is an *ideal for* ${\mathcal{O}}$. It follows that the exceptional divisor $E_{{\mathcal{O}}}$ associated with the minimal resolution of ${\mathcal{O}}$ is a connected subset of the exceptional divisor $E_K$. There is a bijection between the set of irreducible components of $\pi_I^{-1}(O)$ and the set of dicritical points of ${\mathcal{K}}=BP(I)$ (see [@FS4; @Lip69]). This allows to write $\{L_p\}_{p\in {\mathcal{K}}_+}$ for the set of these components on $X$. Because of this, we may think of ${\mathcal{O}}$ as a singularity obtained by contracting a connected curve (which will be called $E_{{\mathcal{O}}}$) of $E_K$ containing no component with self-intersection $-1$ (such a component $E_p$ is necessarily the exceptional divisor of the blowing-up of some maximal point of $K$ and thus, a dicritical point). For any ideal $J= \prod_{p \in {\mathcal{K}}_+} J(p)^{\alpha(p)}$ with positive $\alpha(p)$, we have an analytic isomorphism $X \cong
Bl_J(R)$ (cf. [@Spivak], Corollary I.1.5). Since we are interested in sandwiched singularities modulo analytic isomorphism, the relevant information we need to retain about ${\mathcal{K}}=BP(I)$ is, on one hand, its set of points $K$ and, on the other, knowing which of the points of $K$ are dicritical (the rest being non-dicritical, of excess zero).
Enriques diagrams and dual graphs {#ED-dualG}
---------------------------------
We introduce the Enriques diagrams and the weighted dual graphs related to them. The Enriques diagrams are combinatorial objects that enclose the topological information of the clusters of infinitely near points in $S$, namely they represent the similarity classes of clusters.
A *tree* is a finite graph with a partial order relation $\leq$ between the vertices, without loops, which has a single initial vertex, or *root*, and every other vertex has a unique immediate predecessor. The vertex $q$ is said to be a *successor* of $p$ if $p$ is the immediate predecessor of $q$. If $p$ has no successors then it is an *extremal vertex*. The set of vertices of a graph will be denoted by the same letter as the graph itself. An *Enriques diagram* $D$ ([@EC85] Enriques IV.I, [@Cas00] Casas 3.9; see also [@GSG92] and [@KP99] for a combinatorial presentation) is a tree with a binary relation between vertices, called *proximity* and denoted by $\rightarrow_{D}$, which satisfies:
1. Every vertex but the root is proximate to its immediate predecessor; the root is proximate to no vertex.
2. If $p\rightarrow_D q$, then $p>q$ and there is at most one other vertex in $D$ proximate to both of them.
3. Any vertex is proximate to at most two other vertices. The vertices which are proximate to two points are called *satellite*, the other vertices, but the root, are called *free*. If $q$ is the immediate predecessor of $p$, and $p\rightarrow_D q'$, then $q\rightarrow_D q'$.
If $p$ is a vertex in $D$, we write $r_D(p)$ for the number of vertices in $D$ proximate to $p$. A satellite vertex is said to be *satellite of* the last free vertex that precedes it. In order to express graphically the proximity relation, Enriques diagrams are drawn according to the following rules:
1. If $q$ is a free successor of $p$ then the edge going from $p$ to $q$ is smooth and curved and, if $p$ is not the root, it has at $p$ the same tangent as the edge joining $p$ to its predecessor.
2. The sequence of edges connecting a maximal succession of vertices proximate to the same vertex $p$ are shaped into a line segment, orthogonal to the edge joining $p$ to the first vertex of the sequence.
If $K$ is a cluster, there is an Enriques diagram $D_K$ naturally associated with it by taking one vertex for each point of $K$ and the proximity of the cluster as the proximity of $D_K$; conversely, for any Enriques diagram $D$ there is some cluster $K$ with origin $O$ whose Enriques diagram $D_K$ is $D$. If no confusion may arise, we will label the points in $K$ and their corresponding vertices in $D_K$ with the same symbol. A connected subtree of an Enriques diagram $D$ is a *subdiagram* if it is an Enriques diagram with root some vertex of $D$ and whose proximity is the restriction of the proximity of $D$. Observe that $K'$ is a subcluster of $K$ if and only if the associated Enriques diagram $D_{K'}$ is a subdiagram of $D_K$. If $D$ is the Enriques diagram associated with $K$ and $p \in K$, we denote by $D(p)$ the Enriques diagram of the irreducible cluster $\mathcal{K}(p)$ ending at $p$. If $p$ is extremal, $D(p)$ is called a *branch* of $D$.
By assigning to an Enriques diagram $D$ a *marking map* $\rho: D\rightarrow \{ + , 0 \}$, we obtain a *marked Enriques diagram* ${\mathcal{D}}=(D,\rho)$. Any consistent cluster ${\mathcal{K}}$ induces a marking map $\rho: D_K\rightarrow \{ + , 0 \}$ by taking $\rho(p)=+$ if $p$ corresponds to a dicritical point of ${\mathcal{K}}$ (in this case, $p$ is called a *dicritical vertex*), and $\rho(p)=0$ otherwise. A *marked subdiagram* ${\mathcal{D}}' = (D',
\rho') $ of ${\mathcal{D}}$ is a marked Enriques diagram where $D'$ is a subdiagram of $D$ and $\rho'$ is the restriction of $\rho$ to $D'$. Observe that the extremal vertices of a marked Enriques diagram associated with some ${\mathcal{K}}\in \mathbf{W}$ are always dicritical. If ${\mathcal{O}}$ is a sandwiched surface singularity, we say that $\mathcal{D}$ is an *Enriques diagram for* ${\mathcal{O}}$ if it is the marked Enriques diagram of $BP(I)$, for some ideal $I\in {\mathbf{I}_R}$ for ${\mathcal{O}}$. Under this framework, the goal of this paper is to describe all the Enriques diagrams for a given ${\mathcal{O}}$.
Incidence between the irreducible components of a divisor $E$ on a surface is usually represented by means of the *weighted dual graph* of $E$. It is defined by taking a vertex for each component of $E$, and by joining two vertices by an edge if and only if the corresponding components of $E$ meet; each vertex is weighted by taking minus the self-intersection of the corresponding component. If $D$ is the Enriques diagram of a cluster $K$, the *(weighted) dual graph* of $D$, denoted by $\Gamma _D$, is the weighted dual graph of the exceptional divisor $E_K$ (which has no loops). Since the information enclosed in the weighted dual graph is the same as that contained in the intersection matrix of $K$, this definition is consistent.
The similarity class of a cluster may be represented either by its Enriques diagram or by its weighted dual graph, since from the intersection matrix the ordering $\le$ (of being infinitely near) and the proximity may be inferred. In fact, this is also true for rational surface singularities. From the intersection matrix $\mathbf{A}$ of a rational surface singularity, the fundamental cycle $Z$ may be computed (see [@Lau72] Theorem 4.2) and from it, the order of the blowing-ups performed to resolve the singularity: the negative entries of $\mathbf{A}Z$ correspond to the exceptional components having appeared in the last blowing-up (cf. Theorem 1.14 of [@Reguera97]). It is worth noticing that the proximity of $D$ cannot be recovered in general only from its dual graph without weights (see [@Cas00] 4.4).
A *non-singular graph* is the weighted dual graph of some Enriques diagram (cf. [@Spivak]). The vertex in $\Gamma_{D}$ corresponding to $p$ in $D$ will be denoted by ${\textsl{p}}$, written in *roman* font. By the (weighted) dual graph of a marked Enriques diagram ${\mathcal{D}}=(D,\rho )$ we mean the dual graph of $D$ and it will also be denoted by ${\Gamma}_D$. The vertices of ${\Gamma}_D$ corresponding to dicritical vertices (non-dicritical, respectively) of ${\mathcal{D}}$ will be called *dicritical* (*non-dicritical*, respectively), too. If $I$ is a complete ${\mathfrak{m}}_O$-primary ideal in $R$, we will write ${\mathcal{D}}_I$ and $\Gamma_I$ to mean the marked Enriques diagram and the weighted dual graph of its cluster of base points $BP(I)$, respectively.
\[w=1extrem\] The weighted dual graph ${\Gamma}_{D}$ can be constructed as follows: take one vertex in $\Gamma_{D}$ for each vertex of $D$, and connect two vertices in $\Gamma_{D}$ by an edge if and only if one of the corresponding vertices in $D$ is maximal among the vertices in $D$ proximate to the other. Moreover, if ${\textsl{p}}$ is a vertex of $\Gamma_{D}$, its weight $\omega({\textsl{p}})$ is $r_D(p) + 1$ (cf. §4.4 of [@Cas00] for details). A vertex ${\textsl{p}}\in \Gamma_{D}$ has weight $\omega({\textsl{p}})=1$ if and only if ${\textsl{p}}$ is extremal in $D$.
A *chain* $ch_{{\Gamma}}({\textsl{q}},{\textsl{p}})$ of a graph ${\Gamma}$ without loops is the subgraph composed of all vertices and edges between the vertices ${\textsl{q}},{\textsl{p}}\in {\Gamma}$; it will be described by the ordered sequence of vertices between ${\textsl{q}}$ and ${\textsl{p}}$, and $d_{{\Gamma}}({\textsl{q}},{\textsl{p}})$ will denote its length. Two vertices ${\textsl{q}},{\textsl{p}}\in {\Gamma}$ are *adjacent* if $d_{{\Gamma}}({\textsl{p}},{\textsl{q}})=1$; a vertex is an *end* if it is adjacent to only one vertex. A *weighted subgraph* of a weighted graph ${\Gamma}$ is a subgraph of ${\Gamma}$ whose vertices have the same weights as ${\Gamma}$.
The following result describes the proximity relations between the vertices of a chain:
\[chain\] Let $q \leq p$ be two vertices of an Enriques diagram $D$, and consider the non-singular graph ${\Gamma}$ of $D$.
- If ${\textsl{u}}\in ch_{{\Gamma}}({\textsl{q}},{\textsl{p}})$, then $q \leq u$; if $u \neq
p$, either $u \leq p$ or $p \leq u$. Moreover, all the vertices of $ch_{{\Gamma}}({\textsl{q}},{\textsl{p}})$ correspond to vertices in the same branch of $D$.
- Write $ch_{{\Gamma}}({\textsl{q}},{\textsl{p}})=\{{\textsl{u}}_0={\textsl{q}},{\textsl{u}}_1,\ldots,{\textsl{u}}_n,{\textsl{u}}_{n+1}={\textsl{p}}\}$. There exists some $i_0\in \{0,\ldots,n+~1\}$ satisfying $u_{k+1}
\rightarrow_{D} u_{k}$ for $k\in \{0,\ldots i_0-1\}$, and $u_k
\rightarrow_{D} u_{k+1}$ for $k\in \{i_0,\ldots,n\}$. Furthermore, if $j\geq i_0$, $u_j$ is proximate to some $u_{\sigma(j)}$ with $\sigma(j)\leq i_0-1$.
The first assertion of (a) is just Lemma 3.2 of [@FS2]. Now, if ${\textsl{u}},{\textsl{v}}\in ch_{{\Gamma}}({\textsl{q}},{\textsl{p}})$, either ${\textsl{u}}\in ch_{{\Gamma}}({\textsl{q}},{\textsl{v}})$ or ${\textsl{v}}\in ch_{{\Gamma}}({\textsl{q}},{\textsl{u}})$; in any case, either $u$ is infinitely near to $v$ or viceversa, and hence $u$ and $v$ cannot belong to different branches of $D$. Now, we prove (b). First of all, note that for any $i\in \{0,\ldots,n\}$ either $u_i$ is proximate to $u_{i+1}$ or viceversa (cf. \[w=1extrem\]). By (a), $u_1$ is necessarily infinitely near to $u_0$ and so, proximate to it. If each $u_{i+1}$ is proximate to $u_i$, the first claim is obvious by taking $i_0=n+1$. Assume that there exists some $i\in
\{1,\ldots,n\}$ such that $u_i$ is proximate to $u_{i+1}$, and take $i_0$ to be minimal with this property. We claim that $u_{k+1} \rightarrow_{D} u_{k}$ for $k\in \{0,\ldots i_0-1\}$, and $u_k \rightarrow_{D} u_{k+1}$ for $k\in \{i_0,\ldots,n\}$. To show this, assume that there exists some $j\geq i_0+1$ such that $u_{j+1}\rightarrow_{D} u_j$ and take $j_0$ to be minimal. Then, both $u_{j_0-1}$ and $u_{j_0+1}$ are proximate to $u_{j_0}$ and, since they are adjacent to it, they are maximal among the vertices of $D$ proximate to $u_{j_0}$. However, by (a) they are in the same branch of $D$, so they must be equal, which is impossible. Note that $u_{i_0}$ is the maximal point in $D$ among the vertices belonging to $ch_{{\Gamma}}({\textsl{q}},{\textsl{p}})$. By (a) we know that every $u_j$, $j\geq i_0$ is infinitely near to $q$. Write $u_{\sigma(j)}$ for the maximal vertex in $D$ among the vertices belonging to $ch_{{\Gamma}}({\textsl{q}},{\textsl{u}}_{i_0-1})$ such that $u_j$ is infinitely near to it. By (a) applied to $ch_{{\Gamma}}({\textsl{u}}_{\sigma(j)},{\textsl{u}}_j)$ and the maximality of $u_{\sigma(j)}$, necessarily $u_{\sigma(j)+1}$ is infinitely near to $u_j$ and, because $u_{\sigma(j)+1}$ is proximate to $u_{\sigma(j)}$, so is $u_j$. This completes the proof.
The *resolution graph* of a sandwiched singularity ${\mathcal{O}}$ is the weighted dual graph of the exceptional divisor of the minimal resolution of ${\mathcal{O}}$. These graphs are called *sandwiched graphs* and they are characterized as the weighted subgraphs of some non-singular graph containing no vertices of weight $1$ (see [@Spivak] Proposition II.1.11; cf. forthcoming \[rem\_G\]). In particular, the graph obtained from a sandwiched graph by removing an end is still a sandwiched graph.
\[rem\_G\] If ${\mathcal{D}}$ is an Enriques diagram for ${\mathcal{O}}$ and ${\Gamma}^{0}_{D}$ is the weighted subgraph of ${\Gamma}_D$ comprising only the non-dicritical vertices, then ${\Gamma}_{{\mathcal{O}}}$ equals one of the connected components of ${\Gamma}^{0}_{D}$, whose vertices (and their corresponding vertices in $D$) will be called *non-dicritical vertices relative* to ${\mathcal{O}}$.
Sandwiched singularities and their Enriques diagrams {#section 2}
====================================================
In Remark \[rem\_G\] we have observed that if ${\mathcal{D}}$ is an Enriques diagram for a sandwiched surface singularity ${\mathcal{O}}$, then its dual graph ${\Gamma}_D$ contains the resolution graph ${\Gamma}_{{\mathcal{O}}}$ as a weighted subgraph. Given any Enriques diagram $D$, the following proposition shows that this combinatorial condition is sufficient to infer a result of geometrical nature: suitable marking maps $\rho$ can be chosen so that $(D,
\rho)$ becomes an Enriques diagram for ${\mathcal{O}}$.
\[graph-ideal\] Let ${\mathcal{O}}$ be a sandwiched surface singularity and let ${\mathcal{D}}= (D,
\rho)$ be a marked Enriques diagram. Then, ${\mathcal{D}}$ is an Enriques diagram for ${\mathcal{O}}$ if and only if
1. the dual graph $\Gamma_{D}$ contains $\Gamma_{{\mathcal{O}}}$ as a weighted subgraph.
2. $\rho (p) = 0$ if ${\textsl{p}}\in \Gamma_{{\mathcal{O}}}$; and $\rho (p) = +$ if ${\textsl{p}}\in
\Gamma_{D}\setminus \Gamma_{{\mathcal{O}}}$ and it is adjacent to some vertex of $\Gamma_{{\mathcal{O}}}$.
The “only if” part follows from \[rem\_G\], and, since ${\mathcal{D}}=(D, \rho)$ is an Enriques diagram for ${\mathcal{O}}$, the marking map $\rho$ must satisfy the statement in order to assure that the non-dicritical vertices of $D$ relative to ${\mathcal{O}}$ correspond exactly to the vertices of $\Gamma_{{\mathcal{O}}}$. For the “if” part, consider the reduced exceptional divisor $E_{{\mathcal{O}}}$ of the minimal resolution of ${\mathcal{O}}$. Using plumbing (see [@Lau72], [@Spivak] Remark I.1.10), we can glue smooth rational curves in order to obtain a configuration $E_{\Gamma_{D}}$ containing $E_{{\mathcal{O}}}$, having dual graph $\Gamma_{D}$ and being contained on a smooth surface $S'$. Since $\Gamma_{D}$ is a non-singular graph, by Castelnuovo’s criterion $E_{\Gamma_{D}}$ contracts to a non-singular point $O$ ([@Spivak] II.1.10) on a surface $S$. This contraction factors into a composition of point blowing-ups ([@Lau71] theorem 5.7), so it determines a cluster with origin at $O$ and having Enriques diagram $D$ ([@Cas00] 4.4). It is clear that $S'$ is the surface obtained from $S$ by blowing up all points in $K$. Consider a system of virtual multiplicities $\nu=\nu_K$ for $K$ in such a way that all the points have positive excess except for those corresponding to the vertices of the subgraph $\Gamma_{{\mathcal{O}}}
\subset \Gamma_{D}$, which have excess 0 (such a system exists by [@Cas00] 8.4.1). Notice that no maximal point of $K$ corresponds to a vertex of $\Gamma_{{\mathcal{O}}}$, since any vertex of the resolution graph $\Gamma_{{\mathcal{O}}}$ has weight strictly greater than $1$. Hence ${{\mathcal{K}}}=(K,\nu)$ is a consistent cluster with positive virtual multiplicities. Let $I=H_{{\mathcal{K}}}$ and $X=Bl_I(R)$. The morphism $f:S'\longrightarrow X$ given by the universal property of blowing up is the minimal resolution of the singularities of $X$ ([@Spivak] II.1.4) and its exceptional components correspond to those points of ${{\mathcal{K}}}$ having excess 0. Therefore, $E_{{\mathcal{O}}}$ is the exceptional divisor of $f$ and so, the singularity on $X$ given by its contraction is isomorphic to ${{\mathcal{O}}}$ ([@Lau71] theorem 3.13).
Given a sandwiched surface singularity ${\mathcal{O}}$, there is not a unique non-singular graph ${\Gamma}$ containing ${\Gamma}_{{\mathcal{O}}}$ as a weighted subgraph (see Example \[Ex1\]). In fact, it is possible to construct infinitely many non-singular graphs ${\Gamma}$ containing a given sandwiched graph ${\Gamma}_{{\mathcal{O}}}$, as Example \[Ex2\] shows.
\[Ex2\] If $I\in {\mathbf{I}_R}$ is an ideal for ${\mathcal{O}}$ and $X=Bl_I(R)$, by choosing any non-singular point in the exceptional locus of $X$, and blowing up this point, we obtain a new surface $X'$ containing ${\mathcal{O}}$, as well. This $X'$ is the blowing-up of a complete ${\mathfrak{m}}_O$-primary ideal $J_1=II_1\subset R$, where $I_1$ has codimension one in $I$ (Theorem 3.5 of [@FS1]), and the dual graph $\Gamma_{J_1}$ contains a vertex more than $\Gamma_I$. In this way, an infinite chain of ideals in ${\mathbf{I}_R}$ $$\ldots \subset J_n \subset\ldots\subset J_1 \subset I \subset R$$ for ${\mathcal{O}}$ can be constructed, and each ${\Gamma}_{J_n}$ contains ${\Gamma}_{{\mathcal{O}}}$ as a weighted subgraph. Moreover, for any $n$ the Enriques diagram ${\mathcal{D}}_{J_{n-1}}$ is a marked subdiagram of ${\mathcal{D}}_{J_{n}}$.
\[l.contraction\] Let ${\mathcal{D}}$ be an Enriques diagram for ${\mathcal{O}}$. Consider the set $C$ of non-dicritical vertices of ${\mathcal{D}}$ relative to ${\mathcal{O}}$.
- There is a tree structure on $C$ induced by the natural ordering $\leq$ of $D$.
- For any $p, \, q \in C$ define $p \rightarrow_{C} q$ if and only if $p \rightarrow_{D} q$. Then,$\rightarrow_{C}$ is a proximity, which turns $C$ into an Enriques diagram.
To exhibit the tree structure of $C$, we will prove that
1. there is a unique minimal element of $C$ by $\leq$, which is taken as the root of $C$;
2. for any $p \in C$, its immediate predecessor in $C$ is the maximal element of $\{ q \in C: \ q< p \}$.
Suppose that $p$ and $q$ are two different minimal vertices in $C$, and write $w$ for the maximal vertex in $D(p)\cap D(q)$ (this is, the maximal vertex which both $q$ and $p$ are infinitely near or equal to). Then, as ${\Gamma}_D$ contains no loops, $$ch_{{\Gamma}_D}({\textsl{q}},{\textsl{p}})=ch_{{\Gamma}_D}({\textsl{q}},{\textsl{w}})\cup ch_{{\Gamma}_D}({\textsl{w}},{\textsl{p}}),$$ and, by the connectivity of ${\Gamma}_C$, we infer that $w\in C$, contradicting the minimality of $q$ and $p$. We denote by $O_C$ the minimal vertex of $C$, which is set as the root of $C$. On the other hand, if $p\in C$, $p\neq O_C$, the vertices in $D(p)$ are totally ordered by the natural ordering $\leq$ of $D$. Hence, there exists a unique immediate predecessor of $p$, which is the maximal element of $\{q\in C \mid q<p\}$, and this proves (a).
Now, to prove (b), we show that $\rightarrow_C$ defined as above is a proximity relation for $C$. Since the root is the minimal vertex of $C$, it is clear that it is proximate to no other vertex of $C$. If $p\neq O_C$, its immediate predecessor $q_0$ in $C$ is the maximal element of $\{q\in C \mid q<p\}$; hence $q_0<p$ and $q_0\in C$. Then (b) of \[chain\] says that $p$ is proximate to some vertex ${\textsl{w}}$ of $ch_{{\Gamma}_D}({\textsl{q}}_0,{\textsl{p}})$, and (a) of \[chain\] says that $w$ is infinitely near or equal to $q_0$. Since $ch_{{\Gamma}_D}({\textsl{q}}_0,{\textsl{p}})\subset {\Gamma}_C$, this leads to contradiction. This proves the first condition of the proximity (see §1.3). The conditions 2 and 3 are clearly satisfied.
An Enriques diagram $C$ obtained as in Lemma \[l.contraction\] will be called *a contraction for* ${\mathcal{O}}$ (or *for* ${\Gamma}_{{\mathcal{O}}}$) *associated with* ${\mathcal{D}}$. Reciprocally, we will also say that ${\mathcal{D}}$ is *associated with the contraction* $C$.
\[\] Notice that, by virtue of \[graph-ideal\], any Enriques diagram (respectively, any contraction) for ${\mathcal{O}}$ is in fact an Enriques diagram (respectively, a contraction) for any sandwiched surface singularity whose resolution graph is ${\Gamma}_{{\mathcal{O}}}$. A contraction for ${\mathcal{O}}$ may also be regarded as an enrichment of the resolution graph ${\Gamma}_{{\mathcal{O}}}$ by some proximities between their vertices, these proximities being compatible with the weights of ${\Gamma}_{{\mathcal{O}}}$ in the sense of Lemma \[l.ineq-weights\] below.
\[Ex1\] Figure \[EDs\] provides three distinct Enriques diagrams for the same sandwiched singularity: they are not apparently related, namely one is not a subdiagram of the other, as was the case in Example \[Ex2\]. Notice that ${\mathcal{D}}_1$ and ${\mathcal{D}}_3$ give rise to the same contraction, which is the Enriques subdiagram of $D_1$ comprising the black dots.
![\[EDs\]Three different marked Enriques diagrams for the same singularity ${\mathcal{O}}$ and their corresponding dual graphs. Dicritical vertices are represented with white dots. ](Example23.eps)
\[r.contraction\] If ${\mathcal{D}}$ is an Enriques diagram for ${\mathcal{O}}$ and $C$ is the associated contraction, $C$ is not, in general, an Enriques subdiagram of $D$ (see Enriques diagram ${\mathcal{D}}_3$ of Figure \[EDs\]). In particular, if $I\in {\mathbf{I}_R}$ is an ideal for ${\mathcal{O}}$ with Enriques diagram ${\mathcal{D}}$, the set of points of ${\mathcal{K}}=BP(I)$ corresponding to the vertices of $C$ does not constitute, in general, a subcluster of ${\mathcal{K}}$.
\[l.ineq-weights\] Let $C$ be a contraction for ${\Gamma}_{{\mathcal{O}}}$ associated with an Enriques diagram ${\mathcal{D}}$. Then for any vertex $p \in C$, $$\label{}
\omega_{{\Gamma}_C}({\textsl{p}}) \leq \omega_{\Gamma_{D}}({\textsl{p}}) =
\omega_{\Gamma_{{\mathcal{O}}}}({\textsl{p}}) \, ,$$ and the inequality is strict at the extremal vertices of $C$. In particular, ${\Gamma}_C$ is not a weighted subgraph of ${\Gamma}_{D}$.
The inequality comes from the definition of contraction, since the number of vertices proximate to $p$ in $C$ is less or equal than in $D$. If $p$ is extremal in $C$, $\omega_{{\Gamma}_C}({\textsl{p}})=1$, while $\omega_{{\Gamma}_D}({\textsl{p}})>1$, since $p$ is a non-dicritical vertex of ${\mathcal{D}}$ and hence necessarily non-extremal in $D$ (see \[w=1extrem\]). The last assertion follows by considering the weights at the extremal vertices of $C$.
In [@Spivak] Corollary II.1.14, Spivakovsky introduced a type of birational projection into a plane that could be achieved for any sandwiched singularity. Namely he showed that, once a sandwiched surface singularity ${\mathcal{O}}$ is fixed, an ideal $I\in {\mathbf{I}_R}$ can be chosen in such a way that:
- ${\mathcal{O}}$ is the only singularity of $X=Bl_I(R)$;
- the strict transform (by the minimal resolution of $X$) of any exceptional component of $\pi_{I}^{-1}(O)$ is a curve of the first kind, that is, the strict transform by $f$ (see diagram \[CommDiagr\]) of any $L_p$ with $p \in BP(I)_{+}$ has self-intersection equal to $-1$.
An ideal satisfying the above conditions (i) and (ii) (cf. [@Mohring] 2.3) will be called an *S-ideal for* ${\mathcal{O}}$. A marked Enriques diagram associated with an S-ideal for ${\mathcal{O}}$ will be called an *S-Enriques diagram for* ${\mathcal{O}}$. The following result describes what the equisingularity classes of S-Enriques diagrams look like:
\[idspivak\] An ideal $I\in \textbf{I}_R$ is an S-ideal if and only if the dicritical vertices of ${\mathcal{D}}_I$ are free and extremal.
Write ${\mathcal{D}}$ for the Enriques diagram of ${\mathcal{K}}=BP(I)$. First of all, note that $X=Bl_I(R)$ has only one singularity ${\mathcal{O}}$ if and only if any non-dicritical vertex of ${\Gamma}_D$ belongs to ${\Gamma}_{{\mathcal{O}}}$. Let $p\in {\mathcal{K}}_+$ and assume that there exists some $q\in K$ infinitely near to $p$. We may assume that $q$ is an immediate successor of $p$. Then, $\omega_{{\Gamma}_D}({\textsl{p}})=r_D(p)+1\geq 2$ against condition $(ii)$. Therefore, $p$ must be maximal in $K$. Now, assume that $p$ is satellite, proximate to $u_1$ and $u_2$. Then, ${\textsl{p}}\in
ch_{{\Gamma}_D}({\textsl{u}}_1,{\textsl{u}}_2)$. Necessarily, $u_1$ and $u_2$ are not dicritical points of $K$ and thus, ${\textsl{u}}_1,{\textsl{u}}_2\in {\Gamma}_{{\mathcal{O}}}$. If follows that ${\textsl{p}}\in {\Gamma}_{{\mathcal{O}}}$ against the assumption $p\in {\mathcal{K}}_+$.
Conversely, if the dicritical vertices of ${\mathcal{D}}$ are free and extremal, the union of the non-dicritical vertices of ${\Gamma}_D$ is connected and hence $X$ has only one singularity. Moreover, as above, the self-intersection of the strict transform on $S_K$ of any component $L_p$ with $p\in {\mathcal{K}}_+$ is $-1$.
A contraction $C$ associated with an S-Enriques diagram will be called an *S-contraction*. Contrary to what happened for general contractions (recall \[r.contraction\]), an S-contraction $C$ is a subdiagram of its associated S-Enriques diagram ${\mathcal{D}}$; furthermore, any S-contraction is associated with a unique S-Enriques diagram:
\[P.S-contraction\] If ${\mathcal{D}}$ is an S-Enriques diagram for ${\mathcal{O}}$, then an S-contraction $C$ associated with ${\mathcal{D}}$ satisfies:
1. $C$ is a subdiagram of $D$;
2. $D$ can be recovered from $C$ by adding at any vertex $p \in C$ a number of $\omega_{\Gamma_{{\mathcal{O}}}}({\textsl{p}}) - \omega_{{\Gamma}_C}({\textsl{p}})$ free successors; ${\mathcal{D}}=(D, \rho)$ is recovered by defining the marking map $\rho$ as $\rho (p)= 0$ if $p \in C$, and $\rho(p)=+$ otherwise.
By virtue of \[idspivak\], any dicritical point of $D$ is free and extremal. Therefore, for any $p\in C$, there are exactly $\omega_{\Gamma_{{\mathcal{O}}}}({\textsl{p}}) - \omega_{{\Gamma}_C}({\textsl{p}})$ of these vertices in the first neighborhood of $p$. This gives both claims.
Next result is a sort of converse of Lemma \[l.ineq-weights\]:
\[lem\_2.2\] Let $C$ be an Enriques diagram and assume that the dual graph ${\Gamma}_C$ equals ${\Gamma}_{{\mathcal{O}}}$ and satisfies $\omega_{{\Gamma}_C}({\textsl{p}}) \leq
\omega_{\Gamma_{{\mathcal{O}}}}({\textsl{p}})$ at each vertex $p \in C$. Then $C$ is an S-contraction for ${\Gamma}_{{\mathcal{O}}}$.
To any $p\in C$ add exactly $\omega_{\Gamma_{{\mathcal{O}}}}({\textsl{p}}) -
\omega_{{\Gamma}_C}({\textsl{p}})$ free successors to obtain from $C$ a marked Enriques diagram ${\mathcal{D}}$ whose dicritical vertices are the extremal ones. By construction the dicritical vertices are also free, ${\Gamma}_{{\mathcal{O}}}$ is a weighted subgraph of ${\Gamma}_{D}$. Then invoking \[graph-ideal\] and \[idspivak\] we are done.
Let us end this section by showing that the family of contractions for a sandwiched surface singularity equals the family of S-contractions:
\[GeneralitySpivIdeals\] Any contraction for ${\Gamma}_{{\mathcal{O}}}$ is an S-contraction.
Let ${\mathcal{D}}$ be an Enriques diagram for ${\mathcal{O}}$ and let $C$ be a contraction for ${\Gamma}_{{\mathcal{O}}}$ associated with ${\mathcal{D}}$. Define a new Enriques diagram $D'$ from $C$ by adding to each vertex $p\in C$ as many free successors as the number of vertices in $D \setminus
C$ that are proximate to $p$. Taking $\rho(p)=0$ if $p\in C$ and $\rho(p)=+$ otherwise, the resulting Enriques diagram ${\mathcal{D}}'=(D',
\rho)$ for ${\mathcal{O}}$ is an S-Enriques diagram associated with $C$ (see \[graph-ideal\]).
Contractions for a sandwiched surface singularity
=================================================
In this section we describe all the contractions for a given sandwiched surface singularity ${\mathcal{O}}$. Observe that, by virtue of \[P.S-contraction\] and \[GeneralitySpivIdeals\], this is equivalent to listing all the equisingularity classes of the S-ideals for ${\mathcal{O}}$.
Suppose that the resolution graph $\Gamma _{{\mathcal{O}}}$ has $n$ vertices and that ${\textsl{v}}$ is an end of ${\Gamma}_{{\mathcal{O}}}= {\Gamma}_{n}$. The weighted graph obtained by removing ${\textsl{v}}$ is again a sandwiched graph and will be denoted by $\Gamma_{n-1}$ (see §\[ED-dualG\]). We want to detail a procedure to obtain all the contractions for ${\Gamma}_{n}$ from the contractions for $\Gamma_{n-1}$. Then, by induction on $n$, the whole list of contractions for a given sandwiched singularity will be inferred just from its resolution graph.
The first result of this section describes how the vertex $v$ looks like in any contraction:
\[lem\_2.1\] The vertex $v$ (corresponding to the end ${\textsl{v}}$ of ${\Gamma}_{{\mathcal{O}}}$) in any contraction $C$ for ${\mathcal{O}}$ is either the root or free. Furthermore, if $v$ is not the root of $C$, then either $v$ is extremal or $v$ has a unique successor, which is satellite of $v$.
Assume that $v$ is satellite, proximate to the vertices $u_1$ and $u_2$ in $C$, and suppose that $u_2$ is proximate to $u_1$. We will show that ${\textsl{v}}\in ch_{{\Gamma}_C}({\textsl{u}}_1,{\textsl{u}}_2)$, thus contradicting that ${\textsl{v}}$ is an end, and proving the first claim. Consider the Enriques subdiagram $C(v)$ of $C$ comprising all the points preceding or equal to $v$. In particular, $u_1,u_2$ are both in $C(v)$, and $v$ is maximal among the points of $C(v)$ proximate to $u_1$, and also to $u_2$. Hence, as vertices of $\Gamma_{C(v)}$, ${\textsl{v}}$ is adjacent to both ${\textsl{u}}_1$ and ${\textsl{u}}_2$ and so, ${\textsl{v}}\in ch_{\Gamma_{C(v)}}({\textsl{u}}_1,{\textsl{u}}_2)$. Now, the rest of vertices of $C \setminus C(v)$ all lie after some vertex of $C(v)$, giving rise to blowing-ups of extra points. The combinatorial effect of these blowing-ups is translated in the dual graph by the elementary modifications introduced in I.1.5 of [@Spivak], those of the first kind corresponding to the blowing-ups of free points while those of the second kind to the blowing-ups of satellite points. From their definition, it is immediate that these modifications respect the property of being in the chain determined by two vertices already in the graph.
For the second claim, let ${\textsl{q}}$ be the only vertex in $\Gamma
_{{\mathcal{O}}}$ to which ${\textsl{v}}$ is adjacent, and assume that $v$ is not the root of $C$. We distinguish two cases. The first one is when $v$ is maximal among the points in $C$ proximate to $q$. Since $v$ is an end, there are no vertices in $C$ proximate to $v$ and $v$ is an extremal vertex of $C$. The second case is when $q$ is maximal among the points in $C$ proximate to $v$. Since $v$ is and end, $v$ must have a unique successor, say $w$, preceding $q$. Denote by $u$ the immediate predecessor of $v$. Since $v$ is free and ${\textsl{v}}$ is adjacent only to ${\textsl{q}}$, $v$ cannot be the last point in $C$ proximate to $u$. Hence $w$ must be proximate to $u$ and so $w$ is satellite of $v$. This completes the proof.
The following result shows how to construct a contraction for ${\Gamma}_n$ from a contraction for ${\Gamma}_{n-1}$. Moreover, any contraction for ${\Gamma}_n$ can be obtained in this way.
\[rules\] Suppose ${\Gamma}_{n-1}$ is obtained from a sandwiched graph ${\Gamma}_n$ by removing an end ${\textsl{v}}$. Let ${\textsl{u}}$ be the unique vertex in ${\Gamma}_n$ to which ${\textsl{v}}$ is adjacent, and let $C'$ be a contraction for ${\Gamma}_{n-1}$. Define a new Enriques diagram by taking $C = C' \cup \{ v \}$ and adding to $C'$ extra proximities relating $v$ according to one of the following rules:
1. If $\omega_{{\Gamma}_{C'}}({\textsl{u}}) < \omega_{\Gamma_{n}}({\textsl{u}})$, set $v$ in $C$ as a free successor of $u$.
2. If $u=q_r\rightarrow_{C'} q_{r-1} \rightarrow_{C'} \ldots \rightarrow_{C'} q_1$ are free vertices in $C'$ with $1 \leq r< \omega_{\Gamma_{n}}({\textsl{v}})$ and $q_1\rightarrow_{C'} q_0$, then set $q_i \rightarrow_{C} v $ for all $i \in \{1, \ldots , r \}$, and either set $q_0 \rightarrow_{C} v$, provided $q_0$ is the root of $C'$ and $r< \omega_{\Gamma_{n}}({\textsl{v}})-1$ (with $v$ becoming the root of $C$), or set $v \rightarrow_{C} q_0$, provided $\omega_{{\Gamma}_{n-1}}({\textsl{q}}_{0}) < \omega_{\Gamma_{n}}({\textsl{q}}_{0})$.
Then $C$ is a contraction for $\Gamma_{n}$. Moreover, any contraction $C$ for ${\Gamma}_n$ can be constructed from some contraction $C'$ for ${\Gamma}_{n-1}$ as above.
Clearly $C$ defined as above satisfies ${\Gamma}_C={\Gamma}_{n}$ and $\omega_{{\Gamma}_C}({\textsl{p}}) \leq \omega_{\Gamma_{n}}({\textsl{p}})$ at each vertex $p
\in C$. Thus, invoking \[lem\_2.2\] and \[GeneralitySpivIdeals\], the first claim follows. By virtue of \[lem\_2.1\], the vertex $v$ of $C$ corresponding to ${\textsl{v}}$ is either free or the root of $C$. Since ${\textsl{v}}$ is adjacent to ${\textsl{u}}$ in $\Gamma _{n}$, there are only two possibilities for their corresponding vertices in $C$:
Case 1:
: $v$ is maximal among the vertices in $C$ proximate to $u$. Hence $v$ cannot be the root of $C$, and by \[lem\_2.1\] $v$ is free. Moreover, $v$ is an extremal vertex of $C$: otherwise, \[lem\_2.1\] implies that $v$ has a unique successor, which is satellite of $v$ and thus proximate to $u$, contradicting the maximality of $v$ among the vertices proximate to $u$. Therefore the set of vertices of $C'=
C \smallsetminus \{ v \}$ is connected and has a tree structure. By considering the restriction of the proximity of $C$ to $C'$, $C'$ becomes an Enriques diagram. Clearly the graphs $\Gamma _{C'}$ and $\Gamma _{n-1}$ are equal (disregarding weights). Observe that $r_{C'}(u)=r_C(u)-1$ and that $r_{C'}(q)=r_C(q)-1$ if $q\in C$, $q\neq u$. Applying \[w=1extrem\], we have $$\omega _{\Gamma _{C'}} ({\textsl{q}}) = \omega _{\Gamma _{C}} ({\textsl{q}}) \leq \omega _{\Gamma _{n}} ({\textsl{q}})
= \omega _{\Gamma _{n-1}} ({\textsl{q}}) \, ,$$ for any $q \in C' \smallsetminus \{ u \}$ and $$\omega _{\Gamma _{C'}} ({\textsl{u}}) = \omega _{\Gamma_{C} } ({\textsl{u}})-1 < \omega _{\Gamma _{C}} ({\textsl{u}}) \leq \omega _{\Gamma _{n}} ({\textsl{u}})
= \omega _{\Gamma _{n-1}} ({\textsl{u}}) \, .$$ Thus, invoking \[lem\_2.2\] and \[GeneralitySpivIdeals\], $C'$ is a contraction for ${\Gamma}_{n-1}$. Finally, notice that the Enriques diagram $C$ is obtained from $C'$ by the procedure of the first rule of the statement, and we are done in this case.
Case 2:
: $u$ is maximal among the vertices in $C$ proximate to $v$. Let $p_1, \ldots, p_j=u$ be the vertices in $C$ preceding or equal to $u$ which are proximate to $v$. First of all, we define on the set of vertices of $C'= C
\smallsetminus \{ v \}$ a tree structure. We distinguish two cases:
**2.1.** If $v$ is the root of $C$, then take $p_1$ as the root of $C'$, and for any $q \in C' \smallsetminus \{ p_1 \}$ declare that $p$ is the immediate predecessor of $q$ in $C'$ if and only if $p$ is the immediate predecessor of $q$ in $C$.
**2.2.** Otherwise, take the root of $C$ as the root of $C'$; for any $q \in C' \smallsetminus \{ p_1 \}$ declare that $p$ is the immediate predecessor of $q$ in $C'$ if and only if $p$ is the immediate predecessor of $q$ in $C$; declare that the immediate predecessor of $p_1$ in $C'$ is the immediate predecessor $p_0$ of $v$ in $C$.
Restrict the proximity of $C$ to $C'$, namely, for any $q,q'\in
C'$ set $q \rightarrow_{C'} q'$ if and only if $q \rightarrow_C
q'$. Let us check that it satisfies the properties 1 to 3 of a proximity (see §\[ED-dualG\]). In the first case (where $v$ is the root of $C$) these properties are clearly satisfied. In the second case, the only condition that must be checked is property 1 for the vertex $p_1$, namely, that $p_1 \rightarrow_{C} p_0$. Since $p_1$ is a successor of $v$ in $C$, by \[lem\_2.1\] we infer that $p_1$ is satellite of $v$. Thus $p_1$ is satellite in $C$: $p_1$ is proximate to its immediate predecessor in $C$, which is $v$, and to some point of $C$, say $p$; moreover, $v$ must be proximate to $p$, as well. Since $v$ is proximate to $p_0$, we infer that $p=p_0$ and $p_1 \rightarrow_{C} p_0$, as desired. Therefore $C'$ is an Enriques diagram, whose dual graph $\Gamma _{C'}$ equals $\Gamma _{n-1}$ disregarding weights. Observe that $r_{C'}(p_0)=
r_{C}(p_0)-1$ and $r_{C'}(q)= r_{C}(q)$ if $q\in D$, $q\neq p_0$. Applying \[w=1extrem\], we have $$\omega _{\Gamma _{C'}} ({\textsl{q}}) = \omega _{\Gamma _{C}} ({\textsl{q}}) \leq \omega _{\Gamma _{n}} ({\textsl{q}})
= \omega _{\Gamma _{n-1}} ({\textsl{q}}) \, ,$$ for any $q \in C' \smallsetminus \{ p_0 \}$ and $$\omega _{\Gamma _{C'}} ({\textsl{p}}_0) = \omega _{\Gamma _{C}} ({\textsl{p}}_0)-1 < \omega _{\Gamma _{C}} ({\textsl{p}}_0)
\leq \omega _{\Gamma _{n}} ({\textsl{p}}_0)
= \omega _{\Gamma _{n-1}} ({\textsl{p}}_0) \, .$$ Thus, invoking \[lem\_2.2\] and \[GeneralitySpivIdeals\], $C'$ is a contraction for ${\Gamma}_{n-1}$. Finally, it remains to show that the Enriques diagram $C$ may be obtained from $C'$ by the procedure of the second rule of the statement. Indeed, according to the proximity defined in $C'$, notice first that $\{ u=p_j \rightarrow_{C'} \cdots
\rightarrow_{C'} p_1 \}$ is a chain of free vertices in $C'$ preceding or equal to $u$, and that $p_1$ is the root of $C'$ if and only if $v$ is the root of $C$. On the other hand, recall that the proximity relations in $C$ involving the vertex $v$ are $$p_{1} \rightarrow_{C} v , \ \ldots ,
p_j \rightarrow_{C} v ,$$ and the further proximity relation $v \rightarrow_{C} p_0$ must be added in case $p_1$ is not the root of $C'$. This is exactly what performs the operation of the second rule, and we are done.
The whole list of contractions for ${\Gamma}_{{\mathcal{O}}}$ are obtained by applying recursively the rules of \[rules\]. Let us just sketch the main steps of an implementation of this procedure. Each step of this procedure adds a new vertex keeping the proximities already defined. The idea is that the bigger the weights of ${\Gamma}_{{\mathcal{O}}}$ are, the more Enriques diagrams for ${\mathcal{O}}$ can be found.
**Step 1.** Choose any vertex of ${\Gamma}_{{\mathcal{O}}}$, say ${\textsl{p}}_1$, and take ${\Gamma}_1=({\textsl{p}}_1,\omega_{{\Gamma}_{{\mathcal{O}}}}({\textsl{p}}_1))$ and $C_1={\bullet}_{p_1}$.
**Step i.** Assume that ${\Gamma}_{i-1}$, $C_{i-1}$ have been obtained, where ${\Gamma}_{i-1}$ is a subgraph of ${\Gamma}_{{\mathcal{O}}}$. Choose a vertex ${\textsl{p}}_i$ adjacent to some ${\textsl{q}}\in {\Gamma}_{i-1}$. The graph ${\Gamma}_i$ is obtained by adding ${\textsl{p}}_i$ with weight $\omega_{{\Gamma}_{{\mathcal{O}}}}({\textsl{p}}_i)$ to ${\Gamma}_{i-1}$, adjacent to ${\textsl{q}}$; the new Enriques diagram $C_i$ is obtained from $C_{i-1}$ by adding $p_i$ according to one of the rules of \[rules\]:
1. If $\omega_{{\Gamma}_{i-1}}({\textsl{q}}) < \omega_{\Gamma_{{\mathcal{O}}}}({\textsl{q}})$, $p_i$ can be added to $C_{i-1}$ as a free successor of $q$, $p_i\rightarrow_{C_i}q$.
2. If $q=q_r\rightarrow_{C_{i-1}} q_{r-1} \rightarrow_{C_{i-1}} \ldots \rightarrow_{C_{i-1}} q_1$ are free vertices in $C_{i-1}$ with $1 \leq r< \omega_{\Gamma_{{\mathcal{O}}}}({\textsl{p}}_i)$ and $q_1\rightarrow_{C_{i-1}} q_0$, then set $q_j
\rightarrow_{C_i} p_i $ for all $j \in \{1, \ldots , r \}$, and either set $q_0 \rightarrow_{C_i} p_i$, provided $q_0$ is the root of $C_{i-1}$ and $r< \omega_{\Gamma_{{\mathcal{O}}}}({\textsl{p}}_i)-1$ (with $p_i$ becoming the root of $C_i$), or set $p_i \rightarrow_{C_i} q_0$, provided $\omega_{{\Gamma}_{i-1}}({\textsl{q}}_{0}) < \omega_{\Gamma_{{\mathcal{O}}}}({\textsl{q}}_{0})$.
The procedure stops at step $n$, the number of vertices of ${\Gamma}_{{\mathcal{O}}}$. At this point, the obtained weighted graph ${\Gamma}_n$ equals ${\Gamma}_{{\mathcal{O}}}$, and the Enriques diagram $C_n$ is just a contraction for ${\Gamma}_{{\mathcal{O}}}$.
At any step of the procedure, there may be several choices to add a fixed new vertex (for example, we may apply either rule 1 or 2 to add the vertex $p_i$ to $C_{i-1}$). In order to obtain the whole list of all the contractions for ${\Gamma}_{{\mathcal{O}}}$, all these possibilities must be performed. It might also happen that an Enriques diagram $C_{i-1}$ to which the new vertex cannot be added is reached. This means that no Enriques diagram for ${\mathcal{O}}$ with the subset of proximities of $C_{i-1}$ exists.
Minimal singularities are rational surface singularities whose fundamental cycle is reduced. They are characterized as those sandwiched singularities that can be obtained by blowing up a complete ideal all whose base points are free (see [@Mohring] 2.5; cf. [@Spivak]). As a consequence of our results, a sandwiched surface singularity ${\mathcal{O}}$ is minimal if and only if there exists a contraction for ${\mathcal{O}}$ that is obtained by applying the first rule at each step of the above procedure.
\[exSp\] Let ${\mathcal{O}}$ be a singularity whose resolution graph is shown at the bottom of Figure \[exSp\]. By applying the procedure just described, we obtain the whole list of contractions for ${\mathcal{O}}$. The S-Enriques diagrams shown in Figure \[llistatED\] are obtained by adding free successors to them as explained in (b) of \[P.S-contraction\].
![ \[llistatED\] The complete list of S-Enriques diagrams for a singularity ${\mathcal{O}}$ with resolution graph ${\Gamma}_{{\mathcal{O}}}$ in Example \[exSp\]. The white-filled dots represent the dicritical points, added to the contractions.](LlitatSp2.eps)
Equisingularity classes of the ideals for a sandwiched singularity
==================================================================
In this section we address the problem of describing the equisingularity classes of the ideals for a given sandwiched surface singularity ${\mathcal{O}}$, that is, of describing all the possible Enriques diagrams for ${\mathcal{O}}$. The (finite) family of contractions for ${\mathcal{O}}$ was inferred from the resolution graph ${\Gamma}_{{\mathcal{O}}}$ of ${\mathcal{O}}$ by the procedure explained in the preceding section. It remains to find out all the different Enriques diagrams for ${\mathcal{O}}$ giving rise to the same contraction (an infinite family). Here we will show how to complete contractions in order to describe all the different Enriques diagrams for ${\mathcal{O}}$, thus solving completely the problem we are concerned with.
Given a contraction $C$ for ${\mathcal{O}}$, our aim is to describe all the Enriques diagrams for ${\mathcal{O}}$ associated with $C$. Consider the marked Enriques diagram ${\mathcal{C}}=(C, \rho_{C})$ with $\rho_{C}(p)=0$ for any $p \in C$, and the number $\lambda_{C}=
\sum_{p \in C}(\omega_{\Gamma_{{\mathcal{O}}}}({\textsl{p}}) - \omega_{{\Gamma}_C}({\textsl{p}}))$. By \[l.ineq-weights\], $\lambda_C >0$. Let us describe a procedure to add vertices to ${\mathcal{C}}$ in order to reach an Enriques diagram for ${\mathcal{O}}$. Write ${\mathcal{C}}_0 =(C_0, \rho_{0})={\mathcal{C}}$. For $1 \leq i \leq \lambda_C$, choose a vertex $p_i$ in $C$ such that $\omega_{\Gamma_{{\mathcal{O}}}}({\textsl{p}}_i) >
\omega_{{\Gamma}_{C_{i-1}}}({\textsl{p}}_i)$ and then define inductively ${\mathcal{C}}_i =(C_i, \rho_{i})$ by taking $C_i= C_{i-1} \cup \{q_i
\}$ and ${\rho_{i}}_{|C_{i-1}} = \rho_{i-1}$, where the new vertex $q_i$ is set as a successor of $p_i$ either
- as a free successor of $p_i$, $q_i \rightarrow_{C_i} p_i$, and then set $\rho_i(q_i)=+$;
or, if there is some free successor $p'_i$ of $p_i$ in $C_{i-1}$,
- as a successor preceding $p'_i$, namely $q_i \rightarrow_{C_i} p_i$ and $p'_i \rightarrow_{C_i} q_i$ are the only proximities relating $q_i$, and then set $\rho_i (q_i)= +$ in case $\rho_i (p'_i)= 0$ (otherwise $\rho_i (q_i)$ can be chosen no matter $0$ or $+$).
Notice that at step $i$ the operation of type A may always be performed, independently of the existence of a free successor of $p_i$, which would offer the possibility to choose also an operation of type B. Observe that $\sum_{p \in C}(\omega_{\Gamma_{{\mathcal{O}}}}({\textsl{p}}) -
\omega_{{\Gamma}_{C_i}}({\textsl{p}}))= \lambda_{C}-i$. Thus the procedure performs effectively the $\lambda_{C}$ steps. Any of such marked Enriques diagram ${\mathcal{C}}_{\lambda_{C}}$, obtained from $C$ by the above procedure, will be called an *extension* of the contraction $C$. Clearly any extension of $C$ is an Enriques diagram for ${\mathcal{O}}$ associated with $C$.
Notice that any extension of $C$ all whose vertices have been added performing operation A at each step is an S-Enriques diagram for ${\mathcal{O}}$ (in fact, the unique S-Enriques diagram for ${\mathcal{O}}$ associated with $C$).
The set of all extensions of $C$ forms a family of Enriques diagrams for ${\mathcal{O}}$ associated with $C$ minimal in the following sense:
\[final\] Any Enriques diagram ${\mathcal{D}}$ for a sandwiched singularity ${\mathcal{O}}$ contains, as a marked subdiagram, an extension of the contraction associated with ${\mathcal{D}}$.
Conversely, if a marked Enriques diagram ${\mathcal{D}}$ contains, as a marked subdiagram, an extension ${\mathcal{E}}$ of some contraction $C$ for ${\mathcal{O}}$ and satisfies that any vertex of $D\setminus E$ is proximate to no vertex of $C$, then ${\mathcal{D}}$ is an Enriques diagram for ${\mathcal{O}}$.
For the first assertion, we need to find a marked subdiagram ${\mathcal{E}}$ of ${\mathcal{D}}$ which is an extension of $C$. Take $F= \{ p \in D : p \mbox{ is proximate to some }q \in C\}$, and define ${\mathcal{E}}=(E, \rho_{E})$, where $E= C\cup F$ and $\rho_{E}$ is the restriction of $\rho_{D}$ to $E$. Notice that $E$ is a connected subtree of $D$ since, if $p$ is proximate to some $q\in C$, then any vertex in $D(p)$ infinitely near to $q$ is also proximate to $q$. Hence, $E$ together with the proximities inherited from the proximity of $D$ is an Enriques subdiagram of $D$. Furthermore, ${\mathcal{E}}$ is a marked subdiagram of ${\mathcal{D}}$.
Moreover, by \[w=1extrem\], the cardinality of $F$ equals $\lambda:=\lambda_C$. Denote the vertices of $F$ by $\{ p_1,
\ldots, p_{\lambda}\}$ so that $p_i$ is not infinitely near to $p_j$ if $j>i$. Write ${\mathcal{E}}_{\lambda}:= {\mathcal{E}}$ and for $1\geq
i<\lambda$, define, recursively ${\mathcal{E}}_i$ as the marked Enriques diagram obtained from ${\mathcal{E}}_{i+1}$ by deleting $p_i$ (and keeping the restricted proximity and marking map; the successors of $p_i$ become successors of the immediate predecessor of $p_i$). Notice that the ${\mathcal{E}}_i$ are the marked Enriques diagrams generated by the procedure detailed above to reach ${\mathcal{E}}$, proving that ${\mathcal{E}}$ is an extension of $C$, as wanted.
For the converse, let ${\mathcal{E}}=(E, \rho_{E})$ be an extension of a contraction $C$ for ${\mathcal{O}}$. Thus, by \[graph-ideal\], ${\Gamma}_{E}
\supseteq {\Gamma}_{{\mathcal{O}}} $ as weighted graphs, $\rho_{E}(p)=0$ for any ${\textsl{p}}\in {\Gamma}_{{\mathcal{O}}}$ and $\rho_{E}(p)=+$ for any ${\textsl{p}}\in {\Gamma}_E
\setminus {\Gamma}_{{\mathcal{O}}}$ being adjacent to some vertex of ${\Gamma}_{{\mathcal{O}}}$. If ${\mathcal{D}}=(D, \rho_{D})$ contains ${\mathcal{E}}$ as a marked subdiagram and there any vertex of $D \setminus E$ is proximate to no vertex of $C\subset E$, then ${\Gamma}_{D} \supseteq {\Gamma}_{{\mathcal{O}}} $ as weighted graphs, and $\rho_{D}$ satisfies the marking map hypothesis of \[graph-ideal\], 2: $\rho_{D}(p)=0$ for any ${\textsl{p}}\in {\Gamma}_{{\mathcal{O}}}$ and $\rho_{D}(p)=+$ for any ${\textsl{p}}\in {\Gamma}_D \setminus {\Gamma}_{{\mathcal{O}}}$ adjacent to some vertex of ${\Gamma}_{{\mathcal{O}}}$. Hence, applying \[graph-ideal\] to ${\mathcal{D}}$ we are done.
We have already pointed out that sandwiched singularities are normal birational extensions of the regular ring $R$. If $R\subset
{\mathcal{O}}$ is such an extension, there exists a complete ideal $I\subset
R$ such that ${\mathcal{O}}=R[I/a]_{N_Q}$, where $N_Q$ is a height two maximal ideal in $R[I/a]$ containing $\mathfrak{m}_R$ (the maximal ideal of $R$), and $a$ is a generic element of $I$ (see [@HunSally]). $R$ is said to be *maximally regular* in ${\mathcal{O}}$ if there is no other regular ring $R'$ such that $$R\subsetneq R' \subset {\mathcal{O}}.$$ Write ${\mathcal{D}}_I$ for the marked Enriques diagram of the base points of $I$. Let ${\mathcal{E}}$ and $C$ be the extension and the contraction for ${\mathcal{O}}$ associated with ${\mathcal{D}}_I$. Then, by virtue of \[final\], ${\mathcal{D}}_I$ can be thought as being constructed from ${\mathcal{E}}$ by adding new vertices which are infinitely near to some dicritical vertex of ${\mathcal{E}}$ and not proximate to any vertex of $C$, or preceding the root of $C$ (notice that in any case, the proximities of ${\mathcal{E}}$, and hence also the proximities of $C$, are preserved). Moreover, $R$ is maximally regular in ${\mathcal{O}}$ if and only if the root of ${\mathcal{D}}_I$ equals the root of ${\mathcal{E}}$, i.e. no vertices have been added to ${\mathcal{E}}$ preceding the root.
Let ${\mathcal{O}}$ be a sandwiched singularity whose resolution graph is shown in the top left corner of Figure \[llistatED2\]. The contractions for ${\mathcal{O}}$ are shown at the top of the figure, and below each one of them, a complete list of the associated extensions is drawn. Any Enriques diagram for ${\mathcal{O}}$ contains one of these extensions as a marked subdiagram.
![\[llistatED2\] A complete list of extensions for a sandwiched singularity whose resolution graph is drawn in the top left corner. At the top of the figure the contractions of ${\mathcal{O}}$ are shown. White-filled dots represent dicritical vertices.](LlitatSp3.eps)
\[2\][ [\#2](http://www.ams.org/mathscinet-getitem?mr=#1) ]{} \[2\][\#2]{}
[10]{}
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[^1]: This research has been partially supported by the Spanish Committee for Science and Technology (DGYCIT), projects MTM2005-01518 and MTM2006-14234-C02-02, and the Catalan Research Commission. The second author completed this work as researcher of the program *Juan de la Cierva*.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'It is argued that, in heavy ion collisions, thermal dileptons are good probes of the transport properties of the medium created in such events, and also of its early-time dynamics, usually inaccessible to hadronic observables. In this work we show that electromagnetic azimuthal momentum anisotropies do not only display a sensitivity to the shear relaxation time and to the initial shear-stress tensor profile, but also to the temperature dependence of the shear viscosity coefficient.'
address:
- 'Department of Physics, McGill University, 3600 rue University, Montréal, Québec H3A 2T8, Canada'
- 'Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA'
- 'Physics Department, Brookhaven National Lab, Building 510A, Upton, NY, 11973, USA '
author:
- Gojko Vujanovic
- 'Jean-François Paquet'
- 'Gabriel S. Denicol'
- Matthew Luzum
- Björn Schenke
- Sangyong Jeon
- Charles Gale
bibliography:
- 'references.bib'
title: 'Probing the non-equilibrium dynamics of hot and dense QCD with dileptons'
---
Introduction
============
One of the main goals of Relativistic Heavy Ion Colliders, either the Relativistic Heavy Ion Collider (RHIC, at Brookhaven National Laboratory) or the Large Hadron Collider (LHC, at CERN), is to investigate the thermodynamic and transport properties of the hot and dense phase of QCD. Much work has been concentrated on the determination of an effective value of the shear viscosity coefficient from analyses of relativistic heavy-ion collisions but so far, such investigations haven been performed mostly by comparing to hadrons produced at the final stages of the collision.
Electromagnetic radiation constitutes a class of complementary and penetrating probes that are sensitive to the entire space-time history of nuclear collisions including its very early stages. In this contribution we show that thermal dileptons are affected by the transport properties of the fluid and by the non-equilibrium aspects of the initial state that are usually inaccessible to hadronic probes. We establish that the azimuthal momentum anisotropies of thermal dileptons are particularly sensitive to the temperature dependence of the shear viscosity coefficient. We also show the potential of thermal dileptons in differentiating between possible initial shear-stress tensors and shear relaxation times.
Fluid-dynamical model
======================
We will discuss only Au-Au collisions at $\sqrt{s_{NN}}=200$ GeV. The time evolution of the hot and dense medium created at RHIC is modeled using <span style="font-variant:small-caps;">music</span>, a 3+1D hydrodynamical evolution [@Schenke:2010rr]. The main equations of motion are the conservation laws of energy and momentum, given by the continuity equation for the energy momentum tensor, $T^{\mu \nu }$, i.e., $\partial _{\mu}T^{\mu \nu }=0$. As usual, $T^{\mu \nu }=\varepsilon \,u^{\mu }u^{\nu}-\Delta ^{\mu \nu }P+\pi ^{\mu \nu }$, with $\varepsilon $ being the energy density, $P$ the thermodynamic pressure, $u^{\mu }$ the fluid four-velocity, $\pi ^{\mu \nu }$ the shear-stress tensor, and $\Delta ^{\mu \nu }=g^{\mu\nu }-u^{\mu }u^{\nu }$ the projection operator onto the 3-space orthogonal to the velocity, with a metric tensor $g^{\mu \nu} = {\rm diag}(1, -1, -1, -1).$ The lattice QCD equation of state is used to relate $P$ and $\varepsilon$ [@Huovinen:2009yb].
The conservation laws are complemented by a relaxation equation for the shear-stress tensor, given by a version of Israel-Stewart (I-S) theory [@Israel1976310; @Israel:1979wp], $$\tau _{\pi }\Delta _{\alpha \beta }^{\mu \nu }u^{\lambda }\partial _{\lambda}\pi ^{\alpha \beta }+\frac{4}{3}\tau _{\pi }\pi ^{\mu \nu }\partial_{\lambda }u^{\lambda }=\left( \pi _{\mathrm{NS}}^{\mu \nu }-\pi ^{\mu \nu}\right) \text{,} \label{eq:pi_munu}$$where $\pi _{\mathrm{NS}}^{\mu \nu }=2\eta \,\sigma ^{\mu \nu }=2\eta \Delta_{\alpha \beta }^{\mu \nu }\partial ^{\alpha }u^{\beta }$ is the Navier-Stokes limit of the shear-stress tensor, with $\Delta _{\alpha \beta}^{\mu \nu }=\left( \Delta _{\alpha }^{\mu }\Delta _{\beta }^{\nu }+\Delta_{\beta }^{\mu }\Delta _{\alpha }^{\nu }\right) /2-\Delta _{\alpha \beta}\Delta ^{\mu \nu }/3$ being the double, symmetric, traceless projection operator. In its simplest from, I-S theory has two transport coefficients: the shear viscosity $\eta $, also present in Navier-Stokes theory, and the shear relaxation time, $\tau _{\pi }$, which only exists in I-S theory.
We use a constant value $\eta/s=1/4\pi$ as the default value for the shear viscosity over entropy density ratio. In the QGP phase, i.e. for temperatures above a transition temperature $T_{\mathrm{tr}}=0.18$ GeV, we will also consider an $\eta /s$ with linear temperature dependences of the form $\eta /s(T)=a(T/T_{\mathrm{tr}}-1)+1/4\pi $. The effect of the temperature dependence of $\eta /s$ on hadronic and eletromagnetic flow observables is tested by modifying the slope parameter $a$. The values of $a$ employed in this work are $a=0$, $0.2427$, and $0.5516$, with $a=0$ corresponding to the constant default value.
The shear relaxation time is assumed to be of the form $\tau _{\pi }=b_{\pi}\eta /\left( \varepsilon +P\right) $. The role of $\tau _{\pi }$ is to govern the rate at which $\pi ^{\mu \nu }$ evolves and relaxes towards its Navier-Stokes limit. The default value used in this study is $b_{\pi }=5$. Here, we test the effect of larger relaxation times by also considering $b_{\pi }=$ $10$ and $20$.
The initial energy density profile is determined by the Monte-Carlo Glauber model, with all the free parameters being tuned to describe the multiplicity and elliptic flow of hadronic observables at RHIC’s highest energy. The initial value of the shear-stress tensor is also varied in this work and is parametrized in the following way $\pi _{0}^{\mu \nu }=c\times2\eta \sigma ^{\mu \nu }$. The parameter $c$ controls the deviation of the initial state from local thermodynamic equilibrium. Here, we set $c=0$, 1/2, and 1, with the default value being zero. The initial velocity profile is always set to zero in hyperbolic coordinates.
Thermal dilepton rates
======================
Thermal dilepton rates can generically be expressed as: $$\begin{aligned}
\frac{d^4 R}{d^4 q} = -\frac{\alpha}{12\pi^4} \frac{1}{M^2} \frac{{\rm Im} \Pi^{{\rm R}}_{\gamma^{\ast}}}{e^{ q^0/T}-1}
\label{eq:rate}\end{aligned}$$ where $\alpha $ is the electromagnetic structure constant, $\Pi _{\gamma^{\ast}}^{\mathrm{R}} = \Pi^{\mathrm{R},\,\, \mu}_{\gamma^{\ast},\,\mu}$ is the trace of the retarded virtual photon self-energy, and $M^{2}=q^{2}$, where $M$ is the virtual photon invariant mass. This expression is valid at leading order in $\alpha_{\rm em}$, but is exact at all orders of $\alpha_{\rm s}$ [@kapusta-gale-book]. We have used the Born rate in this paper, which corresponds to the quark-antiquark annihilation rate into dileptons. Viscosity is included via a deviation of the thermal distribution functions $n$ entering in evaluating $\Pi _{\gamma ^{\ast }}^{\mathrm{R}}$ such that $n\rightarrow n+\delta n$, where $\delta n(p)=G(p)n(p)(1\pm n(p))p_{\mu}p_{\nu }\pi ^{\mu \nu }/\left[ 2T^{2}\left( \varepsilon +P\right) \right]$, and $G(p)$ is a function that must be determined through the use of microscopic physics. In order to find $G(p)$, we solved the Boltzmann equation assuming a massless gas of particles with constant $2\rightarrow 2$ cross section.
The general form of the thermal dilepton rates Eq. (\[eq:rate\]) can be applied in the hadronic sector (low temperatures) via the introduction of the Vector Dominance Model (VDM). Through VDM, $\Pi _{\gamma^{\ast }}^{\mathrm{R}}$ is expressed in terms of $D_{V}^{\mathrm{R}}$, the vector meson $(V)$ retarded propagator. A key ingredient in evaluating $D_{V}^{\mathrm{R}}$ is the vector meson self-energy $\Pi _{V}$, the latter being presented in detail in Ref. [@Vujanovic:2013jpa].
Results
=======
[c c]{} ![The effects of varying $\frac{\eta}{s}(T)$ in the QGP phase on the elliptic flow of charged hadrons (left panel) and of virtual photons (right panel), in collisions of Au + Au at 200 $A$ GeV, at a 20–40% centrality class.[]{data-label="fig:v2_eta_s_T"}](charged_hadrons_v2_pt_rms_lin_w_stddev.pdf "fig:"){width="45.00000%"} ![The effects of varying $\frac{\eta}{s}(T)$ in the QGP phase on the elliptic flow of charged hadrons (left panel) and of virtual photons (right panel), in collisions of Au + Au at 200 $A$ GeV, at a 20–40% centrality class.[]{data-label="fig:v2_eta_s_T"}](thermal_v2_M_Matt_lin_w_stddev.pdf "fig:"){width="45.00000%"}
In the left panels of Figures \[fig:v2\_eta\_s\_T\], \[fig:v2\_init\_cond\], and \[fig:v2\_tau\_pi\], we show the differential elliptic flow of charged hadrons as a function of transverse momentum, $v_{2}(p_{T})$. In the right panel of the same figures we show the integrated elliptic flow of thermal dileptons as a function of their invariant mass, $v_{2}(M)$. In Figure \[fig:v2\_eta\_s\_T\] $\eta/s$ was varied, while in Figures \[fig:v2\_init\_cond\] and \[fig:v2\_tau\_pi\] $\pi _{0}^{\mu \nu }$ and $\tau _{\pi }$ were changed, respectively. In each case, the parameters that are not varied are kept at their default values. For each parameter configuration, we computed 200 events, all in the 20–40% centrality class. The color bands in the plots indicate the statistical uncertainties of the calculations. We note that our results for charged hadron $v_2(p_{T})$ are in good agreement with PHENIX data, which corresponds to the points in the left panels of our figures.
It was already shown in Ref. [@Niemi:2011ix] that charged hadrons have a small sensitivity to the $\eta /s(T)$ in the QGP phase at the top RHIC energy. The left panel of Figure \[fig:v2\_eta\_s\_T\] illustrates that this behavior still holds true for the temperature dependence of $\eta /s$ used in this study. In addition, the results plotted in the left panels of Figures \[fig:v2\_init\_cond\] and \[fig:v2\_tau\_pi\] confirm that the elliptic flow of charged hadrons at RHIC’s highest energy has a very small sensitivity also to variations of $\pi _{0}^{\mu \nu }$ and of $\tau _{\pi }$. Even though it is not shown here, we verified that the same is true for the transverse momentum spectra of charged hadrons.
The situation is not the same when it comes to thermal dileptons. The effect of varying $\pi _{0}^{\mu \nu }$ and $\tau _{\pi }$ is visible on $v_{2}(M)$, but it is still relatively modest, as seen in the right panels of Figures \[fig:v2\_init\_cond\] and \[fig:v2\_tau\_pi\]. However, the magnitude of the slope of $\eta /s$ as a function of $T$ has a sizeable influence on the elliptic flow of thermal dileptons; this is shown in the right panel of Figure \[fig:v2\_eta\_s\_T\]. Recall that, unlike the charged hadrons emitted at the freeze-out hyper-surface, thermal dileptons are emitted throughout the collision history and their elliptic flow retains a memory of the $\eta /s(T)$ in the QGP phase. For small and intermediate values of invariant mass ($M<1.2$ GeV), increasing the QGP’s $\eta /s(T)$ leads to an increase in $v_{2}(M)$. Meanwhile, for larger values of invariant mass ($M>1.2$ GeV) the behavior is inverted and a larger $\eta /s(T)$ leads to a smaller elliptic flow coefficient. We have found that this inversion also occurs for the momentum anisotropy, $\epsilon_p=\langle T^{xx}-T^{yy}\rangle/\langle T^{xx}+T^{yy}\rangle$, when plotted against time.
[c c]{} ![The effects of varying $\pi^{\mu\nu}_0$ on charged hadron’s (left panel) and virtual photon’s (right panel) elliptic flow created in collisions of Au + Au at 200 $A$ GeV, in the 20–40% centrality class.[]{data-label="fig:v2_init_cond"}](charged_hadrons_v2_pt_rms_NS_w_stddev.pdf "fig:"){width="45.00000%"} ![The effects of varying $\pi^{\mu\nu}_0$ on charged hadron’s (left panel) and virtual photon’s (right panel) elliptic flow created in collisions of Au + Au at 200 $A$ GeV, in the 20–40% centrality class.[]{data-label="fig:v2_init_cond"}](thermal_v2_M_Matt_NS_w_stddev.pdf "fig:"){width="45.00000%"}
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ --
![The effects of varying $\protect\tau _{\protect\pi }$ on charged hadron’s (left panel) and virtual photon’s (right panel) elliptic flow created in collisions of Au + Au at 200 $A$ GeV, in the 20–40% centrality class.[]{data-label="fig:v2_tau_pi"}](charged_hadrons_v2_pt_rms_w_stddev.pdf "fig:"){width="45.00000%"}![The effects of varying $\protect\tau _{\protect\pi }$ on charged hadron’s (left panel) and virtual photon’s (right panel) elliptic flow created in collisions of Au + Au at 200 $A$ GeV, in the 20–40% centrality class.[]{data-label="fig:v2_tau_pi"}](thermal_v2_M_Matt_w_stddev.pdf "fig:"){width="45.00000%"}
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ --
The change in the hydrodynamical evolution induced by a T-dependent $\eta /s$ is occuring far from the freeze-out surface and therefore is only accessible to electromagnetic probes. At freeze-out (for collisions at RHIC energies), most of the memory of different values of $\eta/s$ in the QGP phase has faded: the charged hadrons $v_2$ is thus mostly unaffected (see left panel of Figure \[fig:v2\_eta\_s\_T\] and Ref. [@Niemi:2011ix]).
Conclusions
===========
In this contribution, we showed that thermal dileptons are affected by the transport properties of the QGP and by non-equilibrium aspects of the initial evolution that are usually inaccessible to hadronic probes. For the first time, we explicitly demonstrate that the invariant mass distribution of dileptons and their azimuthal momentum anisotropy have a small but non-negligible dependence on the magnitude of the shear relaxation time and on the value of initial shear-stress tensor. Importantly, virtual photons may also reveal the temperature dependence of the shear viscosity coefficient. This endeavor reaffirms the potential that penetrating probes, such as dileptons, have in furthering our understanding of QCD at high temperatures and densities. We expect that, as experimental uncertainties become smaller, such probes will play a more dominant role in the extraction of the initial state and transport properties of the bulk QCD matter created in ultrarelativistic heavy ion collisions at RHIC and at the LHC.
Acknowledgements
================
This work was supported in part by the Natural Sciences and Engineering Research Council of Canada, in part by U. S. DOE Contract No. DE-AC02-98CH10886, and in part by the by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of Nuclear Physics, of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. G. Vujanovic acknowledges support by the Canadian Institute for Nuclear Physics, and G.S. Denicol acknowledges support through a Banting Fellowship from the Government of Canada. Computations were performed on the Guillimin supercomputer at McGill University under the auspices of Calcul Québec and Compute Canada. The operation of Guillimin is funded by the Canada Foundation for Innovation (CFI), the National Science and Engineering Research Council (NSERC), NanoQuébec, and the Fonds Québécois de Recherche sur la Nature et les Technologies (FQRNT).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In a noiseless linear estimation problem, one aims to reconstruct a vector $\bf x^*$ from the knowledge of its linear projections ${\bf y} = \Phi {\bf x^*}$. There have been many theoretical works concentrating on the case where the matrix $\Phi$ is a random i.i.d. one, but a number of heuristic evidence suggests that many of these results are universal and extend well beyond this restricted case. Here we revisit this problematic through the prism of development of message passing methods, and consider not only the universality of the $\ell_1$ transition, as previously addressed, but also the one of the optimal Bayesian reconstruction. We observed that the universality extends to the Bayes-optimal minimum mean-squared (MMSE) error, and to a range of structured matrices.'
address: |
$^1$ Laboratoire de Physique de l’Ecole normale supérieure, Université PSL, CNRS, Sorbonne Université, Université Paris-Diderot, Sorbonne Paris Cité, Paris, France\
$^2$ Institut de physique théorique, Université Paris Saclay, CNRS, CEA, 91191 Gif-sur-Yvette, France
author:
- 'Alia Abbara$^1$, Antoine Baker$^1$, Florent Krzakala$^1$ and Lenka Zdeborová$^2$'
bibliography:
- 'refs.bib'
title: |
On the Universality of Noiseless Linear Estimation\
with Respect to the Measurement Matrix
---
Introduction
============
The problem of recovering a signal through the knowledge of its linear projections is ubiquitous in modern information theory, statistics and machine learning. In particular, many applications require to reconstruct an unknown $n-$dimensional signal vector ${\bf x^*}$ from the linear projections $${\bf y} = \Phi {\bf x^*}\, ,\label{def:problem}$$ where ${\bf y}$ is a m-dimensional vector, and $\Phi$ is a $m \times n$ random matrix. For instance, if $ {\bf x^*}$ is sparse, this task of estimating the signal from its linear [*random*]{} projections is at the roots of compressed sensing [@candes2006near]. A fundamental question in the field is how much the algorithmic and the information theoretic performance depends on the choice of the random matrix $\Phi$.
In the present letter, we concentrate on the noiseless and asymptotic, large $n$, regime with a fixed value $\alpha\!=\!m/n$. We consider $\bf x^*$ to be $k$-sparse, i.e. to have only $k$ non-zero values, and we shall work in the limit where $n\to \infty$, $k\to \infty$, and a finite value of $\rho\!=\!k/n$. In such setting, a classical result is the following: for random matrices $\Phi$ with independent standard Gaussian entries, the (convex) reconstruction with $\ell_1$ penalty displays a precisely determined phase transition. For a certain region in the ($\alpha,\rho$)-phase diagram, it typically finds back the vector $\bf x^*$, being the sparsest solution, whereas outside that region, it typically fails. The obundary between these two regions is called the Donoho-Tanner line [@donoho2005sparse]. It has been shown empirically that the very same phase transition location seems to hold for a wider range of random matrix ensembles, see e.g. [@donoho2009observed; @Monajemi1181], suggesting a large universality of the Donoho-Tanner phase transitions. [Another line of work showed that the convex $\ell_1$ reconstruction problem can be treated through conic geometry, and the success probability of signal recovery only depends on a geometric number characterizing a subcone (statistical dimension or Gaussian width) [@chandrasekaran2010; @amelunxen2014].]{}
Here we investigate the universality of the phase transition not only for the $\ell_1$ transition, but also to the performance of the optimal Bayesian reconstruction. We analyze this question through the prism of information theory, message passing methods, and random matrix theory. We shall see that the universality indeed extends to a more generic set of properties than the $\ell_1$ transition, such as the minimum mean-squared (MMSE) error or the easy-hard phase transition for optimal Bayesian learning, and empirically to structured matrices such as the one appearing in [@pennington2017nonlinear; @liao2018spectrum].
We note that investigation of universality are very common to physics problems, and understanding how large is the class of model for which a given result applied is a very fundamental question. The message-passing-based algorithm that we investigate in this paper to demonstrate the universality also has their origin in pysics works, such as [@thouless1977solution].
A short review of results for i.i.d. random matrices {#sec:iid-matrix}
====================================================
A first well-understood case of universality holds for random matrices $\Phi$ where all the elements are generated i.i.d. from a well-behaved distribution -with zero mean and unit variance- which all exhibit the same transitions as Gaussian random matrices. This is known for multiple retrieval problems:
$\ell_1$ recovery
-----------------
Consider for instance the Donoho-Tanner line[@donoho2005sparse] that regulates the $\ell_1$ recovery. Thanks to the approximate message passing solver (see below) that has been shown to be universal with respect to all i.i.d. distributions with finite moments [@bayati2011dynamics; @bayati2015universality], we know that the Donoho-Tanner phase transition is the same for all such random matrices.
Information theoretic optimal reconstruction
--------------------------------------------
There has been a considerable amount of work in the information theory community on the computation of the mutual information and on the MMSE for problems such as (\[def:problem\]) with Gaussian matrices. In particular, following the replica method from statistical physics (the Tanaka formula [@tanaka2001statistical]), a heuristic formula has been postulated in different situations, see e.g. [@tulino2013support; @krzakala2012probabilistic; @krzakala2012statistical; @zhu2013performance]. This heuristic replica result has been recently rigorously proven in a series of papers [@barbier2016mutual; @barbier2016mutual; @reeves2016replica]. In a more recent proof [@barbier2017phase], it has been shown, again, that the formula is not specific to Gaussian i.i.d. matrices, but that any matrix with i.i.d elements of unit variance and zero mean leads to the same exact result for the mutual information and the MMSE.
Hard phase for Bayesian decoders
--------------------------------
A third interesting point is to ask about tractacle decoders that aim to perform the optimal Bayesian estimation, i.e. with a perfect prior knowledge on the distribution of $\bf x^*$. For simplicity, consider for instance the case where each element of $\bf x^*$ has been sampled from a Gauss-Bernoulli distribution: $$x_i \sim (1-\rho) \delta(x) + \rho {\cal N}(0,1)\, .$$ In this case, the best known solver is again AMP, using a Bayesian decoder (instead of the soft thresholding function for $\ell_1$ recovery) [@vila2011expectation; @montanari2012graphical; @krzakala2012probabilistic; @krzakala2012statistical]. Interestingly, it shares with the $\ell_1$ recovery a similar phase transition: for a certain region in the ($\alpha,\rho$) plane it typically finds back the vector $\bf x^*$, whereas outside that region it fails. We shall denote the limit between these regions the “Bayesian hard-phase” transition. The “Bayesian hard-phase” line, that has been precisely computed in [@krzakala2012probabilistic; @krzakala2012statistical] is always better than the Donoho-Tanner line (as it should, since it exploits additional information). Once more, the universality of AMP shows that this phase transition is not restricted to Gaussian matrices, but extends as well to all (well normalized) i.i.d. matrices.
The fact that these three properties (the $\ell_1$, the hard-phase line, as well as the MMSE) are universal for all i.i.d. matrices makes the case for Gaussian computations, as done in theoretical computation, stronger. We shall see that this universality extends well beyond these simple cases.
Random rotationally invariant matrices {#sec:rot-matrix}
======================================
Moving away from the well-known i.i.d. examples, we start by considering a much larger set of random matrices defined through their singular value decomposition (SVD): any real matrix $\Phi$ can be decomposed into ${\Phi = U \Sigma V}$, with $U$ and $V$ orthogonal matrices, and $\Sigma$’s elements being $\Phi$’s singular values. We shall look at the left rotationally invariant random matrix ensemble: these are matrices $\Phi$ that can be written as $$\Phi = U \Sigma V$$ with an arbitrary rotation matrix $U$ and singular values $\Sigma$, but where the matrix $V$ has been randomly (and independently of $\Sigma$ and $U$) generated from the Haar measure (that is, uniformly from all possible rotations).
When the singular values are different from zero, it is straightforward to justify the universality property for matrices from this subclass. We start by the definition of the problem: we wish to find $\boldsymbol{x}$ such that $$\boldsymbol{y}=\Phi \boldsymbol{x}=U\Sigma V \boldsymbol{x}.
\label{def_problem}$$ If $m \le n$, then $\Sigma$ is written as $\Sigma=
\left[
\begin{array}{c|c}
\tilde{\Sigma} & 0 \\
\end{array}
\right]$ and we define $$\Sigma^{inv}=
\left[
\begin{array}{c}
\tilde{\Sigma}^{-1}\\
\hline
0
\end{array}
\right] \text{ such that } \Sigma^{inv} \Sigma=
\left[
\begin{array}{c|c}
I_m & 0\\
\hline
0 & 0
\end{array}
\right].$$ Multiplying on both sides by $U^T$, and then by $\Sigma^{inv}$; one reaches $$\boldsymbol{\tilde y} = \Sigma^{inv} U^T \boldsymbol{y} = \tilde{V} \boldsymbol{x}
\label{eq:magic1}$$ where $\tilde{V}$ is a $m\times n$ matrix composed of the first $m$ lines of $V$. If instead $m > n$, $\Sigma$ is written as $$\Sigma^{inv}=
\left[
\begin{array}{c}
\tilde{\Sigma}\\
\hline
0
\end{array}
\right]$$ and we define $\Sigma^{inv}= \left[
\begin{array}{c|c}
\tilde{\Sigma}^{-1} & 0 \\
\end{array}
\right]$ such that $\Sigma^{inv} \Sigma = I_n$. Multiplying by $U^T$ then $\Sigma^{inv}$, we obtain $$\boldsymbol{\tilde y} = \Sigma^{inv} U^T \boldsymbol{y} = V \boldsymbol{x}.$$ In both cases, we thus see that the problem has been transformed —in a constructive way— into a standard linear system with the sensing matrix $\tilde{V}$ when $m \le n$ being a (sub-sampled) random rotation one, or sensing matrix $V$ when $m > n$. This shows that all rotationally invariant matrices, [which satisfy $U$ and $\Sigma$’s independence on $V$, can be transformed the same way and are in the same universality class as far as noiseless linear recovery is concerned, i.e. they will display the same phase transitions.]{}
Since Gaussian i.i.d. matrices belong among random rotationally invariant matrices (in this case $\Sigma$ follows the Marcenko-Pastur law [@tulino2004random]) this means that all the information theoretic rigorous results [(such as phase transitions and MMSE value)]{} with zero noise for random Gaussian matrices applies verbatim to all rotationally invariant ensemble, [as long as the SVD’s matrices $U$ and $\Sigma$ are independent of $V$.]{} This is a very strong universality, that applies to the three cases (1, 2, 3) from sec. \[sec:iid-matrix\]. Note that the universality of the Donoho-Tanner line with rotationally invariant matrices was already hinted by the replica method [@kabashima2009typical].
Notice, however, that the above construction depends crucially on the fact that we consider here noiseless measurements. It would not work if an additional Gaussian noise were added in eq. (\[def:problem\]): in this case, the transformation would make the i.i.d. Gaussian noise a correlated one. Indeed, the replica formula for noisy measurements underlines that the MMSE depends on the precise set of matrices in noisy reconstruction [@takeda2006analysis; @tulino2013support] (this formula is not yet fully rigorous, but see [@barbier2018mutual] for a proof in a restricted setting). Any differences, however, must go to zero in the noiseless limit.
Approximate Message Passing
===========================
Having discussed the universality with respect to random rotationally invariant matrices, we now wish to discuss its effect on specific solvers, concretely the message passing algorithms.
AMP
---
We first consider the original approximate message passing (AMP) [@donoho2009message] to compute the phase transition between the phase where the algorithm reconstructs $\bf x^*$ perfectly, and the one where reconstruction may be possible but is not achieved by the algorithm. AMP is an iterative algorihm that follows: $$\begin{aligned}
&{\bf \hat x}^{t+1}=\eta_t(\Phi^T {\bf \hat x}^t)\\
&{\bf z}^t={\bf y}-\Phi{\bf \hat x}^t + \frac{1}{\alpha}{\bf z}^{t-1}\langle \eta_{t-1}'(\Phi^T {\bf z}^{t-1}+{\bf \hat x}^{t-1})\rangle.\end{aligned}$$ where $t$ is the iteration index, ${\bf x}^t$ is the current estimate of ${\bf x^*}$, ${\bf z}^t$ the current residual, $\langle \cdot \rangle$ is an averaged sum of components, and $\eta_t$ is a prior-dependent threshold function applied component-wise (the soft thresholding for $\ell_1$, or the Bayesian decoder [@krzakala2012probabilistic; @krzakala2012statistical]).
One of the most interesting features of AMP is that, if $\Phi$ is a Gaussian i.i.d. matrix, its mean squared error (MSE) $\sigma_t$ can be tracked accurately by the state evolution formalism [@donoho2009message; @bayati2011dynamics; @bayati2015universality]. State evolution is a relatively simple recursive equation: $$\sigma_{t+1}^2 = \Psi(\sigma_{t}^2)\, ,~~
\Psi(\sigma^2)={\mathbb E}\left[ \left(
\eta_t(X+\frac{\sigma}{\sqrt{\alpha}}Z)-X
\right)^2 \right] \, ,$$ where the expectation is with respect to independent random variables $Z \sim {\cal N}(0, 1)$ and $X$, whose distribution coincides with the empirical distribution of the entries of $x^*$. Analyzing the evolution of this equation for the $\ell_1$ decoder yields the Donoho-Tanner line[@donoho2009message], while using the Bayesian decoder it yields the hard-phase line for Bayesian decoding [@krzakala2012probabilistic].
It would be interesting to use AMP for rotationally invariant matrices. In order to do this, we follow the construction of sec. \[sec:rot-matrix\]: starting from equation (\[eq:magic1\]) we then multiply by $\Sigma_0$, a $m \times m$ diagonal matrix with singular values sampled from Marcenko-Pastur law (singular values of a Gaussian i.i.d. matrix [^1]), and $U_0$ a $m \times m$ Haar-generated orthogonal matrix, [thus ensuring that $\Sigma_0$ and $U_0$ are generated independently of $V$]{}: $$\begin{aligned}
U_0 \Sigma_0 \tilde{\Sigma}^{-1} U^T \boldsymbol{y} &= U_0 \Sigma_0 \tilde{V} \boldsymbol{x}\\
\boldsymbol{y'} &= \Phi' \boldsymbol{x}.\label{eq:trick}\end{aligned}$$ After this transformation, $\Phi'=U_0 \Sigma_0 \tilde{V}$ is a random matrix that belongs to an ensemble very close to the Gaussian i.i.d. matrices ensemble. [In fact, a recent work showed that AMP applied to a Gaussian matrix follows the same state evolution as matrices such as $\Phi'$ where $U_0, \tilde{V}$ are uniform orthogonal matrices and $\Sigma_0$ diagonal’s elements are singular values sampled from the Marcenko-Pastur law [@takeuchi2019unified].]{} Combining this result with the matrix transformation, we have thus constructively mapped the noiseless reconstruction problem back to the well-understood noiseless compressed sensing case for a Gaussian i.i.d. matrix, where we can safely apply the algorithm, and its state evolution. In the section \[sec:numerics\], we apply this matrix transformation for numerical experiments using AMP.
Vector-AMP
----------
While the transformation trick allows to make AMP work with random rotationally invariant matrices, another alternative is to work directly with a dedicated solver. To this means, different but related approaches were proposed [@takeda2006analysis; @SAMP], in particular, using the general expectation-propagation (EP) [@Minka; @EP] scheme. Ma and Ping proposed a variation of EP called OAMP [@ma2017orthogonal] specially adapted to rotation matrices. Rangan, Schniter and Fletcher introduced a similar approach called VAMP [@VAMP] and proved that it follows state evolution equations corresponding to the fixed point of the replica potential [@takeda2006analysis; @tulino2013support; @barbier2018mutual]. The multi-layer AMP algorithm of[@Manoeletal] also display the same fixed point.
We shall concentrate here on the VAMP (Vector-AMP) approach, and for a moment, put back a small additional random Gaussian i.i.d. noise of variance $\Delta$ in the measurement in eq. (\[def:problem\]) as it is needed for stating the algorithm. VAMP then consists in the following fixed-point iteration: $$\begin{aligned}
&\bm{u}_{\ell}^{t + 1} = \frac{
\bm{\hat x_l}^t}{\langle \operatorname{Var}_\ell^t (\bm{x}) \rangle} -
\bm{u}_{r}^t, \qquad
\rho_{\ell}^{t + 1} = \frac{1}{\langle \operatorname{Var}_\ell^t
(\bm{x}) \rangle} - \rho_r^t, \\
&\bm{u}_r^{t + 1} = \frac{
\bm{\hat x_r}^t}{\langle\operatorname{Var}_r (\bm{x}) \rangle} -
\bm{u}_\ell^t, \qquad
\rho_r^{t + 1} = \frac{1}{\langle \operatorname{Var}_r^t
(\bm{x}) \rangle} - \rho_\ell^t,
\end{aligned}$$ where we denote by $\mathbb{E}_{\ell, r}^t$ the expectation w.r.t. the tilted distributions ${\tilde{Q}_{\ell, r}^t (\bm{x}) \propto P_{\ell, r}
(\bm{x}) Q_{\ell, r}^t (\bm{x})}$, and by $\operatorname{Var}_{r, \ell}^t (\bm{x})$ the variance of these distributions. Here, we have defined $Q_{l,r}({\bf x})=e^{-\frac 12 \rho_{l,r} \bm{x}^T\bm{x} + \bm{u}_{l,t}^T \bm{x}}$, $P_l(\bm{x}) \propto e^{-||{\bf y}-\Phi {\bf x}||_2^2/2\Delta}$ and $P_r(\bm{x})$ is the prior used in the algorithm (i.e. the Laplace prior for the $\ell_1$ model, or the actual distribution of the signal for Bayesian reconstruction). In particular $$\begin{aligned}
\bm{\hat x_l}^t &= (\bm{\Phi}^T \bm{\Phi} + \Delta \rho_r^t I_p)^{-1} (\bm{\Phi}^T \bm{y} + \Delta\bm{u}_r^t),\\
\langle \operatorname{Var}_\ell^t (\bm{x}) \rangle &= \frac{ \Delta}{N} \rm{Tr} (\bm{\phi}^T \bm{\Phi} + \Delta \rho_r^t I_p)^{-1},
\end{aligned}$$ where, as for AMP, we define the denoiser that yields the estimates of $x$ by $z (u, \rho) = \int dx P_r(x) e^{-\frac12 \rho x^2 + u x}$, $$\begin{aligned}
{(\hat x_r)}_j &= \frac{\partial}{\partial u} \log z (u, \rho)
\Big|_{u_{\ell k}^t, \rho_\ell^t},\\
\langle \operatorname{Var}_r^t (\bm{x}) \rangle &= \frac{1}{n} \sum_{j = 1}^p
\frac{\partial^2}{\partial u^2} \log z (u, \rho) \Big|_{u_{\ell k}^t,
\rho_\ell^t}.
\label{eq:ec_avgr}
\end{aligned}$$ Again, the performance of the recursion can be analyzed rigorously through the state evolution [@VAMP]. For simplicity, let us concentrate on the Bayes optimal case in which case the state evolution can be closed on the variables (see [@VAMP]): $$\sigma^t = \langle \operatorname{Var}_r^t (\bm{x}) \rangle~\rm{and}~
\epsilon^t = \langle \operatorname{Var}_l^t (\bm{x}) \rangle \, ,$$ by writing $$\begin{aligned}
\sigma^t (\rho^t_l) &= \Psi((\rho^t_l)^{-1}) \\
\epsilon (\rho^t_r) &= \Delta {\mathbb{E}}\left[ \frac 1{\Sigma^2+ \Delta \rho_r^t}\right] = \Delta S_{\Sigma^2}(-\Delta \rho^t_R)
\end{aligned}$$ where the expectation is above the distribution of the singular values $\Sigma$ of the matrix $\Phi$, and where we recognize the Stieljes transform $S_{X}(r)={\mathbb{E}}\left[ 1/{X - r}\right] $. Though this transform, we see that the performance depends crucially on the distribution of eigenvalues. Let us now go back on the noiseless limit when $\Delta \to 0$ and analyze how the universality shows up. Consider again the Stieltjes transform: out of the $n$ singular values of the $n \times n$ matrix $\bm{\Phi}^T \bm{\Phi}$, we shall have $(1-\alpha)n$ of them to be zero (assuming $\alpha<1$) while the rest are positive (since $m<n$). In this case, the limit $r \to 0$ of the Stieltjes transform will behave as $S_{X}(r) \approx-(1-\alpha)/r$ so that $$\lim_{\Delta \to 0} \epsilon (\rho^t_r) = \frac{1-\alpha}{\rho_r^t}\, .$$ Again, we see that all the complicated dependence on the spectrum of the matrix $\Phi$ has been eliminated. This is a direct, alternative, proof that VAMP will also yield universal results in the zero noise limit for the Bayesian reconstruction. Given that VAMP has the same fixed point as the replica mutual information [@tulino2013support; @barbier2018mutual], this argument applies to the replica prediction for the MMSE as well.
Structured matrices
===================
We now move to very structured matrices, in order to test the universality as well as the quality and the prediction of the state evolution out of its comfort zone. In order to do so, we have considered different matrix ensembles:
![Phase diagram for a DCT matrix (width $n=1000$) in the Bayes-optimal case. The averaged MSE on 50 executions of VAMP is represented by a color-code, displaying a phase transition that matches the theoretical Bayes line for Gaussian i.i.d. matrices (black line). Some finite-size effects can be seen.[]{data-label="fig:sim1"}](DCT_1000_VAMP_50.png){width="75.00000%"}
![Phase diagram in the $\ell_1$ reconstruction case obtained by averaging on 20 to 50 executions on VAMP. The dots indicate the phase transitions for Gaussian i.i.d., DCT (width $n=2000$), Hadamard matrices ($n=4096$), and random feature matrices $\Phi=f(WX)$ with ${f = \text{ReLu}}$, ${f = \text{sign}}$, ${f= \tanh}$ ($W$ and $X$ are Gaussian i.i.d. of size $\alpha n \times n$ and $n \times n$ with $n=2000$). They match the theoretical Donoho-Tanner transition for Gaussian i.i.d. matrices (black line).[]{data-label="fig:sim2"}](Donoho-Tanner_VAMP_50.png){width="75.00000%"}
Tested ensembled of matrices
----------------------------
#### Discrete cosine transform matrices
The first ensemble we consider consists in Fourier-like matrices. A $n \times n$ discrete cosine transform (DCT) matrix $Y$ is defined by: $$Y_{jk}=\sqrt{\dfrac{2}{n}}\epsilon_k \cos\left( \dfrac{\pi (2j+1) k}{2n}\right)\, ,$$ where $j,k \in \llbracket 0, n-1 \rrbracket$, $\epsilon_0=1/ \sqrt{2}$, $\epsilon_i=1$ for $i=1,...,n-1$. We used a sub-sampled version of these matrices in which we picked some rows randomly.
#### Hadamard matrices
A natural variant of DCT is given by the Hadamard matrices. $H$ is a $n \times n$ Hadamard matrix if its entries are $\pm 1$ and its rows are pairwise orthogonal, i.e. $H H^T= n I_n$. For every integer $k$, there exists a Hadamard matrix $H_k$ of size $2^k$. These can be created with Sylvester’s construction: Let $H$ be a Hadamard matrix of order $n$. Then the partitioned matrix $${\begin{bmatrix}H&H\\H&-H\end{bmatrix}}$$ is a Hadamard matrix of order $2n$.
#### Random features maps
Finally, we wanted to consider here random features maps (RFM) as encountered in nonlinear regression problems. In such settings, a random features matrix $\Phi = f(WX)$ is obtained from the raw data matrix $X$ by means of a random projection matrix $W$ and a pointwise nonlinear activation $f$. Kernel regression models, nonlinear in the original data $X$, can then be approximately but efficiently solved by the linear estimation problem (\[def:problem\]), with an appropriate choice for $f$ and the $W$-distribution [@rahimi2008random]. Such matrices, that can be seen as the ouput of a neuron with random weights, have been investigated in particular in the context of neural networks [@pennington2017nonlinear; @liao2018spectrum]. Indeed, in neural networks configurations with random weights play an important role as they define the initial loss landscape. They are also fundamental in the random kitchen sinks algorithm in machine learning [@rahimi2008random] and it is thus of interest to test our understanding of linear reconstructions with AMP and VAMP in this case.
In what follows we will test random features matrices where both $W$ and $X$ are random Gaussian i.i.d. matrices.
Numerical results {#sec:numerics}
-----------------
We provide the codes used to generate the data on github in the repo [http://sphinxteam/Universality-CS-2019](https://github.com/sphinxteam/Universality-CS-2019). To generate Figure \[fig:sim1\] and \[fig:sim2\], we ran VAMP $50$ times on $50\times50$ points spanning the ($\alpha$,$\rho$)-space, and computed the average mean-squared error (MSE) between the signal ${\bf x^*}$ and the reconstructed configuration [**x**]{}. The MSE is represented with a color bar (white means perfect reconstruction). For a DCT and a Hadamard matrix, we observe a phase transition in the Bayes-optimal case that matches the theoretical transition for Gaussian i.i.d. matrices. We also ran VAMP for the $\ell_1$ reconstruction problem. Averaging on 20 executions (or 50 for small $\alpha$ where finite-size effects are more important), we recover again a phase transition matching the theoretical Donoho-Tanner line for Gaussian i.i.d. matrices [@donoho2009observed]. Besides, we compared the MSE obtained by VAMP at each point of the phase diagram for different matrices. In figures \[fig:sim3\] and \[fig:sim4\], we plot the MSE averaged on 20 executions for $\rho$ fixed and $\alpha$ ranging between 0 and 1. We get the same error in reconstruction for all matrices, following the MSE for Gaussian i.i.d. matrix for $\rho =0.25$, $0.5$ and $0.75$. We also checked that AMP, provided one uses the trick eq. (\[eq:trick\]), reproduce these results as well: indeed the two algorithms returned extremely similar results.
![Mean-squared error for $\rho =0.25$, $0.5$ and $0.75$ (bottom to up curves) in the Bayes-optimal case averaged on 20 executions of VAMP for Gaussian i.i.d, DCT, Hadamard, random features matrices $\Phi = f(WX)$ with ${f = \text{ReLu}}$, ${f = \text{sign}}$, ${f= \tanh}$ ($W$ and $X$ are Gaussian i.i.d of size $\alpha n \times n$ and $n \times n$) . The width is $n=2000$ for all matrices.[]{data-label="fig:sim3"}](GaussBernoulli_Bayesoptimal_VAMP_20.png){width="75.00000%"}
![Mean-squared error for $\rho =0.25$, $0.5$ and $0.75$ (bottom to up curves) in the $\ell_1$ reconstruction case averaged on 20 executions of VAMP for Gaussian i.i.d, DCT, Hadamard, random features matrices $\Phi = f(WX)$ with ${f = \text{ReLu}}$, ${f = \text{sign}}$, ${f= \tanh}$ ($W$ and $X$ are Gaussian i.i.d of size $\alpha n \times n$ and $n \times n$). The width is $n=2000$ for all matrices.[]{data-label="fig:sim4"}](GaussBernoulli_L1transition_VAMP_20.png){width="75.00000%"}
Discussion
----------
Figures of the previous section perfectly illustrate our main point: the universality in noiseless compressed sensing is not limited to the $\ell_1$-type reconstruction as in [@donoho2009observed; @Monajemi1181], but extends to other quantities and estimators, such as the hard-phase line in Bayesian reconstruction, and the MMSE. Besides, it is not limited to random orthogonal matrices, but empirically extends to Fourier-type matrices and to the random features maps currently studied in machine learning. It is an open question to extend the proof of state evolution to these challenging matrices. It would be interesting to find a good creterion to identify which matrices satisfy this universality and which do not; this is something that we are yet unable to predict in advance. An example of structured matrices that do not seem to follow these universal phase transitions is given by Haar wavelet matrices, which can be defined recursively by: $$W_2={\begin{bmatrix}1&1\\1&-1\end{bmatrix}}\text{ and } W_{2k}= {\begin{bmatrix}H_k \otimes [1,-1]\\I_k \otimes [1,1]\end{bmatrix}}$$ where $I_k$ is the identity matrix of size $k$ and $\otimes$ is the Kronecker product. In fact, VAMP even fails to converge for these matrices. Investigating this behavior is an interesting direction of research.
Acknowledgment {#acknowledgment .unnumbered}
==============
We thank Andre Manoel and Galen Reeves for useful discussions. We acknowledge funding from the ERC under the European Union’s Horizon 2020 Research and Innovation Program Grant Agreement 714608-SMiLe; and from the French National Research Agency (ANR) grant PAIL.
[^1]: The singular values of a Gaussian matrix are correlated, so in fact we may want to generate $\Sigma_0$ by first generating a random Gaussian matrix, and then calculating its singular values.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Neutron matter is an ideal laboratory for nuclear interactions derived from chiral effective field theory since all contributions are predicted up to next-to-next-to-next-to-leading order (N$^3$LO) in the chiral expansion. By making use of recent advances in the partial-wave decomposition of three- nucleon (3N) forces, we include for the first time N$^3$LO 3N interactions in many-body perturbation theory (MBPT) up to third order and in self-consistent Green’s function theory (SCGF). Using these two complementary many-body frameworks we provide improved predictions for the equation of state of neutron matter at zero temperature and also analyze systematically the many-body convergence for different chiral EFT interactions. Furthermore, we present an extension of the normal-ordering framework to finite temperatures. These developments open the way to improved calculations of neutron-rich matter including estimates of theoretical uncertainties for astrophysical applications.'
author:
- 'C. Drischler'
- 'A. Carbone'
- 'K. Hebeler'
- 'A. Schwenk'
bibliography:
- 'strongint.bib'
title: 'Neutron matter from chiral two- and three-nucleon calculations up to N$^3$LO'
---
\[sec:intro\]Introduction
=========================
Progress in chiral effective field theory (EFT) for nuclear forces [@Epel09RMP; @Mach11PR] and advances in many-body theory [@Soma14GGF2N3N; @Bind14CCheavy; @Holt14Ca; @Lahd13LEFT; @Hage14rev; @Sign14BogCC; @Dikm15NCSMSM; @Herg16IMSRG] offers new paths to systematically improvable calculations of nuclear many-body systems [@Hamm12RMP; @Hebe15ARNPS]. In recent years infinite nuclear matter has been studied based on chiral EFT interactions within various frameworks like many-body perturbation theory (MBPT) [@Hebe11fits; @Cora14nmat; @Well14nmtherm; @Dris16asym], in-medium chiral perturbation theory [@Holt13PPNP], self-consistent Green’s function (SCGF) framework [@Carb13nm], coupled-cluster theory [@Hage14ccnm], the Brueckner-Hartree-Fock approach [@Kohn13gmatchiral; @Isau16pnmdin] and Quantum Monte Carlo methods [@Geze13QMCchi; @Wlaz14QMC; @Rogg14QMC; @Lynn15QMC3nf].
So far, the employed chiral EFT nucleon-nucleon (NN) and three-nucleon (3N) interactions in these calculations were all derived within Weinberg’s power counting scheme [@Epel09RMP; @Mach11PR]. Here the leading 3N forces appear at next-to-next-to-leading order (N$^2$LO) and contain two unknown low-energy couplings, $c_D$ and $c_E$, which need to be determined by fits to few- or many-body observables. In contrast, subleading 3N forces at N$^3$LO do not contain any new low-energy couplings [@Bern083Nlong; @Bern113Nshort] and are thus completely predicted. Hence, including these contributions in calculations offers the possibility to probe systematically the validity of chiral power counting in nuclear systems and to provide estimates of theoretical uncertainties.
Full N$^3$LO calculations of neutron matter were first performed in Refs. [@Tews13N3LO; @Krue13N3LOlong]. These works showed that 3N forces at N$^3$LO provide surprisingly large contributions to the equation of state especially in symmetric matter. Similar results were found for few-body systems in Ref. [@Hebe15N3LOpw]. These findings raise fundamental questions concerning the convergence of the chiral expansion for 3N forces within the employed regularization and power counting scheme.
Generally, the treatment and inclusion of 3N forces is still a challenge in many-body calculations. In particular, due to the complexity and rich analytical structure of 3N forces at N$^3$LO [@Bern083Nlong; @Bern113Nshort] so far it was possible to include effects from 3N interactions only at the Hartree-Fock level in Refs. [@Tews13N3LO; @Krue13N3LOlong]. While this approximation is expected to be reliable for neutron matter, higher-order terms in the many-body expansion are expected to become significant as soon as the proton fraction becomes sufficiently large. In the present paper we address this issue by making use of two recent advances: a) the development of a novel framework that makes it possible to compute matrix elements of 3N interactions in a partial-wave momentum basis [@Hebe15N3LOpw] and the availability of matrix elements up to N$^3$LO and large model spaces, and b) the development of a novel normal-ordering framework based on partial-wave matrix elements [@Dris16asym] that allows to systematically include these 3N interactions in calculations of nuclear matter for arbitray isospin asymmetry. By combining these two advances it is now possible to include general 3N forces that are available in form of plane-wave partial-wave matrix elements and to treat 3N forces on the same footing as NN forces in the many-body expansion. Furthermore, these developments play an important role in view of future calculations that will employ simultaneous evolution of NN and 3N interactions in a momentum basis via similarity renormalization group techniques [@Hebe12msSRG; @Hebe2013nmsrg].
In this paper we will exploit and combine these new capabilities and perform improved calculations of neutron matter up to N$^3$LO in MBPT and SCGF. We benchmark results of these two complementary many-body framework against each other and present a generalization of the normal-ordering framework to finite temperatures. The extension of the present N$^3$LO calculations to arbitrary proton fractions is in principle straightforward but requires reliable fit values for the low-energy couplings $c_D$ and $c_E$ at this order [@Skibi113HN3LO3N; @Gola14n3lo]. This is work in progress. In neutron matter these short-range and mid-range topologies do not contribute within the employed regularization scheme.
The paper is organized as follows. In Sec. \[sec:calc\_details\] we specify the set of employed chiral EFT Hamiltonians and describe the novel normal-ordering framework that allows to include general 3N interactions in calculations of nuclear matter. In addition we briefly discuss the many-body frameworks we used for our calculations. In Sec. \[sec:results\] we present our results based on three different sets of Hamiltonians, with a special focus on the effects of 3N forces beyond the Hartree-Fock approximation. Furthermore, we analyze the many-body convergence in MBPT by comparing with SCGF results. Finally we present a generalization of our normal-ordering framework to finite temperatures and benchmark results for the energy against exact Hartree-Fock results. In Sec. \[sec:outlook\] we conclude with a summary and an outlook.
\[sec:calc\_details\]Calculational details
==========================================
\[subsec:chiral\_ham\]Chiral EFT Hamiltonians
---------------------------------------------
We consider unevolved NN and 3N forces up to N$^3$LO and calculate the energy per particle of infinite neutron matter in the frameworks of MBPT and SCGF. The Hamiltonian takes the form $$\label{eq:ham}
H = T + V_\text{NN} + V_\text{3N} + \ldots \,,$$ where $T$, $V_\text{NN}$ and $V_\text{3N}$ denote the kinetic energy, the NN and 3N intercations, respectively. So far, in most calculations of nuclear matter NN and 3N forces were not included consistently up to the same order in the chiral expansion due to the complex structure of 3N forces at N$^3$LO [@Bern083Nlong; @Bern113Nshort]. Only recently an efficient partial-wave decomposition of these contributions was developed in Ref. [@Hebe15N3LOpw]. In Refs. [@Tews13N3LO; @Krue13N3LOlong] the N$^3$LO 3N contributions were evaluated exactly for neutron matter and symmetric nuclear matter in Hartree-Fock approximation. It was somewhat unexpected that the subleading 3N forces provide significant contributions to the energy. The findings suggest that it is mandatory to investigate these contributions more systematically by including higher-order effects in the many-body expansion.
We note that, considering only NN and 3N forces at N$^3$LO in Eq. (\[eq:ham\]) is still not fully consistent in the chiral expansion. In fact, four-nucleon (4N) forces also contribute at this order. However, Ref. [@Kais124N; @Tews13N3LO; @Krue13N3LOlong] demonstrated that the 4N contributions to the energy in neutron matter in the Hartree-Fock approximation are very small compared to the overall uncertainty, at saturation density. Therefore, 4N contributions only lead to a small shift for all Hamiltonians and do not affect the relative comparison of MBPT and SCGF. Consequently, if not stated otherwise, we neglect 4N (and higher-body) contributions in Hamiltonian and focus on the improvement of subleading 3N forces.
Normal-ordering with respect to a reference state is a well-known method to include 3N contributions in terms of density-dependent effective NN forces, which can then be directly included in NN frameworks. Usually, the remaining residual 3N Hamiltonian leads to small contributions in pure neutron matter and is thus neglected (see, e.g., Ref. [@Hage14ccnm]). Following Refs. [@Hebe10nmatt; @Holt10ddnn] we obtain the effective NN interaction $\overline{V}_\text{3N}^\text{as}$ by summing one particle over the occupied states of the reference state, i.e., $$\label{eq:Veff_formal}
\overline{V}_\text{3N}^\text{as} = \text{Tr}_{\sigma_3} \int \frac{d{\mathbf{k}}_3}{(2\pi)^3} \mathcal{A}_{123} V_\text{3N} \, n_{{\mathbf{k}}_3} \bigg|_{\text{nnn}} \, ,$$ with the momentum-distribution function $n_{\mathbf{k}}$ and $\mathcal{A}_{123}$ is the antisymmetrizer. At zero temperature it is common to approximate the distribution function by the free Fermi gas function $n_{\mathbf{k}} = \Theta \left( k_{\rm F} - |{\mathbf{k}}| \right)$, with Fermi momentum $k_{\rm F}$. It was demonstrated that the inclusion of correlations in the reference state leads to small effects in observables [@Carb14SCGFdd]. In this article, we also discuss the extension of the normal-ordering framework to finite temperatures.
The 3N interactions $V_\text{3N}$ are regularized using non-local regulators of the form $f_\text{R}(p,q)=\exp [-((p^2+ 3 q^2/4)/
\Lambda_{\text{3N}}^2)^{4}]$ with respect to the Jacobi momenta $p,q$. In the literature, Eq. has been first evaluated directly based on the operatorial form of the 3N forces at N$^2$LO [@Holt10ddnn; @Hebe10nmatt; @Hebe11fits]. Since this procedure becomes rather involved for subleading 3N forces, so far only leading 3N interactions could be considered in this approach. One way to solve this is to make use of the recently developed partial-wave decomposition of the 3N interactions [@Hebe15N3LOpw] and evaluate Eq. in a partial-wave momentum basis of the form $$\Ket{pq\alpha} \equiv \Ket{pq; \left[(LS)J \left(l\frac{1}{2}\right)j \right] \mathcal{J} \left(T\frac{1}{2}\right)\mathcal{T}}\, .$$ The quantum numbers $L$, $S$, $J$, and $T=1$ (for neutron matter) denote the relative orbital angular momentum, spin, total angular momentum, and isospin of particles $1$ and $2$ with relative momentum $p$. The quantum numbers $l$ and $j$, respectively, are the orbital angular momentum and total angular momentum of particle $3$ relative to the center of mass of the pair with relative momentum $p$. The quantum numbers $\mathcal{J}$ and $\mathcal{T}=3/2$ define the total 3N angular momentum and isospin. The 3N matrix elements are provided by Ref. [@Hebe15N3LOpw] with total three- and two-body quantum numbers $\mathcal{J} \leq 9/2$ and $J \leq 6$, respectively. The size of this model space is sufficient to ensure convergence for calculations of nuclear matter in the Hartree-Fock approximation [@Hebe15N3LOpw; @Dris16asym] (see also Sec. \[subsec:NO\_finT\]). The resulting effective NN interaction is then added to the NN interactions: $$V_\text{NN+3N}^{\text{as}} = V_\text{NN}^{\text{as}} + \zeta \overline{V}_\text{3N}^{\text{as}} \, .$$ We refer to Ref. [@Hebe10nmatt; @Carb14SCGFdd; @Dris16asym] for detailed discussions on the combinatorial normal-ordering factor $\zeta$. We also note that the summation in Eq. results in a dependence of $\overline{V}_\text{3N}^\text{as}$ on the total momentum ${\mathbf{P}}$ of the two particles, which is not the case for free-space NN forces due to Galilean invariance. This additional momentum makes the effective NN potential computationally involved. Commonly, the approximation ${\mathbf{P}}=0$ is applied, e.g., in Ref. [@Holt10ddnn; @Hebe10nmatt; @Carb13nm]. In Ref. [@Dris16asym], an additional approximation that averages over all directions of ${\mathbf{P}}$ opposed to ${\mathbf{P}}=0$ is studied. It is shown that the resulting 3N Hartree-Fock energies are in reasonable agreement in particular below saturation density. Since the dependence on ${\mathbf{P}}$ is currently not implemented in the SCGF code and since we focus on the benchmark of MBPT to this nonperturbative method we focus here on the ${\mathbf{P}}=0$ approximation for $\overline{V}_\text{3N}^{\text{as}}$. Finally, we note that once reasonable fit values for $c_D$ and $c_E$ are available at N$^3$LO, the described methods can be directly applied beyond neutron matter.
\[subsec:many\_body\_app\]Many-body frameworks
----------------------------------------------
We calculate the energy per neutron at zero temperature up to third order in MBPT. The following notation is used to distinguish interaction energies and total energies at a given order in perturbation theory:
$$\begin{aligned}
\frac{E^{ \text{(HF)} }}{N} &= \frac{T}{N} +\frac{E^{(1)}}{N} \, ,\\
\frac{E_{\text{tot}}^{(2)}}{N} &= \frac{E^{(\text{HF})}}{N} +\frac{E^{(2)}}{N}\, ,\\
\frac{E_{\text{tot}}^{(3)}}{N} &= \frac{E_{\text{tot}}^{(2)}}{N} +\frac{E^{(3)}}{N} \, .\end{aligned}$$
Particle-hole contributions are neglected at third-order similarly to Refs. [@Hebe10PRL; @Krue13N3LOlong; @Dris16asym]. In order to estimate the uncertainties due to neglected higher-order contributions we perform calculations with a free and a Hartree-Fock single-particle spectrum. We refer to Ref. [@Hebe10nmatt; @Dris16asym] for details of the calculation. We assess the many-body convergence order-by-order by comparing to SCGF. In the SCGF method, the energy per neutron is calculated nonperturbatively via knowledge of a dressed one-body Green’s function [@Dick04PPNP]. The energy is obtained in the so-called ladder approximation, where an infinite sum of particle-particle and hole-hole diagrams is performed [@Rios08hotscgf; @Soma093ntherm]. Similar to the MBPT calculations, particle-hole contributions are neglected. The SCGF approach has been recently extended to self-consistently include 3N forces [@Carb13SCGF3B]. In this extension, the ladder resummation and the self-energy are redefined incorporating normal-ordered 3N terms with respect to a dressed reference state. Residual 3N contributions are also neglected in this approach. In this extended approach, the modified sum rule to obtain the total energy per particle in neutron matter reads [@Carb13SCGF3B]: $$\label{eq:gmk}
\frac{E}{N}= \frac{2}{n}\int\frac{{\rm d}{\bf k}}{(2\pi)^3}\int\frac{{\rm d}\omega}{2\pi}\frac{1}{2}\left\{\frac{k^2}{2m}+\omega\right\}A(k,\omega)f(\omega) - \frac{\langle W\rangle}{2} \,,$$ where $n$ the total density of the system and $f(\omega)$ corresponds to the Fermi-Dirac distribution function. $A(k,\omega)$ is the spectral function; this quantity gives the probability of adding or removing a particle with momentum $k$ which causes an excitation in energy ${\rm d}\omega$ in the many-body system. $\langle W\rangle$ is the expectation value of the 3N operator (see Ref. [@Carb14SCGFdd] for details). Throughout the paper we will refer to Eq. as $E_{\rm SCGF}/N$. The present implementation of SCGF is not capable of treating the appearance of pairing below a critical temperature, for this reason calculations are always performed at finite $T$. The pairing instability does not affect the MBPT calculations because the energy diagrams are evaluated directly, for which the pairing singularity is integrable. The zero-temperature results in SCGF are extrapolated using the Sommerfeld expansion [@Rios08hotscgf]. In this expansion, the energy can be written as a quadratic expansion in terms of $T/\varepsilon_{\rm F}$, where $\varepsilon_{\rm F}$ is the Fermi energy, as long as $T/\varepsilon_{\rm F}\ll 1$. A more sophisticated computational method to numerically extrapolate self-energies, spectral functions and thermodynamical properties from finite to zero temperature has been recently presented in Ref. [@Ding16scgfpair].
In order to extend the effective NN interaction $\overline{V}_\text{3N}^\text{as}$ to finite temperatures, we extend the framework presented in Ref. [@Dris16asym] and evaluate Eq. at finite temperature using the general Fermi-Dirac distribution function, $n_{\mathbf{k}} = \left[ \exp(
\beta(\varepsilon_{\mathbf{k}}-\mu) ) + 1 \right]^{-1}$. Given a total density $n$, we compute the chemical potential $\mu(n)$ by solving the non-linear density relation $$n = \frac{1}{\pi^2} \int \limits_0^\infty dk \, k^2 n_{\mathbf{k}}(\mu) \, .$$ We consider here the free single-particle energy, i.e., . Higher-order corrections to the self energy include contributions from the effective NN potential itself and would require thus an involved self-consistent solution for the spectrum. It has been shown in Ref. [@Carb14SCGFdd] that the energy per particle in pure neutron matter shows only at higher densities a dependence on the momentum distribution used in Eq. . Such high densities are not considered in this work, but it will be important to check this approximation at high temperatures.
\[sec:results\]Results
======================
Comparison of MBPT and SCGF
---------------------------
We show in Fig. \[fig:EOS\] the energy per particle as a function of density in neutron matter at zero temperature. From left to right, the first row shows the results for the N$^3$LO NN potentials EM 500 MeV [@Ente03EMN3LO], EGM 450/500 MeV and EGM 450/700 MeV [@Epel05EGMN3LO] with leading N$^2$LO 3N forces. The momentum scales attached to the potentials correspond to different regulator cutoffs: first, the cutoff in the Lippmann-Schwinger equation and second, if not dimensionally regularized, the cutoff in the two-pion-exchange spectral-function regularization. Analogously, the second row shows the results for the same NN potentials but including 3N forces up to N$^3$LO. We consider two sources of uncertainties: from the chiral Hamiltonian and from considering only a finite order in MBPT. As stated in Fig. \[fig:EOS\], the theoretical uncertainties due to the Hamiltonian are estimated by parameter variation in the 3N forces, i.e., the cutoff $\Lambda_\text{3N}$ and the low-energy constants $c_1$ and $c_3$. The $c_i$ values need to be refit at each chiral order, however, to investigate the net effect of N$^3$LO forces, we take here solely the $c_i$-range recommended for N$^3$LO calculations [@Kreb123Nlong]. In addition to the uncertainties in the Hamiltonian, we estimate the neglected higher-order contributions in the many-body expansion by varying the single-particle energies at third order using a free and a Hartree-Fock spectrum. These bands are colored in dark blue in Fig. \[fig:EOS\]. Moreover, following Ref. [@Krue13N3LOlong] we include the results at second order in MBPT using a Hartree-Fock spectrum to the uncertainty estimate. This extension of the pure third-order equation of state is indicated by light-blue bands. In summary, for a given Hamiltonian we perform in total three calculations in MBPT: two third-order calculations using the two single-particle spectra and a second-order calculation using a Hartree-Fock spectrum. Light- and dark-blue bands together characterize the total uncertainty estimate of MBPT in each panel. The actual energy range of MBPT is given in each panel of Fig. \[fig:EOS\] at saturation density $n_0$ (dashed vertical line), with .
Let us focus on the results with leading 3N forces, as shown in the first row of Fig. \[fig:EOS\]. The qualitative description does not change for the calculations with subleading 3N forces (second row in Fig. \[fig:EOS\]). Whereas the results for the two EGM potentials are almost independent of the many-body details, the effects of the variation of spectra and many-body order in MBPT are much more pronounced for EM 500 MeV: at saturation density the many-body uncertainties provide contributions of about $\sim -2.5$ MeV for this Hamiltonian (see light-blue band in Fig. \[fig:EOS\]). Including subleading 3N forces leads basically only to an overall shift of the bands as shown by the given energy range at saturation density. More specifically, the net 3N contribution leads to more attraction for the EGM potentials while the effect on EM 500 MeV is slightly repulsive.
To quantify the many-body convergence in more detail we compare to the results obtained in the SCGF method which are given by the region between the red-dashed lines in Fig. \[fig:EOS\]. The results in SCGF are considered to be converged in the many-body expansion (at the ladder level) and thus include only the uncertainty due to the Hamiltonian (including variations of the low-energy constants $c_1,
c_3$). We focus again on the different NN potentials rather than on discussing the effect of subleading 3N forces. Considering the total uncertainty estimate of MBPT we find for the potentials EM 500 MeV and EGM 450/700 MeV completely overlapping bands and similar trends in density. In the case of EM 500 MeV the extended uncertainty (light-blue band) is however needed to obtain more attraction and consequently fully overlapping bands, whereas for EGM 450/700 MeV the pure third-order energy is already in remarkable agreement. In addition to the above discussion on the size of the light-blue bands this suggests that contributions beyond third-order are small for EGM 450/700 MeV and become significant for EM 500 MeV.
For EGM 450/500 MeV we observe a slightly different density dependence between the MBPT and the SCGF curves, leading to an almost total overlap at saturation density but less agreement in the region around $n \sim 0.1$ fm$^{-3}$. Here, the equation of state in SCGF is slightly more repulsive. We recall that the SCGF results are extrapolated down to zero temperature from calculations performed at $T=2$ MeV for $n \leqslant 0.05$ fm$^{-3}$ and at $T=5$ MeV for densities above. We have tested whether this discrepancy is related to the extrapolation to zero-temperature lowering the temperature down to $T=3,4$ MeV in densities between 0.05 and 0.10 fm$^{-3}$, and have found no dependency on the extrapolation.
Combining the discussions on the size of the additional many-body uncertainty and the comparison of MBPT vs. SCGF we conclude from Fig. \[fig:EOS\] that the perturbativeness improves from EM 500 MeV to EGM 450/500 MeV to EGM 450/700 MeV. It is remarkable that a third-order MBPT calculation compares so well with the nonperturbative case for these chiral NN potentials. We study the many-body convergence as well as the effect of subleading 3N forces in more details in the next section.
Many-body convergence
---------------------
In Fig. \[fig:MBConv\] we address again the many-body convergence and show order-by-order in MBPT the total energy per neutron at $n_0$ (first row) and $n_0/2$ (second row), analogously to Fig. \[fig:EOS\]. More specifically, we show the total energy in Hartree-Fock approximation $E_\text{tot}^{\text{(HF)}}/N$ (“HF"), second order (“2nd") and third order (“3rd"), $E_\text{tot}^{(2)}/N$ and $E_\text{tot}^{(3)}/N$ respectively, in comparison to the results obtained in the SCGF method, $E_{\rm SCGF}/N$ (“SCGF"). The uncertainties are obtained as in Fig. \[fig:EOS\] through variations of the 3N parameters and the single-particle energies. However, to study the many-body convergence the third-order bands do not include here the additional many-body uncertainty (the light-blue bands of Fig. \[fig:EOS\]). The blue (red) data points correspond to N$^2$LO (N$^3$LO) 3N forces.
For all six panels in Fig. \[fig:MBConv\] we observe similar overall patterns: comparing order-by-order to the SCGF method we observe that the second order adds always too much attraction which then is compensated by the third-order repulsion. However, the specific behavior is different for EM 500 MeV and the two EGM potentials. In the case of EM 500 MeV the large third order overcompensates the second-order repulsion. In contrast, the third-order contribution is much smaller and less repulsive for the EGM potentials as can be seen in Fig. \[fig:MBConv\] (second and third column). In particular, this is pronounced in the calculations based on EGM 450/700 MeV, which agree remarkably well with the SCGF result.
As already discussed in the description of Fig. \[fig:EOS\], including N$^3$LO 3N forces has only a small repulsive effect on the energies based on EM 500 MeV, whereas the effect on the EGM potentials is larger but attractive. This behavior can be traced back to NN-3N mixing terms that enter the calculation when including 3N forces beyond the HF level. We also note that the values of the low-energy constants $C_S$ and $C_T$, which enter N$^3$LO 3N contributions, differ for all three potentials. However, the many-body convergence is not altered by including contributions from subleading 3N interactions.
Comparison to previous calculations at N$^3$LO
----------------------------------------------
The authors of Refs. [@Tews13N3LO; @Krue13N3LOlong] performed the first consistent calculations at N$^3$LO including NN, 3N and 4N forces in MBPT. In the cited works N$^3$LO NN and N$^2$LO 3N forces have been considered up to third order in MBPT in terms of effective NN potentials [@Hebe10nmatt], whereas subleading 3N interactions could only be included in the Hartree-Fock approximation since no 3NF partial-wave matrix elements were available at that time. Thanks to the advances discussed in this paper we are now in the position to revisit and systematically improve these calculations.
In Fig. \[fig:fullN3LO\] we show our improved results for the energy of neutron matter (blue bands) for the three Hamiltonians EM 500 MeV, EGM 450/500 MeV and EGM 450/700 MeV (first row) and the total band merged from the previous panels (second row). The uncertainty bands cover again variations of the 3N parameters (as given in the figure), the single-particle spectrum and the additional many-body uncertainty (see also discussion of Fig. \[fig:EOS\]). We furthermore include the 4N Hartree-Fock results, as given in Ref. [@Krue13N3LOlong], and vary the 4N cutoff analogously to the 3N forces. In addition, we show the results of Ref. [@Krue13N3LOlong][^1] depicted by the black solid lines. For a better view we do not fill this region. We give in each panel the energy range at saturation density obtained within the improved calculations presented in this work.
We observe that the effect of adding the N$^3$LO 3N contributions beyond Hartree-Fock varies significantly between the EM 500 MeV and the two EGM potentials. For EM 500 MeV these contributions leave the uncertainty band almost unaffected. For the two EGM potentials the upper uncertainty limits remains the same while the lower increase by $\sim1$ MeV ($\sim0.2$ MeV) for EGM 450/500 MeV (EGM 450/700 MeV), hence decreasing the width of the uncertainty band. These findings are consistent with the observations in Ref. [@Krue13N3LOlong], which stated that the N$^3$LO 3N Hartree-Fock energy is smaller for EM 500 MeV while it is much larger for the two EGM potentials (see Fig. 6 of Ref. [@Krue13N3LOlong]). We emphasize, however, that NN and effective NN forces get mixed at second-order and beyond, and therefore the net effect of these subleading 3N contributions cannot be easily disentangled in the many-body calculation. Combining all bands we find a total uncertainty of $\frac{E}{N} (n_0) = (14.7 -
21.1)~\text{MeV}$ in neutron matter at saturation density. Compared to the corrected total band of Ref. [@Krue13N3LOlong] $\frac{E}{N}
(n_0) = (14.3 - 21.1)~\text{MeV}$, we obtain a slight reduction of the lower limit of the uncertainty band. As suggested in Refs. [@Tews13N3LO; @Krue13N3LOlong], these effects are indeed rather small. However, we expect the effects to be much more important as soon as the proton fraction is finite (see also discussion of symmetric nuclear matter in Ref. [@Krue13N3LOlong]).
\[subsec:NO\_finT\]Normal-ordering at finite temperatures
---------------------------------------------------------
 Comparison of the leading 3N Hartree-Fock energies at saturation density for several temperatures obtained using the effective NN potential in terms of 3N operators (blue) and the partial-wave approach (red). We include 3N matrix elements up to $\mathcal{J}
\leqslant 9/2$ and $J \leqslant 6$. For the uncertainty estimate we use the same parameter variation in the 3N forces as in Fig. \[fig:EOS\].](finiteT_HF.pdf)
We have extended the recently-developed framework for computing effective NN potentials in a partial-wave basis [@Dris16asym] to finite temperatures. Besides being a necessary step in order to include these matrix elements in the SCGF method (due to the extrapolation from finite temperatures), this is also a crucial step for future MBPT calculations of nuclear matter at finite temperatures. In Fig. \[fig:finiteT\_HF\] we show the resulting N$^2$LO 3N Hartree-Fock energies $E^{(1)}/N(n_0,T)$ at six different temperatures in the range of $T=(0-50)$ MeV. We benchmark our new values (red) against previous results (blue) obtained via an operatorial approach [@Carb14SCGFdd]. The uncertainty bands are obtained through 3N parameter variation analogously to Figs. \[fig:EOS\] and \[fig:MBConv\]. The single-particle spectrum does not contribute to the uncertainties since the Fermi-Dirac distribution in Eq. is computed using a free spectrum. A similar benchmark at N$^3$LO is not possible since no matrix elements are currently available based on the operatorial evaluation of 3N forces at N$^3$LO. We note that the 3N interaction energy decreases with temperature as shown in Fig. \[fig:finiteT\_HF\]. Including also kinetic energy contributions would lead to a total increase in energy with increasing temperature. From Fig. \[fig:finiteT\_HF\] we can conclude that the two different methods for the normal-ordering agree very well at zero and finite temperature up to $T=50$ MeV.
 Momentum-space diagonal matrix elements of the density-dependent effective NN potentials at N$^2$LO for a selection of four partial-wave channels and two temperatures.](ME_ddNN_panel.pdf)
In addition to the 3N Hartree-Fock energies, we also benchmark the underlying interaction matrix elements of the effective potential $\overline{V}_\text{3N}^\text{as}$. The results for a selection of four partial-wave channels and two temperatures are shown in Fig. \[fig:Veff\_T\]. These matrix elements contribute to the energy presented in Fig. \[fig:finiteT\_HF\]. The ones obtained in the partial-wave (operatorial) approach are plotted as dashed (solid) lines. We select a representative set of channels, ${}^1\text{S}_0$, ${}^3\text{P}_0$, ${}^3\text{P}_1$, and ${}^3\text{P}_2$, and temperatures $T=10,50$ MeV. We have also compared higher partial waves up to $J=6$ and momentum off-diagonal matrix elements for $\Lambda_\text{3N}=(2.0-2.5)$ fm${}^{-1}$. As in Ref. [@Dris16asym], we find indications of an incomplete partial-wave convergence only for partial-waves channels with $J>4$, We also checked that the agreement can be systematically improved by increasing the 3N model space, i.e. by including channels with $\mathcal J =11/2$ and $13/2$. We found that contributions from these higher partial-wave channels provide $\lesssim 50$ keV to the energy of neutron matter per particle at saturation density. Overall, we find excellent agreement of the two methods at the level of matrix elements and at finite-temperatures. This shows that the computed matrix elements of the effective interactions at finite temperature at N$^2$LO and N$^3$LO are correct and numerically stable and are hence suitable for future calculations of nuclear matter for astrophysical applications [@Hebe13ApJ].
\[sec:outlook\]Summary and Outlook
==================================
In this work we have calculated the zero-temperature equation of state of neutron matter in the framework of MBPT and SCGF based on chiral NN and 3N forces up to N$^3$LO. In addition, we included contributions from 4N interactions at N$^3$LO in the Hartree-Fock approximation. For the inclusion of 3N interactions we have utilized our generalized normal-ordering framework first presented in Ref. [@Dris16asym]. We demonstrated that this framework is able to treat general 3N interactions that are provided in a partial-wave representation and can be extended to finite temperatures.
We have systematically improved previous calculations of neutron matter in MBPT at N$^3$LO [@Tews13N3LO; @Krue13N3LOlong] by including subleading 3N contributions beyond the Hartree-Fock approximation. Specifically, we have obtained the neutron-matter energy based on three different NN plus 3N interactions derived within chiral EFT, comparing calculations including only leading to up-to-subleading 3N forces. For the N$^3$LO NN potentials EGM 450/500 MeV and EGM 450/700 MeV we found additional attractive subleading 3N contributions of about $\sim$2 MeV for the energy per particle at saturation density, while for the EM 500 MeV potential these contributions are smaller in size and repulsive, of the order of $\sim$500 keV. In order to assess the many-body convergence we have benchmarked our MBPT results for three commonly-used N$^3$LO NN potentials plus leading and also subleading 3N forces against results obtained within the SCGF framework. Since the current implementation of SCGF does not account for Cooper pairing, the zero-temperature limit was obtained by extrapolation. We found a systematic convergence of the MBPT results to the SCGF results at third order in MBPT, whereas the detailed convergence pattern depends on details of the NN and 3N interactions.
Finally, we have successfully benchmarked results for the effective NN potential at finite temperature. At order N$^2$LO in the chiral expansion we obtain excellent agreement between results obtained using our novel normal- ordering framework and previous results for 3N Hartree-Fock energy contributions as well as on the level of partial-wave matrix elements. These benchmarks demonstrate that we are now in the position to perform calculations of general isospin-asymmetric matter including all NN and 3N contributions up to N$^3$LO at zero and finite temperatures. Since all 3N topologies contribute for these systems, reliable fits of the 3N low-energy constants $c_D$ and $c_E$ are required. This is currently work in progress. The availability of different sets of Hamiltonians using different regulator choices (see also Refs. [@Tews15QMCPNM; @Dyhd16Regs]) and different fitting strategies (see, e.g., Refs. [@Ekst15sat; @Carl16sim]) will make it possible to probe systematically the order-by-order convergence in the chiral expansion. In turn, this will advance our understanding of the dense matter equation of state.
We thank T. Kr[ü]{}ger, A. Rios, A. Polls and I. Tews for useful discussions. This work was supported by the ERC Grant No. 307986 STRONGINT, the Deutsche Forschungsgemeinschaft through Grant SFB 1245. A.C. acknowledges support by the Alexander von Humboldt Foundation through a Humboldt Research Fellowship for Postdoctoral Researchers.
[^1]: For completeness, we have corrected a small error in the routines of Ref. [@Krue13N3LOlong] for the computation of the second- and third- order contribution of the N$^3$LO NN plus N$^2$LO 3N forces as well as the N$^3$LO 3N Hartree-Fock energy corresponding to the ring topology. Moreover, we are using the typo-corrected values for $\bar{\beta}_{8,9}$ (see Ref. [@Epel15improved] for details).
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{
"pile_set_name": "ArXiv"
}
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abstract: 'At radio frequencies, the current evidence for the microquasar–quasar connection is based on imaging observations showing that relativistic outflows/jets are found in both classes of objects. Some microquasars also display superluminal motion, further strengthening the view that microquasars are in fact Galactic miniatures of quasars. Here we demonstrate that this connection can be extended to incorporate timing and spectral observations. Our argument is based on the striking similarity found in the radio flaring behaviour of the Galactic superluminal source GRO J1655$-$40 and of extragalactic sources, such as the blazar 3C 273. We find that the variability of GRO J1655$-$40 can be explained within the framework of the successful generalised shock model for compact extragalactic radio sources in which the radio emission arises from shocked plasma in relativistic jets. Specifically, the multifrequency flare amplitudes, time delays and radio polarization position angle measurements are consistent with the predictions of the growth stage of this model.'
author:
- 'J. A. Stevens$^{1,2}$, D. C. Hannikainen$^{3,4}$, Kinwah Wu$^{2,5}$, R. W. Hunstead$^{5}$ and'
- |
D. J. McKay$^{6,7}$\
$^{1}$Astronomy Technology Centre, Royal Observatory, Blackford Hill, Edinburgh, EH9 3HJ\
$^{2}$Mullard Space Science Laboratory, University College London, Holmbury St. Mary, Surrey, RH5 6NT\
$^{3}$Department of Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ\
$^{4}$Observatory, PO Box 14, 00014 University of Helsinki, Finland\
$^{5}$School of Physics A28, University of Sydney, NSW 2006, Australia\
$^{6}$Telescope Technologies Ltd, 1 Morpeth Wharf, Birkenhead, Merseyside CH41 1NQ\
$^{7}$Australia Telescope National Facility, Locked Bag 194, Narrabri, NSW 2390, Australia
date: 'Received: '
---
X-rays: binaries — galaxies: jets — quasars: general — black hole physics — radiation mechanism: general — shock waves
Introduction
============
Microquasars, here defined as X-ray binaries containing a black hole accreting material from a companion star, are generally regarded as the Galactic counterparts of more powerful extragalactic radio sources, such as blazars and radio galaxies. [^1] This implies that the underlying physical processes that operate in these systems are the same, and thus are manifested in their radiative properties. At radio frequencies, the main argument for microquasars being Galactic miniatures of quasars is based on imaging observations of relativistic outflows in the former. This is further strengthened by the fact that some microquasars also display superluminal motion analogous to their extragalactic counterparts. For recent reviews of the radio properties of microquasars, see e.g. Mirabel & Rodriguez (1998, 1999) and Fender (2001, 2002).
The two unambiguous Galactic superluminal sources to date are GRS 1915+105 and GRO J1655$-$40. Their radio jets were discovered in the mid-1990s (Hunstead et al. 1994; Reynolds et al. 1994; Mirabel & Rodr[í]{}guez 1994; Tingay et al. 1995; Hjellming & Rupen 1995). They often show radio flares during the X-ray active states, and their radio emission is polarized. The nature of GRS 1915+105 has been controversial but recent observations show that it is likely to be a low mass X-ray binary with a black-hole candidate and K-giant donor star (Greiner et al. 2001). It is certain that GRO J1655$-$40 is a binary system containing an evolved F-type star and a black hole (Orosz & Bailyn 1997; Soria et al. 1998).
Compact extragalactic radio sources usually show power law spectra at high-frequencies ($> 90$ GHz). The radio emission is of synchrotron origin, is often highly polarized, and the power law spectrum implies a non-thermal energy distribution for the relativistic electrons and an optically thin emission region. The spectra often have a turn-over at low frequencies that is generally explained as the consequence of optical-depth effects; the emission is absorbed and the source becomes opaque below a certain critical frequency. The time evolution of the low-frequency (typically less than a few GHz) radio spectra was initially explained by means of a homogeneous cloud composed of relativistic electrons and magnetic field that expands adiabatically and emits synchrotron radiation (Shklovsky 1965; van der Laan 1966; Pauliny-Toth & Kellerman 1966). However, it was later found that a more satisfactory explanation is provided by models invoking shocks propagating along a relativistic jet, and hence accelerating electrons to relativistic energies (e.g. Marscher & Gear 1985; Hughes, Aller & Aller 1985, 1989a, b).
Like extragalactic radio sources, the Galactic superluminal source GRO J1655$-$40 also shows a power law spectrum at radio frequencies that becomes inverted during flaring (Hannikainen et al. 2000). To date the time-dependent radio properties of X-ray binaries have mostly been ascribed to the expanding-cloud (or synchrotron-bubble) model or variations thereof (Hjellming & Johnston 1988; Han & Hjellming 1992; Ball & Vlassis 1993) although an internal shock model has been applied successfully to a radio outburst of GRS 1915+105 (Kaiser, Sunyaev & Spruit 2000). Since the 1994 radio outburst of GRO J1655$-$40 is not readily accommodated by the expanding-cloud model (see Hannikainen et al. 2000 and Section 2 below) we propose alternatively that the processes that generate the radio emission from GRO J1655$-$40 and from extragalactic radio sources are the same. We make use of the shock model of Marscher & Gear (1985) to explain the spectral evolution of GRO J1655$-$40 during the 1994 outburst. The multifrequency variability is discussed in Section 2, basic features of the model are described in Section 3, and the quasar-microquasar connection is discussed in Section 4.
Throughout this paper, spectral index, $\alpha$, is defined as $S_{\rm \nu} \propto \nu^{\alpha}$ where $S_{\rm \nu}$ is the flux density at frequency $\nu$.
Time-dependent properties of GRO J1655$-$40
===========================================
GRO J1655$-$40 flared dramatically in 1994 August, the flux density at 843 MHz increasing from a few hundred mJy to almost 8 Jy in 12 d (Hannikainen et al. 2000). Very Long Baseline Array (VLBA) images at 1.6 GHz taken at several epochs during the outburst show well-collimated relativistic jets. The jets are resolved into several knot-like features which expand outwards from a stationary compact core (Hjellming & Rupen 1995).
The multifrequency flux density light curves and spectra (from Hannikainen et al. 2000) are shown in Figs \[fig:lc\] and \[fig:spec\] respectively. The flux densities include the emission from the entire structure shown in the VLBA images, with the possible exception of the two highest frequencies at which the Australia Telescope Compact Array (ATCA) synthesized beam size is approximately 1 arcsec compared to a source size of 1–2 arcsec at 1.6 GHz. Our flux density measurements involve fitting a point-source model to the data which may lead to a slight systematic underestimate of the flux at 8.6 and 9.2 GHz. However, the source may well be more compact at higher frequencies due to the shorter synchrotron lifetimes of the emitting electrons (see Section 4), so it is unlikely that much/any flux density is missing from our measurements. Indeed, in Section 4, we argue that most of the flux density variability is confined to a region $\ll 1$ arcsec in extent. Since the available observations do not allow separation of emission from the compact variable component(s) and the extended regions we are forced to assume that the radio emission is dominated by a single variable component. The light curves show complex flaring behaviour with several minor events following the initial increase in flux density. The key features of the data are that the initial rise in flux density [*peaks simultaneously*]{} at all frequencies and that the amplitude of the flare, defined as maximum minus minimum flux density, [*increases*]{} towards lower frequency (see Section 4 for further discussion). We quantified the time delays between the data streams with the Discrete Correlation Function (DCF; Edelson & Krolik 1988) and the Interpolated Correlation Function (ICF; Gaskell & Peterson 1987). Both methods give essentially the same result (Fig. \[fig:delay\]); the correlation functions are approximately symmetrical around zero lag in all cases. The flat peak to the curves suggest formal time delays of $\sim 0\pm1$ d.
(3.25,5.0)
(3.25,6.0)
(3.25,3.6)
The first panel of Fig. \[fig:spec\] shows a spectrum from two days before flux density maximum, which is consistent with synchrotron emission that is optically thin at the highest frequencies but is partially self-absorbed at around 1 GHz. As the flux density increases, the entire spectrum becomes optically thin and remains so throughout the subsequent decline in flux density. In Fig. \[fig:spec\] we can see that the spectrum was relatively flat (with a spectral index of $-0.41$ on TJD 9580). It steepened at the time of the maximum (with the spectral indices reaching $-0.65$ on TJD 9582) and then flattened as the flux density declined to a level of $-0.4$ around TJD 9586 and 9587. This behaviour can be explained by models incorporating emission from multiple regions. In the simple core-lobe model suggested by Hannikainen et al. (2000), the observed spectrum is the sum of the emission from a compact region (core), which has a flat spectrum, and the emission from extended regions (lobes/ejecta) with steeper spectra. The outburst can be seen as a consequence of the brightening of the compact region and then the rapid rise in the brightness of the ejecta. The emission from the ejecta eventually dominates as the radio flux densities reach their peak amplitude. At the same time the spectrum steepens. As the flare subsides, the emission from the ejecta drops to a level comparable to, or lower than, the emission from the compact region, and the spectra become flatter, similar to those seen at the onset of the flare.
The time-dependent properties of the radio outbursts of GRO J1655$-$40 and other black-hole X-ray binaries have been attributed to synchrotron emission from expanding clouds of relativistic electrons. Under the usual assumption that all radio frequencies are initially optically thick, the two major characteristics of these models are that the low-frequency emission has a time delay with respect to that at high frequencies and that the flare amplitudes fall off towards lower frequency (see e.g. Han & Hjellming 1992). The observations clearly do not conform to these predictions of the expanding-cloud model (see also Hannikainen et al. 2000).
If electrons are injected into an expanding cloud, and if expansion losses dominate over radiative losses, then it is possible to get a simultaneous peak at optically thin frequencies once the injection stops or once the expansion cooling dominates over the injection. Note that if radiative losses dominate under this scenario we would observe frequency-dependent peak times, assuming the data streams are sufficiently well sampled. However, the first panel of Fig. 3 provides evidence that the synchrotron self-absorption turnover was close to 1 GHz near the peak of the flare, implying that at least some of the radio frequencies were optically thick at earlier epochs, and thus arguing against this possibility. Similar scenarios can be produced by relaxing some of the initial assumptions of the model. For example, Ball & Vlassis (1993) considered cases where electrons are injected into the cloud with a constant energy spectrum. While both of these options can in principle reproduce the observed behaviour (optical depth arguments aside) they require a specific set of conditions that are unlikely to be met in practice. Indeed, Marscher & Gear (1985) found that they could fit the variability of 3C 273 with a uniform expanding source model but only if ‘the injection of relativistic electrons was allowed to vary with radius in a rather ad hoc fashion’. They attributed the success of this model to its large number of free parameters. Arguing along similar lines, we do not consider that such models provide a convincing physical explanation for the phenomena observed in GRO J1655$-$40. Instead, given the obvious analogy between Galactic and extragalactic jet sources, we argue alternatively that the variable radio emission arises in a small region behind a relativistic shock wave, and that the rise of the flare is driven by inverse-Compton cooling (cf. Marscher & Gear 1985). We note that the shock model developed by Kaiser et al. (2000) incorporates adiabatic and synchrotron cooling only and as such is mostly applicable to the decline of radio flares rather than to the rise phase discussed here.
Generalised shock model for extragalactic sources
=================================================
Spectral evolution during the flares of compact extragalactic radio sources can be explained by the generalised shock model of Marscher & Gear (1985). (For specific applications see e.g. Valtaoja et al. 1988 and Stevens et al. 1994, 1996, 1998). In the model, temporal evolution of the radio emission is divided into growth, plateau and decay stages (see the schematic illustration in Fig. \[fig:mg\]). The emission region consists of relativistic jets along which transverse shocks propagate and accelerate electrons to relativistic energies. The shock waves form in response to changes in the conditions in the jet (such as pressure or velocities of the bulk flow). The evolution of the shock is described in terms of the flux density ($S_{\nu}$) at the peak frequency ($\nu_{\rm m}$ where $S_{\nu}[\nu_{\rm m}]=S_{\rm m}$) of the synchrotron spectrum where the opacity is close to unity. This point is fixed at any one time by the dominant energy-loss mechanism. When the emitting region is compact, inverse-Compton losses predominate (Compton or growth stage) but these fall off rapidly with radius as the shock expands and are superseded by synchrotron losses (synchrotron or plateau stage). As the shock expands further, the radiative lifetime of the electrons becomes large with respect to the time needed to traverse the emitting region and losses due to adiabatic expansion (adiabatic or decay stage) become more important. All three stages are approximated by power laws on the logarithmic $(S_{\rm{m}},\nu_{\rm{m}}$) plane. As the emitting region expands, $\nu_{\rm m}$ is predicted to move to lower frequencies with time. $S_{\rm{m}}$ increases rapidly during the Compton stage, remains approximately constant during the synchrotron stage and decreases during the adiabatic stage.
The characteristics of spectral evolution predicted by the generalised shock model are summarised as follows (see Valtaoja et al. 1992 for more details). The maximum flare amplitude for any frequency on the Compton stage occurs when the spectrum transits onto the synchrotron stage. Light curves at such frequencies are thus predicted to peak simultaneously and because of the spectral shape, the flare amplitudes are expected to increase towards lower frequency – specifically, they should have the same power law form as the optically thin portion of the flare spectrum. Flare amplitudes at frequencies commensurate with the later stages are determined by the details of the spectral evolution, and are thus expected to be approximately constant during the synchrotron stage and to decrease during the adiabatic stage. The light curves will display time-lagged behaviour with a delay between any two frequencies equal to the time taken for the spectrum to evolve between them and become optically thin. Note that the development of the flare during the adiabatic stage is very similar to that predicted by the expanding cloud model, although, because the expansion is constrained to occur in a jet, the predicted time delays are shorter and the variation of flare amplitude with frequency is less pronounced.
(3.25,2.6)
Microquasars vs quasars
=======================
The generalised shock model prediction of simultaneously peaking emission at all frequencies coupled with flare amplitudes that increase towards lower frequencies (i.e. the Compton/growth stage) is consistent with the observations of GRO J1655$-$40 in 1994. According to the shock model, we would also expect the flare amplitudes to fall off as a power law since they should have the same frequency dependence as the spectrum of the flare when it reaches the transition point between the growth and plateau stages. In Fig. \[fig:amp\] we plot the flare amplitudes normalised to 4.8 GHz. A power law is clearly a good fit with a slope of $-0.70\pm0.05$, consistent with that expected for the optically thin portion of the flare synchrotron spectrum (cf. Stevens et al. 1996).
(3.25,2.0)
One apparent difference between the flaring behaviour of GRO J1655$-$40 and its extragalactic analogues, the blazars, is that the flare remains in the growth stage to much lower frequencies. For example, many of the sources in the sample considered by Stevens et al. (1994) show time delays between the 37 GHz, or occasionally the 90 GHz, emission and that at higher frequencies, although those authors noted that some BL Lacertae objects remained in the growth stage out to 4.8 GHz. An obvious contributor to this difference is the Doppler effect which will shift the emitted frequency of radiation in the jet frame, $\nu'$, to a higher observed frequency, $\nu = \delta\nu'$. The relativistic Doppler factor, $\delta=\Gamma^{-1}(1-\beta \ {\rm cos} \ \theta)^{-1}$ where $\Gamma=(1-\beta^2)^{-1/2}$ is the bulk Lorentz factor, $\theta$ is the viewing angle and $\beta$ is the jet speed in units of $c$. It is generally accepted that blazars have their jet axes oriented quite closely towards the observer; for example Teräsranta & Valtaoja (1994) find that blazars on average are aligned to within about 20 deg of the line-of-sight (see also Barthel et al. 1989). Estimates of Doppler factors for blazars range from extreme values of 0.005 to 33 with a mean value of around 5 (Guijosa & Daly 1996). The blazars with measured time delays from the sample of Stevens et al. (1994) have ‘inverse-Compton’ Doppler factors in the range 3.4–16 (see Guijosa & Daly 1996). The microquasar GRO J1655$-$40, however, is angled further towards the plane of the sky ($\theta=70$ deg; Orosz & Bailyn 1997) and its estimated jet speed $\beta=0.92$ (Hjellming & Rupen 1995) leads to $\delta\sim0.6$. Higher values of the Doppler factor for blazars compared to that of GRO J1655$-$40 could thus easily lead to an order of magnitude difference in the frequency at which the flare exits the growth stage.
A second contributory factor might arise from physical differences between the jets of Galactic and extragalactic sources. Since the flare evolution is dependent on the photon energy density, the magnetic energy density and the lifetime of the emitting electrons compared to the time they take to cross the shock structure, two important parameters are the size of the emitting region and its magnetic field strength.
We can estimate the magnetic field from the flare decay time by assuming that the decay is radiative rather than expansion-loss driven. For the clear-cut case of a single flare we would expect the 1/e decay times, $t$, of the light curves to vary as $\nu^{-1/2}$ if radiative losses are the dominant energy-loss mechanism, or if adiabatic expansion losses predominate, then $t$ will be the same at all frequencies. Unfortunately, the flaring behaviour that we observe is complex. At the monitoring frequencies presented in Fig. \[fig:lc\] the decay of the flares is interrupted by subsequent events, making it impossible to measure an accurate e-folding timescale. However, the multifrequency light curves presented by Hjellming & Rupen (1995) include data up to 22.5 GHz. For the flare in question, the decay of the 22.5 GHz lightcurve is more rapid than those at lower frequencies, most probably because the monitoring frequency is now sufficiently high that the e-fold timescale is similar to the interval between the bursts. This result provides evidence that synchrotron losses are driving the flux decay. If this is not the case then the implication would be that the synchrotron-loss stage is short lived or non-existent, as observed for 3C 273 (Stevens et al. 1998). The rest of this section assumes that the decay of the flare is driven by synchrotron losses. If this is not the case, and adiabatic expansion losses predominate then we will overestimate the magnetic field strength and underestimate the electron Lorentz factor and source extent.
For an e-fold decay timescale $t$ and a peak frequency $\nu_{\rm m}$, the magnetic field is given by $$B \sim 23\,\delta^{-1/3} \left ( \frac{\nu_{\rm m}}{1~{\rm GHz}} \right )^{-1/3}
\left ( \frac{t}{5~{\rm day}} \right )^{-2/3}~{\rm gauss} \ .$$ We estimated the flare decay time from the best sampled light curve (843 MHz). For the reasons discussed above this estimate is likely to be an upper limit but should not be out by more than a factor of two; using 5 d the estimated magnetic field strength is $\sim$25 gauss. This value is approximately two orders of magnitude higher than typical estimates of 0.1–1 Gauss in extragalactic sources (e.g. Brown et al. 1989).
The Lorentz factor of the electrons emitting at 843 MHz is given by $\gamma\sim(2 \pi m_{\rm e} c \nu/eB)^{1/2}\sim3.5$, where $e$ and $m_{\rm e}$ are the electron charge and mass respectively. This value is approximately that required to explain the observed circular polarization as being intrinsic to the synchrotron radiation (Macquart et al. 2002) although the alternative mechanism, Faraday conversion, could also apply. In any case, we note that the presence of circularly polarized radiation is another characteristic that GRO J1655$-$40 shares with the blazars (see e.g. Wardle et al. 1998). Furthermore, strong evidence for our proposed model comes from the observed position angle of linear polarization at radio wavelengths which is very close to that of the jet direction (Hannikainen et al. 2000). For optically thin synchrotron radiation the implied magnetic field direction is perpendicular to the jet, as observed for many blazars (e.g. Cawthorne et al. 1993a,b), and as predicted by models in which a tangled component of magnetic field in the jet is compressed parallel to a transverse shock front (e.g. Hughes et al. 1989a).
Finally, we can estimate the source extent at the time when the spectrum transits from the growth stage onto the plateau stage. At this transition, the energy densities in particles ($U_{\rm ph}$) and magnetic field are equal and thus so are the synchrotron and inverse-Compton lifetimes $t_{\rm syn}=t_{\rm
IC}$. Using $$t_{\rm IC} \sim {{3\times 10^7} \over {\gamma U_{\rm ph}}} \sim
{{3\times 10^7 cD^2} \over {\gamma \phi L}} \ {\rm seconds} \ ,$$ and assuming that, (1) the transition occurs when the 843 MHz light curve peaks (i.e. the flare amplitudes peak at this frequency), allowing us to use the values $t_{\rm IC}=t_{\rm syn}\sim 5$ d and $\gamma \sim 3.5$ deduced above, (2) the radiation field is produced by a central source of luminosity, $L\sim10^{38}$ ergs$^{-1}$ and (3) that $\phi$, a geometrical constant, is approximately unity, the calculated source size $D \sim 10^{13}$ cm. At a source distance of 3.2 kpc (Hjellming & Rupen 1995) this dimension translates to an angular size of about 0.2 mas. A 1.6 GHz VLBA map taken one day after the peak of the 843 MHz light curve shows a source of about 1 arcsec with a bright core and extended jets (Hjellming & Rupen 1995). Our proposed model requires that the radio flares originate in a small region close to the base of the jet as is observed in AGN.
Final remarks
=============
We have shown that a model incorporating a propagating relativistic shock in a jet can qualitatively reproduce the multifrequency variability of GRO J1655$-$40. This model, which also fits the available polarization observations, is the canonical model used to explain the variability characteristics of extragalactic radio sources such as blazars. A detailed discussion of the applicability of this model to Galactic superluminal sources is beyond the scope of the present work, and in any case is limited by the availability of suitable data sets. It is important that other flares similar to the one discussed here are observed over a broad range of radio frequencies, and that they are first observed on the rising portion of the light curves. In this respect, we point out that the two sources discussed by Ball & Vlassis (1993), Nova Muscae 1991 and the Galactic Centre Transient, for which data were taken during the rise of the flares, both display the behaviour predicted by the Compton stage of the shock model. Both sources were observed to have an optically thin spectrum during the rising phase of the outburst which can be accommodated by the Compton stage if the self-absorption turnover had already passed the lowest observing frequency when the flux monitoring began.
Acknowledgments {#acknowledgments .unnumbered}
===============
J.A.S. and D.C.H. acknowledge support from PPARC, and K.W. acknowledges support from the Australian Research Council through an Australian Research Fellowship. D.C.H also acknowledges travel support from the Academy of Finland, and thanks the University of Sydney and Jodrell Bank for their hospitality and financial support during her visits.
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[^1]: The blazar class comprises the brightest and most variable of the extragalactic radio sources. Accordingly, for the purposes of this study, most observational data for extragalactic sources that allow comparison with the microquasars pertain to blazars. However, the ‘unified scheme’ for radio sources postulates that, to first order, the only difference between radio galaxies, quasars and blazars is one of orientation (e.g. Barthel 1989) so the same physical model should be applicable to all.
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'Traditionally, there are several polynomial algorithms for linear programming including the ellipsoid method, the interior point method and other variants. Recently, Chubanov [@chubanov2015polynomial] proposed a projection and rescaling algorithm, which has become a potentially *practical* class of polynomial algorithms for linear feasibility problems and also for the general linear programming. However, the Chubanov-type algorithms usually perform much better on the infeasible instances than on the feasible instances in practice. To explain this phenomenon, we derive a new theoretical complexity bound for the infeasible instances based on the condition number, which shows that algorithms can indeed run much faster on infeasible instances in certain situations. In order to speed up the feasible instances, we propose a *Polynomial-time Primal-Dual Projection* algorithm (called ${\mathsf{PPDP}}$) by explicitly developing the dual algorithm. The numerical results validate that our ${\mathsf{PPDP}}$ algorithm achieves a quite balanced performance between feasible and infeasible instances, and its performance is remarkably better than previous algorithms.'
author:
- |
Zhize Li\
IIIS, Tsinghua University\
[email protected]
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Wei Zhang\
TLI, National University of Singapore\
[email protected]
- |
Kees Roos\
EEMCS, Delft University of Technology\
[email protected]
bibliography:
- 'bib.bib'
title: 'A Fast Polynomial-time Primal-Dual Projection Algorithm for Linear Programming'
---
Introduction {#sec:intro}
============
Linear programming is a fundamental problem in many areas, such as operations research, network, machine learning, business analysis and finance [@von1947theory; @dantzig1963linear; @luenberger1984linear; @boyd2004convex]. In this paper, we consider the *maximum support* of the linear feasibility problem $$\label{feasibility:primal}
\begin{aligned}
\mathrm{find} ~~&x \in {{\mathbb R}}^n \\
\mathrm{subject~to}~~ &Ax=0, ~x\geq 0,~ x\neq 0,
\end{aligned}$$ with its dual problem $$\label{feasibility:dual}
\begin{aligned}
\mathrm{find} ~~&u \in {{\mathbb R}}^m \\
\mathrm{subject~to}~~ &A^T u> 0,
\end{aligned}$$ where $A\in {{\mathbb R}}^{m\times n}$ is an integer (or rational) matrix and ${\mathrm{rank}}(A) = m$. The *maximum support* means that the set of positive coordinates of the returned solution of (\[feasibility:primal\]) should be inclusion-wise maximum. Actually, for the solution $\hat{x}$ returned by our algorithm, any coordinate $\hat{x}_i=0$ if and only if this coordinate equals to 0 for all feasible solutions of (\[feasibility:primal\]). Thus, our algorithm can be directly used to test the feasibility of the general linear system $Ax=b,x\geq 0$ with the same time complexity, i.e., given the maximum support solution $({\bar{x}},{\bar{x}}')$ to the system $Ax-bx'=0, (x,x')\geq 0$, if ${\bar{x}}'>0$ then the original problem $Ax=b,x\geq0$ has a solution ${\widetilde{x}}= {\bar{x}}/{\bar{x}}'$, otherwise it is infeasible.
There are many (polynomial-time) algorithms for solving linear programming problems, e.g., [@karmarkar1984new], [@wright1997primal] and [@renegar1988polynomial]. Recently, [@chubanov2015polynomial] proposed a polynomial-time projection and rescaling algorithm for solving problem . Due to its simplicity and efficiency, this kind of algorithms has become a potentially *practical* class of polynomial algorithms. See e.g., [@dadush2016rescaled], [@roos2018improved] and [@pena2018computational].
Chubanov’s algorithm [@chubanov2015polynomial] and its variants typically consist of two procedures. The key part is *basic procedure* (BP) and the other part is *main algorithm* (MA). The BP returns one of the following three results:
(i) a feasible solution of ;
(ii) a feasible solution of the dual problem ;
(iii) a cut for the feasible region of .
Note that exactly one of and is feasible according to Farkas’ lemma. Thus is infeasible if BP returns (ii). If BP returns (iii), the other procedure MA rescales the matrix $A$ by using this cut and call BP again on the rescaled matrix $A$. According to [@khachian1979polynomial] which gives a positive lower bound on the entries of a solution of a linear system, after a certain number of rescalings, one can conclude that there is no feasible solution for . So the number of rescaling operations can be bounded, i.e., the number of MA calls can be bounded. Consequently, the algorithm can terminate in finite time no matter whether problem is feasible or infeasible.
To be more precise, we quantify the time complexity. The total time complexity of these Chubanov-type algorithms are typically $O({\mathsf{T_{MA}}}*{\mathsf{T_{BP}}})$, where ${\mathsf{T_{MA}}}$ denotes the number of MA calls (rescaling operations), and ${\mathsf{T_{BP}}}$ denotes the time required by the basic procedure BP. According to the classic lower bound [@khachian1979polynomial], ${\mathsf{T_{MA}}}$ can simply be bounded by $O(nL)$ for these Chubanov-type algorithms, where $L$ denotes the bit size of $A$. However, ${\mathsf{T_{BP}}}$ is the most important and tricky part. Theoretical results and practical performances vary for different BP procedures. The typical BP procedures include the perceptron method, von Neumann’s method, and their variants (see e.g., [@dantzig1992varepsilon; @dunagan2008simple; @dadush2016rescaled; @pena2017solving]). We review more details of these BP in Section \[sec:bp\]. Usually, ${\mathsf{T_{BP}}}$ equals to $O(n^4)$ or $O(n^3m)$ in these BP procedures. In this work, we improve ${\mathsf{T_{BP}}}$ by a factor of $\sqrt{n}$ if (\[feasibility:primal\]) or (\[feasibility:dual\]) is well-conditioned (measured by ), but in the worse case, ${\mathsf{T_{BP}}}$ still equals to $O(n^3m)$ in our algorithm.
[[[**Our Motivation:**]{}]{}]{} In practice, these Chubanov-type projection and rescaling algorithms usually run much faster on the primal infeasible instances (i.e., is infeasible) than on the primal feasible instances (i.e., dual infeasible) no matter what basic procedure (von Neumann, perceptron or their variants) we use (also see Table \[table:comparison\] in Section \[sec:exp\]). In this paper, we try to explain this phenomenon theoretically. Moreover, we try to provide a new algorithm to address this issue.
[[[**Our Contribution:**]{}]{}]{} Concretely, we make the following technical contributions:
1. First, for the theoretical explanation, we provide Lemma \[lm:infeasible\] which shows that the time complexity ${\mathsf{T_{BP}}}$ can be $O(n^{2.5}m)$ rather than $O(n^3m)$ (see Lemma \[lm:tbp\]) in certain situations if is infeasible. This gives an explanation of why these Chubanov-type algorithms usually run much faster if is infeasible.
2. Then, we explicitly develop the *dual algorithm* (see Section \[sec:dual\]) to improve the performance when is feasible. Our dual algorithm is the first algorithm which rescales the row space of $A$ in MA (see Table \[tab:comp\]). As a result, we provide a similar Lemma \[lm:infeasibled\] which shows that the time complexity ${\mathsf{T_{BP}}}$ of our dual algorithm can be $O(n^{2.5}m)$ rather than $O(n^3m)$ in certain situations if is infeasible (i.e. is feasible).
Naturally, we obtain a new fast polynomial primal-dual projection algorithm (called ${\mathsf{PPDP}}$) by integrating our primal algorithm (which runs faster on the primal infeasible instances) and our dual algorithm (which runs faster on the primal feasible instances). See Section \[sec:ppdp\].
3. Finally, the numerical results validate that our primal-dual ${\mathsf{PPDP}}$ algorithm is quite balanced between feasible and infeasible instances, and it runs significantly faster than other algorithms (see Table \[table:comparison\] in Section \[sec:exp\]).
[[[**Remark:**]{}]{}]{} Our algorithms are based on [Dadush-V[é]{}gh-Zambelli ]{}algorithm [@dadush2016rescaled] and the improvements of Roos’s algorithm [@roos2018improved] (see Section \[sec:relatealg\] and Table \[tab:comp\]). Besides, we introduce a new step-size term $c$ for practical consideration (see Line 13 and 14 of Algorithm \[alg:bp\] and \[code:dual Dadush\]). For the maximum support problem , ${\mathsf{T_{BP}}}=O(n^4)$ for Chubanov’s algorithm and Roos’s algorithm, and ${\mathsf{T_{BP}}}=O(n^3m)$ for [Dadush-V[é]{}gh-Zambelli ]{}algorithm. Note that in the worst case ${\mathsf{T_{BP}}}=O(n^3m)$ for our algorithms, but it can be improved by a factor of $\sqrt{n}$ in certain situations. Recall that ${\mathsf{T_{MA}}}=O(nL)$ for these Chubanov-type algorithms as we discussed before. Thus the time complexity of our algorithms (in the worst case) match the result of [Dadush-V[é]{}gh-Zambelli ]{}algorithm [@dadush2016rescaled], i.e., $O({\mathsf{T_{MA}}}*{\mathsf{T_{BP}}})=O(n^4mL)$ (see our Theorems \[thm:primal\]–\[thm:primal\_dual\]). However, we point out that the total time complexity of Chubanov’s algorithm and Roos’s algorithm are $O(n^4L)$, and hence is faster than ours. They speed up it from $O(n^5L)$ to $O(n^4L)$ by using an amortized analysis while we currently do not use. We leave this speedup as a future work.
[[[**Organization:**]{}]{}]{} In Section \[sec:pre\], we introduce some useful notations and review some related algorithms. The details and results for our primal algorithm and dual algorithm are provided in Section \[sec:primal\] and Section \[sec:dual\], respectively. Then, in Section \[sec:ppdp\], we propose the efficient primal-dual ${\mathsf{PPDP}}$ algorithm. Finally, we conduct the numerical experiments in Section \[sec:exp\] and include a brief conclusion in Section \[sec:con\].
Preliminaries {#sec:pre}
=============
In this section, we first review some classic basic procedures and then introduce some notations to review some related algorithms at the end of this section.
Classic Basic Procedures {#sec:bp}
------------------------
Recall that BP returns one of the following three results: (i) a feasible solution of ; (ii) a feasible solution of the dual problem ; (iii) a cut for the feasible region of . Here we focus on the first two outputs, the last one is controlled by an upper bound lemma (similar to Lemma \[lm:roosbound\]).
Letting $y=Ax$, when solving , we know that there is at least an index $k$ such that $a_k^Ty\leq 0$, where $a_k$ is the $k$th-column of $A$ (otherwise $y$ is already a feasible solution for ). On the other hand, to solve , we want to minimize ${\| y \|}$. The goal is to let $y$ go to 0 (in which case $x$ is a feasible solution for ). We review some classic update methods as follows:
[[[**von Neumann’s algorithm:**]{}]{}]{} In each iteration, find an index $k$ such that $a_k^Ty\leq 0$, and then update $x$ and $y$ as $$\label{eq:update}
{\textcolor{red}{y'}}=\alpha y+\beta a_k, \quad x'=\alpha x+\beta e_k \quad (\mathrm{note~ that~}y=Ax \mathrm{~and~} {\textcolor{red}{y'}}=Ax'),$$ where $\alpha,\beta>0$ are chosen such that $\|{\textcolor{red}{y'}}\|$ is smallest and $\alpha+\beta=1$ [@dantzig1992varepsilon].
[[[**Perceptron:**]{}]{}]{} Choose $\alpha=\beta=1$ in at every iteration. See e.g. [@rosenblatt1957perceptron; @novikoff1963convergence].
[[[**Dunagan-Vempala:**]{}]{}]{} Fix $\alpha=1$ and choose $\beta$ to minimize ${\| {\textcolor{red}{y'}} \|}$ [@dunagan2008simple].
Notations
---------
Before reviewing the related algorithms (in the following Section \[sec:relatealg\]), we need to define/recall some useful notations. We use $P_A$ and $Q_A$ to denote the projections of ${{\mathbb R}}^n$ onto the null space ($\mathcal{N}_A$) and row space ($\mathcal{R}_A$) of the $m\times n$ matrix $A$, respectively: $$P_A \triangleq I-A^T (A A^T)^{\dag}A,\qquad Q_A \triangleq A^T (A A^T)^{\dag}A.$$ where $(\cdot)^{\dag}$ denotes the Moore-Penrose pseudoinverse. Particularly, $(A A^T)^{\dag}=(A A^T)^{-1}$ if ${\mathrm{rank}}(A)=m$.
We further define the following notations: $$\label{eq:vec}
v=Q_Ay\in \mathcal{R}_A,\quad z=P_Ay\in \mathcal{N}_A,\quad y=v+z\in {{\mathbb R}}^n.$$ Usually, $z$ is used to denote the feasible solution of and $v$ indicates the feasibility of .
To analyze case (iii) of BP, we note that is feasible if and only if the system $$\label{feasibility:normalized primal}
\begin{aligned}
Ax=0, ~x\in [0,1]^n,~ x\neq 0
\end{aligned}$$ is feasible since is a homogeneous system. From now on, we will consider problem instead of . Similarly, we use a normalized version to replace . Now, we recall a useful lemma which gives an upper bound for the coordinates of any feasible solution. This upper bound will indicate a cut for case (iii).
\[lm:roosbound\] Let $x$ be any feasible solution of , $y$ and $v$ are defined as in (\[eq:vec\]), then every non-zero coordinate $v_j$ of $v$ gives rise to an upper bound for $x_j$, according to $$x_j\leq {\mathrm{bound}}_j(y)\triangleq {\mathbf{1}}^T\Big[\frac{v}{-v_j}\Big]^+,$$ where $x^+\triangleq \max\{0,x\}$ and ${\mathbf{1}}$ denotes the all-ones vector.
This means that we can scale the column $j$ of $A$ by a factor ${\mathrm{bound}}_j(y)$ to make the feasible solutions of (\[feasibility:normalized primal\]) closer to the all-ones vector ${\mathbf{1}}$. Similarly to $x^+$, we denote $x^-\triangleq -(-x)^+$. Furthermore, we need the definition of condition number $\rho(Q)$ for a matrix $Q$ [@goffin1980relaxation]: $$\label{eq:rho}
\rho(Q)\triangleq \max_{x:\|x\|_2=1}\min_{i} \langle x, \frac{q_i}{\|q_i\|_2} \rangle,$$ where $q_i$ is the $i$th-column of $Q$.
Related Algorithms {#sec:relatealg}
------------------
Now, we are able to review some related algorithms for solving based on the BP procedures introduced in Section \[sec:bp\].
[[[**Chubanov’s algorithm:**]{}]{}]{} Instead of updating in the original space $y=Ax$, [@chubanov2015polynomial] updates in the projection space $z={\textcolor{blue}{P_A}}y$, where $P_A = I-A^T (A A^T)^{-1}A$ is a null space projection of $A$. In each BP iteration, it updates $y$ and $z$ in the same way as von Neumann’s update (just replacing $A$ by $P_A$). Intuitively, BP either finds a feasible solution $x^*$ of or finds a cut (i.e., an index $j$ such that $x_j^*\leq 1/2$ for any feasible solution $x^*$ of in $[0,1]^n$). Then the main algorithm MA rescales the null space of $A$ by dividing the $j$th-column of $A$ by 2. According to \[Khachian, 1979\], there is a lower bound for the feasible solutions of . Thus the number of rescaling operations can be bounded. Finally, the algorithm terminates in polynomial-time, where either BP returns a feasible solution or MA claims the infeasibility according to the lower bound.
[[[**Roos’s algorithm:**]{}]{}]{} [@roos2015chubanov; @roos2018improved] provided two improvements of Chubanov’s algorithm:
1. A new cut condition was proposed, which is proved better than the one used by Chubanov.
2. The BP can use *multiple indices* to update $z$ and $y$ ($z=P_Ay$) in each iteration, e.g., a set of indices satisfying $(P_A)_i^Tz\leq 0$. Recall that von Neumann’s update only uses one index $k$ satisfying $a_k^Ty\leq 0$.
[[[**Dadush-V[é]{}gh-Zambelli:**]{}]{}]{} Compared with Chubanov’s algorithm, [@dadush2016rescaled] used the Dunagan-Vempala update instead of von Neumann’s update as its BP, along with Roos’ new cut condition. Besides, the updates are performed in the orthogonal space $v={\textcolor{blue}{Q_A}}y$, where $Q_A = A^T (A A^T)^{-1}A$ is a row space projection matrix of $A$. But the rescaling space in MA is the same, i.e., the null space of $A$.
[[[**Comparison:**]{}]{}]{} To demonstrate it clearly, we provide a comparison of our algorithms with other algorithms in Table \[tab:comp\]. Note that our primal-dual ${\mathsf{PPDP}}$ algorithm is the integration of our primal algorithm and dual algorithm.
\[tab:comp\]
Algorithms Update method Update space Rescaling space \#indices
-------------------------- ----------------- -------------- ----------------- -----------
Chubanov’s algorithm von Neumann Null space Null space One
Roos’ algorithm von Neumann Null space Null space Multiple
Dadush-V[é]{}gh-Zambelli Dunagan-Vempala Row space Null space One
Our primal algorithm Dunagan-Vempala Row space Null space Multiple
Our dual algorithm Dunagan-Vempala Null space Row space Multiple
: Comparison of our algorithms with other algorithms
Our Primal Algorithm {#sec:primal}
====================
In this section, we introduce our primal algorithm which consists of the basic procedure BP and the main algorithm MA. The details of BP and MA are provided in Section \[sec:bpprimal\] and Section \[sec:maprimal\] respectively.
Basic Procedure (BP) {#sec:bpprimal}
--------------------
Our BP is similar to [@dadush2016rescaled] (or [@dunagan2008simple]) (see Table \[tab:comp\]). The details are described in Algorithm \[alg:bp\]. The main difference is that we use multiple indices $K$ to update (see Line 9 of Algorithm \[alg:bp\]) and introduce the step-size $c$ for practical consideration (see Line 13 and 14 of Algorithm \[alg:bp\]).
$Q_A$ $y, z, J, {\mathrm{case}}.$ $r=size(Q_A), {\mathrm{threshold}}=1/2r^{3/2}, c\in (0,2), {\mathrm{case}}=0$ $y={\mathbf{1}}/r, v=Q_Ay, z=y-v=P_Ay$ ${\mathrm{case}}= 1$ ($z$ is primal feasible); [**return**]{} ${\mathrm{case}}= 2$ ($v$ is dual feasible); [**return**]{} find $K=\{k: v_k \leq 0\}$ $q_K=Q_A \sum_{k\in K}e_k$ $\alpha={\langle \frac{q_K}{\|q_K\|_2}, v \rangle}$ $y=y-c(\frac{\alpha}{\|q_K\|_2}\sum_{k\in K} e_k)$ $v=v-c(\frac{\alpha}{\|q_K\|_2} \sum_{k\in K} q_k)$ find a nonempty set $J$ such that $J\subseteq \{j: bound_j(y)\leq \frac{1}{2} \}$ (a cut); [**return**]{}
In the BP (Algorithm \[alg:bp\]), the norm of the iterated vector $v=Q_A y$ is decreasing, while each coordinate of $y$ is increasing. Thus, after a certain number of iterations, we will obtain a feasible solution $z=y-v=P_Ay>0$. Otherwise, it is always possible to find a cut $J$ (Line 16), along with some rescaling operations for the matrix $A$, to make the feasible solutions of (\[feasibility:normalized primal\]) closer to the all-ones vector. The cut is guaranteed by the following lemma.
\[lm:cut\] Let $Q_A$ be the projection matrix at a given iteration of BP (Algorithm \[alg:bp\]). Suppose that $\alpha={\langle \frac{q_K}{\|q_K\|_2}, v \rangle}> -{\mathrm{threshold}}$, then the set $J=\{j: {\mathrm{bound}}_j(y)\leq \frac{1}{2} \}$ is nonempty and every solution $x$ of problem (\[feasibility:normalized primal\]) satisfies $x_j\leq \frac12$ for all $j\in J$.
This lemma is proved with Lemma \[lm:roosbound\] and we defer the proof to Appendix \[app:lmcut\].
For the time complexity of Algorithm \[alg:bp\], i.e. ${\mathsf{T_{BP}}}$, we give the following lemma (the proof is in Appendix \[app:tbp\]).
\[lm:tbp\] The time complexity of Algorithm \[alg:bp\] ${\mathsf{T_{BP}}}=O(n^3m)$. Concretely, it uses at most $O(n^2)$ iterations and each iteration costs at most $O(mn)$ time.
Note that Lemma \[lm:tbp\] holds regardless is feasible or infeasible. However, as we discussed before, the algorithm usually performs much better on the infeasible instances than on the feasible instances. To explain this phenomenon, we give the following lemma. The proof is deferred to Appendix \[app:infeasible\].
\[lm:infeasible\] If is infeasible, the time complexity of Algorithm \[alg:bp\] ${\mathsf{T_{BP}}}=O(n^2m/\rho(Q_A))$, where $\rho(Q_A)$ is the condition number defined in (\[eq:rho\]). In particular, $\rho(Q_A)$ equals to $1/\sqrt{n}$ under well-condition (e.g., $A$ is an identity matrix), then ${\mathsf{T_{BP}}}=O(n^{2.5}m)$ if problem is infeasible.
Main Algorithm (MA) {#sec:maprimal}
-------------------
The details of our MA are described in Algorithm \[alg:ma\]. Particularly, we rescale the null space of $A$ in Line 8.
$A\in{{\mathbb R}}^{m\times n}, d={\mathbf{1}}, \tau=2^{-L}, {\mathrm{case}}=0, H=\varnothing$. $Q_A=A^T (A A^T)^{\dag}A$ $(y,z,J,{\mathrm{case}})\leftarrow$ Basic Procedure for Primal Problem$(Q_A)$ $d_J=d_J /2$
$H=\{i:d_i\leq \tau\}$ $d_H=0$ $A_{J}=A_{J}/2$ $A=A_{\overline{H}}$
$d=d_{\overline{H}}$ $D={\mathrm{diag}}(d)$ Define $x$ as $x_{\overline{H}}=Dz, x_H=0$
Now, we state the complexity of our primal algorithm in the following theorem. The proof is deferred to Appendix \[app:thmprimal\]
\[thm:primal\] The time complexity of the primal algorithm is $O(n^4mL)$.
Our Dual Algorithm {#sec:dual}
==================
The Chubanov-type algorithms all focus on the primal problem (\[feasibility:primal\]) (or the normalized version ), i.e., their MA always rescale the null space of $A$ (see Table \[tab:comp\]). We emphasize that these algorithms usually perform much better on the infeasible instances than on the feasible ones (see our Lemma \[lm:infeasible\] which gives an explanation). Now, we want to address this unbalanced issue by providing a dual algorithm. Our dual algorithm explicitly considers the dual problem (\[feasibility:dual\]) and rescales the *row space* of $A$, unlike the previous algorithms. We already know that the primal algorithm runs faster on the primal infeasible instances. Thus we expect the dual algorithm runs faster on the dual infeasible instances (i.e., primal feasible instances). As expected, our dual algorithm does work. Therefore, in Section \[sec:ppdp\], we integrate our primal algorithm and dual algorithm to obtain a quite *balanced* primal-dual algorithm and its performance is also remarkably better than the previous algorithms.
Similar to our primal algorithm, the dual algorithm also consists of the basic procedure BP and the main algorithm MA. The details of BP and MA are provided in Section \[subsec:bpdual\] and Section \[subsec:madual\] respectively. Similar to , we consider the normalized version of (\[feasibility:dual\]) due to the homogeneity: $$\label{eq:bounded dual}
\mathrm{find }\; u\in {{\mathbb R}}^m \; \mathrm{subject} \; \mathrm{to} \; x=A^T u> 0, x\in (0,1]^n.$$
Basic Procedure for the Dual Problem {#subsec:bpdual}
------------------------------------
The basic procedure for the dual problem is described in Algorithm \[code:dual Dadush\].
$P_A$ $y, z, J, {\mathrm{case}}.$ ${\mathrm{threshold}}=1/2n^{3/2}, c\in (0,2), {\mathrm{case}}=0$ $y = {\mathbf{1}}/n, z=P_Ay, v=y-z=Q_Ay$ ${\mathrm{case}}=2$ (dual feasible); [**return**]{} ${\mathrm{case}}=1$ (primal feasible); [**return**]{} find $K=\{k:\langle z, e_k\rangle\leq 0\}$ $p_K=P_A \sum_{k\in K}e_k$ $\alpha=\langle \frac{p_K}{\|p_K\|_2}, z\rangle $ $y=y-c(\frac{\alpha}{\|p_K\|_2}\sum_{k\in K} e_k)$ $z=z-c(\frac{\alpha}{\|p_K\|_2} \sum_{k\in K} p_k)$ $v=y-z$ find a nonempty set $J$ such that $J\subseteq \{j: {\mathrm{bound}}'_j(y)\leq \frac{1}{2} \}$ (a cut); [**return**]{}
In this basic procedure, either a feasible solution for the primal problem is found, or a dual feasible solution is found, or a cut of the bounded row space is found (which is denoted as $bound'_j(y)$ in Line 17). Now, we need to provide an upper bound in Lemma \[lm:dualcut\], which shows that a cut of the bounded row space can be derived from $z$, instead of $v$ in the case of null space (see Lemma \[lm:roosbound\]).
\[lm:dualcut\] Let $x$ be any feasible solution of and $z=P_A y$ for some $y$. Then every non-zero coordinate $z_j$ of $z$ gives rise to an upper bound for $x_j$, according to $$x_j \leq {\mathrm{bound}}'_j(y) \triangleq \bm{1}^T \Big[\frac{z}{-z_j}\Big]^+.$$
The proof of this lemma is deferred to Appendix \[app:dualcut\]. According to this lemma, we can obtain the following guaranteed cut in Lemma \[lm:cutd\], which is similar to Lemma \[lm:cut\]. The proof is almost the same as Lemma \[lm:cut\] just by replacing Lemma \[lm:roosbound\] with our Lemma \[lm:dualcut\].
\[lm:cutd\] Let $P_A$ be the projection matrix at a given iteration of BP (Algorithm \[code:dual Dadush\]). Suppose that $\alpha={\langle \frac{p_K}{\|p_K\|_2}, z \rangle}> -{\mathrm{threshold}}$, then the set $J=\{j: {\mathrm{bound}}'_j(y)\leq \frac{1}{2} \}$ is nonempty and every solution $x$ of problem (\[feasibility:normalized primal\]) satisfies $x_j\leq \frac12$ for all $j\in J$.
Same to Algorithm \[alg:bp\], for the time complexity ${\mathsf{T_{BP}}}$ of the dual Algorithm \[code:dual Dadush\], we have the following lemma.
\[lm:tbpd\] The time complexity of Algorithm \[code:dual Dadush\] ${\mathsf{T_{BP}}}=O(n^3m)$. Concretely, it uses at most $O(n^2)$ iterations and each iteration costs at most $O(mn)$ time.
Note that Lemma \[lm:tbp\] also holds regardless is feasible or infeasible. Now, we want to point out that our dual algorithm can perform much better on the dual infeasible instances (primal feasible instances) under well-condition as we expected and discussed before. Similar to Lemma \[lm:infeasible\], we have the following lemma for the dual algorithm.
\[lm:infeasibled\] If is infeasible, the time complexity of Algorithm \[alg:bp\] ${\mathsf{T_{BP}}}=O(n^2m/\rho(P_A))$, where $\rho(P_A)$ is the condition number defined in (\[eq:rho\]). In particular, $\rho(P_A)$ equals to $1/\sqrt{n}$ under well-condition (e.g., $A=(I,-I)$, where $I$ is an identity matrix), then ${\mathsf{T_{BP}}}=O(n^{2.5}m)$ if problem is infeasible.
Note that it is easy to see that is feasible (i.e., is infeasible) if $A=(I,-I)$.
Besides, when the dual problem is feasible, we can also utilize the geometry of the problem to bound the iteration complexity instead of Lemma \[lm:tbpd\]. Consider the following kind of condition number of the set $Im(A)\bigcap [0,1]^n$: $$\delta_{\infty}(Im(A)\bigcap [0,1]^n)\triangleq \max_{x}\{\prod_{i} x_i : x\in Im(A)\bigcap [0,1]^n\}.$$
As each rescaling in the basic procedure will at least enlarge the value of $\delta_{\infty}(Im(A)\bigcap [0,1]^n)$ by two times, and the largest possible value of $\delta_{\infty}(Im(B)\bigcap [0,1]^n)$ for all matrices $B$ is 1, it takes at most $-\log_2 \delta_{\infty}(Im(A)\bigcap [0,1]^n)$ basic procedures before getting a feasible solution. This means that the iteration complexity of the whole algorithm is $O(n^2 \log \frac{1}{\delta_{\infty}(Im(A)\bigcap [0,1]^n)})$.
Main Algorithm for the Dual Problem {#subsec:madual}
-----------------------------------
The main algorithm for the dual problem is described in Algorithm \[code:main procedure of dual Dadush\]. Particularly, we rescale the row space of $A$ in Line 6.
$A\in{{\mathbb R}}^{m\times n}, d={\mathbf{1}}, \tau=2^{-L}, {\mathrm{case}}=0$. $P_A=I-A^T (A A^T)^{\dag}A$ $(y,z,J,{\mathrm{case}})\leftarrow$ Basic Procedure for Dual Problem$(P_A)$ $d_J=2d_J$ $A_J=2A_J$ $H=\{i:d_i\geq 2^L\}$ $d_H=0$ $A_H=0$ $D={\mathrm{diag}}(d)$ $x=Dz$
Now, we have the following theorem for our dual algorithm. The proof is provided in Appendix \[app:thmdual\].
\[thm:dual\] The time complexity of the dual algorithm is $O(n^4mL)$.
Our Primal-Dual ${\mathsf{PPDP}}$ Algorithm {#sec:ppdp}
===========================================
In this section, we propose a new polynomial primal-dual projection algorithm (called ${\mathsf{PPDP}}$) to take advantages of our primal algorithm and dual algorithm. Similarly, the ${\mathsf{PPDP}}$ algorithm also consists of two procedures (MA and BP). Intuitively, the BP solves problems (\[feasibility:primal\]) and (\[feasibility:dual\]) simultaneously. Recall that the primal algorithm runs faster on the infeasible instances and the dual algorithm runs faster on the feasible instances (see Table \[table:comparison\]). The MA rescales the matrix (row space or null space) according to the output of BP. The MA and BP are formally described in Algorithms \[code:main procedure of primal-dual\] and \[code:primal dual\] respectively. The details are deferred to Appendix \[app:algo\]. Thus, we have the following theorem.
\[thm:primal\_dual\] The time complexity of our primal-dual ${\mathsf{PPDP}}$ algorithm is $O(n^4mL)$.
Note that the final output of our ${\mathsf{PPDP}}$ algorithm is a feasible solution for either (\[feasibility:primal\]) or (\[feasibility:dual\]). Obviously, the algorithm will stop whenever it finds a solution of or , thus the time complexity of our ${\mathsf{PPDP}}$ algorithm follows easily from Theorems \[thm:primal\] and \[thm:dual\].
Experiments {#sec:exp}
===========
In this section, we compare the performance of our algorithms with Roos’ algorithm [@roos2018improved] and Gurobi (one of the fastest solvers nowadays). We conduct the experiments on the randomly generated matrices. Concretely, we generate $100$ integer matrices $A$ of size $625\times 1250$, with each entry uniformly randomly generated in the interval $[-100,100]$. The parameter $c\in (0,2)$ is the step-size which is a new practical term introduced in this work. The average running time of these algorithms are listed in Table \[table:comparison\].
\[table:comparison\]
Algorithms feasible instances infeasible instances
------------------------------------------- -------------------- ----------------------
Gurobi (a fast optimization solver) 3.08 1.58
Roos’s algorithm [@roos2018improved] 10.75 0.83
Our primal algorithm ($c=1.8$) 9.93
Our dual algorithm ($c=1.8$) 4.57
Our ${\mathsf{PPDP}}$ algorithm ($c=1.8$)
: Running time (sec.) of algorithms wrt. (\[feasibility:primal\]) is feasible or infeasible
Table \[table:comparison\] validates that our new primal-dual ${\mathsf{PPDP}}$ algorithm is quite balanced on the feasible and infeasible instances due to the integration of our primal and dual algorithm. Moreover, it shows that our ${\mathsf{PPDP}}$ algorithm can be a practical option for linear programming since it runs remarkably faster than the fast optimization solver Gurobi.
Conclusion {#sec:con}
==========
In this paper, we try to theoretically explain why the Chubanov-type projection algorithms usually run much faster on the primal infeasible instances. Furthermore, to address this unbalanced issue, we provide a new fast polynomial primal-dual projection algorithm (called ${\mathsf{PPDP}}$) by integrating our primal algorithm (which runs faster on the primal infeasible instances) and our dual algorithm (which runs faster on the primal feasible instances). As a start, we believe more improvements (e.g., the amortized analysis speedup) can be made for the Chubanov-type projection algorithms both theoretically and practically.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank Jian Li (Tsinghua University), Yuanxi Dai (Tsinghua University) and Rong Ge (Duke University) for useful discussions.
Details of ${\mathsf{PPDP}}$ Algorithm {#app:algo}
======================================
In this appendix, we describe the details of our ${\mathsf{PPDP}}$ Algorithm. Concretely, the main procedure is an integration of Algorithms \[alg:ma\] and \[code:main procedure of dual Dadush\], which is formally described in Algorithm \[code:main procedure of primal-dual\]. The basic procedure is an integration of Algorithms \[alg:bp\] and \[code:dual Dadush\], which is formally described in Algorithm \[code:primal dual\].
$A, d={\mathbf{1}}, y={\mathbf{1}}/n, \tau=2^{-L}, {\mathrm{case}}=0$. $A1=A2=A$ $d1=d2=d$ $y_1=y_2=y$ $P_{A1}=I-A1^T (A1 A1^T)^{\dag}A1$ $Q_{A2}=A2^T (A2 A2^T)^{\dag}A2$ $(y_1,y_2,J_1,J_2,{\mathrm{case}}) \leftarrow$ Basic Procedure for Primal-Dual Problem$(P_{A1},Q_{A2},y_1,y_2)$ $d2_{J_2}=d2_{J_2}/2$ $H=\{i:d2_i\leq \tau\}$ $d2_H=0$ $A2_{J_2}=A2_{J_2}/2$ $A2=A2_{\overline{H}}$ $r=size(Q_{A2})$ $y_2={\mathbf{1}}/r$
$d1_{J_1}=2d1_{J_1}$ $A1_{J_1}=2A1_{J_1}$ $y_1=y$ $D={\mathrm{diag}}(d1)$ $x=D z_1$
$D={\mathrm{diag}}(d1)$ $x=Dv_1$ $d=d2_{\overline{H}}$ $D={\mathrm{diag}}(d)$ Define $x$ as $x_{\overline{H}}=Dz_2, x_H=0$ $D={\mathrm{diag}}(d2)$ $x=Dv_2$
$P_{A1},Q_{A2},y_1,y_2$ $y_1, y_2, J_1, J_2, {\mathrm{case}}.$ ${\mathrm{threshold}}_1=1/2n^{3/2}, r=size(Q_{A2}), {\mathrm{threshold}}_2=1/2r^{3/2}, c\in (0,2), {\mathrm{case}}=0$ $z_1=P_{A1}y_1, v_1=y_1-z_1, v_2=Q_{A2}y_2, z_2=y_2-v_2$ ${\mathrm{case}}=2$ ($y_1$ is dual feasible); [**return**]{} ${\mathrm{case}}=1$ ($y_1$ is primal feasible); [**return**]{} ${\mathrm{case}}=4$ ($y_2$ is dual feasible); [**return**]{} ${\mathrm{case}}=3$ ($y_2$ is primal feasible); [**return**]{} find $K_1=\{k:\langle z_1, e_k\rangle\leq 0\}$ $p_{K_1}=P_{A1} \sum_{k\in K_1}e_k$ $\alpha_1=\langle \frac{p_{K_1}}{\|p_{K_1}\|_2}, z_1\rangle $ find $K_2=\{k:\langle v_2, e_k\rangle\leq 0\}$ $q_{K_2}=Q_{A2} \sum_{k\in K_2}e_k$ $\alpha_2=\langle \frac{q_{K_2}}{\|q_{K_2}\|_2}, v_2\rangle $ $y_1=y_1-c(\frac{\alpha_1}{\|p_{K_1}\|_2}\sum_{k\in K_1} e_k)$ $z_1=z_1-c(\frac{\alpha}{\|p_{K_1}\|_2} \sum_{k\in K_1} p_k)$ $v_1=y_1-z_1$ find a nonempty set $J_1$ such that $J_1\subseteq \{j: {\mathrm{bound}}'_j(y_1)\leq \frac{1}{2} \}$ ${\mathrm{case}}=-1$; (the rescaling should be in the row space of $A1$); [**return**]{} $y_2=y_2-c(\frac{\alpha_2}{\|q_{K_2}\|_2}\sum_{k\in K_2} e_k)$ $v_2=v_2-c(\frac{\alpha}{\|q_{K_2}\|_2} \sum_{k\in K_2} q_k)$ $z_2=y_2-v_2$ find a nonempty set $J_2$ such that $J_2\subseteq \{j: {\mathrm{bound}}_j(y_2)\leq \frac{1}{2} \}$ ${\mathrm{case}}=-2$; (the rescaling should be in the null space of $A2$); [**return**]{}
Missing Proofs {#app:pf}
==============
In this appendix, we provide all the proofs for Theorems \[thm:primal\]–\[thm:dual\] and Lemmas \[lm:cut\]–\[lm:dualcut\].
Proof of Lemma \[lm:cut\] {#app:lmcut}
-------------------------
As the basic procedure has not terminated, $z=y-v$ cannot be primal feasible. Initially $y={\mathbf{1}}/n$. For each iteration, the operation $$y=y-c\big(\frac{\alpha}{\|q_K\|_2}\sum_{k\in K} e_k\big)$$ will only increase some components of the vector $y$ by $-c\frac{\alpha}{\|q_K\|_2}\geq c\frac{{\mathrm{threshold}}}{\|q_K\|_2}$. Thus each component of $y$ is at least $1/n$ during the whole procedure， implying that there exists an index $j$ with $v_j\geq 1/n$. Otherwise $z=y-v$ will be primal feasible. For this specific $j$, we have $$\begin{aligned}
{\mathrm{bound}}_j(y)&=\frac{-\sum_{i=1}^{n} v_i^{-}}{v_j}=\frac{-\sum_{k\in K} \langle e_k, v \rangle}{v_j}\\
&=\frac{-\sum_{k\in K} \langle Q_A e_k, v \rangle}{v_j}=\frac{-\langle q_K, v \rangle}{v_j}\\
&\leq n\|q_K\|_2{\mathrm{threshold}}\leq\frac{1}{2},
\end{aligned}$$ where the last inequality follows from $$\|q_K\|_2\leq \|\sum_{k\in K} e_k\|\leq \sqrt{n}.$$ ${\hfill\ensuremath{\square}}$
Proof of Lemma \[lm:tbp\] {#app:tbp}
-------------------------
When $\alpha=\langle \frac{q_K}{\|q_K\|_2}, v\rangle \leq -{\mathrm{threshold}}$, the decrease of $\|v\|_2$ in each iteration has a lower bound: $$\|v-c\frac{\alpha}{\|q_K\|_2} q_K\|_2^2=\|v\|_2^2-(2c-c^2)\alpha^2\leq \|v\|_2^2-(2c-c^2)\frac{1}{4n^3}.$$ Initially, $\|v_0\|_2^2\leq \|y_0\|_2^2=\frac{1}{n}$. After $t$ iterations, $\|v_t\|_2^2\leq \frac{1}{n}-\frac{(2c-c^2)t}{4n^3}$. So it takes at most $O(n^2)$ iterations to obtain a vector $v_t$ with $\|v_t\|_2\leq \frac{1}{n}$, in which case a primal feasible solution $z=y-v$ can be obtained. This means that each basic procedure takes at most $O(n^2)$ iterations before stopping.
Now it remains to bound the time complexity in each iteration. In each basic procedure iteration, we find all indices $k$ such that $v_k\leq 0$ and do the calculation $\sum_{k\in K} q_k $. In the worst case, $|K|$ can be $O(n)$, thus $O(n^2)$ arithmetic operations are needed to calculate this summation. However, there is another way to do this. Recall that ${\mathrm{rank}}(Q_A)={\mathrm{rank}}(A)=m$, thus the number of basis vectors of the $n$ rows are exactly $m$, while the other rows can be represented by a weighted summation of these $m$ basis vectors. This means that we only have to do the naive summation for these $m$ rows, while the other $n-m$ rows can be obtained by doing the weighted summation of these $m$ elements， which cost $O(mn)+O((n-m)m)=O(mn)$ arithmetic operations. Note that the $m$ basis vectors can be computed by the Singular Value Decomposition (SVD). Then the weights of the other $n-m$ rows can be obtained by computing the inverse of an $m\times m$ matrix and multiplying this inverse matrix to the row vectors. These steps cost $O(mn^2+m^3+m^2(n-m))=O(mn^2)$ operations and only needs to be done once at the beginning of the basic procedure. ${\hfill\ensuremath{\square}}$
Proof of Lemma \[lm:infeasible\] {#app:infeasible}
--------------------------------
When (\[feasibility:normalized primal\]) is infeasible, i.e., $\rho(Q_A)>0$. Denoting the vector $w$ as the center which achieves the value $\rho(Q_A)$, we can check the closeness between $v_t$ and $w$ in each iteration: $$\langle v_{t+1},w \rangle =\langle v_{t},w \rangle -c\langle v_{t},\hat{q_K} \rangle \langle w,\hat{q_K} \rangle \geq \langle v_{t},w \rangle+ \frac{c\rho(Q_A)}{2n\sqrt{n}}.$$ On the other hand, as the norm $\|v_t\|\leq \sqrt{\frac{1}{n}-\frac{(2c-c^2)t}{n^3}}$ before the basic procedure stops, we should have $$\rho(Q_A)+\frac{ct\rho(Q_A)}{2n\sqrt{n}}\leq \langle v_{t},w \rangle \leq \|v_{t}\|\|w\|\leq \sqrt{\frac{1}{n}-\frac{(2c-c^2)t}{n^3}}.$$ This implies that the number of the iterations in the basic procedure is $t=O(\min\{\frac{n}{\rho(Q_A)},n^2\})$ when the primal problem (\[feasibility:normalized primal\]) is infeasible. This proof is finished by combining the result of Lemma \[lm:tbp\], i.e., each iteration costs $O(mn)$ time. ${\hfill\ensuremath{\square}}$
Proof of Theorem \[thm:primal\] {#app:thmprimal}
-------------------------------
According to a classic result of [@khachian1979polynomial], there exists a positive number $\tau$ (satisfying $1/\tau=O(2^L)$) such that the positive coordinates of the basic feasible solutions of problem are bounded below by $\tau$. When the coordinate has been rescaled for more than $\tau$ times, the value of this coordinate in all the solutions must be $0$. As a result, the corresponding columns of $A$ can be omitted.
According to Lemma \[lm:cut\], the cut $J$ is nonempty. It means each iteration of MA can rescale at least one coordinate of the feasible solutions by $1/2$. Thus the number of rescaling operations can be bounded by $nL$, i.e., the number of iterations in MA ${\mathsf{T_{MA}}}=O(nL)$. The proof is finished by combining this with Lemma \[lm:tbp\]. ${\hfill\ensuremath{\square}}$
Proof of Lemma \[lm:dualcut\] {#app:dualcut}
-----------------------------
Since $x\in \mathcal{R}_A$, we have $\langle x,z \rangle=0$. Thus we consider the following two cases. For $z_j< 0$, we have $$-z_j x_j=\sum_{i\neq j} z_i x_i \leq \sum_{i, z_i>0}z_i x_i\leq \sum_{i, z_i>0}z_i=\bm{1}^T z^+.$$ On the other hand, for $z_j> 0$, we have $$z_j x_j=-\sum_{i\neq j} z_i x_i \leq \sum_{i, z_i<0}-z_i x_i\leq \sum_{i, z_i<0}-z_i=-\bm{1}^T z^-.$$ ${\hfill\ensuremath{\square}}$
Proof of Theorem \[thm:dual\] {#app:thmdual}
-----------------------------
First, we note that (\[feasibility:dual\]) is feasible if and only if the problem $$\label{eq:dddual}
\begin{aligned}
&P_A x= 0,\quad
x> 0\quad
\end{aligned}$$ is feasible. The reason is that (\[feasibility:dual\]) is feasible if and only if there is a positive vector $x=A^Tu>0$ in the row space of $A$. Further, this system is feasible if and only if the following normalized system $$\label{eq:normalized dual}
\begin{aligned}
P_A x= 0,\quad
x\in (0,1]^n
\end{aligned}$$ is feasible. In problem , the bit length of $P_A$ can be the same as $A$, i.e., $O(L)$. Now, the remaining proof is the same as that for Theorem \[thm:primal\]. According to the classic result of [@khachian1979polynomial], there exists a positive number $\tau$ (satisfying $1/\tau=O(2^L)$) such that the positive coordinates of the basic feasible solutions of problem are bounded below by $\tau$. Also according to Lemma \[lm:cutd\], the cut $J$ is nonempty. It means each iteration of MA can rescale at least one coordinate of the feasible solutions by $1/2$. Thus the number of rescaling operations can be bounded by $nL$, i.e., the number of iterations in MA ${\mathsf{T_{MA}}}=O(nL)$. The proof is finished by combining this with Lemma \[lm:tbpd\]. ${\hfill\ensuremath{\square}}$
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract:
- 'Astrobots are robotic artifacts whose swarms are used in astrophysical studies to generate the map of the observable universe. These swarms have to be coordinated with respect to various desired observations. Such coordination are so complicated that distributed swarm controllers cannot always coordinate enough astrobots to fulfill the minimum data desired to be obtained in the course of observations. Thus, a convergence verification is necessary to check the suitability of a coordination before its execution. However, a formal verification method does not exist for this purpose. In this paper, we instead use machine learning to predict the convergence of astrobots swarm. In particular, we propose a weighted $k$-NN-based algorithm which requires the initial status of a swarm as well as its observational targets to predict its convergence. Our algorithm learns to predict based on the coordination data obtained from previous coordination of the desired swarm. This method first generates a convergence probability for each astrobot based on a distance metric. Then, these probabilities are transformed to either a complete or an incomplete categorical result. The method is applied to two typical swarms including 116 and 487 astrobots. It turns out that the correct prediction of successful coordination may be up to 80% of overall predictions. Thus, these results witness the efficient accuracy of our predictive convergence analysis strategy.'
- 'Observatories involved in the generation of spectroscopic surveys always encounter limited resources to check the throughputs of their planned observations before their executions. The information yielded by an observation directly depend on the convergence rate of the observatory’s astrobots in that particular observation. Namely, if the astrobots’ convergence rate is below a minimum, then the observation has to be revoked and re-planned. So, one may define another observation which fulfills the minimum-information requirement. There has been yet no analytical tool developed to verify the convergence rate of the coordination computed by the state-of-the-art trajectory planners of astrobots swarms. Thus, we propose to use a machine learning scheme to predict the desired convergence rate instead of involving in the infeasible process of finding its exact value. This method is a supervised method which requires the target-to-astrobot assignments table of an observation. The algorithm also needs a dataset including previous coordination results of various observations of a particular swarm. The simulated scenarios manifest magnificent accuracies in the convergence predictions of the some astrobots swarms corresponding to modern spectroscopic surveys such as SDSS-V (including $\sim$500 astrobots). Our strategy is based on the smallest subset of the astrobots’ features which have a pivotal role in convergence rates, say, the projected positions of targets on a hosting telescope’s focal plane. We argue that more explorations have to be considered to find other important features, such as the motion direction of each astrobot, which may even further improve the obtained prediction accuracies.'
author:
- 'Matin Macktoobian$^{a}\footnote{[email protected]}$, Francesco Basciani$^{b}$, Denis Gillet$^{a}$, and Jean-Paul Kneib$^{a}$'
bibliography:
- 'references.bib'
date: |
$^a$EPFL, Lausanne, Switzerland\
$^b$ Turin Polytechnic, Turin, Italy\
nocite: '[@*]'
title: 'Data-Driven Convergence Prediction of Astrobots Swarms[^1]'
---
**keywords**: astrobotics, convergence prediction, machine learning, swarm robotics, spectroscopic surveys, astronomical instrumentation
Introduction[^2]
================
The unknown nature of dark matter and dark energy is among the most major gaps in the modern physics [@arkani2009theory]. Cosmology have actively sought the history of the universe, which is known to be tied with the evolution of dark matter. A unified mathematical model of dark matter has not yet been achieved based on analytical methods [@jungman1996supersymmetric]. Thus, cosmologists have shifted their attention to observational data in various red shift ranges [@zhao2017dynamical; @newman2015spectroscopic]. Each range of redshift represents a particular time interval corresponding to the universe’s lifespan. So, the recent trend in dark matter studies aims to generate the map of the observable universe. Then, the analysis of such a map would eventually reveal new findings about the distribution of dark matter all over the cosmos. Cosmological spectroscopy is the front-runner technique to contribute to the cited goal. In particular, dominant massive objects of the universe, say, galaxies, quasars, etc., all emanate electromagnetic radiations. These radiations can be captured in particular wavelengths by optical fibers mounted on specific ground telescopes. For this purpose, many optical fibers are placed at a particular area of a candidate ground telescope which is called focal plane. The generation of the map of the observable universe is not a trivial task given the huge number of the target objects residing in it. So, a set of observations are defined each of which includes a subset of the all those targets. In this regard, the local map of each observation is a survey. So, the eventual accumulation of many surveys gives rise the complete map of the observable universe. To observe those objects, i.e., capturing their light, each target has to be assigned to one of the optical fibers of a telescope. Then in the course of an observation’s exposure time, the desired rays are collected by the optical fibers. Later, a spectrograph connected to the optical fibers synthesizes the spectroscopic survey corresponding to the planned observation. The number of spectroscopic survey projects has been increased during of the recent decade the most prominent of which are DESI [@flaugher2014dark], MOONS [@cirasuolo2014moons], PFS [@ellis2012extragalactic], SDSS-V [@kollmeier2017sdss], LSST [@mandelbaum2019wide], MegaMapper [@schlegel2019astro2020], etc.
Each observation comprises a unique set of targets. The location of each target obviously differs from those of other targets. Thus from one observation to another, one has to change the configuration of the fibers so that tip of each fiber points its new target associated with a new observation. In the first generation of spectroscopic surveys, these coordination procedures were manually done using various techniques such as magnetic fiber technology [@lewis2014fibre; @fabricant2005hectospec] and slit masks [@dressler2006imacs; @mclean2010design]. However, such manual coordinations are proved to be inefficient in the case of the requirements of the recent advanced spectroscopic projects. First, the current projects are equipped with hundreds to thousands of fibers. The available time to coordinate fibers from one observation to another is limited. In this case, if the fibers are not coordinated on time, their observation’s data would be partially collected in the best case. On the other hand, each observation depends on many celestial factors whose second fulfillment may require very long times. So, planned observations must not be missed according to survey programs. In other words, cosmologists need fast automatic coordination of fibers. Second, the more fibers one places in a focal plane, the larger surveys may be taken into account. Increasing the density of fiber placements makes manual coordination even more challenging because they may disturb the calibration of fibers’ tips. This also magnifies the need to minimize human interventions in the coordination processes.
To resolve the issues above, the idea of astrobotics have been emerged. Each astrobot [@horler2018robotic] is a two-degree-of-freedom rotational-rotational manipulator which contains a fiber. To be specific, a fiber passes through the central axis of its astrobot so that the fiber’s tip is located at the end-effector of the astrobot called ferrule. In this case, the tip of the fiber indeed can reach any point in the circular surface corresponding to the working space of its astrobot’s ferrule. So given any target assigned to a fiber [@morales2011fibre; @macktoobian2020optimal], should the target reside in its astrobot’s working space, the astrobot may be controlled so that its ferrule reaches the projected location of the target on the focal plane. As stated before, one intends to maximize the number of the fibers on a telescope. Thus, astrobots have to be placed, in hexagonal formations, so close to each other that their working spaces unavoidably overlap. These overlapping areas imply the possibility of collisions between various astrobots in the course of their coordination toward their targets. So, the coordination problem of astrobots swarms is inherently safety-critical for which various control strategies were proposed. For example, nonlinear hybrid control was taken into account [@makarem2016collision; @tao2018priority] to realize not only collision avoidance but also coordination priority for the astrobots whose targets are more important than those of other targets in view of the signals they collect. This method cannot generally coordinate all astrobots, so the observational information reflected into surveys are not maximized. In this regard, the formulation of nonlinear hybrid control was revised [@macktoobian2019navigation; @macktoobian2019complete] so that one can check whether or not a particular setup of astrobots can be totally converged to their targets. This method, despite of its merit in completeness determination, is computationally so expensive that its real-time application may not be always feasible if the available times between successive coordination are too short. The convergence rate assessment of coordination may be done using numerical simulations of coordination with respect to various observation settings. This procedure is useful for small and medium surveys but not massive ones. Namely, convergence rate assessment requires the real-time solutions of hundreds to thousands of interdependent differential equations corresponding to distributed navigation functions of astrobots. Such analyses may not be feasible in the case of tight observation schedules in which the available times between observations are not long enough. If such assessment is possible, then inefficient coordination can be re-planned to those whose information throughput satisfy surveys expectations. In particular, a coordination output directly depends on the target-to-astrobot assignments corresponding to its observation. One may revisit an assignment to yield better coordination, thereby achieving higher convergence rates. Supervisory control was also employed to synthesize control commands whose safety and completeness can be formally verified [@macktoobian2019supervisory]. However, this strategy also becomes inefficient because of the curse of dimensionality in the case of crowded astrobots swarm.
Literature Review
-----------------
Machine learning techniques have been partially contributed to the trajectory planning of multi-agent systems. For example in [@su2011using], an anomaly network traffic identification problem is studied for autonomous vehicles. This problem conceptually resembles the collision avoidance aspect of our prediction problem. In this method, the overall working space of the problem is so vast, yet the number of the number of vehicles are relatively small. So, collision avoidance is not a critical issue in the assumed sparse distribution of vehicles. In contrast, our convergence prediction problem indeed implies hazardous interactions in dense formations of astrobots, thereby entailing considerable risk of collisions between them. Additionally, our convergence prediction problem also features noticeable sensitivity to even trivial spatial deviations of configurations in terms of convergence results. On that account, any potential dataset representing our problem needs to encompass sufficient data to cover a wide range of similar configurations. A similar study takes the idea of moving ranges into account to assess neighbors more effectively for the vehicles in crowded urban areas [@lee2015moving]. The predictive model generated by this scheme relaxes the structured assumption by allowing movements of uncertain objects. The aforesaid relaxation, though, complicates the compliance with the safety requirements of this scenario. Instead, our problem enjoys the fully structured dynamics of astrobots swarm. Namely, the extremely constrained dynamics of each astrobot does not exert any uncertain feature to the prediction problem. As another example, learning-based strategies have been employed to predict trajectories of multi-agent systems in unconstrained or loosely constrained systems. For instance, route prediction for ships was investigated [@duca2017k]. This study uses a variation of $k$-NN algorithm which exclusively models each ship as an isolated entity, say, in the absence of any collisions with other peers.
Limited applications of machine learning in trajectory prediction of more complex swarms are also reported. To give an instance, a class of aggregating behaviors in a self-organizing swarm were the subject of a prediction problem [@khaldi2018self] using distance-weighted $k$-NN method [@jin2019improvement; @cataloluk2012diagnostic; @liu2011class; @gou2012new]. The density metric of the swarm is modeled by hydrodynamical particle interpolation. This system seeks predictions through fairly complicated movements scenarios. However, the goal is the classification of collective behaviors while the involved non-interacting agents are subject to no collisions. Collision freeness was interestingly taken into account in a coordination scenario using artificial potential fields [@chen2018collision]. This work is relatively comparable to what we seek in this paper, because the coordination control of astrobots is based on a class of artificial potential fields. However, the prediction application in this method is trivially concentrated on finding the closest point of an obstacle to a robot. Put differently, this strategy only guarantees collision freeness between a single robotic arm and a human’s hand. Thus, in the absence of other agents, the complexity of this scenario is significantly less than what one encounters in the convergence prediction of astrobots.
[.50]{} {width="\textwidth"}
[.4]{}
The machine-learning-based behavioral predictions for multi-agent swarms have not been extensively studied. In particular, a learning system can efficiently train a model of a system if one feeds the data corresponding to all important features of that system. In the case of multi-agent swarms, these feature sets are often so large that final models may not be applicable for various reasons. First of all, training a predictive model requires enough data representing the behavioral patterns of system. The more complicated a system is, the more data of it one needs to effectively synthesize a predictor for it. The complexity of multi-agent swarms then requires huge datasets exhibiting their behaviors. But such amounts of data are often not available specially in the case of heterogeneous swarms. Moreover, a swarm system’s functionalities are generally subject to many constraints whose presence may easily drive any learning model of that swarm toward common machine learning issues like underfitting and overfitting. Accordingly, the complete convergence of astrobots in the course of their coordination has not yet been efficiently resolved for the swarms including thousands of astrobots. On the other hand, partial coordination may lead to small convergence rates according to which the lack of enough data gives rise to the generation of the surveys whose wealth of information and details are not sufficient. Thus, instead of questing after analytical solutions to the completeness checking problem in more efficient ways, we shift our perspective to the prediction of complete coordination. In this framework, we seek to compute some models based on the data obtained from former coordination to predict the convergence rates of future ones in terms of some particular features. For this purpose, we propose a prediction algorithm based on the idea of weighted $k$-NN [@peterson2009k], given the relative simplicity and design intuitions which stems from the geometrical formulation of $k$-NN-driven strategies. Subject to a set of astrobots assigned to their targets, our method predicts whether or not each astrobot would successfully converge to its target spot. The applied evaluations to simulated results using our scheme exhibit high performances in those predictions.
Contributions
-------------
We establish a predictive algorithm which paves the way for assessing the suitability of a particular astrobots-to-targets mapping set in terms of its expected information throughput. In other words, we propose a predictive solution to the decision making problem of whether a particular set of astrobot-to-target pairings would give rise to our expected number of successfully converged astrobots. This achievement is quite important if one takes the notion of observation priority in the definition of a survey plan. In particular, each survey plan may include some targets whose observations have more remarkable impact on the quality of the final survey. In this regard, a successful coordination may be defined as the one through which the astrobots corresponding to high-priority targets can be reached. Our algorithm individually predicts the convergence of each astrobot. Thus, the priority-based decision making process may also be covered using out method by exclusively focusing on the reachability prediction of high-priority targets.
A coordination process is a finite set of movements corresponding to each astrobot of a swarm with respect to many functional and safety requirements. A formal convergence verification tool has to check every single coordination step according to the control signals generated for each astrobot in every step. However, the discussion presented in the previous sections clarified that such exact approach to convergence analysis may be practically infeasible. Thus, among all steps of a coordination process, our algorithm merely works based on the first (i.e., initial) and the last (i.e., final) astrobots-targets configurations of the process. Another challenge raises from the imbalanced nature of the data in our problem. Namely, the convergence rate of large astrobots swarms generally varies between $65\%\sim85\%$ depending on their populations. Thus, the number of the astrobots which converge is noticeably larger than those which don’t converge. So, the data are inherently imbalanced. It is widely observed that imbalance data may adversely impact the output of any naive machine learning algorithm which does not counteract against this issue. The applied simulations of our algorithm with respect to various populations of large astrobots manifest its effectiveness in terms of various performance measures.
Paper Outline
-------------
The remainder of the paper is structured as follows. Section \[sec:rev\] present a brief review on astrobots characterization and the swarms constructed by them. The important elements which play crucial rules in modeling individual astrobots and their swarms through coordination are illustrated. We then shift our attention to the specify the convergence prediction problem in Section \[sec:ps\]. We particularly focus on the features according to which a data-driven solution to the convergence prediction problem is indeed challenging. Section \[sec:cps\] comprises a weighted $k$-NN-based solution to the cited problem. We then present detailed statistical analysis to express the credibility of our algorithm in Section \[sec:sim\]. We indeed apply our algorithm to two complex instances of astrobots swarms which include 116 and 487 astrobots. In the end, Section \[sec:conc\] reflects our conclusions and discusses potential search ideas to improve our results in future.
A Review on Astrobots Swarms {#sec:rev}
============================
Each astrobot is a robotic manipulator with two degrees of freedom whose schematic is depicted in Fig. \[fig:pos\]. It is an active placeholder for the fiber which is passed through its central axis. So, the astrobot has to move its end-effector, called ferrule, so that the fiber may reach any point corresponding to the working space of the astrobot. For this purpose, two rotational arms of the astrobot represent its two degrees of freedom as shown in Fig. \[fig:top\]. The overall length of two arms is long enough to each the centroid of any neighboring astrobot. So, astrobots can overall reach the whole surface of the focal plane which is a particular area of the telescope at which astrobots are mounted. All fibers are connected to a spectrograph, see, Fig. \[fig:spec\], which is located at the back of the focal plane. The spectrograph processes the signals collected by fibers to generate the survey corresponding to each observation.
Because of astronomical requirements, astrobots are densely located in hexagonal formations in their hosting focal plane as illustrated in Fig. \[fig:swarm\]. Such placement paves the way for the proper functionality of each fiber in terms of focal plane coverage and signal capturing. However, many challenges rise in view of coordinating astrobots from their initial configuration to a desired one by a particular observation. First, the dense formation of astrobots severely makes them subject to collisions. Thus, a controller has to plan some trajectories for each astrobot by which its ferrule reaches it desired spot. It is practically observed that if astrobots starts their movements from an arrangement in which their distances from each other is maximum, the planned trajectories would be obtained more efficiently. Thus, to alleviate the trajectory planning complexities in these highly-dense systems, astrobots are always reconfigured to their folded formation in which $\theta = 0$ and $\phi = \pi$ as rendered in Fig. \[fig:full\].
Problem Statement {#sec:ps}
=================
The more astrobots converge to their target spots, the more the throughout of the observation associated with the targets will be. The current trajectory planners are not always able to achieve desired high convergence rates [@makarem2016collision]. If a convergence rate is below a certain threshold, then its corresponding final survey will not represent the expected quality. Thus, one has to assess the performance of a potential coordination process in terms of its final convergence before its execution. The analytical [@macktoobian2019complete] and logical [@macktoobian2019supervisory] tools to verify the results before their execution are often computationally too expensive. In this regard, these methods may not be used in real-time scenarios when the time slots available between observations are too short. The cited tools analyze every coordination step to check the collision freeness of motions which eventually tend to final configurations of astrobots. However, in this research, we only take the initial configuration of astrobots and the locations of their targets into account. We intend to predict whether or not a particular number of astrobots completely converged in the course of an observation surpasses the minimum number of desired convergences. Then, if the predicted convergence rate is larger than the minimum expectation, then we decide to let the trajectory planner coordinate our swarm. Otherwise, we re-plan the unsatisfactory astrobot-to-target assignments to yield better combinations.
The problem statement is graphically shown in Fig. \[fig:arch\] in which we seek the synthesis of a predictor to solve the problem. In particular, we prepare a dataset including many coordination scenarios with respect to multitude of astrobot-to-target assignment pairings which had been already simulated and/or executed. In this dataset, each astrobot in each pairing is labeled by 1 (resp., 0) if it finally reaches (resp., doesn’t reach) its target. The overall set of this results is called *ground truth vector*. We use these data to predict convergence rates using a weighted $k$-NN-based strategy. Since the number of converging astrobots is often larger than that of those which doesn’t converge, our data are inherently biased. Such imbalance data have to become balanced to make predictions reliable. We also only consider safe coordination scenarios in our dataset.
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Convergence Prediction Strategy {#sec:cps}
===============================
In this section, we elaborate on our convergence prediction algorithm, as shown in Fig. \[fig:alg\]. We first compensate the imbalanced data issue using a set of vector weights applied to our data. Then, a distance metric is defined to rank the astrobots neighborhoods with respect to a desired astrobot whose convergence is intended to be predicted. A prediction probability is computed associated with each astrobot. We then note that the prediction problem of each astrobot has to be essentially analyzed in its own neighborhood. Thus, we localize the analysis which is mathematically equivalent to a particular normalization of the quoted prediction probabilities. Next, given a particular *decision filter*, we transform the obtained probabilities to either of two categorical outcomes. Each of these outcomes represents the prediction of our algorithm regrading the successful or the unsuccessful convergence of their corresponding astrobots. We finally perform Monte Carlo cross-validation [@xu2001monte] to assess the reliability of the results of our algorithm.
One notes that the coordinate associated with each astrobot’s initial configuration is fixed (see, Fig. \[fig:full\]), and it does not impact the coordination phase. Thus, in the prediction process, we define the astrobot vector $\bm{\pi}$ according to the location of its projected target on the focal plane of the swarm as follows[^3]. $$\bm{\pi} \coloneqq \begin{bmatrix}
x_{t} & y_{t}
\end{bmatrix}^\intercal$$ Then, the *configuration matrix* $\bm{P}$ of a specific swarm $P$ including $n$ astrobots is indeed the accumulated configurations of its constituting astrobots which is $$\bm{P} \coloneqq \begin{bmatrix}
x_{t^1} & x_{t^2} & \cdots & x_{t^n}\\
y_{t^1} &y_{t^2} & \cdots& y_{t^n}
\end{bmatrix}^\intercal.$$ The ground truth vector corresponding to $\bm{P}$ is $\bm{g^{P}}$. This vector represents the a posteriori information regarding the convergence of its corresponding configuration stored in a dataset. The more configurations exist in the dataset, the more representative the dataset is for its swarm. Since there are infinite number of configurations associated with a swarm, it is impossible to accumulate any possible coordination scenario in the dataset. However, the dataset has to be representative enough because changing the location of a target for just some tenths of millimeters just may change a successful convergence to a deadlock situation or vice versa. The dataset has to be divided into train and test partitions whose division proportion is discussed in Section \[sec:sim\].
Imbalanced Data Compensation
----------------------------
The family of $k$-NN algorithm is very sensitive to the local structure, i.e., the geometry, of data. We particularly enjoy this feature because the convergence prediction problem directly depends on geometrical characteristics of astrobot vectors. As already noted, configuration matrices often include many 1s compared to 0s because the majority of astrobots can be successfully coordinated using a swarm controller. So, their dataset is imbalanced according to which $k$-NN-based algorithms do not properly work [@dubey2013class]. There are two typical approaches to resolving this issue neither of which is effectively applicable to our case. In particular, one may perform an oversampling (resp., undersampling) on the minority class (resp., majority class). This approach is infeasible in our case because an oversampling on the minority class requires the configurations whose ground truth vectors have more 0s than 1s. In the case of huge swarms, such configurations are extremely rare, if not nonexistent. Even if one could find such configurations, the next step would be the generation of a new group of targets which are very close to the targets of that configuration. But, it would be so likely that many 1s are also generated, thereby essentially canceling the purpose of oversampling. On the other hand, any undersampling needs to remove all the configurations whose ground truth vectors include more 1s than 0s. However, it gives rise to the loss of valuable information which are important for potential prediction cases.
Instead, we devise a vector of weights to enhance the impact of 0s in the ground truth vector of a specific configuration. This strategy is similar to the idea of class confidence weights [@liu2011class]. The difference is that we apply the weights to single astrobots, not to data samples, i.e., configurations. Given a configuration $\bm{P}_{i}$ where $i \in \{1,2, \cdots, N\}$, assume that ground truth vector $\bm{g^{P}}_i$ is associated with it. We define *frequency vector* $\bm{u}$ and its complement, say, *pseudo vector* $\bm{v}$ as follows. $$\begin{split}
\bm{u} &\coloneqq \sum_{i} \bm{g^\bm{P}}_i\\
\bm{v} &\coloneqq N\cdot\mathds{1}_{1\times N} - \bm{u}
\end{split}$$ Then, the elements[^4] of *weight vector* $\bm{w} \coloneqq \bigcup\limits_{i}w_{i}$ read as below. $$w_{i} \coloneqq
\begin{dcases*}
u_{i} & if $v_{i} = 0$\\
\dfrac{u_{i}}{v_{i}} & otherwise
\end{dcases*}$$ Each element of $\bm{w}$ has to be applied to the 0s of a particular astrobot of the configuration. We apply different weights to different astrobots because those which are in total neighbourhoods, i.e., surrounded by 6 astrobots, generally don’t reach their target positions as frequent as those which are in partial neighbourhoods configuration. So, the 0s of the astrobots in total neighbourhood configurations have smaller weights compared to those in partial neighborhoods. The notion of weight vector efficiently compensates the problem of imbalanced data. However in our problem, the two classes have not the same importance. In other words, we are more interested in the correct predictions of 1s rather than 0s in an operational point of view. So, we tune the elements of weight vectors according to our prediction requirements using two corrector coefficients $\alpha$ and $\beta$ on which we elaborate in Section \[sec:sim\].
Prediction Probability Computation
----------------------------------
We define a distance metric to quantitatively compare various configurations with each other. Let $\bm{T}$ be a test configuration, say, the one we are interested in predicting its convergence. Let also $\bm{P}_{i}$ be a train configuration. We define *distance metric* $\Delta(\cdot,\cdot)$ which later is used to find the close train configurations to a particular test one as below[^5] $$\Delta(\bm{T},\bm{P}_{i}) \coloneqq \sum\limits_{j}\norm{\bm{T}_{j}-\bm{P}_{i,j}}.$$ Here, $\bm{T}_{j}$ and $\bm{P}_{i,j}$ corresponds to the $j$th columns (i.e., astrobots) of $\bm{T}$ and $\bm{P}_{i}$, respectively.
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Now, we select the *$k$ closest configurations set*, say, $\bm{P}^{\bm{T},k} \subset \bm{P}$, to $\bm{T}$. The specification of $k$ depends on the size of the train dataset and the complexity of the intended swarm. Namely, it must not be too small, otherwise there is some overfitting risk corresponding to the test configuration. On the other hand, if it is too large, one may take some train configurations into account which do not resemble the desired test one. So, it may lead to inaccurate predictions associated with some astrobots. Assume that function $\text{sort}(\text{set},\text{metric})$ sorts its set argument with respect to its metric argument in ascending order. Moreover, fix function $\text{fetch}(\text{set},k)$ which returns the first $k$ elements of its sorted argument set. Then, given a particular $k$, $\bm{P}^{\bm{T},k}$ is defined as follows. $$\bm{P}^{\bm{T},k} \coloneqq \text{fetch}\bigg(\text{sort}\Big(\{\bm{P}_{i} \mid \bm{P}_{i} \in \bm{P} \},\Delta(\bm{T},\bm{P}_{i}) \Big),k\bigg)$$ Now, we use weight vector $\bm{w}$, $\bm{P}^{\bm{T},k}$, and its corresponding ground truth vector $\bm{g}^{\bm{P}^{\bm{T},k}} \coloneqq \bigcup\limits_{i}g^{\bm{P}^{\bm{T},k}}_i$ to compute the predictions corresponding to astrobots which converge to configuration $\bm{T}$. One notes that $\bm{w}$ has to be exclusively applied to the 0s in each element of ${}^{\bm{w}}\bm{g}^{\bm{P}^{\bm{T},k}} \coloneqq \bigcup\limits_{i}{}^{\bm{w}}g^{\bm{P}^{\bm{T},k}}_i$. Then, the result is *weighted ground truth vector* whose elements are defined as below. $${}^{\bm{w}}{g}^{\bm{P}^{\bm{T},k}}_{i} \coloneqq
\begin{dcases*}
1 & if ${g}^{\bm{P}^{\bm{T},k}}_i = 1$\\
{w}_{i} & otherwise
\end{dcases*}$$ Thus, *primary prediction probability vector* $\bm{\hat{\Gamma}}^{\bm{P,T}}$ with respect to test configuration $\bm{T}$ is given by[^6] $$\bm{\hat{\Gamma}}^{\bm{P,T}} \coloneqq \bigg(\sum_{i=1}^{k} \bm{g}^{\bm{P}^{\bm{T},k}}_i\bigg)\oslash\bigg(\sum_{i=1}^{k} {}^{\bm{w}}\bm{g}^{\bm{P}^{\bm{T},k}}_{i}\bigg).$$ One may alternatively plan to apply different weights to each ground truth vector with respect to its distance metric from a particular test configuration. However, it increases the risk of overfitting.
If one deals with very large astrobots swarms, the distance metric $\Delta$ may not be reliable to assess the similarity between two configurations. In fact, once the number of astrobots extremely increases, there may be some astrobots among the closest train configuration whose targets are too far from their corresponding ones in the test configuration. This may be problematic even in the case of small swarms. In the next section, we mitigate this issue by localizing the derived prediction probability vector.
Prediction Probability Localization
-----------------------------------
The global neighborhood analysis of a large astrobots swarm is both inefficient and even problematic in view of the final results. In particular, large swarms geometrically encompass a massive number of neighborhoods. If one checks all available neighborhoods in the course of each lazy evaluation of the algorithm, then the solution would never be obtained after a reasonable amount of time. On the other hand, not all astrobots neighborhoods influence the coordination of a particular astrobot, but only those which are its immediate neighbors. Thus, we have to localize the probability computations of the algorithm. In particular, we perform a local analysis on the neighborhoods of each astrobot. Thereby, the algorithm is exclusively applied to a number of small configurations which includes a maximum number of 7 astrobots. By doing so, it would be less likely to have some astrobots the distances between whose targets and a test configuration are high. For example, Fig. \[fig:neigh\] depicts a neighborhood of astrobots the magnitudes of whose metric distances are reasonable as illustrated in Fig. \[fig:loc\].
Let $\bm{P}$ be a configuration of including $n$ astrobots. We define neighborhood $\bm{\nu_{\pi}} \subset \bm{P}$ associated with a typical astrobot $\pi$ as the central entity of this neighborhood. The dimension of each instance of $\bm{\nu_{\pi}}$ is $2\times r$, where $1\le r \le 7$ denotes the number of the astrobots in the neighborhood. Thus, one has to overall perform $n$ local analyses. To do so, we introduce *counter vector* $\bm{\eta}$ whose dimension is $1\times n$. Element ${\eta}_{i}$ of $\bm{\eta}$ corresponds to the number of the neighborhoods to which the $i$th astrobot of the swarm belongs. The elements of $\bm{\eta}$ are integers varying between 1 and 7. Thus, we yield *neighborhood probability vector* $\widetilde{\bm{\Gamma}}^{\bm{\nu_{\pi},T}}$ with respect to neighborhood $\bm{\nu_{\pi}}$ whose elements are defined as follows. $$\widetilde{\Gamma}^{\bm{\nu_{\pi},T}}_{i} \coloneqq
\begin{dcases*}
{\hat{\Gamma}}^{\bm{\nu_{\pi},T}}_{i} & if ${\pi}_{i} \in \bm{\nu_\pi}$\\
0 & otherwise
\end{dcases*}$$ Now, given, $\widetilde{\bm{\Gamma}}^{\bm{\nu_{\pi},T}}\coloneqq \bigcup\limits_{i} \widetilde{\Gamma}^{\bm{\nu_{\pi},T}}_i$, *final probability vector* $\bm{\Gamma^{P,T}}$ is computed as $$\bm{\Gamma^{P,T}}\coloneqq \left[\sum\limits_{\bm{\pi}}\widetilde{\bm{\Gamma}}^{\bm{\nu_{\pi},T}}\right] \oslash \bm{\eta}$$ Finally, we need to transform these probabilistic entries to categorical 1s and 0s to represent successful or failed convergences, respectively. For this purpose, given a *decision filter* $q$, we define the elements of *output vector* $\bm{Y} \coloneqq \bigcup\limits_{i} y_{i}$ as below. $$y_{i} \coloneqq
\begin{dcases*}
1 & if ${\Gamma}_i^{\bm{P,T}} > q$\\
0 & otherwise
\end{dcases*}$$
![The flow of the convergence prediction algorithm[]{data-label="fig:alg"}](alg.pdf)
The last phase of our convergence prediction algorithm performs Monte Carlo cross-validation to assess the credibility of the results. The rational behind preferring this method over $k$-fold cross-validation is the computational efficiency of the former. Namely, Monte Carlo cross validation enjoys a property that the proportional relation between train/test splits does not depend on the number of validation iterations. Thus, one can perform a series of iterations which are not linked to the dimensions of train and test datasets. The drawback of this method, though, is that some configurations may never be selected as test configurations, whereas others may be selected multiple times. For this reason, it is necessary to put a particular attention to the number of validation iterations. The choice of this number depends on how large a typical test dataset is compared to its corresponding train one. In other words, the smaller the test dataset is, the larger the number of iterations has to be.
Simulations and Results {#sec:sim}
=======================
In this section[^7], we illustrate how our algorithm efficiently predict the convergence of astrobots in massive swarms. We first define our evaluation measures and performance metrics. Then, we take two swarms into account each of which is constituted by 116 and 487 astrobots. The latter is particularly similar to the case of the astrobots swarm associated with the SDSS-V project [@kollmeier2017sdss]. We also present some hints regarding the value selections for the algorithm’s hyperparameters.
Performance Measures
--------------------
Our performance measures are essentially defined based the following four notions.
- A true positive (TP) is an astrobot which is predicted to converge (the predictor predicts 1), and it actually converges to its target position (its corresponding ground truth element is 1).
- A false positive (FP) is an astrobot which is predicted to converge (the predictor predicts 1), but it actually does not converge to its target position (its corresponding ground truth element is 0).
- A true negative (TN) is an astrobot which is not predicted to converge (the predictor predicts 0), and it actually does not converge to its target position (its corresponding ground truth element is 0)
- A false negative (FN) is an astrobot which is not predicted to converge (the predictor predicts 0), but it actually converges to its target position (its corresponding ground truth element is 1).
![The layout of the information represented by a typical confusion matrix[]{data-label="fig:conf"}](matrix.pdf)
Accordingly, we take the standard rates of the above factors, i.e., TPR, FPR, TNR, and FNR, into account. These values are reported in confusion matrices based on Fig. \[fig:conf\]. If a predictor yields good performance, its corresponding confusion matrix has large values in its main-diagonal entries, indicating that the majority of samples have been correctly classified.
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From an engineering point of view, we are more interested in the correct predictions of positives (the astrobots which converge to their target positions). It is because the information regarding the prediction of these astrobots would be crucial to decide whether or not a coordination process should be executed associated with a particular configuration of targets. On the other hand, the number of positives is much greater than that of negatives. If the predictor always predicts 1, the TPR would be perfect. But, the predictor does not indeed predict anything by completely neglecting 0s. So, we track the balanced accuracy measure established as the average of the TPR and the TNR to better assess the predictive essence of the algorithm. We also employ receiver operating characteristic (ROC) curves to illustrate the performance of our predictor in the course of varying one of its hyperparameters.
We also take precision and F1 score (harmonic mean) into the consideration. The precision measure is an index of how accurate the predictor is in predicting positives. Precision is an important measure to look at when FPs have significant impacts on our problem. We intend to maximize precision through minimization of FPs. F1 score indicates the trade-off between precision and the TPR, say, recall. For example, if we increase the TPR, we indeed increase the number of predicted TPs. However, we also increase the number of FPs, thereby decreasing the precision. The bigger the F1 score is, the better the trade-off between precision and recall is.
We include corrector coefficients $\alpha$ and $\beta$, as well. These hyperparameters are used to manually tune the weight vector $\bm{w}$ to obtain better accuracy rates with respect to positives and negatives. In particular, $\alpha$ tunes the ${w}_{i}$s of the astrobots in total neighborhoods, while $\beta$ does the same but for the astrobots residing in partial neighborhoods. In all simulations, we take the decision filter $q = 0.5$.
A swarm including 116 astrobots
-------------------------------
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Our complete dataset is composed of 10100 configurations, where the train and the test datasets include 10049 and 51 configurations, respectively. The algorithm iterations is set to 15. The confusion matrices corresponding to various values of $k$ are depicted in Fig. \[fig:116-confmat\]. We observe that increasing $k$ increases and decreases the TNR and the TPR, respectively. It is reasonable since the more train configurations we take into account for the computation of the output, the higher the likelihood is to consider the train configurations whose astrobots don’t converge. The selection of $k$ depends on how large the train dataset is. The larger the train dataset is, the larger $k$ may be. In this scenario, a proper $k$ may be chosen in the range of 10 to 50. If we increase $k$ too much, the information about the targets locations of the closest train configurations are no longer reliable.
It is also interesting to assess our performance indices for single astrobots. In particular, we obtain the TPR, the TNR, and the balanced accuracy, on the basis of the number of each astrobot’s neighbors. To do so, we take the average of the performances of the astrobots with a specific number of neighbors, as rendered in Fig. \[fig:116-neigh-ana\], where both corrector coefficients are 1. Namely, Fig. \[fig:166-av-13\] indicates that the prediction accuracy bottleneck refers to the astrobots in total neighborhoods. Fig. \[fig:166-av-39\] illustrates how the the balanced accuracy is improved in total neighbourhoods. On the other hand, the astrobots of partial neighborhoods experience the decrement and the increment of the TPR and the TNR, respectively.
Corrector coefficients are expected to impact the qualities of the cases in which total neighborhoods are fairly abundant. In Fig. \[fig:116-ab\], the confusion matrices of three different predictions are reported in which corrector coefficients are varied. In this case, we simply keep their values the same to show the overall effect of magnifying the weights of 0s. In all of the cases, we have $k=13$. In particular, it is evident that increasing the corrector coefficients leads to the increment and the decrement of TNR and TPR, respectively, which is a direct effect of increasing the weight of the minority class.
To tune the hyperparameters, one may find Fig. \[fig:116-ref\] very useful. Fig. \[fig:166-k-ana\] illustrates that any $k>21$ is stable. Specially, $k=39$ realizes the best predictive performance for this swarm. One may note that the right choice of $k$ directly depends on what factor is the main goal of the prediction to be improved. For example, if we one would like to increase the balanced accuracy as much as possible, yet allow the TPR to drop under $80\%$, then $k=39$ seems to be the best choice. But, if the TPR has to be over $80\%$ with the maximum balanced accuracy, one may pick $k=13$. The dynamical trends of the TPR, the TNR, and balanced accuracy are also evident in Fig. \[fig:166-w-ana\] in the course varying the two corrector coefficients while fixing $k=13$. So, since we are more interested in the correct predictions of the positives, $k$ may be chosen large to increase the TPR as much as possible, while assuring that the balanced accuracy does not drop below a certain threshold. Moreover, Fig. \[fig:166-prf1\] shows the trends of the precision, the recall and the F1 score for different values of $k$. Table \[tbl:1\] reflects the best results in the convergence predictions of the 116-astrobots swarm. Finally, we can look at the ROC curve which visualizes the performance of our predictor. Every point on the ROC curve represents the result of a prediction experiment using a different value of $\alpha(=\beta)$ as shown in Fig. \[fig:116-roc\]. Here, we have $k=13$.
![The ROC curve associated with the 116-astrobots swarm[]{data-label="fig:116-roc"}](KNN116_ROC.pdf)
A swarm including 487 astrobots
-------------------------------
The qualities of the results in this case fairly follows the qualities of the 116-astrobots case. So, we observe that our algorithm performance remains relatively acceptable even in the case of very complex swarms. To support this claim, we consider a dataset including 10100 configurations, 10049 of which are used to train the predictor and the remaining 51 ones are test configurations. The number of iterations are 15.
------------------ ---- ---------- --------- -------- -------- ---------------------- -------------- -------
Swarm population K $\alpha$ $\beta$ TPR(%) TNR(%) Balanced accuracy(%) Precision(%) F1(%)
31 1 0.9 79.3 64.7 72 91.51 84.97
39 1 0.85 88.44 63.23 71.83 91.4 85.57
39 1 1 77.2 68.51 72.85 92.22 84.04
39 1 0.9 79.94 60.73 70.33 89.51 84.45
51 1 0.88 80.20 60.97 70.59 89.52 84.61
51 1 1 78.23 63 70.62 89.78 83.61
------------------ ---- ---------- --------- -------- -------- ---------------------- -------------- -------
\[tbl:1\]
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Confusion matrices of Fig. \[fig:487-con-neigh\] reiterate the point that larger $k$ values give rise to the better predictions of the negatives. Fig. \[fig:487-neigh\] witnesses the decrement of the balanced accuracy compared to the 116-astrobots swarm. The reason is that the 487-astrobots swarm comprises more total neighborhoods than the 116-astrobots swarm. The stability analysis of this case, similar to the previous case, also indicates the variations of the accuracy rates with respect to the hyperparameters as shown in Fig. \[fig:487-k-ab\]. In particular, Fig. \[fig:487-k\] exhibits that the algorithm is stable for $k>21$. Moreover, Fig. \[fig:487-prf1\] illustrates the upper bound of the precision which is around $90\%$. Finally, we observe that the algorithm works on this 487-astrobots swarm almost as good as the 116-astrobots one. Namely, Fig. \[fig:487-prf1-comp\] exhibits the ROC curve of the 487-astrobots case which is trivially closer to the random guess line compared to that of the 116-astrobots swarm.
Concluding remarks {#sec:conc}
==================
The first solution to the convergence prediction of populated packs of astrobots is studied. We observe that astrobot-to-target assignments provide a necessary feature subset of an astrobots swarm feature set to reach $\sim$80% of accuracy in predicting the completely-converging set of the pairings. The $k$-NN nature of the proposed algorithm makes the metric design process intuitive enough to exploit the geometrical characteristics of astrobots and their neighborhoods. The presented strategy also enjoys a fairly restricted number of hyperparameters. So, the design process is not only relatively straightforward but tuning processes also require less computational resources.
This research indeed takes only necessary positional features of swarms to predicate convergences. However, it is imperative to look for extra features which obtain better accuracies such as parity, i.e., the motion direction of an astrobot. Needless to say that such feature expansion jeopardizes the computational efficiency of the prediction process as a trade-off. One may also utilize neural networks to train predictors which may noticeably provide noticeable accurate results. However, neural networks include many hyperparameters whose proper setting may be challenging specially if one would like to avoid computationally intensive grid searches.
[^1]: This work was financially supported by the Swiss National Science Foundation (SNF) Grant No. 20FL21\_185771 and the SLOAN ARC/EPFL Agreement No. SSP523.
[^2]: Throughout this paper, scalars and (sets of) matrices are represented by regular and bold symbols, respectively.
[^3]: Unary operator $(\cdot)^\intercal$ represents the transpose of its matrix argument.
[^4]: $n$-ary operator $\bigcup\limits_{i}(\cdot)_{i}$ constructs a vector of the operator argument.
[^5]: Unary operator $\norm{\cdot}$ denotes the Euclidean norm of its vector argument.
[^6]: Binary operator ($\cdot$)$\oslash$($\cdot$) symbolizes Hadamard division [@cyganek2013object].
[^7]: The simulation scripts were all written in Matlab2019 and performed on a Dell Inspiron 15 7000 which is supported by an Intel Core i7-7700HQ processor with 2.80 GHz clockspeed, 16GB RAM, and Windows 10 Home 64 bit.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'A version of the singular Yamabe problem in bounded domains yields complete conformal metrics with negative constant scalar curvatures. In this paper, we study whether these metrics have negative Ricci curvatures. We provide a general construction of domains in compact manifolds and demonstrate that the negativity of Ricci curvatures does not hold if the boundary is close to certain sets of low dimension. The expansion of the Green’s function and the positive mass theorem play essential roles in certain cases. On the other hand, we prove that these metrics indeed have negative Ricci curvatures in bounded convex domains in the Euclidean space.'
address:
- |
Department of Mathematics\
University of Notre Dame\
Notre Dame, IN 46556
- |
Beijing International Center for Mathematical Research\
Peking University\
Beijing, 100871, China
author:
- Qing Han
- Weiming Shen
title: |
On The Negativity of Ricci Curvatures of\
Complete Conformal Metrics
---
[^1]
Introduction {#sec-Intro}
============
Let $(M,g)$ be a compact Riemannian manifold of dimension $n$ without boundary, for $n\ge 3$, and $\Gamma$ be a smooth submanifold in $M$. For $(M,g)=(S^n, g_{S^n})$, Loewner and Nirenberg proved that there exists a complete conformal metric on $S^n\setminus \Gamma$ with a [*negative*]{} constant scalar curvature if and only if dim$(\Gamma)>\frac{n-2}2$. Aviles and McOwen [@AM1988DUKE] proved the same result for the general manifold $(M,g)$. As a consequence, we can take the dimension of the submanifold to be $n-1$ and conclude the following result: In any compact Riemannian manifold with boundary, there exists a complete conformal metric with a negative constant scalar curvature. See [@AM1988DUKE]. For convenience, we always take the constant scalar curvature to be $-n(n-1)$. In this paper, we will study Ricci curvatures of such a metric.
For the case of positive scalar curvatures, the existence and asymptotic behaviors of solutions have been extensively studied over the years. We shall not discuss this case here, but refer to [@Caffarelli1989], [@Korevaar1999], [@Mazzeo; @Pacard1999], [@Mazzeo; @PollackUhlenbeck1996], [@Schoen1988], [@SchoenYau1988].
In the unit ball in the Euclidean space, the complete conformal metric with scalar curvature $-n(n-1)$ is exactly the Poincaré metric of the unit ball model of the hyperbolic space and has sectional curvatures $-1$ and Ricci curvatures $-(n-1)$. In particular, it has negative sectional curvatures and Ricci curvatures. A natural question is whether this remains true for the more general case; namely, whether the complete conformal metric with a negative constant scalar curvature in a compact Riemannian manifold with boundary has negative sectional curvatures or negative Ricci curvatures. We point out that a straightforward calculation based on the polyhomogeneous expansion established in [@ACF1982CMP] and [@Mazzeo1991] yields that such a metric has sectional curvatures asymptotically equal to $-1$ near the boundary and Ricci curvatures asymptotically $-(n-1)$ near $\partial\Omega$. Our main concern is whether the negativity of the sectional curvatures or Ricci curvatures near boundary can be carried over to the entire domain.
In view of the Poincaré metric in the unit ball model of the hyperbolic space, it is reasonable to expect that the complete conformal metric with a negative constant scalar curvature should have negative sectional curvatures in a domain close to the unit ball in the Euclidean space. We will confirm this in this paper. In fact, we will prove an affirmative result for convex domains in the Euclidean space.
It is not clear what to expect if the domain is sufficiently “far from" the unit ball. Results by Aviles and McOwen [@AM1988DUKE] provide a clue. In order to have a complete conformal metric with constant negative scalar curvature in $M\setminus \Gamma$, it is required that dim$(\Gamma)>\frac{n-2}{2}$. Closely related is a result proved by Mazzeo and Pacard [@Mazzeo; @Pacard1999] that there exist complete conformal metrics in $S^n\setminus \Gamma$ with constant [*positive*]{} scalar curvatures if dim$(\Gamma)\le \frac{n-2}{2}$. In view of these results, we can ask what happens to Ricci curvatures of the complete conformal metrics with scalar curvatures fixed at $-n(n-1)$ in domains $\Omega\subset M$ whose $(n-1)$-dimensional boundary is close to a smooth submanifold $\Gamma$ of dimension $\le \frac{n-2}{2}$. Do Ricci curvatures have mixed signs as $\partial\Omega$ becomes close to a low dimensional set?
In this paper, we first construct domains where complete conformal metrics have large positive Ricci curvatures in domains in compact Riemannian manifolds.
Let $(M,g)$ be a Riemannian manifold of dimension $n$ and $W$ be the Weyl tensor. Define the set $$\widetilde{W}=\{x\in M:\,
\lim_{r\to0}r^{2-n-i}\int_{B_{r}(x)}|W|^2dV_{g}>0\text{ for some } i\text{ with } 2\leq i \leq n-4\}.$$ We note the sign of the above limit depends only on the class $[g]$. Moreover, $\widetilde{W} \neq \emptyset$ if $n\geq 6$ and $M$ is not locally conformally flat.
We will prove the following theorem in this paper.
\[thrm-large-positive-Ricci\] Let $(M, g)$ be a compact Riemannian manifold of dimension $n\geq 3$ without boundary and $\Gamma$ be a disjoint union of finitely many closed smooth embedded submanifolds in $M$ of varying dimensions, between $0$ and $\frac{n-2}{2}$. Consider the following cases:
$\text{Case 1.}$ $\Gamma$ contains a submanifold of dimension $j$, with $1\le j\leq \frac{n-2}{2}$.
$\text{Case 2.}$ If $(M, g)$ is not conformally equivalent to the standard sphere $S^n$, $\Gamma$ consists of finitely many points $\{p_i\}$, with one of the following additional assumptions: (1) the Yamabe invariant of $M$ is nonpositive, (2) $3\le n\leq 7$, (3) $M$ is locally conformally flat, (4) $M$ is spin, or (5) $\{p_i\}\subseteq \widetilde{W}$.
$\text{Case 3.}$ If $(M, g)$ is conformally equivalent to $S^{n}$, $\Gamma$ consists of at least two finitely many points.
Suppose that $\Omega_i$ is a sequence of increasing domains with smooth boundary in $M$ which converges to $M \setminus\
\Gamma$ and that $g_i$ is the complete conformal metric in $\Omega_i$ with the constant scalar curvature $-n(n-1)$. Then, for sufficiently large $i$, $g_i$ has a positive Ricci curvature component somewhere in $\Omega_{i}$. Moreover, the maximal Ricci curvature in $\Omega_{i}$ diverges to $\infty$ as $i\to\infty$.
By the convergence of $\Omega_i$ to $M \setminus\Gamma$, we mean $\cup_{i=1}^\infty \Omega_i
=M\setminus \Gamma$ and, for any $\varepsilon>0$, $\partial\Omega_i$ is in the $\varepsilon$-neighborhood of $\Gamma$ for all large $i$. By convention, a zero dimensional submanifold is simply a point.
The difference between Case 2 and Case 3 in Theorem \[thrm-large-positive-Ricci\] lies on the number of isolated points when closed smooth embedded submanifolds of positive dimension are absent from $\Gamma$. On manifolds conformally equivalent to the standard sphere, the number of the isolated points has to be at least two; while on manifolds not conformally equivalent to the standard sphere, we can allow one point. Theorem \[thrm-large-positive-Ricci\] does not necessarily hold if $\Gamma$ consists of one point on manifolds conformally equivalent to the standard sphere. See Remark \[rmk-single-point-sphere\].
We point out that the additional assumptions in Case 2 in Theorem \[thrm-large-positive-Ricci\] is associated with the application of the positive mass theorem. With the validity of the positive mass theorem in all dimensions, these extra assumptions are not needed and Case 2 can be presented in a clean form. In other words, we only need to assume that $\Gamma$ consists of finitely many points if $(M, g)$ is not conformally equivalent to the standard sphere $S^n$. On the other hand, the proof of Theorem 1.1 needs few changes. A similar remark holds for Theorem \[them-rigidity\]. Refer to [@Lohkamp1], [@Lohkamp2], [@Lohkamp3], [@Lohkamp4], and [@SchoenYau2017] for the present status of the positive mass theorem.
As a consequence of Case 2, with $\Gamma$ consisting of just one point, we have the following rigidity result.
\[them-rigidity\] Let $(M, g)$ be a compact Riemannian manifold of dimension $n\geq3$ without boundary and $x_0$ be a point in $M$, with one of the following additional assumptions: (1) $3\leq n\leq 7$, (2) $(M, g)$ is locally conformally flat, (3) $M$ is spin. Suppose that there exists a sequence $\Omega_i$ of increasing domains with smooth boundary in $M$ which converges to $M \setminus\
\{x_0\}$, such that the complete conformal metric in $\Omega_i$ with the constant scalar curvature $-n(n-1)$ has uniformly bounded Ricci curvatures in $\Omega_{i}$. Then, $M$ is conformally equivalent to the standard sphere $S^n$.
As a consequence, we have the following result.
\[them-rigidity2\] Let $(M, g)$ be a compact Riemannian manifold of dimension $n\geq 3$ without boundary. Suppose that for every point $x$ in $M$, there exist a sequence $\Omega_{x}^{i}$ of increasing domains with smooth boundary in $M$ which converges to $M \setminus\
\{x\}$, such that the complete conformal metric in $\Omega_{x}^{i}$ with the constant scalar curvature $-n(n-1)$ has uniformly bounded Ricci curvatures in $\Omega_{x}^i$. Then, $M$ is conformally equivalent to the standard sphere $S^n$.
The set $\Gamma$ in Theorem \[thrm-large-positive-Ricci\] resembles that in [@Mazzeo; @Pacard1999]. The metric $g_i$ in $\Omega_i$ as in Theorem \[thrm-large-positive-Ricci\] is assumed to have a [*negative*]{} constant scalar curvature, $-n(n-1)$. As $\Omega_i$ becomes close to $M\setminus \Gamma$, Ricci curvatures are expected to split in sign. Some components become negatively large, while some others positively large.
The proof of Theorem \[thrm-large-positive-Ricci\] relies on a careful analysis of the Ricci curvatures of the complete conformal metrics near boundary. The polyhomogeneous expansion provides correct values near boundary for applications of the maximum principle. The Yamabe invariant of $(M,g)$ plays a crucial role and determines behaviors of the convergence of the conformal factors. Among the three cases listed in Theorem \[thrm-large-positive-Ricci\], Case 2 is the most difficult to prove, especially when the Yamabe invariant is between zero and that of the standard sphere. In such a case with $\Gamma$ consisting of one point $x_0$, we need expansions of Green’s functions. If $n=3,4,5$, or $M$ is conformally flat in a neighborhood of $x_0$, we need to employ the positive mass theorem. If $n\ge 6$ and $M$ is not conformally flat in a neighborhood of $x_0$, we need to distinguish the two cases $W(x_0)\neq 0$ and $W(x_0)=0$. Discussions for the latter case is much more complicated than the former case. The proof here seems to resemble the solution of the Yamabe problem with one twist. In solving the Yamabe problem, we can choose a point where the Weyl tensor is not zero in the case that $M$ is not conformally flat. In our case, $x_0$ is a given point and the Weyl tensor can be zero even if $M$ is not conformally flat in a neighborhood of $x_0$. Different vanishing orders of $W$ at $x_0$ requires different methods. In fact, we also need to employ the positive mass theorem if the Weyl tensor vanishes at $x_0$ up to a sufficiently high order.
Now, we turn our attention to bounded smooth domains in the Euclidean space. Closely related to the negativity of the Ricci curvatures is whether there is a constant rank theorem for metrics with negative Ricci curvatures, since it is already known that the Ricci curvatures are negative near boundary. Caffarelli, Guan and Ma [@Caffarelli2007] proved a constant rank theorem for the $\sigma_k$-curvature equations under certain positivity conditions on curvatures. However, their result is not applicable in our case. Our strategy is to connect directly boundary curvatures of domains with the interior curvature tensors of the complete conformal metrics.
We will prove the following theorem in this paper.
\[main reslut\] Let $\Omega\subset \mathbb{R}^{n}$ be a bounded convex domain, for $n\ge 3$, and $g_\Omega$ be the complete conformal metric in $\Omega$ with the constant scalar curvature $-n(n-1)$. Then, $g_\Omega$ has negative sectional curvatures in $\Omega$. Moreover, $g_\Omega$ has Ricci curvatures strictly less than $-{n}/{2}$ in $\Omega$.
The convexity assumption of the domain $\Omega$ is crucial. It allows us to apply a convexity theorem by Kennington [@Kennington1985] directly to conformal factors. Theorem \[main reslut\] does not hold for general bounded domains in $\mathbb R^n$, even for bounded star-shaped domains. In fact, in certain bounded star-shaped domains, the conformal metrics may have arbitrarily large positive Ricci curvature components. This is a simple consequence of Theorem \[thrm-large-positive-Ricci\] for the case $(M, g)=(S^{n}, g_{S^{n}})$, with the help of the stereographic projection.
The paper is organized as follows. In Section \[An Equivalent Form\], we discuss some preliminary identities. In Section \[sec-compact-manifolds\], we study the Ricci curvatures of complete conformal metrics in domains in compact manifolds and prove Theorem \[thrm-large-positive-Ricci\]. In Section \[sec-Conv\], we study the Ricci curvatures of complete conformal metrics in bounded convex domains in the Euclidean space and prove Theorem \[main reslut\].
We would like to thank Matthew Gursky for suggesting the problem studied in this paper and many helpful discussions. Gursky graciously shared many of his stimulating computations with us. We would also like to thank Yuguang Shi for helpful discussions.
Preliminaries {#An Equivalent Form}
=============
Let $(M,g)$ be a smooth Riemannian manifold of dimension $n$, for some $n\ge 3$, either compact without boundary or noncompact and complete. Assume $\Omega\subset M$ is a smooth domain, with an $(n-1)$-dimensional boundary. If $(M, g)$ is noncompact, we assume, in addition, that $\Omega$ is bounded. We consider the following problem: $$\begin{aligned}
\label{eq-MEq} \Delta_{g} u -\frac{n-2}{4(n-1)}S_gu&= \frac14n(n-2) u^{\frac{n+2}{n-2}} \quad\text{in }\,\Omega,\\
\label{eq-MBoundary}u&=\infty\quad\text{on }\partial \Omega,\end{aligned}$$ where $S_g$ is the scalar curvature of $M$. According to Loewner and Nirenberg for $(M,g)=(S^n, g_{S^n})$ and Aviles and McOwen [@AM1988DUKE] for the general case, and admits a unique positive solution. We note that $u^{\frac{4}{n-2}}g$ is the complete metric with a constant scalar curvature $-n(n-1)$ on $\Omega$. Andersson, Chruściel and Friedrich [@ACF1982CMP] and Mazzeo [@Mazzeo1991] established the polyhomogeneous expansions for the solutions. For the first several terms, we have $$u=d^{-\frac{n-2}{2}}\left[1+\frac{n-2}{4(n-1)}H_{\partial\Omega}d+O(d^2)\right],$$ where $d$ is the distance to $\partial\Omega$ and $H_{\partial\Omega}$ is the mean curvature of $\partial\Omega$ with respect to the interior unit normal vector of $\partial\Omega$. Set $$\label{eq-def-v}v=u^{-\frac{2}{n-2}}.$$ Then, $$\begin{aligned}
\label{eq-MEq-v}
v\Delta_{g} v +\frac{1}{2(n-1)}S_{ g} v^2&= \frac{n}{2}(|\nabla_{g} v|^2-1)\quad\text{in }\Omega, \\
\label{eq-MBoundary-v}v&=0\quad\text{on }\partial \Omega.\end{aligned}$$ Moreover, $$\label{boundary expansion v}v=d-\frac{1}{2(n-1)}H_{\partial\Omega}d^2+O(d^3).$$ This implies $$\label{boundary expansion v-gradient}|\nabla_gv|=1\quad\text{on }\partial\Omega.$$ We will use this repeatedly in the next section.
Consider the conformal metric $$\label{eq-conformal-metric}
g_\Omega=u^{\frac{4}{n-2}}g=v^{-2}g.$$ For a unit vector $X$ of $g$, $vX$ is a unit vector of $g_\Omega$. Let $R_{ij}$ be the Ricci components of $g$ in a local frame for the metric $g$ and $R^\Omega_{ij}$ be the Ricci components of $g_\Omega$ in the corresponding frame for the metric $g_\Omega$. Then, $${R}^\Omega_{kl}=v^2R_{kl}+(n-2)\big[vv_{,kl}-\frac12g_{kl}|\nabla _gv|^2\big]
+g_{kl}\big[v\Delta_gv-\frac{n}{2}|\nabla_gv|^2\big].$$ By , we have $$\label{Ricci cur in v Manifold-1}
{R}^\Omega_{kl}=v^2R_{kl}+(n-2)\big[vv_{,kl}-\frac12g_{kl}|\nabla _gv|^2\big]
-g_{kl}\big[\frac{1}{2(n-1)}v^2S_g+\frac{n}{2}\big],$$ or $$\label{Ricci cur in v Manifold}
{R}^\Omega_{kl}=v^2R_{kl}-\frac{1}{2(n-1)}v^2g_{kl}S_g+(n-2)vv_{,kl}-\frac{n-2}{2}g_{kl}|\nabla_gv|^2-\frac{n}{2}g_{kl}.$$ We emphasize that and play important roles in the rest of the paper. By and , we obtain $${R}^\Omega_{kl}=-(n-1)g_{kl}+O(d).$$ In other words, the Ricci curvatures of conformal metrics $g_\Omega$ are asymptotically equal to $-(n-1)$ near boundary. We note that this holds in arbitrary smooth domains.
If $(M,g)=(\mathbb R^n, g_E)$, then and reduce to $$\begin{aligned}
\label{eq-MainEq} \Delta u &= \frac14n(n-2) u^{\frac{n+2}{n-2}} \quad\text{in }\,\Omega,\\
\label{eq-MainBoundary}u&=\infty\quad\text{on }\partial \Omega.\end{aligned}$$ In this case, the function $v$ given by satisfies $$\label{eq-transf} v\Delta v=\frac{n}{2}(|\nabla v |^2-1).$$ Let $g_\Omega$ be the metric given by with $g=g_E$, i.e., $g_\Omega=v^{-2}g_E$. Denote by $R^\Omega_{ijij}$ and $R^\Omega_{ij}$ the sectional curvatures and Ricci curvatures of $g_\Omega$ in the orthonormal coordinates of $g_\Omega$, respectively. Then, for $i\neq j$, $$\label{Sectional cur in v}
R^\Omega_{ijij}=vv_{ii}+vv_{jj}-|\nabla v|^{2},$$ and, for any $i, j$, $$\label{Ricci cur in v}
R^\Omega_{ij}=(n-2)vv_{ij}-\big[\frac{n-2}{2}|\nabla v|^{2}+\frac{n}{2}\big]\delta_{ij}.$$ Hence, for any $i\neq j$, $$R^\Omega_{ijij}=-1+O(d),$$ and, for any $i, j$, $$R^\Omega_{ij}=-(n-1)\delta_{ij}+O(d).$$ Note $$v_i=-\frac{2}{n-2}u^{-\frac{2}{n-2}-1}u_i,$$ and $$\label{eq-relation-2derivatives}
v_{ij}=-\frac{2}{n-2}u^{-\frac{2}{n-2}}\bigg(\frac{u_{ij}}{u}-\frac{n}{n-2}\frac{u_iu_j}{u^2}\bigg).$$ We can also express $R^\Omega_{ijij}$ and $R^\Omega_{ij}$ in terms of $u$.
Domains in Compact Manifolds {#sec-compact-manifolds}
============================
In this section, we discuss domains in compact Riemannian manifolds without boundary. We construct domains with boundary close to certain sets of low dimension such that the complete conformal metrics with a negative constant scalar curvature have positive Ricci components somewhere. Throughout this section, the Yamabe invariant plays a crucial role. It determines convergence behaviors of conformal factors and, as a consequence, the methods to be employed. In certain cases, we need to employ expansions of the Green’s function, and also the positive mass theorem.
Suppose $(M, g)$ is a compact Riemannian manifold of dimension $n\geq 3$ without boundary. The Yamabe invariant of $M$ is given by $$\lambda(M, [g])=\inf\left\{
\frac{\int_M ( |\nabla_g \phi|^2 +\frac{n-2}{4(n-1)} S_g\phi^2 )dV_g }{(\int_M \phi^{\frac{2n}{n-2}}dV_g)^{\frac{n-2}{n}}}
\bigg|\,\phi \in C^{\infty}(M),\phi>0 \right\}.$$ The conformal Laplacian of $(M,g)$ is given by $$L_g=-\Delta_g +\frac{n-2}{4(n-1)}S_g.$$ For any function $\psi$ in $M$, we have $$L_g(u\psi)= u^{\frac{n+2}{n-2}}L_{u^{\frac{4}{n-2}}g}(\psi ).$$
We first prove a convergence result which plays an important role in this section. According to signs of Yamabe invariants, conformal factors exhibit different convergence behaviors. We note that the maximum principle is applicable to the operator $L_g$ if $S_g\ge 0$.
\[lemme-convergence\] Suppose $(M, g)$ is a compact Riemannian manifold of dimension $n\geq 3$ without boundary, with a constant scalar curvature $S_g$, and $\Gamma$ is a closed smooth submanifold of dimension $d$ in $M$, $0\le d\leq \frac{n-2}{2}$. Suppose $\Omega_i$ is a sequence of increasing domains with smooth boundary in $M$ which converges to $M \setminus\Gamma$. Let $u_i$ be the solution of and in $\Omega_i$. Then, for any positive integer $m$, if $S_g\ge 0$, $$\label{u_i go to zero-plus}
u_i\rightarrow 0\quad\text{in }C^{m}_{\mathrm{loc}}( M \setminus\Gamma)
\text{ as }i\rightarrow\infty,$$ and, if $S_g<0$, $$\label{u_i go to zero-minus}
u_i\rightarrow \left(\frac{-S_g}{n(n-1)}\right)^{\frac{n-2}{4}}
\quad\text{in }C^{m}_{\mathrm{loc}}( M \setminus\Gamma)
\text{ as }i\rightarrow\infty.$$
We first consider the case $S_g\ge 0$. By the maximum principle, we have $u_i \geq u_{i+1}$ in $\Omega_i$. It is straightforward to verify, for any $m$, $$u_i\rightarrow u\quad\text{in }C^{m}_{\mathrm{loc}}( M \setminus\Gamma)
\text{ as }i\rightarrow\infty,$$ where $u$ is a nonnegative solution of in $M \setminus\Gamma$. By the second part of [@AM1988DUKE] (Page 398), $u$ is bounded. Let $\rho(x)$ be a positive smooth function in $ M \setminus\Gamma$ which equals to $\text{dist}(x, \Gamma)$ in a neighborhood of $\Gamma$ in $M$. Then, $$\rho(x)\Delta_g \rho(x)\rightarrow n-d-1\quad\text{as }x\rightarrow\Gamma.$$ Take $\epsilon_0>0$ sufficiently small. Then, $$\Delta_g \rho^{-\frac{n-2}{2}+\epsilon_0}=\left(-\frac{n-2}{2}+\epsilon_0\right)\rho^{-\frac{n+2}{2}+\epsilon_0}
\left(\rho\Delta_g \rho -\big(\frac{n}{2}-\epsilon_0\big)|\nabla_g\rho|^2\right).$$ Since $d\leq\frac{n-2}{2}$, we have $\Delta \rho^{-\frac{n-2}{2}+\epsilon_0}<0$ near $\Gamma$. For any $\epsilon>0$, we can find $\delta <\epsilon$ sufficiently small such that $$\Delta_g(\delta \rho^{-\frac{n-2}{2}+\epsilon_0}+\epsilon)
-S_g (\delta \rho^{-\frac{n-2}{2}+\epsilon_0}+\epsilon)
\leq \frac{n-2}{4}(\delta \rho^{-\frac{n-2}{2}+\epsilon_0}+\epsilon)^{\frac{n+2}{n-2}}\quad\text{in }M\setminus\Gamma.$$ By the maximum principle, we have $$u \leq \delta \rho^{-\frac{n-2}{2}+\epsilon_0}+\epsilon\quad\text{in }M\setminus\Gamma.$$ This implies $u\equiv 0$. In conclusion, we obtain .
We now consider the case $S_g<0$. We first prove $\Delta_g u_i\geq 0$ in $\Omega_i,$ or equivalently $$\label{eq-AlgebraicRelation}u_i\geq
\left(\frac{-S_g}{n(n-1)}\right)^{\frac{n-2}{4}}
\quad\text{in }\Omega_i.$$ If is violated somewhere, then $u_i$ must assume its minimum at some point $x_0$ in the set $$\left\{x\in \Omega_i :\, \frac14n(n-2) u_i^{\frac{n+2}{n-2}} +\frac{n-2}{4(n-1)}S_gu_i< 0 \right\}.$$ On the other hand, we have $
\Delta_g u_i(x_0)\geq 0$, which leads to a contradiction. By taking a difference, we have $$\Delta_g (u_{i+1}-u_{i})=c_i(u_{i+1}-u_{i})\quad \text{in } \Omega_{i},$$ where $c_i$ is a nonnegative function in $\Omega_{i}$ by . The maximum principle implies $u_{i+1}\leq u_{i}$ in $\Omega_i$. Then, for any $m$, $$u_i\rightarrow u\quad\text{in }C^{m}_{\mathrm{loc}}( M \setminus\Gamma)
\text{ as }i\rightarrow\infty,$$ where $u$ is a solution of in $M \setminus\Gamma$. By , we have $$\label{eq-u lower bounded} u\geq
\left(\frac{-S_g}{n(n-1)}\right)^{\frac{n-2}{4}}\quad\text{in }M\setminus\Gamma.$$ For $\epsilon>0$ sufficient small, let $u^{\epsilon}_i$ be the solution of $$\begin{aligned}
\label{eq-MEq2} \Delta_g u^{\epsilon}_i &= \epsilon\frac{n(n-2)}{4}( u^{\epsilon}_i)^{\frac{n+2}{n-2}}
\quad\text{in }\Omega_i,\\
\label{eq-MBoundary2}u^{\epsilon}_i &=\infty\quad\text{on }\partial \Omega_i.\end{aligned}$$ The existence of $u^{\epsilon}_i$ can be obtained by the standard method. By the same method as in the proof of the case $S_g\geq0$, we obtain, for any $m$, $$u^{\epsilon}_i\rightarrow 0\quad\text{in }C^{m}_{\mathrm{loc}}( M \setminus\Gamma)
\text{ as }i\rightarrow\infty.$$ Next, we can verify $$\begin{aligned}
\Delta_g \bigg[u^{\epsilon}_i+ \bigg(\frac{-S_g}{(1-\epsilon)n(n-1)}\bigg)^{\frac{n-2}{4}} \bigg]
&\le\frac{n-2}{4(n-1)}S_g
\bigg[u^{\epsilon}_i+ \bigg(\frac{-S_g}{(1-\epsilon)n(n-1)}\bigg)^{\frac{n-2}{4}} \bigg] \\
&\qquad +\frac{n(n-2)}{4} \bigg[
u^{\epsilon}_i+ \bigg(\frac{-S_g}{(1-\epsilon)n(n-1)}\bigg)^{\frac{n-2}{4}} \bigg]^{\frac{n+2}{n-2}}.\end{aligned}$$ To prove this, we simply split the last term according to $1=\epsilon+(1-\epsilon)$. Then, $$\begin{aligned}
&\frac{n-2}{4(n-1)}S_g
\bigg[u^{\epsilon}_i+ \bigg(\frac{-S_g}{(1-\epsilon)n(n-1)}\bigg)^{\frac{n-2}{4}} \bigg]\\
&\qquad\qquad+\frac{n(n-2)}{4} \bigg[
u^{\epsilon}_i+ \bigg(\frac{-S_g}{(1-\epsilon)n(n-1)}\bigg)^{\frac{n-2}{4}} \bigg]^{\frac{n+2}{n-2}}
-\epsilon\frac{n(n-2)}{4}( u^{\epsilon}_i)^{\frac{n+2}{n-2}}\\
&\qquad\geq\frac{n(n-2)}{4} \bigg[
u^{\epsilon}_i+ \bigg(\frac{-S_g}{(1-\epsilon)n(n-1)}\bigg)^{\frac{n-2}{4}} \bigg] \cdot\\
&\qquad\qquad \bigg\{(1-\epsilon)
\bigg[u^{\epsilon}_i+ \bigg(\frac{-S_g}{(1-\epsilon)n(n-1)}\bigg)^{\frac{n-2}{4}} \bigg]^{\frac{4}{n-2}}
+\frac{1}{n(n-1)}S_g\bigg\},\end{aligned}$$ which is nonnegative. By the maximum principle, we have $$u_i \leq u^{\epsilon}_i+ \bigg(\frac{-S_g}{(1-\epsilon)n(n-1)}\bigg)^{\frac{n-2}{4}}
\quad\text{in }\Omega_i,$$ where we can verify the boundary condition by the polyhomogeneous expansions of $u_i$ and $u^{\epsilon}_i$. Therefore, we have $$\label{eq-u upper bounded}
u\leq \left(\frac{-S_g}{(1-\epsilon)n(n-1)}\right)^{\frac{n-2}{4}}\quad\text{in }M\setminus\Gamma.$$ This holds for any $\epsilon\in (0,1)$. Combining and , we obtain $$u=\left(\frac{-S_g}{n(n-1)}\right)^{\frac{n-2}{4}}.$$ In conclusion, we obtain .
A similar result holds if the scalar curvature has a fixed sign, not necessarily constant.
Now, we study the case that the boundary is close to a closed smooth submanifold of low dimension. The result below holds for all compact manifolds without boundary, but different signs of the Yamabe invariants require different methods, mostly due to the different convergence behaviors as in Lemma \[lemme-convergence\].
\[prop-general mainfold\] Suppose $(M, g)$ is a compact Riemannian manifold of dimension $n\geq 3$ without boundary and $\Gamma$ is a closed smooth submanifold of dimension $d$ in $M$, $1\le d\leq \frac{n-2}{2}$. Suppose $\Omega_i$ is a sequence of increasing domains with smooth boundary in $M$ which converges to $M \setminus\Gamma$ and $g_i$ is the complete conformal metric in $\Omega_i$ with the scalar curvature $-n(n-1)$. Then, for sufficiently large $i$, $g_{i}$ has a positive Ricci curvature component somewhere in $\Omega_{i}$. Moreover, the maximal Ricci curvature of $g_i$ in $\Omega_{i}$ diverges to $\infty$ as $i\to\infty$.
Let $u_i$ be the solution of and in $\Omega_i$ and set $v_i=u_i^{-\frac{2}{n-2}}$. Then, $$g_{i}=u_i^{\frac{4}{n-2}}g=v_i^{-2}g.$$ By the solution of the Yamabe problem, we can assume the scalar curvature $S_g$ of $M$ is the constant $\lambda(M,[g])$. Since $M$ is compact, we can take $\Lambda>0$ such that $$|R_{ij}|\leq \Lambda g_{ij}.$$ We now discuss two cases according to the sign of $S_g$.
[*Case 1.*]{} We first consider the case $S_g\geq0$. By Lemma \[lemme-convergence\], for any $m$, $$u_i\rightarrow 0\quad\text{in }C^{m}_{\mathrm{loc}}( M \setminus\Gamma)
\text{ as }i\rightarrow\infty,$$ and hence $$v_i \text{ diverges to $\infty$ locally uniformly in $M \setminus\Gamma$
as $i\rightarrow\infty$}.$$ We now consider two subcases.
[*Case 1.1.*]{} $\Gamma$ is not totally geodesic. For any $\epsilon>0$, there exist two points $p,q \in \Gamma$, such that the length of the shortest geodesic $\sigma_{pq}$ connecting $p$ and $q$ is less than $\epsilon$ and $\sigma_{pq}\bigcap \Gamma=\{p, q\}$. When $\epsilon$ is sufficiently small, we can assume $q$ is located in a small neighborhood of $p$ covered by normal coordinates. Without loss of generality, we assume $p=0$ and $q=Le_n$.
For $i$ large, set $p_i=\widehat{t}_ie_n$ and $q_i=\widetilde{t}_ie_n$, where $$\begin{aligned}
\widehat{t}_i&=\min \{t'\in[0,{L}/{2}] |te_n \in\Omega_i , \text{ for any } t\in (t',{L}/{2}]\},\\
\widetilde{t}_i&= \max \{t'\in[{L}/{2},L]|te_n \in\Omega_i , \text{ for any } t\in [{L}/{2},t')\}.\end{aligned}$$ Then, $p_i, q_i\in \partial\Omega_i$. By the convergence of $\Omega_i$ to $M\setminus \Gamma$, we have $$p_i\rightarrow p,\quad q_i\rightarrow q.$$ By the polyhomogeneous expansions of $v_i$, we have $$|\partial_n v_i(p_i)|\leq C_i \quad\text{and}\quad |\partial_n v_i(q_i)|\leq C_i,$$ where $C_i$ is some positive constant which converges to 1 as $i\rightarrow\infty$ and $\epsilon\rightarrow 0$.
Since $v_i(Le_n/2)\rightarrow\infty$ as $i\rightarrow\infty$, for $i$ large, we can take $t_i \in (\widehat{t}_i,\widetilde{t}_i)$ such that, for any $t\in (\widehat{t}_i,\widetilde{t}_i)$, $$\partial_nv_i(te_n)\leq \partial_nv_i(t_ie_n).$$ Then, $$\partial_nv_i(t_ie_n)>\frac{v_i(\frac{L}{2}e_n)-0}{\frac{L}{2}}\geq \frac{2}{L}v_i(\frac{L}{2}e_n),$$ and $$\partial_{nn}v_i( t_ie_n)=0.$$ We also have $$\label{eq-v_icontrol}|v_i( t_ie_n)|\leq \frac{L}{2}\partial_nv_i(t_ie_n).$$ Denote by $R^i_{nn}$ the Ricci curvature of $g_i$ acting on the unit vector $v_i \frac{\partial}{\partial x^n}$ with respect to the metric $g_i$. By , we have, at $t_ie_n$, $$R^i_{nn}\le v_i^2|R_{nn}|-\frac{n-2}{2}(\partial_nv_i)^2\le
\big[\frac{L^2}{4}|R_{nn}|-\frac{n-2}{2}\big](\partial_nv_i)^2\to-\infty,$$ if $L$ is sufficiently small. Hence, some component of the Ricci curvature of $g_i$ at the point $t_ie_n$ diverges to $\infty$ as $i\rightarrow\infty$.
[*Case 1.2.*]{} $\Gamma$ is totally geodesic. Fix a point $x_0 \in \Gamma$ and choose normal coordinates near $x_0$ such that $x_{0}=0$ and $\Gamma$ near $x_0$ is given by $x_i=0$, $i=1,..,n-d$. Consider the curve $\sigma$ given by $$\sigma(t)=(\sqrt{R^2-t^2}-\sqrt{R^2-\epsilon^2}, 0, \cdots, 0, t)\quad\text{for }t \in [-\epsilon,\epsilon],$$ where $R$ is some sufficiently large constant and $\epsilon$ is some sufficiently small constant such that $\sigma\bigcap \Gamma=\{ \sigma(-\epsilon), \sigma(\epsilon)\}$.
For $i$ large, set $p_i=\sigma(\widehat{t}_i)$ and $q_i=\sigma( \widetilde{t}_i)$, where $$\begin{aligned}
\widehat{t}_i&=\min \{t'\in[-\epsilon,0] |\sigma(t) \in \Omega_i,
\text{ for any } t\in ( t',0]\},\\
\widetilde{t}_i&=\max \{t'\in [0,\epsilon] |\sigma(t) \in \Omega_i, \text{ for any } t\in [0,t')\}.\end{aligned}$$ Then, $p_i, q_i\in\partial\Omega_i$ and $$p_i\rightarrow \sigma(-\epsilon),
\quad q_i\rightarrow \sigma(\epsilon).$$ By the polyhomogeneous expansion of $v_i$, we have $$|\partial_{n} v_i(\sigma(\widehat{t}_i))|\leq C_i\quad\text{and}\quad
|\partial_{n} v_i(\sigma(\widetilde{t}_i))|\leq C_i,$$ where $C_i$ is some positive bounded constant independent of $i$.
Consider the single variable function $(v_i\circ\sigma)(t)$. Since $(v_i\circ\sigma)(0)\rightarrow\infty$ as $i\rightarrow\infty$, for $i$ large, we can take $t_i \in (\widehat{t}_i,\widetilde{t}_i)$ such that, for any $t\in (\widehat{t}_i,\widetilde{t}_i)$, $$\partial_{t}(v_i\circ\sigma)(t)\leq \partial_{t} (v_i\circ\sigma)(t_i).$$ Then, $$\partial_{t} (v_i\circ\sigma)(t_i)>\frac{1}{\epsilon} (v_i\circ\sigma)(0),$$ and $$\label{eq-second deriv}\partial_{tt} (v_i\circ\sigma)(t_i)=0.$$ We also have $$\label{eq-v_icontrol2}|(v_i\circ\sigma)(t_i)|\leq
\epsilon\partial_{t} (v_i\circ\sigma)(t_i).$$ Note that $$\partial_{t}(v_i\circ\sigma)(t_i)=
(\partial_n v_i)(\sigma(t_i))-\frac{t_i}{\sqrt{R^2-t_i^2}}(\partial_1 v_i)(\sigma(t_i)).$$ Set $$\nu_i=\frac{\partial}{\partial x_n}-\frac{t_i}{\sqrt{R^2-t_i^2}}\frac{\partial}{\partial x_1}.$$ By , we have $$( \partial_{\nu_i\nu_i}v_i)(\sigma(t_i))
=\bigg(-\frac{1}{\sqrt{R^2-t_i^2}}+\frac{t_i^2}{(R^2-t_i^2)^{\frac{3}{2}}}\bigg)\partial_1 v_i(\sigma(t_i)).$$ Hence, $( \partial_{\nu_i\nu_i}v_i)(\sigma(t_i))$ is sufficiently small compared with $( |\nabla v_i|)(\sigma(t_i))$, for $R$ sufficiently large and $\epsilon$ sufficiently small. Write $g_{\nu_i\nu_i}=g (\nu_i, \nu_i)$ and denote by $R^i_{\nu_i\nu_i}$ the Ricci curvature of $g_i$ acting on the unit vector $\frac{v_i\nu_i}{\sqrt{g_{\nu_i\nu_i}}}$ with respect to the metric $g_i$. Similarly as in Case 1.1, we can verify at the point $\sigma(t_i)$, $R^i_{\nu_i\nu_i}$ diverges to $ -\infty$ as $i\rightarrow\infty$. Hence, some component of the Ricci curvature of $g_i$ at the point $\sigma(t_i)$ diverges to $\infty$ as $i\rightarrow\infty$.
We now consider the case $S_g<0$. By Lemma \[lemme-convergence\], for any $m$, $$u_i\rightarrow \left(\frac{-S_g}{n(n-1)}\right)^{\frac{n-2}{4}}
\quad\text{in }C^{m}_{\mathrm{loc}}( M \setminus\Gamma)
\text{ as }i\rightarrow\infty,$$ and hence $$\label{eq-v convergence}
v_i \rightarrow
\left(\frac{-S_g }{n(n-1)}\right)^{-\frac{1}{2}} \quad\text{in }C^{m}_{\mathrm{loc}}( M \setminus\Gamma)
\text{ as }i\rightarrow\infty.$$ Fix a point $x_0 \in \Gamma$ and choose normal coordinates in a small neighborhood of $x_{0}$ such that $x_{0}=0$ and $x_n$-axis is a normal geodesic of $\Gamma$ near $x_0$. Take $\epsilon>0$ sufficiently small. For $i$ large, set $p_i=t_ie_n$, where $$t_i=\min \{t'\in[0,\epsilon] |te_n \in \Omega_i, \text{ for any }t\in (t', \epsilon]\}.$$ Then, $p_i\in\partial\Omega_i$ and $$p_i\rightarrow 0.$$ By the polyhomogeneous expansion of $v_i$, we have $$|\partial_n v_i(p_i)|\leq C_i,$$ where $C_i$ is some positive constant which converges to 1 as $i\rightarrow\infty$. By , $$\frac{\partial v_i}{\partial x_n}(\epsilon e_n )\rightarrow0\quad\text{as }i\rightarrow\infty.$$ For $i$ large, we take $\widetilde{t}_i \in (t_i,\epsilon)$ such that, for any $t\in (t_i,\epsilon)$, $$\partial_nv_i(te_n)\leq \partial_nv_i(\widetilde{t}_ie_n).$$ Then, $$\partial_nv_i(\widetilde{t}_ie_n)>\frac{v_i(\epsilon e_n)-0}{\epsilon-t_i}>
\frac{1}{2}\epsilon^{-1}\left(\frac{-S_g }{n(n-1)}\right)^{-\frac{1}{2}},$$ and $$\partial_{nn}v_i(\widetilde{ t}_ie_n)=0.$$ Denote by $R^i_{nn}$ the Ricci curvature of $g_i$ acting on the unit vector $v_i \frac{\partial}{\partial x^n}$ with respect to the metric $g_i$. Similarly, by at the point $\widetilde{t}_ie_n$, $R_{nn}^i\le -C\epsilon^{-2}$, for all large $i$, for some positive constant $C$ independent of $i$ and $\epsilon$. By choosing appropriate $\epsilon$, we have the desired result.
Next, we discuss the case that the boundary is close to a point $x_0$. The proof of the next result is rather delicate if the Yamabe invariant is positive, in which case expansions of the Green’s function play an essential role. We need to employ the positive mass theorem if the manifold has a dimension 3, 4, or 5, or is locally conformally flat. In the case that $n\ge 6$ and $M$ is not conformally flat in a neighborhood of $x_0$, we need to analyze Weyl tensors and distinguish two cases $W(x_0) \neq 0$ and $W(x_0)=0$. The proof for the case $W(x_0)=0$ is quite delicate. It is worth to emphasize that the Weyl tensor can be zero at $x_0$ even if $M$ is not conformally flat in a neighborhood of $x_0$. Different vanishing orders of $W$ at $x_0$ requires different methods. In fact, we also need to employ the positive mass theorem if the Weyl tensor vanishes at $x_0$ up to a sufficiently high order.
\[blow up-one point\] Let $(M, g)$ be a compact Riemannian manifold of dimension $n\geq 3$ without boundary, with $\lambda(M,[g])< \lambda(S^n,[g_{S^n}])$, where $S^n$ is the sphere with its standard metric $g_{S^n}$, and let $x_0$ be a point in $M$. Suppose that $\Omega_i$ is a sequence of increasing domains with smooth boundary in $M$ which converges to $M \setminus\
\{x_0\}$ and $g_{i}$ is the complete conformal metric in $\Omega_i$ with the scalar curvature $-n(n-1)$. Then, for $i$ sufficiently large, $g_i$ has a positive Ricci curvature component somewhere in $\Omega_{i}$. Moreover, the maximal Ricci curvature of $g_i$ in $\Omega_{i}$ diverges to $\infty$ as $i\to\infty$.
Let $u_i$ be the solution of and in $\Omega_i$ and set $v_i=u_i^{-\frac{2}{n-2}}$. Then, $$g_{i}=u_i^{\frac{4}{n-2}}g=v_i^{-2}g.$$ We consider several cases according to the sign of the Yamabe invariant $\lambda(M,[g])$.
We first consider $\lambda(M,[g])<0$. We point out that the proof of Case 2 of Theorem \[prop-general mainfold\] can be adapted to yield the conclusion.
Next, we consider $\lambda(M,[g])=0$. As in the proof of Theorem \[prop-general mainfold\], we assume the scalar curvature of $M$ is 0 and $$|R_{ij}|\leq \Lambda g_{ij}.$$ Let $\delta$ be some small positive constant such that $\Lambda\delta<{1}/{10}$ and there exist normal coordinates in $B_{\delta}(x_0 )$.
Take a sufficiently small $r>0$ with $r\leq \delta$. Since $\Omega_i\rightarrow M \setminus\ \{x_0\}$, we have $M \setminus\ B_{r}(x_0 )\subset\subset \Omega_i$ for $i$ large. For such $i$, by the Harnack inequality, we have $$\max u_i \leq C \min u_i
\quad\text{in }M \setminus\ B_{r}(x_0 ),$$ where $C$ is some positive constant depending only on $n$, $M$ and $r$. Then for $i$ sufficiently large, by , we have $$\label{u_i grad estimate} |\nabla_g u_i| \leq C(u_i+ u_i^{\frac{n+2}{n-2}})\leq Cu_i
\quad\text{in }M \setminus\ B_{r}(x_0 ).$$ We denote by $m_i$ the minimum of $u_i$ in $\Omega_i$. With the definition of $v_i$, we have, for $i$ sufficient large, $$\label{v_i grad estimate} |\nabla_g v_i| \leq C v_i \leq C m_i^{-\frac{2}{n-2}}
\quad\text{in }M \setminus\ B_{r}(x_0 ),$$ where $C$ is some positive constant depending only on $n$, $M$ and $\delta$. Set $$A_i=\{x\in \Omega_i|\, u_i(x)<2m_i\}.$$ Then, for any fixed $ r>0$ with $r\leq \delta$ and any $i$ sufficiently large, we have $A_i\bigcap
B_{r}(x_0 ) \neq \emptyset.$ Otherwise, by the maximum principle, we have $$u_i(x) \geq 2m_i-Cm_i^{\frac{n+2}{n-2}}\quad\text{in }A_i,$$ where $C$ is some constant depending only on $n$, $M$ and $r$. Hence, $$m_i \geq 2m_i-Cm_i^{\frac{n+2}{n-2}}.$$ Note that $m_i\rightarrow0$ as $i\rightarrow\infty$, which leads to a contradiction.
By $v_i=0$ on $\partial \Omega_i$, we have, for any fixed $ r>0$ with $r\leq \delta$ and for any $i$ sufficiently large, $$|\nabla_g v_i| \geq \frac{1}{r}(2m_i)^{-\frac{2}{n-2}}\quad\text{somewhere in }\Omega_i\bigcap B_{r}(x_0 ).$$ Therefore, for $i$ sufficiently large, $|\nabla_g v_i|$ must assume its maximum at $p_i \in \Omega_i\bigcap B_{\delta}(x_0 ) $. Write $\nu_i=\frac{\nabla_g v_i}{|\nabla_g v_i|}$ and denote by $R^i_{\nu_i\nu_i}$ the Ricci curvature of $g_i$ acting on the unit vector $v_i\nu_i$ with respect to the metric $g_i$. Then, we can proceed as in the proof of Theorem \[prop-general mainfold\] to verify at the point $p_i$, $R^i_{\nu_i\nu_i}$ diverges to $ -\infty$ as $i\rightarrow\infty$. Hence, some component of the Ricci curvature of $g_i$ at the point $p_i$ diverges to $\infty$ as $i\rightarrow\infty$.
We now consider the case $\lambda(M,[g])>0$. In this case, there exists $G_{x_0}\in C^{\infty}(M \setminus\
\{x_0\})$, the Green’s function for the conformal Laplacian $L_g$, such that $$L_gG_{x_0}=(n-2)\omega_{n-1}\delta_{x_0},\,\, G_{x_0}>0,$$ where $\omega_{n-1}$ is the volume of $S^{n-1}$. Up to a conformal factor, we can assume $(M, g)$ has conformal normal coordinates near $x_0$. See [@LeeParker1987] (Page 69) or [@SchoenYau1994], chapter 5. We can perform a conformal blow up at $x_0$ to obtain an asymptotic flat and scalar flat manifold by using $G_{x_0}$. Specifically, if we define the metric $\widetilde{g}
= G_{x_0}^{\frac{4}{n-2}}g$ on $\widetilde{M} =M \setminus\
\{x_0\}$, then, $(\widetilde{M}, \widetilde{g})$ is an asymptotically flat and scalar flat manifold, and $\widetilde{g}$ has an asymptotic expansion near infinity. See [@LeeParker1987] (Page 64-65), or [@SchoenYau1994], chapter 5.
Set $\widetilde{u}_i=u_i/ G_{x_0}$. Then, $\widetilde{u}_i$ satisfies $$\begin{aligned}
\label{eq u_i} \Delta_{\widetilde{g}} \widetilde{u}_{i} &=
\frac14n(n-2) \widetilde{u}_{i}^{\frac{n+2}{n-2}} \quad\text{in }\,\Omega_i,\\
\label{eq-u i Boundary}\widetilde{u}_{i}&=\infty\quad\text{on }\partial \Omega_i,\end{aligned}$$ and for any $m$, $$\label{u_i go to zero-a}
\widetilde{u}_i\rightarrow 0\quad\text{in }C^{m}_{\mathrm{loc}}( M \setminus\{x_0\})
\text{ as }i\rightarrow\infty.$$
Fix a point $p_0\in M \setminus\{x_0\}$. Then, $p_0 \in \Omega_i$, for $i$ sufficiently large. Set $\widetilde{w}_i={\widetilde{u}_i}/{\widetilde{u}_i(p_0)}$. Then, $\widetilde{w}_i(p_0)=1$ and $\widetilde{w}_i$ satisfies $$\begin{aligned}
\label{eq w_i} \Delta_{\widetilde{g}} \widetilde{w}_{i} &=
\frac14n(n-2)u_i(p_0)^{\frac{4}{n-2}} \widetilde{w}_{i}^{\frac{n+2}{n-2}} \quad\text{in }\,\Omega_i,\\
\label{eq-w i Boundary}\widetilde{w}_{i}&=\infty\quad\text{on }\partial \Omega_i.\end{aligned}$$ By interior estimates, there exists a positive function $\widetilde{w}\in \widetilde{M}$ such that, for any $m$, $$\widetilde{w}_i\rightarrow \widetilde{w}\quad\text{in }C^{m}_{\mathrm{loc}}( \widetilde{M})
\text{ as }i\rightarrow\infty,$$ and $$\Delta_{\widetilde{g}} \widetilde{w}=0 \quad\text{in } \widetilde{M}.$$ Hence, $$L_g( G_{x_0}\widetilde{w})=0 \quad\text{in } M \setminus\{x_0\}.$$ By the expansion of $G_{x_0}$ near $x_0$ and Proposition 9.1 in [@LiZhu1999], we conclude that $ \widetilde{w}$ converges to some constant as $x\to x_0$. Therefore, $\widetilde{w}\equiv1$ in $\widetilde{M}$. Hence, for any $m$, $$\label{u_i behavioer2}\frac{u_i}{\widetilde{u}_i (p_0) G_{x_0}}\rightarrow 1
\quad\text{in }C^{m}_{\mathrm{loc}}( M \setminus\{x_0\})
\text{ as }i\rightarrow\infty.$$ In the following, we always discuss in the conformal normal coordinates near $x_0$. Set $$v_i= u_i^{-\frac{2}{n-2}}=
(\widetilde{u}_i (p_0))^{-\frac{2}{n-2}}
(\frac{u_i}{\widetilde{u}_i (p_0) G_{x_0}})^{-\frac{2}{n-2}}G_{x_0}^{-\frac{2}{n-2}}.$$ We will fix a direction appropriately, which we call $x_1$. Denote by $R^i_{11}$ the Ricci curvature of $g_i$ acting on the unit vector $v_i\frac{\partial}{\partial x_1}$ with respect to the metric $g_i$. To study $R^i_{11}$ given by , we need to analyze the expansion of $G_{x_0}^{-\frac{2}{n-2}}.$ See [@LeeParker1987] or [@SchoenYau1994] for details.
Now we discuss several cases.
$n=3,4,5$, or $M$ is conformally flat in a neighborhood of $x_0$. In this case, we have $$G_{x_0}=r^{2-n}+A+O(r),$$ where $A$ is a constant. Since $\lambda(M,[g])< \lambda(S^n,[g_{S^n}])$, we have $A>0$ when $3\leq n \leq 7$, or $M$ is locally conformally flat, or $M$ is spin. We also have $A>0$ when $M$ is just conformally flat in a neighborhood of $x_0$ under the assumption that the positive mass theorem holds. Then, $$\begin{aligned}
G_{x_0}^{-\frac{2}{n-2}}&=r^2-\frac{2}{n-2}Ar^n+O( r^{n+1}),\\
\partial_rG_{x_0}^{-\frac{2}{n-2}}&=2r-\frac{2n}{n-2}Ar^{n-1}+O( r^{n}),\\
\partial_{rr}G_{x_0}^{-\frac{2}{n-2}}&=2-\frac{2n(n-1)}{n-2}Ar^{n-2}+O( r^{n-1}),\end{aligned}$$ and hence $$\begin{aligned}
G_{x_0}^{-\frac{2}{n-2}} \partial_{rr}G_{x_0}^{-\frac{2}{n-2}}
-\frac12( \partial_r G_{x_0}^{-\frac{2}{n-2}})^2
=-2(n-1)Ar^n+ O( r^{n+1}).\end{aligned}$$
For $n=3,4,5$, by [@LeeParker1987] (Page 61), $R_{11}(x_0)=0$, $R_{11,1}(x_0)=0$ and $R_{11,11}(x_0)\leq 0$, we have $$R_{11}\leq C|x_1|^3\quad\text{on the $x_1$-axis near }x_0=0.$$ We also have $S_{g}(x_0)=0$ and $S_{g,1}(x_0)=0$, and hence $$|S_g |\leq Cx_1^2\quad\text{on the $x_1$-axis near }x_0=0.$$ Take any $x_1>0$ small. Then, at the point $x_1 e_1$, $$\begin{aligned}
\label{eq-Ricci-order}v_i^2R_{11}\le C(\widetilde{u}_i(p_0))^{-\frac{4}{n-2}}x_1^7, \end{aligned}$$ and $$\begin{aligned}
\label{eq-scalar-order}v_i^2|S_g|\le C(\widetilde{u}_i(p_0))^{-\frac{4}{n-2}}x_1^6.\end{aligned}$$ For $i$ large, by , we have, at the point $x_1 e_1$, $$\begin{aligned}
\label{Ricci comp}
R^i_{11}\le (\widetilde{u}_i(p_0))^{-\frac{4}{n-2}}\big[-2(n-1)(n-2)Ax_1^n+Cx_1^6+o(1)\big],\end{aligned}$$ where $o(1)$ denotes terms converging to zero as $i\to \infty$, uniformly for small $x_1$ away from 0. The dominant term in is the $x_1^n$-term, with a negative coefficient. Hence, the expression inside the bracket in is strictly less than 0, for a fixed small $x_1\neq0$ and $i$ large. Therefore, at the point $x_1e_1$, $R^i_{11}$ diverges to $ -\infty$ as $i\rightarrow\infty$. Hence, some component of the Ricci curvature of $g_i$ at the point $x_1e_1$ diverges to $\infty$ as $i\rightarrow\infty$.
If $M$ is conformally flat in a neighborhood of $x_0$, then $R_{11}=0$ and $S_{g}=0$ on the $x_1$-axis and near $x_0=0$. The $x_1^6$-term in is absent. Similarly, at the point $x_1e_1$ for $x_1>0$ sufficient small, $R^i_{11}$ diverges to $ -\infty$ as $i\rightarrow\infty$.
If we denote by $R^i_{rr}$ the Ricci curvature of $g_i$ acting on the unit vector $v_i\frac{\partial}{\partial r}$ with respect to the metric $g_i$, then we conclude similarly that $R^i_{rr}$ at $x$ diverges to $ -\infty$ as $i\rightarrow\infty$, for some $x$ sufficiently close to $x_0$.
$n=6$ and $M$ is not conformally flat in a neighborhood of $x_0$. In this case, $$G_{x_0}(x)=r^{2-n}-\frac{n-2}{1152(n-1)} |W(x_0)|^2\log r -\frac{1}{96}S_{g,ij}(x_0)\frac{x^{i}x^{j}}{r^{2}}
+P(x)\log r +\alpha(x),$$ where $W$ is the Weyl tensor, $P(x)$ is a polynomial with $P(0)=0$, and $\alpha$ is a $C^{2,\mu}$-function. We note that $W_{ijkl}$ is given by $$\begin{aligned}
\label{Weyl tensor}\begin{split}
W_{ijkl} &=R_{ijkl}-\frac{1}{n-2}\big(R_{ik}g_{jl}-R_{il}g_{jk}+R_{jl}g_{ik}-R_{jk}g_{il}\big)\\
&\qquad+\frac{S_{g}}{(n-1)(n-2)}\big(g_{ik}g_{jl}-g_{il}g_{jk}\big).
\end{split}\end{aligned}$$
[*Case 3.2.1.*]{} If $W(x_0) \neq 0$, then, $$\begin{aligned}
G_{x_0}^{-\frac{2}{n-2}}&=r^2+\frac{1}{2880}|W(x_0)|^2r^6\log r + O(r^7\log r) ,\\
\partial_rG_{x_0}^{-\frac{2}{n-2}}&=2r+\frac{1}{480}|W(x_0)|^2r^5\log r+O( r^{5}),\\
\partial_{rr}G_{x_0}^{-\frac{2}{n-2}}&=2+\frac{1}{96}|W(x_0)|^2r^4\log r+O( r^{4}),\end{aligned}$$ and hence $$\begin{aligned}
G_{x_0}^{-\frac{2}{n-2}} \partial_{rr}G_{x_0}^{-\frac{2}{n-2}}
-\frac12( \partial_r G_{x_0}^{-\frac{2}{n-2}})^2
=\frac{1}{144}|W(x_0)|^2r^6 \log r+ O( r^{6}).\end{aligned}$$ Take any $x_1>0$ small. Then, at the point $x_1 e_1$, and still hold. For $i$ large, instead of , we have, at the point $x_1 e_1$, $$\begin{aligned}
R^i_{11}\le (\widetilde{u}_i(p_0))^{-1}\big[\frac{1}{36}|W(x_0)|^2x_1^6 \log x_1+Cx_1^6+o(1)\big].\end{aligned}$$ Similarly as in Case 3.1, at the point $x_1e_1$ for $x_1>0$ sufficiently small, $R^i_{11}$ diverges to $ -\infty$ as $i\rightarrow\infty$.
Similarly, $R^i_{rr}$ at $x$ diverges to $ -\infty$ as $i\rightarrow\infty$, for some $x$ sufficiently close to $x_0$.
[*Case 3.2.2.*]{} We now consider the case $W(x_0)=0$. By , we have $R_{ijkl}(x_0)=0$. Hence, $(\widetilde{M}, \widetilde{g})$ is asymptotically flat of order 3. Using the spherical coordinates, we set $$\phi_4(\theta)=\frac{1}{96}S_{g,ij}(x_0)\frac{x^{i}x^{j}}{r^{2}},$$ and denote by $r^2g_2(\theta )$ the degree two part of the Taylor expansion of $S_{g}$ at $x_0$. Since $$\sum_{i=1}^{n}S_{g,ii}(x_0)=-\frac{1}{6}|W|^2(x_0)=0,$$ then, $$\int_{S^{n-1}}g_2(\theta)d\theta=0.$$ By the positive mass theorem, see [@LeeParker1987] (Page 79, 80), we have $$\int_{S^{n-1}}\big(\phi_4(\theta)+\alpha(0)\big)d\theta>0.$$ Along a radial geodesic $\{(r,\theta):\,0\le r\le \delta\}$, for a small constant $\delta$, we have $$\begin{aligned}
G_{x_0}^{-\frac{2}{n-2}}&=r^2-\frac{1}{2}\big(\phi_4(\theta)+\alpha(0) \big) r^6 +o( r^{6}),\\
\partial_rG_{x_0}^{-\frac{2}{n-2}}&=2r-3\big(\phi_4(\theta)+\alpha(0) \big) r^{5}+o( r^{5}),\\
\partial_{rr}G_{x_0}^{-\frac{2}{n-2}}&=2-15\big(\phi_4(\theta)+\alpha(0) \big) r^{4}+o( r^{4}),\end{aligned}$$ and hence $$\begin{aligned}
G_{x_0}^{-\frac{2}{n-2}} \partial_{rr}G_{x_0}^{-\frac{2}{n-2}}
-\frac12( \partial_r G_{x_0}^{-\frac{2}{n-2}})^2
=-10\big(\phi_4(\theta)+\alpha(0) \big) r^n+ o( r^{n}).\end{aligned}$$ By [@LeeParker1987] (Page 61), along the radial geodesic $(\cdot,\theta)$, $R_{rr,rrr}(x_0)=0$ and $R_{rr,rrrr}(x_0)\leq 0$. Therefore, for $i$ large, by , we have, along the radial geodesic $(\cdot,\theta)$, $$\begin{aligned}
R^i_{rr}|_{(r,\theta)}\le (\widetilde{u}_i(p_0))^{-1}\big[-\frac{1}{10}g_2(\theta)r^6
-40\big(\phi_4(\theta)+\alpha(0) \big)r^6 +o(r^6)+o(1)\big],\end{aligned}$$ where $o(1)$ denotes terms converging to zero as $i\to \infty$, uniformly for small $x$ away from 0. Hence, $$\begin{aligned}
\int_{S^{n-1}}R^i_{rr}d\theta\le (\widetilde{u}_i(p_0))^{-1}
\big[-40r^6\int_{S^{n-1}}\big(\phi_4(\theta)+\alpha(0) \big)d\theta
+o(r^6)+o(1)\big].\end{aligned}$$ Therefore, we can find $\theta_0\in S^{n-1}$ that $R^i_{rr}$ at $(r,\theta_0)$ diverges to $ -\infty$ as $i\rightarrow\infty$, for some $r$ sufficiently small.
[*Case 3.3.*]{} $n\geq7$ and $M$ is not conformally flat in a neighborhood of $x_0$. In this case, $$\begin{aligned}
G_{x_0}(x)=r^{2-n}\bigg[1+\sum_{i=4}^{n}\psi_{5}\bigg]+c\log r+P(x)\log r+\alpha(x),\end{aligned}$$ where $\psi_{i}$ is a homogeneous polynomial of degree $i$, $c$ is a constant, $P(x)$ is a polynomial with $P(0)=0$, and $\alpha$ is a $C^{2,\mu}$-function. We note that $c=0$ and $P\equiv 0$ if $n$ is odd. Moreover, $$\psi_4(x)=\frac{n-2}{48(n-1)(n-4)}\bigg(\frac{r^4}{12(n-6)} |W(x_0)|^2-S_{g,ij}(x_0)x^i x^jr^2\bigg),$$ where $W$ is the Weyl tensor.
[*Case 3.3.1.*]{} First, we consider the case $|W(x_0)| \neq 0$. Note that $S_{g}(x_0)=0$, $\nabla_gS_{g}(x_0)=0$ and $$\Delta_{g}S_{g}(x_0)=-\frac{1}{6}|W(x_0)|^2.$$ Without loss of generality, we assume $S_{g,11}(x_0)<0$. Take any $x_1>0$ small. Then, at the point $x_1 e_1$, still holds. Set $$A=\frac{n-2}{48(n-1)(n-4)}\bigg[\frac{1}{12(n-6)} |W(x_0)|^2
-S_{g,11}(x_0)\bigg].$$ Then, on the positive $x_1$-axis near $x_0=0$, we have $$\begin{aligned}
G_{x_0}^{-\frac{2}{n-2}}&=x_1^2-\frac{2}{n-2}Ax_1^6 +O(x_1^7) ,\\
\partial_{x_1}G_{x_0}^{-\frac{2}{n-2}}&=2x_1-\frac{12}{n-2}Ax_1^5 +O(x_1^6),\\
\partial_{x_1x_1}G_{x_0}^{-\frac{2}{n-2}}&=2-\frac{60}{n-2}Ax_1^4 +O(x_1^5) ,\end{aligned}$$ and $$\begin{aligned}
&(n-2)[G_{x_0}^{-\frac{2}{n-2}} \partial_{x_1x_1}G_{x_0}^{-\frac{2}{n-2}}
-\frac12( \partial_{x_1} G_{x_0}^{-\frac{2}{n-2}})^2]-\frac{1}{2(n-1)}S_{g} G_{x_0}^{-\frac{4}{n-2}}\\
&\quad =-40Ax_1^6
-\frac{1}{4(n-1)}S_{g,11}(x_0)x_1^6+O(x_1^7) .\end{aligned}$$ By the definition of $A$, we obtain $$\begin{aligned}
&(n-2)[G_{x_0}^{-\frac{2}{n-2}} \partial_{x_1x_1}G_{x_0}^{-\frac{2}{n-2}}
-\frac12( \partial_{x_1} G_{x_0}^{-\frac{2}{n-2}})^2]-\frac{1}{2(n-1)}S_{g} G_{x_0}^{-\frac{4}{n-2}}\\
&\quad =-\frac{1}{12(n-1)(n-4)}\bigg[\frac{5(n-2)}{6(n-6)}|W(x_0)|^2
-(7n-8)S_{g,11}(x_0)\bigg]x_1^6+O(x_1^7).\end{aligned}$$ For $i$ large, instead of , we have, at the point $x_1 e_1$, $$\begin{aligned}
R^i_{11}\le (\widetilde{u}_i(p_0))^{-\frac{4}{n-2}}\big[-Bx_1^6+Cx_1^7+o(1)\big],\end{aligned}$$ for some positive constant $B$. Then, we conclude ${R}_{11}^i$ at the point $x_1e_1$ diverges to $ -\infty$ as $i\rightarrow\infty$, for $x_1>0$ sufficiently small.
[*Case 3.3.2.*]{} We now consider the case $W(x_0)=0$. By , we have $R_{ijkl}(x_0)=0$. Using the spherical coordinates, we set $$\psi_i=r^i\phi_i(\theta),$$ and denote by $r^ig_i(\theta )$ the $i$-th Taylor expansion of $S_{g}$ at $x_0$. Let $r^lg_l(\theta )$ be the first nonzero term in the Taylor expansion of $S_{g}$ at $x_0$.
[*Subcase 3.3.2(a).*]{} $2\leq l \leq n-5$. By [@LeeParker1987] or [@SchoenYau1994], we have $\psi_{i}=0$, $i=4,...,l-1$, and $$\mathcal{L}\psi_{l+2} =-\frac{n-2}{4(n-1)} r^{l+2}g_l(\theta),$$ where $$\mathcal{L}=-r^2\Delta+2(n-2)r\partial_r.$$ Here, $\Delta$ is the standard Laplacian on the Euclidean space, i.e., $$\Delta=\frac{\partial^2}{\partial r^2}+\frac{n-1}{r}\frac{\partial}{\partial r}+\frac{1}{r^2}\Delta_{S^{n-1}}.$$ Then, we have $$(l+2)(n-4-l)\int_{S^{n-1}}\psi_{l+2}=\int_{S^{n-1}}\mathcal{L}\psi_{l+2} =-\frac{n-2}{4(n-1)} \int_{S^{n-1}}r^{l+2}g_l(\theta).$$ Hence, $$\label{psi-g}\int_{S^{n-1}}\phi_{l+2}=-\frac{n-2}{4(n-1)(l+2)(n-4-l)}\int_{S^{n-1}}g_l(\theta).$$ We also have $$\begin{aligned}
\label{intergral indentity1}\begin{split}
\int_{S^{n-1}}g_l(\theta)&=\frac{r^{2-n-l}}{l}\int_{0}^{r}\int_{S^{n-1}}\Delta\bigg(s^l g_l(\theta)\bigg)s^{n-1}dr d\theta \\
&=\lim_{r\rightarrow0}\frac{r^{2-n-l}}{l}\int_{B_{r}(x_0)}\Delta_{g}S_{g}dV_{g}\\
&=\lim_{r\rightarrow0}\frac{-r^{2-n-l}}{6l}\int_{B_{r}(x_0)}|W|^2dV_{g}\leq 0.
\end{split}\end{aligned}$$ Note that the sign of $\lim_{r\rightarrow0}r^{2-n-l}\int_{B_{r}(x_0)}|W|^2dV_{g}$ is independent of $g\in[g]$. Hence, if for some $ i$ with $2\leq i \leq n-5$, $$0<\lim_{r\rightarrow0}r^{2-n-i}\int_{B_{r}(x_0)}|W|^2dV_{g}<\infty,$$ then the $i$-th Taylor expansion of $S_{g}$ at $x_0$ must not be identical to zero.
Note $$\begin{aligned}
G_{x_0}^{-\frac{2}{n-2}}&=r^2-\frac{2}{n-2} r^{l+4}\phi_{l+2} +o( r^{l+4}),\\
\partial_rG_{x_0}^{-\frac{2}{n-2}}&=2r-\frac{2}{n-2}(l+4) r^{l+3}\phi_{l+2} +o( r^{l+3}),\\
\partial_{rr}G_{x_0}^{-\frac{2}{n-2}}&=2-\frac{2}{n-2}(l+4)(l+3)r^{l+2}\phi_{l+2}+o( r^{l+2}),\end{aligned}$$ and hence, $$\begin{aligned}
&(n-2)[G_{x_0}^{-\frac{2}{n-2}} \partial_{x_1x_1}G_{x_0}^{-\frac{2}{n-2}}
-\frac12( \partial_{x_1} G_{x_0}^{-\frac{2}{n-2}})^2]-\frac{1}{2(n-1)}S_{g} G_{x_0}^{-\frac{4}{n-2}}\\
&\quad =\bigg[-2(l+2)(l+3)\phi_{l+2}(\theta)-\frac{1}{2(n-1)}g_{l}(\theta) \bigg]r^{l+4}+o(r^{l+4}).\end{aligned}$$
By [@LeeParker1987] (Page 61), along a radial geodesic $(\cdot,\theta)$, $\frac{\partial^i}{\partial r^i}R_{rr}(x_0)=0$, $i=1,...,l-1$ and $\frac{\partial^l}{\partial r^l}R_{rr}(x_0)\leq 0$. Therefore, for $i$ large, by , we have, along a radial geodesic $(\cdot,\theta)$, $$\begin{aligned}
\label{Ricci comp3}\begin{split}
R^i_{rr}|_{(r,\theta)}&\le (\widetilde{u}_i(p_0))^{-\frac{4}{n-2}}\bigg\{ \bigg[-2(l+2)(l+3)\phi_{l+2} (\theta)
-\frac{1}{2(n-1)}g_{l}(\theta) \bigg]r^{l+4}\\
&\qquad+o(r^{l+4})+o(1)\bigg\},
\end{split}\end{aligned}$$ where $o(1)$ denotes terms converging to zero as $i\to \infty$, uniformly for small $x$ away from 0. By and , we have $$\begin{aligned}
&-2(l+2)(l+3)\int_{S^{n-1}}\phi_{l+2}(\theta)d\theta-\frac{1}{2(n-1)}\int_{S^{n-1}}g_{l}(\theta)d\theta\\
&\qquad=\frac{(n-2)(l+3)-(n-4-l)}{2(n-1)(n-4-l)}\int_{S^{n-1}}g_l(\theta) d\theta
\leq 0.\end{aligned}$$ By [@LeeParker1987] Lemma 5.3 or [@SchoenYau1994] chapter 5, $$2(l+2)(l+3)\phi_{l+2} \neq \frac{1}{2(n-1)}g_{l}.$$ Hence, we can find $\theta_0 \in S^{n-1}$ such that $$-2(l+2)(l+3)\phi_{l+2} (\theta_0)-\frac{1}{2(n-1)}g_{l}(\theta_0) \leq -\epsilon_0,$$ for some positive constant $\epsilon_0$. Therefore, along the radial geodesic $(r,\theta_0)$, $$\begin{aligned}
R^i_{rr}|_{(r,\theta_0)}\le& (\widetilde{u}_i(p_0))^{-\frac{4}{n-2}}\bigg[-\epsilon_0r^{l+4}+o(r^{l+4})+o(1)\bigg].\end{aligned}$$ Then, we conclude ${R}_{rr}^i$ at the point $(r,\theta_0)$ diverges to $ -\infty$ as $i\rightarrow\infty$, for $r>0$ sufficiently small.
[*Subase 3.3.2(b).*]{} $l\geq n-4$. When $n$ is even, we have $$\mathcal{L}(\psi_{n-2}+cr^{n-2} \log r)=\mathcal{L}\psi_{n-2}-(n-2)cr^{n-2} =-\frac{n-2}{4(n-1)} r^{n-2}g_{n-4}(\theta).$$ Then, $$\begin{aligned}
\label{intergral indentity2}\begin{split}
(n-2)c w_{n-1} &=-r^{2-n}\int_{S^{n-1}}\mathcal{L}(\psi_{n-2}+cr^{n-2} \log r)\\
&=\frac{(n-2)r^{6-2n}}{4(n-1)(n-4)}\int_{0}^{r}\int_{S^{n-1}}\Delta\bigg(s^{n-4}g_l(\theta)\bigg)s^{n-1}dr d\theta \\
&=\lim_{r\rightarrow0}\frac{(n-2)r^{6-2n}}{4(n-1)(n-4)}\int_{B_{r}(x_0)}\Delta_{g}S_{g}dV_{g}\\
&=\lim_{r\rightarrow0}\frac{-(n-2)r^{6-2n}}{24(n-1)(n-4)}\int_{B_{r}(x_0)}|W|^2dV_{g}\leq 0.
\end{split}\end{aligned}$$ We note $$\lim_{r\rightarrow0}r^{6-2n}\int_{B_{r}(x_0)}|W|^2dV_{g}=0,$$ when $n$ is odd, since $\int_{S^{n-1}}g_{n-2}(\theta) d\theta=0$ when $n$ is odd.
If $c>0$, we can proceed as the proof of Case 3.2.1, $n=6$ and $|W(x_0)|\neq 0$, and conclude that $R^i_{rr}$ at $x$ diverges to $ -\infty$ as $i\rightarrow\infty$, for some $x$ sufficiently close to $x_0$.
In general, we first consider the case that there exist a pair $(i,j)\in\{1\cdots n\}\times\{1\cdots n\}$ and a constant $k<[\frac{n-4}{2}]$ such that $R_{ij}\neq0$ and $k$ is the order of the first nonzero term in the Taylor expansion of $R_{ij}$ at $x_0$.
Without loss of generality, we assume the order of the first nonzero term in the Taylor expansion of some $ R_{pq}$ at $x_0$ is $k$, $k<[\frac{n-4}{2}]$, and all other $R_{ij}$ vanish up to order $k$ at $x_0$. Then, by , all $R_{ijkl}$ vanish up to order $k$, and hence, all $g_{ij}-\delta_{ij}$ vanish up to order $k+2$. By a rotation, we can assume $$\frac{\partial^k}{\partial x_1^k}R_{pq}|_{x_0}\neq0.$$ By [@LeeParker1987] (Page 61) $(p,q)\neq (1,1)$. By [@LeeParker1987] (Page 61), $(p,q)\neq(1,1)$. If $p \neq1$ and $q \neq 1$, by a rotation, we can assume $p=q=2$. Otherwise, we can assume $(p,q)=(1,2)$.
We consider the case $(p,q)=(2,2)$. By Guass Lemma, we have $$x_j=\sum_{i=1}^n g_{ji}x_i.$$ Then, we have, on the $x_1$-axis near $x_0=0$, $$\begin{aligned}
\label{g12g22}1&=x_1\frac{\partial}{\partial x_2 } g_{12}+g_{22},\\
\label{g12g11}0&=x_1\frac{\partial}{\partial x_2 } g_{11}+g_{12},\end{aligned}$$ and $$0=x_1\frac{\partial^2 }{\partial x_2^2}g_{22}+2\frac{\partial }{\partial x_2}g_{12}.$$ We also have, at $x_0=0$, $$(k+2)\frac{\partial^{k+2} }{\partial x_1^{k+1} \partial x_2}g_{21}+\frac{\partial^{k+2} }{\partial x_1^{k+2} }g_{22}=0,$$ and $$(k+1)\frac{\partial^{k+2} }{\partial x_1^{k} \partial^2 x_2}g_{11}
+2\frac{\partial^{k+2} }{ \partial x_1^{k+1} \partial x_2}g_{12}=0.$$ Hence, we have, at $x_0$, $$\begin{aligned}
\frac{\partial^{k} }{\partial x_1^{k} }R_{1212}&=\frac{1}{2}
\bigg( 2\frac{\partial^{k+2} }{\partial x_1^{k+1} \partial x_2}g_{21}- \frac{\partial^{k+2} }{\partial x_1^{k+2} }g_{22}
- \frac{\partial^{k+2}}{\partial x_1^{k} \partial^2 x_2}g_{11}\bigg)\\
&=-\frac{k+3}{k+1}\frac{\partial^{k+2} }{\partial x_1^{k+2} }g_{22}(x_0).\end{aligned}$$ Therefore, by , we have $$\label{r22 expansion}\frac{\partial^{k} }{\partial x_1^{k} }|_{x_0}R_{22}
=(n-2)\frac{\partial^{k} }{\partial x_1^{k} }|_{x_0}R_{1212}
=-(n-2)\frac{k+3}{k+1}\frac{\partial^{k+2} }{\partial x_1^{k+2} }g_{22}(x_0)<0.$$ Then, for $i$ large, by , we have, at the point $x_1 e_1$, $$\begin{aligned}
R^i_{22}= (\widetilde{u}_i(p_0))^{-\frac{4}{n-2}}&\bigg[\frac{1}{k!}\frac{\partial^{k}}{\partial x_1^{k} }|_{x_0}R_{22}x_1^{k+4}
+(n-2)\bigg(-2\Gamma_{22}^1 x_1^3-2(g_{22}-1)x_1^2\bigg)\\
&\quad+O(x_1^{k+5} )+o(1)\bigg],\end{aligned}$$ where $o(1)$ denotes terms converging to zero as $i\to \infty$, uniformly for small $x_1$ away from 0. At the point $x_1 e_1$, $$\Gamma_{22}^1 =\frac{1}{2}\bigg(2\frac{\partial}{\partial x_2}g_{12}-\frac{\partial}{\partial x_1}g_{22}\bigg).$$ Combining with , we get, at the point $x_1 e_1$, $$\begin{aligned}
R^i_{22}&= (\widetilde{u}_i(p_0))^{-\frac{4}{n-2}}\bigg[\frac{1}{k!}\frac{\partial^{k}}{\partial x_1^{k} }|_{x_0}R_{22}x_1^{k+4}
+(n-2)\frac{\partial}{\partial x_1}g_{22}x_1^3+O(x_1^{k+5} )+o(1)\bigg]\\
&=(\widetilde{u}_i(p_0))^{-\frac{4}{n-2}}\bigg[-(n-2)\frac{k+2}{(k+1)!}
\frac{\partial^{k+2} }{\partial x_1^{k+2} }g_{22}(x_0)x_1^{k+4}+O(x_1^{k+5} )+o(1)\bigg].\end{aligned}$$ Then, we conclude ${R}_{22}^i$ at the point $x_1e_1$ diverges to $ -\infty$ as $i\rightarrow\infty$, for $x_1>0$ sufficiently small.
If $(p,q)=(1,2)$, we can argue similarly to conclude $|{R}_{12}^i|$ at the point $x_1e_1$ diverges to $ \infty$ as $i\rightarrow\infty$, for $x_1>0$ sufficiently small.
We now consider the case that the order of the first nonzero term in the Taylor expansion of all $R_{pq}$ are greater or equal to $[\frac{n-4}{2}]$ at $x_0$, and $$\lim_{r\rightarrow0}r^{6-2n}\int_{B_{r}(x_0)}|W|^2dV_{g}=0.$$ Then, by , the order of the first nonzero term in the Taylor expansion of $R_{ijkl}$ at $x_0$ are greater or equal to $[\frac{n-4}{2}]$, and hence, the order of the first nonzero term in the Taylor expansion of $g_{ij}-\delta_{ij}$ at $x_0$ are greater or equal to $[\frac{n-4}{2}]+2$. Hence, $(\widetilde{M}, \widetilde{g})$ is asymptotically flat of order $[\frac{n-4}{2}]+2$. Hence, the ADM-mass of $(\widetilde{M}, \widetilde{g})$ is well defined. By the positive mass theorem, we have $$\int_{S^{n-1}}\big(\phi_{n-2}(\theta)+\alpha(0)\big)d\theta>0.$$ Then, we can proceeds as the proof of Case 2.2, $n=6$ and $|W(x_0)|= 0$, and find $\theta_0\in S^{n-1}$ such that $R^i_{rr}$ at $(r,\theta_0)$ diverges to $ -\infty$ as $i\rightarrow\infty$, for some $r$ sufficiently small.
\[rmk-single-point-sphere\] The blow up phenomena in Theorem \[blow up-one point\] are significantly different from those for the case that the underlying manifold is $S^n$. For example, take $\Omega_i=S^n\setminus B_{1/i}(e_n)$, where $B_{1/i}(e_n)$ is a small ball on $S^n$ centered at the north pole. Then, $\Omega_i\rightarrow S^n\backslash \{e_n\}$ and the complete conformal metric $g_{i}$ in $\Omega_i$ with the constant scalar curvature $-n(n-1)$ has a constant sectional curvature $-1$! This can be verified by the stereographic projection, as the image of $S^n\setminus B_{1/i}(e_n)$ under the stereographic projection from the north pole is a ball in $\mathbb R^n$ centered at the origin.
We note that Theorem \[them-rigidity\] follows easily from Theorem \[blow up-one point\]. Now we are ready to prove Theorem \[thrm-large-positive-Ricci\].
Let $u_i$ be the solution of and in $\Omega_i$. Then, $g_{i}=u_i^{\frac{4}{n-2}}g.$
The proof of Theorem \[prop-general mainfold\] can be adapted to prove Case 1, i.e., $\Gamma$ contains a submanifold of dimension $j$, with $1\le j\leq \frac{n-2}{2}$.
Next, we consider Case 2, i.e., $(M, g)$ is not conformally equivalent to the standard sphere $S^n$ and $\Gamma$ consists of finitely many points. If $\lambda(M,[g])\leq 0$, the proof of Theorem \[prop-general mainfold\] and Theorem \[blow up-one point\] can be adapted to yield the conclusion here. Hence, we only need to discuss the case $\lambda(M,[g])> 0$. For simplicity, we assume $\Gamma$ consists of two points $x_0$ and $y_0$. The general case can be discussed similarly.
Let $G_{x_0}\in C^{\infty}(M \setminus\
\{x_0\})$ and $G_{y_0}\in C^{\infty}(M \setminus\
\{y_0\})$ be the Green’s functions for the conformal Laplacian $L_g$ with the pole at $x_0$ and $y_0$, respectively; namely, $$\begin{aligned}
L_gG_{x_0}&=(n-2)\omega_{n-1}\delta_{x_0},\,\, G_{x_0}>0,\\
L_gG_{y_0}&=(n-2)\omega_{n-1}\delta_{y_0},\,\, G_{y_0}>0,\end{aligned}$$ where $\omega_{n-1}$ is the volume of $S^{n-1}$. Up to conformal factors, we assume $(M, g)$ has conformal normal coordinates in small neighborhoods of $x_0$ and $y_0$. Consider the metric $$\widetilde{g}
= (G_{x_0}+G_{y_0})^{\frac{4}{n-2}}g\quad\text{on }\widetilde{M} =M \setminus\
\{x_0, y_0\}.$$ Then, $(\widetilde{M}, \widetilde{g})$ is an asymptotically flat and scalar flat manifold, and $\widetilde{g}$ has an asymptotic expansion near infinity.
Set $u_i= (G_{x_0}+G_{y_0})\widetilde{u}_i$. Then, $\widetilde{u}_i$ satisfies $$\begin{aligned}
\label{eq u_i2} \Delta_{\widetilde{g}} \widetilde{u}_{i} &=
\frac14n(n-2) \widetilde{u}_{i}^{\frac{n+2}{n-2}} \quad\text{in }\,\Omega_i,\\
\label{eq-u i Boundary2}\widetilde{u}_{i}&=\infty\quad\text{on }\partial \Omega_i,\end{aligned}$$ and, for any $m$, $$\label{u_i go to zero2}
\widetilde{u}_i\rightarrow 0\quad\text{in }C^{m}_{\mathrm{loc}}( M \setminus\{x_0, y_0\})
\text{ as }i\rightarrow\infty.$$ Fix a point $p_0\in M \setminus\{x_0, y_0\} $. Then, for $i$ sufficiently large, $p_0 \in \Omega_i$. Set $\widetilde{w}_i={\widetilde{u}_i}/{\widetilde{u}_i(p_0)}$. Then, $\widetilde{w}_i(p_0)=1$ and $\widetilde{w}_i$ satisfies $$\begin{aligned}
\label{eq w_i2} \Delta_{\widetilde{g}} \widetilde{w}_{i}
&= \frac14n(n-2)u_i(p_0)^{\frac{4}{n-2}} \widetilde{w}_{i}^{\frac{n+2}{n-2}} \quad\text{in }\,\Omega_i,\\
\label{eq-w i Boundary2}\widetilde{w}_{i}&=\infty\quad\text{on }\partial \Omega_i.\end{aligned}$$ By interior estimates, there exists a positive function $\widetilde{w}\in \widetilde{M}$ such that, for any $m$, $$\widetilde{w}_i\rightarrow \widetilde{w}\quad\text{in }C^{m}_{\mathrm{loc}}( \widetilde{M})
\text{ as }i\rightarrow\infty,$$ and $$\Delta_{\widetilde{g}} \widetilde{w}=0 \quad\text{in } \widetilde{M}.$$ Hence, $$L_g( (G_{x_0}+G_{y_0})\widetilde{w})=0 \quad\text{in } M \setminus\{x_0, y_0\}.$$ By the expansions of $G_{x_0}$ and $G_{y_0}$ near $x_0$ and $y_0$, respectively, and Proposition 9.1 in [@LiZhu1999], we conclude that $ \widetilde{w}$ converges to some constant $\alpha$ as $x\to x_0$ and converges to some constant $\beta$ as $x\to y_0$. Without loss of generality, we assume $\alpha \geq \beta$. Then, $\alpha \geq1$. By Proposition 9.1 in [@LiZhu1999] and the maximum principle, we have $$\widetilde{w}=\alpha+Br^{n-2}+O(r^{n-1}),$$ for some nonpositive constant $B$. Then, the proof follows similarly as that of Theorem \[blow up-one point\].
Next, we consider Case 3, i.e., $(M,g)$ is conformally equivalent to the standard sphere $S^n$ and $\Gamma$ consists of at least two points. We can assume $(M,g)=(S^n,g_{S^n})$. By Lemma \[lemme-convergence\], we have, for any $m$, $$u_i\rightarrow 0\quad\text{in }C^{m}_{\mathrm{loc}}(S^n \setminus \Gamma)
\text{ as }i\rightarrow\infty.$$ Set $v_i=u_i^{-\frac{2}{n-2}}$. Then, $$v_i \text{ diverges to $\infty$ locally uniformly in $S^n \setminus\Gamma$
as $i\rightarrow\infty$}.$$ Take two different points $p ,q \in \Gamma$ and let $\sigma_{pq}$ be the shorter geodesic connecting $p$ and $q$. Up to a conformal transform if necessary, we assume $|\sigma_{pq}|=2\epsilon$, which is less than $\frac{1}{100n}$, and $\sigma_{pq}\bigcap\Gamma=\{p,q\}$. We parametrize $\sigma_{pq}$ by its arc length $t\in[0, 2\epsilon]$, with $p$ corresponding to $t=0$ and $q$ to $t=2\epsilon$.
For $i$ large, let $p_i$ and $q_i$ be the points parametrized by $\widehat{t}_i$ and $\widetilde{t}_i$, respectively, where $$\begin{aligned}
\widehat{t}_i&=\min \{t'\in[0, \epsilon] |te_n \in \Omega_i, \text{ for any } t\in ( t',\epsilon]\},\\
\widetilde{t}_i&=\max \{t'\in [\epsilon, 2\epsilon] |t' \in \Omega_i ,\text{ for any } t\in [\epsilon,t')\}.\end{aligned}$$ Then, $p_i, q_i\in \partial\Omega_i$ and $$p_i\rightarrow p,\quad q_i\rightarrow q.$$ For convenience, we denote by $v_i(t)$ the function $v_i$ restricted to the geodesic $\sigma_{pq}$. By the polyhomogenous expansion of $v_i$, we have $|\partial_t v_i(p_i)|\leq 1$ and $|\partial_t v_i(q_i)|\leq 1$.
Since $v_i(\epsilon)\to \infty$, for $i$ large, we take $t_i \in (\widehat{t}_i,\widetilde{t}_i)$ such that, for any $t\in (\widehat{t}_i,\widetilde{t}_i)$, $$\partial_tv_i(t)\leq \partial_tv_i(t_i).$$ Then, $$\partial_tv_i(t_i)>\frac{v_i(\epsilon)-0}{\epsilon-\widehat{t}_i}\geq\frac{ v_i(\epsilon)}{\epsilon},$$ and $$\partial_{tt}v_i( t_i)=0.$$ Denote by $R^i_{tt}$ the Ricci curvature of $g_i$ acting on the unit vector $v_i\frac{\partial}{\partial t}$ with respect to the metric $g_i$. Then, we can verify at the point $t_ie$, $R^i_{tt}$ diverges to $ -\infty$ as $i\rightarrow\infty$.
Domains in Euclidean Spaces {#sec-Conv}
===========================
In this section, we study Ricci curvatures and sectional curvatures of the complete conformal metrics associated with the Loewner-Nirenberg problem in bounded domains in the Euclidean space. There are two classes of results. First, we will prove that the complete conformal metrics in bounded convex domains have negative sectional curvatures. Second, we will construct bounded star-shaped domains in which the complete conformal metrics have positive Ricci components at some points. This shows that pure topological conditions are not sufficient for domains in which the complete conformal metrics have negative Ricci curvatures.
We first discuss the complete conformal metrics in convex domains and prove Theorem \[main reslut\]. The convexity assumption allows us to apply a convexity theorem by Kennington [@Kennington1985] directly to conformal factors.
Let $u$ be the solution of and in $\Omega$ and $v$ be given by . Then, $g=v^{-2}g_E$ is the complete conformal metric in $\Omega$ with a constant scalar curvature $-n(n-1)$. Denote by $R_{ijij}$ and $R_{ij}$ the sectional curvatures and Ricci curvatures of $g$ in the orthonormal coordinates of $g$, given by and , respectively. Here, we surpress $\Omega$ from the notations $g$, $R_{ij}$ and $R_{ijij}$.
By applying the Laplacian operator to , we get $$v\Delta (\Delta v)+(2-n)\nabla v \nabla(\Delta v)= n|\nabla^2v|^2-(\Delta v)^2 \geq 0.$$ First, we assume that the boundary of $\Omega$ is smooth. By , we have $$\Delta v=-H_{\partial\Omega}-\frac{1}{n-1}H_{\partial\Omega}+O(d)=-\frac{n}{n-1}H_{\partial\Omega}+O(d).$$ Since $\Omega$ is convex, we have $\Delta v\le 0$ on $\partial \Omega$. By the strong maximum principle, we obtain $\Delta v<0$ in $\Omega$. Therefore, $|\nabla v| <1$ in $\Omega$ by . Next, we apply Theorems 3.1 and 3.2 [@Kennington1985] in $\Omega$ and conclude that $v$ is concave. For general bounded convex domains, we can obtain the concavity of $v$ by approximations.
Since $v_{ii}\leq 0$, by and , we get, for any $i\neq j$, $$R_{ijij}\leq 0,$$ and, for any $i$, $$R_{ii} \leq -\frac{n}{2}.$$ By , we also have, for any $i$, $$\label{v-concave}\frac{u_{ii}}{u}-\frac{n}{n-2}\frac{u_i^2}{u^2} \geq 0.$$
Next, we prove that $R_{ijij}$ does not vanish in $\Omega$ for any $i\neq j$. If $R_{ijij}= 0$ at some point $x_0\in\Omega$ for some $i\neq j$, then $$\bigg(\frac{u_{ii}}{u}-\frac{n}{n-2}\frac{u_i^2}{u^2}\bigg)(x_0)=0,$$ and $$\nabla u(x_0)=0.$$ Hence, $u_{ii}(x_0)=0$. Applying $\partial_i$ twice to the equation , we get $$\Delta u_{ii}=\frac14n(n+2)u^{\frac{n+2}{n-2}}\frac{u_{ii}}{u}+\frac{n(n+2)}{n-2}u^{\frac{n+2}{n-2}}\frac{u_i^2}{u^2}.$$ Combining with , we have $$\Delta u_{ii}-\frac14(n+2)(n+4)u^{\frac{4}{n-2}}u_{ii}
=(n+2)u^{\frac{n+2}{n-2}}\bigg(\frac{n}{n-2}\frac{u_i^2}{u^2}-\frac{u_{ii}}{u}\bigg)\leq0.$$ By the strong maximum principle, we have $u_{ii}\equiv0 $ in $\Omega$. On the other hand, by $u_{i}(x_0)=0$, we get $u_{i}\equiv0$ on $\Omega \cap \{x_0+te_i|t \in \mathbb{R}\}$. Therefore, $u$ is constant on $\Omega \cap \{x_0+te_i|t \in \mathbb{R}\}$. This leads to a contradiction. Therefore, we have, for any $i\neq j$, $$R_{ijij}< 0.$$ Similarly, we have, for any $i$, $$R_{ii} <-\frac{n}{2}.$$ This completes the proof.
We now construct bounded domains in which the complete conformal metrics have positive Ricci components at some points. The most straightforward way to do this is to combine Theorem \[thrm-large-positive-Ricci\] for the case $(M,g)=(S^n,g_{S^n})$ and the stereographic projections.
We identify $\mathbb{R}^{n}$ in $\mathbb{R}^{n+1}$ as $\mathbb{R}^{n}\times \{0\}$ and write $x=(x_1, \cdots, x_n)\in\mathbb R^n$. Then, $$S^n=\{(x,x_{n+1}):\,|x|^{2}+x_{n+1}^{2}=1\}.$$ Consider the transform $T: \mathbb R^{n}\to S^n$ given by $$T(x)=\left(\frac{2 x}{1+|x|^2},\frac{|x|^2-1}{1+|x|^2}\right).$$ Then, $T$ is the inverse transform of the stereographic projection which lifts $\mathbb{R}^{n}\times \{0\}$ to $S^n$.
\[prop-general example\] Let $\Gamma$ be a set in $S^n$ as in Theorem \[thrm-large-positive-Ricci\], containing the north pole. Suppose $\widetilde{\Omega}_i$ is a sequence of increasing smooth domains in $S^n$ which converges to $S^{n}\setminus \Gamma$, with $\partial \widetilde{\Omega}_i$ not containing the north pole, and set $\Omega_i=T^{-1}\widetilde{\Omega}_i$. Assume $g_i$ is the complete conformal metric in $\Omega_i$ with the constant scalar curvature $-n(n-1)$. Then, for sufficiently large $i$, $g_i$ has a positive Ricci curvature component somewhere in $\Omega_{i}$. Moreover, the maximal Ricci curvature of $g_i$ in $\Omega_{i}$ diverges to $\infty$ as $i\to\infty$.
Proposition \[prop-general example\] follows easily from Theorem \[thrm-large-positive-Ricci\] for the case $(M,g)=(S^n,g_{S^n})$.
We point out that notations in Proposition \[prop-general example\] is slightly different from those in Theorem \[thrm-large-positive-Ricci\]. In Proposition \[prop-general example\], $\widetilde {\Omega}_i$ is a domain in $S_n$ and $\Omega_i$ is a domain in $\mathbb R^n$. We also note that $\Omega_i$ is a bounded domain in $\mathbb R^n$ if the north pole is not in the closure of $\widetilde {\Omega}_i$.
As the first application, let $\{p_1,..,p_k \}$ be a collection of finitely many points in $\mathbb R^n$, with $k\ge 1$, and set $\Omega_{R,r}=B_R(0)\backslash \bigcup_{i=1}^{k} B_r(p_i)$. Then, for $R$ sufficiently large and $r$ sufficiently small, the complete conformal metric in $\Omega_{R,r}$ with the constant scalar curvature $-n(n-1)$ has a positive Ricci curvature component somewhere. Note that the corresponding $\Gamma$ in $S^n$ is given by $\Gamma=\{e_{n+1}, T(p_1), \cdots, T(p_k)\}$, which consists of at least two points. If $k=1$ and $p_1=0$, then $\Omega_{R, r}$ is an annular region.
As another application, we construct bounded star-shaped domains in which the complete conformal metrics have positive Ricci components somewhere. For $n\ge 4$, set $$\gamma=\{(0, \cdots, 0, x_n)|\, |x_n|\ge 1\}\subset\mathbb R^n.$$ Let $\Omega_i$ be a sequence of increasing bounded smooth domains in $\mathbb R^n$, star-shaped with respect to the origin, which converges to $\mathbb R^n\setminus \gamma$. Then, for $i$ sufficiently large, the complete conformal metric in $\Omega_i$ with the constant scalar curvature $-n(n-1)$ has a positive Ricci curvature component somewhere. Note that the corresponding $\Gamma$ in $S^n$ is given by the equator in the $x_n$-$x_{n+1}$ plane minus the image under $T$ of the segment $(-1,1)$ on $x_n$-axis. Hence, the dimension of $\Gamma$ is 1. This is the reason we require $n\ge 4$.
[DG]{}
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[^1]: The first author acknowledges the support of NSF Grant DMS-1404596. The second author acknowledges the support of NSFC Grant 11571019.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Acoustic fields scattered by poroelastic materials contain key information about the materials’ pore structure and elastic properties. Therefore, such materials are often characterised with inverse methods that use acoustic measurements. However, it has been shown that results from many existing inverse characterisation methods agree poorly. One reason is that inverse methods are typically sensitive to even small uncertainties in a measurement setup, but these uncertainties are difficult to model and hence often neglected. In this paper, we study characterising poroelastic materials in the Bayesian framework, where measurement uncertainties can be taken into account, and which allows us to quantify uncertainty in the results. Using the finite element method, we simulate measurements where ultrasonic waves are incident on a water-saturated poroelastic material in normal and oblique angles. We consider uncertainties in the incidence angle and level of measurement noise, and then explore the solution of the Bayesian inverse problem, the posterior density, with an adaptive parallel tempering Markov chain Monte Carlo algorithm. Results show that both the elastic and pore structure parameters can be feasibly estimated from ultrasonic measurements.'
author:
- 'Matti Niskanen$^{a,b,}$[^1], Olivier Dazel$^{b}$, Jean-Philippe Groby$^{b}$, Aroune Duclos$^{b}$, Timo Lähivaara$^{a}$'
date: |
$^{a}$ Department of Applied Physics, University of Eastern Finland, Yliopistonranta 1, FIN-70211 Kuopio, Finland\
$^{b}$ Laboratoire d’Acoustique de l’Université du Mans – UMR CNRS 6613, Avenue Olivier Messiaen,\
F-72085 Le Mans Cedex, France\
title: 'Characterising poroelastic materials in the ultrasonic range - A Bayesian approach'
---
Introduction
============
Modelling the propagation of sound in poroelastic media is motivated by two complementary aims. Firstly, it is needed to explain the acoustic absorption and transmission properties of structures. Secondly, the sound waves interact with the material in a non-destructive manner and carry information about its micro-structure and elastic behaviour. Knowledge of these properties is highly valuable in many areas such as geophysical exploration [@slatt2013stratigraphic], design of materials for noise treatment [@allard2009propagation; @cox2016acoustic], industrial filtration applications [@espedal2007filtration], and characterisation of bone tissues [@baroncelli2008quantitative].
Inverse characterisation methods are based on conducting measurements that can be related to the unknowns of interest through a physical model. It is possible to measure some material properties directly, but the use of such methods is limited and they are not considered in this paper. The attractiveness of inverse methods for porous materials stems partly from the fact that they can rely on relatively simple acoustic measurements. In the audible frequency range, such experiments are usually done with the impedance tube. The tested material is cut into the shape of the tube and faces normally to the incident wave field. For an overview of these methods we refer to recent review papers by Bonfiglio and Pompoli [@bonfiglio2013inversion] and Horoshenkov [@horoshenkov2017review] (see also [@zielinski2015normalized] and the references therein). Another option is to do the measurements in an open field, typically in the ultrasonic frequency range, although audible frequencies can be used with large enough samples. In the open field, the sound source can be pointed at the tested object either in normal or oblique incidence [@jocker2007minimization; @groby2010analytical].
Currently, numerous direct and inverse characterisation methods are in use, and the results they give do not agree very well. This phenomenon has been noted in the case of rigid frame porous materials in two inter-laboratory tests by Horoshenkov *et al.* in 2007 [@horoshenkov2007reproducibility] and Pompoli *et al.* in 2017 [@pompoli2017reproducible]. The reproducibility of the results was poor, especially with materials having a high flow resistivity. Recently, Bonfiglio *et al.* [@bonfiglio2018reproducible] reported a result where fourteen independent laboratories measured the elastic properties of specimens of the same porous materials. Again the finding was that the reproducibility of the experimental methods is poor, and in the worst case the results differed by two orders of magnitude. This poor reproducibility can be attributed to a lack of standardised measurement and calibration procedures, and to underlying uncertainties in measurement setup, which are difficult to model and therefore often neglected. However, it has been shown that, in inverse problems, neglecting modelling errors typically yields meaningless estimates [@kaipio2013approximate; @nicholson2018estimation].
Most of the inverse methods found in literature for characterisation of porous media are deterministic and based on iteratively minimising a cost function to find a point estimate for the parameters. A problem with the deterministic approach is that it does not provide a simple framework for quantifying uncertainty in the estimates. Another issue is that the cost function often exhibits several minima, and finding the global minimum can be difficult [@aster2018parameter]. Several papers have discussed ways to circumvent this problem, by the use of multiple cost functions formed from different frequency data [@ogam2011non], normalising the estimated parameters before doing inversion [@zielinski2015normalized], or using differential evolution algorithms [@atalla2005inverse], among others. A promising recently proposed approach is to minimise the cost function in a stepwise manner [@goransson2019parameter].
In practice the models and measurement data always include uncertainties, and so the material parameters cannot be known exactly. Therefore, we adopt the Bayesian framework (see for example [@kaipio2006statistical; @calvetti2007introduction; @gelman2013bayesian]), where the parameters are treated as random variables, all prior information on the parameters is coded into a prior probability density function (pdf), and the solution to the inverse problem is the posterior pdf. The Bayesian approach can account for the structure and level of measurement noise as well as the effect of modelling errors. The posterior density then allows the calculation of various point estimates but also credibility intervals, which represent the uncertainty in the estimates. These can be calculated by sampling the posterior with Markov chain Monte Carlo (MCMC) methods. Based on such analysis we can decide if the accuracy of the estimates is high enough for our application, or if more information is needed (for example in the form of additional measurements).
For the reasons outlined above, the Bayesian approach has gained popularity in the recent years. To characterise porous media using the impedance tube, Bayesian methods have been considered both in the case where the solid part of the material (the frame) is modelled as rigid [@niskanen2017deterministic], and where it is modelled as elastic [@chazot2012acoustical]. The rigid frame case has also been considered in the free field ultrasonic regime [@roncen2018bayesian]. In geoacoustics, Bayesian methods have been used to characterise the sediment structure of seabed, using Bayesian model selection and parameter inference [@dettmer2009model; @dettmer2010trans; @dosso2012parallel; @bonomo2018comparison].
In this work, we consider a slab of ceramic-like poroelastic material and study the feasibility of fully characterising it using ultrasonic measurements. The frame of the considered material is relatively stiff, and a problem is that sound waves in air would not be able to deform it and we would not see any elastic behaviour. Using an impedance tube to measure such materials is therefore not practical, and instead we consider free field ultrasound measurements in water. To model the material and its acoustic response we use Biot’s theory of poroelasticity [@biot1956theoryLow; @biot1956theoryHigh; @biot1962mechanics]. The goal is then to estimate all the model parameters that are related to the poroelastic object. We also emphasize the importance of taking sources of uncertainty in the measurements into account in producing reliable estimates of the material parameters. For example, the measurement noise level is taken as an additional parameter to be estimated. Since this paper deals mainly with the data analysis and inversion, we simulate the (noisy) measurement data numerically. This allows us to compare the inverse solution to the known material parameters.
The degree of difficulty of sampling the posterior using MCMC is partly related to the dimensionality of the problem. In the case of a rigid frame material, the posterior pdf is relatively low-dimensional and easy to sample. This is because such materials can be modelled as fluids, and relatively few parameters are needed to describe them. Sampling the posterior pdf is a lot more challenging when we use the full poroelastic model, primarily due to the added model complexity and number of parameters. Unknowns related to the uncertainty in the measurements further increase the problem’s dimensionality. Therefore, in the present work, we use state of the art MCMC techniques, namely adaptive methods [@haario2001adaptive; @andrieu2008tutorial] and parallel tempering [@earl2005parallel] to explore the posterior efficiently. In addition, we develop a computationally fast and stable method to solve the poroelastic model. The solution is based on the global matrix method [@knopoff1964matrix; @lowe1995matrix]. To avoid an inverse crime [@kaipio2006statistical], we simulate the measurement data with the finite element method.
The paper is organised as follows. In Section \[sec:Biot\_numsolution\] we describe the Biot model and our approach to solving it. In Section \[sec:inverse\_problem\] we formulate the inverse problem and outline the used MCMC sampling methods. Next, in Section \[sec:numerical\_experiments\] we describe the numerical experiment and apply the inversion method. Results are shown in Section \[sec:results\], followed by a discussion in Section \[sec:discussion\] before a conclusion in Section \[sec:conclusion\].
The forward model {#sec:Biot_numsolution}
=================
![A schematic of the model geometry.[]{data-label="fig:geometry_2D"}](fig1){width="0.5\linewidth"}
We consider two fluid (water) domains $\Omega^{[0]}$ and $\Omega^{[2]}$, separated by a poroelastic slab $\Omega^{[1]}$ of thickness $L$ (see Fig. \[fig:geometry\_2D\]). The actual geometry extends to infinity in the $x_1$- and $x_3$-directions, and the boundaries surrounding the model are there only to truncate the computational domain for the finite element method calculations, described in Section \[sec:numerical\_experiments\]. The forward problem consists of predicting the fields reflected and transmitted by the poroelastic layer in response to an incident plane wave impinging the structure with an angle $\varphi_{\mathrm{inc}}$ and initially propagating in $\Omega^{[0]}$.
The model is assumed to have circular symmetry around the $x_2$ axis, so that the coordinate system can be rotated to make the $(x_1,x_2)$ plane coincide with the sagittal plane defined by the incident wave vector. Hence there is no propagation in the $x_3$-direction and the problem reduces to two spatial dimensions.
Biot model of porous media {#ssec:Biot_model}
--------------------------
Porous materials are made of a solid phase (called the frame) and of a fluid phase that is an interconnected network of pores inside the solid. Here we assume that all of the pore volume is occupied by water. When a flow of the fluid is able to cause the solid to deform, the material is called poro*elastic*. Poroelastic materials are most of the time modelled using the Biot theory.
The original Biot model [@biot1956theoryLow; @biot1956theoryHigh] expresses the poroelastic medium in terms of the displacements of its homogenised solid phase $\mathbf{u}$ and fluid phase $\mathbf{U}$. We adopt here an alternative formulation [@biot1962mechanics], which describes the medium with the solid displacement and a fluid/solid relative displacement $\mathbf{w} = \phi(\mathbf{U} - \mathbf{u})$, where $\phi$ denotes the material’s open porosity. The alternative formulation is chosen because in our case it simplifies writing the interface conditions, which now do not include porosity, and because it is valid for inhomogeneous materials. We consider the equations in the frequency domain, so an arbitrary time-dependent field $\bar{\chi}(\mathbf{x};t)$, $\mathbf{x} = (x_1,x_2)$, is related to its counterpart in the frequency domain with the Fourier transform $\chi(\mathbf{x};\omega) = \int_{-\infty}^{\infty}\bar{\chi}(\mathbf{x};t)e^{-\I \omega t} dt$.
In the alternative formulation Biot’s motion equations read
$$\begin{aligned}
\label{eq:Biot_motioneqn1}
& \omega^2\rho_f\mathbf{w} + \omega^2\rho\mathbf{u} = -\nabla\cdot\bm{\sigma},\\
& \omega^2\rho_f\mathbf{u} + \omega^2\tilde{\rho}_{\text{eq}}\mathbf{w} = \nabla\cdot p,\label{eq:Biot_motioneqn2}\end{aligned}$$
where $\rho_f$ is the density of the fluid, $\rho = (1 - \phi)\rho_s + \phi\rho_f$ is the bulk density of the medium (with $\rho_s$ density of the solid), $\tilde{\rho}_{\mathrm{eq}}$ is the equivalent density of the porous material, $\bm{\sigma}$ is the total stress tensor, $p$ is the pressure, $\omega = 2\pi f$ is the angular frequency, and $f$ is the frequency. For an isotropic porous material the stress-strain relations can be written as
$$\begin{aligned}
\bm{\sigma} & = 2N\bm{\epsilon} + (\lambda_c\nabla\cdot\mathbf{u} +\alpha_{\mathrm{B}} M\nabla\cdot\mathbf{w})\mathbf{I}, \label{eq:stress-strain1}\\
p & = -M(\nabla\cdot\mathbf{w} + \alpha_{\mathrm{B}}\nabla\cdot\mathbf{u}), \label{eq:stress-strain2}\end{aligned}$$
where $N$ is the shear modulus, $\bm{\epsilon} = \frac{1}{2}\left(\nabla\mathbf{u} + (\nabla\mathbf{u})^T\right)$ is the strain tensor, and $\mathbf{I}$ is the identity tensor. The parameter $\alpha_{\mathrm{B}}$ is the Biot-Willis coefficient [@biot1957elastic], and $M$ and $\lambda_c = \lambda + \alpha_{\mathrm{B}}^2M$ are elastic parameters, where $\lambda$ is the first Lamé’s coefficient of the elastic frame. These elastic parameters used by Biot can be stated as a function of $\phi, K_f, K_s$ (the bulk modulus of the elastic solid from which the frame is made of), and $K_b$ (the bulk modulus of the porous frame in a vacuum) with the relations
$$\begin{aligned}
\alpha_{\mathrm{B}} & = 1 - \frac{K_b}{K_s}, \\
M & = \left[\alpha_{\mathrm{B}} + \left(\frac{K_s}{K_f} - 1\right)\phi\right]^{-1}K_s, \\
\lambda_c & = \left[(1 + \phi)\alpha_{\mathrm{B}} + \left(\frac{K_b}{K_f} - 1\right)\phi\right]M - \frac{2}{3}N.\end{aligned}$$
The set $\phi, K_f, K_s$, and $K_b$ can be measured directly, and we therefore consider them as fundamental properties of the material.
Viscous losses in the medium are accounted for in the equivalent density by the model of dynamic tortuosity introduced by Johnson *et al.* [@johnson1987theory]. Using this model we can write the equivalent density as
$$\label{eq:JKD_rhoeq}
\tilde{\rho}_{\mathrm{eq}} = \frac{\rho_f}{\phi}\tilde{\alpha}(\omega),$$
where the dynamic tortuosity is
$$\tilde{\alpha}(\omega) = \alpha_\infty + \frac{\I\nu}{\omega}\frac{\phi}{k_0}\sqrt{1 - \frac{\I\omega}{\nu}\left(\frac{2\alpha_\infty k_0}{\phi\Lambda}\right)^2} \:\:,$$
where $\alpha_\infty$ is the geometric tortuosity, $k_0$ is the viscous static permeability, $\Lambda$ is the viscous characteristic length, and $\nu = \eta/\rho_f$ is the kinematic viscosity of the saturating fluid with $\eta$ the dynamic viscosity.
In addition to viscous losses introduced through the dynamic tortuosity model, we let the elastic parameters $K_b,K_s$, and $N$ be complex to allow for damping in the frame. The parameters are assumed to be constant with respect to frequency, which corresponds to a slightly inelastic Biot model [@turgut1991investigation]. This assumption is simpler than a fully poro-viscoelastic model, where the elasticity parameters are a function of frequency, but the constant parameter model has been shown to be slightly non-causal. It can be shown, however, that when attenuation in the frame (corresponding to the ratio of imaginary to real part) is small, the elastic parameters can be approximated as constants [@bourbie1987acoustics].
We seek to express the Biot motion equations and as a system of first order ordinary differential equations (ODEs) that can then be solved in several ways. A common way is to use the transfer matrix method (TMM) which, while straightforward to implement, suffers from known numerical instabilities [@lowe1995matrix] at high frequencies and certain parameter combinations. These issues make the TMM unsuitable for our use because the inversion algorithm solves the model over a wide range of parameters, and the numerical problems are often present. Instead, we use the global matrix method (GMM) [@knopoff1964matrix], which is stable even for high frequencies (as shown for example in [@lowe1995matrix; @chin1984matrix; @schmidt1986efficient]), and has also proved stable in our tests. In the GMM, each backward and forward propagating wave inside the poroelastic material is explicitly written out as a function of their amplitudes, and their origin is individually chosen to be on the boundary from which they originate.
State vector formalism
----------------------
We perform a Fourier transform of the field along $x_1$, which due to the plane wave nature of the excitation can be written as $\chi(\mathbf{x};\omega) = \hat{\chi}(k_1,x_2;\omega)e^{\I k_1x_1}$, where $k_1 = -k^{[0]}\sin (\varphi_{\mathrm{inc}})$, and $k^{[0]} = \omega/c_f$. For the remainder of this paper we omit the $\omega$ and $k_1$ dependence in the expressions for the fields and write $\hat{\chi}(k_1,x_2;\omega) \equiv \hat{\chi}(x_2)$.
To write out the motion equations as an ODE system we introduce a state vector $\hat{\mathbf{s}}(x_2)$. Components of the state vector can be selected arbitrarily as long as they completely describe the state of the system at $x_2$. A convenient choice is to use the normal components of the stress tensor, $\hat{\sigma}_{12}^{[1]}$ and $\hat{\sigma}_{22}^{[1]}$, the pressure $\hat{p}^{[1]}$, the solid displacements $\hat{u}_1^{[1]}$ and $\hat{u}_2^{[1]}$, and the normal component of the total velocity $\hat{v}_2^{[1]} = -\I\omega(\hat{w}_2^{[1]} + \hat{u}_2^{[1]})$. This choice simplifies the coupling conditions between interfaces in the current formulation of the Biot equations. We therefore have $\hat{\mathbf{s}}(x_2) = [\hat{\sigma}_{12}^{[1]},\: \hat{\sigma}_{22}^{[1]},\: \hat{p}^{[1]},\: \hat{u}_1^{[1]},\: \hat{u}_2^{[1]},\: \hat{v}_2^{[1]}]$, where the $x_2$-dependence of the parameters has been dropped for clarity.
After performing the spatial Fourier transform, the motion equations - and the stress-strain relations - can be cast in the form
$$\label{eq:ODEsystem}
\frac{\mathrm{d} \hat{\mathbf{s}}(x_2)}{\mathrm{d} x_2} + \mathbf{A}\hat{\mathbf{s}}(x_2) = 0.$$
The matrix $\mathbf{A}$ is called the state matrix, and it is a function of the material parameters, components of the incident wave, and frequency. The state matrix for this specific state vector was derived by Gautier *et al.* [@gautier2011propagation] and is provided in \[app:statematrix\].
Four boundary conditions are needed to couple waves that propagate between a fluid and a poroelastic material [@allard2009propagation]. Let $\bar{x}$ be a location of the interface separating the two mediums and let the superscript $0$ represent here either of the fluid domains $\Omega^{[0]}$ or $\Omega^{[2]}$. The coupling conditions are continuity of pressure, $\hat{p}^{[1]}(\bar{x}) = \hat{p}^{[0]}(\bar{x})$, continuity of the normal component of velocity, $\hat{v}_2^{[1]}(\bar{x}) = \hat{v}_2^{[0]}(\bar{x}) = -\frac{\I}{\omega\rho_f}\frac{\mathrm{d} \hat{p}^{[0]}(x_2)}{\mathrm{d} x_2}\Bigr|_{\substack{x_2 = \bar{x}}}$, and continuity of the normal components of the stress tensor, $\hat{\sigma}_{12}^{[1]}(\bar{x}) = 0$ and $\hat{\sigma}_{22}^{[1]}(\bar{x}) = -\hat{p}^{[0]}(\bar{x})$.
In the fluid domains we can represent the incident, reflected, and transmitted waves as
$$\begin{aligned}
\hat{p}^{[0]}(x_2) & = e^{-\I k_2x_2} + Re^{\I k_2x_2}, \\
\hat{p}^{[2]}(x_2) & = Te^{-\I k_2(x_2 - L)},
\end{aligned}$$
where $k_2 = k^{[0]}\cos (\varphi_{\mathrm{inc}})$. Substituting these into the coupling relations we get on the first interface
$$\begin{aligned}
\hat{p}^{[1]}(0) & = 1 + R, \\
\hat{v}_2^{[1]}(0) & = -\frac{1}{Z_f}\cos(\varphi_{\mathrm{inc}}) + \frac{R}{Z_f}\cos(\varphi_{\mathrm{inc}}), \\
\hat{\sigma}_{12}^{[1]}(0) & = 0, \\
\hat{\sigma}_{22}^{[1]}(0) & = -1 - R,
\end{aligned}$$
where $Z_f = \rho_fc_f$ is the characteristic impedance of the fluid. A similar system is written on the second interface.
Global matrix solution of the Biot equations
--------------------------------------------
In an isotropic poroelastic layer, there are three forward propagating waves and three backward propagating waves. The idea in the GMM is to express the physical wave fields as a function of the wave amplitudes. Then we can choose to represent the origin of the wave at the interface it originates from and hence avoid the numerical issues that occur in the TMM. After applying the coupling conditions between layers, we can solve the transmission and reflection coefficients.
The solution proceeds as follows [@dazel2013stable]. We first calculate the eigendecomposition of $\mathbf{A}$:
$$\label{eq:alpha_eigDecomposition}
\mathbf{A} = \mathbf{\Phi}\mathbf{\Gamma}\mathbf{\Phi}^{-1},$$
where the column $\mathbf{\Phi}_j$ of the $(6\times6)$ eigenvector matrix $\mathbf{\Phi}$ corresponds to the polarisation of the $j$-th wave, and the eigenvalue $\mathbf{\Gamma}_{jj}$ on the diagonal corresponds to $\I k^{[1]}_j$, $j = 1,\dots,6$ [@dazel2013stable; @de2009materials]. The eigenvalues are ordered in pairs of a forward and a backward propagating wave. A classical solution to is $\hat{\mathbf{s}}(x_2) = \exp\{-(x_2 - \mathbf{x}_*)\mathbf{A}\}\hat{\mathbf{s}}(\mathbf{x}_*)$, which, using decomposition , can be written as
$$\hat{\mathbf{s}}(x_2) = \mathbf{\Phi}\mathbf{\Lambda}(x_2)\mathbf{\Phi}^{-1}\hat{\mathbf{s}}(\mathbf{x}_*),$$
where $\mathbf{\Lambda}(x_2) = \exp\{-(x_2 - \mathbf{x}_*)\mathbf{\Gamma}\}$ is the propagation factor along $x_2$. The reference location $\mathbf{x}_*$ is set at the first interface $(\mathbf{x}_* = 0)$ in the TMM, whereas the GMM allows us to change the basis of the waves so that their origin is on the interface they originate from. Let us define the new origin as $(j=1,\dots,6)$
$$x_{*,j} = \begin{cases}
0, \quad\: j \:\: \mathrm{odd} \\
L, \quad j \:\: \mathrm{even}.
\end{cases}$$
The odd index refers to a forward, and even to a backward propagating wave. Next, let $\mathbf{q} = \mathbf{\Phi}^{-1}\hat{\mathbf{s}}(\mathbf{x}_*)$, so we can express the state vector as
$$\label{eq:eigenvalue_amplitudes}
\begin{split}
\hat{\mathbf{s}}(x_2) & = \mathbf{\Phi}\mathbf{\Lambda}(x_2)\mathbf{q} \\
& =: \mathbf{M}(x_2)\mathbf{q}.
\end{split}$$
Note that the $(6\times1)$ vector $\mathbf{q}$ does not depend on $x_2$. This allows us to combine two systems of the form , with a common $\mathbf{q}$ but the state vector evaluated at different interfaces, to create the global matrix.
Before stacking the equations together, we reduce redundant degrees of freedom to speed up the calculations. Each row in the $(6\times6)$ matrix $\mathbf{M}(x_2)$ is related to the corresponding state vector variable. Therefore, to relate the four coupling conditions on both sides of the material, we do not need the fourth and fifth rows corresponding to $u_1$ and $u_2$. Let us denote the $(4\times6)$ matrix with these rows removed by $\mathbf{M}^{-}(x_2)$, and a reduced state vector (without $u_1$ and $u_2$) by $\hat{\mathbf{s}}^{-}(x_2)$.
Now we can write out the state vectors and corresponding matrices on both interfaces, and concatenate the system as
$$\begin{bmatrix}
\hat{\mathbf{s}}^{-}(0) \\
\hat{\mathbf{s}}^{-}(L) \\
\end{bmatrix}
=
\begin{bmatrix}
\mathbf{M}^{-}(0) \\
\mathbf{M}^{-}(L)
\end{bmatrix}
\mathbf{q}.$$
Substituting in the coupling conditions we get the following linear system with 8 equations and 8 unknowns
$$\label{eq:RTsystem}
\begin{bmatrix}
0\\
-1\\
1\\
-\frac{\cos(\varphi_{\mathrm{inc}})}{Z_f}\\
0\\
0\\
0\\
0
\end{bmatrix}
=
\left[
\begin{array}{c;{2pt/3pt}c;{2pt/3pt}c}
0 & & 0 \\
1 & \mathbf{M}^{-}(0) & 0 \\
-1 & & 0 \\
-\frac{\cos(\varphi_{\mathrm{inc}})}{Z_f} & & 0 \\ \hdashline[2pt/3pt]
0 & & 0 \\
0 & \mathbf{M}^{-}(L) & 1 \\
0 & & -1 \\
0 & & \frac{\cos(\varphi_{\mathrm{inc}})}{Z_f}
\end{array}
\right]
\begin{bmatrix}
R \\
\mathbf{q} \\
T
\end{bmatrix},$$
from which the reflection and transmission coefficients are easy to solve.
The system must be solved for each frequency at a time, and one loop over the frequencies constitutes as one ”forward solution”. To loop over the frequency range faster, we can stack the 8x8 matrices of each frequency on the diagonal of a bigger matrix. This block diagonal matrix is typically very sparse and fast algorithms exist to solve the system for all the frequencies at once. Computing one forward solution with 100 frequencies takes between 5 and 10 milliseconds using MATLAB^^ R2018a on a laptop with an Intel i7-4710MQ processor.
The inverse problem {#sec:inverse_problem}
===================
In the Bayesian framework all unknown parameters are modelled as random variables, and the inverse problem is seen as a problem of statistical inference [@kaipio2006statistical; @calvetti2007introduction]. The solution of a Bayesian inverse problem is the posterior pdf, which is constructed based on measured data, a model relating the measurements to the unknowns, and any prior information that might be available. The posterior pdf represents all the information we have on the unknowns.
Let us denote the observable quantities by $\mathbf{y}$, the unknowns by $\bm{\theta}$, and measurement error (noise) by $\mathbf{e}$. To keep the notations simple, we will denote random variables and their fixed realisations with the same symbol. Our data vector $\mathbf{y}\in\mathbb{C}^{2n_\omega}$ consists of the reflection and transmission coefficients solved over $n_\omega$ frequencies. For the exact definition of $\mathbf{y}$, see Sec. \[ssec:FEM\_data\]. Further, let us denote the number of unknowns by $n_\theta$ so that $\bm{\theta}\in\mathbb{R}^{n_\theta}$. Assuming additive noise, we can model the measurement as
$$\label{eq:observation model}
\mathbf{y} = h(\bm{\theta}) + \mathbf{e} \:\:,$$
where $h: \mathbb{R}^{n_\theta} \rightarrow \mathbb{C}^{2n_\omega}$ is the forward model (the GMM solution over multiple frequencies) that maps the unknown parameters to measurable data, and $\mathbf{e}\in\mathbb{C}^{2n_\omega}$ denotes measurement noise.
According to Bayes’ formula, the posterior density $\pi(\bm{\theta}|\mathbf{y})$ is proportional (up to a normalising constant) to the product of a likelihood $\pi(\mathbf{y}|\bm{\theta})$ and prior $\pi(\bm{\theta})$ pdfs
$$\label{eq:Bayes}
\pi(\bm{\theta}|\mathbf{y}) = \frac{\pi(\mathbf{y}|\bm{\theta})\pi(\bm{\theta})}{\pi(\mathbf{y})} \propto \pi(\mathbf{y}|\bm{\theta})\pi(\bm{\theta}) \:\:.$$
The prior density is constructed to give a high probability to parameter values we expect to find, and a low probability for those we think are unlikely. Such information can be based on for example other experiments or an expert’s beliefs. We can also use the prior to rule out values inconsistent with physical reality, by setting the probability of such parameter values to zero.
When the measurement noise and the unknowns are mutually independent, the likelihood density can be written as $\pi(\mathbf{y}|\bm{\theta}) = \pi_e(\mathbf{y} - h(\bm{\theta}))$, where $\pi_e$ denotes the pdf of $\mathbf{e}$. We assume that the measurement noise is normally distributed with an expected value of zero and a covariance $\mathbf{\Gamma}_e$. Then, the posterior density becomes
$$\label{eq:posterior}
\pi(\bm{\theta}|\mathbf{y}) \propto \exp\left\{-(\mathbf{y} - h(\bm{\theta}))^T\mathbf{\Gamma}_e^{-1}(\mathbf{y} - h(\bm{\theta}))\right\}\pi(\bm{\theta}).$$
We further assume that the covariance $\mathbf{\Gamma_e}$ is diagonal, i.e. the measurements in $\mathbf{y}$ are statistically independent. If we do not know the noise covariance $\mathbf{\Gamma}_e$, it is possible in the Bayesian framework to consider $\mathbf{\Gamma}_e$ as an additional unknown (or unknowns) to be estimated. In such a case the posterior takes the form $\pi(\bm{\theta},\mathbf{\Gamma}_e|\mathbf{y})\propto\pi(\mathbf{y}|\bm{\theta},\mathbf{\Gamma}_e)\pi(\bm{\theta})\pi(\mathbf{\Gamma}_e)$, where $\pi(\mathbf{\Gamma}_e)$ represents a prior for the noise covariance. Now, the normalising factor of the noise distribution is no longer constant and we have to include it in the expression of the likelihood. By denoting $\mathbf{L}_e^T\mathbf{L}_e^{} = \mathbf{\Gamma}_e^{-1}$, the posterior is written as
$$\label{eq:posterior_withnoise}
\pi(\bm{\theta},\mathbf{\Gamma}_e|\mathbf{y}) \propto \exp\left\{-\left\Vert \mathbf{L}_e(\mathbf{y} - h(\bm{\theta}))\right\Vert^2 -\log(\det(\mathbf{\Gamma}_e))\right\}\cdot\pi(\bm{\theta})\pi(\mathbf{\Gamma}_e).$$
Calculating the parameter estimates
-----------------------------------
In order to explore the posterior density to calculate parameter and uncertainty estimates, we need a way to draw samples from it. For this purpose we employ MCMC methods [@brooks2011handbook], which are a widely used family of algorithms that generate correlated samples from the posterior. MCMC methods are especially suitable for computing the conditional mean (CM) estimate
$$\label{eq:CMestimate}
\hat{\bm{\theta}}_{\mathrm{CM}} = \int_{\mathbb{R}^{n_\theta}}\bm{\theta}\pi(\bm{\theta}|\mathbf{y})d\bm{\theta} \approx \frac{1}{n}\sum_{i=1}^{n}\bm{\theta}^{(i)},$$
where $\{\bm{\theta}^{(i)}\}$ is an ergodic Markov chain produced by the MCMC sampler, and $n$ is the sample size. The approximation in becomes exact in the limit $n\rightarrow\infty$. From the posterior samples we can also compute uncertainty estimates, such as credible intervals. A 95 % credible interval $I_k(95) = [a_I,b_I]\subset\mathbb{R}$ for the parameter $\theta_k$ is defined as
$$\int_{a_I}^{b_I}\pi(\theta_k|\mathbf{y})d\theta_k = 0.95,$$
where $\pi(\theta_k|y) = \int_{\mathbb{R}^{n_\theta-1}}\pi(\theta_1,\dots,\theta_{n_\theta}|\mathbf{y})d\theta_1\cdots\theta_{k-1}\theta_{k+1}\cdots\theta_{n_\theta}$ is the marginal density of the $k$-th component $\theta_k$ of $\bm{\theta}$. In this paper we consider the narrowest interval $I_k(95)$. The Bayesian credible intervals can be directly interpreted as probability statements about the likely values of the parameters given the data [@gelman2013bayesian].
A downside to MCMC is that it is computationally very demanding since generating posterior samples relies on repeatedly solving the forward model. In high-dimensional cases, and with computationally demanding forward models, running MCMC quickly becomes infeasible and often in such a case the only option is to solve the maximum a posteriori (MAP) estimate, which is an optimization problem. To make sampling and the calculation of the CM estimate viable in our problem, we make use of adaptive MCMC methods and parallel tempering. In general, the reason for using adaptive methods and parallel tempering is to improve the sampling efficiency by reducing autocorrelation times and facilitating easier jumping between posterior modes. We describe the algorithm next, and then later use it for the parameter estimation problem.
An adaptive parallel tempered MCMC algorithm
--------------------------------------------
The Metropolis-Hastings (M-H) algorithm [@metropolis1953equation; @hastings1970monte] is the standard tool for drawing samples from a target probability distribution $\pi(\cdot)$. In the random walk Metropolis algorithm, given the current location $\bm{\theta}^{(i)}$ of the chain, a candidate $\bm{\theta}^*$ for the next step is drawn from a proposal probability distribution that is symmetric and centered around $\bm{\theta}^{(i)}$. Then the proposed location is either accepted or rejected with an acceptance probability $P(\bm{\theta}^{(i)},\bm{\theta}^*) = \min\{1,\pi(\bm{\theta}^*)/\pi(\bm{\theta}^{(i)})\}$. For reasons of numerical stability, it is preferable to work with the logarithm of the target distribution so that the acceptance ratio becomes $\log\pi(\bm{\theta}^*)-\log\pi(\bm{\theta}^{(i)})$.
A usual choice for the proposal density is a multivariate Gaussian. The candidate for the next move is thus drawn from $\bm{\theta}^* \sim \mathcal{N}(\bm{\theta}^{(i)},\mathbf{\Sigma})$, where $\mathbf{\Sigma}$ is the proposal covariance. In an $n_\theta$ dimensional problem $\mathbf{\Sigma}$ has up to $n_\theta(n_\theta+1)/2$ tunable variables. Correct tuning of the proposal covariance is crucial for the efficiency of the M-H algorithm, but tuning the proposal for a low-dimensional problem by hand is difficult and for large $n_\theta$ it is practically impossible.
An alternative to hand-tuning the covariance is to learn it on the go, using adaptive algorithms such as the Adaptive Metropolis (AM) proposed by Haario *et al.* [@haario1999adaptive; @haario2001adaptive]. The idea of AM is that, during sampling, the covariance $\mathbf{\Sigma}$ is continuously updated based on the MCMC samples accumulated so far. In this way the proposal size and orientation is automatically adapted to the shape of the target density, which is beneficial especially when some parameters are correlated. At iteration $i$, we denote the covariance by $\mathbf{\Sigma}_i$, and the proposal density is of the form $\mathcal{N}(\bm{\theta}^{(i)},s_d\mathbf{\Sigma}_i)$, where $\mathbf{\Sigma}_i = \mathrm{cov}(\bm{\theta}^{(1)},...,\bm{\theta}^{(i)})$ and $s_d$ is a scaling parameter. As a rule of thumb one can set $s_d = 2.4^2/n_\theta$ [@gelman1996efficient]. With non-Gaussian target densities and at the beginning of the MCMC run when the estimate of the posterior covariance is uncertain, the scaling parameter $s_d$ can be too large which results in a low acceptance ratio and efficiency. To improve this, Andrieu and Thoms [@andrieu2008tutorial] proposed to also adapt $s_d$ to achieve the desired acceptance rate. We include this adaptation with a target acceptance ratio of 0.2.
In general, the M-H algorithm works well when $\pi(\cdot)$ is close to a unimodal distribution, but when the target distribution consists of multiple separated modes it tends to get stuck in one of them and fail to sample the other modes. We observed this behaviour in the current parameter estimation problem, where the random walk chains adapted to a local maximum and in practice never found the main mode (whose location is known in simulated problems) unless the chain was started in it.
Tempering, i.e. considering a target density $\pi(\cdot)^{1/T}$, where $T\geq 1$ is called the temperature, flattens and widens the modes thus enabling a Markov chain to move around the density more freely. In parallel tempering (PT) [@swendsen1986replica; @geyer1991markov; @earl2005parallel] we use $m$ Markov chains to sample in parallel $m$ tempered densities with temperatures $1 = T_1 < T_2 < \cdots < T_m$, and couple the chains together to exchange information on their location in parameter space. In this way, the (possibly very isolated) high-probability locations found by the hot chains eventually propagate all the way to the chain at $T_1$, and as a result the cold chain is able to effectively explore the whole parameter space. Note that the additional chains are introduced only to improve sampling efficiency of the chain at $T_1$, and at the end of the run only the chain at $T_1$ is retained.
In this work we consider a particular form of tempering where the likelihood is tempered but the prior is not:
$$\pi_{T}(\bm{\theta}|\mathbf{y}) \propto \pi(\mathbf{y}|\bm{\theta})^{\frac{1}{T}}\pi(\bm{\theta}),$$
and set $T_m = \infty$. At the highest temperature the target density corresponds to the prior only, which by our construction is unimodal and thus easy to sample. Restricting the hottest chain to parameter values supported by the prior is a natural limit for the volume of parameter space we want to search in, because values outside the prior support we have, by definition, deemed impossible.
Coupling the chains together is achieved by proposing at pre-defined intervals to swap the locations of pairs of chains in the parameter space. The swap is accepted with a probability that preserves the joint distribution of the chains. This probability can be derived directly from the Metropolis update criterion, and in the case where we only temper the likelihood, it can be stated as [@vousden2015dynamic]
$$P_{i,j} = \min\left\{1, \left(\frac{\pi(\mathbf{y}|\bm{\theta}_i)}{\pi(\mathbf{y}|\bm{\theta}_j)}\right)^{1/T_j - 1/T_i}\right\},$$
where $\bm{\theta}_i$ is the location in the parameter space of the $i$th chain and $T_i$ is the chain’s temperature. Usually the pairs that are adjacent in temperature are selected, because the swap acceptance probability decays quickly with a growing temperature difference.
Simulating $m$ chains obviously increases the computational demand $m$-fold. However, this can be offset by a more than $m$-fold increase in sampling efficiency in strongly nonlinear and multimodal problems. The efficiency of the PT sampler is greatly affected by the between chain acceptance probabilities, in a way similar to the within-chain acceptance probability issues of the M-H algorithm. The between chain acceptance ratio can be tuned by adjusting the temperatures. The closer any two chains are in temperature, the higher the probability of swapping them is, and vice versa. The optimal acceptance rate for temperature swaps turns out to be the same as the optimal M-H acceptance rate, around 0.23 [@kone2005selection]. To achieve this we start with $m$ geometrically spread out temperatures, and fix $T_1 = 1$ and $T_m = \infty$. Temperatures $T_2,...,T_{m-1}$ are continuously adapted as the simulation goes on, following the procedure in Vousden *et al.* [@vousden2015dynamic]. This method changes the temperatures in such a way that an equal acceptance rate is achieved between each temperature pair.
Finally, we note that the number of temperatures $m$ could be automatically adjusted during an MCMC run, but to simplify the algorithm we have found it is enough to set $m$ manually before the start of the run. If all acceptance rates between chains are too low ($< 0.15$), we restart the run with more temperatures. In the case that the overall temperature swap rates are too high ($> 0.4$) we reduce the number of temperatures to avoid redundant sampling.
The following summarises the MCMC algorithm we have developed for use in this problem. For clarity the considerations regarding burn-in and frequency of the adaptations have been omitted. Swapping of the states is proposed at every iteration.
For $T=1,\dots,m$, initialise $\bm{\theta}^{(i,T)}$, $\mathbf{\Sigma}_i^{(T)}$, and $s_d^{(T)}$. Set $i=1$. - Generate a candidate $\bm{\theta}^{(*,T)}$ and accept it with probability based on the Metropolis ratio. - Adapt $s_d^{(T)}$ based on the acceptance probability, and $\mathbf{\Sigma}_i^{(T)}$ based on the new sample. - Swap the states between chains in adjacent temperatures with probability based on the Metropolis ratio. - Adapt the temperatures $T_2,\dots,T_{m-1}$ based on the swap probabilities. - Check convergence - Set $i = i + 1$
Convergence and Monte Carlo error
---------------------------------
Diagnosing convergence of a Markov chain is a subtle issue. To avoid terminating the MCMC run prematurely or running it for an unnecessarily long amount of time, we implement an objective stopping criterion. The chosen criterion is based on comparing Monte Carlo error in the CM estimate at iteration $n$, to the overall variability of the parameters. However, we must keep in mind that no criterion can guarantee convergence and it is good practise to go through the usual checks (autocorrelation times, visual inspection of stationarity of the chains) every time as well.
Based on the central limit theorem (CLT) [@brooks2011handbook], we can estimate the Monte Carlo standard error of $\hat{\bm{\theta}}_{\mathrm{CM}}$ with $\hat{\bm{\sigma}}_{\theta_{CM}}/\sqrt{n}$. The estimate $\hat{\bm{\sigma}}_{\theta_{CM}}^2$ of the asymptotic variance in the CLT can be calculated via the consistent batch means method (see Flegal *et al.* [@flegal2008markov]). Half width of the $(1 - \alpha)\cdot 100$ % confidence interval for the CM estimate is given by
$$\mathrm{\bf{CI}}_{1-\alpha} = t_{a_n-1}\left(1-\frac{\alpha}{2}\right)\frac{\hat{\bm{\sigma}}_{\theta_{CM}}}{\sqrt{n}} \:,$$
where $t_{a_n-1}$ denotes the suitable quantile from Student’s $t$-distribution with $a_n-1$ degrees of freedom, and $a_n$ is the number of batches used for the estimate $\hat{\bm{\sigma}}_{\theta_{CM}}^2$.
Our stopping criteria is then the following. After a burn-in period, we first run the chain for long enough so that $\hat{\bm{\sigma}}_{\theta_{CM}}$ can be reliably estimated, and then at every $k$ iterations ($k = 1000$ for example), we compute $\mathrm{CI}_{95}$ for the CM estimate and the standard deviation $\bm{\sigma}_{\theta}$ of the samples accumulated so far. When the uncertainty in the CM estimate gets below 10 % of the parameter standard deviation, $\mathrm{\bf{CI}}_{95} < \bm{\sigma}_{\theta}/10$, for each parameter, the simulation is stopped.
Numerical experiments {#sec:numerical_experiments}
=====================
To investigate the performance of the inversion algorithm we conduct numerical experiments with data from a finite element simulation. We simulate a material whose properties resemble those of commercially available filter materials made by sintering glass powder. Properties of such a material can be found for example in Jocker *et al.* [@jocker2009ultrasonic] (material S3) and they are shown in Table \[tab:inversion\_results\]. We set the imaginary parts of the elastic coefficients to two percent of the real parts. Properties of the fluid saturating the porous frame are: density $\rho_f = 1000$ kg$\cdot$m$^{-3}$, dynamic viscosity $\eta = 1.14\cdot 10^{-3}$ Pa$\cdot$s, speed of sound $c_f = 1500$ m$\cdot$s$^{-1}$, and bulk modulus $K_f = 2.19\cdot 10^9$ Pa. For the considered material the viscous-inertial regime transition frequency is $\omega_c = \eta\phi/(\rho_f\alpha_\infty k_0) \approx 45 $ kHz. To be well within the inertial regime we choose to work in the range 100 - 400 kHz.
The finite element model and data {#ssec:FEM_data}
---------------------------------
The finite element model is constructed using the Pressure Acoustics and Poroelastic Waves interfaces in COMSOL Multiphysics^^ software v. 5.3. The model configuration is similar to Fig. \[fig:geometry\_2D\]. Boundary $\Gamma_1$ is given a plane wave radiation condition, and the opposing boundary $\Gamma_2$ an absorbing impedance condition. The impedance boundary is known to work perfectly only at normal incidence, and reflect some of the energy back when hit by a wave propagating at an oblique angle. We use the impedance boundary condition as a way to introduce model discrepancy that is not accounted for in the inversion. To approximate plane wave propagation, we take a 15 mm wide (in the $x_1$-direction) slice of the material as a ”unit cell”, and extend it to infinity by giving the upper and lower sides of the geometry a periodic Floquet condition. The interior boundaries from left to right have the acoustic-poroelastic and poroelastic-acoustic condition, respectively. The mesh consists of 2,760 quadrilateral elements. All elements have the same size 0.5 mm, which corresponds to 6 quadratic elements per wavelength in water at 500 kHz. We consider the frequency range 100 - 400 kHz with a 5 kHz resolution, so in total there are 61 frequencies in the data. We simulate measurements where the incident wave is at an angle between $0^{\circ}$ and $45^{\circ}$.
The parametrisation of the Biot model in COMSOL is otherwise similar to our implementation, except for the way the equivalent density is written. We choose to use different viscous loss models to further avoid using a too similar model in the inversion. COMSOL implements the original loss function of Biot [@biot1956theoryHigh], whereas we use the nowadays more common Johnson *et al.* model [@johnson1987theory]. Instead of the viscous characteristic length $\Lambda$, the Biot loss function is defined with a characteristic pore size parameter $a$. By making use of the high-frequency limit of the Biot loss function (equation (3.23) in [@biot1956theoryHigh]) we can relate $\Lambda$ to $a$ approximately as
$$\label{eq:atoLambda}
\Lambda \approx \dfrac{8k_0\alpha_\infty}{a\phi}.$$
In this paper, we will regard $\Lambda$ from as the true value of the viscous length for examining the results, although like mentioned, this is not exact.
To construct a measurement in the frequency domain, we simulate acoustic fields in two configurations for every incidence angle. The first configuration includes the poroelastic plate, but in the second one the plate is removed and substituted with water to record a reference field. The transmission coefficient is then obtained as
$$\label{eq:transmission}
T^{\mathrm{exp}}(\omega) = \dfrac{p_{T}(\omega)}{p_{\mathrm{ref},T}(\omega)},$$
where $p_{T}$ is the complex pressure recorded at a point on the transmission side of the plate (dot 3 in Fig. \[fig:geometry\_2D\]), and $p_{\mathrm{ref},T}(\omega)$ is the pressure in the same location without the plate. To account for the phase difference introduced when the plate is removed and substituted with water, we should multiply equation with a correction factor $\exp\{\I k^{[0]}L\cos\varphi_{\mathrm{inc}}\}$ [@jocker2007minimization]. However, the factor includes parameters $L$ and $\varphi_{\mathrm{inc}}$ that we want to estimate and that are unknown prior to carrying out the inversion. Therefore during each MCMC iteration, we instead multiply the modelled transmission coefficient by an inverse correction factor $\exp\{-\I k^{[0]}L\cos\varphi_{\mathrm{inc}}\}$ and use for $L$ and $\varphi_{\mathrm{inc}}$ the current value of the MCMC chain.
In the configuration with the plate, a microphone on the reflection side records both the incoming and the reflected waves. Therefore, to calculate the reflection coefficient we need to record two reference locations:
$$R^{\mathrm{exp}}(\omega) = \dfrac{p_{R}(\omega) - p_{\mathrm{inc}}(\omega)}{p_{\mathrm{ref},R}(\omega)},$$
where $p_{R}(\omega)$ is the pressure recorded at a point on the reflection side (dot 1 in Fig. \[fig:geometry\_2D\]), $p_{\mathrm{inc}}(\omega)$ is the pressure in the same location in the reference configuration, and $p_{\mathrm{ref},R}(\omega)$ is the pressure recorded at a point mirrored on the other side of the reflection surface (dot 2 in Fig. \[fig:geometry\_2D\]).
For each incidence angle $\varphi_{\mathrm{inc}}$, the simulated reflection and transmission coefficients are solved over $n_\omega$ frequencies, and combined to form a single full measurement $\mathbf{R}^{\mathrm{exp}}(\varphi_{\mathrm{inc}}) = [R^{\mathrm{exp}}(\omega_1;\varphi_{\mathrm{inc}}),\dots,R^{\mathrm{exp}}(\omega_{n_\omega};\varphi_{\mathrm{inc}})]$, and $\mathbf{T}^{\mathrm{exp}}(\varphi_{\mathrm{inc}}) = [T^{\mathrm{exp}}(\omega_1;\varphi_{\mathrm{inc}}),\dots,T^{\mathrm{exp}}(\omega_{n_\omega};\varphi_{\mathrm{inc}})]$. The measurement vector then reads
$$\mathbf{y}(\varphi_{\mathrm{inc}}) = [\mathbf{R}^{\mathrm{exp}}(\varphi_{\mathrm{inc}}), \mathbf{T}^{\mathrm{exp}}(\varphi_{\mathrm{inc}})]^{T}.$$
Since we mostly deal with inversion from one angle at a time, the $\varphi_{\mathrm{inc}}$-dependence is dropped to improve readability.
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -- -- -- -- -- -- -- -- -- -- --
**Parameter & $\Lambda$ & $\alpha_\infty$ & $\log_{10}k_0$ & $\phi$ & Re $K_b$, $N$ & Im $K_b$, $N$ & Re $K_s$ & Im $K_s$ & $\rho_s$ & $L$ & $\varphi_{\mathrm{inc}}$ & $\sigma_{e_R}$, $\sigma_{e_T}$\
unit & & & & & & & & & -\
**Mean & 30 & 2 & $-12$ & 0.4 & 15 & 0 & 30 & 0 & 2500 & 16 & $\varphi_{\mathrm{inc}}^{\mathrm{True}}$ & 0.05\
**Std & 20 & 1 & 2 & 0.1 & 10 & 0.5 & 20 & 1 & 1000 & 0.2 & 2 & 0.1\
******
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -- -- -- -- -- -- -- -- -- -- --
The noise model {#ssec:noisemodel}
---------------
We add Gaussian noise with zero mean to the simulated reflection and transmission coefficients. Let us denote the noise variance of the reflection and transmission measurements by $\sigma_{e_R}^2$ and $\sigma_{e_T}^2$, respectively. Level of the additive noise is set to 5 % of the peak amplitude of the simulated data. This corresponds to setting the standard deviation $\sigma_{e_R} = 0.05\max\{|\mathbf{R}^{\mathrm{exp}}|\}$. To achieve the correct noise level for a complex-valued measurement, noise with standard deviation $\sigma_{e_R}/\sqrt{2}$ is added to both the real and imaginary parts of $\mathbf{R}^{\mathrm{exp}}$ (and similarly for $\mathbf{T}^{\mathrm{exp}}$).
The noise level stays constant over the frequency range, which means that the noise covariance can be written as
$$\mathbf{\Gamma}_e = \begin{bmatrix}
\sigma_{e_R}^2 \mathbf{I} & 0 \\
0 & \sigma_{e_T}^2 \mathbf{I} \\
\end{bmatrix},$$
where $\mathbf{I}$ denotes the $n_\omega\times n_\omega$ identity matrix. The noise level may be estimated from repeated measurements, but sometimes conducting many measurements is not possible or the measurement accuracy is not good enough. In such a case we often have to make a conservative guess about the noise level, because underestimating the noise variance might mean fitting the model to noise and so the spread of the estimates could be misleading. The downside is that an overestimated noise level means we are not using all the information the measurements contain. Therefore we include the noise variances as unknown parameters and estimate them at the same time as the other unknowns.
Now the noise covariance matrix consists of two variables, $\sigma_{e_R}^2$ and $\sigma_{e_T}^2$, and the logarithm of the likelihood density in can be written as
$$\label{eq:log_likelihood}
\log\pi(\mathbf{y}|\bm{\theta},\sigma_{e_R},\sigma_{e_T}) \propto -\left\Vert \mathbf{L}_e(\mathbf{y} - h(\bm{\theta}))\right\Vert^2 -2n_\omega(\log\sigma_{e_R} + \log\sigma_{e_T}).$$
The parameter vector and prior {#ssec:prior}
------------------------------
We assume the properties of the saturating fluid to be known, and let the real and imaginary parts of the elastic moduli be independent of each other. As was mentioned in Sec. \[ssec:Biot\_model\], using a model where the real and imaginary parts of the elastic moduli are constant with frequency and inverted independently of each other, leads to slightly non-causal solutions. Keeping the ratio of imaginary to real part small limits the error coming from the non-causal model.
Including the material thickness, the model now has 12 unknown parameters. In addition, a measurement of the angle of the incident wave field has some uncertainty as well, so we estimate it with the material parameters. Hence we have $\bm{\theta} = [\Lambda,\alpha_\infty,k_0,\phi,\mathrm{Re}\:K_b,\mathrm{Im}\:K_b,\mathrm{Re}\:K_s,\mathrm{Im}\:K_s,\mathrm{Re}\:N,\mathrm{Im}\:N,\rho_s,L,\varphi_{\mathrm{inc}}]$. Adding the noise level parameters, we have 15 unknowns to estimate in total. Let us denote the parameter vector augmented with the noise level parameters as $\tilde{\bm{\theta}} = [\bm{\theta},\sigma_{e_R},\sigma_{e_T}]$.
We construct the prior using a multivariate normal distribution, i.e. $\tilde{\bm{\theta}}\sim\mathcal{N}(\tilde{\bm{\theta}}_*,\mathbf{\Gamma}_{\tilde{\theta}})$, where $\tilde{\bm{\theta}}_*$ is the prior mean and $\mathbf{\Gamma}_{\tilde{\theta}}$ the covariance. No correlations between the parameters are assumed, so the prior covariance $\mathbf{\Gamma}_{\tilde{\theta}}$ is diagonal. To enforce physical limits of the parameters (such as non-negativity), we simply multiply the prior with an indicator function
$$B(\tilde{\theta}_k) =
\begin{cases}
1, & \text{if } \tilde{\theta}_{\mathrm{min},k} \le \tilde{\theta}_k \le \tilde{\theta}_{\mathrm{max},k} \\
0, & \text{otherwise},
\end{cases}$$
where $\tilde{\bm{\theta}}_{\mathrm{min}}$ stands for the minimum, and $\tilde{\bm{\theta}}_{\mathrm{max}}$ for the maximum, admissible values. Let $\mathbf{L}_{\tilde{\theta}}^T\mathbf{L}_{\tilde{\theta}}^{} = \mathbf{\Gamma}_\theta^{-1}$. The log posterior can now be stated as
$$\label{eq:log_posterior}
\begin{split}
\log\pi(\tilde{\bm{\theta}}|\mathbf{y}) \propto &-\left\Vert \mathbf{L}_e(\mathbf{y} - h(\bm{\theta}))\right\Vert^2 - 2n_\omega(\log\sigma_{e_R} + \log\sigma_{e_T}) \\
& - \frac{1}{2}\Vert \mathbf{L}_{\tilde{\theta}}(\tilde{\bm{\theta}} - \tilde{\bm{\theta}}_*)\Vert^2 + \log B(\tilde{\bm{\theta}}).
\end{split}$$
Most of the parameters have $\tilde{\theta}_{\mathrm{min},k} = 0$ and $\tilde{\theta}_{\mathrm{max},k} = \infty$, corresponding to a non-negativity constraint. The exceptions to this are the imaginary parts of the elastic moduli which are non-positively constrained, and porosity, tortuosity, and permeability. Porosity is, by definition, constrained between $\phi\in[0,1]$, and tortuosity is only defined for $\alpha_\infty \ge 1$. Permeability on the other hand has possible values that span multiple orders of magnitude. The range between the least and most permeable porous media [@schon2015physical] can be up to $k_0\in [10^{-16},10^{-8}]$. For the inversion we therefore prefer to transform $k_0$ to the logarithmic scale, $\log_{10}(k_0)\in[-16,-8]$, and only transform it back to the real scale during the forward model evaluation.
In a numerical study the true parameter values are known, and we need to be careful not to exploit this information when constructing the prior density. We have attempted to select the prior mean and standard deviation conservatively, reflecting the kind of values we could reasonably expect in the given type of materials (see Table \[tab:prior\]). However, we assume that we can measure thickness and the incidence angle reasonably accurately and reflect that by a smaller standard deviation.
In addition, we impose a constraint on $K_b$ based on the Biot-Willis coefficient $\alpha_{\mathrm{B}}$ [@biot1957elastic], which is bounded by $\phi~\le~\alpha_{\mathrm{B}}~\le~1$. The lower limit corresponds to a rigid frame, and the upper limit to a soft or limp frame. The upper limit is enforced by the positivity constraint of $K_b$, but for the lower limit we require $K_b \le K_s(1 - \phi)$. This constraint has little effect on the posterior density, but considerably limits the volume of the prior space, and thus we need fewer temperatures to bridge the gap between the coldest and hottest chains. We implement this constraint using the indicator function as well.
{width="\linewidth"}
MCMC sampling
-------------
Before presenting the results let us briefly comment on the sampling process. We show the case $\varphi_{\mathrm{inc}} = 0 ^{\circ}$ as an example, and just note that the MCMC algorithm works similarly at other angles of incidence unless stated otherwise. Results for measurements at normal incidence are probably the most interesting since normal incidence corresponds to the easiest (and hence the most common) measurement setup.
A starting location for each chain is drawn from the prior. The algorithm is insensitive to the starting location, i.e. we are able to find convergence no matter what the initial point is. The first 25,000 samples are discarded as burn-in, during which we can observe the posterior chain (the chain with $T = 1$) reaching a more or less stable state, see Fig. \[fig:TS\_0deg\]. The big jumps that occur in the beginning of the simulation, and bring the chain quickly to the main posterior mode, are typical for the parallel tempering algorithm. We can also observe how the proposal covariance adapts to the target density during the burn-in. To achieve a good between-chain swap acceptance rate, we used 11 temperatures. The stopping criterion was reached at 75,000 iterations.
As a model feasibility check, we plot the posterior predictive distribution (pdd) in Fig. \[fig:0deg\_PE3\_CM\]. It shows the noisy measurement data, the CM estimate, and the pdd as standard deviations of the model predictions. Standard deviations are a good way to summarise the pdd since we found it approximately normally distributed. The graphical check shows that there are no systematic discrepancies between the Comsol data and predictions from our model at normal incidence.
{width="\linewidth"}
Lastly we note that sampling the posterior at 20 and 25 degrees of incidence turned out to be more difficult than at other angles. We needed 75,000 samples and 13 temperatures to reach a stable state, and convergence was achieved at 150,000 samples. One explanation for this is that the ratio between the width of the posterior and prior density is more extreme at these angles, and thus the temperature steps need to be smaller.
Results {#sec:results}
=======
Let us now examine the inversion results, again starting from normal incidence. Fig. \[fig:0deg\_PE3\_1D\] shows the marginal distributions of the posterior, along with the prior densities. The shaded areas denote the 95 % credible posterior intervals. Comparing the width of the marginal posterior and prior distributions tells about the amount of information provided by the measurement data (the narrower the posterior, the more informative the measurement). A visual inspection shows that the posterior densities of $\Lambda, \alpha_\infty, \phi, \rho_s$, and $L$ are the most accurate by this criterion.
The posterior density of permeability has a peak at its true value, but the distribution continues to the specified upper limit of $10^{-8}$ m$^{2}$. However, there seems to be a clear lower bound below which $k_0$ cannot admit values. Hence the CM estimate tends to overestimate permeability. Moreover, this shows why it is advisable to express $k_0$ on a log scale. On a linear scale the values near the upper bound would completely dominate the distribution and we would not be able to see the peak at the true parameter value.
The elastic parameters are not as well identified as are the other unknowns. We can even see several peaks in the distribution of $K_b$ and $N$. The real and imaginary parts of $K_s$ are almost not identified at all and their posterior distribution mainly follows the prior. The real and imaginary parts of $K_b$ and $N$ are better identified, although they have lots of uncertainty and the 95 % credible interval even admits values down to zero.
{width="\linewidth"}
Checking the joint marginal distributions provides complementary information about the problem of identifiability and non-identifiability. These distributions are shown in Fig. \[fig:0deg\_PE3\_2D\]. We can immediately point out a strong negative linear correlation between the real (and to an extent the imaginary) parts of $K_b$ and $N$. The strong negative correlation between $K_b$ and $N$ explains why the parameters are able to have values near zero. To understand where the correlation comes from, we recall that at normal incidence no shear waves are produced and therefore we do not see their effect. However, the fast longitudinal, or the P-, wave is controlled by both the bulk and shear modulus as illustrated [@jocker2009ultrasonic] by the P-wave speed of a drained porous frame ($K_f \ll K_b,N$):
$$\label{eq:drainedPspeed}
V_p = \sqrt{\dfrac{K_b + \frac{4}{3}N}{(1 - \phi)\rho_s}}.$$
While our material is saturated with fluid that has $K_f \approx N$, the difference between and the saturated material P-wave speed (see for example [@johnson1994probing]) is less than 1 % in the considered frequency range and we can use the simple expression to give physical intuition. At the same time the only expression where $N$ can be found without $K_b$ is the expression for the shear wave velocity:
$$V_{sh} = \sqrt{\frac{N}{\rho - \rho_f^2/\tilde{\rho}_{\mathrm{eq}}}}.$$
This implies that we need to have shear waves to find both $K_b$ and $N$, and in normal incidence can only find a linear combination of $K_b$ and $N$.
Tortuosity is another parameter that is highly correlated with the elastic constants. This is also probably due to a lack of information about the wave speeds, because tortuosity determines the speed of the slow longitudinal wave. The good news is that the true parameter values are all found on the narrow and long support of the joint densities. This means that given accurate prior information on one of the highly correlated parameters, we can expect to estimate the other correlated parameters more accurately as well.
![Joint marginal posterior distributions, $\varphi_{\mathrm{inc}} = 0^{\circ}$.[]{data-label="fig:0deg_PE3_2D"}](fig5){width="1\linewidth"}
Oblique angles
--------------
Fig. \[fig:params\_angles\] shows the CM estimates and 95 % credible intervals of each parameter as a function of the incidence angle from 0 to 45 degrees. We can see that the credible intervals contain the true parameter values at each angle, with the exception of characteristic viscous length. The marginal posterior densities for the two extreme angles, 0 and 45 degrees, are summarised numerically in Table \[tab:inversion\_results\].
The results for oblique angles are as expected in that the estimation of the bulk and shear modulus of the frame is remarkably more accurate than at normal incidence. Now the measurements include effects of the shear wave, and it is thus possible to estimate $N$ and $K_b$ separately. In addition, tortuosity is estimated more accurately at higher angles of incidence. Accuracy of the other parameters stays approximately unchanged, except that, interestingly, the imaginary part of $N$ gets more accurate but the imaginary part of $K_b$ gets less accurate as $\varphi_{\mathrm{inc}}$ increases.
Three parameters, porosity, viscous length, and bulk modulus of the solid, seem to be underestimated slightly at most of the angles. For porosity and solid bulk modulus the prior mean is set lower than the true value, and it might be pulling the posterior below the true value. They are also positively correlated, so that a lower porosity can be compensated by a lower bulk modulus. However, the fact that $\Lambda$ is consistently underestimated shows that the expression used to find the ”true” value is not exact. We confirmed this by considering a noiseless measurement, and maximising the posterior with a Gauss-Newton iteration that was started at the true values. We found that while other parameters stayed at their starting location, the best fit was achieved with $\Lambda = 11.6$ $\mu$m, a value that also corresponds to the CM estimates.
The only parameter that clearly gets less accurate as the angle increases is the thickness of the plate. This also makes sense because the amplitudes of reflections inside the plate, which inform about the thickness, are smaller at higher angles and thus more easily masked by noise.
![The estimated parameters and 95 % credible intervals as a function of the measurement angle.[]{data-label="fig:params_angles"}](fig6){width="\linewidth"}
----------------------------------------------------------------------------------- -- -- -- -- -- -- --
**Parameter & **Truth & &\
$\Lambda$ & 12.8 \* & \[10.5,&**11.5**,&12.6\] & \[10.4,&**11.4**,&12.5\]\
$\alpha_\infty$ & 1.6 & \[1.50,&**1.61**,&1.72\] & \[1.53,&**1.58**,&1.64\]\
$\log_{10}k_0$ & -11.1 & \[-11.4,&**-10.8**,&-9.3\] & \[-11.4,&**-10.6**,&-8.7\]\
$\phi$ & 0.5 & \[0.48,&**0.50**,&0.51\] & \[0.47,&**0.49**,&0.52\]\
Re $K_b$ & 6 & \[2.51,&**5.89**,&9.44\] & \[4.47,&**5.33**,&6.37\]\
Im $K_b$ & -0.12 & \[-0.17,&**-0.09**,&-0.00\] & \[-0.25,&**-0.09**,&-0.00\]\
Re $K_s$ & 50 & \[20.3,&**40.4**,&63.0\] & \[12.7,&**30.9**,&53.9\]\
Im $K_s$ & -1 & \[-1.61,&**-0.63**,&-0.00\] & \[-2.39,&**-1.02**,&-0.00\]\
Re $N$ & 2 & \[0.01,&**2.22**,&4.40\] & \[1.91,&**1.97**,&2.03\]\
Im $N$ & -0.04 & \[-0.12,&**-0.06**,&-0.00\] & \[-0.04,&**-0.04**,&-0.03\]\
$\rho_s$ & 2220 & \[2113,&**2200**,&2279\] & \[2081,&**2156**,&2226\]\
$L$ & 16 & \[15.99,&**16.01**,&16.04\] & \[15.93,&**16.10**,&16.27\]\
****
----------------------------------------------------------------------------------- -- -- -- -- -- -- --
: Conditional mean (CM) and 95 % credible interval estimates.[]{data-label="tab:inversion_results"}
\
\*Calculated from with $a = 16 \:\mu$m.
Let us take another look at the joint marginal distributions (Fig. \[fig:15deg\_PE3\_2D\]), but with an oblique angle of 15 degrees this time. The first observation is that $K_b$ and $N$ are no longer nearly as correlated, and specifically, the shear modulus does not exhibit any of the previous correlations. However, since some of the parameters are more accurately resolved, we start to see correlations that were not visible in the normal incidence case. For example $N$ and $\rho_s$ show moderate correlation, which is reasonable since they both control the speed of the shear wave. In addition, strong correlations can be seen between $K_s$ and tortuosity, porosity, and $K_b$. This shows that the solid bulk modulus is not very well identified even from an oblique incidence data, and that its effect can be compensated by tuning several other parameters. To achieve an accurate result, we would need to either measure $K_s$ separately and use it as prior information, or devise a new kind of measurement set up altogether. The comparison in the lower left corner of Fig. \[fig:15deg\_PE3\_2D\] shows how much smaller the joint posterior is for $N$ and $K_b$ in the 15 degree situation.
![Joint marginal posterior distributions for $\varphi_{\mathrm{inc}} = 15\:^{\circ}$, and a comparison to the normal incidence case.[]{data-label="fig:15deg_PE3_2D"}](fig7){width="\linewidth"}
----------------------------------------------------------------------------------- -- -- -- -- -- -- --
**Parameter & **Truth & &\
$\Lambda$ & 12.8 \* & \[10.3,&**11.0**,&11.5\] & \[10.1,&**11.2**,&12.5\]\
$\alpha_\infty$ & 1.6 & \[1.56,&**1.59**,&1.62\] & \[1.57,&**1.66**,&1.75\]\
$\log_{10}k_0$ & -11.1 & \[-11.2,&**-10.1**,&-8.4\] & \[-11.4,&**-10.3**,&-8.6\]\
$\phi$ & 0.5 & \[0.48,&**0.50**,&0.51\] & \[0.45,&**0.47**,&0.49\]\
Re $K_b$ & 6 & \[5.81,&**6.16**,&6.52\] & \[5.92,&**6.62**,&7.28\]\
Im $K_b$ & -0.12 & \[-0.14,&**-0.10**,&-0.05\] & \[-0.25,&**-0.17**,&-0.08\]\
Re $K_s$ & 50 & \[27.8,&**46.1**,&68.0\] & \[17.4,&**30.8**,&52.5\]\
Im $K_s$ & -1 & \[-1.98,&**-0.93**,&-0.00\] & \[-2.04,&**-0.86**,&-0.00\]\
Re $N$ & 2 & \[1.98,&**2.01**,&2.04\] & \[1.91,&**1.96**,&2.00\]\
Im $N$ & -0.04 & \[-0.05,&**-0.05**,&-0.04\] & \[-0.03,&**-0.02**,&-0.01\]\
$\rho_s$ & 2220 & \[2172&**2229**,&2282\] & \[1995,&**2071**,&2151\]\
$L$ & 16 & \[15.99,&**16.00**,&16.02\] & \[15.91,&**15.93**,&15.95\]\
****
----------------------------------------------------------------------------------- -- -- -- -- -- -- --
: Conditional mean (CM) and 95 % credible interval estimates.[]{data-label="tab:inversion_results_v2"}
\
\*Calculated from with $a = 16 \:\mu$m.
Discussion {#sec:discussion}
==========
To extract the maximum amount of information possible from acoustic reflection and transmission measurements, it is recommended to measure the poroelastic object at an oblique angle. This way the shear wave effects are recorded. At the same time, measurements at normal incidence are the simplest to perform and are enough to estimate many parameters, such as porosity, permeability, density of the solid part, and viscous length, with accuracy equal to the oblique angle measurements. Normal incidence may be all that is needed for example in reservoir characterisation, where the two main microscopic-scale properties of interest are porosity and permeability [@slatt2013stratigraphic].
We can of course join several measurements together for the inversion. This way complementary information from different angles can be used at the same time. Results from inversion using normal incidence and 15 degrees together as data are shown in Table \[tab:inversion\_results\_v2\]. We can see that the results are very accurate.
In the case we consider in this paper, the minimum angle required to include the shear effects into inversion is only five degrees. However, this is not a universal truth, and for several reasons will likely be case dependent. In real world experiments, we may have to deal with a larger model error, i.e. there will be bigger differences between the (ideal, noiseless) measurement and the inverse solution. In this work, sources of difference between our model and the Comsol implementation include distortions from the truncation of the computational domain, and a different viscosity model. However, the solutions returned by the forward model agree well with the finite element model, and we conclude that the modelling errors in this case are relatively small.
On the other hand, the level of additive noise used in this work, five percent, is quite large compared to what can be achieved by repeated measurements in laboratory settings. The advantage of having a noise dominated measurement as opposed to model error dominated one in the inversion is that we know (or at least can reasonably assume) the structure of the measurement noise. The noise level naturally affects the minimum angle needed to resolve shear effects. For example, from a simulation with ten percent additive noise, we found that in order to estimate both $K_b$ and $N$ accurately, the incident wave angle has to be at least ten degrees. We also achieved similar results as in the ten percent noise case by keeping the noise level at five percent but halving the amount of measurement points to 31 frequencies. Another observation regarding the noise level is that the more there is noise, the easier the inverse problem is to solve from a sampling point of view. With small noise levels the likelihood is narrower and more peaky, which is obviously harder to sample and requires more temperatures.
In this ultrasonic measurement setup, we can estimate, for example, porosity to within five percent of its true value, and the true value happens to be always included within the 95 percent credible intervals. Permeability is more difficult to find accurately, even with the transformation to a logarithmic scale. For the considered material we do find a lower bound, which also corresponds closely to the true value. However, the likelihood does not seem to provide an upper bound for permeability. The undefined upper bound is a symptom of the fact that in the current model, reflection and transmission coefficients do not change when permeability is higher than its true value.
In summary, the preferred angle of incidence is dependent on the parameters in which we are the most interested. Most of the parameters, and a linear combination of $K_b$ and $N$ can be estimated reliably from just the normal incidence measurement, but using a measurement from an oblique angle allows us to separate $N$ from $K_b$. Another option is to measure the shear modulus separately, in which case normal incidence would be enough to invert everything else, but this requires another experimental system.
Uncertainty of measurement conditions
-------------------------------------
Estimation accuracy of the material parameters depends on several factors, such as noise level, accuracy of prior knowledge, validity of models, and sampling accuracy. An often overlooked source of error is the uncertainty in the measurement setup. This includes variables such as the angle of the incident wave, any dimensions of the measurement system and the measured object that we use, and possibly even the level of measurement noise. We have not found attempts to control for these in the literature concerning characterisation of porous media.
Ignoring uncertainty in the angle of incidence produces misleading estimates. The incidence angle exhibits correlation with several parameters, such as material thickness, density, and some of the elastic parameters. This means that if the angle uncertainty is not accounted for, distributions of the parameters that are correlated with the angle will be too narrow and, as a result, the credible intervals may not include the true parameter values. To see the effect of a slightly wrong fixed value for the incidence angle, we ran a test at $\varphi_{\mathrm{inc}} = 10$ degrees, and set the incidence angle slightly wrong in the inversion, to 8 degrees. The results are shown in Table \[tab:inversion\_results\_v2\]. We notice that $\rho_s$, $L$, and $\phi$ are already underestimated by so much that the true value is not within the 95 percent CI.
![Estimated noise standard deviations as a function of incidence angle. Also included is the relative error between the outputs of Comsol and GMM models.[]{data-label="fig:errors_angles"}](fig8){width="\linewidth"}
Another parameter not directly related to the material properties is variance of the noise, in both the reflection and transmission measurements. Inversion results for the noise parameters are shown in Fig. \[fig:errors\_angles\]. We find that $\sigma_{e_T}$ is correctly estimated for all considered angles of incidence, but the estimate of $\sigma_{e_R}$ seems to be too big for angles past 30 degrees. However, we also notice that at higher angles the reflection part of the GMM model does not fit the COMSOL measurements as well as it does in the lower angles. This can be seen by plotting the relative model error, defined by (resp. for $\mathbf{T}$)
$$\mathrm{Rel.\:model\:error} = \frac{\left\Vert \mathbf{R}^{\mathrm{exp}} - \mathbf{R}^{\mathrm{GMM}}\right\Vert}{\left\Vert \mathbf{R}^{\mathrm{exp}}\right\Vert},$$
where the coefficients are calculated with the same parameter values and no noise is added. The model error that increases with the incidence angle is caused by our choice of an imperfect absorbing boundary condition in the finite element model. We use an impedance boundary, which is known to work the best only when the boundary is normal to an incoming wave.
We can see in Fig. \[fig:errors\_angles\] that the overestimated $\sigma_{e_R}$ seems to be compensating for the model error. In this way, the model error is approximated with the same distribution that the measurement noise has, which is not necessarily a good approximation. Nevertheless, when we do not estimate the noise level and instead fix it at the correct value of five percent, we find that the estimates at angles where there is model error are wrong. For example, the estimate for permeability was found to be much narrower than in other cases and the 95 percent credible interval did not even contain the true value.
Conclusion {#sec:conclusion}
==========
In this paper we examined the feasibility of the inverse characterisation of poroelastic media based on simulated ultrasonic measurement data. We considered a situation where a slab of poroelastic material is submerged in water and measured from different angles. First we solved the alternative formulation of the Biot equations for poroelastic media to find the acoustic reflection and transmission coefficients of the object of interest. To increase stability of the solution, we used the global matrix method instead of the usual Thomson-Haskell transfer matrix method.
The inverse problem was formulated in the Bayesian framework to take advantage of prior information, account for measurement uncertainties, and to be able to straightforwardly assess the uncertainty in the parameter estimates. We used an adaptive parallel tempering MCMC algorithm to sample the solution of the inverse problem, the posterior probability density. Such a sampling approach was necessary because the posterior density was found to be highly peaked, and standard single-chain samplers tended to get stuck and not explore the posterior in a satisfactory way.
We showed that the proposed approach is able to extract reliably the information contained by the acoustic transmission and reflection measurements. Measurements at normal incidence do not provide information on the shear properties of the material, but small oblique incidences can be enough to get that information. It is preferable to account for uncertainty in the angle of the incident wave to correctly estimate the parameters. In addition, we found that it is possible to estimate accurately the level of measurement noise, and small modelling errors can also be compensated for using this approach.
Acknowledgment {#acknowledgment .unnumbered}
==============
This work has been supported by the strategic funding of the University of Eastern Finland, by the Academy of Finland (Finnish Centre of Excellence of Inverse Modelling and Imaging), and by the RFI Le Mans Acoustique (Pays de la Loire) Decimap project. This article is based upon work initiated under the support from COST Action DENORMS CA-15125, funded by COST (European Cooperation in Science and Technology).
The state matrix {#app:statematrix}
================
The state matrix is of the form $\mathbf{A} = \mathbf{B}^{-1}\mathbf{D}$, with (after Gautier *et al.* [@gautier2011propagation])
$$\mathbf{B} = \begin{bmatrix}
1 & 0 & 0 & 0 & \I k_1(\lambda_c - \alpha_{\mathrm{B}} M) & -\dfrac{k_1\alpha_{\mathrm{B}} M}{\omega} \\
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & -1 & 0 & 0 & 0 \\
0 & 0 & 0 & N & 0 & 0 \\
0 & 0 & 0 & 0 & \lambda_c + 2N - \alpha_{\mathrm{B}} M & \I\dfrac{\alpha_{\mathrm{B}} M}{\omega} \\
0 & 0 & 0 & 0 & (\alpha_{\mathrm{B}} - 1)M & \I\dfrac{M}{\omega} \\
\end{bmatrix},$$
and
$$\mathbf{D} = \begin{bmatrix}
0 & 0 & \I k_1\left(\dfrac{\rho_f}{\tilde{\rho}_{\mathrm{eq}}} - \dfrac{k_1^2\alpha_{\mathrm{B}} M}{\tilde{\rho}_{\mathrm{eq}}\omega^2}\right) & \omega^2\left(\rho - \dfrac{\rho_f^2}{\tilde{\rho}_{\mathrm{eq}}}\right) - k_1^2\left(\lambda_c + 2N - \dfrac{\rho_f}{\tilde{\rho}_{\mathrm{eq}}}\alpha_{\mathrm{B}} M\right) & 0 & 0 \\
\I k_1 & 0 & 0 & 0 & (\rho - \rho_f)\omega^2 & \I\rho_f\omega \\
0 & 0 & 0 & 0 & (\rho_f - \tilde{\rho}_{\mathrm{eq}})\omega^2 & \I\tilde{\rho}_{\mathrm{eq}}\omega \\
-1 & 0 & 0 & 0 & \I k_1N & 0 \\
0 & -1 & -\dfrac{k_1^2\alpha_{\mathrm{B}} M}{\tilde{\rho}_{\mathrm{eq}}\omega^2} & \I k_1\left(\lambda_c - \dfrac{\rho_f}{\tilde{\rho}_{\mathrm{eq}}}\alpha_{\mathrm{B}} M\right) & 0 & 0 \\
0 & 0 & 1 - \dfrac{k_1^2M}{\tilde{\rho}_{\mathrm{eq}}\omega^2} & \I k_1M\left(\alpha_{\mathrm{B}} - \dfrac{\rho_f}{\tilde{\rho}_{\mathrm{eq}}}\right) & 0 & 0 \\
\end{bmatrix}.$$
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[^1]: Corresponding author. email: *[email protected]*
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The current paper is devoted to the study of integral curves of constant type in generalized flag varieties. We construct a canonical moving frame bundle for such curves and give a criterion when it turns out to be a Cartan connection. Generalizations to parametrized curves, to higher-dimensional submanifolds and to parabolic geometries are discussed.'
address:
- 'Belarusian State University, Nezavisimosti Ave. 4, Minsk 220030, Belarus; E-mail: [email protected]'
- 'Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA; E-mail: [email protected]'
author:
- Boris Doubrov
- Igor Zelenko
title: Geometry of curves in generalized flag varieties
---
Introduction
============
This paper is devoted to the local geometry of curves in generalized flag varieties. We provide the uniform approach to this problem based on algebraic properties of so-called distinguished curves and their symmetry algebras. In particular, we develop the methods that allow to deal with arbitrary curves without any additional non-degeneracy assumptions. Although the paper deals with smooth curves in real homogeneous spaces, the whole theory applies equally to the case of holomorphic curves in complex generalized flag varieties.
Unlike the classical moving frame methods and techniques [@cartan; @griffits; @green; @jensen; @olver1; @olver2], we avoid branchings as much as possible treating all curves at once. In fact, the only branching used in our paper is done at a very first step based on the type of the tangent line to a curve, or, in other words on its 1-st jet.
For each curve type there is a class of most symmetric curves, also known as *distinguished curves* in a generalized flag variety [@slovak; @capslovak]. There are simple algorithms to compute the symmetry algebra of these curves [@dk]. We try to approximate any curve by the distinguished curves of the same type and construct the smallest possible moving frame bundle for all curves of a given type. It has the same dimension as the symmetry algebra of the distinguished curve, and its normalization conditions as well as the number of fundamental invariants (or curvatures) is defined by the simple algebraic properties of this symmetry algebra.
We are mainly interested in curves that are integral curves of the vector distribution defined naturally on any generalized flag variety. All distinguished curves of this kind admit the so-called *p*rojective parameter [@dou:05]. This means that there is a natural 3-dimensional family of parameters on each such curve related by projective transformations. One of the questions we answer in this paper is when such projective parameter exists for any (non-homogeneous) curve of a given type. For example, it is well-known [@cartan-proj] that an arbitrary non-degenerate curve in projective space can be naturally equipped with a family projective parameters, even if this curve is no longer homogeneous and, unlike rational normal curves (the distinguished curves in case of the projective space), is not invariant under the global action of $SL(2,{\mathbb R})$.
It appears that the existence of such projective parameter in case of arbitrary generalized flag variety and any curve depends on some specific algebraic properties of the symmetry algebra of the distinguished curve approximating a given curve. In particular, such parameter always exists when this symmetry algebra is reductive.
However, this is not always the case, and we give one of the smallest examples (curves in the flag variety $F_{2,4}({\mathbb R}^5)$) when there are no invariant normalization conditions for the moving frame, and there is no uniform way of constructing the projective parameter on all integral curves of generic type in $F_{2,4}({\mathbb R}^5)$.
Let us introduce the notation we shall use throughout the paper. Let $M=G/P$ be an arbitrary generalized flag variety, where ${\mathfrak g}=\sum_{i\in{\mathbb Z}}{\mathfrak g}_i$ is a graded semisimple Lie algebra of the Lie group $G$ and ${\mathfrak p}=\sum_{i\ge 0}{{\mathfrak g}}$ is a parabolic subalgebra of ${\mathfrak g}$. It is well-known that there is a so-called *grading element* $e\in{\mathfrak g}_0$ such that ${\mathfrak g}_i=\{u\in{\mathfrak g}\mid [e,u]=iu\}$ for all $i\in{\mathbb Z}$. Denote by $G_0$ the stabilizer of $e$ with respect of the adjoint action of $G$ on ${\mathfrak g}$. Then $G_0$ is a reductive, but not necessarily connected subgroup of $G$ with the Lie algebra ${\mathfrak g}_0$.
Each generalized flag variety is naturally equipped with a bracket-generating vector distribution $D$, which can be described as follows. Let $o=eP$ be the “origin” in $M$. We can naturally identify $T_oM$ with a quotient space ${\mathfrak g}/{\mathfrak p}$. Then the vector distribution $D\subset TM$ is defined as a $G$-invariant distribution equal to ${\mathfrak g}_{-1}\mod {\mathfrak p}$ at $o$.
We are interested in local differential invariants of unparametrized integral curves of $D$. The first natural invariant of such curve $\gamma$ is a type of its tangent line. Namely, let $PD$ be the projectivization of the distribution $D$. The action of $G$ is naturally lifted to $PD$ and is, in general, no longer transitive. The orbits of this action are in one-to-one correspondence with the orbits of the action of $P$ (or, in fact, of $G_0$) on the projective space $P({\mathfrak g}_{-1})$. Each integral curve $\gamma$ is naturally lifted to the curve in $PD$. We say that $\gamma$ is *of constant type*, if its lift to $PD$ lies in a single orbit of the action of $G$ on $PD$.
As proved by Vinberg [@vinberg], the action of the group $G_0$ on $P({\mathfrak g}_{-1})$ has a finite number of orbits. So, the above definition of a curve of constant type automatically holds for an open subset of $\gamma$ and essentially means that we exclude singular points, where the tangent lines to $\gamma$ degenerate to the boundary of the orbit of tangent lines at generic points.
The goal of this article is to provide a universal method for constructing *moving frames* for integral curves of constant type on $M=P/G$. This study is motivated by a number of examples of curves in flag varieties, which appear in the geometry of non-linear differential equations of finite type via the so-called linearization procedure [@dou:08; @quasi], non-holonomic vector distributions [@douzel2; @douzel3] and more general structures coming from the Control Theory [@agrzel2; @agrzel3; @zel-che; @quasi] via the so-called symplectification/linearization procedure. Note that our setup covers many classical works on the projective geometry of curves and ruled surfaces. Particular cases of curves in generalized flag varieties considered earlier in the literature also include the geometry of curves in Lagrangian Grassmannains [@agrzel1; @agrzel2; @agrzel3; @zel05; @zel-che], generic curves in $|1|$-graded flag varieties [@beffa1; @beffa2], curves in Grassmann varieties ${\operatorname{Gr}}(r,kr)$ that correspond to systems of $r$ linear ODEs of order $k$ [@se-ashi].
The paper is organized as follows. In Section \[sec:2\] we formulate the main result of the paper and prove it in Section \[sec:3\]. Section \[sec:4\] is devoted to various examples including the detailed description of all curve types for $G_2$ flag varieties.
In Section \[sec:app\] we apply the developed techniques for constructing conformal, almost-symplectic and $G_2$ structures on the solution space of a single ODE of order 3 and higher. These results can be considered a generalization of the classical Wunschmann result on the existence of a natural conformal structure on the solution space of a 3rd order ODE provided that a certain invariant of the equation (known today as Wunschmann invariant) vanishes identically. This results was generalized in [@dou:08; @dun-tod; @godl; @dun-godl; @dou:dun] to the case of higher order ODEs which have all higher order Wunschmann invariants identically equal to $0$. In Theorem \[thm:2\] we find necessary and sufficient conditions for such geometric structures to exist.
Finally, in Section \[sec:5\] we discuss the generalizations of our main result to the cases of parametrized curves, curves in curved parabolic geometries and the case of higher dimensional integral submanifolds.
Acknowledgments {#acknowledgments .unnumbered}
---------------
The first author would like to thank the International Center for Theoretical Physics in Trieste and the Mathematical Sciences Institute at the Australian National University for their hospitality and financial support during the work on this paper. We also would like to thank Mike Eastwood, Tohru Morimoto, Maciej Dunajski, Dennis The for many stimulating discussions on the topic.
Formulation of the main result {#sec:2}
==============================
Let us fix an element $x\in {\mathfrak g}$ of degree $-1$. We say that an integral curve $\gamma$ is *of type $x$*, if the lift of $\gamma$ to $PD$ lies in the orbit of the line ${\mathbb R}x$ under the action of $G$ on $PD$. Here we treat the one-dimensional space ${\mathbb R}x$ as a point in $PD_o$. In other words, $\gamma$ is of type $x$ if for each point $p\in \gamma$ there exists an element $g\in G$ such that $g.o=p$ and $g_*(x)\subset T_p\gamma$.
Define a curve $\gamma_0\subset G/P$ as a closure of the trajectory of the one-parameter subgroup $\exp(tx)\subset G$ through the origin $o=eP$. It is known [@dou:05] that we can complete $x$ to the basis $\{x,h,y\}$ of the subalgebra ${\mathfrak{sl}}(2,{\mathbb R})\subset {\mathfrak g}$, such that $\deg h = 0$ and $\deg y = 1$. Then $\gamma_0$ is exactly the orbit of the corresponding subgroup in $G$ and, thus, is a rational curve or its finite covering. As we shall see, it is the most symmetric curve of type $x$. We call $\gamma_0$ *a flat curve of type $x$*.
Let $S$ be the symmetry group of $\gamma_0$, and let ${\mathfrak s}\subset {\mathfrak g}$ be the corresponding subalgebra. It is known (see [@dk]) that ${\mathfrak s}$ is a graded subalgebra of ${\mathfrak g}$, which can be also defined as follows: $$\label{symalg}
\begin{aligned}
{\mathfrak s}_{i} & = 0 \textrm{ for } i\le -2;\\
{\mathfrak s}_{-1} &= \langle x \rangle;\\
{\mathfrak s}_{i} &= ({\operatorname{ad}}x)^{-1}({\mathfrak s}_{i-1}) = \{ u \in {\mathfrak g}_i \mid [x,u]\in{\mathfrak s}_{i-1}\} \textrm{ for all } i\ge 0.
\end{aligned}$$ Denote by $S^{(0)}$ the intersection of $S$ and $P$ and by ${\mathfrak s}^{(0)}$ the corresponding subalgebra of ${\mathfrak g}$. It is clear that we have ${\mathfrak s}^{(0)}=\sum_{i\ge 0}{\mathfrak s}_i$.
Let us define two sequences of subgroups in $P$:
- $P^{(k)}$ is a subgroup of $P$ with the subalgebra ${\mathfrak p}^{(k)}=\sum_{i\ge k}{\mathfrak g}_i$. It is equal to $P$ for $k=0$ and to $\prod_{i\ge k}\exp({\mathfrak g}_i)$ for $k\ge 1$. It is easy to see that $P^{(k)}$ is trivial for sufficiently large $k$.
- $H^{(k)}$ is a subgroup of $P$ with the subalgebra ${\mathfrak h}^{(k)}=\sum_{i=0}^{k}{\mathfrak s}_i + \sum_{i>k}{\mathfrak g}_i$. Denote by $S_0$ the intersection of $S$ and $G_0$. Then we have: $$H^{(k)} = S_0 \prod_{i=1}^k \exp({\mathfrak s}_i) \prod_{i>k} \exp({\mathfrak g}_i), \quad \text{for any }k\ge 0.$$ Lemma \[lem:jet\] below shows that $H^{(k)}$ is indeed a subgroup of $G$ and ${\mathfrak h}^{(k)}$ is a subalgebra of ${\mathfrak g}$ for any $k\ge 0$. The subgroup $H^{(k)}$ becomes equal to $S^{(0)}=S\cap P$ for sufficiently large $k$. Note that $H^{(k)}$ contains $P^{(k+1)}$ for any $k\ge 0$.
The sequence of subgroups $H^{(k)}$ has a very natural geometric meaning:
\[lem:jet\] The subgroup $H^{(k)}$ is exactly the stabilizer of the $(k+1)$-th jet of the flat curve $\gamma_0$ of type $x$ at the origin $o=eP\in G/P$.
First, assume $k=0$. The 1-jet of the curve $\gamma_0$ at the origin is determined by its tangent line. Under the natural identification of $T_o(G/P)$ with ${\mathfrak g}/{\mathfrak p}$ this line is identified with the subspace $\langle x \rangle + {\mathfrak p}$. The stabilizer of the 1-jet coincides with the stabilizer of this subspace under the isotropic action of $P$ on ${\mathfrak g}/{\mathfrak p}$. As $\deg x = -1$, it is clear that this stabilizer contains $P^{(1)}$. As $P=G_0P^{(1)}$ and $P^{(1)}$ is a normal subgroup in $P$, it remains to show that the intersection of the stabilizer with $G_0$ coincides with $S_0=S \cap G_0$.
It is clear that $S_0$ lies in the stabilizer of the 1-jet of $\gamma_0$ at the origin. Let now $g\in G_0$ and ${\operatorname{Ad}}g (x) \subset \langle x \rangle + {\mathfrak p}$. It implies that ${\operatorname{Ad}}g (x) =\alpha x$ for some non-zero constant $\alpha$, and, hence, $g\in S_0$.
The case $k\ge 1$ can be treated in the similar manner. We only need to note that the $(k+1)$-th jet of $\gamma_0$ is completely determined by the tangent line to the lift of $\gamma_0$ to the space of $k$-jets.
Let $\pi\colon G\to G/P$ be the canonical principal $P$-bundle over the generalized flag variety $G/P$ and let ${\omega}\colon TG\to {\mathfrak g}$ be the left-invariant Maurer–Cartan form on $G$. By a *moving frame bundle* $E$ along $\gamma$ we mean any subbundle (not necessarily principal) of the P-bundle $\pi^{-1}(\gamma)\to \gamma$.
Let us recall the basic notions of filtered and graded vector spaces as they will be used throughout the paper. Let $V=\sum_{i\in{\mathbb Z}} V_i$ be a (finite-dimensional) graded vector space. Then $V$ has a natural decreasing filtration $\{V^{(k)}\}$, where $V^{(k)}=\sum_{i\ge k}V_i$. If $U$ is any linear subspace in $V$, then we define $U^{(k)}=U\cap V^{(k)}$ and denote by ${\operatorname{gr}}U=\sum_{i\in{\mathbb Z}}U_i$ the corresponding graded subspace of $V$. Here $U_i=U^{(i)}/U^{(i+1)}$ is naturally identified with a subspace of $V_i$.
The role of “normalization conditions” for the canonical moving frame is played by a graded subspace $W\subset{\mathfrak p}$ complementary to ${\mathfrak s}^{(0)} + [x,{\mathfrak p}]$. In other words, let $W=\sum_{i\ge 0} W_i$, where $W_i$ is a subspace in ${\mathfrak g}_i$ complementary to ${\mathfrak s}_i + [x,{\mathfrak g}_{i+1}]$.
\[def1\] We say that a subspace $U\subset {\mathfrak g}$ is *$W$-normal (with respect to ${\mathfrak s}$),* if the following two conditions are satisfied:
1. ${\operatorname{gr}}U ={\mathfrak s}$;
2. $U$ contains an element $\bar x = \sum_{i\ge -1}x_i$, where $x_{-1}=x$ and $x_{i}\in W_i=W\cap{\mathfrak g}_i$ for all $i\ge 0$.
Note that the element $\bar x$ (and, hence, all $x_i$, $i\ge 0$) is defined uniquely for any $W$-normal subspace $U\subset{\mathfrak g}$. Indeed, suppose there is another element $\bar x'=\sum_{i\ge -1}x_i'$ with this property. Then the element $\bar x - \bar x' = \sum_{i\ge 0} (x_i-x_i')$ also lies in $U$. As ${\operatorname{gr}}U={\mathfrak s}$, we get $x_0-x_0'\in {\mathfrak s}_0$. On the other hand, both $x_0$ and $x_0'$ lie in $W_0$, which is complementary to ${\mathfrak s}_0+[x,{\mathfrak g}_1]$ and, in particular, has zero intersection with ${\mathfrak s}_0$. Hence, $x_0=x_0'$. Proceeding by induction by $i$, we prove in the similar manner that $x_i=x_i'$ for all $i\ge 0$.
The main results of this paper can be formulated as follows.
\[thm1\] Fix any graded subspace $W\subset{\mathfrak p}$ complementary to ${\mathfrak s}^{(0)} + [x,{\mathfrak p}]$. Then for any curve $\gamma$ of type $x$ there is a unique frame bundle $E\subset G$ along $\gamma$ such that the subspace ${\omega}(T_pE)$ is $W$-normal for each point $p\in E$ .
If, moreover, $W$ is $S^{(0)}$-invariant, then $E$ is a principal $S^{(0)}$-bundle, and ${\omega}|_{E}$ decomposes as a sum of the $W$-valued 1-form ${\omega}_W$ and the ${\mathfrak s}$-valued 1-form ${\omega}_{{\mathfrak s}}$. The form ${\omega}_{\mathfrak s}$ is a flat Cartan connection on $\gamma$ modeled by $S/S^{(0)}$, and ${\omega}_W$ is a vertical $S^{(0)}$-equivariant 1-form defining a fundamental set of differential invariants of $\gamma$.
The construction of thus moving frame bundle will be done in the next section.
The homogeneous space $S/S^{(0)}$ is exactly the model for the flat (or distinguished) curve of type $x$. As the it is (locally) isomorphic to ${{\mathbb R}\mathrm{P}}^1$, the Cartan connection ${\omega}_{\mathfrak s}$ defines the projective parameter on $\gamma$ in the canonical way.
Here by *a fundamental set of invariants* we mean that all the (relative or absolute) differential invariants of $\gamma$ can be obtained from the 1-form ${\omega}_W$ and all its covariant derivatives defined by the connection ${\omega}_{\mathfrak s}$. In particular, the curve $\gamma$ is locally flat if and only if ${\omega}_W$ vanishes identically.
Note that the $S^{(0)}$-invariant complement $W\subset{\mathfrak p}$ always exists, if the Lie algebra ${\mathfrak s}$ is reductive, or, what is equivalent, if the restriction of the Killing form of ${\mathfrak g}$ to ${\mathfrak s}$ is non-degenerate. Indeed, let ${\mathfrak s}^{\perp}$ be its orthogonal complement with respect to the Killing form of ${\mathfrak g}$. As Killing form $K$ of ${\mathfrak g}$ is compatible with the grading (i.e., $K({\mathfrak g}_i,{\mathfrak g}_j)=0$ for all $i+j\ne 0$), the subspace ${\mathfrak s}^{\perp}$ is also graded: ${\mathfrak s}^{\perp} = \sum_{i\in {\mathbb Z}} {\mathfrak s}^{\perp}_i$, where ${\mathfrak s}^{\perp}_i = {\mathfrak s}^{\perp} \cap {\mathfrak g}_i$.
Define the subspace $W\subset{\mathfrak p}$ as: $$\label{eq:w}
W = \{ u\in {\mathfrak s}^{\perp}\cap{\mathfrak p}\mid [u,y] = 0 \},$$ where $y$ is an element of degree $+1$ in the ${\mathfrak{sl}}_{2}$-subalgebra containing $x$.
The space $W$ is a ${\mathfrak s}^{(0)}$-invariant complement to ${\mathfrak s}^{(0)}+[x,{\mathfrak g}_{+}]$ in ${\mathfrak p}$.
First, note that ${\mathfrak s}^{\perp}$ is ${\mathfrak s}$-invariant and complementary to ${\mathfrak s}$. In particular, it is also stable with respect to the action of the ${\mathfrak{sl}}_2$-subalgebra containing $x$. From the representation theory of ${\mathfrak{sl}}_2$, it immediately follows that the space $W$ defined above is spanned by highest weight vectors (of non-negative degree) in the decomposition of ${\mathfrak s}^{\perp}$ into the direct sum of irreducible ${\mathfrak{sl}}_2$-modules. This immediately implies that $W$ is complementary to $[x,{\mathfrak s}^{\perp}]$ in ${\mathfrak s}^{\perp}\cap{\mathfrak p}$.
As the restriction of the Killing form $K$ to ${\mathfrak s}$ is non-degenerate and ${\mathfrak s}_{-1}={\mathbb R}x$, we see that ${\mathfrak s}_{+1}={\mathbb R}y$ and, in particular, ${\mathbb R}{y}$ is an ideal in ${\mathfrak s}^{(0)}={\mathfrak s}_0+{\mathfrak s}_1$. This proves that $W$ is ${\mathfrak s}^{(0)}$-invariant.
The invariant complement $W$ may exist even if the Lie algebra ${\mathfrak s}$ is not reductive. See Subsection \[g2-ex\] for such example.
Canonical frame bundle {#sec:3}
======================
General reduction procedure
---------------------------
Let $\gamma$ be an arbitrary unparametrized curve in $G/P$ of type $\gamma_0$ and let $\pi\colon E\to \gamma$ be a moving frame bundle along the curve $\gamma$. For each point $p\in E$ the image ${\omega}(T_pE)$ is a subspace in the Lie algebra ${\mathfrak g}$. We say that $E$ is an $H$-bundle, where $H$ is a subgroup of $P$, if $E$ is preserved by the right action of $H$ on $G$. Note that we do not assume that $E$ is a principal $H$-bundles, i.e. the fibers of $\pi\colon E\to H$ are unions of orbits of $H$, but each fiber may in general be of larger dimension than $H$.
From definition of the Maurer–Cartan form we see that ${\omega}(T_pE)$ contains ${\mathfrak h}$ for all $p\in E$, where ${\mathfrak h}$ is a subalgebra of ${\mathfrak g}$ corresponding to the subgroup $H$. Thus, we can define a *structure function* of the moving frame $H$-bundle $E$ as follows: $$c\colon E \to {\operatorname{Gr}}_r({\mathfrak g}/{\mathfrak h}),\quad p\mapsto {\omega}(T_pE) /{\mathfrak h}\subset {\mathfrak g}/{\mathfrak h},$$ where $r=\dim E - \dim H$ and ${\operatorname{Gr}}_r({\mathfrak g}/{\mathfrak h})$ is a Grassmann variety of all $r$-dimensional subspaces in ${\mathfrak g}/{\mathfrak h}$.
As the Maurer–Cartan form ${\omega}$ satisfies the condition $R_g^*{\omega}= {\operatorname{Ad}}g^{-1}\circ{\omega}$ for any $g\in P$, we see that the structure function $c$ is $H$-equivariant: $$c(ph) = h^{-1}.c(p), \quad p\in E, h\in H,$$ where the action of $H$ on ${\operatorname{Gr}}_r({\mathfrak g}/{\mathfrak h})$ is induced from the adjoint action of $H$ on ${\mathfrak g}$.
One of the main ideas of the reduction procedure is to impose certain *normalization* conditions on the structure function $c$ and reduce $E$ to a subbundle $E'$ consisting of all points of $E$, such that $c(p)$ satisfies these imposed conditions. In general, $E'$ might not be a smooth submanifold of $E$, as $c(E)$ may not be transversal to the orbits of $H$ on ${\operatorname{Gr}}_r({\mathfrak g}/{\mathfrak h})$. However, in our case we are always able to choose the normalization conditions in such a way that $E'$ is a regular submanifold in $E$ and, thus, is also a frame bundle along $\gamma$.
Zero order reduction {#ss:zero}
--------------------
Let $\pi\colon G\to G/P$ be the canonical projection. Define the moving frame bundle $E_0$ as a set of all elements $g\in G$ satisfying the properties $g.o\in \gamma$ and $g_*(x)\in T_{g.o}\gamma$. Let $\pi_0$ be the restriction of $\pi$ to $E_0$. It is clear that $\pi_0\colon E_0\to \gamma$ is a principal $H^{(0)}$-bundle, as $H^{(0)}$ is exactly the subgroup of $P$ preserving the 1-jet of $\gamma_0$ at the origin.
Let us describe the structure function of the frame bundle $E_0$. By construction ${\omega}(T_pE_0) \subset \langle x \rangle + {\mathfrak p}$ for all $p\in E_0$. As $r = \dim E_0 - \dim H^{(0)} = 1$, the structure function $c_0$ of the moving frame bundle $E_0$ takes values in the projectivization of the vector space $({\mathbb R}x + {\mathfrak p})/{\mathfrak h}^{(0)}\equiv ({\mathbb R}x + {\mathfrak g}_0)/{\mathfrak s}_0$. In particular, for each $p\in E_0$ we can uniquely define an element $\chi_0(p)\in {\mathfrak g}_0/{\mathfrak s}_0$ such that $x+\chi_0(p)\in c(p)={\omega}(T_pE_0)$. The structure function $c$ of $E_0$ is completely determined by this function $\chi_0\colon E_0 \to {\mathfrak g}_0/{\mathfrak s}_0$.
Further reductions and the canonical moving frame
-------------------------------------------------
From now on we assume that we fixed a graded subspace $W\subset {\mathfrak p}$ complementary to ${\mathfrak s}^{(0)} + [x,{\mathfrak p}]$. Set $W_i=W\cap {\mathfrak g}_i$ for any $i\ge 0$.
Let us introduce a slight generalization of the above notion of $W$-normality.
We say that a subspace $U\subset {\mathfrak g}$ is *$W$-normal up to order $k$*, if it satisfies the following conditions:
1. $U \supset {\mathfrak p}^{(k)}$;
2. ${\operatorname{gr}}U = {\mathfrak s}+{\mathfrak p}^{(k)}$;
3. $U$ contains an element $\bar x = \sum_{i=-1}^{k-1} x_i$, where $x_{-1}=x$ and $x_{i}\in W_i=W\cap{\mathfrak g}_i$ for all $i=0,\dots,k-1$.
Since ${\mathfrak g}$ has a finite grading, ${\mathfrak p}^{(N)}=0$ for sufficiently large $N$ and the condition of $W$-normality up to order $N$ is equivalent to the notion of $W$-normality from the previous section.
We construct a canonical moving frame bundle for a given curve $\gamma$ via the sequence of reductions $E_k$ of the frame bundle $E_0$, where each $E_k$ is a moving frame $P^{(k+1)}$-bundle along $\gamma$.
For $k=0$ we can take as $E_0$ the zero-order frame bundle constructed above. Indeed, as $E_0$ is a principal $H^{(0)}$-bundle and $H^{(0)}$ contains $P^{(1)}$, it is invariant with respect to the right action of $P^{(1)}$ on $G$.
Assume by induction that for some $k\ge1$ we have constructed a moving frame $P^{(k)}$-bundle $E_{k-1}$, such that if $c_{k-1}$ is its structure function, then $c_{k-1}(p)$ is $W$-normal up to order $k-1$ for any point $p\in E_{k-1}$. Then we define $E_{k}$ as the following reduction of $E_{k-1}$: $$\label{reduct}
E_{k} = \{ p\in E_{k-1} \mid c_{k-1}(p) \textrm{ is $W$-normal up to order $k$} \}.$$
1. The frame bundle $E_{k}$ is well-defined and is stable with respect to the right action of the group $P^{(k+1)}$.
2. The structure function $c_k(p)={\omega}(T_pE_k)$, $p\in E_k$, satisfies the conditions: $c_k(p)\supset {\mathfrak p}^{(k+1)}$ and ${\operatorname{gr}}c_k(p) = {\mathfrak s}+{\mathfrak p}^{(k+1)}$ for any $p\in E_k$.
3. If, in addition, $W$ is $S^{(0)}$-invariant, then $E_k$ is stable with respect to the right action of the subgroup $H^{(k)}$.
Consider the structure function $c_{k-1}$ of the frame bundle $E_{k-1}$. By induction, the subspace $c_{k-1}(p)$ is $W$-normal up to order $k-1$. Thus, $c_{k-1}(p)$ contains an element: $$\label{eq:chi}
\chi(p) = x + \sum_{i=0}^{k-1}\chi_i(p),$$ where $\chi_i(p)\in W_i$ are uniquely defined for $i=0,\dots,k-2$, and $\chi_{k-1}(p)$ is a certain element of ${\mathfrak g}_{k-1}$ uniquely defined modulo ${\mathfrak s}_{k-1}$. Note that the condition is equivalent to: $$\label{reduct1}
\chi_{k-1}(p)\in W_{k-1}+{\mathfrak s}_{k-1}.$$
Further, we see that: $$\begin{aligned}
\chi(pg) &= \chi(p) \mod {\mathfrak s}_{k-1}\quad\text{for any }g\in P^{(k+1)},\\
\chi(p\exp(u)) &= \chi(p)-[u,x] \mod {\mathfrak s}_{k-1} \quad\text{for any }u\in{\mathfrak g}_k.\end{aligned}$$ The latter equation follows from the equivariance of Maurer–Cartan form: $$c_{k-1}(p\exp(u)) = {\operatorname{Ad}}(\exp(-u) )c_{k-1}(p),\quad \text{for any }p\in E_{k-1}, u\in{\mathfrak g}_{k},$$ and the equality: $${\operatorname{Ad}}\exp(-u) = \sum_{i=0}^{\infty} \frac{{\operatorname{ad}}^i(-u)}{i!},$$ where this sum is actually finite for any element $u\in {\mathfrak g}_k$, $k>0$.
As $P^{(k)}=\exp({\mathfrak g}_k)P^{(k+1)}$, it follows that the condition can always be achieved by an appropriate choice of the element $u\in{\mathfrak g}_k$. Moreover, as $W_{k-1}$ is complementary to ${\mathfrak s}_{k-1}+[x,{\mathfrak g}_k]$, the set of all elements $u\in{\mathfrak g}_k$, which satisfy the condition $[x,u]\in{\mathfrak s}_{k-1}+W_{k-1}$ coincides with ${\mathfrak s}_k=\{u\in{\mathfrak g}_k \mid [x,u]\in{\mathfrak s}_{k-1}\}$. This proves that the reduction $E_k$ is well-defined and is stable with respect to $\exp({\mathfrak s}_k)P^{(k+1)}$.
If $W$ is not $S^{(0)}$-invariant, we just reduce this group to $P^{(k+1)}$. However, if $W$ is $S^{(0)}$-invariant, then the decomposition is stable with respect to the subgroup $H^{(k-1)}$, and, hence, the condition defines the reduction of the principal $H^{(k-1)}$-bundle $E_{k-1}$ to the subgroup $H^{(k)}$.
Thus, we see that $E_k$ is a moving $P^{(k+1)}$-bundle. If, in addition, the subspace $W$ is $S^{(0)}$-invariant, we treat $E_k$ as a principal $H^{(k)}$-bundle. As ${\mathfrak h}^{(k)}={\mathfrak s}^{(0)}$ for sufficiently large $k$ and ${\mathfrak s}=\langle x \rangle + {\mathfrak s}^{(0)}$ we see that we finally get the moving frame bundle $E$ along $\gamma$ that satisfies conditions of Theorem \[thm1\].
The case of an invariant complementary subspace
-----------------------------------------------
Suppose now that the subspace $W$ is in addition $S^{(0)}$-invariant. As it was shown above, at each step $\pi\colon E_k\to \gamma$ is a principal $H^{(k-1)}$-bundle. For sufficiently large $k$ we get $H^{(k-1)}=S^{(0)}$ and $E=E_k$ is a principal $S^{(0)}$-bundle equipped with a 1-form ${\omega}|_{E}$. This implies that ${\omega}(T_pE)$ contains ${\mathfrak s}^{(0)}$ for all $p\in E$. Hence, from $W$-normality condition we immediately get: $${\mathfrak s}^{(0)}\subset {\omega}(T_pE) \subset W\oplus{\mathfrak s}\quad\text{for all }p\in E.$$ Let us decompose ${\omega}|_{E}$ into the sum: $${\omega}|_{E} = {\omega}_W + {\omega}_{{\mathfrak s}},$$ where the 1-forms ${\omega}_W$ and ${\omega}_{{\mathfrak s}}$ take values in $W$ and ${\mathfrak s}$ respectively.
The 1-form ${\omega}_{{\mathfrak s}}$ is a flat Cartan connection with model $S/S^{(0)}$.
From the properties of the Maurer–Cartan ${\omega}$ it follows that ${\omega}_{{\mathfrak s}}$ is $S^{(0)}$-equivariant and takes the fundamental vector fields on $E$ to the corresponding elements of the subalgebra ${\mathfrak s}^{(0)}$. By construction ${\omega}_{{\mathfrak s}}(T_pE) + {\mathfrak s}^{(0)} = {\mathfrak s}$ for all $p\in E$, and, hence ${\omega}_{{\mathfrak s}}$ is an isomorphism of $T_pE$ and ${\mathfrak s}$ for all $p\in E$. Thus, ${\omega}_{{\mathfrak s}}$ is indeed a Cartan connection on $E$ modeled by the homogeneous space $S/S^{(0)}$.
As $\gamma$ is one-dimensional, this Cartan connection is necessarily flat.
Consider now the 1-form $\omega_W$. It is clearly $S^{(0)}$-equivariant. It is also vertical, as ${\omega}|_{E}$ takes all vertical vector fields to ${\mathfrak s}$. Finally, it is easy to see that $\gamma$ is locally equivalent to $\gamma_0$ (or, in other words, $\gamma$ is locally flat), if and only if ${\omega}$ takes values in ${\mathfrak s}$, or, in other words, when ${\omega}_W$ vanishes identically. Hence, ${\omega}_W$ constitutes a fundamental system of invariants of $\gamma$.
Note that there are many more local differential invariants of $\gamma$. For example, we can construct new invariants by taking total derivatives of ${\omega}_W$ with respect to the Cartan connection ${\omega}_{{\mathfrak s}}$.
Examples {#sec:4}
========
Curves in projective spaces {#ssec:41}
---------------------------
As a very first example, consider a non-degenerate curve $\gamma\subset {{\mathbb R}\mathrm{P}}^k$. Non-degeneracy means that $\gamma$ does not lie in any proper linear subspace of ${{\mathbb R}\mathrm{P}}^k$. Taking the osculating flag of $\gamma$, we can also consider $\gamma$ as a curve in the complete flag manifold $F_{1,\dots,k-1}(k)$. We consider this flag manifold as a generalized flag variety $G/P$, where $G=SL(k+1,{\mathbb R})$ and $P$ is a Borel subgroup of $G$ consisting of all non-degenerate upper-triangular matrices. As a homogeneous curve $\gamma_0$ we take the closure of the trajectory of $\exp(tx)$, where $$x = \begin{pmatrix} 0 & 0 & \dots & 0 & 0 \\ 1 & 0 & \dots & 0 & 0 \\ 0 & 1 & \ddots & 0 & 0 \\
0 & 0 & \ddots & 0 & 0 \\ 0 & 0 & \dots & 1 & 0
\end{pmatrix}$$ It is easy to see that $\gamma_0$ is exactly the osculating variety of the rational normal curve in ${{\mathbb R}\mathrm{P}}^k$.
It is clear that $\gamma$ can be approximated by $\gamma_0$ in the flag variety up to the first order. The symmetry algebra ${\mathfrak s}$ of $\gamma_0$ is isomorphic to ${\mathfrak{sl}}(2,{\mathbb R})$ embedded irreducibly into ${\mathfrak{sl}}(k+1,{\mathbb R})$. As ${\mathfrak s}$ is reductive, the restriction of the Killing form to ${\mathfrak s}$ is non-degenerate, and we can choose $W$ as the set of all $y$-invariant elements of the orthogonal complement to ${\mathfrak s}$ with respect to the Killing form $K(A,B)=\operatorname{tr}(AB)$. Simple computation shows that $W = \langle y^i \mid i = 2, \dots, k\rangle$.
Thus, for any non-degenerate curve $\gamma$ in ${{\mathbb R}\mathrm{P}}^k$ we can construct the canonical frame bundle $E\to \gamma$ such that ${\omega}_{\mathfrak s}$ takes values in ${\mathfrak{sl}}(2,{\mathbb R})$ and defines a (flat) projective connection on it, while the 1-form ${\omega}_W$ is equal to: $$\label{eq:omw}
{\omega}_W = \sum_{i=2}^k \theta_{i+1}y^i.$$ Here $\theta_i$ are the equivariant $1$-forms on the bundle $E$ defining a set of fundamental invariants of the curve $\gamma$.
These invariants can be constructed explicitly as follows. Let as choose an arbitrary local parameter $t$ on $\gamma$ and a curve $v_0(t)$ in ${\mathbb R}^{k+1}$ such that $\gamma(t) = \langle v_0(t) \rangle$ for all $t$. Define $v_i(t)=v_0^{(i)}(t)$. The osculating flag of $\gamma$ is given by subspaces $\gamma^{(i)} = \langle v_0(t), \dots, v_i(t) \rangle$ for all $0\le i \le k$. By regularity condition the family of vectors $\{ v_0(t), v_1(t),\dots, v_k(t) \}$ forms a basis in ${\mathbb R}^{k+1}$ for all $t$. In particular, this defines a section $s\colon \gamma \to GL(k+1,{\mathbb R})$. Express the derivative of $v_k(t)$ in this basis: $$v_k'(t) = v_0^{(k+1)}(t) = p_0(t) v_0(t) + \dots + p_k(t) v_k(t).$$ The pull-back of the Maurer-Cartan form ${\omega}$ to the curve $\gamma$ has the form: $$\label{frame_pk}
s^*\omega =
\begin{pmatrix}
0 & 0 & \dots & 0 & p_0(t) \\
1 & 0 & \dots & 0 & p_1(t) \\
\vdots & \ddots & \vdots & \vdots \\
0 & 0 & \dots & 1 & p_k(t)
\end{pmatrix}.$$ We see that the section $s$ takes values in $SL(k+1,{\mathbb R})$ if and only if ${\operatorname{tr}}s^*\omega = p_k(t) = 0$. Multiplying the vector $v_0(t)$ by the function $\lambda(t)$, where $\lambda' = (k+1) p_k(t)\lambda$, we can always make the coefficient $p_k(t)$ vanish identically. Moreover, this defines the vector $v_0(t)$ uniquely up to the non-zero constant multiplier. Thus, for every choice of a local parameter $t$ we get a well-defined section $s\colon \gamma \to SL(k+1,{\mathbb R})$. Moreover, from we see that these sections take values already in the zero-order reduction $E_0$ in terms of Subsection \[ss:zero\].
In order to construct the section further reductions $E_i$, we can adjust the section $s$ by recalibrating it to $s\cdot A(t)$, where $A\colon \gamma \to S^{(0)}$ is any smooth function. It is easy to see that the connected component of the group $S^{(0)}$ has the form: $$S^{(0)} = \{{\operatorname{diag}}(a^k,a^{k-2},\dots,a^{-k})\mid a\in {\mathbb R}^*\} N(k+1,{\mathbb R}),$$ where $N(k+1,{\mathbb R})$ is the subgroup of all unipotent matrices in $SL(k+1,{\mathbb R})$. The first multiplier is a subgroup in $S_0$, the symmetry group of the rational normal curve $\gamma_0$. So, we can consider only functions $h$ taking values in $N(k+1,{\mathbb R})$. Let $A(t) = (A_{ij}(t))_{0\le i,j \le k}$, where $A_{ii} = 1$, $i=0,\dots, k$ and $h_{ij} = 0 $ for $i>j$. Then the condition that $sA(t)$ takes values in the canonical frame is equivalent to the condition that $A^{-1} (s^*\omega) A + A^{-1}A'$ is of the form: $$x + \sum_{i=2}^{k} \theta_{i+1}(t) y^i \mod \langle h, y\rangle.$$
The normalization procedure of Section \[sec:3\] guarantees that such matrix $A(t)$ can always be found. Moreover, each reduction step involves only solving *linear* equations for a part of the functions $A_{ij}$. In particular, the invariants $\theta_i(t)$ are polynomial functions of the coefficients $p_i(t)$, $i=0,\dots,k$ and their derivatives. For example, the invariant $\theta_3$ is explicitly given as: $$\theta_3 = p_{k}''-\frac6{k+1}p_kp_k'+\frac4{(k+1)^2}p_k^3+\frac{6}{k}p_{k-1}'
+\frac{12}{k(k+1)}p_kp_{k-1}+\frac{12}{k(k-1)}p_{k-2}.$$
Invariants $\theta_i(t)$ were first constructed in the classical book by Wilczynski [@wilch] in 1905 using the correspondence between the non-degenerate curves in ${{\mathbb R}\mathrm{P}}^k$ and linear homogeneous differential equations of order $k+1$. See also [@ovs-tab; @se-ashi] for the definition of these invariants in the modern language. In the following we shall call them *Wilczynski invariants* of the projective curve. As an easy consequence of our theory we immediately see that a curve $\gamma$ is an open part of a rational normal curve if and only if all its Wilczynski invariants vanish identically.
In the simplest case of curves in on the projective plane $P^2$ we recover this way the classical construction of Élie Cartan [@cartan-proj] of the projective parameter and the projective invariants of plane curves considered up to projective transformations.
Curves in Grassmann and flag varieties
--------------------------------------
Let $\gamma$ be a curve in Grassmann variety ${\operatorname{Gr}}_r(V)$. In general, the tangent line $T_t\gamma$ for any $t\in\gamma$ can be identified with a one-dimensional subspace in ${\operatorname{Hom}}(\gamma(t),V/\gamma(t))$. Take any element $\phi_t$ in this line and denote by $\gamma'$ the subspace $\gamma(t) + \operatorname{Im}\phi_t$. We define $k$-th derivative $\gamma^{(k)}$ in a similar manner. Thus, for each $t\in \gamma$ we get a flag: $$0\subset \gamma(t) \subset \gamma'(t) \subset \dots \subset V,$$ which is called the osculating flag of $\gamma$. Assume that $\dim \gamma^{(i)}(t)=r_i$ for all $t\in\gamma$. Then the local geometry of $\gamma$ is equivalent to the local geometry of the corresponding curve of osculating flags. It is easy to see that this curve is an integral curve in the flag variety $F_{r_0,r_1,\dots,r_k}(V)$, the flag variety of the group $G=SL(V)$.
More generally, if $$0 \subset V_1(t)\subset V_2(t) \subset \dots \subset V_k(t)\subset V$$ is any curve in the flag variety $F_{r_1,\dots,r_k}$, $r_i=\dim V_i(t)$, then it is an integral curve if and only if it satisfies $V_i'(t)\subset V_{i+1}(t)$ for all $i=1,\dots,k-1$. Such curves appear naturally in the geometry of system of differential equations of mixed order (i.e. with in general non-constant highest order of derivatives for different unknown functions) via the linearization procedure [@quasi; @mix1].
Consider now the simplest example when the symmetry algebra ${\mathfrak s}$ of the flat curve is not reductive in ${\mathfrak g}$. Let $\gamma$ be a curve in Grassmann variety ${\operatorname{Gr}}_2({\mathbb R}^5)$. We assume that $\gamma$ satisfies two additional non-degeneracy conditions: $$\dim\gamma'(x)=4,\ \dim\gamma''(x)=5\quad\text{for all }x\in\gamma.$$ These conditions mean that $\gamma$ can be lifted to the curve (we also denote it by $\gamma$) in the flag variety $F_{2,4}({\mathbb R}^5)$, that can be approximated by the flat curve $\gamma_0$ corresponding to the following element $x\in {\mathfrak{sl}}(5,{\mathbb R})$: $$x = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{pmatrix}.$$ Direct computation shows that the subalgebra ${\mathfrak s}$ has the form: $$\left\{
\left(\begin{smallmatrix}
b & 0 & c & 0 & 0 \\ 0 & 2b & 0 & 2c & 0 \\ a & 0 & -b & 0 & 0 \\
0 & a & 0 & 0 & 2c \\ 0 & 0 & 0 & a & -2b
\end{smallmatrix}\right)
+
\left(\begin{smallmatrix} 3d & 0 & 0 & 0 & 0 \\ 0 & -2d & 0 & 0 & 0 \\ 0 & 0 & 3d & 0 & 0 \\
0 & 0 & 0 & -2d & 0 \\ 0 & 0 & 0 & 0 & -2d \end{smallmatrix}\right)
+
\left(\begin{smallmatrix} 0 & e_0 & 0 & e_1 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & e_0 & 2e_1 \\
0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{smallmatrix}\right)\right\}.$$ Here the first summand is an ${\mathfrak{sl}}_2({\mathbb R})$-subalgebra containing an element $x$, the second summand is its centralizer and the third summand is the nilradical of ${\mathfrak s}$. Note that this is a particular case of Theorem 8.1 and Remark 8.2 of [@flags] and it is based on the theory of ${\mathfrak{sl}}_2$ representations.
Let us show that there is no ${\mathfrak s}^{(0)}$-invariant subspace $W\subset{\mathfrak p}$ complementary to ${\mathfrak s}^{(0)} + [x,{\mathfrak p}]$. Indeed, it is clear that such subspace $W$ should satisfy $W_2={\mathfrak g}_2$ and $\dim W_1=2$. Decomposing ${\mathfrak g}_1$ into the direct sum of eigenspaces with respect to the action of the 2-dimensional diagonalizable subalgebra in ${\mathfrak s}_0$, we see that $W_1$ would necessarily be spanned by the following two elements: $$w_1 = \left(\begin{smallmatrix} 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{smallmatrix}\right),\quad
w_2 = \left(\begin{smallmatrix} 0 & 0 & \alpha_1 & 0 & 0 \\ 0 & 0 & 0 & \alpha_2 & 0 \\ 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & \alpha_3 \\ 0 & 0 & 0 & 0 & 0 \end{smallmatrix}\right).$$ However, the action of ${\mathfrak s}_0$ on these 2 elements generates at least 3-dimensional space in degree 1.
Thus, in general, we can not equip a non-degenerate curve in ${\operatorname{Gr}}_2({\mathbb R}^5)$ with a natural Cartan connection as it was done for non-degenerate curves in projective spaces. However, by choosing a non-invariant complementary subspace $W$ we still can build a natural frame bundle along $\gamma$ satisfying the conditions of Theorem \[thm1\].
Curves of isotropic and coisotropic subspaces
---------------------------------------------
Let $V$ be a vector space equipped with a non-degenerate symmetric or antisymmetric form $b$. Denote by ${\mathfrak{sl}}(V,b)$ the subalgebra in ${\mathfrak{sl}}(V)$ preserving $b$. It is either $\mathfrak{sp}(V)$ or $\mathfrak{so}(V)$, but we prefer to use the unifying notation.
We shall denote by ${\operatorname{Gr}}_r(V,b)$ the variety of all $r$-dimensional isotropic subspaces in $V$. More generally, denote by $F_{r_1,r_2,\dots,r_k}(V,b)$ the variety of flags of isotropic subspaces $V_1\subset V_2 \subset \dots \subset V_k$ of dimensions $r_1,r_2,\dots, r_k$.
We shall consider curves $\gamma$ in $F_{r_1,r_2,\dots,r_k}(V,b)$ satisfying the following additional conditions: $$\begin{aligned}
V_i' & \subset V_{i+1}, i =1,\dots,k-1;\\
V_k' & \subset V_{k-1}^\perp,\quad \text{if $V_k$ is Lagrangian};\\
V_k' & \subset V_{k}^\perp,\quad \text{if $V_k$ is not Lagrangian}.\end{aligned}$$ Here by $V_i$ we mean the curve in ${\operatorname{Gr}}_{r_i}(V)$ defined by $\gamma$, and by $V_i'$ its derivative defined above.
In fact, these are exactly the types of curves that can be approximated up to first order by a flat curves $\gamma_0$ corresponding to elements $x$ of degree $-1$ of ${\mathfrak{sl}}(V,b)$ equipped with an appropriate grading. In the case of $\mathfrak{sp}(V)$ such curves appear in the geometry of nonholonomic distributions [@douzel2; @douzel3] and more general structures coming from the Control Theory [@agrzel2; @agrzel3; @zel-che; @quasi].
It is easy to check that ${\mathfrak s}$ is reductive if $r_i = is$ and $r_k = ks = \dim V/2$ for an appropriate $s$. In particular, we can find an $S^{(0)}$-invariant subspace $W$ in all these cases and equip the curve $\gamma$ with a Cartan connection (and a projective parameter) in a natural way.
In [@flags] we provide the complete classification of all possible curve types and compute their symmetry algebras for all flag varieties of classical simple Lie algebras.
$G_2$ examples {#g2-ex}
--------------
Let $G$ be a split real Lie group of type $G_2$ and ${\mathfrak g}$ be the corresponding Lie algebra. Up to conjugation there are three different connected parabolic subgroups of $G$, which can be easily described using the realization of $G$ as an automorphism group of split octonions. Let ${{\mathbb O}_s}$ be the algebra of split octonions as defined in [@humph §19.3], and let $V$ be the 7-dimensional subspace of all imaginary octonions in ${{\mathbb O}_s}$. The multiplication of imaginary octonions split into scalar and imaginary parts defines the non-degenerate symmetric bilinear form $B$ on $V$ and a skew-symmetric bilinear map $\Omega\colon \wedge^2 V \to V$. Identifying $V$ with $V^*$ by means of $B$, we can identify $\Omega$ with an element of $\wedge^2 V^*\otimes V^*$, which turns to be totally antisymmetric. Thus, the multiplication of ${{\mathbb O}_s}$ defines a 3-form $\Omega\in \wedge^3 V^*$. In fact, it is also well-known that the form $\Omega$ defines $B$ uniquely up to a constant via the following identity: $$i_{v_1}\Omega \wedge i_{v_2}\Omega \wedge \Omega = B(v_1,v_2)\operatorname{vol},$$ where $\operatorname{vol}$ is a volume form on $V$.
We identify the Lie group $G$ of type $G_2$ with the automorphism group of ${{\mathbb O}_s}$. It preserves $V$, the symmetric form $B$ and the 3-form $\Omega$ on $V$. We denote by $\Delta$ the root system of type $G_2$, which has the following Dynkin diagram: $$\cir[,\alpha_1]\lllar\cir[,\alpha_2]$$ and the set of all positive roots: $$\alpha_1, \alpha_2, \alpha_1+\alpha_2, \alpha_1+2\alpha_2,\alpha_1+3\alpha_2,2\alpha_1+3\alpha_2.$$ We denote the basis of ${\mathfrak g}$ by $\{H_1,H_2,X_\alpha : \alpha\in\Delta\}$, where $\{H_1,H_2\}$ is the basis of the Cartan subalgebra of ${\mathfrak g}$ dual to the basis $\{\alpha_1,\alpha_2\}$ of the root system $\Delta$ and $X_{\alpha}$ is any non-zero element in ${\mathfrak g}_{\alpha}$ for $\alpha\in\Delta$. We shall not need the exact values of the structure constants of ${\mathfrak g}$ in this basis.
The space $V$ of imaginary octonions in ${{\mathbb O}_s}$ is 7-dimensional, is preserved by $G$ and is a minimal non-trivial representation of $G$. We say that a subspace $W$ is an *null subalgebra* in $V$, if the product of any two elements in $W$ is equal to $0$. It is easy to see that all one-dimensional null subalgebras are exactly the subalgebras of the form $\langle c\rangle$, where $c$ is an isotropic vector in $V$ with respect to the symmetric form $B$ (or, equivalently, $i_c\Omega\wedge\i_c\Omega\wedge \Omega = 0$). It is known that $G$ acts transitively on the set of all such vectors.
Fix an isotropic vector $c\in V$ and consider its annihilator $A(c) = \{x\in V \mid xc = 0\}$. It is known to be a $3$-dimensional subalgebra of ${{\mathbb O}_s}$, which lies in $V$ and is no longer null. In terms of the 3-form $\Omega$ it can be described as: $$A(c) = \ker i_c\Omega = \{ v\in V \mid \Omega(c,v,\cdot)=0 \}.$$
In fact, we can always find an element $g_c\in G$ such that $g_c.c=c_3$, so that $g_c.A(c)$ will become the subalgebra with the basis $\langle c_3, c_7, c_8\rangle$ and the multiplication $c_7c_8=-c_8c_7 =c_3$, $c_3^2=c_7^2=c_8^2=0$. (Here we use the notation $\{c_1,\dots, c_8\}$ for the basis in ${{\mathbb O}_s}$ as given in [@humph].) Thus, we see that there are no 3-dimensional null subalgebras in $V$, and any isotropic element $c$ is contained in a one-parameter family of $2$-dimensional null subalgebras of the form $g_c^{-1}.\langle c_3, \lambda_7 c_7 + \lambda_8 c_8\rangle$, $[\lambda_1:\lambda_2]\in{{\mathbb R}\mathrm{P}}^1$.
It is easy to see that $G$ acts transitively on the set of all flags $V_1\subset V_2$, where $V_1$ and $V_2$ are both null subalgebras of dimension $1$ and $2$ respectively. Fix any of such flags. Than its stabilizer under the action of $G$ is exactly the Borel subgroup of $G$, while the stabilizers of $V_1$ and $V_2$ are two different parabolic subgroups $P_1$ and $P_2$ of $G$.
Both homogeneous spaces $M_i=G/P_i$, $i=1,2$ are 5-dimensional, while $G/B$ is 6-dimensional. Let us describe the distribution $D$ and the notion of integral curves of constant type in each of these cases.
Let us start with the homogeneous space $G/B$. First, note that any flag of null subalgebras $V_1\subset V_2$ is naturally extended to the complete flag in $V$: $$0\subset V_1\subset V_2 \subset V_3 = A(V_1) \subset V_4 = V_3^{\perp} \subset V_5 = V_2^{\perp}\subset V_6 = V_1^{\perp} \subset V.$$ As ${\mathfrak g}_{-1} = \langle X_{-\alpha_1}, X_{-\alpha_2} \rangle$, we see that $\dim D=2$, and $D = \ker d\pi_1 \oplus \ker d\pi_2$, where $\pi_i \colon G/B \to G/P_i$, $i=1,2$ are natural projections. This implies that a curve $\gamma$ of flags of null subalgebras: $$\label{curveB}
0 \subset V_1(t)\subset V_2(t) \subset V$$ is an integrable curve of $D$ if and only if it satisfies the following conditions: $$\label{intcurve}
V_1' \subset V_2,\quad V_2' \subset A(V_1).$$ It is easy to see that the action of $G$ on $PD$ has exactly 3 orbits: two are the kernels of $d\pi_i$, $i=1,2$ and one open orbit is complementary to the first two. So, all integral curves of constant type consist of the following:
1. fibers of the projection $\pi_1\colon G/B\to G/P_1$, or, in other words, curves with constant $V_1(t)$;
2. fibers of the projection $\pi_2\colon G/B\to G/P_2$, or, in other words, curves with constant $V_2(t)$;
3. *non-degenerate* curves , satisfying $V_1'=V_2$ and $V_2'=A(V_1)$.
Elementary calculations show that a non-degenerate curve additionally satisfies: $V_1'''=V_3'=V_3^{\perp}$, $V_1^{(4)}=V_2^\perp$, $V_1^{(5)}=V_1^\perp$ and $V_1^{(6)}=V$. Thus, the complete osculating flag of the curve $V_1(t)$, considered as a projective curve in $PV$, is completely determined by $V_1$ and $V_2(t)=V_1'$ via algebraic operations only.
Let us now describe how the developed theory applies to the non-degenerate integral curves in $G/B$. In this case we can take $x=X_{-\alpha_1}+X_{-\alpha_2}$. Simple computation shows that
- the subalgebra ${\mathfrak s}$ is isomorphic to ${\mathfrak{sl}}(2,{\mathbb R})$ with the basis $x$, $h=H_1+H_2$ and $y=X_{\alpha_1}+X_{\alpha_2}$;
- the space $({\mathfrak s}+[x,{\mathfrak p}])\cap {\mathfrak p}$ coincides with $\sum_{i=1}^4 {\mathfrak g}_i$;
- the subspace $W={\mathfrak g}_5$ is ${\mathfrak s}^{(0)}$-invariant and is complementary to ${\mathfrak s}+[x,{\mathfrak p}]$.
Hence, applying Theorem \[thm1\] in this case, we see that we can canonically associate a projective connection and a W-valued fundamental invariant to each non-degenerate integral curve $\gamma$ on $G/B$. The projective connection defines a canonical projective parametrization $\tau$ on the curve $\gamma$, while fundamental invariant can be written as $f(\tau)(d\tau)^6$ and completely determines the equivalence class of $\gamma$.
Next, consider curves $\gamma$ in the homogeneous space $G/P_1$ of all $1$-dimensional null subalgebras in $V$. We shall write $\gamma=V_1(t)$. The distribution $D$ is 2-dimensional in this case and can be written as $D_{V_1} = A(V_1)/V_1$ for each point $V_1\in G/P_1$. Thus, we see that a curve $V_1(t)$ is integrable if and only if it satisfies $V_1'\subset A(V_1)$. We assume that $V_1(t)$ does not degenerate to a point. Then $V_2(t) = V_1'(t)$ is a 2-dimensional subspace in $A(V_1(t))$ for each $t\in\gamma$. As it also contains $V_1(t)$, we see that $V_2(t)$ is necessarily an null subalgebra. Simple computations show that $V_2'\subset A(V_1)$, and, hence, the curve $\gamma\subset G/P_1$ is naturally lifted to $G/B$. If, moreover, $V_2'=A(V_1)$, then the lifted curve is non-degenerate, and we can apply the above constructions to equip $\gamma$ with a canonical projective invariant and a single fundamental invariant.
Finally, consider curves $\gamma$ in the homogeneous space $G/P_2$ of all $2$-dimensional null subalgebras in $V$. In this case ${\mathfrak g}_0\equiv{\mathfrak{gl}}(2,{\mathbb R})$, and the ${\mathfrak g}_0$-module ${\mathfrak g}_{-1}$ is equivalent to $S^3({\mathbb R}^2)$. In particular, $\dim D = 4$, and for each $V_2\in G/P_2$ the space $D_{V_2}$ can be identified with a 4-dimensional subspace in ${\operatorname{Hom}}(V_2, V_2^{\perp}/V_2)$. A curve $V_2(t)$ in $G/P_2$ is integral if and only if $V_2'\subset V_2^\perp$.
The action of $G$ on $PD$ has 4 different orbits corresponding to the orbits of $GL(2,{\mathbb R})$ action on $S^3({\mathbb R}^2)$, that is on the space of cubic polynomials:
1. polynomials with triple root;
2. polynomials with one simple and one double root;
3. polynomials with one real and 2 complex roots;
4. polynomials with 3 different real roots.
We shall call integral curves, whose tangent lines belong to the first orbit *special* curves. Let us describe them in more detail. Namely, for each null subalgebra $V_2\subset V$ we define the embedding of $PV_2$ into $PD_{V_2}$ as follows: $$PV_2\to PD_{V_2},\quad V_2 \supset V_1 \mapsto (V_2/V_1)^* \otimes A(V_1)/V_2.$$
This embedding defines a field of rational normal curves in $PD$. This cone is exactly the orbit (1) of the action of $G$ on $PD$. In particular, we see that a curve $\gamma=V_2(t)$ is special, if and only if $V_2'$ is 3-dimensional. In this case it is equal to $A(V_1(t))$, where $V_1 \subset V_2(t)$ is defined uniquely by the condition $V_1'\subset V_2$. Thus, as in the case of curves in $G/P_1$, any special curve in $G/P_2$ is naturally lifted to the curve in $G/B$. If, in addition, we suppose that $V_1'=V_2$, then this lift is a non-degenerate curve, and we can naturally equip $\gamma$ with canonical projective parameter and a single fundamental differential invariant.
Consider now the case, when $\gamma\subset G/P_2$ is an integral curve of constant type, and its tangent lines belong to the orbit (2) above. Then we can no longer lift this curve to an integral curve on $G/B$ and have to deal with it on the homogeneous space $G/P_2$ itself. We can take the corresponding element $x\in{\mathfrak g}_{-1}$ as $X_{-\alpha_1-\alpha_2}$. Direct computation shows that:
- the subalgebra ${\mathfrak s}$ is isomorphic to ${\mathfrak{sl}}(2,{\mathbb R})+{\mathfrak{st}}(2,{\mathbb R})$ with the basis $x$, $H_1, H_2$, $X_{\alpha_1+\alpha_2}$ and $X_{\alpha_1+3\alpha_2}$ (here by ${\mathfrak{st}}(2,{\mathbb R})$ we denote the 2-dimensional subalgebra of ${\mathfrak{sl}}(2,{\mathbb R})$ consisting of upper-triangular matrices);
- the space $({\mathfrak s}+[x,{\mathfrak p}])\cap {\mathfrak p}$ has codimension 2 in ${\mathfrak p}$ and is concentrated in degrees $0$ and $1$;
- the subspace $W=\langle X_{\alpha_1}, X_{2\alpha_1+3\alpha_2}\rangle$ is ${\mathfrak s}^{(0)}$-invariant and is complementary to ${\mathfrak s}+[x,{\mathfrak p}]$.
Thus, we see that in this case we get a natural projective connection and a $W$-valued fundamental invariant on $\gamma$. Note that the ${\mathfrak s}^{(0)}$-invariant subspace $W$ exists in this case, even though the subalgebra ${\mathfrak s}$ is not reductive in ${\mathfrak g}$.
Assume now that $\gamma\subset G/P_2$ is an integral curve of constant type, and its tangent lines belong to the orbit (3) above. We can take the corresponding element $x\in{\mathfrak g}_{-1}$ as $X_{-\alpha_1}+X_{-\alpha_1-3\alpha_2}$. Direct computation shows that:
- the subalgebra ${\mathfrak s}$ is isomorphic to ${\mathfrak{sl}}(2,{\mathbb R})$;
- the space $({\mathfrak s}+[x,{\mathfrak p}])\cap {\mathfrak p}$ has codimension 3 in ${\mathfrak p}$ and is concentrated in degrees $0$ and $1$;
- the subspace $W=\langle X_{2\alpha_1+3\alpha_2}, X_{\alpha_1+\alpha_2}, X_{\alpha_1+2 \alpha_2}\rangle$ is ${\mathfrak s}^{(0)}$-invariant and is complementary to ${\mathfrak s}+[x,{\mathfrak p}]$.
Thus, we see that we can again associate a natural projective connection on $\gamma$ and 3 scalar fundamental invariants on top of it.
The case of the orbit of type (4) can be considered in the same way and leads to the same results, as in case of orbit (3).
Conformal, symplectic and $G_2$-structures on the solution space of an ODE {#sec:app}
==========================================================================
Compatibility of projective curves with symmetric and skew-symmetric bilinear forms
-----------------------------------------------------------------------------------
As one of the applications of the theory we find the explicit conditions when the solution space of an ODE of order $k\ge 3$ carries a natural symplectic, conformal, or a $G_2$-structure.
As in Subsection \[ssec:41\] consider a non-degenerate curve $\gamma$ in ${{\mathbb R}\mathrm{P}}^k$. Its dual curve $\gamma^* \subset {{\mathbb R}\mathrm{P}}^{k,*}$ is defined as a curve of osculating hyperplanes $\gamma^{(k-2)}$. The curve $\gamma$ is called self-dual, if there is a projective transformation from ${{\mathbb R}\mathrm{P}}^k$ to ${{\mathbb R}\mathrm{P}}^{k,*}$ that maps $\gamma$ to $\gamma^*$. Wilczynski [@wilch] proved that a curve is self-dual if and only if its Wilczynski invariants $\theta_{2i+1}$, $i=1,\dots,[k/2]$ vanish identically.
Let as give a simple proof of this result using the developed techniques as well as establish when a projective curve $\gamma$ in ${{\mathbb R}\mathrm{P}}^6$ defines a unique $G_2$ subalgebra in ${\mathfrak{gl}}(7,{\mathbb R})$.
Let $V$ be an arbitrary finite-dimensional vector space of dimension $k+1$ and let $P(V)$ denote its projectivization. We say that a non-degenerate curve $\gamma\in P(V)$ is compatible with a non-degenerate bilinear form $b\colon V\times V\to {\mathbb R}$ if: $$\label{b}
b(\gamma^{(i)},\gamma^{(k-2-i)}) = 0\quad\text{for all }i=0,\dots,k-2.$$
Let us show that the existence of such $b$ is equivalent to the above definition of self-duality. Indeed, if a curve $\gamma$ is compatible with a bilinear form $b$, then this form $b$ considered as a map from $V$ to $V^*$ defines a projective transformation that maps $\gamma$ to its dual curve $\gamma^{(k-2)}$.
Similarly, any projective transformation from $P(V)$ to $P(V^*)$ mapping $\gamma$ to $\gamma^{(k-2)}$ defines a bilinear from on $V$ such that $b(\gamma,\gamma^{(k-2)})=0$. This also implies that $b(\gamma,\gamma^{(k-3)})=0$. Differentiating this equality, we immediately get $b(\gamma',\gamma^{(k-3)})=0$. Proceeding in the same way we prove that the curve $\gamma$ satisfies .
\[lem:dual\] A non-degenerate curve $\gamma\subset P(V)$ is compatible with a non-degenerate symmetric ($\dim V$ is odd) or skew-symmetric form ($\dim V$ is even) if and only if all Wilczynski invariants $\theta_{2i+1}$, $i=1,\dots, [(\dim V-1)/2]$ of $\gamma$ vanish identically.
The condition can be interpreted as saying that the flag $$0\subset\gamma\subset \gamma' \subset \dots \subset \gamma^{([\frac{k-1}{2}])}$$ is isotropic with respect to $b$ and the spaces $\gamma^{(i)}$ with $i>[\frac{k-1}{2}]$ are exactly the orthogonal complements to $\gamma^{k-2-i}$ with respect to $b$.
Denote by $SL(V,b)$ the subgroup of $SL(V)$ preserving the form $b$. That is $SL(V,b)$ is either the group of orthogonal or symplectic matrices depending on the symmetry index of $b$. It is well-known that the group $SL(V,b)$ acts transitively on flags of isotropic subspaces. Thus, if the curve is compatible with a bilinear form $b$, then we can consider it as a curve in the flag variety of the group $SL(V,b)$. Denote by ${\mathfrak{sl}}(V,b)$ the subalgebra of ${\mathfrak{sl}}(V)$ corresponding to $SL(V,b)$.
Let $P\subset SL(V,b)$ be the canonical moving frame for this curve corresponding to the subspace $\widetilde W\subset {\mathfrak{sl}}(V,b)$ defined by . Note that the Killing form of ${\mathfrak{sl}}(V,b)$ coincides up to a constant with the restriction of the Killing form of ${\mathfrak{sl}}(V)$. Therefore, the same formula in case of ${\mathfrak{sl}}(V)$ defines the subspace $W$ containing $\widetilde W$. Hence, the moving frame $P$ is at the same time the canonical frame of $\gamma$ viewed as a curve in the corresponding (larger) flag variety of the group $SL(V)$.
Recall (see Subsection \[ssec:41\]) that the space $W$ is spanned by $y^i$, $i=2,\dots,k$, where $\{x,h,y\}$ is the standard basis of ${\mathfrak{sl}}(2,{\mathbb R})$ embedded irreducibly into ${\mathfrak{sl}}(V)$. Decomposing ${\mathfrak{sl}}(2,{\mathbb R})$-module ${\mathfrak{sl}}(V,b)$ into the sum of irreducible representations, we see that $y^i$ lies in ${\mathfrak{sl}}(V,b)$ if and only if $i$ is odd. Then equation implies that the canonical moving frame for the curve $\gamma$ takes values in ${\mathfrak{sl}}(V,b)$ if and only if all Wilczynski invariants $\theta_{2i+1}$ vanish identically.
Compatibility of projective curves with $G_2$-structures
--------------------------------------------------------
As in Subsection \[g2-ex\], let $V\subset {{\mathbb O}_s}$ be the 7-dimensional vector space of imaginary split octonions, $G_2$ a subgroup of $GL(V)$ and let $\Omega\in \wedge^3 V^*$ be the $G_2$-invariant 3-form on $V$ that corresponds to the imaginary part of multiplication of imaginary octonions. In fact, $G_2$ coincides with the stabilizer of $\Omega$ under the natural action of $GL(V)$ on $\wedge^3 V^*$.
It is well-known that the action of $GL(7,{\mathbb R})$ on the space of all $3$-forms has 2 open orbits. One of these orbits has a compact form of $G_2$ as a stabilizer (it corresponds to the multiplication tensor of the imaginary octonions) and another has a split real form of $G_2$ as a stabilizer (it corresponds to the multiplication tensor of split octonions).
We shall call a 3-form on ${\mathbb R}^7$ *a $G_2$-structure* if it lies in the orbit of $\Omega$ under the standard action of $GL(V)$ on $\wedge^3 V^*$, or, what is the same, if its stabilizer is conjugate to the split real form of $G_2$ in $GL(V)$. Abusing notation, we shall denote $G_2$-structures by the same letter $\Omega$.
We say that a non-degenerate curve $\gamma\subset P^6=P(V)$ is compatible with a $G_2$-structure $\Omega$, if $$\Omega(\gamma, \gamma'', \cdot) = 0.$$ In terms of Subsection \[g2-ex\] this means that $\gamma$ is a non-degenerate integral curve on the generalized flag variety $G_2/B$, where $B$ is a Borel subgroup of $G_2$.
Using the same argument as in Lemma \[lem:dual\], we easily prove that the curve $\gamma$ is compatible with a 3-form $\Omega$, if and only if the canonical frame of $\gamma$ lies in the stabilizer of $\Omega$. It is easy to see that the irreducible subalgebra ${\mathfrak{sl}}(2,{\mathbb R})\subset {\mathfrak{sl}}(7,{\mathbb R})$ is contained in a unique split form of the Lie algebra $G_2$. Moreover, this real form of $G_2$ viewed as an ${\mathfrak{sl}}(2,{\mathbb R})$-module is decomposed into the 3-dimensional and 11-dimensional irreducible submodules. The first one is ${\mathfrak{sl}}(2,{\mathbb R})$ itself, and the 11-dimensional submodule has $y^5$ as its highest weight vector. (Here, as above, we denote by $\{x,h,y\}$ the standard basis of ${\mathfrak{sl}}(2,{\mathbb R})$ viewed as an irreducible subalgebra of ${\mathfrak{sl}}(7,{\mathbb R})$.) Thus, similar to Lemma \[lem:dual\] we get:
A non-degenerate curve $\gamma\subset P^6$ is compatible with a $G_2$-structure if and only if all its Wilczynski invariants except for $\theta_6$ vanish identically.
Geometric linearization of ordinary differential equations
----------------------------------------------------------
Classical linearization procedure associates a linear differential equation with each solution of a non-linear equation $\mathcal E$. This equation, called the linearization along a given solution, describes all possible first order deformations of the solution that still satisfy the original differential equation up to the first order of deformation parameter. In more detail, let $$\label{ode}
y^{(k+1)}=f(x,y,y',\dots,y^{(k)})$$ be an arbitrary ordinary differential equation ${\mathcal E}$ of order $(k+1)$ solved with respect to the highest derivative. Suppose $\bar y(x)$ is an arbitrary solution of . Consider its deformation $y(x) = \bar y(x) + {\varepsilon}z(x)$, substitute it to and expand it with respect to ${\varepsilon}$. As $\bar y$ is a solution, the zero-order term will vanish identically. The first order term will define the linear equation on $z(x)$: $$\label{linode}
z^{(k+1)} = \frac{\partial f}{\partial y} z + \frac{\partial f}{\partial y'} z' + \dots + \frac{\partial f}{\partial y^{(k)}} z^{(z)},$$ where all partial derivatives $\partial f/\partial y^{(i)}$, $i=0,\dots,k$, are evaluated at the solution $\bar y(x)$. Equation is called *the linearization* of the equation along the solution $\bar y$. The solution space of the linearization can be considered as the tangent space to the solution space $\operatorname{sol}({\mathcal E})$ of the original equation ${\mathcal E}$ given by .
As it was originally discovered by Wilczynski [@wilch], the geometry of linear differential equations is equivalent to the projective geometry of curves. In the modern language this equivalence can be formalized as follows. Let $M\to {\operatorname{Gr}}_k(V)$ be an embedding of a smooth manifold $M$ into the Grassmann variety ${\operatorname{Gr}}_k(V)$. Let $\pi\colon \mathbb E\to {\operatorname{Gr}}_k(V)$ be the hyperplane vector bundle over ${\operatorname{Gr}}_k(V)$, which is dual to the canonical one, i.e. $\mathbb E_W=W^*$ for each $W\in {\operatorname{Gr}}_k(V)$. Each element $\alpha\in V^*$ naturally defines the global section $s_\alpha$ of $\mathbb E$. Let $E$ be the restriction of $\mathbb E$ on $M$, and let $S\subset\Gamma(E)$ be the restriction of all sections $s_\alpha$ to $M$. Then $S$ is a well-defined finite-dimensional subspace in the space of sections of $E$. Prolonging elements of $S$ to the jet spaces $J^i(E)$, we can construct a differential equation $\mathcal E\subset J^r(E)$ whose solution space coincides with $S$.
In other direction, let $\pi\colon E\to M$ be an arbitrary $k$-dimensional vector bundle over the manifold $M$ and let $\mathcal E\subset J^r(E)$ be a finite-type linear differential equation on sections of $E$. Let $S=\operatorname{sol}(\mathcal E)$ be the solution space of $\mathcal E$. Let $V=S^*$ and define the embedding $M\to {\operatorname{Gr}}_k(V)$ as follows: $$x\mapsto \{s\in S \mid s(x) = 0\}^{\perp} \subset V.$$ This correspondence between finite-type linear differential equations and embedded submanifolds in Grassmann varieties is extensively used in work of Se-ashi [@se-ashi] for describing the invariants of linear systems of ordinary differential equations.
Geometric structures on the solution space of an ODE
----------------------------------------------------
Let $M={\operatorname{sol}}({\mathcal E})$ be the solution space of the equation . For any solution $\bar y\in M$ we get the curve $\gamma\colon {\mathbb R}\to PT_{\bar y}(M)$ defined by the linearization of the equation at the solution $\bar y$. Wilczynski invariants $\theta_i$, $i=3,\dots,k+1$ for each such linearization define the so-called *generalized Wilczynski invariants* $\Theta_i$, $=3,\dots,k+1$ of the initial non-linear ODE. See [@dou:08] for more details.
Recall that a *pseudo-conformal structure* on a smooth manifold $M$ is defined as a family of pseudo-Riemannian metrics on $M$ proportional to each other by non-zero functional factor. Similarly, we define the *pseudo-symplectic structure* on $M$ as a family of non-degenerate 2-forms on $M$ proportional to each other by non-zero functional factor. Finally, *a $G_2$-structure* on $M$ is defined as a differential 3-form $\Omega$ such that $\Omega_p$ defines a $G_2$-structure on each tangent space $T_pM$.
We say that a pseudo-symplectic, pseudo-conformal or $G_2$-structure on the solution space $M$ of the given ODE is *natural*, if it is compatible with linearizations $\gamma \to PT_\gamma(M)$ for each solution $\gamma$. Using the above results on the compatibility of projective curves with symmetric and skew-symmetric bilinear forms and $G_2$-structures, we immediately get the following result.
\[thm:2\] Let $y^{(k+1)}=f(x,y,y',\dots,y^{(k)})$ be an ordinary differential equation of order $k+1\ge 3$. Its solution space is admits a natural:
1. pseudo-conformal structure if and only if $k$ is odd and all generalized Wilczynski invariants $\Theta_{2i+1}$ vanish identically;
2. pseudo-symplectic structure if and only if $k$ is even and all generalized Wilczynski invariants $\Theta_{2i+1}$ vanish identically;
3. $G_2$-structure if and only if $\Theta_3=\Theta_4=\Theta_5=\Theta_7=0$.
In all these cases such structure is uniquely defined.
Generalizations and discussion {#sec:5}
==============================
Parametrized curves
-------------------
The paper is easily generalized to the case of parametrized curves. In this case one needs to change the definition of the subalgebra ${\mathfrak s}$ as follows: $$\begin{aligned}
{\mathfrak s}_{-i} & = 0,\quad i \ge 0;\\
{\mathfrak s}_{-1} & = \langle x \rangle;\\
{\mathfrak s}_0 &= \{ u \in{\mathfrak g}_0 \mid [x,u] = 0 \};\\
{\mathfrak s}_{i+1} &= \{ u \in {\mathfrak g}_{i+1} \mid [x,u]\subset {\mathfrak s}_i \}.\end{aligned}$$ We also have to consider tangent spaces to a parametrized curve $\gamma$ as elements of ${\mathfrak g}/{\mathfrak h}$ instead of one-dimensional subspaces. As a result, we get a result similar to Theorem \[thm1\], which is applicable to all parametrized curves $\gamma$ such that their images (treated as unparametrized curves) can be approximated by $\gamma_0$ up to the first order.
Curves in general parabolic geometries
--------------------------------------
By a parabolic geometry modeled by $G/P$ we understand the principal $P$-bundle $\pi\colon{\mathcal G}\to M$ equipped with a 1-form $\omega\colon T{\mathcal G}\to {\mathfrak g}$ such that:
1. $\omega_p$ is an isomorphism of vector spaces for any $p\in {\mathcal G}$;
2. $\omega (X^*) = X$ for any fundamental vector field $X^*$ on ${\mathcal G}$ corresponding to an element $X\in {\mathfrak p}$;
3. $R_g^*\omega = {\operatorname{Ad}}g^{-1}\omega$ for any right shift $R_g$ defined by the action of an element $g\in P$.
The classical examples of parabolic geometries include the projective and conformal structures on smooth manifolds. The tangent space $TM$ is naturally identified with ${\mathcal G}\times_P ({\mathfrak g}/{\mathfrak p})$ and, as in the case of a homogeneous space $G/P$, carries a natural distribution $D\subset TM$ corresponding to the subspace $({\mathfrak g}_{-1}+{\mathfrak p})/{\mathfrak p}$. In particular, we can also introduce the notion of an integral curve $\gamma$ in $M$. Using again the identification $TM\equiv {\mathcal G}\times_P({\mathfrak g}/{\mathfrak p})$ we can define the notion of a curve of constant type $x$ for any $x\in {\mathfrak g}_{-1}$.
The notion of a moving frame along $\gamma$ can be naturally generalized to any parabolic geometry as an arbitrary subbundle of a $P$-bundle $\pi^{-1}(\gamma)\to\gamma$.
Theorem \[thm1\] is also immediately generalized as follows:
\[thm1b\] Let $(M,{\mathcal G},{\omega})$ be a parabolic Cartan geometry modeled by a generalized flag variety $G/P$ and let $x$ be an arbitrary non-zero element in ${\mathfrak g}_{-1}$. Fix any graded subspace $W\subset{\mathfrak p}$ complementary to ${\mathfrak s}^{(0)} + [x,{\mathfrak p}]$.
Then for any integral curve $\gamma$ in $M$ of type $x$ there is a canonical frame bundle $E\subset {\mathcal G}$ along $\gamma$ such that for each point $p\in E$ the subspace ${\omega}(T_pE)$ is $W$-normal.
If, moreover, $W$ is $S^{(0)}$-invariant, then $E$ is a principal $S^{(0)}$-bundle and ${\omega}|_{E}$ decomposes as a sum of the $W$-valued 1-form ${\omega}_W$ and the ${\mathfrak s}$-valued 1-form ${\omega}_{{\mathfrak s}}$, where ${\omega}_{\mathfrak s}$ is a flat Cartan connection on $\gamma$ defining a canonical projective parametrization of $\gamma$, and ${\omega}_W$ is a vertical $S^{(0)}$-equivariant 1-form.
Submanifolds of higher dimension in generalized flag varieties
--------------------------------------------------------------
We briefly outline how the results of this paper are generalized to the submanifolds of arbitrary dimension in generalized flag varieties (but not in curved parabolic geometries).
For this purpose the element $x\in{\mathfrak g}_{-1}$ needs to be replaced by an arbitrary commutative subalgebra ${\mathfrak{x}}\subset {\mathfrak g}_{-1}$. We say that a submanifold $X\subset G/P$ has *constant type ${\mathfrak{x}}$*, if for any point $p\in X$ there exists an element $g\in G$, such that $g.o=p$ and $g_*({\mathfrak{x}})=T_pX$. In other words, $X$ is of type ${\mathfrak{x}}$, if all tangent spaces $T_pX$, $p\in X$ belong to the orbit of the subspace ${\mathfrak{x}}\in T_oM$ under the natural action of $G$ on ${\operatorname{Gr}}_r(TM)$, where $r=\dim {\mathfrak{x}}$.
As in case of unparametrized curves, we can define a graded subalgebra ${\mathfrak s}\subset{\mathfrak g}$ by $$\begin{aligned}
{\mathfrak s}_{-1} &= {\mathfrak{x}};\\
{\mathfrak s}_{i} &= \{ u \in {\mathfrak g}_{i} \mid [u,{\mathfrak{x}}]\subset {\mathfrak s}_{i-1} \}\quad\text{for all }i\ge 0.\end{aligned}$$ Then according to [@dk] ${\mathfrak s}$ is exactly the symmetry algebra of the orbit of the commutative subgroup $\exp({\mathfrak{x}})\subset G$ through the origin $o=eP$. We call this orbit the *flat* submanifold of type ${\mathfrak{x}}$.
Denote by $C^k({\mathfrak s}_{-1},{\mathfrak g}/{\mathfrak s})$ the standard cochain complex corresponding to the ${\mathfrak s}$-module ${\mathfrak g}/{\mathfrak s}$. The grading of ${\mathfrak g}$ naturally extends to the gradings of all spaces $C^k({\mathfrak s}_{-1},{\mathfrak g}/{\mathfrak s})$, as well as their cochain and coboundary subspaces $Z^k({\mathfrak s}_{-1},{\mathfrak g}/{\mathfrak s})$ and $B^k({\mathfrak s}_{-1},{\mathfrak g}/{\mathfrak s})$. Denote by $Z_{+}^k({\mathfrak s}_{-1},{\mathfrak g}/{\mathfrak s})$ and $B_{+}^k({\mathfrak s}_{-1},{\mathfrak g}/{\mathfrak s})$ their subspaces spanned by elements of positive degree.
Let $W$ be an arbitrary graded subspace in $Z^1_+({\mathfrak s}_{-1},{\mathfrak g}/{\mathfrak s})$ complementary to $B^1_+({\mathfrak s}_{-1},{\mathfrak g}/{\mathfrak s})$. Generalizing the above notion of $W$-normality, we say that a subspace $U\subset {\mathfrak g}$ is $W$-normal (with respect to ${\mathfrak s}$), if
1. ${\operatorname{gr}}U = {\mathfrak s}$;
2. there is a map $\chi\colon {\mathfrak s}_{-1}\to {\mathfrak g}$ such that the corresponding quotient map $\bar \chi\colon {\mathfrak s}_{-1}\to {\mathfrak g}/{\mathfrak s}$ lies in $W$ and $x+\chi(x)\in U$ for all $x\in{\mathfrak s}_{-1}$.
As in the case $\dim {\mathfrak s}_{-1}=1$, it is easy to check that the element $\bar\chi\in W$ is uniquely defined for any $W$-normal subspace $U\in {\mathfrak g}$.
Then we can prove that for any submanifold $X$ of type ${\mathfrak{x}}={\mathfrak s}_{-1}$ there is a unique moving frame $E\subset G$ along $X$ such that $\omega(E_p)$ is a $W$-normal subspace for any point $p\in X$. In particular, we see that if $H^1_+({\mathfrak s}_{-1},{\mathfrak g}/{\mathfrak s})=0$, then any submanifold of type ${\mathfrak s}_{-1}$ is locally equivalent to the flat one.
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{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We propose two types of universal codes that are suited to two asymptotic regimes when the output alphabet is possibly continuous. The first class has the property that the error probability decays exponentially fast and we identify an explicit lower bound on the error exponent. The other class attains the epsilon-capacity the channel and we also identify the second-order term in the asymptotic expansion. The proposed encoder is essentially based on the packing lemma of the method of types. For the decoder, we first derive a Rényi-divergence version of Clarke and Barron’s formula the distance between the true distribution and the Bayesian mixture. This result is of independent interest and was not required in the author’s previous paper. The universal decoder is stated in terms of this formula and quantities used in the information spectrum method. The methods contained herein allow us to analyze universal codes for channels with continuous and discrete output alphabets in a unified manner, and to analyze their performances in terms of the exponential decay of the error probability and the second-order coding rate.'
author:
- 'Masahito Hayashi[^1][^2]'
title: Universal channel coding for general output alphabet
---
Universal coding; information spectrum; Bayesian; method of types
Introduction
============
In wireless communication, the channel is described with continuous output alphabet, e.g., additive white Gaussian noise (AWGN) channel and Gaussian fading channel. In these cases, it is not so easy to identify the channel even although the channel is stationary and memoryless. Then, it is needed to make a code that achieves good performances for any channel in a set of multiple stationary and memoryless channels, e.g. a set of MIMO (multiple-input and multiple-output) Gaussian channels (e.g. [@A; @de4; @B; @C; @D]). More precisely, it is desired to construct a code that works well for any stationary and memoryless channel in a given parametric family of possible single-antenna/multi-antenna AWGN channels for a real wireless communication alphabet.
In the discrete case, to resolve this problem, Csiszár and Körner[@CK] proposed universal channel coding by employing the method of types. Since their code construction depends only on the input distribution and the coding rate, it does not depend on the form of the channel, which is a remarkable advantage. They also provide an explicit form of a lower bound of the exponential decreasing rate of the decoding error probability. However, their method works only when input and output alphabets have finite cardinality. Hence, their method cannot be applied to any continuous output alphabet while several practical systems have a continuous output alphabet. Indeed, even in the continuous output case, universal channel codes have been discussed for MIMO Gaussian channels [@C; @D; @E; @F], in which, this problem was often discussed in the framework of compound channel. Although the studies [@Do; @D; @E] did not cover the general discrete memoryless case, the paper [@F] covered the MIMO Gaussian channels as well as the general discrete memoryless case[^3]. However, they did not provide any explicit form of a lower bound of the exponential decreasing rate of the decoding error probability. Therefore, it is desired to invent a universal channel code satisfying the following two conditions. (1) The universal channel code can be applied to the discrete memoryless case and the continuous case in a unified way. (2) The universal channel code has an explicit form of a lower bound of the exponential decreasing rate of the decoding error probability.
Even in the discrete case, Csiszár and Körner[@CK]’s analysis is restricted to the case when the transmission rate is strictly smaller than the capacity. When the transmission rate equals the capacity, the asymptotic minimum error depends on the second-order coding rate [@Strassen; @Hsec; @Pol]. For the fixed-length source coding and the uniform random number, this kind of analysis was done in [@H-source]. Recently, with the second order rate, Polyanskiy [@G] addressed a universal code in the framework of compound channel. Also the papers [@YN1; @YN2; @H; @I] addressed the optimal second order rate for the mixed channel. However, no study discussed the universal channel coding with the second-order coding rate for the continuous case although the case of fixed continuous channels was discussed with the higher order analysis including the second-order analysis by Moulin [@Moulin]. Hence, it is also desired to propose a universal channel code working with the second-order coding rate even with the continuous case. Further, the universal coding with the second order analysis has another problem as follows. The conventional second order analysis [@Strassen; @Hsec; @Pol] has meaning only when the mutual information is the first order coding rate. However, the set of such channels has measure zero. So, such a analysis might be not so useful when we do not know the transition matrix of the channel.
In this paper, we deal with the universal coding with a general output alphabet (including the continuous case) and derive the exponential decreasing rate of the average error probability and the second order analysis. Further, to resolve the above problem for the second order analysis, we introduce the perturbation with the order $O(\frac{1}{n^{\frac{1}{2}}})$ of the channel. Under this perturbation, we derive the second order analysis. This order is the maximum order to achieve the asymptotic constant average error probability. Indeed, even if the input alphabet is continuous, we usually use only a finite subset of the input alphabet for encoding. Hence, it is sufficient to realize a universal channel code for the case when the input alphabet is finite and the output alphabet is continuous. In a continuous alphabet, we need an infinite number of parameters to identify a distribution when we have no assumption for the true distribution. In statistics, based on our prior knowledge, we often assume that the distribution belongs to a certain parametric family of distributions as in [@F]. In particular, an exponential family is often employed as a typical example. This paper adopts this typical assumption. That is, we consider several assumptions. One is that the output distribution $P_{Y|X=x}$ belongs to an exponential family on a general set ${\cal Y}$ for each element $x$ of a given finite set ${\cal X}$. As explained in Example \[ex1\], this assumption covers the usual setting with finite-discrete case because the set of all distributions on a given finite-discrete set forms an exponential family. This assumption also covers the Gaussian fading channel and the multi-antenna Gaussian channel as addressed in Examples \[ex2\] and \[ex3\]. Under this assumption, we provide a universal code with an explicit form of a lower bound of the exponential decreasing rate of the decoding error probability.
To construct our universal encoder, we employ the method given for the quantum universal channel coding [@Ha1], in which, the packing lemma [@CK] is employed independently of the output alphabet. That is, the paper [@Ha1] showed that the encoder given by the packing lemma can simulate the average of the decoding error probability under the random coding. Since the method of [@Ha1] does not depend on the output alphabet, it well works with a continuous alphabet.
To construct our universal decoder, we focus on the method given for the quantum universal channel coding [@Ha1]. The paper [@Ha1] considered a universally approximated output distribution by employing the method of types in the sense of the maximum relative entropy. Then, it employed the decoder constructed by the information spectrum method based on the approximating distributions. However, in a continuous alphabet, we cannot employ the method of types so that it is not so easy to give a universally approximated output distribution in the sense of maximum relative entropy. That is, we cannot directly employ the method in the paper [@Ha1].
To resolve this problem, we focus on Clarke and Barron formula [@CB] that shows that the Baysian average distribution well approximates any independent and identical distribution in the sense of Kullback-Leibler divergence. Their original motivation is rooted in universal data compression. Its quantum extension was shown in [@Ha2]. However, they did not discuss the $\alpha$-Rényi divergence. In this paper, we evaluate the quality of this kind of approximation in terms of the $\alpha$-Rényi divergence. Then, we apply the decoder constructed by the information spectrum method based on the Baysian average distributions. Modifying the method given in [@Ha1], we derive our lower bound of the error exponent of our universal code. We also derive the asymptotic error of our universal code with the second-order coding rate.
The remaining part of this paper is the following. Section \[s2\] explains notations, our assumptions, our formulation and obtained results. In this section, we provide three examples for the channels whose output distributions form an exponential family. In Section \[s3\], we give notations for the method of types. This part is the same as the previous paper [@Ha1]. In Section \[s4\], based on the result by Clarke and Barron [@CB], we derive an $\alpha$-Rényi divergence version of Clarke and Barron formula as another new result. In Section \[s5\], we prove our lower bound of our error exponent. In Section \[s8\], we prove the universal achievability for the second order sense.
Notations and Main result
=========================
Information quantities
----------------------
We focus on an input alphabet ${\cal X}:=\{1, \ldots,d\}$ with finite cardinality and an output alphabet ${\cal Y}$ that may have infinite cardinality and is a general measurable set. Then, we consider a parametric family of channels $\{W_\theta\}_{\theta\in \Theta }$ from ${\cal X}$ to ${\cal Y}$ with a finite-dimensional parameter set $\Theta \subset \mathbb{R}^k$. Firstly, we assume that a channel $W_{\theta}=(W_{\theta,x})$ from ${\cal X}$ to ${\cal Y}$, where $W_{\theta,x}$ are absolutely continuous with respect to a measure $dy$.
When a distribution on ${\cal X}$ is given by a probability distribution $P$, and a conditional distribution on a set ${\cal Z}$ with the condition on ${\cal X}$ is given by $V$, we define the joint distribution $V \times P$ on ${\cal X} \times {\cal Z}$ by $V \times P(x,z):=V(z|x)P(x)$, and the distribution $V \cdot P$ on ${\cal Z}$ by $V \cdot P(z):=
\sum_x V(z|x)P(x)$. We denote the expectation under the distribution $P$ by $E_P[~]$. Throughout this paper, the base of the logarithm is chosen to be $e$. For two distributions $P$ and $Q$ on ${\cal Y}$, we define the Kullback-Leibler divergence $D(P\|Q):= E_{P}[\log \frac{P(X)}{Q(X)}]$, and the value $s D_{1+s}(P\|Q):= \log E_{P}[(\frac{P(X)}{Q(X)})^{s}]$ for $s >-1 $ when these expectations $E_{P}[\log \frac{P(X)}{Q(X)}]$ and $E_{P}[(\frac{P(X)}{Q(X)})^{s}]$ exist. The function $s \mapsto s D_{1-s}(P\|Q)$ is a concave function for $s \in [0,1]$ because $-s D_{1-s}(P\|Q)$ can be regarded as a cummulant generating function of $- \log \frac{P(X)}{Q(X)}$. For $s \in [-1,0) \cup (0,\infty)$, the Rényi divergence $D_{1+s}(P\|Q)$ is defined as $D_{1+s}(P\|Q):=\frac{s D_{1+s}(P\|Q)}{s}$. Here, the function $s D_{1+s}(P\|Q)$ is defined for $s=0$, but the Rényi divergence $D_{1+s}(P\|Q)$ is not necessarily defined for $s=0$.
Given a channel $W$ from ${\cal X}$ to ${\cal Y}$ and a distribution $P$ on ${\cal X}$, we define the value $s I_{1-s}(P,W)$ for $s \in [0,1]$ as $$\begin{aligned}
s I_{1-s}(P,W)
:= - \sup_Q \log \sum_x P(x) e^{-s D_{1-s}(W_x \|Q)}
=\inf_Q s D_{1-s}(W \times P\| Q \times P).\end{aligned}$$ Since the minimum of concave functions is a concave function, the function $s\mapsto s I_{1-s}(P,W)$ is also a concave function. In fact, Hölder inequality guarantees that $$\begin{aligned}
s I_{1-s}(P,W) = -(1-s) \log \int
(\sum_{x}P(x) W_x(y)^{1-s} )^{\frac{1}{1-s}} dy
\Label{eq20}\end{aligned}$$ when the RHS exists [@Sibson][@q-wire (34)]. For $s \in (0,1]$, when $s I_{1-s}(P,W)$ is finite, $ I_{1-s}(P,W)$ is defined as $\frac{s I_{1-s}(P,W)}{s}$. The quantity $I_{1-s}(P,W)$ with (\[eq20\]) is the same as the Gallager function [@Gal] with different parametrization for $s$. We also define the mutual information $$\begin{aligned}
I(P,W):=
\sum_{x} P(x) D(W_x \| W\cdot P)
=
\sum_{x} P(x)
\int_{{\cal Y}}
W_x (y) \log \frac{W_x (y)}{W\cdot P(y)} d y\end{aligned}$$ and its variance $$\begin{aligned}
V(P,W)
:=&
\sum_{x}P(x) E_{W_x}
\Big[\log \frac{W_x(Y)}{W \cdot P(Y)} - I(P,W)\Big]^2 \nonumber \\
=& \sum_{x}P(x)
\int_{{\cal Y}}
W_x (y)
\Big(\log \frac{W_x(y)}{W \cdot P(y)} - I(P,W)\Big)^2
d y.\end{aligned}$$ When the channel satisfies some suitable conditions, we have $$\begin{aligned}
\lim_{s \to 0}I_{1-s}(P,W)
=I(P,W).\Label{2-26-A}\end{aligned}$$ For example, when the alphabet ${\cal Y}$ has a finite cardinality, the relation holds by choosing the measure $dy$ to be the counting measure.
Assumptions
-----------
Now, we clarify our assumption for family for the channels $\{W_\theta\}_{\theta\in \Theta }$. A parametric family of channels $\{W_{\theta}\}_{\theta\in\Theta}$ is called an [*exponential family of channels*]{} with a parametric space $\Theta \subset \mathbb{R}^k$ if a channel $W_{0}=(W_{0,x})$ from ${\cal X}$ to ${\cal Y}$, where $W_{0,x}$ are absolutely continuous with respect to a measure $dy$, and it can be written as $$\begin{aligned}
W_{\theta,x}(y)=
W_{0,x}(y) e^{\sum_{j=1}^{k_x} \theta^{j,x} g_{j,x}(y)- \phi_x(\theta)}\end{aligned}$$ for any $x \in {\cal X}$ with generators $g_{j,x}(y)$ and satisfies the following conditions.
A1
: $\sum_{x \in {\cal X}}k_x=k$.
A2
: The potential function $\phi_x(\theta)$ equals the cummulant generating function of $g_{j,x}$, i.e., $$\begin{aligned}
e^{\phi_x(\theta)}=
\int_{{\cal Y}} W_{0,x}(y) e^{\sum_{j=1}^{k_x} \theta^j g_{j,x}(y)}
d y< \infty\end{aligned}$$ for $\theta \in \Theta$.
A3
: $\phi_x$ is a $C^2$ function on $\Theta$, i.e., the Hessian of $\phi_x$ is continuous on $\Theta$.
When ${\cal Y}$ equals ${\cal X}$, this definition of exponential family has been discussed in many papers in the context of Markovian processes [@Bhat1; @Bhat2; @Hudson; @KM; @Feigin]. Here, for convenience, we say that a family of channels satisfies Condition A when all of the above conditions hold, i.e., it is an exponential family of channels.
When the input element $x\in {\cal X}$ is fixed, the output obeys the distribution $W_{\theta,x}(y)\mu(dy):=W_{\theta}(y|x)dy$ on ${\cal Y}$ with a fixed measure $\mu$ on ${\cal Y}$. For example, for a discrete and finite set ${\cal Y}$, the family of channels $\{W_\theta\}_{\theta\in \Theta }$ is exponential when $\{W_{\theta,x}\}_{\theta\in \Theta }$ is the set of all distributions on ${\cal Y}$ for any input $x$. Hence, this assumption covers the discrete memoryless stationary (DMS) case.
However, the assumption for exponential family of channels is slightly strong. To relax this condition, we introduce the following condition for a family of channels $\{W_\theta\}$ from a discrete alphabet ${\cal X}$ to a general alphabet ${\cal Y}$.
B1
: $\sum_{x \in {\cal X}}k_x=k$.
B2
: The parametric space $\Theta$ is compact.
B3
: For any $s >0$ and $x\in {\cal X}$, the map $(\theta,\theta')\mapsto D_{1+s}(W_{\theta,x}\| W_{\theta',x})$ is continuous, the likelihood ratio derivative $l_{\theta,x}(y):= \frac{d}{d\theta}\log W_{\theta,x}(y)$ exists, and the convergence $2\lim_{\epsilon}\frac{D_{1+s}(W_{\theta,x}\| W_{\theta+\epsilon,x})}{\epsilon}
=J_{\theta,x}:=
\int_{{\cal Y}} W_{\theta,x}(y) l_{\theta,x}(y)^2 d y$ holds and is uniform for $\theta$. Also, the map $\theta \mapsto J_{\theta,x}$ is continuous.
Hence, we say that a family of channels satisfies Condition B when all of the above conditions hold.
The compactness of Condition B can be relaxed as follows.
C1
: $\sum_{x \in {\cal X}}k_x=k$.
C2
: There exists a sequence of compact subsets $\{\Theta_i\}_{i=1}^{\infty}$ satisfying the following conditions. (1) $\Theta_i \subset \Theta_{i+1}$. (2) For any $\theta \in \Theta$, there exists $\Theta_i$ such that $\theta \in \Theta_i$.
C3
: For any $s >0$ and $x\in {\cal X}$, the map $(\theta,\theta')\mapsto D_{1+s}(W_{\theta,x}\| W_{\theta',x})$ is continuous, the likelihood ratio derivative $l_{\theta,x}(y)$ exists, and the convergence $2\lim_{\epsilon}\frac{D_{1+s}(W_{\theta,x}\| W_{\theta+\epsilon,x})}{\epsilon}
=J_{\theta,x}$ holds and is compact uniform for $\theta$. Also, the map $\theta \mapsto J_{\theta,x}$ is continuous.
We say that a family of channels satisfies Condition C when all of the above conditions hold. Conditions A and B are special cases of Condition C. Since discussions with Conditions A or B is simpler that of Condition C, we discuss Conditions A and B as well as Condition C.
Results
-------
In this paper, we address the $n$-fold stationary memoryless channel of $W_\theta$, i.e., we focus on the channel $W_{\theta,x^n}^n(y^n):=\prod_{i=1}^n W_{\theta,x_i}(y_i)$ with $x^n:=(x_1,\ldots,x_n)\in {\cal X}^n$ and $y^n:=(y_1,\ldots,y_n)\in {\cal Y}^n$. When the set of messages is ${\cal M}_n:=
\{1, \ldots, M_n\}$, the encoder is given as a map $E_n$ from ${\cal M}_n $ to ${\cal X}^n$, and the decoder is given as a map $D_n$ from ${\cal Y}^n$ to ${\cal M}_n $. The triple $\Phi_n:=(M_n,E_n,D_n)$ is called a code, and the size $M_n$ is often written as $|\Phi_n|$. The decoding error probability is $e_\theta(\Phi_n):= \frac{1}{M_n}\sum_{i=1}^{M_n}\sum_{j \neq i} W_{\theta,E_n(i)}^n( D_n^{-1}(j))$.
Given real numbers $R$ and $R_1$, a distribution $P$ on ${\cal X}$, and a family of channels $\{W_\theta\}_\theta$, we assume that the family of channels $\{W_\theta\}_\theta$ satisfies Condition A, B, or C, and that $s I_{1-s}(P,W_\theta )$ can be defined for any $s \in [0,1]$. Then, there exists a sequence of codes $\Phi_n$ with the size $|\Phi_n|=e^{nR}$ satisfying that $$\begin{aligned}
\lim_{n \to \infty}\frac{-1}{n}
\log
e_\theta (\Phi_n)
\ge
\min(\max_{s \in [0,1]}(s I_{1-s}(P,W_\theta ) -s R_1), R_1-R)
\Label{eq2} \end{aligned}$$ for any $\theta \in \Theta$. $\square$
The proof of Theorem \[Th1\] is given in Section \[s6\]. There are two methods to choose the rate $R_1$ and the distribution $P$ to identify our code for a given transmission rate $R$ when we assume that the true channel parameter $\theta$ belongs to a subset $\Theta_0 \subset \Theta$.
(M1)
: When $\argmax_P \inf_{\theta \in \Theta_0} I(P, W_\theta)$ is a non-empty set, we fix an element $P_1$ in this set. Otherwise, we choose a distribution $P_1$ such that $\inf_{\theta \in \Theta_0} I(P_1, W_\theta)$ is sufficiently close to $\sup_P \inf_{\theta \in \Theta_0} I(P, W_\theta))$. Next, we choose $R_1 \in (R, \inf_{\theta \in \Theta_0} I(P_1, W_\theta))$. Then, the exponential decreasing rate is greater than $\min(\max_{s\in [0,1]}(s I_{1-s}(P,W_\theta ) -s R_1) ,R-R_1)>0$ when the true parameter is $\theta \in \Theta_0$.
(M2)
: When $\argmax_P
\inf_{\theta\in \Theta_0}
\max_{s\in [0,1]}\frac{1}{1+s}(s I_{1-s}(P,W_\theta ) -s R)$ is a non-empty set, we fix an element $P_1$ in this set. Otherwise, we choose a distribution $P_1$ such that $\inf_{\theta\in \Theta_0}
\max_{s\in [0,1]}\frac{1}{1+s}(s I_{1-s}(P_1,W_\theta ) -s R)$ is arbitrarily close to $\sup_P \inf_{\theta\in \Theta_0}
\max_{s\in [0,1]}\frac{1}{1+s}(s I_{1-s}(P,W_\theta ) -s R)$. Then, due to in Lemma \[L11-27-1\], the exponential decreasing rate is greater than $\inf_{\theta\in \Theta_0}
\max_{s\in [0,1]}\frac{1}{1+s}(s I_{1-s}(P_1,W_\theta ) -s R)>0$, which maximizes the minimum lower bound of the exponential decreasing rate of the average error.
In both cases, it is enough to know that the true channel belongs to an exponential family of channels $\{W_\theta\}_{\theta \in \Theta_0}$. That is, we do not need to specify the true parameter $\theta$ perfectly. Although the method (M2) optimizes the worst case among $\Theta_0$, other cases also have the same lower bound of the exponent. However, the method (M1) produces a better exponent when the true parameter is not the worst case.
When the function $s \mapsto s I_{1-s}(P,W_\theta )$ is a $C^1$ function for any $\theta \in \Theta$, we have $$\begin{aligned}
& \max_{R_1} \inf_{\theta\in \Theta_0}\min (\max_{s \in [0,1]}
(s I_{1-s}(P,W_\theta ) -s R_1),R_1-R) \nonumber \\
&=
\inf_{\theta\in \Theta_0}\max_{R_1} \min
(\max_{s \in [0,1]}( s I_{1-s}(P,W_\theta ) -s R_1),R_1-R)\nonumber \\
&=\inf_{\theta\in \Theta_0}
\max_{s\in [0,1]}\frac{1}{1+s}(s I_{1-s}(P,W_\theta ) -s R) .\Label{eq11-27-1}\end{aligned}$$ The maximum value $\max_{R_1} \min (\max_{s \in [0,1]}
(s I_{1-s}(P,W_\theta ) -s R_1),R_1-R)$ is attained when $R_1= R+\max_{s\in [0,1]}\frac{1}{1+s}(s I_{1-s}(P,W_\theta ) -s R)$. In particular, the maximum value $\max_{R_1} \inf_{\theta\in \Theta_0}\min (\max_{s \in [0,1]}(s I_{1-s}(P,W_\theta ) -s R_1),R_1-R)$ is attained when $R_1= R+
\inf_{\theta\in \Theta_0}\max_{s\in [0,1]}\frac{1}{1+s}(s I_{1-s}(P,W_\theta ) -s R)$. $\square$
The proof of Lemma \[L11-27-1\] is given in Appendix \[A1\]. Note that Gallager’s exponent is $\max_{s \in [0,1]}\frac{1}{1-s}(s I_{1-s}(P,W_\theta ) -s R)$[@Gal], which is strictly larger than ours.
When we need to realize the transmission rate close to the capacity, we need to employ the second order analysis.
Given real numbers $R_1^*$ and $R_2^*$, a distribution $P$ on ${\cal X}$, and a family of channels $\{W_\theta\}_\theta$, we assume that the family of channels $\{W_\theta\}_\theta$ satisfies Condition A, B, or C, and that $0<I(P,W_\theta )< \infty$ and $0<V(P,W_\theta)<\infty$ for any parameter $\theta\in \Theta$. Then, there exists a sequence of codes $\Phi_n$ with the size $|\Phi_n|=e^{nR_1^*+\sqrt{n} R_2^*- n^{\frac{1}{4}}}$ satisfying the following conditions. (1) When $I(P,W_\theta ) > R_1^*$, we have $$\begin{aligned}
\lim_{n \to \infty}
e_\theta (\Phi_n)=0.\Label{11-27-6b}\end{aligned}$$ (2) When $I(P,W_\theta )=R_1^*$, we have $$\begin{aligned}
\lim_{n \to \infty}
e_\theta (\Phi_n)
\le
\int_{-\infty}^{\frac{R_2^*}{\sqrt{V(P,W_\theta )}}}
\frac{1}{\sqrt{2\pi}}
\exp( - \frac{x^2}{2}) dx
\Label{eq2-2} .\end{aligned}$$ $\square$
The proof of Theorem \[Th2\] is given in Section \[s8\].
However, most of channels $W_{\theta}$ do not satisfy the condition $I(P,W_\theta )=R_1^*$ for a specific first order rate $R_1^*$. That is, Theorem \[Th2\] gives the asymptotic average error probability $0$ or $1$ as the coding result for most of channels $W_{\theta}$. This argument does not reflect the real situation properly because many channels have their average error probability between $0$ and $1$, i.e., $0$ nor $1$ is not an actual value of average error probability. To overcome this problem, we introduce the second order parameterization $\theta_1 +\frac{1}{\sqrt{n}}\theta_2\in \Theta$. This parametrization is applicable when the unknown parameter belongs to the neighborhood of $\theta_1$. Given a parameter $\theta$, we choose the parameter $\theta_2:= (\theta-\theta_1)\sqrt{n}$. This parametrization is very conventional in statistics. For example, in statistical hypothesis testing, when we know that the true parameter $\theta$ belongs to the neighborhood of $\theta_1$, using the new parameter $\theta_2$, we approximate the distribution family by the Gaussian distribution family [@Lehmann]. In this sense, the $\chi^2$-test gives the asymptotic optimal performance. As another example, this kind of parametrization is employed to discuss local minimax theorem [@Vaart 8.11 Theorem].
The range of the neighborhood of this method depends on $n$ and $\theta_1$. In the realistic case, we have some error for our guess of channel and the number $n$ is finite. When the range of the error of our prior estimation of the channel is included in the neighborhood, this method effectively works. When our estimate of channel has enough precision as our prior knowledge, we can expect such a situation. In this scenario, we may consider the following situation. We fix the parameter $\theta_1$ as a basic (or standard) property of the channel. We choose the next parameter $\theta_2$ as a fluctuation depending on the daily changes or the individual specificity of the channel. Here, $n$ is chosen depending on the our calculation ability of encoding and decoding. Hence, it is fixed priorly to the choice of $\theta_2$. Then, Theorem \[Th2\] is refined as follows.
Given real numbers $R_1^*$ and $R_2^*$, a distribution $P$ on ${\cal X}$, and a family of channels $\{W_{\theta_1 +\frac{1}{\sqrt{n}}\theta_2}\}$ with the second order parameterization, we assume the following conditions in addition to the assumptions of Theorem \[Th3\]. The function $\theta \mapsto I(P,W_\theta )$ is a $C^1$ function on $\Theta$, and the function $\theta \mapsto V(P,W_\theta )$ is a continuous function on $\Theta$. Under these conditions, there exists a sequence of codes $\Phi_n$ with the size $|\Phi_n|=e^{nR_1^*+\sqrt{n} R_2^*- n^{\frac{1}{4}}}$ satisfying the following conditions. (1) When $I(P,W_{\theta_1} ) > R_1^*$, we have $$\begin{aligned}
\lim_{n \to \infty}
e_{\theta_1 +\frac{1}{\sqrt{n}}\theta_2} (\Phi_n)=0.\Label{11-27-6C}\end{aligned}$$ (2) When $I(P,W_{\theta_1} )=R_1^*$, we have $$\begin{aligned}
\lim_{n \to \infty}
e_{\theta_1 +\frac{1}{\sqrt{n}}\theta_2} (\Phi_n)
\le
\int_{-\infty}^{\frac{
R_2^*-f(\theta_2)}{\sqrt{V(P,W_\theta )}}}
\frac{1}{\sqrt{2\pi}}
\exp( - \frac{x^2}{2}) dx
\Label{eq2-2C} ,\end{aligned}$$ where $f(\theta_2):=
\sum_i\frac{\partial I(P,W_\theta )}{\partial \theta^i}|_{\theta=\theta_1}
\theta_2^i$. Further, when the convergence is compact uniform with respect to $\theta_2$ there exists an upper bound $\bar{e}_{\theta_1 +\frac{1}{\sqrt{n}}\theta_2} (\Phi_n)$ of $e_{\theta_1 +\frac{1}{\sqrt{n}}\theta_2} (\Phi_n)$ such that the bound $\bar{e}_{\theta_1 +\frac{1}{\sqrt{n}}\theta_2} (\Phi_n)$ converges to the RHS of compact uniformly for $\theta_2$. $\square$
The proof of Theorem \[Th3\] is given in Section \[s8\].
Now, we consider how to realize the transmission rate close to the capacity, In this case, we need to know that the true parameter $\theta_0$ belongs to the neighborhood of $\theta_1$. More precisely, given a real number $C>0$ and the parameter $\theta_1$, we need to assume that the true parameter $\theta_0$ belongs to the set $\{ \theta_1+\frac{1}{\sqrt{n}}\theta_2 | \|\theta_2\| \le C \}$, where $ \|\theta\|:=\sum_{i=1}^{k} (\theta^i)^2$. To achieve the aim, we choose the distribution $P$ such that $I(P,W_{\theta_1})$ equals the capacity of the channel $W_{\theta_1}$. Then, we choose $R_1^*$ to be the capacity, i.e., $I(P,W_{\theta_1})$. To keep the average error probability approximately less than $\epsilon$, we choose $R_2^*$ as the maximum number satisfying $\max_{\theta_2: \|\theta_2\| \le C}
\int_{-\infty}^{\frac{
R_2^*-f(\theta_2)}{\sqrt{V(P,W_\theta )}}}
\frac{1}{\sqrt{2\pi}}
\exp( - \frac{x^2}{2}) dx \le \epsilon$. Then, using Theorem \[Th3\], we can realize a rate close to the capacity and the average error probability approximately less than $\epsilon$. When we have larger order perturbation than $\frac{1}{\sqrt{n}}$, our upper bound of $\lim_{n \to \infty}e_{\theta} (\Phi_n)$ becomes $0$ or $1$ because such a case corresponds to the case when $\|\theta_2\|$ goes to infinity. Hence, we find that $\frac{1}{\sqrt{n}}$ is the maximum order of the perturbation to discuss the framework of the second order.
One might doubt the validity of the above approximation, however, it can be justified as follows. When the true parameter belongs to the neighborhood of $\theta_1$ satisfying $I(P,W_{\theta_1} )=R_1^*$, the above theorem gives the approximation of average error probability. Theorem \[Th3\] reflects such a real situation. Indeed, the convergence of the upper bound $\bar{e}_{\theta} (\Phi_n)$ is not uniform for $\theta$ in the neighborhood of such a parameter $\theta_1$ because the limit is discontinuous at $\theta_1$. However, it is possible that the convergence of $e_{\theta_1 +\frac{1}{\sqrt{n}}\theta_2} (\Phi_n)$ is not compact uniform for $\theta_2$. At least, the convergence of our upper bound is compact uniform for $\theta_2$. The reason for the importance of the second order asymptotics is that the limiting error probability can be used for the approximation of the true average error probability because the convergence is compact uniform for the second order rate [@Haya]. Due to the same reason, the RHS of can be used for the approximation of our upper bound of the true average error probability.
Here, we list typical examples as follows. All of the below examples satisfy the assumptions of these theorems.
Consider a finite input set ${\cal X}=\{1, \ldots, d\}$ and a finite output set ${\cal Y}=\{0,1, \ldots, m\}$. Then, the measure $\mu$ is chosen to be the counting measure. For parameters $\theta=(\theta^{j,i})$ with $i=1, \ldots, d, j=1, \ldots,m$, we define the output distributions $$\begin{aligned}
W_{\theta}(y|x)
:=
\left\{
\begin{array}{ll}
\frac{e^{\theta^{y,x}}}{1+\sum_{j=1}^{m} e^{\theta^{j,x}}}
& \hbox{ when } y \ge 1 \\
\frac{1}{1+\sum_{j=1}^{m} e^{\theta^{j,x}}}
& \hbox{ when } y =0 .
\end{array}
\right.\end{aligned}$$ Then, the set of output distributions $\{W_{\theta,x}\}_{\theta}$ forms an exponential family for $x \in {\cal X}$. $\square$
Assume that the output set ${\cal Y}$ is the set of real numbers $\mathbb{R}$ and the input set ${\cal X}$ is a finite set $\{1, \ldots, d\}$. We choose $d$ elements $x_1, \ldots, x_d \in \mathbb{R}$ as input signals. The additive noise $Z$ is assumed to be subject to the Gaussian distribution with the expectation $b$ and the variance $v$. Then, we assume that the received signal $Y$ is given by the scale parameter $a$ as $$\begin{aligned}
Y= a x_i +Z.\end{aligned}$$ The class of these channels is known as Gaussian fading channels, and its compound channel is discussed in the paper [@E]. By using the three parameters $\theta^1:=\frac{1}{v}$, $\theta^2:=\frac{a}{v}$, and $\theta^3:=\frac{b}{v}$, the channel is given as $$\begin{aligned}
& W_\theta(y|i)
:=
\frac{1}{\sqrt{2\pi v}}
e^{-\frac{(y-a x_i-b)^2}{2v} } \nonumber \\
=&
\frac{\theta^1}{\sqrt{2\pi}}
e^{-\frac{-y^2}{2} \theta^1+ y (\theta^2 x_i+\theta^3)
-\frac{(\theta^2 x_i+\theta^3)^2}{2 \theta^1}}.\end{aligned}$$ Hence, for each input $i \in {\cal X}$, although this parametrization has redundancy, the set of output distributions $\{W_{\theta,i}\}_{\theta}$ forms an exponential family. $\square$
We consider the constant multi-antenna (MIMO) Gaussian channel when the sender has $t$ antennas and the receiver has $r$ antennas. Then, the output set ${\cal Y}$ is given as the set of $r$-dimensional real numbers $\mathbb{R}^r$ and the input set ${\cal X}$ is given as a finite set $\{1, \ldots, d\}$. In this case, the sender chooses $d$ elements $\vec{x}_1, \ldots, \vec{x}_d \in \mathbb{R}^t$ as input signals. The $r$-dimensional additive noise $\vec{Z}$ is assumed to be subject to the $r$-dimensional Gaussian distribution with the expectation $\vec{b}\in \mathbb{R}^d$ and the covariance matrix $v_{i,i'}$. We assume that the received signal $\vec{Y}$ is written by a $r \times t$ matrix $(a_{i,j})$ as $$\begin{aligned}
\vec{Y}= \sum_{j=1}^t a_{i,j} x^j_i +\vec{Z}.\end{aligned}$$ The class of these channels is known as constant multi-antenna Gaussian channels [@de4]. Then, we define three kinds of parameters: The symmetric matrix $(\theta^{1,i,i'})_{i,i'}$ is defined as the inverse matrix of the covariance matrix $v_{i,i'}$. The matrix $(\theta^{2,i,i'})_{i,i'} $ is defined as $\theta^{2,i,j}:=\sum_{i'} \theta^{1,i,i'} a_{i',j}$. The vector $\theta^{3,i}:=\sum_{i'} \theta^{1,i,i'} b^{i'}$. Then, the channel is given as $$\begin{aligned}
& W_\theta(\vec{y}|i)
:=
\frac{ \det (\theta^1)}{\sqrt{2\pi}}
e^{-\frac{1}{2}
\sum_{i,i'} \theta^{1,i,i'}
(y^i- \sum_j a_{i,j} x^j_i-b^i)
(y^{i'}- \sum_{j'} a_{i',j'} x^{j'}_{i'}-b^{i'})
} \nonumber \\
=&
\frac{ \det (\theta^1)}{\sqrt{2\pi}}
e^{-\frac{1}{2}
\sum_{i,i'} \theta^{1,i,i'} y^i y^{i'}
+
\sum_{i} y^{i} (\theta^{3,i}+ \sum_{j}\theta^{2,i,j} x^j )
-\frac{1}{2}
\sum_{i,i'}
((\theta^{1})^{-1})^{i,i'}
(\theta^{3,i}+ \sum_{j}\theta^{2,i,j} x^j )
(\theta^{3,i'}+ \sum_{j}\theta^{2,i',j} x^j )}.\end{aligned}$$ Hence, for each input $i \in {\cal X}$, although this parametrization has redundancy, the set of output distributions $\{W_{\theta,i}\}_{\theta}$ forms an exponential family. $\square$
Method of types
===============
In this section, we define our universal encoder for a given distribution $P$ on ${\cal X}$ and a real number $R_1 <H(P)$ by the same way as [@Ha1]. Although the contents of this section is the same as a part of [@Ha1], since the paper [@Ha1] is written with quantum terminology, we repeat the same contents as a part of [@Ha1] for readers’ convenience.
First, we prepare notations for method of types. Given an element ${x}^n\in {\cal X}^n$, we define the integer $n_x:=| \{ i| x_i=x\} |$ for $x\in {\cal X}$ and the empirical distribution $T_Y(x^n):= (\frac{n_1}{n},\ldots, \frac{n_d}{n})$, which is called a type. The set of types is denoted by $T_n({\cal X})$. For ${P} \in T_n({\cal X})$, a subset of ${\cal X}^n$ is defined by: $$\begin{aligned}
T_{P}:= \{{x}^n \in {\cal X}^n| \hbox{The empirical distribution of } \vec{x}
\hbox{ is equal to }P\}.\end{aligned}$$ Hence, we define the distribution $$\begin{aligned}
P_{T_P}(x^n):=
\left\{
\begin{array}{cc}
\frac{1}{|T_P|} & {x}^n \in T_P \\
0 & {x}^n \notin T_P.
\end{array}
\right.\end{aligned}$$ Then, we define a constant $c_{n,P}$ such that $$\begin{aligned}
\frac{1}{c_{n,P}} P_{T_P}({x'}^n) = P^{n}({x'}^n)
=e^{\sum_{i=1}^d n_i \log P(i) } =e^{-nH(P)}
\Label{eq3-2}.\end{aligned}$$ for ${x'}^n \in T_{P}$. So, the constant $c_{n,P}$ is bounded as $$\begin{aligned}
c_{n,P} \le |T_n({\cal X})|
\Label{eq3}.\end{aligned}$$
Further, the sequence of types ${V}=({v}_1, \ldots, {v}_d)\in
T_{n_1}({\cal X})\times \cdots \times T_{n_d}({\cal X})$ is called a conditional type for ${x}^n$ when the type of $x^n$ is $(\frac{n_1}{n}, \ldots, \frac{n_d}{n})$ [@CK]. We denote the set of conditional types for ${x}^n$ by $V({x}^n,{\cal X})$. For any conditional type $V$ for ${x}^n$, we define the subset of ${\cal X}^n$: $$\begin{aligned}
T_{{V}}({x}^n)
:= \left\{{x'}^n \in {\cal X}^n\left|
T_Y((x_1,x_1'), \ldots, (x_n,x_n'))
={V} \cdot P
\right.\right\},\end{aligned}$$ where ${P}$ is the empirical distribution of ${x}^n$.
According to Csiszár and Körner[@CK], the proposed code is constructed as follows. The main point of this section is to establish that Csiszár-Körner’s Packing lemma [@CK Lemma 5.1] provides a code whose performance is essentially equivalent to the average performance of random coding in the sense of (\[8\]).
For a positive number $\delta>0$, a type ${P} \in T_n({\cal X})$, and a real positive number $R< H(P)$, there exist $M_n:=e^{n(R-\delta)}$ distinct elements $$\begin{aligned}
\hat{\cal M}_n:=\{E_{n,P,R}(1),\ldots, E_{n,P,R}({M_n})\} \subset T_P\end{aligned}$$ such that their empirical distributions are ${P}$ and the inequality $$\begin{aligned}
|T_{{V}}({x}^n) \cap (\hat{\cal M}_n\setminus \{{x}^n\})|
\le |T_{{V}}({x}^n)|
e^{-n(H(P)-R)} $$ holds for any ${x}^n \in \hat{\cal M}_n \subset T_{{P}}$ and any ${V} \in V({x}^n,{\cal X})$. $\square$
This lemma can be shown by applying Lemma 5.1 in Csiszár and Körner[@CK] to the case when $\hat{V}(x|x')=\delta_{x,x'}$ with replacing $\delta$ by $n^{-\frac{3}{4}}$. That is, there exist $M_n:= e^{n R- n^{\frac{1}{4}}}$ distinct elements $$\begin{aligned}
\hat{\cal M}_n:=\{E_{n,P,R}(1),\ldots, E_{n,P,R}({M_n})\} \subset T_{P}\end{aligned}$$ such that the empirical distributions of $E_{n,P,R}(1),\ldots, E_{n,P,R}({M_n})$ are ${P}$ and the inequality $$\begin{aligned}
|T_{{V}}({x}^n) \cap (\hat{\cal M}_n\setminus \{{x}^n\})|
\le |T_{{V}}({x}^n)|
e^{-n(H({P})-R)} \Label{20}\end{aligned}$$ holds for any ${x}^n \in \hat{\cal M}_n \subset T_{{P}}$ and any ${V} \in V({x}^n,{\cal X})$. Using the argument, we define our universal encoder $E_{n,P,R}$. Note that this encoder $\hat{\cal M}_n$ does not depend on the output alphabet because the employed Packing lemma treats the conditional types from the input alphabet to the input alphabet. Now, we transform the property (\[20\]) to a more useful form.
Using the encoder $\hat{\cal M}_n$, we define the distribution $P_{\hat{\cal M}_n}$ as $$\begin{aligned}
P_{\hat{\cal M}_n}({x}^n):=
\left\{
\begin{array}{cc}
\frac{1}{{M}_n} & {x}^n \in \hat{\cal M}_n \\
0 & {x}^n \notin \hat{\cal M}_n.
\end{array}
\right.\end{aligned}$$ Now, we focus on the permutation group $S_n$ on $\{1, \ldots, n\}$. For any ${x}^n \in {\cal X}^n$, we define an invariant subgroup $S_{{x}^n}\subset S_n$: $$\begin{aligned}
S_{{x}^n}: = \{g \in S_n | g({x}^n)={x}^n \}.\end{aligned}$$ For $V \in V({x}^n,{\cal X})$, the probability $\sum_{g \in S_{{x}^n}}
\frac{1}{|S_{{x}^n}|}
P_{\hat{\cal M}_n}\circ g ({x'}^n)$ does not depend on the element ${x'}^n \in T_{{V}}({x}^n) \subset T_{P}$. Since $
\sum_{{x'}^n \in T_{{V}}({x}^n)}
\sum_{g \in S_{{x}^n}}
\frac{1}{|S_{{x}^n}|}
P_{\hat{\cal M}_n}\circ g ({x'}^n)
=
\sum_{{x'}^n \in T_{{V}}({x}^n)}
P_{\hat{\cal M}_n} ({x'}^n)
=
|T_{{V}}({x}^n) \cap (\hat{\cal M}_n\setminus \{{x}^n\})|
\cdot \frac{1}{{M}_n}$, any element ${x'}^n \in T_{{V}}({x}^n) \subset T_{P}$ satisfies $
\sum_{g \in S_{{x}^n}}
\frac{1}{|S_{{x}^n}|}
P_{\hat{\cal M}_n}\circ g ({x'}^n)
=
\frac{|T_{{V}}({x}^n) \cap \hat{\cal M}_n|}{|T_{{V}}({x}^n)|}
\cdot
\frac{1}{{M}_n}$. Thus, any element ${x'}^n \in T_{{V}}({x}^n) \subset T_{P}$ satisfies $$\begin{aligned}
&\sum_{g \in S_{{x}^n}}
\frac{1}{|S_{{x}^n}|}
P_{\hat{\cal M}_n}\circ g ({x'}^n)
=
\frac{|T_{{V}}({x}^n) \cap \hat{\cal M}_n|}{|T_{{V}}({x}^n)|}
\cdot
\frac{1}{{M}_n} \nonumber \\
= &
\frac{|T_{{V}}({x}^n) \cap (\hat{\cal M}_n\setminus \{{x}^n\})|}
{|T_{{V}}({x}^n)|{M}_n} \nonumber \\
\stackrel{(a)}{\le} &
e^{-nH(P)}
e^{n^{\frac{1}{4}} }
\stackrel{(b)}{=}
{P}^{n}({x'}^n) e^{n^{\frac{1}{4}} }
=\frac{1}{c_{n,P}} P_{T_P} ({x'}^n) e^{n^{\frac{1}{4}} }
\Label{8}\end{aligned}$$ when the conditional type ${V}$ is not identical, where $(a)$ and $(b)$ follow from and , respectively. Relation (\[8\]) holds for any ${x'}^n(\neq {x}^n) \in T_P$ because there exists a conditional type ${V}$ such that ${x'}^n \in T_{{V}}({x}^n)$ and ${V}$ is not identical.
$\alpha$-Rényi divergence version of Clarke-Barron formula
==========================================================
In this section, we discuss an $\alpha$-Rényi divergence version of Clarke-Barron formula. For simplicity, we discuss only the exponential family of distribution $\{P_\theta(y)\}$. We fix a distribution $P_0$ absolutely continuous with respect to a measure $dy$ on a measurable set ${\cal Y}$. A parametric family of distributions $\{P_\theta\}_{\theta\in \Theta }$ is called an [*exponential family*]{} with a parametric space $\Theta \subset \mathbb{R}^k$ when it written as [@AN] $$\begin{aligned}
P_{\theta}(y)=
P_{0}(y) e^{\sum_{j=1}^{k} \theta^j g_{j}(y)- \phi(\theta)}\end{aligned}$$ with generators $g_{j}(y)$ and satisfies the following conditions.
D1
: The potential function $\phi(\theta)$ equals the cummulant generating function of $g_{j}$, i.e., $$\begin{aligned}
e^{\phi(\theta)}=
\int_{{\cal Y}} P_{0}(y) e^{\sum_{j=1}^{k} \theta^j g_{j}(y)}
d y< \infty\end{aligned}$$ for $\theta \in \Theta$.
D2
: $\phi$ is a $C^2$ function on $\Theta$, i.e., the Hessian of $\phi$ is continuous on $\Theta$.
For a given distribution $w$ on the parameter space $\Theta$, we consider the mixture distribution $Q_{w}^n(y^n):= \int_{\Theta} P_{\theta}^n(y^n) w(d \theta)$. As an $\alpha$-Rényi divergence version of Clarke-Barron formula, we have the following lemma.
Assume that $w(\theta)$ is continuous for $\theta$. Then, an exponential family $\{P_{\theta}\}$ satisfies that $$\begin{aligned}
D_{1+n}(P_{\theta}^n\|Q_{w}^n)
\le & \frac{k}{2}\log n+O(1)
\Label{3-28-Ab} \\
D_{1+s}(P_{\theta}^n\|Q_{w}^n)
\le & \frac{k}{2}\log \frac{n}{2\pi}+\frac{1}{2}\log \det J_{\theta}
+\log \frac{1}{w(\theta)}- \frac{k}{2s}\log (1+s)+o(1)
\Label{3-28-A} \end{aligned}$$ for $s>0$, as $n$ goes to infinity. When the continuity for $w(\theta)$ is compact uniform for $\theta$, the constant on the RHS of can be chosen compact uniformly for $\theta$. $\square$
Clarke-Barron show the relation $$\begin{aligned}
D(P_{\theta}^n\|Q_{w}^n)
= \frac{k}{2}\log n +O(1)
\Label{3-28-AT} \end{aligned}$$ as $n$ goes to infinity. Since $D(P_{\theta}^n\|Q_{w}^n)
\le D_{1+s}(P_{\theta}^n\|Q_{w}^n) $ for $s>0$, we have $$\begin{aligned}
D_{1+s}(P_{\theta}^n\|Q_{w}^n)
= \frac{k}{2}\log n +O(1)
\Label{3-28-AT2} \end{aligned}$$ as $n$ goes to infinity.
The proof of Lemma \[l2-A\] is given in Appendix \[A2\]. Here, it is better to explain a relation of and with maximum relative entropy. Since $$\begin{aligned}
& P_{\theta}^n(
\{y^n| P_{\theta}^n(y^n) > Q_{w}^n(y^n) e^{n\delta}\})
\nonumber \\
\le & e^{-n s' \delta+ s' D_{1+s'}( P_{\theta}^n\| Q_{w}^n)},\end{aligned}$$ (\[3-28-A\]) implies that $$\begin{aligned}
& -\frac{1}{n}\log
P_{\theta}^n (
\{y^n| P_{\theta}^n(y^n) > Q_{w}^n(y^n) e^{n\delta}\})
\nonumber \\
\ge &
s' \delta- \frac{s'}{n} D_{1+s'}( P_{\theta}^n\| Q_{w}^n) \nonumber \\
\ge &
s' \delta- \frac{s'}{n}
\bigg(\frac{k}{2}\log \frac{n}{2\pi}
+\frac{d}{2}\log \det J_\theta \nonumber \\
& +d \log \frac{1}{w(\theta)}- \frac{kd}{2s'}\log (1+s')+o(1)\bigg) \nonumber \\
\to & s' \delta \Label{eq8B}\end{aligned}$$ for $s'>0$. Since $s'$ is arbitrary, we have $$\begin{aligned}
\lim_{n\to \infty}
-\frac{1}{n}\log
P_{\theta}^n(
\{y^n| P_{\theta}^n(y^n) > Q_{w}^n (y^n)e^{n\delta}\})
=
\infty \Label{eq1B}.\end{aligned}$$
The exponent in seems too large. However, this is not so unnatural if we consider the following way. To discuss this issue, we consider the maximum relative entropy $D_{\max}( P\|Q):= \max_{y: P(y)>0 }\log \frac{P(y)}{Q(y)}$. When ${\cal Y}$ is a finite alphabet, we can find a distribution $Q^{(n)}$ such that $$\begin{aligned}
D_{\max}( P_{\theta}^{n}\| Q^{(n)}) =O(\log n)\Label{17-1}\end{aligned}$$ for any $\theta$. Such a distribution $Q^{(n)}$ can be constructed as the uniform mixture of the uniform distributions $P_{T_P}$, i.e., $\sum_{P} \frac{1}{|T_n({\cal Y})|}P_{T_P}$[@TH]. Hence, the relation $P_{\theta}^n(y^n) \le Q_{w}^n (y^n)e^{n\delta}$ holds for any $y^n$ when we choose $\delta$ to be sufficiently large. Therefore, and can be regarded as the refinement of .
Now, we consider other conditions for family of distributions $\{P_\theta\}_{\theta\in \Theta}$ as follows.
E1
: The parametric space $\Theta$ is compact.
E2
: For any $s >0$, the map $(\theta,\theta')\mapsto D_{1+s}(P_\theta\| P_{\theta'})$ is continuous, the likelihood ratio derivative $l_{\theta}(y):= \frac{d}{d\theta}\log P_{\theta}(y)$ exists, and the convergence $2\lim_{\epsilon}\frac{D_{1+s}(P_{\theta}\| P_{\theta+\epsilon})}{\epsilon}
=J_{\theta,x}:=
\int_{{\cal Y}} P_{\theta}(y) l_{\theta}(y)^2 d y$ holds and is uniform for $\theta$. Also, the map $\theta \mapsto J_{\theta}$ is continuous.
Hence, we say that a family of channels satisfies Condition E when all of the above conditions hold.
In this case, we choose the discrete mixture distribution $Q_E^n$ as follows. Since $\Theta$ is compact, the subset $\Theta_{[n]}:= \frac{1}{\sqrt{n}} \mathbb{Z}^k \cap \Theta$ has finite cardinality, which increases with the order $O( n^{\frac{k}{2}})$. Then, we define $Q_E^n(y^n):= \sum_{\theta\in \Theta_{[n]}}
\frac{1}{|\Theta_{[n]}|} P_{\theta}^n(y^n)$ and have the following lemma.
When a family $\{P_{\theta}\}$ satisfies Condition E, we have $$\begin{aligned}
D_{1+s}(P_{\theta}^n\|Q_{E}^n)
\le & \frac{k}{2}\log n+O(1)
\Label{3-28-AE} \end{aligned}$$ for $s>0$, as $n$ goes to infinity. More precisely, there is an upper bound $A_{\theta,n} $ of $D_{1+s}(P_{\theta}^n\|Q_{E}^n)$ satisfying the following. $A_{\theta,n} $ does not depend on $s>0$, and the quantity $A_{\theta,n} -\frac{k}{2}\log n$ converges uniformly for $\theta$. Hence, the above inequality contains the statement like . $\square$
We fix $\epsilon>0$. Due to the assumption and the definition of $\Theta_{[n]}$, we can choose an integer $N$ satisfying the following conditions. For $n \ge N$ and $\theta \in \Theta$, we can choose $\theta' \in \Theta_{[n]}$ such that $D_{1+s}(P_{\theta}\|P_{\theta'}) \le \frac{J_\theta+\epsilon}{2 n} $. Since $ \frac{1}{|\Theta_{[n]} |}P_{\theta'}^n (y^n)
\le Q_{E}^n(y^n)$, we have $$\begin{aligned}
e^{sD_{1+s}(P_{\theta}^n\|Q_{E}^n)}
\le
|\Theta_{[n]} |^s
e^{sD_{1+s}(P_{\theta}^n\|P_{\theta'}^n )}
\le
|\Theta_{[n]} |^s
e^{sn \frac{J_\theta+\epsilon}{2 n} }
=|\Theta_{[n]} |^s
e^{s \frac{J_\theta+\epsilon}{2 } }.\end{aligned}$$ That is, $$\begin{aligned}
D_{1+s}(P_{\theta}^n\|Q_{E}^n)
\le
\log |\Theta_{[n]} | +\frac{J_\theta+\epsilon}{2 } .\end{aligned}$$ Since the compactness of $\Theta$ implies $\log |\Theta_{[n]} |=
\frac{k}{2}\log n +O(1)$, we obtain the desired argument.
The compactness of Condition E can be relaxed as follows.
F1
: There exists a sequence of compact subsets $\{\Theta_i\}_{i=1}^{\infty}$ satisfying the following conditions. (1) $\Theta_i \subset \Theta_{i+1}$. (2) For any $\theta \in \Theta$, there exists $\Theta_i$ such that $\theta \in \Theta_i$.
F2
: For any $s >0$, the map $(\theta,\theta')\mapsto D_{1+s}(P_\theta\| P_{\theta'})$ is continuous, the likelihood ratio derivative $l_{\theta,x}(y)$ exists, and the convergence $2\lim_{\epsilon}\frac{D_{1+s}(P_{\theta}\| P_{\theta+\epsilon})}{\epsilon}
=J_{\theta}$ holds and is compact uniform for $\theta$. Also, the map $\theta \mapsto J_{\theta}$ is continuous.
We say that a family of channels satisfies Condition F when all of the above conditions hold.
In this case, we choose the discrete mixture distribution $Q_F^n$ as follows. Since $\Theta_i$ is compact, the subset $\Theta_{[n,i]}:= \frac{1}{\sqrt{n}} \mathbb{Z}^k \cap \Theta_i$ has finite cardinality, which increases with the order $O( n^{\frac{k}{2}})$. Then, we define $Q_F^n(y^n):=
\sum_{i=1}^{\infty}\frac{6}{ \pi^2 i^2}
\sum_{\theta\in \Theta_{[n,i]}}
\frac{1}{|\Theta_{[n,i]}|} P_{\theta}^n(y^n)$ and have the following lemma.
When a family $\{P_{\theta}\}$ satisfies Condition F, we have the same inequality as with $Q_{F}^n$. More precisely, there is an upper bound $B_{\theta,n} $ of $D_{1+s}(P_{\theta}^n\|Q_{F}^n)$ satisfying the similar condition as $A_{\theta,n} $. The difference is that the quantity $B_{\theta,n} -\frac{k}{2}\log n$ converges not uniformly but compact uniformly for $\theta$ because $\Theta$ is not necessarily compact. Hence, the obtained inequality contains the statement like . $\square$
For any compact subset $\Theta'\subset \Theta$ and $\epsilon>0$, we choose $i$ such that $\Theta' \subset \Theta_i$. Due to the assumption and the definition of $\Theta_{[n,i]}$, we can choose an integer $N$ satisfying the following conditions. For $n \ge N$ and $\theta \in \Theta'$, we can choose $\theta' \in \Theta_{[n,i]}$ such that $D_{1+s}(P_{\theta}\|P_{\theta'}) \le \frac{J_\theta+\epsilon}{2 n} $. Since $ \frac{6}{\pi^2 i^2|\Theta_{[n]} |}P_{\theta'}^n (y^n)
\le Q_{E}^n(y^n)$, we have $$\begin{aligned}
e^{sD_{1+s}(P_{\theta}^n\|Q_{E}^n)}
\le
\frac{\pi^{2s} i^{2s}}{6^s}
|\Theta_{[n]} |^s
e^{sD_{1+s}(P_{\theta}^n\|P_{\theta'}^n )}\end{aligned}$$ That is, $$\begin{aligned}
D_{1+s}(P_{\theta}^n\|Q_{E}^n)
\le
\log \frac{\pi^{2} i^{2}}{6}
+\log |\Theta_{[n]} | +\frac{J_\theta+\epsilon}{2 } .\end{aligned}$$ So, we obtain the desired argument.
Universal decoder
=================
The aim of this section is to make a universal decoder, which does not depend on the parameter $\theta$ and depends only on the distribution $P$, the family of channel $\{W_\theta\}$, prior distributions on $\Theta$, the coding rate $R$, and another rate $R_1$. The other rate $R_1$ is decided by the distribution $P$, the family $\{W_\theta\}$, and $R$ as discussed in Lemma \[L11-27-1\](large deviation) and Section \[s8\] (second order). The required property holds for any choice of the prior distributions on $\Theta$.
In this section, we construct a decoder only under Condition A. However, under Condition B or C, we can construct it in the same way by replacing $Q_w^{n}$ by $Q_E^{n}$ or $Q_F^{n}$ due to Lemmas \[l2-E\] and \[l2-F\]. For this purpose, we need to extract the message that has large correlation with detected output signal like the maximum mutual information decider [@CK]. However, since the cardinality of the output alphabet ${\cal Y}$ is not necessarily finite, we cannot directly apply the maximum mutual information decider because it can be defined only with finite input and output alphabets. Here, to overcome this problem, we need to extract the message that has large correlation with detected output signal. To achieve this aim, we firstly construct output distributions that universally approximate the true output distribution and the mixture of the output distribution. Then, we apply the decoder for information spectrum method [@Verdu-Han] to these distributions. Using this idea, we construct our universal decoder. That is, for a given number $R>0$ and a distribution $P$ on ${\cal X}$, we define our universal decoder for our universal encoder $E_{n,P,R}$ given in Section \[s3\] by using the type $P=
(\frac{n_1}{n}, \ldots, \frac{n_d}{n})$.
We choose prior distributions $w_x$ and $w_P$ on $\Theta$ for $x \in {\cal X}=\{1, \ldots,d\}$ and a distribution $P$ on ${\cal X}$, respectively so that $w_x(\theta)$ and $w_P(\theta)$ are continuous for $\theta \in \Theta$. $Q_{P}^{(n)}$ is defined to be $Q_{w_P,P}^n$. For a given $x^n\in T_P$, we define the distribution $Q_{x^n}^{(n)}$ on ${\cal Y}^n$ as follows. First, for simplicity, we consider the case when ${x'}^n=(\underbrace{1,\ldots,1}_{n_1},\underbrace{2,\ldots,2}_{n_2},\ldots,
\underbrace{d,\ldots, d}_{n_d}) \in T_P$. In this case, $Q_{{x'}^n}^{(n)}(y^n)$ is defined as $Q_{w_1,1}^{n_1}\times Q_{w_1,2}^{n_2} \times \cdots \times Q_{w_d,d}^{n_d}$. For a general element $x^n \in T_P$, the distribution $Q_{{x}^n}^{(n)}$ on ${\cal Y}^n$ is defined by the permuted distribution of $Q_{{x'}^n}^{(n)}$ by the application of the permutation converting ${x'}^n$ to $x^n$.
Now, to make our decoder, we apply the decoder for information spectrum method [@Verdu-Han] to the case when the output distribution is given by $Q_{{x}^n}^{(n)}$ and the mixture of the output distribution is given by $Q_{P}^{(n)}$. Then, we define the subset $$\begin{aligned}
\hat{{\cal D}}_{x^n}:= \{y^n|
Q_{{x}^n}^{(n)} (y^n) \ge e^{n R_1} Q_{P}^{(n)} (y^n)
\}.
\Label{11-27-6}\end{aligned}$$ Given a number $R_1 >R$ and an element $i \in {\cal M}_n$, we define the subset ${\cal D}_{i}:=
\hat{{\cal D}}_{E_{n,P,R}(i)} \setminus
\cup_{j=1}^{i-1}\hat{{\cal D}}_{E_{n,P,R}(i)}$, inductively. Finally, we define our universal decoder $D_{n,P,R,R_1}$ as $D_{n,P,R,R_1}^{-1}(i):= {\cal D}_{i}$ and our universal code $\Phi_{n,P,R,R_1}:=(e^{nR- n^{\frac{1}{4}}},E_{n,P,R},D_{n,P,R,R_1})$.
Now, we discuss several important properties related to our universal decoder by using Lemma \[l2-A\]. For this purpose, we rewrite Lemma \[l2-A\] in terms of an exponential family of channels as follows.
Assume that $w(\theta)$ is continuous for $\theta$. Then, an exponential family $\{W_{\theta,x}\}$ of channels satisfies that $$\begin{aligned}
D_{1+n}(W_{\theta,(x,\ldots,x )}^n\|Q_{w,x}^n)
\le & \frac{k}{2}\log n+O(1)
\Label{3-28b} \\
D_{1+s}(W_{\theta,(x,\ldots,x )}^n\|Q_{w,x}^n)
\le & \frac{k}{2}\log \frac{n}{2\pi}+\frac{1}{2}\log \det J_{\theta,x}
+\log \frac{1}{w(\theta)}- \frac{k}{2s}\log (1+s)+o(1)
\Label{3-28} \end{aligned}$$ for $s>0$ and for each $x\in {\cal X}$, as $n$ goes to infinity. When the above continuity is compact uniform for $\theta$, the constant on the RHS of can be chosen compact uniformly for $\theta$. $\square$
For $x^n \in T_P$ and $s>0$, of Lemma \[l2\] guarantees that $$\begin{aligned}
& D_{1+s}( W^{n}_{\theta,x^n}\| Q_{{x}^n}^{(n)}) \nonumber \\
\le & \sum_{x \in {\cal X}}
\Bigl(\frac{k}{2}\log \frac{n P(x)}{2\pi}\Bigr)
+\frac{d}{2}\log \det J_\theta \nonumber \\
&+d \log \frac{1}{w(\theta)}- \frac{kd}{2s}\log (1+s)+o(1).
\Label{3-28-1}\end{aligned}$$ Further, by defining $Q_{P}^{(n)} := \sum_{x^n \in T_P} P_{T_P} Q_{{x}^n}^{(n)}$, the information processing inequality for relative Rényi entropy yields that $D_{1+s}( W^{n}_{\theta}\cdot T_{T_P}\| Q_{P}^{(n)})
\le
\sum_{x^n \in T_P} P_{T_P} (x^n) D_{1+s}( W^{n}_{\theta,x^n}\| Q_{{x}^n}^{(n)})
=
D_{1+s}( W^{n}_{\theta,x^n}\| Q_{{x}^n}^{(n)})$, which implies $$\begin{aligned}
& D_{1+s}( W^{n}_{\theta}\cdot P_{T_P}\| Q_{P}^{(n)}) \nonumber \\
\le & \sum_{x \in {\cal X}}
\Bigl(\frac{k}{2}\log \frac{n P(x)}{2\pi}\Bigr)
+\frac{d}{2}\log \det J_\theta \nonumber \\
&+d \log \frac{1}{w(\theta)}- \frac{kd}{2s}\log (1+s)+o(1).
\Label{3-28-1b}\end{aligned}$$
Hence, similar to , (\[3-28-1\]) implies that $$\begin{aligned}
-\frac{1}{n}\log
W^{n}_{\theta,x^n}(
\{y^n| W^{n}_{\theta,x^n}(y^n) > Q_{{x}^n}^{(n)}(y^n) e^{n\delta}\})
\to s' \delta \Label{eq8}\end{aligned}$$ for $s'>0$. Since $s'$ is arbitrary, we have $$\begin{aligned}
\lim_{n\to \infty}
-\frac{1}{n}\log
W^{n}_{\theta,x^n}(
\{y^n| W^{n}_{\theta,x^n} (y^n) > Q_{{x}^n}^{(n)}(y^n) e^{n\delta}\})
=
\infty \Label{eq1}.\end{aligned}$$
Error exponent
==============
In this section, we will prove Theorem \[Th1\], i,e., show that the code $\Phi_{n,P,R,R_1}$ satisfies that $$\begin{aligned}
\lim_{n \to \infty}\frac{-1}{n}
\log
e_\theta (\Phi_{n,P,R,R_1})
\ge
\min( \max_{s \in [0,1]}
(s I_{1-s}(P,W_\theta ) -s R_1),
R_1-R ). \Label{eq5}\end{aligned}$$
First, we have $$\begin{aligned}
& \sum_{i=1}^{M_n}\frac{1}{M_n}
W^{n}_{\theta,E_{n,P,R}(i)}({\cal D}_{i}^c) \nonumber \\
\le &
\sum_{i=1}^{M_n}\frac{1}{M_n} \Big(
(W^{n}_{\theta,E_{n,P,R}(i)}(\hat{{\cal D}}_{E_{n,P,R}(i)}^c) \nonumber \\
&\qquad \qquad+
\sum_{j\neq i}W^{n}_{\theta,E_{n,P,R}(i)}(\hat{{\cal D}}_{E_{n,P,R}(j)}))\Big)
\nonumber \\
= &
\sum_{i=1}^{M_n}\frac{1}{M_n}
\Big( W^{n}_{\theta,E_{n,P,R}(i)}(\hat{{\cal D}}_{E_{n,P,R}(i)}^c) \nonumber \\
&+
\sum_{j=1}^{M_n} \frac{1}{M_n}
\sum_{i \neq j} W^{n}_{\theta,E_{n,P,R}(i)}(\hat{{\cal D}}_{E_{n,P,R}(j)})\Big) .
\Label{eq6}\end{aligned}$$
In the following, we evaluate the first term of (\[eq6\]). Any element $x^n \in T_P$ satisfies that $$\begin{aligned}
& W^{n}_{\theta,x^n}(\hat{{\cal D}}_{x^n}^c) \nonumber \\
=&
W^{n}_{\theta,x^n}(
\{y^n| Q_{{x}^n}^{(n)} (y^n) < e^{n R_1} Q_{P}^{(n)} (y^n) \}) \nonumber \\
\le &
W^{n}_{\theta,x^n}(
\{y^n| W^{n}_{\theta,x^n}(y^n) < e^{n (R_1+\delta)} Q_{P}^{(n)} (y^n) \})
\nonumber \\
&+
W^{n}_{\theta,x^n}(
\{y^n| W^{n}_{\theta,x^n}(y^n) > e^{n\delta} Q_{{x}^n}^{(n)}(y^n) \})
\Label{eq7}\end{aligned}$$ because when $W^{n}_{\theta,x^n}(y^n) \le e^{n\delta} Q_{{x}^n}^{(n)}(y^n)$, the condition $Q_{{x}^n}^{(n)} (y^n) < e^{n R_1} Q_{P}^{(n)} (y^n)$ implies $W^{n}_{\theta,x^n}(y^n) < e^{n (R_1+\delta)} Q_{P}^{(n)} (y^n)$. Due to , the exponential decreasing rate of $W^{n}_{\theta,x^n}(\hat{{\cal D}}_{x^n}^c)$ equals that of $W^{n}_{\theta,x^n}(\{y^n| W^{n}_{\theta,x^n}(y^n) < e^{n (R_1+\delta)} Q_{P}^{(n)} (y^n) \})$. To get the latter exponent, we choose any elements $s\in [0,1]$ and $x^n \in T_P$. So, we have $$\begin{aligned}
& W^{n}_{\theta,x^n}(
\{y^n| W^{n}_{\theta,x^n}(y^n) < e^{n (R_1+\delta)} Q_{P}^{(n)} (y^n) \})\nonumber \\
\le &
\int_{{\cal Y}}
W^{n}_{\theta,x^n}(y^n)^{1-s}
e^{n s(R_1+\delta)} Q_{P}^{(n)} (y^n)^s d y^n
\nonumber \\
\stackrel{(a)}{=} &
\sum_{x^n \in T_P}\frac{1}{|T_P|}
\int_{{\cal Y}} W^{n}_{\theta,x^n}(y^n)^{1-s}
e^{n s(R_1+\delta)} Q_P^{(n)} (y^n)^s
d y^n
\nonumber \\
\stackrel{(b)}{\le} &
|T_n({\cal X})|
\nonumber \\
& \cdot
\sum_{x^n \in {\cal X}^n}P^n(x^n)
\int_{{\cal Y}} W^{n}_{\theta,x^n}(y^n)^{1-s}
e^{n s(R_1+\delta)} Q_P^{(n)} (y^n)^s d y^n
\nonumber \\
= &
|T_n({\cal X})|
e^{n s(R_1+\delta)}
\nonumber \\
& \cdot
\int_{{\cal Y}} \Big(\sum_{x^n \in {\cal X}^n}P^n(x^n)
W^{n}_{\theta,x^n}(y^n)^{1-s}\Big)
Q_P^{(n)} (y^n)^s d y^n
\nonumber \\
\stackrel{(c)}{\le} &
|T_n({\cal X})|
e^{n s(R_1+\delta)}
\nonumber \\
& \cdot
\Big(\int_{{\cal Y}} \Big(\sum_{x^n \in {\cal X}^n}P^n(x^n)
W^{n}_{\theta,x^n}(y^n)^{1-s}\Big)^{\frac{1}{1-s}}
d y^n \Big)^{1-s} \nonumber \\
& \cdot
\Big(\int_{{\cal Y}} Q_P^{(n)} (y^n)^\frac{s}{s} d y^n\Big)^{\frac{1}{s}}
\nonumber \\
=&
|T_n({\cal X})|
e^{n s(R_1+\delta)}
\nonumber \\
& \cdot
\Big(\int_{{\cal Y}} \Big(\sum_{x^n \in {\cal X}^n}P^n(x^n)
W^{n}_{\theta,x^n}(y^n)^{1-s}\Big)^{\frac{1}{1-s}}
d y^n \Big)^{1-s}
\nonumber \\
= &
|T_n({\cal X})|
e^{n s(R_1+\delta)}
\nonumber \\
& \cdot
\Big(\int_{{\cal Y}} \Big(\sum_{x \in {\cal X}}P (x)
W_{\theta,x}(y)^{1-s}\Big)^{\frac{1}{1-s}}
d y^n\Big)^{n(1-s)} \nonumber \\
=& |T_n({\cal X})|
e^{n (s(R_1+\delta)-s I_{1-s}(P,W_\theta ) )},\Label{eq9}\end{aligned}$$ where $(a)$ follows from the relation $
\int_{{\cal Y}} W^{n}_{\theta,x^n}(y^n)^{1-s}
e^{n s(R_1+\delta)} Q_{P}^{(n)} (y^n)^s dy^n=
\int_{{\cal Y}} W^{n}_{\theta,{x^n}'}(y^n)^{1-s}
e^{n s(R_1+\delta)} Q_{P}^{(n)} (y^n)^s dy^n$ with ${x^n}'\neq {x^n}\in T_P $, and $(b)$ and $(c)$ follow from (\[eq3\]) and the Hölder inequality, respectively. Thus, we have $$\begin{aligned}
\lim_{n\to \infty}
-\frac{1}{n}\log W^{n}_{\theta,x^n}(\hat{{\cal D}}_{x^n}^c)
\ge
-s(R_1+\delta) +s I_{1-s}(P,W_\theta ).\end{aligned}$$ Since $\delta>0$ is arbitrary, we have $$\begin{aligned}
\lim_{n\to \infty}
-\frac{1}{n}\log W^{n}_{\theta,x^n}(\hat{{\cal D}}_{x^n}^c)
\ge
-s R_1+ s I_{1-s}(P,W_\theta ).
\Label{eq10}\end{aligned}$$
Next, we proceed to the second term of (\[eq6\]). In the following, we simplify $W_{\theta}\cdot P $ to be $W_{\theta,P} $. $$\begin{aligned}
& \frac{1}{M_n} \sum_{j=1}^{M_n} \sum_{i \neq j}
W^{n}_{\theta,E_{n,P,R}(i)}(\hat{{\cal D}}_{E_{n,P,R}(j)}) \nonumber\\
\stackrel{(a)}{=} &
\sum_{j=1}^{M_n} \sum_{i\neq j}
\sum_{g\in S_{E_{n,P,R}(j)}}\frac{1}{|S_{E_{n,P,R}(j)}|}
\frac{1}{M_n}
W^{n}_{\theta,g (E_{n,P,R}(i))}(\hat{{\cal D}}_{E_{n,P,R}(j)}) \nonumber\\
= &
\sum_{j=1}^{M_n}
\sum_{g\in S_{E_{n,P,R}(j)}}\frac{1}{|S_{E_{n,P,R}(j)}|}
\sum_{{x'}^n (\neq E_{n,P,R}(j) )\in T_P}
P_{\hat{\cal M}_{n}} (g({x'}^n))
W^{n}_{\theta,{x'}^n}(\hat{{\cal D}}_{E_{n,P,R}(j)})
\nonumber\\
\stackrel{(b)}{\le} &
\frac{e^{n^{\frac{1}{4}}}}{c_{n,P}}
\sum_{j=1}^{M_n}
\sum_{x^n \in {\cal X}^n}
P_{T_P}(x^n)
W^{n}_{\theta,x^n}(\hat{{\cal D}}_{E_{n,P,R}(j)}) \nonumber\\
=&
\frac{e^{n^{\frac{1}{4}}}}{c_{n,P}}
\sum_{j=1}^{M_n}
W_{\theta}^n\cdot P_{T_P}(\hat{{\cal D}}_{E_{n,P,R}(j)}) \nonumber\\
=&
e^{n^{\frac{1}{4}}}
\sum_{j=1}^{M_n}
W_{\theta}^n\cdot P_{T_P}
\{y^n|
Q_{E_{n,P,R}(j)}^{(n)} (y^n) \ge e^{n R_1} Q_P^{(n)} (y^n)
\} \nonumber\\
\le &
\frac{e^{n^{\frac{1}{4}}}}{c_{n,P}}
\sum_{j=1}^{M_n}
W_{\theta}^n\cdot P_{T_P}
\{y^n|
Q_{E_{n,P,R}(j)}^{(n)} (y^n) \ge e^{n (R_1-\delta)} W_{\theta}^n\cdot P_{T_P} (y^n)
\} \nonumber\\
& +
\frac{e^{n^{\frac{1}{4}}}}{c_{n,P}}
\sum_{j=1}^{M_n}
W_{\theta}^n\cdot P_{T_P}
\{y^n|
W_{\theta}^n\cdot P_{T_P}(y^n) > e^{n \delta} Q_P^{(n)}(y^n)
\} \nonumber\\
\le &
\frac{e^{n^{\frac{1}{4}}}}{c_{n,P}}
\sum_{j=1}^{M_n}
e^{-n (R_1-\delta)}
Q_{E_{n,P,R}(j)}^{(n)}
\{y^n|
Q_{E_{n,P,R}(j)}^{(n)} (y^n) \ge e^{n (R_1-\delta)} W_{\theta}^n\cdot P_{T_P} (y^n)
\}
\nonumber \\
& +
\frac{e^{n^{\frac{1}{4}}}}{c_{n,P}}
\sum_{j=1}^{M_n}
W_{\theta}^n\cdot P_{T_P}
\{y^n|
Q_P^{(n)} (y^n) > e^{n \delta} W_{\theta}^n\cdot P_{T_P} (y^n)
\} \nonumber \\
\le &
\frac{e^{n^{\frac{1}{4}}}}{c_{n,P}}
\sum_{j=1}^{M_n}
e^{-n (R_1-\delta)}
+
\frac{e^{n^{\frac{1}{4}}}}{c_{n,P}}
M_n
W_{\theta}^n\cdot P_{T_P}
\{y^n|
Q_P^{(n)} (y^n) > e^{n \delta} W_{\theta}^n\cdot P_{T_P} (y^n)
\} \nonumber \\
= &
\frac{e^{n^{\frac{1}{4}}}}{c_{n,P}}
M_n
e^{-n (R_1-\delta)} +
e^{\sqrt{n}}
M_n
W_{\theta}^n\cdot P_{T_P}
\{y^n|
Q_P^{(n)} (y^n) > e^{n \delta} W_{\theta}^n\cdot P_{T_P} (y^n)
\}
\Label{eq11}\end{aligned}$$ where $(a)$ follows from the relation $
W^{n}_{\theta,g (E_{n,P,R}(i))}(\hat{{\cal D}}_{E_{n,P,R_1}(j)})
=W^{n}_{\theta,E_{n,P,R}(i)}(\hat{{\cal D}}_{E_{n,P,R_1}(j)}) $ for $g\in S_{E_{n,P,R}(j)}$ and $(b)$ follows from (\[8\]).
Since $W_{\theta}^n\cdot P_{T_P}
\{y^n|
Q_P^{(n)} (y^n) > e^{n \delta} W_{\theta}^n\cdot P_{T_P} (y^n)
\}
$ satisfies the same condition as (\[eq1\]) due to , $$\begin{aligned}
\lim_{n \to \infty} \frac{-1}{n}\log
\frac{e^{n^{\frac{1}{4}}}}{c_{n,P}}
M_n
W_{\theta}^n\cdot P_{T_P}
\{y^n|
Q_P^{(n)} (y^n) > e^{n \delta} W_{\theta}^n\cdot P_{T_P} (y^n)
\}
=\infty.\Label{eq13}\end{aligned}$$ Since $ \lim_{n \to \infty} \frac{-1}{n}\log
\frac{e^{n^{\frac{1}{4}}}}{c_{n,P}}
M_n e^{-n (R_1-\delta)}
=R_1-\delta-R$, and imply that $$\begin{aligned}
\lim_{n \to \infty} \frac{-1}{n}\log
\frac{1}{M_n} \sum_{j=1}^{M_n} \sum_{i \neq j}
W^{n}_{\theta,E_{n,P,R}(i)}(\hat{{\cal D}}_{E_{n,P,R}(j)})
\ge R_1-\delta-R.\end{aligned}$$ Since $\delta>0$ is arbitrary, we have $$\begin{aligned}
\lim_{n \to \infty} \frac{-1}{n}\log
\frac{1}{M_n} \sum_{j=1}^{M_n} \sum_{i \neq j}
W^{n}_{\theta,E_{n,P,R}(i)}(\hat{{\cal D}}_{E_{n,P,R}(j)})
\ge R_1-R.\Label{11-27-3}\end{aligned}$$ Combining (\[eq6\]), (\[eq10\]), and (\[11-27-3\]), we obtain (\[eq5\]).
Indeed, the above discussion is valid only under Condition A. However, under Condition B or C, we can construct it in the same way by replacing $Q_w^{n}$ by $Q_E^{n}$ or $Q_F^{n}$ due to Lemmas \[l2-E\] and \[l2-F\].
Second order
============
In this section, we show Theorems \[Th2\] and \[Th3\]. First, we show Theorem \[Th2\]. Next, we show Theorem \[Th3\] by modifying the proof of Theorem \[Th2\]. To satisfy the condition of Theorem \[Th2\], we modify the encoder given in Section \[s3\] and the decoder given in Section \[s5\] as follows. For our encoder, we choose $R$ to be $R_1^*+\frac{R_2^*}{\sqrt{n}}$, and for our decoder, we choose $R_1$ to be $R_1^*+\frac{R_2^*}{\sqrt{n}}+ \frac{1}{n^{2/3}}$. The modified code is denoted by $\Phi_{n,P,R_1^*,R_2^*}$. We evaluate the first and the second terms in , separately. First, we evaluate the first term in , which is upper bounded by two terms in . So, we evaluate the first and the second terms in . For an arbitrary $\delta_2>0$, we substitute $ \delta_2/\sqrt{n} $ into $\delta$ in . Similar to , we can show $$\begin{aligned}
\lim_{n \to \infty} -\frac{1}{\sqrt{n}}\log
W^{n}_{\theta,x^n}(
\{y^n| W^{n}_{\theta,x^n}(y^n) > Q_{{x}^n}^{(n)}(y^n) e^{\sqrt{n}\delta_2}\})
\ge & s' \delta_2. \Label{16-10}\end{aligned}$$ which implies that the second term in goes to zero. Further, for an arbitrary $\delta_3>0$ $$\begin{aligned}
& W^{n}_{\theta,x^n}(
\{y^n| W^{n}_{\theta,x^n}(y^n) <
e^{n R_1^*+ \sqrt{n}(R_2^*+\delta_2)} Q_{P}^{(n)} (y^n) \})
\nonumber \\
\le &
W^{n}_{\theta,x^n}(
\{y^n| W^{n}_{\theta,x^n}(y^n) <
e^{n R_1^*+ \sqrt{n}(R_2^*+\delta_2+\delta_3)} W^{n}_{\theta,P}(y^n) \})
+
W^{n}_{\theta,x^n}(
\{y^n| W^{n}_{\theta,P}(y^n) < e^{-\sqrt{n} \delta_3 } Q_{P}^{(n)} (y^n) \}) \Label{12-25-10}.\end{aligned}$$ We also have $$\begin{aligned}
& W^{n}_{\theta,x^n}(
\{y^n| W^{n}_{\theta,P}(y^n) < e^{-\sqrt{n} \delta_3 } Q_{P}^{(n)} (y^n) \}) \nonumber \\
=&
\sum_{{x^n}' \in T_P} \frac{1}{|T_P|}
W^{n}_{\theta,{x^n}'}(
\{y^n| W^{n}_{\theta,P}(y^n) < e^{-\sqrt{n} \delta_3 } Q_{P}^{(n)} (y^n) \}) \nonumber \\
\le &
|T_n({\cal X})|
\sum_{{x^n}' \in {\cal X}^n}
P^n(x)
W^{n}_{\theta,{x^n}'}(
\{y^n| W^{n}_{\theta,P}(y^n) < e^{-\sqrt{n} \delta_3 } Q_{P}^{(n)} (y^n) \}) \nonumber \\
= &
|T_n({\cal X})|
W^{n}_{\theta,P}
(
\{y^n| W^{n}_{\theta,P}(y^n) < e^{-\sqrt{n} \delta_3 } Q_{P}^{(n)} (y^n) \}) \nonumber \\
\le &
|T_n({\cal X})|
e^{-\sqrt{n} \delta_3 } Q_{P}^{(n)}
(\{y^n| W^{n}_{\theta,P}(y^n) < e^{-\sqrt{n} \delta_3 } Q_{P}^{(n)} (y^n) \}) \nonumber \\
\le &
|T_n({\cal X})|
e^{-\sqrt{n} \delta_3 }
\to 0 \Label{12-25-11}.\end{aligned}$$ Therefore, due to , , and , the limit of the first term in with $\delta= \delta_2/\sqrt{n}$ not larger than the limit of $\lim_{n \to \infty}
W^{n}_{\theta,x^n}(
\{y^n| W^{n}_{\theta,x^n}(y^n) < e^{n R_1^*+ \sqrt{n}(R_2^*+\delta_2+\delta_3)} W^{n}_{\theta,P}(y^n) \})$. So, and guarantee that $$\begin{aligned}
\lim_{n \to \infty}
W^{n}_{\theta,x^n}(\hat{{\cal D}}_{x^n}^c)
\le
\lim_{n \to \infty}
W^{n}_{\theta,x^n}(
\{y^n| W^{n}_{\theta,x^n}(y^n) < e^{n R_1^*+ \sqrt{n}(R_2^*+\delta_2+\delta_3)} W^{n}_{\theta,P}(y^n) \})\end{aligned}$$ for arbitrary $\delta_2>0$ and $\delta_3>0$.
When $ I(P,W_{\theta})>R_1^*$, $$\begin{aligned}
W^{n}_{\theta,x^n}(
\{y^n| W^{n}_{\theta,x^n}(y^n) < e^{n R_1^*+ \sqrt{n}(R_2^*+\delta_2+\delta_3)} W^{n}_{\theta,P}(y^n) \})
\to 0
\Label{12-25-12B}.\end{aligned}$$ When $ I(P,W_\theta)=R_1^*$, due to the central limit theorem with the assumptions of Theorem \[Th2\], any element $x^n \in T_P$ satisfies that $$\begin{aligned}
W^{n}_{\theta,x^n}(
\{y^n| W^{n}_{\theta,x^n}(y^n) < e^{n R_1^*+ \sqrt{n}(R_2^*+\delta_2+\delta_3)} W^{n}_{\theta,P}(y^n) \})
\to
\int_{-\infty}^{\frac{R_2^*+\delta_2+\delta_3}{\sqrt{V(P,W_\theta )}}}
\frac{1}{\sqrt{2\pi}}
\exp( - \frac{x^2}{2}) dx.\Label{12-25-12}\end{aligned}$$ That is, $$\begin{aligned}
\lim_{n \to \infty}
W^{n}_{\theta,x^n}(\hat{{\cal D}}_{x^n}^c)
\le
\int_{-\infty}^{\frac{R_2^*+\delta_2+\delta_3}{\sqrt{V(P,W_\theta )}}}
\frac{1}{\sqrt{2\pi}}
\exp( - \frac{x^2}{2}) dx. \Label{17-4}\end{aligned}$$ In summary, since $\delta_2$ and $\delta_3$ are arbitrary in , we have $$\begin{aligned}
\begin{array}{lll}
\lim_{n \to \infty}
W^{n}_{\theta,x^n}(\hat{{\cal D}}_{x^n}^c)
&= 0 &\hbox{when } I(P,W_{\theta})> R_1^*
\\
\lim_{n \to \infty}
W^{n}_{\theta,x^n}(\hat{{\cal D}}_{x^n}^c)
&\le
\int_{-\infty}^{\frac{R_2^*}{\sqrt{V(P,W_{\theta} )}}}
\frac{1}{\sqrt{2\pi}}
\exp( - \frac{x^2}{2}) dx&\hbox{when } I(P,W_{\theta})=R_1^*.
\end{array}
\Label{eq24}\end{aligned}$$
Now, we proceed to the second term in . Similar to , we have $$\begin{aligned}
& \frac{1}{M_n} \sum_{j=1}^{M_n} \sum_{i \neq j}
W^{n}_{\theta,E_{n,P,R}(i)}(\hat{{\cal D}}_{E_{n,P,R}(j)}) \nonumber\\
=&
\frac{e^{n^{\frac{1}{4}}}}{c_{n,P}}
\sum_{j=1}^{M_n}
W_{\theta}^n\cdot P_{T_P}(\hat{{\cal D}}_{E_{n,P,R}(j)}) \nonumber\\
=&
\frac{e^{n^{\frac{1}{4}}}}{c_{n,P}}
\sum_{j=1}^{M_n}
\int_{\hat{{\cal D}}_{E_{n,P,R}(j)}}
W_{\theta}^n\cdot P_{T_P}(y^n) Q_P^{(n)} (y^n)^{s-1} Q_P^{(n)} (y^n)^{1-s}
d y^n
\nonumber\\
\stackrel{(a)}{\le} &
\frac{e^{n^{\frac{1}{4}}}}{c_{n,P}}
\sum_{j=1}^{M_n}
\Big(\int_{\hat{{\cal D}}_{E_{n,P,R}(j)}}
W_{\theta}^n\cdot P_{T_P}(y^n)^{\frac{1}{s}} Q_P^{(n)} (y^n)^{\frac{s-1}{s}}
d y^n\Big)^{s}
\Big(\int_{\hat{{\cal D}}_{E_{n,P,R}(j)}}
Q_P^{(n)} (y^n)^{\frac{1-s}{1-s}}
d y^n\Big)^{1-s}
\nonumber\\
\le &
\frac{e^{n^{\frac{1}{4}}}}{c_{n,P}}
\sum_{j=1}^{M_n}
\Big(\int_{{\cal Y}^n}
W_{\theta}^n\cdot P_{T_P}(y^n)^{\frac{1}{s}} Q_P^{(n)} (y^n)^{\frac{s-1}{s}}
d y^n\Big)^{s}
Q_P^{(n)} ( \hat{{\cal D}}_{E_{n,P,R}(j)})^{1-s}
\nonumber\\
= &
\frac{e^{n^{\frac{1}{4}}}}{c_{n,P}}
\sum_{j=1}^{M_n}
e^{s(\frac{1}{s}-1)D_{\frac{1}{s}} (W_{\theta}^n\cdot P_{T_P}
\|Q_P^{(n)} ) }
Q_P^{(n)} ( \hat{{\cal D}}_{E_{n,P,R}(j)})^{1-s}
\nonumber\\
= &
\frac{e^{n^{\frac{1}{4}}}}{c_{n,P}}
\sum_{j=1}^{M_n}
e^{(1-s)D_{\frac{1}{s}} (W_{\theta}^n\cdot P_{T_P}
\|Q_P^{(n)} ) }
Q_P^{(n)} ( \hat{{\cal D}}_{E_{n,P,R}(j)})^{1-s}
\nonumber\\
=&
\frac{e^{n^{\frac{1}{4}}}}{c_{n,P}}
M_n
e^{(1-s)D_{\frac{1}{s}} (W_{\theta}^n\cdot P_{T_P}
\|Q_P^{(n)} ) }
Q_P^{(n)} ( \hat{{\cal D}}_{E_{n,P,R}(j)})^{1-s}
\nonumber\\
\le &
\frac{e^{n^{\frac{1}{4}}}}{c_{n,P}}
e^{n R_1^* + \sqrt{n}R_2^*-n^{\frac{1}{4}} }
e^{(1-s)D_{\frac{1}{s}} (W_{\theta}^n\cdot P_{T_P} \|Q_P^{(n)} ) }
e^{-n R_1^* - \sqrt{n}R_2^* - n^{\frac{1}{3}} }
Q_{ E_{n,P,R}(j) }^{(n)} ( \hat{{\cal D}}_{E_{n,P,R}(j)})^{1-s}
\nonumber\\
= &
\frac{1}{c_{n,P}}
e^{- n^{\frac{1}{3}} }
e^{(1-s)D_{\frac{1}{s}} (W_{\theta}^n\cdot P_{T_P} \|Q_P^{(n)} ) },\Label{eq22}\end{aligned}$$ where $(a)$ follows from Hölder inequality.
Here, we choose $s$ to be $\frac{1}{n+1}$. Since implies $$\begin{aligned}
& (1-s)D_{\frac{1}{s}} (W_{\theta}^n\cdot P_{T_P} \|Q_P^{(n)} )
=
(1-\frac{1}{n+1})D_{1+n} (W_{\theta}^n\cdot P_{T_P} \|Q_P^{(n)} )
\nonumber\\
\le & (1-\frac{1}{n+1}) \frac{k}{2}\log \frac{n}{2\pi}+O(1),
\Label{eq11b}\end{aligned}$$ yields $$\begin{aligned}
\lim_{n \to \infty} \frac{1}{M_n} \sum_{j=1}^{M_n} \sum_{i \neq j}
W^{n}_{\theta,E_{n,P,R}(i)}(\hat{{\cal D}}_{E_{n,P,R}(j)}) =0.\Label{eq23}\end{aligned}$$ That is, the second term of goes to zero. Relations , , and yield . Therefore, we complete our proof of Theorem \[Th2\].
Now, we proceed to the proof of Theorem \[Th3\]. Since the function $\theta \mapsto I(P,W_\theta )$ is a $C^1$ function on $\Theta$ and the function $\theta \mapsto V(P,W_\theta )$ is a continuous function on $\Theta$, the likelihood ratio $\frac{1}{n} \log \frac{W^{n}_{\theta,x^n}(Y^n)}{W^{n}_{\theta,P}(Y^n) }$ has the expectation $I(P,W_{\theta_1})+\frac{1}{\sqrt{n}} f(\theta_2)+o(\frac{1}{\sqrt{n}})$ and the variance $\frac{1}{n}V(P,W_\theta )+o(\frac{1}{n}) $ with $\theta=\theta_1 +\frac{1}{\sqrt{n}}\theta_2$ for $x^n \in T_P$. This property yields the relation $$\begin{aligned}
&\lim_{n \to \infty}
W^{n}_{\theta,x^n}(
\{y^n| W^{n}_{\theta,x^n}(y^n) < e^{n R_1^*+ \sqrt{n}R_2^*} W^{n}_{\theta,P}(y^n) \})\nonumber \\
=&
\left\{
\begin{array}{ll}
0
& \hbox{when } I(P,W_{\theta_1} ) > R_1^* \\
\displaystyle \int_{-\infty}^{\frac{R_2^*-f(\theta_2)}{\sqrt{V(P,W_{\theta_1} )}}}
\frac{1}{\sqrt{2\pi}}
\exp( - \frac{x^2}{2}) dx
& \hbox{when } I(P,W_{\theta_1} ) = R_1^*
\end{array}
\right.
\Label{12-25-12x}\end{aligned}$$ for $x^n \in T_P$. Now, we show Theorem \[Th3\] by using this condition.
Similarly, we evaluate the average error probability by discussing the first and second terms of , separately. Fortunately, the upper bound in the RHS of goes to zero compact uniformly with respect to $\theta$ due to Lemma \[l2\]. So, the second term of goes to zero. Now, we evaluate the first term of , which is upper bounded by two terms in . The second term of goes to zero due to . Then, the remaining term is the first term of , which is upper bounded by two terms in . The second term of goes to zero due to . Therefore, it is enough to evaluate $\lim_{n \to \infty}
W^{n}_{\theta,x^n}(
\{y^n| W^{n}_{\theta,x^n}(y^n) < e^{n R_1^*+ \sqrt{n}(R_2^*+\delta_2+\delta_3)} W^{n}_{\theta,P}(y^n) \})$. When $I(P,W_{\theta_1})>R_1^*$, we can discuss it in the same way. When $I(P,W_{\theta_1})=R_1^*$, fortunately, this term can be calculated due to the condition . Since $\delta_2$ and $\delta_3$ are arbitrary, we obtain the desired argument. The compact uniformity also holds from the above discussion.
Indeed, the above discussion is valid only under Condition A. However, under Condition B or C, we can construct it in the same way by replacing $Q_w^{n}$ by $Q_E^{n}$ or $Q_F^{n}$ due to Lemmas \[l2-E\] and \[l2-F\].
Conclusion
==========
We have proposed a universal channel coding for general output alphabet including continuous output alphabets. Although our encoder is the same as the encoder given in the previous paper [@Ha1], we cannot directly apply the decoder given in [@Ha1] because it is not easy to make a distribution that universally approximates any independent and identical distribution in the sense of maximum relative entropy in the continuous alphabet. To overcome the difficulty, we have invented an $\alpha$-Rényi divergence version of Clarke and Barron’s formula for Bayesian average distribution. That is, we have shown that the Bayesian average distribution well approximates any independent and identical distribution in the sense of $\alpha$-Rényi divergence. Then, we have made our universal decoder by applying the information spectrum method to the Bayesian average distribution. We have lower bounded the error exponent of our universal code, which implies that our code attains the mutual information rate. Since our approach covers the discrete and continuous cases and the exponential and the second-order type evaluations for the decoding error probability, our method provides a unified viewpoint for the universal channel coding, which is an advantage over the existing studies [@CK; @de4; @C; @D; @E; @F; @G; @H; @Do].
Further, we have introduced the parametrization $\theta_1+\frac{\theta_2}{\sqrt{n}}$ for our channel, which is commonly used in statistics for discussing the asymptotic local approximation by normal distribution family [@Lehmann; @Vaart]. This parametrization matches the second order parameterization of the coding rate. So, we can expect that this parametrization is applicable to the case when we have an unknown small disturbance in the channel.
Here, we compare our analysis on universal coding with the compound channel [@BBT; @Wolfowitz; @G]. In the compound channel, we focus on the worst case of the average error probability for the unknown channel parameter. Hence, we do not evaluate the average error probability when the channel parameter is not the worst case. However, in the universal coding [@CK], we evaluate the error probability for all possible channels. Hence, we can evaluate how better the average error probability of each case is than the worst case. In particular, our improved second order analysis in Theorem \[Th3\] clarifies its dependence of the unknown parameter $\theta_2$.
Acknowledgment {#acknowledgment .unnumbered}
==============
The author is grateful for Dr. Hideki Yagi to explaining MIMO Gaussian channels and informing the references [@de4; @A; @B; @C; @D; @E; @F; @G; @H]. He is grateful to the referee for Dr. Vincent Tan for helpfu comments. He is also grateful to the referee of the first version of this paper for helpful comments. The author is partially supported by a MEXT Grant-in-Aid for Scientific Research (A) No. 23246071. The author is also partially supported by the National Institute of Information and Communication Technology (NICT), Japan. The Centre for Quantum Technologies is funded by the Singapore Ministry of Education and the National Research Foundation as part of the Research Centres of Excellence programme.
Proof of Lemma \[L11-27-1\]
===========================
When $\inf_{\theta\in \Theta_0}I(P,W_\theta ) \le R$, all terms in are zero. So, we can assume that $\inf_{\theta\in \Theta_0}I(P,W_\theta ) > R$ without loss of generality.
Firstly, we show that $$\begin{aligned}
&\max_{R_1} \min (\max_{s \in [0,1]}(s I_{1-s}(P,W_\theta ) -s R_1),R_1-R)
\nonumber \\
=&\max_{s\in [0,1]}\frac{1}{1+s}(sI_{1-s}(P,W_\theta ) -sR) .\Label{2-15-1}\end{aligned}$$ Since the function $R_1 \mapsto \max_{s \in [0,1]}(sI_{1-s}(P,W_\theta ) -sR_1)$ is monotone decreasing and continuous and the function $R_1 \mapsto R_1-R$ is monotone increasing and continuous, there exists a real number $R_1^*>R$ such that $\max_{s \in [0,1]}(sI_{1-s}(P,W_\theta ) -s R_1^*) =R_1^*-R$. We also choose $s^*:=\argmax _{s \in [0,1]}(sI_{1-s}(P,W_\theta ) -sR_1^*)$. Here, we assume that $s^*\in (0,1)$. $s^*$ satisfies $\frac{d s I_{1-s}(P,W_\theta )}{ds}|_{s=s^*}=R_1^*$. Since $(s^* I_{1-s^*}(P,W_\theta ) -s^* R_1^*) =R_1^*-R$, we have $R_1^*= \frac{R-s^* I_{1-s^*}(P,W_\theta )}{1+s^*}$. So, $\max_{s \in [0,1]}(s I_{1-s}(P,W_\theta ) -s R_1^*) =
\frac{ (s^* I_{1-s^*}(P,W_\theta )-s^* R)}{1+s^*}$ and $\frac{d s I_{1-s}(P,W_\theta )}{ds}|_{s=s^*}=\frac{R-s^* I_{1-s^*}(P,W_\theta )}{1+s^*}$.
Since the first derivative of $\frac{ (s I_{1-s}(P,W_\theta )-s R)}{1+s}$ with respect to $s$ is $\frac{(1+s)\frac{d s I_{1-s}(P,W_\theta )}{ds} -s I_{1-s}(P,W_\theta )-R}{(1+s)^2}$ and $(1+s)\frac{d s I_{1-s}(P,W_\theta )}{ds} -s I_{1-s}(P,W_\theta )-R$ is monotone decreasing for $s \in [0,1]$, $s_*:=\argmax_{s \in [0,1]}
\frac{ (s I_{1-s}(P,W_\theta )-s R)}{1+s}$ satisfies the same condition $\frac{d s I_{1-s}(P,W_\theta )}{ds}|_{s=s_*}=\frac{R-s_* I_{1-s_*}(P,W_\theta )}{1+s_*}$. So, we find that $\max_{s \in [0,1]}\frac{ (s I_{1-s}(P,W_\theta )-s R)}{1+s}=
\frac{ (s^* I_{1-s^*}(P,W_\theta )-s^* R)}{1+s^*}$. Thus, we obtain when $s^*\in (0,1)$. When $s^*=0$, we can show $s_*=0$, which implies . Similarly, we can show when $s^*=1$.
Since $$\begin{aligned}
\max_{R_1} \inf_{\theta\in \Theta_0}\min (\max_{s \in [0,1]}(s I_{1-s}(P,W_\theta ) -s R_1),R_1-R)
\le
\inf_{\theta\in \Theta_0}\max_{R_1} \min (\max_{s \in [0,1]}(s I_{1-s}(P,W_\theta ) -s R_1),R_1-R) ,\end{aligned}$$ it is sufficient to show there exists $R_1$ such that $$\begin{aligned}
\inf_{\theta\in \Theta_0}\min (\max_{s \in [0,1]}(sI_{1-s}(P,W_\theta ) -s R_1),R_1-R)
\ge
\inf_{\theta\in \Theta_0} \max_{s\in [0,1]}\frac{1}{1+s}(sI_{1-s}(P,W_\theta ) -s R) .\Label{2-15-2}\end{aligned}$$ We choose $R_1$ to be $R+ \inf_{\theta\in \Theta_0} \max_{s\in [0,1]}\frac{1}{1+s}(sI_{1-s}(P,W_\theta ) -sR) $. Given a parameter $\theta \in \Theta_0$, using the function $f(s,\theta):=\frac{1}{1+s}(sI_{1-s}(P,W_\theta ) -sR)$ and $s_{\theta}:= \argmax_{s \in [0,1]} f(s,\theta)$, we have $$\begin{aligned}
f(s_{\theta},\theta)
\ge
\inf_{\theta'\in \Theta_0} \max_{s'\in [0,1]} f(s',\theta'),\end{aligned}$$ which implies that $$\begin{aligned}
&
\max_{s \in [0,1]}(sI_{1-s}(P,W_\theta ) -sR_1)
\ge
(s_{\theta} I_{1-s_{\theta}}(P,W_\theta ) -s_{\theta} R_1)
\nonumber \\
=&
(s_{\theta} I_{1-s_{\theta}}(P,W_\theta ) -s_{\theta} R)+ s_{\theta}
\inf_{\theta'\in \Theta_0} \max_{s'\in [0,1]}f(s',\theta')
\nonumber \\
=&
f(s_{\theta},\theta) + s_{\theta}
(f(s_{\theta},\theta)-\inf_{\theta'\in \Theta_0} \max_{s'\in [0,1]} f(s',\theta'))
\ge f(s_{\theta},\theta)
\nonumber \\
\ge &
\inf_{\theta\in \Theta_0} \max_{s\in [0,1]}\frac{1}{1+s}(sI_{1-s}(P,W_\theta ) -sR)
=R_1-R.
\Label{2-15-3}\end{aligned}$$ Thus, $$\begin{aligned}
\min (\max_{s \in [0,1]}(sI_{1-s}(P,W_\theta ) -sR_1),R_1-R)
=
\inf_{\theta\in \Theta_0}
\max_{s\in [0,1]}\frac{1}{1+s}(sI_{1-s}(P,W_\theta ) -sR),\end{aligned}$$ which implies .
Proof of $\alpha$-Rényi divergence version of Clarke-Barron formula
===================================================================
Now, we show Lemma \[l2-A\]. Under the assumption of exponential family, we consider the logarithmic derivatives $l_{\theta}(y):=(l_{\theta,1}(y), \ldots, l_{\theta,k}(y))$, where $$\begin{aligned}
l_{\theta,j}(y)
:=\frac{\partial}{\partial \theta^j} \log P_{\theta}(y)
= g_{j}(y)-\frac{\partial}{\partial \theta^j} \phi(\theta).\end{aligned}$$ The Fisher information matrix $J_{\theta,i,j}$ is given as $$\begin{aligned}
J_{\theta,i,j}
:=
\frac{\partial^2}{\partial \theta^i \partial \theta^j} \phi(\theta)
=
\int_{{\cal Y}}
\frac{\partial}{\partial \theta^i} l_{\theta,j}(y)
P_{\theta}(y)\mu(d y).\end{aligned}$$ Due to the above assumption, $J_{\theta,i,j}$ is continuous for $\theta\in \Theta$. Hence, we have $$\begin{aligned}
\frac{\partial^2}{\partial \theta^i \partial \theta^j} \log P_{\theta}(y)
=-J_{\theta,i,j}
$$ which is independent of $y$.
We firstly prove . Define $$\begin{aligned}
l_{\theta,j;n}(y^n) &:=
\frac{1}{\sqrt{n}}
\sum_{i=1}^n \frac{\partial}{\partial \theta^j}
\log P_{\theta}(y_i) .\end{aligned}$$ Since $P_{\theta}$ is an exponential family, we have $$\begin{aligned}
J_{\theta,j,j';n} :=
-\frac{1}{n}
\sum_{i=1}^n \frac{\partial^2}{\partial\theta^j \partial\theta^{j'}}
\log P_{\theta}(y_i).\end{aligned}$$
In the following discussion, we employ the Laplace approximation (Laplace method of approximation). To use the Taylor expansion at $\theta_0$, we choose $\theta_1(\theta) $ between $\theta $ and $\theta_0$. Then, we have $$\begin{aligned}
&
\frac{P_{\theta}^n(y^n)}{P_{\theta_0}^n(y^n)}
w(\theta) \nonumber \\
=&
\frac{e^{\sum_{i=1}^n \log P_{\theta}(y_i)}
}{P_{\theta_0}^n(y^n)}
w(\theta) \nonumber \\
=&
\frac{e^{
\sum_{i=1}^n \log P_{\theta_0}(y_i)
+ \sum_{j=1}^k (\theta-\theta_0)^j
\sum_{i=1}^n \frac{\partial}{\partial \theta^j}
\log P_{\theta}(y_i)|_{\theta=\theta_0}
+
\sum_{j,j'=1}^k
\frac{(\theta-\theta_0)^j (\theta-\theta_0)^{j'}}{2}
\sum_{i=1}^n \frac{\partial^2}{\partial\theta^j \partial\theta^{j'}}
\log P_{\theta}(y_i)|_{\theta=\theta_1}
}
}{P_{\theta_0}^n(y^n)}
w(\theta) \nonumber \\
=&
e^{
-n (\theta-\theta_0)^T \frac{J_{\theta_1(\theta)}}{2}(\theta-\theta_0)
+\sqrt{n} (\theta-\theta_0)^T l_{\theta_0;n} (y^n)
}
w(\theta) ,\Label{12-25-9}\end{aligned}$$ where the final equation follows from the properties of $l_{\theta;n}(y^n):=(l_{\theta,j;n}(y^n))$ and the matrix $J_{\theta}$.
Next, for an arbitrary $\epsilon>0$, we choose a neighborhood $
U_{\theta_0,\delta}:=\{\theta | \|\theta-\theta_0\| \le \delta\} $ such that $J_{\theta} \le J_{\theta_0} (1+\epsilon)$ and $w(\theta) \ge w(\theta_0) (1-\epsilon)$ for $\theta \in U_{\theta_0,\delta}$. For $\theta \in U_{\theta_0,\delta}$, we have $$\begin{aligned}
&
e^{
-n (\theta-\theta_0)^T \frac{J_{\theta_1(\theta)}}{2}(\theta-\theta_0)
+\sqrt{n} (\theta-\theta_0)^T l_{\theta_0;n} (y^n)
}
w(\theta) \nonumber \\
\ge &
e^{
-n (\theta-\theta_0)^T \frac{J_{\theta_0}(1+\epsilon)}{2}(\theta-\theta_0)
+\sqrt{n} (\theta-\theta_0)^T l_{\theta_0;n} (y^n)
}
w(\theta_0) (1-\epsilon) \Label{12-25-8}\\
= &
e^{\frac{1}{2}
l_{\theta_0;n}(y^n)^T
J_{\theta_0}^{-1} (1+\epsilon)^{-1}
l_{\theta_0;n}(y^n) }
e^{
-
(\sqrt{n} (\theta-\theta_0 ) - (J_{\theta_0}(1+\epsilon ))^{-1} l_{\theta_0;n}(y^n))^T
\frac{J_{\theta_0}(1+\epsilon )}{2}
(\sqrt{n} (\theta-\theta_0 ) - (J_{\theta_0}(1+\epsilon ))^{-1} l_{\theta_0;n}(y^n))
}
w(\theta_0) (1-\epsilon) .\Label{12-25-7}\end{aligned}$$ Then, we focus on the set $B_{y^n}:=\{z \in \mathbb{R}^k| z^T l_{\theta_0;n} (y^n) \ge 0,
\|z\|\le 1\}$ for $y^n$. For $n \ge \frac{1}{\delta^2}$, we have $$\begin{aligned}
& \frac{Q_{w}^n(y^n)}{P_{\theta_0}^n(y^n)}
=\int_{\Theta}
\frac{P_{\theta}^n(y^n)}{P_{\theta_0}^n(y^n)}
w(\theta) d \theta \nonumber \\
\ge & \int_{U_{\theta_0,\delta}}
\frac{P_{\theta}^n(y^n)}{P_{\theta_0}^n(y^n)}
w(\theta) d \theta \nonumber \\
\stackrel{(a)}{\ge} &
\int_{U_{\theta_0,\delta}}
e^{
-n (\theta-\theta_0)^T \frac{J_{\theta_0}(1+\epsilon)}{2}(\theta-\theta_0)
+\sqrt{n} (\theta-\theta_0)^T l_{\theta_0;n} (y^n)
}
w(\theta_0) (1-\epsilon)
d \theta \nonumber \\
\stackrel{(b)}{=} &
\frac{1}{n^{\frac{k}{2}}}
\int_{\| z\|\le \sqrt{n} \delta}
e^{
-z^T \frac{J_{\theta_0}(1+\epsilon)}{2}z
+z^T l_{\theta_0;n} (y^n)
}
w(\theta_0) (1-\epsilon)
d z \nonumber \\
\stackrel{(c)}{\ge} &
\frac{1}{n^{\frac{k}{2}}}
\int_{B_{y^n}}
e^{
-z^T \frac{J_{\theta_0}(1+\epsilon)}{2}z
+z^T l_{\theta_0;n} (y^n)
}
w(\theta_0) (1-\epsilon)
d z \nonumber \\
\stackrel{(d)}{\ge} &
\frac{1}{n^{\frac{k}{2}}}
e^{-\frac{ \|J_{\theta_0}\|(1+\epsilon)}{2}}
\int_{B_{y^n}}
w(\theta_0) (1-\epsilon)
d z \nonumber \\
\stackrel{(e)}{\ge} &
\frac{1}{n^{\frac{k}{2}}}
e^{-\frac{ \|J_{\theta_0}\|(1+\epsilon)}{2}}
\frac{\pi^{\frac{k}{2}}}{2\Gamma(\frac{k}{2}+1)}
(1-\epsilon) ,\end{aligned}$$ where $(a)$, $(b)$, $(c)$, and $(d)$ follow from and , the relation $z= \sqrt{n} (\theta-\theta_0)$, the relation $n \ge \frac{1}{\delta^2}$, and the relation $ \|J_{\theta_0} \| \ge z^T J_{\theta_0}z$ for $\|z\| \le 1$, respectively. The inequality $(e)$ is shown because the volume of $B_{y^n}$ is $\frac{\pi^{\frac{k}{2}}}{2\Gamma(\frac{k}{2}+1)}$. That is, we have $$\begin{aligned}
\frac{1}{n^{\frac{k}{2}}}\cdot
\frac{P_{\theta_0}^n(y^n)}
{Q_{w}^n(y^n)}
\le
e^{\frac{ \|J_{\theta_0}\|(1+\epsilon)}{2}}
\frac{2\Gamma(\frac{k}{2}+1)}{( 1-\epsilon ) \pi^{\frac{k}{2}}}
\Label{12-25-2}.\end{aligned}$$
Therefore, for $n \ge \frac{1}{\delta^2}$, using we have $$\begin{aligned}
& n^{-s\frac{k}{2}}
e^{s D_{1+s}(P_{\theta_0}^n\|Q_{w}^n)}\nonumber \\
=&
n^{-s\frac{k}{2}}
E_{P_{\theta_0}^n}
[(\frac{P_{\theta_0}^n(Y^n)}{Q_{w}^n(Y^n)})^s]
\nonumber \\
\le &
E_{P_{\theta_0}^n}
[
e^{s\frac{ \|J_{\theta_0}\|(1+\epsilon)}{2}}
(\frac{2\Gamma(\frac{k}{2}+1)}{( 1-\epsilon ) \pi^{\frac{k}{2}}})^s
]
\nonumber \\
\le &
e^{s\frac{ \|J_{\theta_0}\|(1+\epsilon)}{2}}
(\frac{2\Gamma(\frac{k}{2}+1)}{( 1-\epsilon ) \pi^{\frac{k}{2}}})^s.\Label{16-b}\end{aligned}$$ Substituting $n$ into $s$ in , we have $$\begin{aligned}
D_{1+n}(P_{\theta}^n\|Q_{w}^n) - \frac{k}{2}\log n
=
\frac{1}{n}\log n^{-n\frac{k}{2}}
e^{n D_{1+n}(P_{\theta_0}^n\|Q_{w}^n)}
\le
\log [e^{\frac{ \|J_{\theta_0}\|(1+\epsilon)}{2}}
(\frac{2\Gamma(\frac{k}{2}+1)}{( 1-\epsilon ) \pi^{\frac{k}{2}}})],\end{aligned}$$ which implies .
When the continuity in the assumption is compact uniform for $\theta$, we can choose a common constant $\delta>0$ in a compact subset in $\Theta$. Since the constant $\delta$ decides the range for $n$ in the above discussion, the constant on the RHS of can be chosen compact uniformly for $\theta$.
Next, for a deeper analysis for proving , we fix an arbitrary small real number $\epsilon>0$. Then, we choose a sufficiently large real number $R$ and a large integer $N_1$ such that $$\begin{aligned}
P_{\theta_0} (C_n^c) \le \epsilon\Label{12-25-3}\end{aligned}$$ for $n \ge N_1$, where $C_n:= \{ y^n | \|l_{\theta_0;n}(y^n) \| < R\}$. Then, we evaluate $\frac{Q_{w}^n(y^n)}{P_{\theta_0}^n(y^n)}$ under the assumption $\|l_{\theta_0;n}(y^n) \|< R$ as follows. Then, we can choose sufficiently large $N_2$ such that $$\begin{aligned}
& \int_{ \|z\| \le \sqrt{n} \delta}
e^{
-
(z - (J_{\theta_0}(1+\epsilon ))^{-1} l_{\theta_0;n}(y^n))^T
\frac{J_{\theta_0}(1+\epsilon )}{2}
(z - (J_{\theta_0}(1+\epsilon ))^{-1} l_{\theta_0;n}(y^n))
}
d z \nonumber \\
\ge &
\frac{ (2\pi)^{\frac{k}{2}}}{ (\det (J_{\theta_0}))^{\frac{1}{2}}
(1+\epsilon )^{\frac{k}{2}}}
(1-\epsilon) \Label{12-25-a}\end{aligned}$$ for $n \ge N_2$ and $y^n$ satisfying that $\|l_{\theta_0;n}(y^n) \|< R$ because the limit of LHS of is $\frac{ (2\pi)^{\frac{k}{2}}}{ (\det (J_{\theta_0}))^{\frac{1}{2}}
(1+\epsilon )^{\frac{k}{2}}}$. Thus, when $n \ge N_2$, we have $$\begin{aligned}
& \frac{Q_{w}^n(y^n)}{P_{\theta_0}^n(y^n)}
=\int_{\Theta}
\frac{P_{\theta}^n(y^n)}{P_{\theta_0}^n(y^n)}
w(\theta) d \theta \nonumber \\
\ge & \int_{U_{\theta_0,\delta}}
\frac{P_{\theta}^n(y^n)}{P_{\theta_0}^n(y^n)}
w(\theta) d \theta \nonumber \\
\stackrel{(a)}{\ge} &
e^{\frac{1}{2}
l_{\theta_0;n}(y^n)^T
J_{\theta_0}^{-1} (1+\epsilon)^{-1}
l_{\theta_0;n}(y^n) } \nonumber\\
& \cdot \int_{U_{\theta_0,\delta}}
e^{
-
(\sqrt{n} (\theta-\theta_0 ) - (J_{\theta_0}(1+\epsilon ))^{-1} l_{\theta_0;n}(y^n))^T
\frac{J_{\theta_0}(1+\epsilon )}{2}
(\sqrt{n} (\theta-\theta_0 ) - (J_{\theta_0}(1+\epsilon ))^{-1} l_{\theta_0;n}(y^n))
}
w(\theta_0) (1-\epsilon)
d \theta \nonumber \\
\stackrel{(b)}{=} &
n^{-\frac{k}{2}}
e^{\frac{1}{2}
l_{\theta_0;n}(y^n)^T
J_{\theta_0}^{-1} (1+\epsilon)^{-1}
l_{\theta_0;n}(y^n) }\nonumber \\
& \cdot \int_{ \|z\| \le \sqrt{n} \delta}
e^{
-
(z - (J_{\theta_0}(1+\epsilon ))^{-1} l_{\theta_0;n}(y^n))^T
\frac{J_{\theta_0}(1+\epsilon )}{2}
(z - (J_{\theta_0}(1+\epsilon ))^{-1} l_{\theta_0;n}(y^n))
}
w(\theta_0) (1-\epsilon)
d z \nonumber \\
\stackrel{(c)}{\ge} &
n^{-\frac{k}{2}}
e^{\frac{1}{2}
l_{\theta_0;n}(y^n)^T
J_{\theta_0}^{-1} (1+\epsilon)^{-1}
l_{\theta_0;n}(y^n) }
w(\theta_0) (1-\epsilon)^2
\frac{ (2\pi)^{\frac{k}{2}}}{ (\det (J_{\theta_0}))^{\frac{1}{2}}
(1+\epsilon )^{\frac{k}{2}}}\Label{12-25-1},\end{aligned}$$ where $(a)$, $(b)$, and $(c)$ follow from and , the relation $z= \sqrt{n} (\theta-\theta_0)$, and .
Now, we introduce a notation. For a distribution $P$, a subset $S$ of the probability space, and a random variable $X$, we denote the value $\int_{S} X(\omega) P(d \omega)$ by $E_{P|S}[X]$. Therefore, using this notation, we have $$\begin{aligned}
& n^{-s\frac{k}{2}}
e^{s D_{1+s}(P_{\theta_0}^n\|Q_{w}^n)}\nonumber \\
=&
n^{-s\frac{k}{2}}
E_{P_{\theta_0}^n}
\Bigl[(\frac{P_{\theta_0}^n(Y^n)}{Q_{w}^n(Y^n)})^s
\Bigr]
\nonumber \\
= &
n^{-s\frac{k}{2}}
E_{P_{\theta_0}^n|C_n}
\Bigl[(\frac{P_{\theta_0}^n(Y^n)}{Q_{w}^n(Y^n)})^s\Bigr]
+
n^{-s\frac{k}{2}}
E_{P_{\theta_0}^n|C_n^c}
\Bigl[(\frac{P_{\theta_0}^n(Y^n)}{Q_{w}^n(Y^n)})^s\Bigr]
\nonumber \\
\stackrel{(a)}{\le} &
E_{P_{\theta_0}^n|C_n}
\Bigl[
e^{-\frac{s}{2}
l_{\theta_0;n}(y^n)^T
J_{\theta_0}^{-1} (1+\epsilon)^{-1}
l_{\theta_0;n}(y^n) }
w(\theta_0)^{-s} (1-\epsilon)^{-2s}
\frac
{ (\det (J_{\theta_0}))^{\frac{s}{2}}(1+\epsilon )^{\frac{sk}{2}}}
{ (2\pi)^{\frac{sk}{2}}}
\Bigr]
\nonumber \\
& +
E_{P_{\theta_0}^n|C_n^c}
\Bigl[
e^{s\frac{ \|J_{\theta_0}\|(1+\epsilon)}{2}}
(\frac{2\Gamma(\frac{k}{2}+1)}{( 1-\epsilon ) \pi^{\frac{k}{2}}})^s
\Bigr]
\nonumber \\
\stackrel{(b)}{\le} &
E_{P_{\theta_0}^n}
\Bigl[
e^{-(1+\epsilon)^{-1}
\frac{s}{2}
l_{\theta_0;n}(y^n)^T
J_{\theta_0}^{-1}
l_{\theta_0;n}(y^n) }
w(\theta_0)^{-s} (1-\epsilon)^{-2s}
\frac
{ (\det (J_{\theta_0}))^{\frac{s}{2}}(1+\epsilon )^{\frac{sk}{2}}}
{ (2\pi)^{\frac{sk}{2}}}
\Bigr]
\nonumber \\
& +
e^{s\frac{ \|J_{\theta_0}\|(1+\epsilon)}{2}}
(\frac{2\Gamma(\frac{k}{2}+1)}{( 1-\epsilon ) \pi^{\frac{k}{2}}})^s
\epsilon,\end{aligned}$$ where $(a)$ and $(b)$ follow from , and , respectively.
Due to the central limit theorem, the random variable $ l_{\theta_0;n}(Y^n)^T J_{\theta_0}^{-1} l_{\theta;n}(Y^n)$ asymptotically obeys the $\chi^2$ distribution with the degree $k$. Hence, the expectation $E_{P_{\theta_0}^n}
[e^{-(1+\epsilon)^{-1} \frac{s}{2}
l_{\theta_0;n}(Y^n)^T J_{\theta_0}^{-1} l_{\theta_0;n}(Y^n)}]$ converges to $\frac{1}{(1+s(1+\epsilon)^{-1})^{k/2}}
$. Thus, $$\begin{aligned}
& \lim_{n \to \infty} n^{-s\frac{k}{2}}
e^{s D_{1+s}(P_{\theta_0}^n\|Q_{w}^n)}\nonumber \\
\le &
\frac{1}{(1+s(1+\epsilon)^{-1})^{k/2}}
w(\theta_0)^{-s} (1-\epsilon)^{-2s}
\frac
{ (\det (J_{\theta_0}))^{\frac{s}{2}}(1+\epsilon )^{\frac{sk}{2}}}
{ (2\pi)^{\frac{sk}{2}}}
\nonumber \\
& +
e^{s\frac{ \|J_{\theta_0}\|(1+\epsilon)}{2}}
(\frac{2\Gamma(\frac{k}{2}+1)}{( 1-\epsilon ) \pi^{\frac{k}{2}}})^s
\epsilon.\end{aligned}$$ Since $\epsilon>0$ is arbitrary, we have $$\begin{aligned}
& \lim_{n \to \infty} n^{-s\frac{k}{2}}
e^{s D_{1+s}(P_{\theta_0}^n\|Q_{w}^n)}\nonumber \\
\le &
\frac{1}{(1+s)^{k/2}}
w(\theta_0)^{-s}
\frac
{ (\det (J_{\theta_0}))^{\frac{s}{2}}}
{ (2\pi)^{\frac{sk}{2}}},\end{aligned}$$ which implies .
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[^1]: This research was partially supported by the MEXT Grant-in-Aid for Scientific Research (A) No. 23246071, and was also partially supported by the National Institute of Information and Communication Technology (NICT), Japan. The Centre for Quantum Technologies is funded by the Singapore Ministry of Education and the National Research Foundation as part of the Research Centres of Excellence programme. This paper was presented in part at 2014 the International Symposium on Information Theory and Its Applications, Melbourne, Australia from 26 to 29 October 2014.
[^2]: M. Hayashi is with Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, 464-8602, Japan, and Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117542. (e-mail: [email protected])
[^3]: While the review paper [@C] discussed a universal code in the both cases, their treatments were separated.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We study a small piece of two dimensional Toda lattice as a complex dynamical system. In particular the Julia set, which appears when the piece is deformed, is shown analytically how it disappears as the system approaches to the integrable limit.'
---
:[$$}
\def\e:{$$]{} :[$$\begin{aligned}
}
\def\ee:{\end{aligned}$$]{}
=-0.5cm =-0.5cm =0.7cm
Introduction
============
The two dimensional Toda lattice is one of soliton equations which has become more and more important as a key object in theoretical physics. It was first formulated by Hirota in 1981[@Hirota] as a discrete version of two continuous time Toda lattice and shown by Miwa[@Miwa] its equivalence to the KP hierarchy. When quasi periodic solutions are substituted, it is nothing but the identity known as Fay’s trisecant formula which characterizes algebraic curves.
This equation has become known in other fields of physics in the last ten years. It was shown being satisfied by the string amplitudes in particle physics[@S]. More recently there appeared papers demonstrating unexpected correlation of this equation with other topics in physics. The transfer matrix of the solvable lattice model with $A_l$ symmetry, for example, was shown to satisfy this equation[@KLWZ][@Kuniba]. This equation has been also proven to unify discrete Painlevé equations[@Ramani]. The connection of solvable cellular automata to this equation offers another example[@TTMS].
Completely integrable nonlinear systems must play fundamental roles in various phenomena in physics. It is remarkable that many integrable systems in different fields are unified into the single equation. We are interested in clarifying ultimate notion of integrability of the systems. Investigation of such systems themselves, however, will not reveal all of features of the systems. The real meaning of integrability will be clarified only in comparison with nonintegrable systems.
An arbitrary deformation of the two dimensional Toda lattice will destroy integrability and create chaos. Since the system contains infinite number of degrees of freedom it is extremely difficult to study analytically the behaviour of transition from nonintegrable to integrable phases. It should be recalled that a very little is known about analytical properties of nonintegrable systems. The main part of the studies of complex dynamical systems were limited to simple systems with one degree of freedom.
Very recently we pointed out[@SS; @toda30] that a set of lattice points, which form a parallelogram in the two dimensional lattice space, constitute a piece of the Toda lattice. We call it a Toda molecule[@Toda; @molecule] since it is essentially what is intended to be called by this name, but used in a bit different context in the literature. The remarkable fact is that the small pieces can be separated from other parts without loosing any properties of the original Toda lattice.
The purpose of this paper is to study in detail analytical properties of the smallest piece of Toda molecules. The smallest Toda molecule is a smallest parallelogram of four lattice points.We will call it a Toda atom for convenience. Since every Toda molecule preserves properties possessed by the Toda lattice, we can study analytical properties of the system from the knowledge of a Toda atom. In the first part of this paper we show that the time evolution of a Toda atom is described by an iterative Möbius map. The form invariance of this map certificates integrability of this system. In the second part of this paper we will consider a deformation of this piece. Under generic deformation a chaos will be generated through the time evolution. We are especially concerned with analytical property of the Julia set as the system approaches to the integrable map. We will show how the Julia set converges to the points on the orbit of Möbius map as a parameter, which interpolates between integrable and nonintegra
Pieces of Toda Lattice
======================
In this section we like to show that the two dimensional Toda lattice can be cut into small pieces without loosing any properties possessed by the original system. To begin with let us write down the equation which was derived by Hirota as a discrete version of the two continuous time Toda lattice[@Hirota]: : && g\_n(l+1,m)g\_n(l,m+1) + g\_n(l,m)g\_n(l+1,m+1)&&- (+) g\_[n+1]{}(l+1,m)g\_[n-1]{}(l,m+1)=0,, ,l,m,n. [\[eqn:HBDE\]]{} : We called this equation Hirota bilinear difference equation[^1] and abbreviated as HBDE. This is a nonlinear system defined on the three dimensional lattice space. Our key observation is the following. For a fixed point of the lattice $(l,m, n)=(\bar l,\bar m,\bar n)$ , we denote by the set of points $(\bar l,\bar m,\bar n), (\bar l+1,\bar
m,\bar n), (\bar l,\bar m+1,\bar n),(\bar l+1,\bar m+1,\bar n) , (\bar l+1,\bar
m,\bar n+1), (\bar l,\bar m+1,\bar n-1)$ . Then if $g_n(l,m)$ is a solution of $({\ref{eqn:HBDE}})$, : f(l,m,n)= [\[eqn:Toda atom\]]{} : is also a solution of $({\ref{eqn:HBDE}})$. This is the smallest piece of the Toda lattice.
The proof is simple. Because is surrounded by zero, every equation on other pieces is automatically satisfied. The result can be easily generalized to larger parallelogram prism when it is surrounded by zero. We call it a Toda molecule according to ref.[@Toda; @molecule]. Then it will be natural to call $({\ref{eqn:Toda atom}})$ a Toda atom. If there are many Toda molecules in the three dimensional lattice space separated by zeros from each other it is again a solution of $({\ref{eqn:HBDE}})$. A slice perpendicular to the $l$ axis of such example is presented in Fig. 1.
(70,75) (0,20)[(1,0)[60]{}]{}(62,18)[(3,3)\[l\][$n$]{}]{} (30,5)[(0,1)[55]{}]{}(28,62)[(3,3)\[l\][$m$]{}]{} (25,70)[(10,3)\[c\][Fig. 1]{}]{} (5,55)(10,55)(15,55) (20,55)(25,55) (30,55)(35,55)(40,55) (45,55)(50,55)(55,55) (5,50)(10,50)(15,50) (20,50)(25,50)(30,50) (35,50)(40,50)(45,50) (50,50)(55,50) (5,50)[(1,0)[30]{}]{}(45,50)[(1,0)[10]{}]{} (5,50)[(1,-1)[5]{}]{}(10,50)[(1,-1)[5]{}]{} (15,50)[(1,-1)[5]{}]{}(20,50)[(1,-1)[5]{}]{} (25,50)[(1,-1)[5]{}]{}(30,50)[(1,-1)[5]{}]{} (35,50)[(1,-1)[5]{}]{}(45,50)[(1,-1)[5]{}]{} (50,50)[(1,-1)[5]{}]{}(55,50)[(1,-1)[4]{}]{} (5,45)(10,45)(15,45) (20,45)(25,45)(30,45) (35,45)(40,45)(45,45) (50,45)(55,45) (5,45)[(1,0)[35]{}]{}(50,45)[(1,0)[7]{}]{} (5,40)(10,40)(15,40) (20,40)(25,40) (30,40)(35,40)(40,40) (45,40)(50,40)(55,40) (5,35)(10,35)(15,35) (20,35)(25,35)(30,35) (35,35)(40,35)(45,35) (50,35)(55,35) (20,35)[(1,0)[5]{}]{}(35,35)[(1,0)[10]{}]{} (20,35)[(1,-1)[28]{}]{}(25,35)[(1,-1)[28]{}]{} (35,35)[(1,-1)[10]{}]{}(40,35)[(1,-1)[10]{}]{} (45,35)[(1,-1)[10]{}]{} (5,30)(10,30)(15,30) (20,30)(25,30) (30,30)(35,30)(40,30) (45,30)(50,30)(55,30) (5,30)[(1,0)[10]{}]{}(25,30)[(1,0)[5]{}]{}(40,30)[(1,0)[10]{}]{} (5,30)[(1,-1)[5]{}]{}(10,30)[(1,-1)[5]{}]{} (15,30)[(1,-1)[5]{}]{}(5,25)(10,25)(15,25) (20,25)(25,25)(30,25) (35,25)(40,25)(45,25) (50,25)(55,25) (10,25)[(1,0)[10]{}]{}(30,25)[(1,0)[5]{}]{} (45,25)[(1,0)[10]{}]{} (5,20)(10,20)(15,20) (20,20)(25,20) (30,20)(35,20)(40,20) (45,20)(50,20)(55,20) (35,20)[(1,0)[5]{}]{} (5,15)(10,15)(15,15) (20,15)(25,15) (30,15)(35,15)(40,15) (45,15)(50,15)(55,15) (15,15)[(1,0)[5]{}]{}(40,15)[(1,0)[5]{}]{}(15,15)[(1,-1)[5]{}]{} (20,15)[(1,-1)[5]{}]{} (5,10)(10,10)(15,10) (20,10)(25,10) (30,10)(35,10)(40,10) (45,10)(50,10)(55,10) (20,10)[(1,0)[5]{}]{}(45,10)[(1,0)[5]{}]{}
For an illustration let us consider the one soliton state localized on the smallest parallelogram specified by $(m,n)=(0,0), (0,1), (1,-1), (1,0)$ on the $(m,n)$ lattice plane, but allowed to range all integers along $l$. Now we recall that in the usual lattice space the one soliton solution is given by[@Miwa][@SS3] : f\^[1sol]{}(l,m,n)=\_j(1-az\_j)\^[-k\_j]{}+\_j(1-bz\_j)\^[-k\_j]{}. [\[eqn:Miwa1soliton\]]{} : Here $a,b$ are arbitrary constants and $\{z_j\}$ are parameters which determine velocity of the soliton. $\{k_j\}$ are variables taking values on integers. We can choose any three among $\{k_j\}$ to relate them to our variables $(l,m,n)$. Let $k_1, k_2, k_3$ be such three and relate them according to : k\_1=m+n-[12]{},k\_2=-m-[12]{},k\_3=l-n-[12]{}. : Writing $({\ref{eqn:Miwa1soliton}})$ explicitly we find : f\^[1sol]{}(l,0,0)&=&A(1-az\_3)\^[-l]{}+B(1-bz\_3)\^[-l]{}f\^[1sol]{}(l,1,0)&=&A[1-az\_21-az\_1]{}(1-az\_3)\^[-l]{}+B[1-bz\_21-bz\_1]{}(1-bz\_3)\^[-l]{}f\^[1sol]{}(l,0,1)&=&A[1-az\_31-az\_1]{}(1-az\_3)\^[-l]{}+B[1-bz\_31-bz\_1]{}(1-bz\_3)\^[-l]{}f\^[1sol]{}(l,1,-1)&=&A[1-az\_21-az\_3]{}(1-az\_3)\^[-l]{}+B[1-bz\_21-bz\_3]{}(1-bz\_3)\^[-l]{} [\[eqn:1soliton\]]{} : where $$A:=\sqrt{(1-az_1)(1-az_2)(1-az_3)},\si B:=\sqrt{(1-bz_1)(1-bz_2)(1-bz_3)}.$$ We see from $({\ref{eqn:1soliton}})$ that all points belonging to the same piece behave similarly. The parameters are related to $\alpha,\ \beta$ of $({\ref{eqn:HBDE}})$ by $$\alpha=z_1(z_2-z_3),\si \beta=z_2(z_3-z_1),$$ for $({\ref{eqn:1soliton}})$ to satisfy HBDE. If we define the amplitude $\varphi_{mn}(l)$ by : \_[mn]{}(l):=[f(l+1,m,n)f(l-1,m,n)f\^2(l,,m,n)]{} - 1 [\[eqn:amplitude\]]{} : it behaves as : \_[00]{}\^[1sol]{}(l)=[\^2p\^2 (pl+)]{},p:=[12]{},:=[12]{}. [\[eqn:soliton peak\]]{} : This represents a localized peak along the $l$ axis. The other amplitudes $\varphi_{mn}(l)$ behave almost the same but different by the values of the phase $\chi$. If we consider an evolution of the system in variable $l$, a Toda atom is composed of four lattice points. Since there is only one equation of motion $({\ref{eqn:HBDE}})$, they are not independent variables. Three of them can be chosen as we like leaving one to be determined by the equation. Let $z_l$ be $f(l,0,0)$. The other three could be either dependent or independent of $z_l$. If they are independent of $z_l$, the equation of motion is linear in $z_l$. On the other hand if they do depend on $z_l$ they are allowed at most linear in $z_l$, for the equation to remain Hirota bilinear form. Namely we can write : f(l,m,n)=A\_[m,n]{}z\_l+B\_[m,n]{},(m,n)=(0,0),(1,0),(0,1),(1,-1), [\[eqn:linear transf\]]{} : with $A_{0,0}=1,\ B_{0,0}=0$. Upon substituting them together into HBDE, it is easy to see that we obtain an equation of the form : z\_[l+1]{}=[Az\_l+BCz\_l+D]{}. [\[eqn:general Moebius\]]{} : $$A=-\beta B_{1,0}+(\alpha+\beta)B_{0,1}A_{1,-1},\si
B=(\alpha+\beta)B_{0,1}B_{1,-1},$$ $$C=(\alpha+\beta)(A_{1,0}-A_{0,1}A_{1,-1}),\si D=\alpha
B_{1,0}-(\alpha+\beta)A_{0,1}B_{1,-1}.$$ $({\ref{eqn:general Moebius}})$ is a Möbius map. Thererfore the map is integrable.
If we remember that HBDE is invariant under the transformation of $f(l,m,n)\rightarrow e^{al+bm+cn}f(l,m,n)$, the one soliton solution $({\ref{eqn:1soliton}})$ offers an example of $({\ref{eqn:linear transf}})$.
The solution of $({\ref{eqn:general Moebius}})$ can be obtained as follows. A Möbius map has three fixed points. By an appropriate transformation : $z_l\rightarrow
\phi\circ z_l\circ \phi^{-1}$, one of the fixed points can be transformed into 0. After the transformation the map will have the form : z\_[l+1]{}=. [\[eqn:ILM\]]{} : $({\ref{eqn:ILM}})$ is easily solved for an arbitrary initial value $z_0$ to get : z\_l=[\^l z\_01+z\_0]{}. [\[eqn:Moebius\]]{} : Applying to this the inverse transformation : $z_l\rightarrow \phi^{-1}\circ
z_l\circ \phi$ , the general solution to $({\ref{eqn:general Moebius}})$ is obtained. We call the map $({\ref{eqn:ILM}})$ the integrable logistic map (ILM). The meaning of this name will become clear later.
We notice that $({\ref{eqn:ILM}})$ corresponds to the case in which one of the lattice point is fixed constant and the other three points behave the same: : f\^[ILM]{}(l,0,0)&=&f\^[ILM]{}(l,0,1)=f\^[ILM]{}(l,1,-1)=:z\_l,f\^[ILM]{}(l,1,0)&=&[-1]{},=-. [\[eqn:simple case\]]{} :
How does the amplitude look like in this case? To see it we substitute $({\ref{eqn:Moebius}})$ into $({\ref{eqn:amplitude}})$ and get : \^[ILM]{}=[\^2p\^2(pl+)-\^2 p]{},p:=[12]{},:=[12]{}. : The similarity of this result to the one soliton solution $({\ref{eqn:soliton peak}})$ must be apparent.
We may further simplify the equation by : f\^[lin]{}(l,0,0)=:z\_l,f\^[lin]{}(l,1,-1)=f\^[lin]{}(l,1,0)=1-[1]{},f\^[lin]{}(l,0,1)=c (c: [const]{}). : The map turns to be linear : z\_[l+1]{}=z\_l +(1-)c : and yields the solution : z\_l=\^l (z\_0-c)+c. : The corresponding amplitude is : \^[lin]{}(l)=[\^2p\^2(pl+)]{},p=[12]{},=[12]{}. : which is again the form of $({\ref{eqn:soliton peak}})$.
Generalized Logistic Map
========================
As we have learned in the preceeding section the smallest piece of Toda lattice already possesses useful informations of the integrable dynamical systems. In this section we study a deformation of the Toda atom. There could be many different ways of deformation, some of which preserve integrability and some others destroy it. Since we are interested in studying the transition between integrable and nonintegrable maps, we must break integrability of the Toda atom.
For this purpose we recall that the Toda molecules have a characteristic form as seen in Fig. 1. Their cross sections in the $(m,n)$ plane are parallelogram declined to the same direction. It owes to the property of the Toda atom defined in $({\ref{eqn:Toda atom}})$. The very reason of this asymmetry comes from the asymmetry[^2] of HBDE under the exchange of $l$ and $m$ as seen in $({\ref{eqn:HBDE}})$. The equation in which the role of $l$ and $m$ in $({\ref{eqn:HBDE}})$ are exchanged is also integrable. In fact we could start from it without changing none of the results.
From this argument we are tempted to consider the following deformation of HBDE. : && g\_n(l+1,m)g\_n(l,m+1)+ g\_n(l,m)g\_n(l+1,m+1)&&- (+) && =0. [\[eqn:deformed HBDE\]]{} : We notice that this equation is integrable when $\gamma=\gamma'=0$ and $\delta\delta'=1$ ,or $\gamma=\gamma'=1$. Integrability of other cases is not known at this point. Moreover we are not able to separate some small part of lattice independently from the rest as it was done to get a Toda atom. Nevertheless it is worthwhile to study $({\ref{eqn:deformed HBDE}})$ defined on a portion of the lattice shown in Fig. 2a.
(60,50) (5,5)(15,5)(25,5) (35,5)(45,5) (5,15)(15,15)(25,15) (35,15)(45,15) (5,25)(15,25)(25,25) (35,25)(45,25) (5,35)(15,35)(25,35) (35,35)(45,35) (15,25)[(1,0)[20]{}]{}(15,25)[(0,-1)[10]{}]{} (15,15)[(1,0)[20]{}]{}(35,25)[(0,-1)[10]{}]{} (25,25)(30,20)(34,16)(15,25)(20,20)(24,16) (20,45)[(10,3)\[c\][Fig. 2a]{}]{}
(60,50) (5,5)(15,5)(25,5) (35,5)(45,5) (5,15)(15,15)(25,15) (35,15)(45,15) (5,25)(15,25)(25,25) (35,25)(45,25) (5,35)(15,35)(25,35) (35,35)(45,35) (15,25)[(1,0)[20]{}]{}(15,25)[(1,-1)[10]{}]{} (35,25)[(0,-1)[10]{}]{}(25,15)[(1,0)[10]{}]{} (25,25)(30,20)(34,16) (20,45)[(10,3)\[c\][Fig. 2b]{}]{}
In order to proceed further we have to specify the model so that we can study analytical properties of the map explicitly. We will consider, in the following discussion, the map given by $({\ref{eqn:ILM}})$ and its deformation. We also restrict our argument to the case of $\gamma'=0$, $\delta=\delta'^{-1}={\nu\over\mu}$ in $({\ref{eqn:deformed HBDE}})$ for simplicity and define (Fig. 2b) $$f^{GLM}(l,0,0)=f^{GLM}(l,0,1)=f^{GLM}(l,1,-1)=f^{GLM}(l,1,1)=:z_l,$$ : f\^[GLM]{}(l,1,0)=[-1]{} (: [const]{}). : The dynamics of this model is described by the map : z\_[l+1]{}=f(z\_l):=;z\_l ,l. [\[eqn:GLM\]]{} : We call this map a generalized logistic map (GLM). Some properties are listed below :
1. When $\gamma=1$, the map becomes the ordinary logistic map studied in the literature extensively.
2. GLM becomes the logistic equation for all values of the parameters $\gamma,\mu
,\nu$ when the continuous limit of the variable $l$ is taken. To show it let us introduce new variables $u$ and new parameters $a$ and $h$ by : u(l):=[+--1]{}z\_l,ah:=-1. : We replace $z_{l+1}$ by $z_{l+h}$ and take the limit $h\rightarrow 0$. We will find that $({\ref{eqn:GLM}})$ reduces to the logistic equation: : [dudl]{}=au(1-u). [\[eqn:logistic equation\]]{} :
3. GLM includes $({\ref{eqn:ILM}})$ as the special case with $\gamma=0$. This explaines the name of ILM used for $({\ref{eqn:ILM}})$.
4. GLM generates Julia set as long as $\gamma\ne 0$. Hence it is not integrable except for $\gamma=0$. This will be discussed later.
The most important feature of GLM is that it interpolates nonintegrable map to integrable map in the limit of continuous deformation. This fact enables us to study analytically the transition between two phases. The problem we concern in what follows is the analytical properties of the map $({\ref{eqn:GLM}})$. To proceed further it is more convenient to convert the map $({\ref{eqn:GLM}})$ into the standard form of rational map of degree 2: : F(z)=f\^[-1]{}(z)=[z(z+)1+’ z]{}e\^[i]{}[\[eqn:F(z)\]]{} : by the Möbius transformation : (x)=[(1-)x(--)x+(--)e\^[-i]{}]{}. : where : =e\^[-i]{},’=[--2+\^2(--)]{} e\^[i]{}. : The corresponding integrable map turns to be the following case : F(z)=z=e\^[i]{}z,=0 ’=1. [\[eqn:linear map\]]{} :
The main feature of a dynamical system is determined by the nature of fixed points of the map. Namely the multiplier $\Lambda$ at fixed point $a$ of the map $\varphi(z)$ is defined by the derivative of the map at $a$: : :=.[d(z)dz]{}|\_[z=a]{}. : The fixed point $a$ is an attractor of the map if $|\Lambda|<1$, a repeller if $|\Lambda|>1$, and neutral if $|\Lambda|=1$.
In the case of GLM, the fixed points are easily found as : 0,p=--[1-’’-e\^[i]{}]{},. : The corresponding multipliers are : \_0=e\^[i]{},\_p=[2-e\^[i]{}-’e\^[-i]{}1-’]{},\_=’ e\^[-i]{}. :
In the integrable limit $\lambda\lambda'\rightarrow 1$, we observe the following characteristic features:
1. Since : |\_0\_|1, : the map converges either to $0$ or to $\infty$ depending on $|\lambda|=|\mu|<1$ or $>1$.
2. The fixed point $p$ approaches to $-\lambda$ and it turns to a super repeller : |\_p| . :
Julia Sets
==========
In the complex dynamical systems, chaos appears from a Julia set. Given the map $f(z)$ on a Riemann sphere $\bar{\mbox{\boldmath $C$}}=\mbox{\boldmath $C$}\cup \{ \infty \}$, the Riemann sphere is devided into two parts depending on whether the orbits converge or not. A set of initial values whose orbits,together with their neiborhood, converge is called Fatou set $F(f)$. On the other hand, a set which does not is called Julia set $J(f)$. This definition leads to the fact that the Julia set does not contain any attractive periodic cycle. In this sense the orbit in Julia set is chaotic.
By definiton, Fatou set and Julia set are invariant of the map, that is $$f(F)=f^{-1}(F)=F\, ,\: f(J)=f^{-1}(J)=J.$$ It is easy to understand that attractive fixed points belong to Fatou set. Contrary it is known that repulsive fixed points belong to Julia set [@chaos]. Then we can compute Julia set by inversely mapping a repulsive fixed point as an initial value. We show some of their examples in Fig. 3 for the map of $({\ref{eqn:F(z)}})$.
The Julia set does not exist if the map is completely integrable. Integrable maps converge to orbits predictable for any given initial values. Conversely if there exists an orbit not predictable for some initial values, the map is not integrable. Therefore a Julia set appears in nonintegrable maps, but not in integrable maps.
In our standard map of degree 2 given by $({\ref{eqn:F(z)}})$, a Julia set is known to exists except for at the integrable point $\lambda\lambda'=1$. We like to know how it disappears from the complex plane of the variable when the parameters approach to the limit $\lambda\lambda'\ \rightarrow\ 1$. We have given in [@SSSY] an argument about this problem for some limited range of parameters. The purpose of this section is to present another argument which should supplement our previous one.
The inverse map of $({\ref{eqn:F(z)}})$ is easily obtained as : z\_l=F\^[-1]{}(z\_[l+1]{})=[12]{}(z\_[l+1]{}-), [\[eqn:inverse map\]]{} : where we defined : :=’e\^[-i]{}. : From this expression it is apparent that the inverse map is not unique but double valued at every step. As we pointed out in above the inverse map generates points of the Julia set if it starts from a point on the Julia set. Substituting one value of the Julia set into $({\ref{eqn:inverse map}})$, we get two points every time. After $n$ steps the number of points of the Julia set increases as many as $2^{n+1}-1$. This explains the nature of the Julia set. Some of the points could be those of periodic maps. They must be subtracted from the number.
In the integrable limit $\lambda\lambda'\ \rightarrow\ 1$ the inverse map $({\ref{eqn:inverse map}})$ is still double valued. They are : z\_l=. [\[eqn:ILM set\]]{} : We notice that the second solution does not depend on $z_{l+1}$, hence is the same at every step of the map. For $({\ref{eqn:ILM set}})$ to generate the Julia set we must start from a repulsive fixed point. When $|\lambda|>1$, the origin is such a point. Thence we find from $({\ref{eqn:ILM set}})$ the ‘Julia set’[^3] : J\^[ILM]{}={. -\^n | n}: for the integrable map. The number of the ‘Julia set’ increases proportional to the number of the steps $n$. Moreover the element of $J^{ILM}$ is equal to $-\mu^{-n}\lambda$, which is nothing but the solution exactly expected from the map $({\ref{eqn:linear map}})$, if it started from $-\lambda$.
The next problem we concern is to explore how the Julia set of GLM turns into those points of $({\ref{eqn:ILM set}})$ in the limit $\lambda\lambda'\ \rightarrow\
1$. Since we are interested in the transition from a nonintegrable map to the integrable map, we are to consider small values of $|\lambda\lambda'-1|$. The inverse map $({\ref{eqn:inverse map}})$ can be rewritten as : F\^[-1]{}(z)={}E(z) [\[eqn:F\^-1\]]{} : where we put : E(z):=[12]{}(z+)( -1),:=’-1. [\[eqn:E(z)\]]{} : Note that $E(z)$ vanishes for small values of $\epsilon$. To see the behaviour of $E(z)$ for small $\epsilon$ we first observe the inequality which is true for all $\epsilon$: : |E(z)|< 3,\^. [\[eqn:|E(z)|<\]]{} : The proof of this inequality owes to the following facts.
1. If $|w|<1$, : |[1w]{}(-1)|&=&[1|w|]{}|1-| (1-)&&[1|w|]{}(1-(1-|w|))=|w|1. :
2. If $|w|>1$, : |[1w]{}(-1)|&=&|-|< 3. :
Substituting : w:=[4ze\^[-i]{}(z+)\^2]{}, [\[eqn:w\]]{} : into $E(z)$ of $({\ref{eqn:E(z)}})$, we can write : |E(z)|=|[1w]{}(-1)|,: from which $({\ref{eqn:|E(z)|<}})$ follows.
We can perform the inverse map $({\ref{eqn:F^-1}})$ iteratively. Let us denote the map $({\ref{eqn:F^-1}})$ as : A(z):=z+E(z),B(z):=--E(z). : Then the second map becomes : F\^[-2]{}(z)==. : After $n$ steps we obtain : F\^[-n]{}(z)={. A\^[\_1]{}B\^[\_2]{}A\^[\_3]{}B\^[\_n]{}(z)| \_1+\_2++\_n=n}. :
If we had started from the repeller the above maps have produced the Julia set of GLM. In the following we consider the case $|\lambda|>1,\ |\lambda'|<1$, so that the origin is a repulsive fixed point and the infinity is an attractive fixed point. Since $E(0)=0$ the origin is mapped to : A(0)=0,B(0)=-: by the first iteration. The second iteration yields : A\^2(0)=0, AB(0)=-+E(-), BA(0)=-,B\^2(0)=--E(-). [\[eqn:2 iteration\]]{} : We notice that since $E(-\lambda)$ is the order of $\epsilon$ from $({\ref{eqn:|E(z)|<}})$, all of the points after the second iteration are in the neighbourhood of $J^{ILM}$. Proceeding similarly we obtain the Julia set as follows: : J\^[GLM]{}={ . A\^[\_1]{}B\^[\_2]{}A\^[\_3]{}B\^[\_]{}(0)| \_1,\_2, }. [\[eqn:J\^GLM\]]{} :
We remark some important properties which result from this expression.
1. The invariance of the Julia set under the map.
It is obvious from $({\ref{eqn:J^GLM}})$ that : J\^[GLM]{}=A(J\^[GLM]{})B(J\^[GLM]{})=F\^[-1]{}(J\^[GLM]{}). :
2. An element of the form $B\circ X$ for any $X\in J^{GLM}$ belongs to the neibourhood of $-\lambda$, as seen from : BX=--E(X). :
3. An element of the form $A^s\circ B\circ X$ maps $B\circ X$ to the neighbourhood of $-\rho^s\lambda$.
In fact after applying $A$’s $s$ times we get : A\^s(BX)&=&A\^[s-1]{}(BX+E(BX))&=&A\^[s-2]{}(\^2 BX+E(BX)+E(ABX))&=&\^s BX +\_[k=0]{}\^[s-1]{}\^k E(A\^[s-k-1]{}BX)&=&-\^s-\^s E(X) +\_[k=0]{}\^[s-1]{}\^k E(A\^[s-k-1]{}BX). [\[eqn:A\^s(BX)\]]{} :
Since every element of $J^{GLM}$, beside 0, is either the form of $B\circ X$ or $A^s\circ B\circ X$, we conclude that every element of $J^{GLM}$ is in the neighbourhood of $J^{ILM}$.
We now proceed to show that $J^{GLM}$ approaches uniformly to $J^{ILM}$ as $\epsilon$ goes to 0. Since the infinity is an attractive fixed point the Julia set must be in a finite region of the complex plane. We assume that they are inside of the disc of radius $R$, [*i.e.*]{}, $|z|< R,\ ^\forall z\in J^{GLM}$. Therefore we can bound $|E(z)|$ by : |E(z)|< 3R,zJ\^[GLM]{}. :
The summation of $({\ref{eqn:A^s(BX)}})$ can be estimated as : \_[k=0]{}\^[s-1]{}|\^kE(A\^[s-k-1]{}BX)|< (\_[k=0]{}\^[s-1]{}||\^k)3R=[1-||\^s1-||]{}3R, : which vanishes as $|\epsilon|$ approaches to 0 for all integer $s$ because we assume $|\rho|^s=|\lambda'|^s < 1$. This proves that all points in $J^{GLM}$ approach to $J^{ILM}$ uniformly in the integrable limit.
In the above we considered the case of $|\lambda|>1,|\lambda'|<1$. The other case $|\lambda|>1,|\lambda'|<1$ can be also treated similarly if $z_l$ is transformed into $w_l=1/z_l$ in ([\[eqn:F(z)\]]{}). Since this transformation is equivalent to the exchange of the role of $\lambda$ and $\lambda'$ (and replacement of $\theta$ by $-\theta$ ) in ([\[eqn:F(z)\]]{}), we can replay on the $w$-plane the same argument to the above.
We conclude this paper by showing pictures which represent the convergence of the Julia set to the points of iterative maps of the integrable system. The parameters of the map are fixed at $\lambda=4$ and $\theta=0.03\pi$. Under the choice of these parameters, $\lambda'$, hence $\epsilon$, can be changed freely. In the integrable limit $\epsilon=0$, $J^{ILM}=\left\{-4{1\over
4^n}e^{-i0.03\pi n},\ n=0,1,2,\cdots\right\}$.It is shown in Fig. 3c. As $\epsilon$ differs from 0 the Julia set expand from these points as seen in other pictures. The real and imaginary axes are not drawn except in Fig. 3a, so that the points in the neighbourhoods of $z=-4$ and 0 are visible in other figures.
(80,80) (0,0) (35,70)[(10,3)\[c\][Fig. 3a]{}]{} (10,35)[(4,4)\[c\][$- 4$]{}]{} (67,35)[(4,4)\[c\][0]{}]{} (35,5)[(10,3)\[c\][$\epsilon=-1$]{}]{}
(80,80) (0,0) (35,70)[(10,3)\[c\][Fig. 3b]{}]{} (10,35)[(4,4)\[c\][$- 4$]{}]{} (67,35)[(4,4)\[c\][0]{}]{} (35,5)[(10,3)\[c\][$\epsilon=-0.5$]{}]{}
(80,80) (0,0) (35,70)[(10,3)\[c\][Fig. 3c]{}]{} (10,35)[(4,4)\[c\][$- 4$]{}]{} (67,35)[(4,4)\[c\][0]{}]{} (35,5)[(10,3)\[c\][$\epsilon=0$]{}]{}
By studying analytical property of a piece of Toda lattice we attempted to clarify how a nonintegrable system approaches to the integrable one. Our argument is based on the fact that the two dimensional Toda lattice can be disjoined into small pieces, which are integrable by themselves and are called Toda molecules. A Toda molecule is composed from smaller pieces, which we called Toda atoms. Hence the two dimensional Toda lattice is a crystal consisting of Toda atoms. For such a macroscopic system being integrable every piece must be joined very carefully not to create a Julia set.
In this onnection it will be worth while recalling that a similar property is possessed commonly in other integrable models. In the solvable lattice models the partition function is factorizable into a product of Boltzmann weights. The Yang-Baxter equation is a condition imposed on the factors to be connected properly. Another exampele is the factorizability condition imposed on the string amplitudes which led us to the $\tau$ function of the KP hierarchy. In any case the connection rule must be such that the symmetry characterizing the unit blocks is preserved under the coupling.
We will be interested in studying analytically properties of the compound system of two GLM pieces in the forth comming paper.
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[^1]: This equation is also called Hirota-Miwa equation in recent literature.
[^2]: If we had chosen other set of variables, HBDE looked more symmetric and the corresponding Toda atom could be either cubic or octahedron[@DYB]. We have used asymmetric variables such that deformations can be discussed.
[^3]: This set does not possess properties expected for the ordinary Julia set. We call it ‘Julia set’ only in the sense that it is generated by the inverse map starting from a repeller.
|
{
"pile_set_name": "ArXiv"
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abstract: 'Let $X$ be a smooth separated geometrically connected variety over ${\mathbb{F}}_q$ and $f:Y\to X$ a smooth projective morphism. We compare the invariant dimensions of the $\ell$-adic representation $V_\ell$ and the ${\mathbb{F}}_\ell$-representation $\bar V_\ell$ of the geometric étale fundamental group of $X$ arising from the sheaves $R^wf_*{\mathbb{Q}}_\ell$ and $R^wf_*{\mathbb{Z}}/\ell{\mathbb{Z}}$ respectively. These invariant dimension data is used to deduce a maximality result of the geometric monodromy action on $V_\ell$ whenever $\bar V_\ell$ is semisimple and $\ell$ is sufficiently large. We also provide examples for $\bar V_\ell$ to be semisimple for $\ell\gg0$.'
title: Invariant dimensions and maximality of geometric monodromy action
---
[^1]
Chun Yin Hui
Mathematics Research Unit, University of Luxembourg,\
6 rue Richard Coudenhove-Kalergi, L-1359 Luxembourg\
Email: [[email protected]]([email protected])
Introduction
============
Consider a smooth projective ${\mathbb{F}}_q$-morphism $f:Y\to X$, where $X$ is a smooth separated geometrically connected ${\mathbb{F}}_q$-variety. Fix a geometric point $\bar x_0:\mathrm{Spec}(\bar{\mathbb{F}}_q)\to X$. For any prime $\ell\nmid q$ and integer $w$, $\mathscr{F}_\ell:=R^w f_*{\mathbb{Q}}_\ell$ is a *lisse*, *pure of weight* $w$, ${\mathbb{Q}}_\ell$-sheaf on $X$ [@De80] inducing an $\ell$-adic representation of the *étale fundamental group* $\pi_1^{et}(X):=\pi_1^{et}(X,\bar x_0)$ on the stalk $\mathscr{F}_{\ell,\bar x_0}\cong H^w(Y_{\bar x_0},{\mathbb{Q}}_\ell)=:V_\ell$, $$\label{1}
\Phi_\ell:\pi_1^{et}(X)\to {\mathrm{GL}}(V_\ell);$$ $\bar{\mathscr{F}}_\ell:=R^w f_*{\mathbb{Z}}/\ell{\mathbb{Z}}$ is a locally constant sheaf on $X$ inducing an ${\mathbb{F}}_\ell$-representation on the stalk $\bar{\mathscr{F}}_{\ell,\bar x_0}\cong H^w(Y_{\bar x_0},{\mathbb{Z}}/\ell{\mathbb{Z}})=:\bar V_\ell$, $$\label{1'}
\phi_\ell:\pi_1^{et}(X)\to {\mathrm{GL}}(\bar V_\ell).$$ The *geometric étale fundamental group* of $X$, $\pi_1^{et}(X_{\bar{\mathbb{F}}_q}):=\pi_1^{et}(X_{\bar {\mathbb{F}}_q},\bar x_0)$, is a normal subgroup of $\pi_1^{et}(X)$ satisfying the exact sequence $$\label{2}
1\to \pi_1^{et}(X_{\bar{\mathbb{F}}_q})\to \pi_1^{et}(X)\to {\operatorname{Gal}}(\bar {\mathbb{F}}_q/{\mathbb{F}}_q)\to 1$$ so that any $x\in X({\mathbb{F}}_q)$ induces a splitting $i_x$ of (\[2\]). The *monodromy group* $\Gamma_\ell$ (resp. $\bar\Gamma_\ell$) and the *geometric monodromy group* $\Gamma_\ell^{{\operatorname{geo}}}$ (resp. $\bar{\Gamma}_\ell^{{\operatorname{geo}}}$) are defined to be the images of $\pi_1^{et}(X)$ and $\pi_1^{et}(X_{\bar{\mathbb{F}}_q})$ respectively in ${\mathrm{GL}}(V_\ell)$ (resp. ${\mathrm{GL}}(\bar V_\ell)$); their Zariski closures in ${\mathrm{GL}}_{V_\ell}$, denoted respectively by $\mathbf{G}_\ell$ and $\mathbf{G}_\ell^{{\operatorname{geo}}}$, are called the *algebraic monodromy group* and the *algebraic geometric monodromy group* of $\Phi_\ell$.
Since ${\operatorname{Gal}}(\bar {\mathbb{F}}_q/{\mathbb{F}}_q)\cong\hat{{\mathbb{Z}}}$ is abelian, the geometric monodromy groups $\Gamma_\ell^{{\operatorname{geo}}}$ and $\mathbf{G}_\ell^{{\operatorname{geo}}}$ are of particular interest by (\[2\]). Deligne has proved that the identity component of $\mathbf{G}_\ell^{{\operatorname{geo}}}$ is a semisimple subgroup of ${\mathrm{GL}}_{V_\ell}$ [@De80 Cor. 1.3.9, Thm. 3.4.1(iii)]. Determining $\Gamma_\ell^{{\operatorname{geo}}}$ (or $\bar{\Gamma}_\ell^{{\operatorname{geo}}}$) and $\mathbf{G}_\ell^{{\operatorname{geo}}}$ for families of curves (elliptic [@Ha08]; hyperelliptic [@La90s],[@Yu96],[@AP07]; trielliptic [@AP07]) is of independent interest and also has applications to the arithmetic of function fields (see [@Yu96],[@Ac06]) and arithmetic geometry (see [@Ch97],[@Ko06a; @Ko06b; @Ko06c; @Ko08]) over function fields. A crucial point is that for all sufficiently large $\ell$, the geometric monodromy $\Gamma_\ell^{{\operatorname{geo}}}$ is a *large* compact subgroup of $\mathbf{G}_\ell^{{\operatorname{geo}}}({\mathbb{Q}}_\ell)$. The motivation of this paper is to investigate the following large geometric monodromy conjecture. Let $\pi:\mathbf{G}_\ell^{{\operatorname{sc}}}\to \mathbf{G}_\ell^{{\operatorname{geo}}}$ be the natural morphism such that $\mathbf{G}_\ell^{{\operatorname{sc}}}$ is the universal cover of the identity component of $\mathbf{G}_\ell^{{\operatorname{geo}}}$.
[1]{}\[conj\] Let $\Phi_\ell$ be the $\ell$-adic representation defined in (\[1\]). Then $\pi^{-1}(\Gamma_\ell^{{\operatorname{geo}}})$ is a hyperspecial maximal compact subgroup of $\mathbf{G}_\ell^{{\operatorname{sc}}}({\mathbb{Q}}_\ell)$ whenever $\ell$ is sufficiently large.
Let us make a detour to the characteristic zero case. Suppose $f:Y\to X$ is not defined over ${\mathbb{F}}_q$, but over a subfield $K$ of ${\mathbb{C}}$. Denote the $w$th Betti cohomology $H^w(Y_{\bar x_0}({\mathbb{C}}),{\mathbb{Q}})$ by $V$, which is acted on by the topological fundamental group $\pi_1(X({\mathbb{C}}))$. Since the geometric representation $\Phi_\ell:\pi_1^{et}(X_{\bar K})\to {\mathrm{GL}}(V_\ell)$ is arising from $\Phi:\pi_1(X({\mathbb{C}}))\to {\mathrm{GL}}(V)$ by the comparison theorem between Betti and étale cohomologies [@SGA1 XII],[@SGA4 XVI] and the identity component of algebraic monodromy group of $\Phi$ is semisimple over ${\mathbb{Q}}$ [@De71 Cor. 4.2.9], the geometric monodromy $\Gamma_\ell^{{\operatorname{geo}}}$ is large in $\mathbf{G}_\ell^{{\operatorname{geo}}}({\mathbb{Q}}_\ell)$ for $\ell\gg0$, thanks to [@MVW84]. On the other hand, $\pi_1^{et}(X)$ satisfies (\[2\]) with ${\mathbb{F}}_q$ replaced with $K$. Since ${\operatorname{Gal}}(\bar K/K)$ is non-abelian, the monodromy groups $\Gamma_\ell\subset\mathbf{G}_\ell({\mathbb{Q}}_\ell)$ are complicated and carry a lot of arithmetic information. If $K$ is a number field and $X=\mathrm{Spec}(K)$, then $\Phi_\ell$ is a Galois representation of $K$ arising from the smooth projective variety $Y/K$ and the largeness of $\Gamma_\ell$ in $\mathbf{G}_\ell({\mathbb{Q}}_\ell)$ for $\ell\gg0$ follows from the remarkable conjectures of Hodge, Grothendieck, Tate, Mumford-Tate, and Serre [@Se94 $\mathsection11$], see also [@HL15a $\mathsection5$]. The prototypical result in this direction is due to Serre [@Se72], which states that for any non-CM elliptic curve $Y$, the monodromy $\Gamma_\ell$ on $H^1$ is ${\mathrm{GL}}_2({\mathbb{Z}}_\ell)$ for all sufficiently large $\ell$, see also [@Ri76; @Ri85],[@Se85],[@BGK03; @BGK06; @BGK10],[@Ha11] for certain abelian varieties; [@HL15b] for arbitrary abelian varieties; [@Se98] for abelian representations; [@HL14] for type A representations; and partial results [@La95a],[@Hu14] for arbitrary varieties. To get large Galois monodromy, one always needs handles on the invariants of $V_\ell$ and $\bar V_\ell$. For example when $Y$ is an abelian variety and $w=1$, Faltings has proved that the Galois invariants of $V_\ell\otimes V_\ell^*$ and $\bar V_\ell\otimes \bar V_\ell^*$ depend essentially on the endomorphism ring ${\operatorname{End}}(Y_{\bar K})$ if $\ell$ is sufficiently large [@Fa83],[@FW84] (the Tate conjecture). Since the Tate conjecture remains largely open, large Galois monodromy is presumably difficult.
Back to our setting $f:Y\to X$ over ${\mathbb{F}}_q$, the main idea of this paper is that there is a *cohomological* way, without resorting to the Tate conjecture, to compare the geometric invariant dimensions of $V_\ell^{\otimes m}$ and $\bar V_\ell^{\otimes m}$ for sufficiently large $\ell$ and sufficiently many $m$.
[2]{} For any $m\in{\mathbb{N}}$, if $\ell$ is sufficiently large, then $$\dim_{{\mathbb{F}}_\ell}(\bar V_\ell^{\otimes m})^{\pi_1^{et}(X_{\bar{\mathbb{F}}_q})}=\dim_{{\mathbb{Q}}_\ell}(V_\ell^{\otimes m})^{\pi_1^{et}(X_{\bar{\mathbb{F}}_q})}.$$
This is accomplished in $\mathsection2$ first assuming $X$ is a curve by étale cohomology theory [@SGA4; @SGA4.5; @SGA5],[@Mi80],[@FK87],[@Fu11] and the remarkable theorems of Deligne [@De74b; @De80], Gabber [@Ga83], and de Jong [@dJ96], the general case then follows from that by space filling curves [@Ka99] and $\ell$-independence of $\mathbf{G}_\ell$ [@Ch04].
[3]{} If $\phi_\ell$ is semisimple for all sufficiently large $\ell$, then Conjecture \[conj\] holds.
Theorem $3$ is proved in $\mathsection3$ by a recent result of Cadoret and Tamagawa on $\bar{\Gamma}_\ell^{{\operatorname{geo}}}$ [@CT15], the group theoretic techniques we employed and developed in [@HL14], and exploiting the invariant dimension data (Theorem $2$ and Corollary \[sym\]). The ${\mathbb{F}}_\ell$-semisimplicity hypothesis of Theorem $3$ holds if $X$ is a curve and the fibers of $f$ are curves or abelian varieties [@Za74a; @Za74b]. It is suggestive that the hypothesis holds in general because the invariant dimensions of $\Gamma_\ell^{{\operatorname{geo}}}$ and $\bar\Gamma_\ell^{{\operatorname{geo}}}$ are alike (Theorem $2$) and $\Gamma_\ell^{{\operatorname{geo}}}$ is semisimple on $V_\ell$. Nevertheless, we provide in $\mathsection4$ some examples for the hypothesis to hold.
Invariant dimensions
====================
The notation in $\mathsection1$ remains in force. Embed $\bar{\mathbb{Z}}[\frac{1}{q}]$ into $\bar{\mathbb{Z}}_\ell$ with unique maximal ideal $\mathfrak{m}_\ell$. The common dimension of $V_\ell$ for all $\ell$ (not dividing $q$) is also equal to the common dimension of $\bar V_\ell$ for all sufficiently large $\ell$ [@Ga83]. Then whenever $\dim_{{\mathbb{F}}_\ell}\bar V_\ell=\dim_{{\mathbb{Q}}_\ell}V_\ell$, one obtains $$\label{geq}
\dim_{{\mathbb{F}}_\ell}(\bar V_\ell^{\otimes m})^{\pi_1^{et}(X_{\bar{\mathbb{F}}_q})}\geq\dim_{{\mathbb{Q}}_\ell}(V_\ell^{\otimes m})^{\pi_1^{et}(X_{\bar{\mathbb{F}}_q})}$$ for any $m\in{\mathbb{N}}$ by identifying $\Gamma_\ell^{{\operatorname{geo}}}$ as a subgroup of ${\mathrm{GL}}(H^i(Y_{\bar x_0},{\mathbb{Z}}_\ell))$, the reduction map ${\mathrm{GL}}(H^i(Y_{\bar x_0},{\mathbb{Z}}_\ell))\to{\mathrm{GL}}(\bar V_\ell)$, and Lemma \[summand\].
\[summand\] Let $F$ be a characteristic $0$ non-Archimedean local field with $\mathscr{O}_F$ the ring of integers. Let $M$ be a free $\mathscr{O}_F$-module of finite rank. If $W$ is an $F$-subspace of $M\otimes_{\mathscr{O}_F} F$, then $W\cap M$ is a direct summand of $M$.
Since $\mathscr{O}_F$ is a PID and $\mathscr{O}_F/I$ is finite for any non-zero ideal $I$, the finitely generated module $M/W\cap M$ is a direct sum of a free submodule and a torsion submodule. If $x\in M$ maps to a torsion element in $M/W\cap M$, then $k\cdot x\in W\cap M$ for some $k\in{\mathbb{N}}$. This implies $x\in W$ because $F$ is of characteristic $0$. Hence, $M/W\cap M$ is free and $W\cap M$ is a direct summand of $M$.
Let $d$ be the dimension of $X$. Since $X$ is smooth, $\mathscr{F}_\ell$ is lisse, and $\bar{\mathscr{F}}_\ell$ is locally constant, we obtain perfect pairings by Poincaré duality [@Mi80 VI Thm. 11.1] which is compatible with the action of *geometric Frobenius* ${\mathrm{Fr}}_q$: $$\begin{aligned}
\label{Poincare}
\begin{split}
H^i(X_{\bar {\mathbb{F}}_q},\mathscr{F}_\ell)&\times H^{2d-i}_c(X_{\bar {\mathbb{F}}_q},\mathscr{F}_\ell^\vee)\to {\mathbb{Q}}_\ell(-d);\\
H^i(X_{\bar {\mathbb{F}}_q},\bar{\mathscr{F}}_\ell)&\times H^{2d-i}_c(X_{\bar {\mathbb{F}}_q},\bar{\mathscr{F}}_\ell^\vee)\to {\mathbb{F}}_\ell(-d).
\end{split}\end{aligned}$$ The geometric invariants admit the following descriptions: $$\begin{aligned}
\label{des}
\begin{split}
V_\ell^{\pi_1^{et}(X_{\bar{\mathbb{F}}_q})}&=H^0(X_{\bar {\mathbb{F}}_q},\mathscr{F}_\ell);\\
\bar V_\ell^{\pi_1^{et}(X_{\bar{\mathbb{F}}_q})}&=H^0(X_{\bar {\mathbb{F}}_q},\bar{\mathscr{F}}_\ell).
\end{split}\end{aligned}$$ Without loss of generality, assume $x_0$ is an ${\mathbb{F}}_q$-rational point of $X$ that induces a *splitting* of (\[2\]). Then the *multiset* $A'$ of the ${\mathrm{Fr}}_q$-eigenvalues on $V_\ell:=H^w(Y_{\bar x_0},{\mathbb{Q}}_\ell)$ are independent of $\ell$ [@De74b] and pure of weight $w$ [@De80]. It follows that the eigenvalues on $H^0(X_{\bar {\mathbb{F}}_q},\mathscr{F}_\ell)$ belong to those on $V_\ell$ by the splitting and (\[des\]). One also sees by the same token that the eigenvalues on $H^0(X_{\bar {\mathbb{F}}_q},\bar{\mathscr{F}}_\ell)$ belong to the reduction modulo $\mathfrak{m}_\ell$ of the eigenvalues on $V_\ell$ whenever $\dim_{{\mathbb{F}}_\ell}\bar V_\ell=\dim_{{\mathbb{Q}}_\ell}V_\ell$. Define $A$ to be the following multiset: $$\label{setA}
A:=\{q^d\alpha^{-1}:~\alpha\in A'\}.$$ We conclude by (\[Poincare\]) and above that the numbers in $A$ are pure of weight $2d-w$, the eigenvalues of $H^{2d}_c(X_{\bar {\mathbb{F}}_q},\mathscr{F}_\ell^\vee)$ is a sub-multiset of $A$, and the eigenvalues of $H^{2d}_c(X_{\bar {\mathbb{F}}_q},\bar{\mathscr{F}}_\ell^\vee)$ is a sub-multiset of the reduction modulo $\mathfrak{m}_\ell$ of $A$ for $\ell\gg0$.
[2]{}\[inv\] For any $m\in{\mathbb{N}}$, if $\ell$ is sufficiently large, then $$\dim_{{\mathbb{F}}_\ell}(\bar V_\ell^{\otimes m})^{\pi_1^{et}(X_{\bar{\mathbb{F}}_q})}=\dim_{{\mathbb{Q}}_\ell}(V_\ell^{\otimes m})^{\pi_1^{et}(X_{\bar{\mathbb{F}}_q})}.$$
***Step I***. Assume $X$ is a (geometrically connected) curve, i.e., $d=1$. If $U$ is an affine open subscheme of $X$ containing $x_0$, then $\pi_1^{et}(U)$ surjects onto $\pi_1^{et}(X)$ [@Fu11 Prop. 3.3.4(i)] and we obtain a commutative diagram: $$\label{square}
\begin{aligned}
\xymatrix{
\pi_1^{et}(U) \ar@{->>}[d] \ar[r] & {\mathrm{GL}}(V_\ell) \ar[d]^{=} \\
\pi_1^{et}(X) \ar[r] & {\mathrm{GL}}(V_\ell)}
\end{aligned}$$ Hence, we may further assume $X$ is an affine curve.
***Step II***. Let $e>0$ be the relative dimension of $f:Y\to X$. Then the dimension of $Y$ is $e+1$. Let $Y^c$ be a compactification of $Y_{\bar{\mathbb{F}}_q}$. Then $Y^c$ admits a *simplicial scheme* $Y.$ projective and smooth over $\bar{\mathbb{F}}_q$ and an augmentation $Y.\to Y^c$ which is a *proper hypercovering* of $Y^c$ (see [@dJ96 $\mathsection1$]). This induces a spectral sequence $$\label{ss1}
E^{i,j}_1:=H^j(Y_i,{\mathbb{Z}}/\ell{\mathbb{Z}})\Rightarrow H^{i+j}(Y^c,{\mathbb{Z}}/\ell{\mathbb{Z}})$$ by [@Co03 (6.3), Thm. 7.9] (see also [@De74a]). Let $B'$ be the multiset consisting of all the ${\mathrm{Fr}}_q$-eigenvalues on $H^j(Y_i,{\mathbb{Q}}_\ell)$ for all $(i,j)\in{\mathbb{Z}}_{\geq 0}^2$ satisfying $i+j=2(e+1)-(1+w)$. Since $Y_i$ is smooth projective for all $i$, the multiset $B'$ is mixed of weight $\leq 2(e+1)-(1+w)$ and is independent of $\ell$ [@De74b; @De80]. Since there are only finitely many such $(i,j)$, the ${\mathrm{Fr}}_q$-eigenvalues on $$H^{2(e+1)-(1+w)}(Y^c,{\mathbb{Z}}/\ell{\mathbb{Z}})=:H^{2(e+1)-(1+w)}_c(Y_{\bar{\mathbb{F}}_q},{\mathbb{Z}}/\ell{\mathbb{Z}})$$ belong to the reduction (modulo $\mathfrak{m}_\ell$) of $B'$ for $\ell\gg0$ by [@Ga83] and the biregular spectral sequence (\[ss1\]). Since $Y$ is smooth, the reduction (modulo $\mathfrak{m}_\ell$) of the multiset (mixed of weight $\geq 1+w$) $$B'':=\{q^{e+1}\beta^{-1}:~\beta\in B'\}$$ contains all the ${\mathrm{Fr}}_q$-eigenvalues on $H^{1+w}(Y_{\bar{\mathbb{F}}_q},{\mathbb{Z}}/\ell{\mathbb{Z}})$ for $\ell\gg0$ by Poincare duality. Since the spectral sequence $$E^{i,j}_2:=H^i(X_{\bar{\mathbb{F}}_q},R^jf_*{\mathbb{Z}}/\ell{\mathbb{Z}})\Rightarrow H^{i+j}(Y_{\bar{\mathbb{F}}_q},{\mathbb{Z}}/\ell{\mathbb{Z}})$$ degenerates on page $2$ (as $X$ is an affine curve), $E^{1,w}_2=H^1(X_{\bar{\mathbb{F}}_q},\bar{\mathscr{F}}_\ell)$ is a sub-quotient of $H^{1+w}(Y_{\bar{\mathbb{F}}_q},{\mathbb{Z}}/\ell{\mathbb{Z}})$. Thus, the eigenvalues on $H^1(X_{\bar{\mathbb{F}}_q},\bar{\mathscr{F}}_\ell)$ belong to the reduction of $B''$ for $\ell\gg0$. Then we conclude that the multiset (mixed of weight $\leq 1-w$) $$\label{setB}
B:=\{q\beta^{-1}:~\beta\in B''\}$$ after reduction contains all the eigenvalues on $H^1_c(X_{\bar{\mathbb{F}}_q},\bar{\mathscr{F}}_\ell^\vee)$ for $\ell\gg0$ by $X$ smooth and Poincare duality again.
***Step III***. By the Lefschetz trace formula on the lisse sheaf $\mathscr{F}_\ell^\vee$ and the locally constant sheaf $\bar{\mathscr{F}}_\ell^\vee$ on $X_{\bar{\mathbb{F}}_q}$ [@Mi80 VI Thm. 13.4], we obtain $$\begin{aligned}
\label{trace}
\begin{split}
\sum_{x\in X({\mathbb{F}}_{q^k})}\mathrm{Tr}({\mathrm{Fr}}_{q}^k:H^w(Y_{\bar x},{\mathbb{Q}}_\ell)^\vee)
&=\sum_{i=0}^2(-1)^i\mathrm{Tr}({\mathrm{Fr}}_{q}^k:H^i_c(X_{\bar{\mathbb{F}}_q},\mathscr{F}_\ell^\vee));\\
\sum_{x\in X({\mathbb{F}}_{q^k})}\mathrm{Tr}({\mathrm{Fr}}_{q}^k:H^w(Y_{\bar x},{\mathbb{Z}}/\ell{\mathbb{Z}})^\vee)
&=\sum_{i=0}^2(-1)^i\mathrm{Tr}({\mathrm{Fr}}_{q}^k:H^i_c(X_{\bar{\mathbb{F}}_q},\bar{\mathscr{F}}_\ell^\vee))
\end{split}\end{aligned}$$ for all $k\in{\mathbb{N}}$. Since $Y_{\bar x}$ is smooth projective, the Frobenius action on $H^w(Y_{\bar x},{\mathbb{Z}}/\ell{\mathbb{Z}})^\vee$ factors through $H^w(Y_{\bar x},{\mathbb{Q}}_\ell)^\vee$ for $\ell\gg0$ [@Ga83]. Hence, the reduction of the first local sum is equal to the second local sum for $\ell\gg0$. Since $H^0_c=0$ by $X$ affine, we obtain $$\label{trace2}
\sum_{i=1}^2(-1)^i\overline{\mathrm{Tr}({\mathrm{Fr}}_{q}^k:H^i_c(X_{\bar{\mathbb{F}}_q},\mathscr{F}_\ell^\vee))}=\sum_{i=1}^2(-1)^i\mathrm{Tr}({\mathrm{Fr}}_{q}^k:H^i_c(X_{\bar{\mathbb{F}}_q},\bar{\mathscr{F}}_\ell^\vee))$$ for $\ell\gg0$ by reduction. Denote the reductions of $A$ (\[setA\]) and $B$ (\[setB\]) by $\bar A$ and $\bar B$ respectively and the following:
- $\{\alpha_1,...,\alpha_r\}\subset\bar A$ the multiset of the ${\mathrm{Fr}}_q$-eigenvalues on $H^2_c(X_{\bar{\mathbb{F}}_q},\bar{\mathscr{F}}_\ell^\vee)$;
- $\{\beta_1,...,\beta_s\}\subset\bar B$ the multiset of the ${\mathrm{Fr}}_q$-eigenvalues on $H^1_c(X_{\bar{\mathbb{F}}_q},\bar{\mathscr{F}}_\ell^\vee)$;
- $\{a_1,...,a_t\}\subset\bar A$ the multiset of reduction of the ${\mathrm{Fr}}_q$-eigenvalues on $H^2_c(X_{\bar{\mathbb{F}}_q},\mathscr{F}_\ell^\vee)$;
- $\{b_1,...,b_u\}$ the multiset of reduction of the ${\mathrm{Fr}}_q$-eigenvalues on $H^1_c(X_{\bar{\mathbb{F}}_q},\mathscr{F}_\ell^\vee)$.
Note that $r,t\leq |A|$, $s\leq |B|$, and the number $u$ is independent of $\ell$ [@Ka83]. It follows from (\[trace2\]) that the above eigenvalues (in $\bar{\mathbb{F}}_\ell^*$) satisfy $$\label{trace3}
a_1^k+\cdots+a_t^k+\beta_1^k+\cdots+\beta_s^k=\alpha_1^k+\cdots+\alpha_r^k+b_1^k+\cdots+b_u^k$$ for all $k\in{\mathbb{N}}$. If $\ell>|A|+\mathrm{max}\{|B|,u\}$, then by Lemma \[trick\] the two multisets coincide: $$\label{equal}
\{a_1,...,a_t,\beta_1,...,\beta_s\}=\{\alpha_1,...,\alpha_r,b_1,...,b_u\}.$$ Since $A$ is pure of weight $2-w$ and $B$ is mixed of weight $\leq 1-w$ (Step II), $\bar A\cap\bar B= \emptyset$ for $\ell\gg0$ which implies $t\geq r$ by (\[equal\]). Together with (\[geq\]),(\[Poincare\]), and (\[des\]), we obtain $$\label{m=1}
\dim_{{\mathbb{F}}_\ell}\bar V_\ell^{\pi_1^{et}(X_{\bar{\mathbb{F}}_q})}=\dim_{{\mathbb{Q}}_\ell}V_\ell^{\pi_1^{et}(X_{\bar{\mathbb{F}}_q})}$$ for all sufficiently large $\ell$.
***Step IV***. Since $f:Y\to X$ is smooth projective, the natural morphism $$Y^{[m]}:=\overbrace{Y\times_X Y\times_X \cdots\times_X Y}^{m~\mathrm{terms}} \to X$$ is still smooth projective with the fiber $$\label{fiber}
(Y^{[m]})_{\bar x_0}=\prod^m Y_{\bar x_0}$$ inducing the representations $W_\ell:=H^{mw}((Y^{[m]})_{\bar x_0},{\mathbb{Q}}_\ell)$ and $\bar W_\ell:=H^{mw}((Y^{[m]})_{\bar x_0},{\mathbb{Z}}/\ell{\mathbb{Z}})$ of $\pi^{et}_1(X)$. For all sufficiently large $\ell$, we have $$\label{invw}
\dim_{{\mathbb{F}}_\ell}\bar W_\ell^{\pi_1^{et}(X_{\bar{\mathbb{F}}_q})}=\dim_{{\mathbb{Q}}_\ell}W_\ell^{\pi_1^{et}(X_{\bar{\mathbb{F}}_q})}$$ by (\[m=1\]). Since the representation $V_\ell^{\otimes m}$ (resp. $\bar V_\ell^{\otimes m}$) is a direct summand of the representation $W_\ell$ (resp. $\bar W_\ell$) by (\[fiber\]) and the K$\mathrm{\ddot{u}}$nneth isomorphism, we obtain by (\[invw\]) that $$\label{mgen}
\dim_{{\mathbb{F}}_\ell}(\bar V_\ell^{\otimes m})^{\pi_1^{et}(X_{\bar{\mathbb{F}}_q})}=\dim_{{\mathbb{Q}}_\ell}(V_\ell^{\otimes m})^{\pi_1^{et}(X_{\bar{\mathbb{F}}_q})}$$ holds for all sufficiently large $\ell$. This proves Theorem \[inv\] when $X$ is a curve.
***Step V***. For general smooth geometrically connected $X$, it suffices to prove Theorem \[inv\] for quasi-projective $X$ (see (\[square\])). If $C\subset X$ (containing $x_0$) is a smooth geometrically connected curve over ${\mathbb{F}}_q$, then $$\Psi_{\ell}:\pi_1^{et}(C)\to{\mathrm{GL}}(V_{\ell})$$ factors through $\Phi_\ell$ for all $\ell$. Denote by $\Lambda_\ell^{{\operatorname{geo}}}$ and $\mathbf{H}_{\ell}^{{\operatorname{geo}}}$ respectively the geometric monodromy group and the algebraic geometric monodromy group of $\Psi_\ell$. Choose $\ell_0$ such that the dimension of $\mathbf{G}_{\ell_0}^{{\operatorname{geo}}}$ is the largest. By [@Ka99 Cor. 7, Thm. 8], there exists a space filling curve $C\subset X$ (smooth, geometrically connected, containing $x_0$, over ${\mathbb{F}}_q$) satisfying $$\label{monoeq}
\mathbf{H}_{\ell_0}^{{\operatorname{geo}}}= \mathbf{G}_{\ell_0}^{{\operatorname{geo}}}.$$ Since the system $\{\Psi_\ell\}$ is pure of weight $w$ and is semisimple on the geometric étale fundamental group $\pi_1^{et}(C_{\bar{\mathbb{F}}_q})$ (Deligne), the identity component $(\mathbf{H}_{\ell}^{{\operatorname{geo}}})^\circ$ (semisimple) is isomorphic to the derived group of the identity component of the algebraic monodromy group of the semisimplification of $\Psi_\ell$ for all $\ell$ by (\[2\]). This implies $(\mathbf{H}_{\ell}^{{\operatorname{geo}}})^\circ\times{\mathbb{C}}$ is independent of $\ell$ by applying [@Ch04 Thm. 1.4] to the semisimplification of the system $\{\Psi_\ell\}$. In particular, the dimension of $\mathbf{H}_{\ell}^{{\operatorname{geo}}}$ is independent of $\ell$. Since we have $$\dim \mathbf{H}_{\ell}^{{\operatorname{geo}}}=\dim \mathbf{H}_{\ell_0}^{{\operatorname{geo}}}=\dim \mathbf{G}_{\ell_0}^{{\operatorname{geo}}}\geq \dim\mathbf{G}_{\ell}^{{\operatorname{geo}}}$$ and the groups $\mathbf{H}_{\ell}^{{\operatorname{geo}}}\subset \mathbf{G}_{\ell}^{{\operatorname{geo}}}$ (by $\Psi_\ell$ factors through $\Phi_\ell$) have the same number of connected components (by (\[monoeq\]) and [@LP95 Prop. 2.2(iii)]) for all $\ell$, we obtain $\mathbf{H}_{\ell}^{{\operatorname{geo}}}=\mathbf{G}_{\ell}^{{\operatorname{geo}}}$ for all $\ell$. Since (a) $\Lambda_\ell^{{\operatorname{geo}}}$ (resp. $\Gamma_\ell^{{\operatorname{geo}}}$) is Zariski dense in $\mathbf{H}_{\ell}^{{\operatorname{geo}}}$ (resp. $\mathbf{G}_{\ell}^{{\operatorname{geo}}}$) and (b) $\Psi_\ell$ factors through $\Phi_\ell$, we obtain $$\begin{aligned}
\begin{split}
\dim_{{\mathbb{F}}_\ell}(\bar V_\ell^{\otimes m})^{\pi_1^{et}(X_{\bar{\mathbb{F}}_q})}&\stackrel{(b)}{\leq}\dim_{{\mathbb{F}}_\ell}(\bar V_\ell^{\otimes m})^{\pi_1^{et}(C_{\bar{\mathbb{F}}_q})}
\stackrel{(\ref{mgen})}{=}\dim_{{\mathbb{Q}}_\ell}(V_\ell^{\otimes m})^{\pi_1^{et}(C_{\bar{\mathbb{F}}_q})}\\
\stackrel{(a)}{=}\dim_{{\mathbb{Q}}_\ell}(V_\ell^{\otimes m})^{\mathbf{H}_{\ell}^{{\operatorname{geo}}}({\mathbb{Q}}_\ell)}&=
\dim_{{\mathbb{Q}}_\ell}(V_\ell^{\otimes m})^{\mathbf{G}_{\ell}^{{\operatorname{geo}}}({\mathbb{Q}}_\ell)}\stackrel{(a)}{=}
\dim_{{\mathbb{Q}}_\ell}(V_\ell^{\otimes m})^{\pi_1^{et}(X_{\bar{\mathbb{F}}_q})}
\end{split}\end{aligned}$$ for $\ell\gg0$. We are done by (\[geq\]).
\[trick\] Suppose $a_1,...,a_m,b_1,...,b_n\in\bar{\mathbb{F}}_\ell^*$ satisfying $\mathrm{max}\{m,n\}<\ell$ and $$\label{sums}
a_1^k+\cdots +a_m^k=b_1^k+\cdots +b_n^k$$ for all $1\leq k\leq \mathrm{max}\{m,n\}$. Then the two multisets $\{a_1,...,a_m\}$ and $\{b_1,...,b_n\}$ coincide.
First assume $m=n$. Let $x_1,...,x_m$ be indeterminate variables. Denote the elementary symmetric polynomials in $x_1,...,x_m$ by $e_1,...,e_m$ and $x_1^k+\cdots +x_m^k$ by $p_k$. The *Newton’s identities* imply $$e_1,...,e_m\in{\mathbb{Z}}[\frac{1}{m!}](p_1,...,p_m).$$ Hence, $e_k(a_1,...,a_m)=e_k(b_1,...,b_m)$ for all $1\leq k\leq m$ by (\[sums\]) and $m<\ell$. We conclude that $\{a_1,...,a_m\}=\{b_1,...,b_m\}$ by constructing a degree $m$ polynomial in $\bar{\mathbb{F}}_\ell[t]$ whose roots are exactly $a_1,...,a_m$ (resp. $b_1,...,b_m$).
Suppose $m>n$. Let $b_{n+1},...,b_m$ be all zeros. Then some $a_i$ is zero by the above case, which contradicts $a_i\in\bar{\mathbb{F}}_\ell^*$.
\[sym\] For any $m\in{\mathbb{N}}$, if $\ell$ is sufficiently large, then $$\begin{aligned}
\begin{split}
\dim_{{\mathbb{F}}_\ell}(\mathrm{Sym}^m\bar V_\ell)^{\pi_1^{et}(X_{\bar{\mathbb{F}}_q})}&=\dim_{{\mathbb{Q}}_\ell}(\mathrm{Sym}^m V_\ell)^{\pi_1^{et}(X_{\bar{\mathbb{F}}_q})};\\
\dim_{{\mathbb{F}}_\ell}(\mathrm{Alt}^m\bar V_\ell)^{\pi_1^{et}(X_{\bar{\mathbb{F}}_q})}&=\dim_{{\mathbb{Q}}_\ell}(\mathrm{Alt}^m V_\ell)^{\pi_1^{et}(X_{\bar{\mathbb{F}}_q})}.
\end{split}\end{aligned}$$
Since the left hand side is always greater than or equal to the right hand side of the equation and the representations $\bar V_\ell^{\otimes m}$ and $V_\ell^{\otimes m}$ contain respectively $\mathrm{Sym}^m \bar V_\ell$ and $\mathrm{Sym}^m V_\ell$ (resp. $\mathrm{Alt}^m \bar V_\ell$ and $\mathrm{Alt}^m V_\ell$) as direct summands, the corollary follows from Theorem \[inv\].
Maximality
==========
If $X'_{\bar{\mathbb{F}}_q}$ is a connected finite étale cover of $X_{\bar{\mathbb{F}}_q}$, then $\pi_1^{et}(X'_{\bar{\mathbb{F}}_q})$ is a finite index subgroup of $\pi_1^{et}(X_{\bar{\mathbb{F}}_q})$. Since $X'_{\bar{\mathbb{F}}_q}\to X_{\bar{\mathbb{F}}_q}$ is always defined over some finite extension ${\mathbb{F}}_{q'}$ of ${\mathbb{F}}_q$ (e.g., $X'_{{\mathbb{F}}_{q'}}\to X_{{\mathbb{F}}_{q'}}$) which does not affect the geometric monodromy and the restriction of a semisimple representation to a normal subgroup is still semisimple, it suffices to prove Theorem \[main\] by considering the base change $$Y\times_X X'_{{\mathbb{F}}_{q'}}\to X'_{{\mathbb{F}}_{q'}}$$ of $f:Y\to X$ by a connected finite Galois étale cover $X'_{{\mathbb{F}}_{q'}}\to X_{{\mathbb{F}}_{q'}}\to X$. Hence, we assume from now on that the algebraic geometric monodromy group $\mathbf{G}_\ell^{{\operatorname{geo}}}$ is *connected* for all $\ell$ by taking a connected finite étale cover of $X$ [@LP95 Prop. 2.2(ii)]. Let $n$ be the common dimension of $V_\ell$ for all $\ell$, which is also the common dimension of $\bar V_\ell$ for $\ell\gg0$.
[3]{}\[main\] If $\phi_\ell$ is semisimple for all sufficiently large $\ell$, then Conjecture \[conj\] holds.
***Step I***. For any subgroup $\bar\Gamma$ of ${\mathrm{GL}}_n({\mathbb{F}}_\ell)$, denote by $\bar\Gamma^+$ the (normal) subgroup of $\bar\Gamma$ that is generated by $\bar\Gamma[\ell]$, the subset of order $\ell$ elements of $\bar\Gamma$. By taking some connected finite Galois étale cover of $X$, we may assume $\bar\Gamma_\ell^{{\operatorname{geo}}}=(\bar\Gamma_\ell^{{\operatorname{geo}}})^+$ [@CT15 Prop. 3.2, Thm. 1.1], $\bar\Gamma_\ell^{{\operatorname{geo}}}$ is semisimple on $\bar V_\ell$, and $\mathbf{G}_\ell^{{\operatorname{geo}}}$ is connected for all sufficiently large $\ell$. Since $n=\dim_{{\mathbb{F}}_\ell}\bar V_\ell$ for $\ell\gg0$, there exists an *exponentially generated* subgroup $\bar{\mathbf{S}}_\ell$ of ${\mathrm{GL}}_{\bar V_\ell}$ such that $\bar\Gamma_\ell^{{\operatorname{geo}}}=\bar{\mathbf{S}}_\ell({\mathbb{F}}_\ell)^+$ for all $\ell\gg 0$ by Nori [@No87 Thm. B]. The Nori subgroup $\bar{\mathbf{S}}_\ell$ is connected and an extension of semisimple by unipotent. Since $\bar\Gamma_\ell^{{\operatorname{geo}}}$ is semisimple on $\bar V_\ell$ for $\ell\gg0$, $\bar{\mathbf{S}}_\ell$ is connected semisimple for $\ell\gg0$. Let $\bar{\mathbf{S}}_\ell^{{\operatorname{sc}}}\to \bar{\mathbf{S}}_\ell$ be the universal covering of $\bar{\mathbf{S}}_\ell$. The representation $$\bar{\mathbf{S}}_\ell^{{\operatorname{sc}}}\times\bar{\mathbb{F}}_\ell\to \bar{\mathbf{S}}_\ell\times\bar{\mathbb{F}}_\ell\hookrightarrow{\mathrm{GL}}_{\bar V_\ell\times\bar{\mathbb{F}}_\ell}$$ can be lifted to a representation of some simply-connected *Chevalley scheme* over ${\mathbb{Z}}$ for all $\ell\gg0$ [@Se86] (see [@EHK12 Thm. 27]), $$\label{Chevalley}
\rho_{\ell,{\mathbb{Z}}}:\mathbf{H}_{\ell,{\mathbb{Z}}}\to {\mathrm{GL}}_{V_{\mathbb{Z}}}.$$
***Step II***. We would like to study the invariants of $\bar{\mathbf{S}}_\ell$ on $\bar V_\ell^{\otimes m}\otimes\bar{\mathbb{F}}_\ell$. Let us recall the construction of $\bar{\mathbf{S}}_\ell$. Define $\mathrm{exp}(x)$ and $\mathrm{log}(x)$ by $$\label{explog1}
\mathrm{exp}(x)=\sum_{i=0}^{\ell-1}\frac{x^i}{i!}\hspace{.1in}\mathrm{and}\hspace{.1in}
\mathrm{log}(x)=-\sum_{i=1}^{\ell-1}\frac{(1-x)^i}{i}.$$ For all sufficiently large $\ell$, $\bar{\mathbf{S}}_\ell$ is the Zariski closure in ${\mathrm{GL}}_{\bar V_\ell}\cong{\mathrm{GL}}_{n,{\mathbb{F}}_\ell}$ of the subgroup generated by the one-parameter subgroup $$\label{explog2}
t\mapsto x^t:=\mathrm{exp}(t\cdot\mathrm{log}(x))$$ for all $x\in\bar\Gamma_\ell^{{\operatorname{geo}}}[\ell]$ (the order $\ell$ elements) [@No87]. When $\ell>n$, $x$ is unipotent and $\mathrm{log}(x)$ is nilpotent by (\[explog1\]). Identify $\bar V_\ell\otimes\bar{\mathbb{F}}_\ell$ with $\bar{\mathbb{F}}_\ell^n$, then every entry of the matrix $x^t\in {\mathrm{GL}}_n(\bar{\mathbb{F}}_\ell[t])$ is a polynomial of degree less than $n^2$ by (\[explog1\]) and (\[explog2\]). Similarly, the action of $x^t$ on $\bar V_\ell^{\otimes m}\otimes\bar{\mathbb{F}}_\ell$ can be identified with an element of ${\mathrm{GL}}_{n^m}(\bar{\mathbb{F}}_\ell[t])$ whose entries are polynomials of degree less than $n^2m$. Consider an invariant $v\in (\bar V_\ell^{\otimes m})^{\pi_1^{et}(X_{\bar{\mathbb{F}}_q})}=(\bar V_\ell^{\otimes m})^{\bar\Gamma_\ell^{{\operatorname{geo}}}}$, then the equation in $\bar{\mathbb{F}}_\ell[t]^{n^m}$ below $$x^t\cdot v=v$$ has at least $\ell$ distinct roots $t=0,1,...,\ell-1$ because $id,x,...,x^{\ell-1}\in \bar\Gamma_\ell^{{\operatorname{geo}}}$. This implies $x^t\cdot v\equiv v$ when $\ell\geq n^2m$. Hence, we obtain $v\in (\bar V_\ell^{\otimes m}\otimes\bar{\mathbb{F}}_\ell)^{\bar{\mathbf{S}}_\ell}$ when $\ell\geq n^2m$ by the construction of $\bar{\mathbf{S}}_\ell$ . Since $\bar\Gamma_\ell^{{\operatorname{geo}}}=(\bar\Gamma_\ell^{{\operatorname{geo}}})^+$ [@CT15] is a subgroup of $\bar{\mathbf{S}}_\ell$ for $\ell\gg0$, we obtain $$\dim_{{\mathbb{F}}_\ell} (\bar V_\ell^{\otimes m})^{\pi_1^{et}(X_{\bar{\mathbb{F}}_q})} =\dim_{\bar{\mathbb{F}}_\ell}(\bar V_\ell^{\otimes m}\otimes\bar{\mathbb{F}}_\ell)^{\bar{\mathbf{S}}_\ell}$$ for $\ell\gg0$. It follows that $$\dim_{{\mathbb{F}}_\ell} ((\oplus^n\bar V_\ell)^{\otimes m})^{\pi_1^{et}(X_{\bar{\mathbb{F}}_q})} =\dim_{\bar{\mathbb{F}}_\ell}((\oplus^n\bar V_\ell)^{\otimes m}\otimes\bar{\mathbb{F}}_\ell)^{\bar{\mathbf{S}}_\ell}$$ for $\ell\gg0$. By Corollary \[sym\] and embedding ${\mathbb{Q}}_\ell$ into ${\mathbb{C}}$, we conclude for $\ell\gg0$ that $$\label{Noriinv}
\dim_{{\mathbb{C}}} (\mathrm{Sym}^m (\oplus^n V_\ell)\otimes {\mathbb{C}})^{\mathbf{G}_\ell^{{\operatorname{geo}}}}=\dim_{\bar{\mathbb{F}}_\ell}(\mathrm{Sym}^m(\oplus^n\bar V_\ell)\otimes\bar{\mathbb{F}}_\ell)^{\bar{\mathbf{S}}_\ell}.$$
***Step III***. Denote the base change of (\[Chevalley\]) to ${\mathbb{C}}$ by $\rho_{\ell,{\mathbb{C}}}:\mathbf{H}_{\ell,{\mathbb{C}}}\to {\mathrm{GL}}_{V_{\mathbb{C}}}$. For fixed $m\in{\mathbb{N}}$, we obtain by Step I and (\[Noriinv\]) that for $\ell\gg0$, $$\label{Ceqt}
\dim_{{\mathbb{C}}} (\mathrm{Sym}^m (\oplus^n V_\ell)\otimes {\mathbb{C}})^{\mathbf{G}_\ell^{{\operatorname{geo}}}} =\dim_{{\mathbb{C}}}(\mathrm{Sym}^m (\oplus^n V_{\mathbb{C}}))^{\mathbf{H}_{\ell,{\mathbb{C}}}}.$$ Since there are finitely many connected semisimple subgroup of ${\mathrm{GL}}_{n,{\mathbb{C}}}$ (up to isomorphism), (\[Ceqt\]) holds for all $m\in{\mathbb{N}}$ when $\ell$ is sufficiently large. Identify $V_\ell\otimes {\mathbb{C}}$ with $V_{\mathbb{C}}$. Then the (Noetherian) graded rings $$R={\mathbb{C}}[\oplus^n V_{\mathbb{C}}]^{\mathbf{G}_\ell^{{\operatorname{geo}}}}\hspace{.1in}\mathrm{and}\hspace{.1in} R'={\mathbb{C}}[\oplus^n V_{\mathbb{C}}]^{\mathbf{H}_{\ell,{\mathbb{C}}}}$$ have the same *Hilbert polynomial*, hence the same *Krull dimension* for $\ell\gg0$. Since $\dim_{Krull} R=n^2-\dim \mathbf{G}_\ell^{{\operatorname{geo}}}$ and $\dim_{Krull} R'=n^2-\dim \rho_{\ell,{\mathbb{C}}}(\mathbf{H}_{\ell,{\mathbb{C}}})$ (for example [@LP90 $\mathsection0$]), we conclude by the lifting (\[Chevalley\]) that for all $\ell\gg0$, $$\label{dim}
\dim \mathbf{G}_\ell^{{\operatorname{geo}}} = \dim\bar{\mathbf{S}}_\ell.$$
***Step IV***. Suppose $\ell\geq 5$. For any compact subgroup $\Gamma\subset{\mathrm{GL}}_n({\mathbb{Q}}_\ell)$ (resp. $\bar\Gamma\subset{\mathrm{GL}}_n({\mathbb{F}}_\ell)$), we defined the *$\ell$-dimension* $\dim_\ell\Gamma$ (resp. $\dim_\ell\bar\Gamma$) in [@HL14 $\mathsection2$] satisfying the following properties:
(i) $\dim_\ell$ is additive on short exact sequences;
(ii) $\dim_\ell$ vanishes for pro-solvable groups and finite simple groups that are not of Lie type in characteristic $\ell$;
(iii) if $\bar\Gamma$ is a finite simple group of Lie type in characteristic $\ell$, then there exists some connected adjoint semisimple group $\bar{\mathbf{S}}/{\mathbb{F}}_\ell$ such that $\bar\Gamma$ is isomorphic to the derived group of $\bar{\mathbf{S}}({\mathbb{F}}_\ell)$ and we define $\dim_\ell\bar\Gamma:=\dim \bar{\mathbf{S}}$.
We obtain for $\ell\gg0$ that $$\label{ldim}
\dim_\ell\bar\Gamma_\ell^{{\operatorname{geo}}}=\dim_\ell\bar{\mathbf{S}}_\ell({\mathbb{F}}_\ell)^+=\dim_\ell\bar{\mathbf{S}}_\ell({\mathbb{F}}_\ell)=\dim\bar{\mathbf{S}}_\ell=\dim \mathbf{G}_\ell^{{\operatorname{geo}}}$$ by Step I, [@No87 3.6(v)], [@HL14 Prop. 3(iii)], and (\[dim\]) respectively for each equality. Recall the universal covering $\pi:\mathbf{G}_\ell^{{\operatorname{sc}}}\to \mathbf{G}_\ell^{{\operatorname{geo}}}$. Since (a) the kernel and cokernel of $\pi^{-1}(\Gamma_\ell^{{\operatorname{geo}}})\to \Gamma_\ell^{{\operatorname{geo}}}$ are abelian and (b) the kernel of $\Gamma_\ell^{{\operatorname{geo}}}\twoheadrightarrow\bar\Gamma_\ell^{{\operatorname{geo}}}$ is pro-solvable (via the reduction map ${\mathrm{GL}}(H^i(Y_{\bar x_0},{\mathbb{Z}}_\ell))\to{\mathrm{GL}}(\bar V_\ell)$ for $\ell\gg0$), we obtain by the properties of $\dim_\ell$ that for $\ell\gg0$, $$\label{hopedim}
\dim_\ell\pi^{-1}(\Gamma_\ell^{{\operatorname{geo}}})\stackrel{(a)}{=}\dim_\ell\Gamma_\ell^{{\operatorname{geo}}}\stackrel{(b)}{=}\dim_\ell\bar\Gamma_\ell^{{\operatorname{geo}}}\stackrel{(\ref{ldim})}{=}\dim \mathbf{G}_\ell^{{\operatorname{geo}}}=:g.$$
***Step V***. Let $\Delta_\ell$ be a maximal compact subgroup of $\mathbf{G}_\ell^{{\operatorname{sc}}}({\mathbb{Q}}_\ell)$ that contains $\pi^{-1}(\Gamma_\ell^{{\operatorname{geo}}})$. By [@Ti79 3.2], $\Delta_\ell$ is the stabilizer $\mathbf{G}_\ell^{{\operatorname{sc}}} ({\mathbb{Q}}_\ell)^x$ of a vertex $x$ in the *Bruhat-Tits building* of $\mathbf{G}_\ell^{{\operatorname{sc}}} /{\mathbb{Q}}_\ell$. There exists a smooth affine group scheme ${\mathcal{G}}$ over ${\mathbb{Z}}_\ell$ and an isomorphism $\iota$ from the generic fiber of ${\mathcal{G}}$ to $\mathbf{G}_\ell^{{\operatorname{sc}}}$ such that $\iota({\mathcal{G}}({\mathbb{Z}}_\ell)) = \mathbf{G}_\ell^{{\operatorname{sc}}} ({\mathbb{Q}}_\ell)^x$ [@Ti79 3.4.1]. As $\mathbf{G}_\ell^{{\operatorname{sc}}}$ is simply-connected semisimple, the special fiber ${\mathcal{G}}_{{\mathbb{F}}_\ell}$ is connected [@Ti79 3.5.2]. The maximal compact subgroup $\Delta_\ell$ is *hyperspecial* if and only if ${\mathcal{G}}_{{\mathbb{F}}_{\ell}}$ is reductive [@Ti79 3.8.1], in which case it has the same root datum as the generic fiber [@SGA3 XXII, 2.8]. Since (c) the kernel of the reduction map $r:{\mathcal{G}}({\mathbb{Z}}_\ell)\to {\mathcal{G}}({\mathbb{F}}_\ell)$ is pro-solvable and (d) ${\mathcal{G}}$ is smooth over ${\mathbb{Z}}_\ell$, we obtain by the properties of $\dim_\ell$ that for $\ell\gg0$, $$\label{moddim}
\dim_\ell r(\pi^{-1}(\Gamma_\ell^{{\operatorname{geo}}}))\stackrel{(c)}{=}\dim_\ell \pi^{-1}(\Gamma_\ell^{{\operatorname{geo}}})\stackrel{(\ref{hopedim})}{=}g=
\dim \mathbf{G}_\ell^{{\operatorname{sc}}}\stackrel{(d)}{=}\dim {\mathcal{G}}_{{\mathbb{F}}_\ell}.$$ Since the special fiber ${\mathcal{G}}_{{\mathbb{F}}_\ell}$ is connected, ${\mathcal{G}}_{{\mathbb{F}}_\ell}$ is semisimple for $\ell\gg0$ by (\[moddim\]) and [@HL14 Thm. 4(iv)]. It follows from above that $\Delta_\ell$ is hyperspecial and ${\mathcal{G}}_{{\mathbb{F}}_\ell}$ is simply-connected semisimple for $\ell\gg0$. For any connected algebraic group $\bar{\mathbf{G}}$ of dimension $g$ defined over ${\mathbb{F}}_\ell$, the order of $\bar{\mathbf{G}}({\mathbb{F}}_\ell)$ satisfies $$\label{Norder}
(\ell-1)^g\leq |\bar{\mathbf{G}}({\mathbb{F}}_\ell)|\leq (\ell+1)^g$$ by [@No87 Lem 3.5]. Hence, there exists a constant $c(g)\geq 1$ depending only on $g$ such that for $\ell\gg0$, $$\label{compare}
\frac{(\ell-1)^g}{c(g)}\leq |r(\pi^{-1}(\Gamma_\ell^{{\operatorname{geo}}}))| \stackrel{subgp}{\leq}
|{\mathcal{G}}({\mathbb{F}}_\ell)|\stackrel{(\ref{Norder})}{\leq} (\ell+1)^g,$$ where the first inequality follows by considering (\[moddim\]), (\[Norder\]), the properties of $\dim_\ell$ in Step IV, and the orders of finite simple groups of Lie type in characteristic $\ell$ [@St67 $\mathsection9$] (e.g., $|\mathrm{PSL}_k(\ell)|=\frac{1}{(k,\ell-1)}\ell^{k(k-1)/2}(\ell^2-1)(\ell^3-1)\cdots(\ell^k-1)$). Since $g:=\dim\mathbf{G}_\ell^{{\operatorname{geo}}}\leq n^2$ for all $\ell$, the index $[{\mathcal{G}}({\mathbb{F}}_\ell):r(\pi^{-1}(\Gamma_\ell^{{\operatorname{geo}}}))]\leq C(n)$ (a constant depending only on $n$) for $\ell\gg0$. Since ${\mathcal{G}}_{{\mathbb{F}}_\ell}$ is simply-connected semisimple, ${\mathcal{G}}({\mathbb{F}}_\ell)$ is generated by the subset of order $\ell$ elements ${\mathcal{G}}({\mathbb{F}}_\ell)[\ell]$ when $\ell\gg0$ (see the proof of [@HL14 Thm. 4]). Since ${\mathcal{G}}({\mathbb{F}}_\ell)[\ell]$ belongs to $r(\pi^{-1}(\Gamma_\ell^{{\operatorname{geo}}}))$ for $\ell\gg C(n)$, the equality $r(\pi^{-1}(\Gamma_\ell^{{\operatorname{geo}}}))={\mathcal{G}}({\mathbb{F}}_\ell)$ holds for $\ell\gg C(n)$. Therefore, the subgroup $\pi^{-1}(\Gamma_\ell^{{\operatorname{geo}}})\subset {\mathcal{G}}({\mathbb{Z}}_\ell)$ surjects onto ${\mathcal{G}}({\mathbb{F}}_\ell)$ under the reduction map $r$ for $\ell\gg0$. By the main theorem of [@Va03], this implies $\pi^{-1}(\Gamma_\ell^{{\operatorname{geo}}})={\mathcal{G}}({\mathbb{Z}}_\ell)=\Delta_\ell$ for $\ell\gg 0$, which is hyperspecial maximal compact in $\mathbf{G}_\ell^{{\operatorname{geo}}}({\mathbb{Q}}_\ell)$.
\[unramified\] For all sufficiently large $\ell$, the identity component of the algebraic geometric monodromy group $\mathbf{G}_\ell^{{\operatorname{geo}}}$ is unramified over ${\mathbb{Q}}_\ell$.
Since a connected reductive group $\mathbf{G}/{\mathbb{Q}}_\ell$ is unramified if and only if $\mathbf{G}({\mathbb{Q}}_\ell)$ contains a hyperspecial maximal compact subgroup [@Mi92 $\mathsection1$], $\mathbf{G}_\ell^{{\operatorname{sc}}}$ is unramified for $\ell\gg0$ by Theorem \[main\]. Since $\pi:\mathbf{G}_\ell^{{\operatorname{sc}}}\twoheadrightarrow (\mathbf{G}_\ell^{{\operatorname{geo}}})^\circ$ is surjective, the identity component $(\mathbf{G}_\ell^{{\operatorname{geo}}})^\circ$ is unramified for $\ell\gg0$.
Assuming $\Phi_\ell$ is semisimple for all $\ell$, then $\mathbf{G}_\ell^\circ\times{\mathbb{C}}\subset{\mathrm{GL}}_{V_\ell\times{\mathbb{C}}}$ is independent of $\ell$ by [@Ch04] and [@Ka99] (see Step V of Theorem \[inv\]). Corollary \[unramified\] is a necessary condition for the existence of a common ${\mathbb{Q}}$-form of $\{\mathbf{G}_\ell^\circ\subset{\mathrm{GL}}_{V_\ell}\}_\ell$.
Semisimplicity
==============
In this section, we give two examples of $\{\Phi_\ell\}$ such that the hypothesis of Theorem \[main\] holds. It suffices to show by the lemma below that the restriction of $\phi_\ell$ to a normal subgroup of $\pi_1^{et}(X_{\bar{\mathbb{F}}_q})$ (i.e., by taking a connected Galois étale cover of $X$) is semisimple for $\ell\gg0$.
[@HL15b Lemma 3.6] Let $F$ be a field, $G$ a finite group, $H$ a normal subgroup of $G$ such that $[G:H]$ is non-zero in $F$, and $V$ a finite dimensional $F$-representation of $G$. Then $V$ is semisimple if and only if its restriction to $H$ is so.
**Example 1.** Suppose the fibers of $f:Y\to X$ are curves or abelian varieties. Then the hypothesis of Theorem \[main\] holds.
***Step I.*** When $X$ is a curve, $\phi_\ell$ is factored through by a Galois representation of $K(X)$, the function field of $X$. When the fibers of $f$ are abelian varieties, the conclusion follows directly from the Tate conjecture of abelian varieties over function fields [@Za74a; @Za74b] (see also [@FW84 Ch. VI$\mathsection3$],[@LP95 Thm. 3.1(iii)]). When the fibers are curves, the conclusion follows from above and the fact that a smooth curve and its Jacobian variety have isomorphic $H^1$ representations.
***Step II.*** For general $X$, we may first assume $\mathbf{G}_\ell^{{\operatorname{geo}}}$ is connected for all $\ell$ by taking a connected Galois étale cover [@LP95 Prop. 2.2(i)]. By [@Ka99] and [@Ch04] (see Step V of Theorem \[inv\]), there exists a smooth geometrically connected curve $C$ of $X$ such that the algebraic geometric monodromy group associated to $Y\times_X C\to C$ is also equal to $\mathbf{G}_\ell^{{\operatorname{geo}}}$ for all $\ell$. Denote the images of $\pi_1^{et}(C_{\bar{\mathbb{F}}_q})$ and $\pi_1^{et}(X_{\bar{\mathbb{F}}_q})$ in respectively ${\mathrm{GL}}(V_\ell)$ and ${\mathrm{GL}}(\bar V_\ell)$ by $$\begin{aligned}
\begin{split}
\Lambda_\ell^{{\operatorname{geo}}}\subset\Gamma_\ell^{{\operatorname{geo}}}\subset{\mathrm{GL}}(V_\ell);\\
\bar\Lambda_\ell^{{\operatorname{geo}}}\subset\bar\Gamma_\ell^{{\operatorname{geo}}}\subset{\mathrm{GL}}(\bar V_\ell).
\end{split}\end{aligned}$$ We may assume $\bar\Gamma_\ell^{{\operatorname{geo}}}$ is generated by its order $\ell$ elements for $\ell\gg0$ by [@CT15]. Since $\pi^{-1}(\Lambda_\ell^{{\operatorname{geo}}})\subset\pi^{-1}(\Gamma_\ell^{{\operatorname{geo}}})$ are compact subgroups of $\mathbf{G}_\ell^{{\operatorname{sc}}}({\mathbb{Q}}_\ell)$ and $\pi^{-1}(\Lambda_\ell^{{\operatorname{geo}}})$ is hyperspecial maximal compact in $\mathbf{G}_\ell^{{\operatorname{sc}}}({\mathbb{Q}}_\ell)$ for $\ell\gg0$ by Step I and Theorem \[main\], we have $\pi^{-1}(\Lambda_\ell^{{\operatorname{geo}}})=\pi^{-1}(\Gamma_\ell^{{\operatorname{geo}}})$ for $\ell\gg0$. Hence, the index $[\Gamma_\ell^{{\operatorname{geo}}}:\Lambda_\ell^{{\operatorname{geo}}}]$ is bounded by some constant $C$ (depending on $n=\dim V_\ell$) for $\ell\gg0$. It follows that $[\bar\Gamma_\ell^{{\operatorname{geo}}}:\bar\Lambda_\ell^{{\operatorname{geo}}}]\leq C$ for $\ell\gg0$ via the reduction map ${\mathrm{GL}}(H^i(Y_{\bar x_0},{\mathbb{Z}}_\ell))\to{\mathrm{GL}}(\bar V_\ell)$. This implies that the order $\ell$ elements of $\bar\Gamma_\ell^{{\operatorname{geo}}}$ belong to $\bar\Lambda_\ell^{{\operatorname{geo}}}$ when $\ell\gg C$. Since $\bar\Gamma_\ell^{{\operatorname{geo}}}$ is generated by its order $\ell$ elements and $\bar\Lambda_\ell$ is semisimple on $\bar V_\ell$ for $\ell\gg0$, $\bar\Gamma_\ell^{{\operatorname{geo}}}$ is semisimple on $\bar V_\ell$ for $\ell\gg0$.
**Example 2.** Identify $\Gamma_\ell^{{\operatorname{geo}}}$ as a subgroup of ${\mathrm{GL}}(H^i(Y_{\bar x_0},{\mathbb{Z}}_\ell))={\mathrm{GL}}_n({\mathbb{Z}}_\ell)$ for $\ell\gg0$. Suppose there exists a connected semisimple subgroup $\mathbf{G}\subset{\mathrm{GL}}_{n,{\mathbb{Q}}}$ such that $(\mathbf{G}_\ell^{{\operatorname{geo}}})^\circ=\mathbf{G}\times{\mathbb{Q}}_\ell$ in ${\mathrm{GL}}_{n,{\mathbb{Q}}_\ell}$ and $$\Gamma_\ell^{{\operatorname{geo}}}\cap(\mathbf{G}_\ell^{{\operatorname{geo}}})^\circ\subset \mathbf{G}({\mathbb{Z}}_\ell)\subset {\mathrm{GL}}_n({\mathbb{Z}}_\ell)$$ for $\ell\gg0$. Then the hypothesis of Theorem \[main\] holds.
***Step I.*** By taking a connected Galois étale cover, we may assume $\mathbf{G}_\ell^{{\operatorname{geo}}}$ is connected for all $\ell$ [@LP95 Prop. 2.2(i)] and $\bar\Gamma_\ell^{{\operatorname{geo}}}=(\bar\Gamma_\ell^{{\operatorname{geo}}})^+$ for $\ell\gg0$ [@CT15]. The closed subgroup $\mathbf{G}\subset{\mathrm{GL}}_{n,{\mathbb{Q}}}$ can be extended to a closed subgroup scheme $\mathbf{G}_{{\mathbb{Z}}[\frac{1}{N}]}\subset {\mathrm{GL}}_{n,{\mathbb{Z}}[\frac{1}{N}]}$ smooth over ${\mathbb{Z}}[\frac{1}{N}]$ for some sufficiently divisible integer $N$. Let $\mathbf{G}_{{\mathbb{F}}_\ell}\subset{\mathrm{GL}}_{n,{\mathbb{F}}_\ell}$ be the base change to ${\mathbb{F}}_\ell$ for $\ell\gg0$. Since $\bar\Gamma_\ell^{{\operatorname{geo}}}\subset\mathbf{G}_{{\mathbb{F}}_\ell}$ for $\ell\gg0$, we obtain $$\label{compare2}
\dim_{\bar{\mathbb{Q}}_\ell}(V_\ell^{\otimes m}\otimes\bar{\mathbb{Q}}_\ell)^{\mathbf{G}}\stackrel{Lem.~\ref{summand}}{\leq}
\dim_{\bar{\mathbb{F}}_\ell}(\bar V_\ell^{\otimes m}\otimes\bar{\mathbb{F}}_\ell)^{\mathbf{G}_{{\mathbb{F}}_\ell}}\leq
\dim_{\bar{\mathbb{F}}_\ell}(\bar V_\ell^{\otimes m}\otimes\bar{\mathbb{F}}_\ell)^{\bar\Gamma_\ell^{{\operatorname{geo}}}}$$ for $\ell\gg0$. Since $\dim_{{\mathbb{Q}}_\ell}(V_\ell^{\otimes m})^{\pi_1^{et}(X_{\bar{\mathbb{F}}_q})}=\dim_{\bar{\mathbb{Q}}_\ell}(V_\ell^{\otimes m}\otimes\bar{\mathbb{Q}}_\ell)^{\mathbf{G}}$ as $\Gamma_\ell^{{\operatorname{geo}}}$ is Zariski dense in $\mathbf{G}$, we obtain $$\label{eq1}
\dim_{\bar{\mathbb{F}}_\ell}(\bar V_\ell^{\otimes m}\otimes\bar{\mathbb{F}}_\ell)^{\mathbf{G}_{{\mathbb{F}}_\ell}}=
\dim_{\bar{\mathbb{F}}_\ell}(\bar V_\ell^{\otimes m}\otimes\bar{\mathbb{F}}_\ell)^{\bar\Gamma_\ell^{{\operatorname{geo}}}}$$ for $\ell\gg0$ by (\[compare2\]) and Theorem \[inv\]. Since $\mathbf{G}_{{\mathbb{F}}_\ell}$ is connected semisimple for $\ell\gg0$, the natural representation $i_\ell:\mathbf{G}_{{\mathbb{F}}_\ell}\to{\mathrm{GL}}(\bar V_\ell\otimes\bar{\mathbb{F}}_\ell)$ is semisimple for $\ell\gg0$ [@La95b]. Hence, it suffices to prove that for all $\ell\gg0$, the restriction of any irreducible $\bar{\mathbb{F}}_\ell$-subrepresentation $W_{\bar{\mathbb{F}}_\ell}$ of $i_\ell$ to $\bar\Gamma_\ell^{{\operatorname{geo}}}$ is still irreducible as $\bar\Gamma_\ell^{{\operatorname{geo}}}\subset \mathbf{G}_{{\mathbb{F}}_\ell}$.
***Step II.*** Suppose $W_{\bar{\mathbb{F}}_\ell}$ is a direct summand of $i_\ell$. Then for any $m\in{\mathbb{N}}$, we have $$\label{eq2}
\dim_{\bar{\mathbb{F}}_\ell}(W_{\bar{\mathbb{F}}_\ell}^{\otimes m})^{\mathbf{G}_{{\mathbb{F}}_\ell}}=
\dim_{\bar{\mathbb{F}}_\ell}(W_{\bar{\mathbb{F}}_\ell}^{\otimes m})^{\bar\Gamma_\ell^{{\operatorname{geo}}}}$$ when $\ell$ is sufficiently large by (\[eq1\]). Suppose $\bar\Gamma_\ell^{{\operatorname{geo}}}$ is not irreducible on $W_{\bar{\mathbb{F}}_\ell}$. Then there exists a $k$-dimensional subrepresentation $U_{\bar{\mathbb{F}}_\ell}$ of $\bar\Gamma_\ell^{{\operatorname{geo}}}$ and $k<\dim W_{\bar{\mathbb{F}}_\ell}\leq n$ holds. By (\[eq2\]) and (the proof of) Corollary \[sym\], $$\label{eq3}
\dim_{\bar{\mathbb{F}}_\ell}(\mathrm{Alt}^k W_{\bar{\mathbb{F}}_\ell})^{\mathbf{G}_{{\mathbb{F}}_\ell}}=
\dim_{\bar{\mathbb{F}}_\ell}(\mathrm{Alt}^k W_{\bar{\mathbb{F}}_\ell})^{\bar\Gamma_\ell^{{\operatorname{geo}}}}$$ holds when $\ell\gg0$. Since $\bar\Gamma_\ell^{{\operatorname{geo}}}\subset \mathbf{G}_{{\mathbb{F}}_\ell}$ for $\ell\gg0$, $$\label{eq4}
(\mathrm{Alt}^k W_{\bar{\mathbb{F}}_\ell})^{\mathbf{G}_{{\mathbb{F}}_\ell}}=
(\mathrm{Alt}^k W_{\bar{\mathbb{F}}_\ell})^{\bar\Gamma_\ell^{{\operatorname{geo}}}}$$ holds when $\ell\gg0$. Since $\bar\Gamma_\ell^{{\operatorname{geo}}}$ is generated by its order $\ell$ elements for $\ell\gg0$, $\mathrm{Alt}^k U_{\bar{\mathbb{F}}_\ell}$ is one-dimensional and belongs to $(\mathrm{Alt}^k W_{\bar{\mathbb{F}}_\ell})^{\bar\Gamma_\ell^{{\operatorname{geo}}}}$ when $\ell\gg0$ by construction. Thus, $\mathrm{Alt}^k U_{\bar{\mathbb{F}}_\ell}\subset (\mathrm{Alt}^k W_{\bar{\mathbb{F}}_\ell})^{\mathbf{G}_{{\mathbb{F}}_\ell}}$ by (\[eq4\]) which is impossible. Indeed, let $\{v_1,...,v_k\}$ be a basis of $U_{\bar{\mathbb{F}}_\ell}$ and $Z_{\bar{\mathbb{F}}_\ell}\neq 0$ a complement of $U_{\bar{\mathbb{F}}_\ell}$ in $W_{\bar{\mathbb{F}}_\ell}$. Since $\mathbf{G}_{{\mathbb{F}}_\ell}$ is irreducible on $W_{\bar{\mathbb{F}}_\ell}$, there exists $x\in \mathbf{G}_{{\mathbb{F}}_\ell}(\bar{\mathbb{F}}_\ell)$ that does not preserve $U_{\bar{\mathbb{F}}_\ell}$. Then we have the following equations $$\begin{aligned}
\begin{split}
x\cdot v_1 &= u_1+z_1\\
x\cdot v_2 &= u_2+z_2\\
&\vdots \\
x\cdot v_k &= u_k+z_k,
\end{split}\end{aligned}$$ where the notation is defined so that $u_i\in U_{\bar{\mathbb{F}}_\ell}$ and $z_i\in Z_{\bar{\mathbb{F}}_\ell}$ for $1\leq i\leq k$. We may assume $\{z_1,...,z_h\}$ is a non-empty maximal linearly independent subset of $\{z_1,...,z_k\}$. If $\mathrm{Alt}^k U_{\bar{\mathbb{F}}_\ell}\subset (\mathrm{Alt}^k W_{\bar{\mathbb{F}}_\ell})^{\mathbf{G}_{{\mathbb{F}}_\ell}}$, then $$\label{eq5}
x\cdot (v_1\wedge\cdots\wedge v_k)=(u_1+z_1)\wedge \cdots\wedge (u_k+z_k)\in \mathrm{Alt}^k U_{\bar{\mathbb{F}}_\ell}.$$ Since $z_1\neq 0$, we obtain $k>1$ by (\[eq5\]). Since we have the decomposition $$\label{decomp}
\mathrm{Alt}^k(U_{\bar{\mathbb{F}}_\ell}\oplus Z_{\bar{\mathbb{F}}_\ell})=\bigoplus_{i+j=k}\mathrm{Alt}^iU_{\bar{\mathbb{F}}_\ell}\otimes \mathrm{Alt}^jZ_{\bar{\mathbb{F}}_\ell},$$ we have $z_1\wedge\cdots\wedge z_k=0$ by (\[eq5\]), which is the same as $h<k$. We may assume $z_{h+1},...,z_k$ are all equal to zero by the fact that $\{z_1,...,z_h\}$ is a maximal linearly independent subset of $\{z_1,...,z_k\}$ and replacing $\{v_1,...,v_h,v_{h+1},...,v_k\}$ with a suitable basis $\{v_1,...,v_h,v_{h+1}',...,v_k'\}$. It follows that $\{z_1,...,z_h,u_{h+1},...,u_k\}$ is linearly independent because $x$ is invertible. Therefore, $ u_{h+1}\wedge\cdots\wedge u_k\wedge z_1\wedge\cdots\wedge z_h$ is non-zero, which is absurd by $z_{h+1}=\cdots=z_k=0$, (\[eq5\]), and (\[decomp\]). This implies that $\bar\Gamma_\ell^{{\operatorname{geo}}}$ cannot have a $k$-dimensional subrepresentation of $W_{\bar{\mathbb{F}}_\ell}$ when $\ell$ is sufficiently large. Since there are finitely many $k$ less than $\dim W_{\bar{\mathbb{F}}_\ell}\leq n$, we conclude that $\bar\Gamma_\ell^{{\operatorname{geo}}}$ is irreducible on $W_{\bar{\mathbb{F}}_\ell}$ and thus semisimple on $\bar V_\ell\otimes\bar{\mathbb{F}}_\ell$ if $\ell$ is sufficiently large.
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[^1]: Chun Yin Hui is supported by the National Research Fund, Luxembourg, and cofunded under the Marie Curie Actions of the European Commission (FP7-COFUND)
|
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|
[Tiled Image Convention for Storing Compressed Images]{} [in FITS Binary Tables]{}\
Richard L. White, STScI\
Perry Greenfield, STScI\
William Pence, NASA/GSFC\
Doug Tody, NRAO\
Rob Seaman, NOAO\
Version 2.2\
May 4, 2011\
General Description
===================
This document describes a convention for compressing n-dimensional images and storing the resulting byte stream in a variable-length column in a FITS binary table. The FITS file structure outlined here is independent of the specific data compression algorithm that is used. The implementation details for 4 widely used compression algorithms are described here, but any other compression technique could also be supported by this convention.
The general principle used in this convention is to first divide the n-dimensional image into a rectangular grid of subimages or ‘tiles’. Each tile is then compressed as a block of data, and the resulting compressed byte stream is stored in a row of a variable length column in a FITS binary table. By dividing the image into tiles it is generally possible to extract and uncompress subsections of the image without having to uncompress the whole image. The default tiling pattern treats each row of a 2-dimensional image (or higher dimensional cube) as a tile, such that each tile contains [NAXIS1]{} pixels. This default many not be optimal for some applications or compression algorithms, so any other rectangular tiling pattern may be defined using the [ZTILEn]{} keywords that are described below. In the case of relatively small images it may be sufficient to compress the entire image as a single tile, resulting in an output binary table with 1 row. In the case of 3-dimensional data cubes, it may be advantageous to treat each plane of the cube as a separate tile if application software typically needs to access the cube on a plane by plane basis.
Keywords
========
The following keywords are defined by this convention for use in the header of the FITS binary table extension to describe the structure of the compressed image.
- [ZIMAGE]{} (required keyword) This keyword must have the logical value T. It indicates that the FITS binary table extension contains a compressed image, and that logically this extension should be interpreted as an image and not as a table.
- [ZCMPTYPE]{} (required keyword) The value field of this keyword shall contain a character string giving the name of the algorithm that must be used to decompress the image. Currently, values of [GZIP\_1]{}, [RICE\_1]{}, [PLIO\_1]{}, and [HCOMPRESS\_1]{} are reserved, and the corresponding algorithms are described in a later section of this document.
- [ZBITPIX]{} (required keyword) The value field of this keyword shall contain an integer that gives the value of the [BITPIX]{} keyword in the uncompressed FITS image.
- [ZNAXIS]{} (required keyword) The value field of this keyword shall contain an integer that gives the value of the [NAXIS]{} keyword in the uncompressed FITS image.
- [ZNAXISn]{} (required keywords) The value field of these keywords shall contain a positive integer that gives the value of the [NAXISn]{} keywords in the uncompressed FITS image.
- [ZTILEn]{} (optional keywords) The value of these indexed keywords (where [n]{} ranges from 1 to [ZNAXIS]{}) shall contain a positive integer representing the number of pixels along axis [n]{} of the compression tiles. Each tile of pixels is compressed separately and stored in a row of a variable-length vector column in the binary table. The size of each image dimension (given by [ZNAXISn]{}) is not required to be an integer multiple of ZTILEn, and if it is not, then the last tile along that dimension of the image will contain fewer image pixels than the other tiles. If the [ZTILEn]{} keywords are not present then the default ’row by row’ tiling will be assumed such that [ZTILE1 = ZNAXIS1]{}, and the value of all the other [ZTILEn]{} keywords equals 1.
The compressed image tiles are stored in the binary table in the same order that the first pixel in each tile appears in the FITS image; the tile containing the first pixel in the image appears in the first row of the table, and the tile containing the last pixel in the image appears in the last row of the binary table.
- [ZNAMEn and ZVALn]{} (optional keywords) These pairs of optional array keywords (where n is an integer index number starting with 1) supply the name and value, respectively, of any algorithm-specific parameters that are needed to compress or uncompress the image. The value of [ZVALn]{} may have any valid FITS datatype. The order of the compression parameters may be significant, and may be defined as part of the description of the specific decompression algorithm.
- [ZMASKCMP]{} (optional keyword) Used to record the name of the image compression algorithm that was used to compress the optional null pixel data mask. See the“Preserving undefined pixels with lossy compression” section for more details.
- The following 8 optional keywords are defined to store a verbatim copy of the the value and comment fields of the corresponding keywords in the original uncompressed FITS image. These keywords can be used to reconstruct an identical copy of the original FITS file when the image is uncompressed.
- [ZSIMPLE]{} - preserves the original [SIMPLE]{} keyword
- [ZTENSION]{} - preserves the original [XTENSION]{} keyword
- [ZEXTEND]{} - preserves the original [EXTEND]{} keyword
- [ZBLOCKED]{} - preserves the original [BLOCKED]{} keyword
- [ZPCOUNT]{} - preserves the original [PCOUNT]{} keyword
- [ZGCOUNT]{} - preserves the original [GCOUNT]{} keyword
- [ZHECKSUM]{} - preserves the original [CHECKSUM]{} keyword
- [ZDATASUM]{} - preserves the original [DATASUM]{} keyword
The [ZSIMPLE]{}, [ZEXTEND]{}, and [ZBLOCKED]{} keywords may only be used if the original uncompressed image was contained in the primary array of the FITS file. The [ZTENSION]{}, [ZPCOUNT]{}, and [ZGCOUNT]{} keywords may only be used if the original uncompressed image was contained in in IMAGE extension.
- [ZQUANTIZ]{} (optional keyword) This keyword records the name of the algorithm that was used to quantize floating-point image pixels into integer values which are then passed to the compression algorithm, as discussed further in section 4 of this document.
- [Other Keywords]{} The FITS header of the compressed image may contain other optional keywords. If a FITS primary array or IMAGE extension is compressed using the convention described here, it is recommended that all the keywords in the header of the original image, except for the mandatory keywords mentioned above, be copied verbatim and in the same order into the header of the binary table extension that contains the compressed image. All these keywords will have the same meaning and interpretation as they did in the original image, even in cases where the keyword is not normally expected to occur in the header of a binary table extension (e.g., the [BSCALE]{} and [BZERO]{} keywords, or the World Coordinate System keywords such as [CTYPEn, CRPIXn]{} and [CRVALn]{}).
Columns
=======
The following columns in the FITS binary table are defined by this convention. The order of the columns in the table is not significant. The column names (given by the [TTYPEn]{} keyword) are shown here in upper case letters, but the case is not significant.
- [COMPRESSED\_DATA]{} (required column) Each row of this variable-length column contains the byte stream that was generated as a result of compressing the corresponding image tile. The datatype of the column (as given by the [TFORMn]{} keyword) will generally be either [’1PB’, ’1PI’]{}, or [’1PJ’]{}, depending on whether the compression algorithm generates an output stream of 8-bit bytes, 16-bit integers, or 32-bit integers, respectively. If it is not possible to efficiently compress a particular image tile, then the [COMPRESSED\_DATA]{} vector in the corresponding row will have a length of zero, and the uncompressed tile pixels will be written instead to the [UNCOMPRESSED\_DATA]{} or [GZIP\_COMPRESSED\_DATA]{} columns, as described below.
- [UNCOMPRESSED\_DATA]{} (optional column) This variable length column contains the uncompressed pixels for any tiles that cannot be compressed. The datatype of this column will usually correspond to the datatype of the original image as shown in the following table:
Datatype [BITPIX]{} [TFORMn]{}
----------- ------------ ------------
byte 8 ’1PB’
short int 16 ’1PI’
long int 32 ’1PJ’
float -32 ’1PE’
double -64 ’1PD’
If all the tiles in an image are able to be compressed, then the [UNCOMPRESSED\_DATA]{} column is not required. A tile compressed image may only contain either the [UNCOMPRESSED\_DATA]{} column or the [GZIP\_COMPRESSED\_DATA]{} column (or neither), but not both.
- [GZIP\_COMPRESSED\_DATA]{} (optional column) The lossy quantization method that is often used to compress floating-point images, as described in Section 4, can fail in certain cases (for example, when all the pixels within a tile have the same value and hence have a calculated RMS noise = 0). In such cases, the [GZIP\_COMPRESSED\_DATA]{} column may be used to store the original floating-point pixel values after compressing them with the gzip algorithm. This is almost always more efficient than storing the uncompressed pixel values in the [UNCOMPRESSED\_DATA]{} column. This optional column was introduced in version 2.2 of this convention in May 2011.
If all the tiles in an image are able to be compressed, then the [GZIP\_COMPRESSED\_DATA]{} column is not required. A tile compressed image may only contain either the [UNCOMPRESSED\_DATA]{} column or the [GZIP\_COMPRESSED\_DATA]{} column (or neither), but not both.
- [ZSCALE and ZZERO]{} (optional columns) These columns give the linear scale factor and zero point offset which may be needed to transform the raw uncompressed values back to the original image pixel values (or at least a close approximation to the original values) using the following formula:
image\_pixel\_value = (uncompressed\_value \* [ZSCALE]{}) + [ZZERO]{}\
[ZSCALE]{} and [ZZERO]{} generally have double precision values and have default values of 1.0 and 0.0, respectively. If the same values of ZSCALE and ZZERO apply to every tile in the image, then they may be given as header keywords rather than as table columns.
[ZSCALE]{} and [ZZERO]{} are typically used to scale floating-point images (with [BITPIX]{} = -32 or -64) into integers before compression, since most compression algorithms are not very efficient with floating-point data. One particularly effective scaling algorithm is described in the next section.
These 2 parameters should not be confused with the reserved [BSCALE]{} and [BZERO]{} keywords which may be present in integer FITS images (which have BITPIX = 8, 16, or 32). Any such integer images should normally be compressed without any further scaling, and the [BSCALE]{} and [BZERO]{} keywords should be copied verbatim into the header of the binary table containing the compressed image.
- [ZBLANK]{} (optional column) In cases where floating-point images are converted to integers before being compressed, this column gives the the integer value that is used to represent undefined pixels (if any) in the image. These pixels would have an IEEE NaN (Not a Number) value in the uncompressed floating-point FITS image. If every tile uses the same null value, then [ZBLANK]{} may be given as a keyword instead of as a table column. If there are no undefined pixels in the image then [ZBLANK]{} is not required. If the uncompressed image has an integer datatype ([ZBITPIX]{} $>$ 0) then the reserved [BLANK]{} keyword which already serves this purpose should be used instead of [ZBLANK]{}.
- [NULL\_PIXEL\_MASK]{} (optional column) In cases where the image contains undefined pixels and a lossy compression algorithm is used (and hence the pixel values are not exactly preserved) then this column is used to store a compressed image mask that records the location of any undefined pixels. See the “Preserving undefined pixels with lossy compression” section for more details.
- [Other Columns]{} Any number of other columns may be present in the table to supply other parameters that relate to each image tile.
Quantization of Floating-Point Data
===================================
Images that have floating-point data type pixels often do not compress very effectively due to the presence of noise in the least significant bits of the pixel values. In order to achieve a higher degree of compression, one can effectively discard some of the noise bits by linearly scaling the image into integer pixel values, so that
$F_i$ = ($I_i$ \* [ZSCALE]{}) + [ZZERO]{}\
where $I_i$ and $F_i$ are the integer and floating-point values, respectively.
Note that the tiled image compression convention does not require that floating point images be scaled to integers before compressing them, but if linear scaling is performed, then the ZSCALE and ZZERO columns in the FITS binary table should be used to record the 2 scaling coefficients, as described in the previous section.
The maximum amount of numerical precision will be preserved if the ZSCALE and ZZERO values are calculated such that the scaled pixel values span the full range of the integer datatype (e.g., from -32768 to +32767 for 16-bit integers). This may also preserve an undesirable amount of non-significant noise, which can adversely affect the amount of compression that can be achieved.
A more effective scaling algorithm that preserves a specified amount of noise in each pixel value is described by White and Greenfield (in the Proceedings of the 1998 ADASS VIII conference) and by Pence, Seaman, and White, PASP 121, 414 (2009). With this method, the ZSCALE value (which is numerically equal to the spacing between adjacent quantization levels) is calculated to be some fraction, Q, of the RMS noise as measured in background regions of the image. It can be shown that the number of binary bits of noise that are preserved in each pixel value is given by $log_2(Q) + 1.792$. For example, using Q = 8 (so that the quantized levels have a spacing of 1/8th of the background RMS noise value) produces a quantized image that preserves about 4.8 bits of noise in each pixel. Specifying the quantization level relative to the amount of noise in the image in this way produces comparable quality images regardless of the noise level. Q is directly related to the compressed file size: decreasing Q by a factor of 2 will decrease the file size by about 1 bit/pixel. In order to achieve the greatest amount of compression, one should use the smallest value of Q that still preserves the required amount of photometric and astrometric precision in the image.
As the Q value is decreased, the spacing between the quantized levels in the image increases, which can have the undesirable effect of significantly biasing the pixel values in the faint regions the image (i.e., the ’sky’ level in typical astronomical images). This bias can be mitigated by adding noise during the quantization process. So instead of simply scaling every pixel value using the equation:
$I_i$ = ROUND(($F_i$ - [ZZERO]{}) / [ZSCALE]{})\
(where the ROUND function rounds the result to the nearest integer value) one can randomize the quantized levels by using this slightly modified equation:
$I_i$ = ROUND((($F_i$ - [ZZERO]{}) / [ZSCALE]{}) + $R_i$ - 0.5)\
where $R_i$ is a random number between 0 and 1, and the 0.5 is subtracted so that the mean quantity is equal to 0. Then when restoring the floating-point value, the same random number is used with the inverse formula
$F_i$ = (($I_i$ - $R_i$ +0.5) \* [ZSCALE]{}) + [ZZERO]{}\
This technique, which is referred to as ‘subtractive dithering’ in the signal processing literature (e.g., “Quantization Noise” by Widrow and Kollar) has the effect of dithering the zero-point of the quantization grid on a pixel by pixel basis without introducing any additional noise in the image. The net effect of this is that the mean (and median) pixel value in faint regions of the image more closely approximate the value in the original unquantized image than if all the pixels are scaled without randomization. This can significantly increase the precision when measuring the net flux from faint sources in the compressed image.
The key requirement when using this technique is that the exact same random number sequence must be used when quantizing the pixel values to integers, and when restoring them to floating point values. While most computer languages supply a function for generating random numbers, these functions are not guaranteed to generate the same sequence of numbers every time. Accordingly, we define a specific algorithm here for generating a repeatable sequence of pseudo random numbers. The steps in the algorithm for quantizing (or unquantizing) each tile of the image are as follows:
1. Generate a sequence of 10000 random numbers using the algorithm given in Appendix A. Since it would be computationally expensive to generate a unique random number for every pixel of large images, we repeatedly recycle through this ‘look up table’ of random numbers.
2. The above sequence of random numbers is used when quantizing or unquantizing each tile of the floating point image. In order to avoid possible ‘banding’ effects if one were to use exactly the same sequence of random numbers for every tile, we calculate a unique, random offset to the first random number in the sequence to use as a function of the tile number using the formula:
offset = INT ( 500. \* R(N) ) + 1\
where offset is the ones-based index to the first random number in the sequence to use, INT is the floating-point to integer truncation function, and R(N) is the Nth random number in the sequence where N is the tile number. If N exceeds 10000, then one should use ((N - 1) modulo 10000) + 1. So for example, when compressing the 2nd tile in an image, the 2nd random number in the sequence has a value of 0.131538, and thus the offset value is 66. For reference, the 66th random number should have a value of 0.493977.
This random number is then used to quantize (or unquantize) the first pixel of this tile using the subtractive dithering function given above. The next random number in the sequence is then used for next pixel in the tile, and so on.
3. If one reaches the end of the sequence of 10000 random numbers while quantizing or unquantizing the pixels in tile N, then one should cycle back through the random number sequence, using a new random starting offset calculated using the Nth + 1 random number. For example, if one is quantizing tile number 9 of the image, the original starting offset values would be calculated by multiplying the 9th random number (0.679296) in the sequence by 500 (plus 1). Then if one reaches the end of the random number sequence again, the next starting offset value is calculated using the 10th random number (0.934693). If necessary, this process is repeated using the next random number each time (starting over at 1 if one reaches 10000).
4. Repeat Steps 2 and 3 for each tile of the image.
The above algorithm is clearly not unique, but we present it here as a well defined method that should be easy to implement in almost any computer language. If this particular ’subtractive dithering’ algorithm is used when quantizing a floating point image, then the following keyword should be recorded in the compressed image header:
ZQUANTIZ= 'SUBTRACTIVE_DITHER_1'
Other values for this keyword may be defined in the future to identify other quantizing methods. If this keyword is not present in the header of a tile-compressed, quantized, floating-point image, then it should be assumed that only simple linear scaling was applied when quantizing the image.
It should be noted that an image that is quantized using this technique can still be unquantized using the simple linear scaling function. The only side effect in this case is to introduce slightly more noise in the image than if the full subtractive dithering algorithm were applied.
Preserving undefined pixels with lossy compression
==================================================
Any undefined pixels in a FITS image are flagged with a special pixel value: the BLANK keyword specifies the value in integer data type FITS images, and an IEEE NaN (Not a Number) value is used in single or double precision floating point FITS images. Floating point images are often converted to scaled integers prior to compression (as described previously) in which case the undefined pixel value is then given by the ZBLANK keyword (or column).
The null pixel values in the image will be preserved if a lossless compression algorithm is used. If the image is compressed with a lossy algorithm (e.g., H-Compress with a scale factor greater than 1), then some other technique must be used to identify the null pixels in the image.
The recommended method of recording the null pixels when a lossy compression algorithm is used is to create an integer data mask with the same dimensions as the image tile. Set the null pixels to 1 and all the other pixels to 0, then compress the mask array using a lossless algorithm such as PLIO or GZIP. Store the compressed byte stream in a variable-length array column called ’NULL\_PIXEL\_MASK’ in the row corresponding to that image tile. The ZMASKCMP keyword should be used to record the name of the algorithm used to compress the data mask (e.g., RICE\_1). The data mask array pixels will be assumed to have the shortest integer datatype that is supported by the compression algorithm (i.e., usually 8-bit bytes).
When uncompressing the image tile, the software must check if the corresponding compressed data mask exists with a length greater than 0, and if so, then uncompress the mask and set the corresponding undefined pixels in the image array to the appropriate value (as given by the BLANK or ZBLANK keyword).
Currently Implemented Compression Algorithms
============================================
This section describes the 4 compression algorithms that are currently supported in the CFITSIO implementation of this tiled image compression convention (available from the HEASARC web site). This does not imply that other implementations of this convention must support these same algorithms, nor does it limit other implementations from supporting other compression algorithms.
Rice compression algorithm
--------------------------
The Rice algorithm (Rice, R. F., Yeh, P.-S., and Miller, W. H. 1993, in Proc. of the 9th AIAA Computing in Aerospace Conf., AIAA-93-4541-CP, American Institute of Aeronautics and Astronautics) is simple and very fast, compressing or decompressing $10^7$ pixels/sec on modern workstations. It requires only enough memory to hold a single block of 16 or 32 pixels at a time. It codes the pixels in small blocks and so is able to adapt very quickly to changes in the input image statistics (e.g., Rice has no problem handling cosmic rays, bright stars, saturated pixels, etc.).
The block size that is used should be recorded in the compressed image header with
ZNAMEn = 'BLOCKSIZE'
ZVALn = 16 or 32
If these keywords are absent, then a default blocksize of 32 should be assumed.
The number of 8-bit bytes in each original integer pixel value should be recorded in the compressed image header with
ZNAMEn = 'BYTEPIX'
ZVALn = 1, 2, 4, or 8
If these keywords are absent, then the default value of 4 bytes per pixel (32 bits) should be assumed..
GZIP compression algorithm
--------------------------
Gzip is the compression algorithm used in the widely distributed GNU free software utility of the same name. It was created by Jean-loup Gailly and Mark Adler. Version 0.1 was first publicly released on October 31, 1992. Version 1.0 followed in February 1993. It is based on the DEFLATE algorithm, which is a combination of LZ77 and Huffman coding. DEFLATE was intended as a replacement for LZW and other patent-encumbered data compression algorithms which, at the time, limited the usability of compress and other popular archivers. Further information about this compression technique is readily available on the Internet.
The gzip algorithm has no associated parameters that need to be specified with the [ZNAMEn and ZVALn]{} keywords.
IRAF PLIO compression algorithm
-------------------------------
The IRAF PLIO (pixel list) algorithm was developed to store integer-valued image masks in a compressed form. Typical uses of image masks are to segment images into regions, or to mark bad pixels. Such masks often have large regions of constant value hence are highly compressible. The compression algorithm used is based on run-length encoding, with the ability to dynamically follow level changes in the image, allowing a 16-bit encoding to be used regardless of the image depth. The worst case performance occurs when successive pixels have different values. Even in this case the encoding will only require one word (16 bits) per mask pixel, provided either the delta intensity change between pixels is usually less than 12 bits, or the mask represents a zero floored step function of constant height. The worst case cannot exceed npix\*2 words provided the mask depth is 24 bits or less.
A good compromise between storage efficiency and efficiency of runtime access, while keeping things simple, is achieved if we maintain the compressed line lists as variable length arrays of type short integer (16 bits per list element), regardless of the mask depth. A line list consists of a series of simple instructions which are executed in sequence to reconstruct a line of the mask. Each 16 bit instruction consists of the sign bit (not used at present), a three bit opcode, and twelve bits of data, i.e.:
+--+-----------+-----------------------------+
|16|15 13|12 1|
+--+-----------+-----------------------------+
| | opcode | data |
+--+-----------------------------------------+
The significance of the data depends upon the instruction. The instructions currently implemented are summarized in the table below.
Instruction Opcode Description
ZN 00 Output N zeros
HN 04 Output N high values
PN 05 Output N-1 zeros plus one high value
SH 01 Set high value, absolute
IH,DH 02,03 Increment or decrement high value
IS,DS 06,07 Like IH-DH, plus output one high value
In order to reconstruct a mask line, the application executing these instructions is required to keep track of two values, the current high value and the current position in the output line. The detailed operation of each instruction is as follows:
- Zero the next N (=data) output pixels.
- Set the next N output pixels to the current high value.
- Zero the next N-1 output pixels, and set pixel N to the current high value.
- Set the high value (absolute rather than incremental), taking the high 15 bits from the next word in the instruction stream, and the low 12 bits from the current data value.
- Increment (IH) or decrement (DH) the current high value by the data value. The current position is not affected.
- Increment (IS) or decrement (DS) the current high value by the data value, and step, i.e., output one high value.
The high value is assumed to be set to 1 at the beginning of a line, hence the IH,DH and IS,DS instructions are not normally needed for Boolean masks. If the length of a line segment of constant value or the difference between two successive high values exceeds 4096 (12 bits), then multiple instructions are required to describe the segment or intensity change.
H-Compress algorithm
--------------------
Hcompress is an the image compression package written by Richard L. White for use at the Space Telescope Science Institute ([email protected]). Hcompress was used to compress the STScI Digitized Sky Survey and has also been used to compress the preview images in the Hubble Data Archive. Briefly, the method used is:
1. a wavelet transform called the H-transform (a Haar transform generalized to two dimensions), followed by
2. quantization that discards noise in the image while retaining the signal on all scales, followed by
3. quadtree coding of the quantized coefficients.
The technique gives very good compression for astronomical images and is relatively fast. The calculations are carried out using integer arithmetic and are entirely reversible. Consequently, the program can be used for either lossy or lossless compression, with no special approach needed for the lossless case (e.g. there is no need for a file of residuals.)
There are 2 user-defined parameters associated with the H-Compress algorithm: an integer scale factor that determines the amount of compression, and a Boolean parameter the specifies whether the image should be smoothed during the decompression operation, to reduce residual artifacts in the image.
- [**Scale Factor.**]{} The integer scale parameter determines the amount of compression. Scale = 0 or 1 leads to lossless compression, i.e. the decompressed image has exactly the same pixel values as the original image. If the scale factor is greater than 1 then the compression is lossy: the decompressed image will not be exactly the same as the original. For astronomical images, lossless compression is generally rather ineffective because the images have a good deal of noise, which is inherently incompressible. However, if some of this noise is discarded then the images compress very well. The scale factor determines how much of the noise is discarded. Setting scale to 2 times sigma, the RMS noise in the image, usually results in compression by about a factor of 10 (i.e. the compressed image requires about 1.5 bits/pixel), while producing a decompressed image that is nearly indistinguishable from the original. In fact, the RMS difference between the decompressed image and the original image will be only about 1/2 sigma. Experiments indicate that this level of loss has no noticeable effect on either the visual appearance of the image or on quantitative analysis of the image (e.g. measurements of positions and brightnesses of stars are not adversely affected.)
Using a larger value for scale results in higher compression at the cost of larger differences between the compressed and original images. A rough rule of thumb is that if scale equals N sigma, then the image will compress to about 3/N bits/pixel, and the RMS difference between the original and the compressed image will be about N/4 sigma. This crude relationship is inaccurate both for very high compression ratios and for lossless compression, but it does at least give an indication of what to expect of the compressed images.
For images in which the noise varies from pixel to pixel (e.g. CCD images, where the noise is larger for brighter pixels), the appropriate value for scale is determined by the RMS noise level in the sky regions of the image. For images that are essentially noiseless, any lossy compression is noticeable under sufficiently close inspection of the image, but some loss is nonetheless acceptable for typical applications. Note that the quantization scheme used in Hcompress is not designed to give images that appear as much like the original as possible to the human eye, but rather is designed to produce images that are as similar as possi- ble to the original under quantitative analysis. Thus, the emphasis is on discarding noise without affecting the signal rather than on discarding components of the image that are not very noticeable to the eye (as may be done, for example, by JPEG compression.) The resulting compression scheme is not ideal for typical terrestrial images (though it is still a reasonably good method for those images), but is believed to be close to optimal for astronomical images.
It is not necessary to know what scale factor was used when compressing the image in order to uncompress it, but it is still useful to record the value that was used. It is recommended that the [ZNAMEn]{} and [ZVALn)]{} pair of keywords be used for this purpose, with
ZNAMEn = 'SCALE'
ZVALn = I
where [*I*]{} is the integer scale value.
- [**Smoothing Flag.**]{} At high compressions factors the decompressed image begins to appear blocky because of the way information is discarded. This blockiness ness is greatly reduced, producing more pleasing images, if the image is smoothed slightly during decompression. When done properly, the smoothing will not affect any quantitative photometric or astrometric measurements derived from the compressed image. Of course, the smoothing should never be applied when the image has been losslessly compressed with a scale factor (defined above) of 0 or 1.
The smoothing option only needs to be specified when uncompressing the image, however, in many cases, this can best be determined by the person or project that creates the compressed image files. Thus it is recommended that the smoothing flag be specified in the compressed image header with the [ZNAMEn]{} and [ZVALn]{} keywords with
ZNAMEn = 'SMOOTH'
ZVALn = 0 or 1
A value of 0 means no smoothing, and any other value means smoothing is recommended. This should be regarded as only a recommendation which the image decompression program may override.
A paper describing Hcompress was published in the Proceedings of the NASA Space and Earth Science Data Compression Workshop, ed. James C. Tilton, Snowbird, Utah, March 1992. This paper is reproduced in the Appendix B of this document.
Random Number Generator
=======================
This portable random number generator algorithm comes from the publication “Random number generators: good ones are hard to find", Communications of the ACM, Volume 31 , Issue 10 (October 1988) Pages: 1192 - 1201 which is available on the Web. This algorithm basically just repeatedly evaluates the function seed = (a \* seed) mod m, where the values of a and m are shown below, but it is implemented in a way to avoid integer overflow problems.
int random_generator(void) {
/* initialize an array of random numbers */
int ii;
double a = 16807.0;
double m = 2147483647.0;
double temp, seed;
float rand_value[10000];
/* initialize the random numbers */
seed = 1;
for (ii = 0; ii < N_RANDOM; ii++) {
temp = a * seed;
seed = temp -m * ((int) (temp / m) );
rand_value[ii] = seed / m; /* divide by m to get value between 0 and 1 */
}
}
If implemented correctly, the 10000th value of seed must equal 1043618065.
High-Performance Compression of Astronomical Images
===================================================
*Richard L. White*
*Joint Institute for Laboratory Astrophysics, University of Colorado*
*Campus Box 440, Boulder, CO 80309*
*and*
*Space Telescope Science Institute*
*3700 San Martin Drive, Baltimore, MD 21218*
*[email protected]*
**Summary**
Astronomical images have some rather unusual characteristics that make many existing image compression techniques either ineffective or inapplicable. A typical image consists of a nearly flat background sprinkled with point sources and occasional extended sources. The images are often noisy, so that lossless compression does not work very well; furthermore, the images are usually subjected to stringent quantitative analysis, so any lossy compression method must be proven not to discard useful information, but must instead discard only the noise. Finally, the images can be extremely large. For example, the Space Telescope Science Institute has digitized photographic plates covering the entire sky, generating 1500 images each having $14000\times14000$ 16-bit pixels. Several astronomical groups are now constructing cameras with mosaics of large CCDs (each $2048\times2048$ or larger); these instruments will be used in projects that generate data at a rate exceeding 100 MBytes every 5 minutes for many years.
An effective technique for image compression may be based on the H-transform/ (Fritze et al. 1977). The method that we have developed can be used for either lossless or lossy compression. The digitized sky survey images can be compressed by at least a factor of 10 with no noticeable losses in the astrometric and photometric properties of the compressed images. The method has been designed to be computationally efficient: compression or decompression of a $512\times512$ image requires only 4 seconds on a Sun SPARCstation 1. The algorithm uses only integer arithmetic, so it is completely reversible in its lossless mode, and it could easily be implemented in hardware for space applications.
**1. Introduction**
Astronomical images consist largely of empty sky. Compression of such images can reduce the volume of data that it is necessary to store (an important consideration for large scale digital sky surveys) and can shorten the time required to transmit images (useful for remote observing or remote access to data archives.)
Data compression methods can be classified as either “lossless” (meaning that the original data can be reconstructed exactly from the compressed data) or “lossy” (meaning that the uncompressed image is not exactly the same as the original.) Astronomers often insist that they can accept only lossless compression, in part because of conservatism, and in part because the familiar lossy compression methods sacrifice some information that is needed for accurate analysis of image data. However, since all astronomical images contain noise, which is inherently incompressible, lossy compression methods produce much better compression results.
A simple example may make this clear. One of the simplest data compression techniques is run-length coding, in which runs of consecutive pixels having the same value are compressed by storing the pixel value and the repetition factor. This method is used in the standard compression scheme for facsimile transmissions. Unfortunately, it is quite ineffective for lossless compression of astronomical images because even though the sky is [*nearly*]{} constant, the noise in the sky ensures that only very short runs of equal pixels occur. The obvious way to make run-length coding more effective is to force the sky to be exactly constant by setting all pixels below a threshold (chosen to be just above the sky) to the mean sky value. However, then one has lost any information about objects close to the detection limit. One has also lost information about local variations in the sky brightness, which severely limits the accuracy of photometry and astrometry on faint objects. Worse, there may be extended, low surface brightness objects that are not detectable in a single pixel but that are easily detected when the image is smoothed over a number of pixels; such faint structures are irretrievably lost when the image is thresholded to improve compression.
**2. The H-transform**
Fritze et al. (1977; see also Richter 1978 and Capaccioli et al. 1988) have developed a much better compression method for astronomical images based on what they call the [*H-transform*]{} of the image. A similar transform called the S-transform has also been used for image compression (Blume & Fand 1989). The H-transform is a two-dimensional generalization of the Haar transform (Haar 1910). The H-transform/ is calculated for an image of size $2^N\times 2^N$ as follows:
- Divide the image up into blocks of $2\times2$ pixels. Call the 4 pixels in a block $a_{00}$, $a_{10}$, $a_{01}$, and $a_{11}$.
- For each block compute 4 coefficients:
$
h_0 = (a_{11}+a_{10}+a_{01}+a_{00})/2 \\
h_x = (a_{11}+a_{10}-a_{01}-a_{00})/2 \\
h_y = (a_{11}-a_{10}+a_{01}-a_{00})/2 \\
h_c = (a_{11}-a_{10}-a_{01}+a_{00})/2 \\
$
- Construct a $2^{N-1}\times 2^{N-1}$ image from the $h_0$ values for each $2\times2$ block. Divide that image up into $2\times2$ blocks and repeat the above calculation. Repeat this process $N$ times, reducing the image in size by a factor of 2 at each step, until only one $h_0$ value remains.
This calculation can be easily inverted to recover the original image from its transform. The transform is exactly reversible using integer arithmetic if one does not divide by 2 for the first set of coefficients. It is straightforward to extend the definition of the transform so that it can be computed for non-square images that do not have sides that are powers of 2. The H-transform can be performed in place in memory and is very fast to compute, requiring about $16M^2/3$ (integer) additions for a $M\times M$ image.
The H-transform is a simple 2-dimensional wavelet transform. It has several advantages over some other wavelet transforms that have been applied to image compression (e.g., Daubechies 1988). First, the transform can be performed entirely with integer arithmetic, making it exactly reversible. Consequently it can be used for either lossless or lossy compression (as indicated below) and one does not need a special technique for the case of lossless compression (as was required, e.g., for the JPEG compression standard.)
A second major advantage is that the H-transform is a native 2-dimensional wavelet transform. The standard 1-dimensional wavelet transforms are extended to two dimensions by transforming the image first along the rows, then along the columns. Unfortunately, this generates many wavelet coefficients that are high frequency (hence localized) in the $x$-direction but low frequency (hence global) in the $y$-direction. Such coefficients are counter to the philosophy of the wavelet transform: high-frequency basis functions should be confined to a relatively small area of the image. Discarding these mixed-scale terms, which may be negligible compared to the noise, generates very objectionable artifacts around point sources and edges in the image. The H-transform, on the other hand, is a fully 2-dimensional wavelet transform, with all high frequency terms being completely localized. It is consequently more suitable for image compression and produces fewer artifacts.
A possible disadvantage of the H-transform is that other wavelet transforms take better advantage of the continuity of pixel values within images, so that they can produce higher compressions for very smooth images. However, for astronomical images (which are mostly flat sky sprinkled with point sources) the smoothness built into higher-order transforms can actually reduce the effectiveness of compression, because one must keep more coefficients to describe each point source.
**3. Compression Using the H-transform**
If the image is nearly noiseless, the H-transform is somewhat easier to compress than the original image because the differences of adjacent pixels (as computed in the H-transform) tend to be smaller than the original pixel values for smooth images. Consequently fewer bits are required to store the values of the H-transform coefficients than are required for the original image. For very smooth images the pixel values may be constant over large regions, leading to transform coefficients that are zero over large areas.
Noisy images still do not compress well when transformed, though. Suppose there is noise $\sigma$ in each pixel of the original image. Then from simple propagation of errors, the noise in each of the H-transform coefficients is also $\sigma$. To compress noisy images, divide each coefficient by $S\sigma$, where $S\sim1$ is chosen according to how much loss is acceptable. This reduces the noise in the transform to $0.5/S$, so that large portions of the transform are zero (or nearly zero) and the transform is highly compressible.
Why is this better than simply thresholding the original image? As discussed above, if we simply divide the image by $\sigma$ then we lose all information on objects that are within $1\sigma$ of sky in a [*single*]{} pixel, but that are detectable by averaging a [*block*]{} of pixels. On the other hand, in dividing the H-transform by $\sigma$, we preserve the information on any object that is detectable by summing a block of pixels! The quantized H-transform preserves the mean of the image for every block of pixels having a mean significantly different than that of neighboring blocks of pixels.
As an example, Figure 1 shows a $128\times128$ section ($3.6\times3.6$ arcmin) from a digitized version of the Palomar Observatory–National Geographic Society Sky Survey plate containing the Coma cluster of galaxies. Figures 2, 3, and 4 show the resulting image for $S \simeq 0.5$, 1, and 2. These images are compressed by factors of 10, 20, and 60 using the coding scheme described below. In all cases a logarithmic gray scale is used to show the maximum detail in the image near the sky background level; the noise is clearly visible in Figure 1. The image compressed by a factor of 10 is hardly distinguishable from the original. In quantizing the H-transform we have adaptively filtered the original image by discarding information on some scales and keeping information on other scales. This adaptive filtering is most apparent for high compression factors (Fig. 4), where the sky has been smoothed over large areas while the images of stars have hardly been affected.
The adaptive filtering is, in itself, of considerable interest as an analytical tool for images (Capaccioli et al. 1988). For example, one can use the adaptive smoothing of the H-transform to smooth the sky without affecting objects detected above the (locally determined) sky; then an accurate sky value can be determined by reference to any nearby pixel.
The blockiness that is visible in Figure 4 is the result of difference coefficients being set to zero over large areas, so that blocks of pixels are replaced by their averages. It is possible to eliminate the blocks by an appropriate filtering of the image. A simple but effective filter can be derived by simply adjusting the H-transform coefficients as the transform is inverted to produce a smooth image; as long as changes in the coefficients are limited to $\pm S\sigma/2$, the resulting image will still be consistent with the thresholded H-transform.
**4. Efficient Coding**
The quantized H-transform has a rather peculiar structure. Not only are large areas of the transform image zero, but the non-zero values are strongly concentrated in the lower-order coefficients. The best approach we have found to code the coefficient values efficiently is quadtree coding of each bitplane of the transform array. Quadtree coding has been used for many purposes (see Samet 1984 for a review); the particular form we are using was suggested by Huang and Bijaoui (1991) for image compression.
- Divide the bitplane up into 4 quadrants. For each quadrant code a ‘1’ if there are any 1-bits in the quadrant, else code a ‘0’.
- Subdivide each quadrant that is not all zero into 4 more pieces and code them similarly. Continue until one is down to the level of individual pixels.
This coding (which Huang and Bijauoi call “hierarchic 4-bit one” coding) is obviously very well suited to the H-transform image because successively lower orders of the H-transform coefficients are located in successively divided quadrants of the image.
We follow the quadtree coding with a fixed Huffman coding that uses 3 bits for quadtree values that are common (e.g., 0001, 0010, 0100, and 1000) and uses 4 or 5 bits for less common values. This reduces the final compressed file size by about 10% at little computational cost. Slightly better compression can be achieved by following quadtree coding with arithmetic coding (Witten, Bell, and Cleary 1987), but the CPU costs of arithmetic coding are not, in our view, justified for 3–4% better compression. We have also tried using arithmetic coding directly on the H-transform, with various contexts of neighboring pixels, but find it to be both computationally inefficient and not significantly better than quadtree coding.
For completely random bitplanes, quadtree coding can actually use more storage than simply writing the bitplane directly; in that case we just dump the bitplane with no coding.
Note that by coding the transform one bitplane at a time, the compressed data can be viewed as an incremental description of the image. One can initially transmit a crude representation of the image using only the small amount of data that is required for the sparsely populated, most significant bit planes. Then the lower bit planes can be added one by one until the desired accuracy is required. This could be useful, for example, if the data is to be retrieved from a remote database — one could examine the crude version of the image (retrieved very quickly) and abort the transmission of the rest of the data if the image is judged to be uninteresting.
**5. Astrometric and Photometric Properties of Compressed Images**
Astronomical images are not simply subjected to visual examination, but are also subjected to careful quantitative analysis. For example, for the image in Figure 1 one would typically like to do astrometric (positional) measurements of objects to an accuracy much better than 1 pixel, photometric (brightness) measurements of objects to an accuracy limited only by the detector response and the noise, and accurate measurements of the surface brightness of extended sources.
We have done some experiments to study the degradation of astrometry and photometry on the compressed images compared to the original images (White, Postman, and Lattanzi 1991). Even the most highly compressed images have very good photometric properties for both point sources and extended sources; indeed, photometry of extended objects can be improved by the adaptive filtering of the H-transform (Capaccioli et al. 1988). Astrometry is hardly affected by the compression for modest compression factors (up to about a factor of 20 for our digitized photographic plates), but does begin to degrade for the most highly compressed images.
These results are based on tests carried out with tools optimized for the original images; it is likely the best results will be obtained for highly compressed images only with analysis tools specifically adapted to the peculiar noise characteristics of the compressed images.
**6. Conclusions**
In order to construct the Guide Star Catalog for use in pointing the Hubble Space Telescope, the Space Telescope Science Institute scanned and digitized wide-field photographic plates covering the entire sky. The digitized plates are of great utility, but to date it has been impossible to distribute the scans because of the massive volume of data involved (a total of about 600 Gbytes). Using the compression techniques described in this paper, we plan to distribute our digital sky survey on CD-ROMs; about 100 CD-ROMs will be required if the survey is compressed by a factor of 10.
The algorithm described in this paper has been shown to be capable of producing highly compressed images that are very faithful to the original. Algorithms designed to work on the original images can give comparable results on object detection, astrometry, and photometry when applied to the images compressed by a factor of 10 or possibly more. Further experiments will determine more precisely just what errors are introduced in the compressed data; it is possible that certain kinds of analysis will give more accurate results on the compressed data than on the original because of the adaptive filtering of the H-transform (Capaccioli et al. 1988).
This compression algorithm can be applied to any image, not just to digitized photographic plates. Experiments on CCD images indicate that lossless compression factors of 3–30 can be achieved depending on the CCD characteristics (e.g., the readout noise). A slightly modified algorithm customized to the noise characteristics of the CCD will do better. This application will be explored in detail in the future.
We gratefully acknowledge grant from NAGW-2166 from the Science Operations Branch of NASA headquarters which supported this work. The Space Telescope Science Institute is operated by AURA with funding from NASA and ESA.
**References**
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Blume, H., and Fand, A. 1989, [*SPIE Vol. 1091, Medical Imaging III: Image Capture and Display*]{}, p. 2.
Capaccioli, M., Held, E. V., Lorenz, H., Richter, G. M., and Ziener, R. 1988, [*Astronomische Nachrichten*]{}, [**309**]{}, 69.
Daubechies, I. 1988, [*Comm. Pure and Appl. Math.*]{}, [**41**]{}, 909.
Fritze, K., Lange, M., Möstl, G., Oleak, H., and Richter, G. M. 1977, [*Astronomische Nachrichten*]{}, [**298**]{}, 189.
Haar, A. 1910, [*Math. Ann.*]{} [**69**]{}, 331.
Huang, L, and Bijaoui, A. 1991, [*Experimental Astronomy*]{}, [**1**]{}, 311.
Richter, G. M. 1978, [*Astronomische Nachrichten*]{}, [**299**]{}, 283.
Samet, H. 1984, [*ACM Computing Surveys*]{}, [**16**]{}, 187.
White, R. L., Postman, M., and Lattanzi, M. 1991, in [*Proceedings of the Edinburgh Meeting on Digital Sky Surveys*]{}, in press.
Witten, I. H., Radford, M. N., and Cleary, J. G. 1987, [*Communications of the ACM*]{}, [**30**]{}, 520.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We have investigated shot noise and conduction of graphene field effect nanoribbon devices at low temperature. By analyzing the exponential $I-V$ characteristics of our devices in the transport gap region, we found out that transport follows variable range hopping laws at intermediate bias voltages $1 < V_{bias} < 12$ mV. In parallel, we observe a strong shot noise suppression leading to very low Fano factors. The strong suppression of shot noise is consistent with inelastic hopping, in crossover from one- to two-dimensional regime, indicating that the localization length $l_{loc} < W$ in our nanoribbons.'
author:
- 'R. Danneau'
- 'F. Wu'
- 'M.Y. Tomi'
- 'J.B. Oostinga'
- 'A.F. Morpurgo'
- 'P.J. Hakonen'
title: Shot Noise Suppression and Hopping Conduction in Graphene Nanoribbons
---
Graphene, a two-dimensional crystal of carbon atoms, has shown some amazing electrical properties [@geim2007] attracting the interest of both scientific community and microelectronic industry. However, graphene is a zero-gap semiconductor with a minimum conductivity way too large to be utilized as base material for high on-off ratio field effect transistor. One way to circumvent this problem would be to open a gap in graphene’s band-structure. It is possible in bilayer graphene by the means of doping (either chemical [@otah2006] or electrostatic [@oostinga2008]). Another way to create an efficient graphene transistor is to build a constriction and/or to form a nanoribbon. Early theoretical studies have predicted that a gap could be opened in graphene nanoribbons (GNR) depending on the edges being either zigzag or armchair [@nakada1996].
However, the first studies of GNRs were performed on etched graphene leading to ribbon width down to around 20 nm [@chen2007; @han2007]. These experiments demonstrated the presence of a transport gap inversely proportional to the width and independent on the crystallographic orientation [@han2007]. It was also estimated that part of the ribbons at the edges were probably not conducting (around 14 nm at $T$ = 4.2 K), suggesting that edge roughness is significant. Similar transport gaps were observed for much smaller ribbon width in GNRs fabricated using sonication of intercalated graphite in solution, indicating smoother edges than the etched GNRs [@li2008]. Indeed, experiments performed on GNRs [@chen2007; @han2007; @ozyilmaz2007; @han2009; @oostinga2010] tend to prove that the origin of the gap may be more complex than the early theoretical studies suggested [@nakada1996]. Despite several models based on Anderson localization, Coulomb blockade or percolation phenomenon [@son2006], there is not yet a consensus as to the origin of the gap in GNRs.
In this work, we report the first shot noise measurements on etched GNRs performed at low temperature. Our results show a strong shot noise reduction while $I-V$ characteristics measured follow variable range hopping (VRH) laws [@shklovskii1984] in the gap region. Such shot noise suppression is the consequence of inelastic hopping conduction from a localized state to an adjacent one, localized states arising from the rough edges and disorder due to residues and defects from the fabrication process. We also find that relaxation of electrons is stronger than expected in our ribbons.
The GNRs have been fabricated from the same graphene monolayer (identified using the RGB green shift as described in [@oostinga2008; @craciun2009; @oostinga2010]) using Scotch tape micromechanical cleavage on natural graphite. The graphene sheets were deposited on a heavily *p*-doped substrate with 300 nm SiO$_2$ layer (see Fig. 1(a)). The graphene sheet was first connected using standard e-beam lithography followed by a Ti(10 nm)/Au(40 nm) bilayer deposition with lift-off in acetone. A second lithography step allowed the patterning of the GNRs. The resist (PMMA) was used as mask in this step and GNRs were etched using an Ar plasma. We present the measurements on two GNRs: Sample A with a length $L \sim 600$ nm and a nominal width $W \sim 90$ nm, and sample B with a length $L \sim 200$ nm and a nominal width $W \sim 70$ nm. After the experiments, the GNRs were observed using scanning electron microscope at 0.5 kV (see Fig. 1(b)).
The measurements were performed in a similar fashion as described in Ref. , from room temperature down to $T$ = 4.2 K. The differential conductance $\frac{dI}{dV}$ was measured using standard low-frequency ac lock-in technique with an excitation amplitude from 0.38 mV up to 0.8 mV ($\sim$ 4 K to $\sim$ 8 K) at *f*= 63.5 Hz. A tunnel junction was used for calibration of the shot noise [@danneau2008; @wu2006].
Fig. 2(a) and (b) display the gate voltage $V_{gate}$ dependence of the zero bias conductance $G$ for different temperatures $T$ of sample A and B, respectively. In both cases, we observe a drop of $G$ when $T$ is lowered, and a high impedance region emerges as $T \rightarrow 4.2$ K. Clear conductance oscillations at zero bias are visible at the lowest temperatures. However, no periodicity is detectable in a Fourier analysis. Far away from the charge neutrality point $G \sim 2e^2/h$, i.e. twice the conductance quantum $g_0$. On Fig 2(c) and (d), we show a color map of the scaled differential conductance $\frac{dI}{dV}/g_0$ as a function of bias voltage $V_{bias}$ and $V_{gate}$ at liquid helium temperature, for sample A and B, respectively. These measurements highlight the formation of a “large impedance region” or a “gap” as previously observed [@chen2007; @han2007; @ozyilmaz2007; @han2009; @oostinga2010]. This region can be viewed in different ways. In the Anderson picture, it arises from localization due to the rough edges and the disorder resulting in to the high impedance region (around the original Dirac point) at zero bias. Out of equilibrium measurements, on the other hand, illuminate the Coulombic aspects of the transport suppression in GNRs: a “source and drain” gap is modulated by the “Coulomb diamond-like” structures which could originate from the formation of a series of dots, all contributing their share to the “gap”. We found “source drain gap” of about 5 meV and 15 meV from our $\frac{dI}{dV}-V_{bias}$ data, and a “transport gap” of about 14 V and 18 V from the $\frac{dI}{dV}-V_{gate}$ curves for sample A and B respectively. We observe clear irregular “Coulomb diamond”-like structures comparable to previous studies [@chen2007; @han2007; @ozyilmaz2007; @han2009; @oostinga2010], suggesting that Coulomb interactions are significant.
VRH generally describes electronic transport in the presence of disorder [@shklovskii1984]. Temperature dependence of the conductance $G(T)$ is conventionally used to identify the regime. In the case of GNRs, the minimum conductance can vary in gate voltage $V_{gate}$ as the temperature is lowered even under vacuum condition [@han2007], leading to uncertainties in the data analysis. The uncontrolled doping by adsorbed molecules may move the minimum conduction region during the cool down. However, G(T) study has been recently successfully performed [@han2009]. An alternative way is to analyze $I-V$ curves at a temperature $T$. At high bias, below a certain $V_0$, the following equation can be used to describe VRH: $$\begin{aligned}
%I(E,T) = VG_{0}(T) \exp \left(\frac{B(E)E_0}{E}\right)^{1/(d+1)} \label{equ2}
I(E,T) = VG_{0}(T) \exp \left\{-\left(\frac{V_0}{V} \right)^{1/(d+1)} \right\} \label{equ1}\end{aligned}$$ where $d$ is the dimensionality of hopping (for the effect of interactions, see below) and $G_{0}$ is the zero bias conductance. Eq. \[equ1\] transforms to Mott’s law by replacement of $e V_0= k_B T_0$ and $e V = k_B T$ in the exponent ($V_0$ being the upper most value for which the formula is valid) which provides the basic motivation for using this functional form [@pollak; @likharev; @fogler].
Fig. 3 displays $I-V$ curves for sample A and B measured in the gap region. Following Eq. \[equ1\], we see that the conduction follows variable range hopping law in the gap region. The data are plotted using $d=1$ which describes VRH for one-dimensional (1-d) systems with or without interactions or two-dimensional (2-d) systems with interactions. We obtain $V_0 \sim$ 8 and 12 mV for sample A and B, respectively. Here, $\frac{a}{L}e V_0$ describes the bias needed to overcome the potential barrier of the localized state with radius $a$. The fact that we obtain a larger $V_0$ for sample B which has a width 20 nm smaller (and is even shorter) than sample A indicates an enhanced influence of the rough edges on the conduction. Consequently, our results show that the appearance of the high impedance region in GNRs is also affected by defects like localized states at the edges and, likewise, by the local doping due to contaminants. This is in agreement with the recent works on temperature dependence of GNR conductance [@han2009]. Han *et al.* have shown that for various GNR geometries $l \gtrsim W$ indicating 1-d VRH transport in the high impedance region of GNRs; the origin of the transport gap would then be due to localized states [@han2009]. This has recently been confirmed by magneto-transport measurements [@oostinga2010]. Our value for $V_0 \simeq 10$ meV is close to the value $k_B T_0/e \simeq 6$ meV given in Ref. .
In order to gain more information on the hopping in GNRs, we have studied shot noise. Shot noise denotes current fluctuations arising from the granular nature of the charge carriers (see Ref. for a review). It provides a powerful tool to probe mesoscopic systems and it is usually regarded as a complementary technique to conductance measurements. The Fano factor $F$, given by the ratio of shot noise and mean current, is commonly employed to quantify shot noise. The noise power spectrum then reads $S(I) = F \times 2eI$. In the case of phase coherent transport in GNRs, shot noise strongly depends on the boundary conditions, i.e. whether the edges are zigzag or armchair [@cresti2007]. However, phase coherent length in etched GNRs have been estimated to be at most 175 nm [@oostinga2010] and it is clearly less in our experiment due to higher temperature and a finite bias that enhances energy relaxation. While in the case of phase coherent transport, shot noise can be described simply by the scattering matrix theory, it can be treated using semiclassical means in the incoherent regime. When inelastic processes dominate (inelastic length $l_{in} <L$), shot noise starts to decrease and it becomes dependent on the details of the relaxation processes that govern the ensuing non-equilibrium state. In inelastic hopping conduction with short hopping length ($l_{hop}<<L$), strong suppression of shot noise takes place as observed in [@kuznetsov2000].
Assuming strongly inelastic behavior, classical addition of uncorrelated noise sources can be employed and networks of resistors with shunting current noise generators become an appealing choice for noise modeling in GNRs. Within this classical limit, the internal topology of the ribbon becomes relevant. If hopping is 2-d in GNRs, then part of the noise current of individual noise generators is shunted via the conduction paths inside the ribbon and the noise coupled to an outside load becomes reduced. Consequently, we expect that the Fano factor is reduced a bit further down from the 1-d classical limit given by $l_{hop}/L$.
We have performed our shot noise measurements at frequency around 800 MHz. This frequency is high enough so that all noise due to slow fluctuations of resistance (transmission coefficients) can be neglected. On the other hand, the frequency is low compared with internal charge relaxation time scales and high frequency effects can be neglected. Fig. 4 displays the current noise per unit bandwidth $S_I$ versus current $I$ in the high impedance region for samples A and B, respectively. Both curves are fitted using the formula defined previously [@danneau2008] with $F$ as the only fitting parameter. We find a rather low Fano factor for both GNRs $F \sim 0.1$ at low bias (the results involve a correction due to non-linear $I-V$ curves as discussed in Ref. ). With increasing bias, we find a further reduction of the Fano factor, which signals a strong role of inelastic processes as the localized states become delocalized.
Why such a low shot noise? The observed conductance modulation in the high impedance regime suggests that a series/array of dots is formed in GNRs. Quantum dots often show super-Poissonian noise instead of low noise level (see, for example, the work done on carbon nanotubes [@onac2006; @wu2007]) and as theoretically expected for a series of quantum dots [@aghassi2006]. However, a series of $N$ quantum dots without inelastic effects should lead to a Fano factor of $\frac{1}{3}$ [@golubev2004]. We note that shot noise suppression could be seen in asymmetric, open quantum cavity [@blanter2000], but the resistance of one or two open quantum cavities (regions at the ends of the ribbon) is too small to account for our results. There will, however, be a small contribution by the end reservoirs on the shot noise.
The main contribution to the shot noise suppression can only come from hopping conduction via so small localized states that the nature of hopping conduction is likely to be almost 2-d. $F$ for a series of $N$ sites with inelastic hopping is approximately $1/N \sim l_{hop}/L$, and this remains as a good approximation also in the 2-d situation where N then denotes the number of hops *along the voltage bias*. In order to explain the observed suppression, the hopping length has to be in the range of $l_{hop} \sim 20-60$ nm; As the localization length $l_{loc} \sim l_{hop}$ is less than the width of the GNR, we conclude that the hopping conduction in our ribbons is not 1-d in nature but rather it falls in the crossover regime between 1-d and 2-d (or quasi 1-d). Our shot noise results thus indicate even a slightly smaller hopping length than was found in [@han2009; @oostinga2010].
The shot noise crossover from VRH region to high bias regime without localized states in Fig. 4 points to strong relaxation of electrons: otherwise an increase of the Fano factor would be expected across the crossover as the number of hops decreases and $l_{loc}$ increases [@kuznetsov2000; @likharev]. Indeed, even in the VRH regime, the apparent Fano factor could be formed by other means, for example by noise from the graphene islands at the ends, and that the actual shot noise from the ribbon nearly vanishes. This would be reminiscent to carbon nanotubes where very small $F$ have been observed in various configurations [@roche2002; @tsuneta2009]. Nearly total suppression of shot noise indicates very effective energy relaxation at finite bias which could be realized by disorder-enhanced electron-phonon coupling [@sergeev2000] or by relaxation via new degrees of freedom provided by the edges of the ribbon.
To conclude, we have measured shot noise and conductance in GNRs. While the dc transport shows characteristic behavior of GNRs, we clearly observe a strong shot noise suppression. We were able to fit the $I-V$ curves with VRH laws in the high impedance region. We have shown that shot noise suppression could be explained by inelastic hopping conduction in the quasi-1-d limit. Our results are consistent with the strong effect of rough edges and local contaminants in the conduction and shot noise of GNRs. Shot noise being a limited factor for electrical devices, our findings enlighten the great potential of GNRs as building blocks for future electronics.
We thank F. Evers, D. Golubev, T. Heikkilä, M. Hettler, A. Ioselevich, P. Pasanen, M. Laakso, G. Metalidis and M. Paalanen for fruitful discussions. This work was funded by the Academy of Finland, by EU Commission (FP6-IST-021285-2) and by Nokia NRC (NANOSYSTEMS project). R.D.’s Shared Research Group SRG 1-33 received financial support by the “Concept for the future” of the Karlsruhe Institute of Technology within the framework of the German Excellence Initiative.
A.H. Castro Neto *et al.*, Rev. Mod. Phys. **81**, 109 (2009); S. Das Sarma *et al.*, arXiv:1003.4731. T. Otah *et al.*, Science **313**, 951 (2006). J.B. Oostinga *et al.*, Nature Mater. **7**, 151 (2008). K. Nakada *et al.*, Phys. Rev. B **54**, 17954 (1996). Z. Chen *et al.*, Physica E **40**, 228 (2007). M.Y. Han *et al.*, Phys. Rev. Lett. **98**, 206805 (2007). X. Li *et al.*, Science **319**, 1229 (2008); X. Wang *et al.*, Phys. Rev. Lett. **100**, 206803 (2008); J.M. Poumirol *et al.*, arXiv:1002.4571. B. Özyilmaz *et al.*, Appl. Phys. Lett. **91**, 192107 (2007); K. Todd *et al.*, Nano Lett. **9**, 416 (2009); C. Stampfer *et al.*, Phys. Rev. Lett. **102**, 056403 (2009); F. Molitor *et al.*, Phys. Rev B **79**, 075426 (2009); X. Liu *et al.*, Phys. Rev B **80**, 121407(R) (2009); P. Gallagher, K. Todd, and D. Goldhaber-Gordon, Phys. Rev. B **81**, 115409 (2010). M.Y. Han, J.C. Brant, and P. Kim, Phys. Rev. Lett. **104**, 056801 (2010). J.B. Oostinga *et al.*, Phys. Rev B **81**, 193408 (2010). Y.-W. Son, M.L. Cohen, and S.G. Louie, Phys. Rev. Lett. **97**, 216803 (2006); D. Gunlycke, D.A. Areshkin, and C.T. White, Appl. Phys. Lett. **90**, 142104 (2007); F. Sols, F. Guinea, and A.H. Castro Neto, Phys. Rev. Lett. **99**, 166803 (2007); A. Lherbier *et al.*, Phys. Rev. Lett. **100**, 036803 (2008); D. Querlioz et al., Appl. Phys. Lett. **92**, 042108 (2008); S. Adam et al., Phys. Rev. Lett. **101**, 046404 (2008); M. Evaldsson et al., Phys. Rev. B **78**, 161407(R) (2008); I. Martin and Y. M. Blanter, Phys. Rev. B **79**, 235132 (2009). B.I. Shklovskii and A.L. Efros, *Electronic properties of doped semiconductors* (Springer-Verlag, Berlin, 1984). M.F. Craciun *et al.*, Nat. Nanotechnol. **4**, 383 (2009). R. Danneau *et al.*, Phys. Rev. Lett. **100**, 196802 (2008); J. Low Temp. Phys. **153**, 374 (2008); Solid State Commun. **149**, 1050 (2009). F. Wu *et al.*, AIP Conf. Proc. **850**, 1482 (2006). M. Pollak and I. Riess, J. Phys. C **9**, 2339 (1976). Y.A. Kinkhabwala, V.A. Sverdlov, K.K. Likharev, J. Phys.: Condens. Matter **18**, 2013 (2006). A. Rodin and M. Fogler, Phys. Rev. B **80**, 155435 (2009). Ya. Blanter, and M. Büttiker, Phys. Rep. **336**, 1 (2000). A. Cresti, G. Grosso, and G.P. Parravicini, Phys. Rev. B **76**, 205433 (2007); E.R. Mucciolo, A.H. Castro Neto, and C.H. Lewenkopf, Phys. Rev. B **79**, 075407 (2009); R.L. Dragomirova, D.A. Areshkin, and B.K. Nicolić, Phys. Rev. B **79**, 241401(R) (2009). V.V. Kuznetsov *et al.*, Phys. Rev. Lett. **85**, 397 (2000); S.H. Roshko *et al.*, Physica E **12**, 861 (2002); F.E. Camino *et al.*, Phys. Rev. B **68**, 073313 (2003). G.-E. Onac *et al.*, Phys. Rev. Lett. **96**, 026803 (2006). F. Wu *et al.*, Phys. Rev. B **75**, 125419 (2007). J. Aghassi *et al.*, Appl. Phys. Lett. **89**, 052101 (2006). See, *e.g*., D.S. Golubev and A.D. Zaikin, Phys. Rev. B **70**, 165423 (2004). P.E. Roche *et al.*, Eur. Phys. J. B **28**, 217 (2002). T. Tsuneta *et al.*, Europhys. Lett. **85**, 37004 (2009). A. Sergeev and V. Mitin, Phys. Rev. B **61**, 6041 (2000).
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{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Based on an estimate of the knot entropy of a worm-like chain we predict that the interplay of bending energy and confinement entropy will result in a compact metastable configuration of the knot that will diffuse, without spreading, along the contour of the semi-flexible polymer until it reaches one of the chain ends. Our estimate of the size of the knot as a function of its topological invariant (ideal aspect ratio) agrees with recent experimental results of knotted dsDNA. Further experimental tests of our ideas are proposed.'
author:
- 'Alexander Y. Grosberg$^{\dag}$ and Yitzhak Rabin$^{\ddag}$'
bibliography:
- 'TightKnots.bib'
title: 'Metastable tight knots in a worm-like polymer '
---
While everyday experience suggests that knots are very common in long linear strings of any kind, and they get quite tight whenever a string is not carefully handled, the study of knots in polymers concentrated almost exclusively on closed loops. It is understandable in the sense that knots are not mathematically well defined for open strings. However, if the string is long enough, while the knot occupies a short fragment of it far from the ends, then distinguishing between different knots, or between knots and no knots, becomes sufficiently unambiguous.
The recent achievement in the theory of knotted loops [@Orlandini1; @Metzler_Kardar_Knot_Localization; @Orlandini2] is the idea of knot localization. As simulations show [@Orlandini2], when a polymer loop with a knot is placed in a good or $\theta$-solvent, it typically adopts a conformation in which most of the polymer forms a long unknotted loop, while the knot gets somewhat tightened in a small part of the contour. A theoretical explanation of the knot localization phenomenon is given [@Metzler_Kardar_Knot_Localization] for the so-called flat knots, confined in a thin slit between two planes. In this case, one can consider a two-dimensional network which corresponds to any configuration of the knot by identifying chain crossings as the cross-links. This mapping allows one to resort to the sophisticated theory of 2D networks with excluded volume [@Duplantier_2D_network; @Duplantier_2D_network_details] and conclude that the most entropically favorable network is obtained when one of the knot arcs is made long at the expense of all others, thus localizing the knot.
In the present note, we would like to use much simpler hand-waving arguments to show that, at least for a worm-like polymer, there exists a local (metastable) minimum of free energy corresponding to a tight state of the knot even when the chain itself is open (not a loop). That means that if we intentionally tie a sufficiently tight knot somewhere on a very long polymer chain, the knot will spontaneously shrink or expand to a well-defined size and then, on much longer time scales, it will diffuse along the polymer (by polymer self-reptating through the knot) until, finally, the knot is released through the chain end. We expect that this knot will diffuse along the polymer as a soliton, in the sense that its size will remain relatively stable as it diffuses over large distances. The size of such a solitary knot depends on the complexity of the knot and on the persistence length of the polymer; we estimate that the knot will tighten to a size smaller than the persistence length of the polymer.
To put forward our argument, we employ the knot entropy estimates based on the idea [@Flory_Knot], similar to that in reptation theory [@Doi_Edwards] , that non-crossing constraints imposed on the chain in the knot can be self-consistently described by confining the chain in an effective tube. Physically most natural construction of such a tube corresponds to making maximally inflated or, equivalently, shortest length tube consistent with the given knot topology. This approach to estimate knot entropy was invented by us [@Flory_Knot]. It was also used in a number of other contexts and widely popularized under the name “ideal knots” [@Ideal_Knots]. The configuration and aspect ratio, $p$, of the “ideal” tube represent topological invariants of the knot. This is sketched in the figure \[fig:cartoon\].
Within the framework of this approach, we imagine that a tight knot, characterized by ideal aspect ratio $p$, has been tied in a very long polymer, and that currently the degree of tightening of the knot is such that the diameter of its self-consistently confining tube is $D$. The length of the tube is $pD$ and the size of the knot in space is $R$, such that $pD\times
D^{2}\sim R^{3}$ or $$R\sim p^{1/3}D\ .\label{eq:densely_packed_tube}$$ Throughout this paper, we drop all numerical coefficients, emphasizing only the scaling aspects of our considerations. We want to estimate the free energy of the knot as a function of $D$ or $R$.
Although the original estimates of knot entropy were designed for a Gaussian chain [@Flory_Knot], they can be readily adapted to a worm-like polymer. Consider a knot with $D<\ell$, where $\ell$ is the persistence length of the polymer. In this case, the polymer is quite tightly confined in the tube and can only wiggle a little bit around its centerline. Therefore, the length of polymer within the knot is close to $pD$, and its free energy consists of bending energy and confinement entropy. We estimate the bending energy as $TpD\ell/R^{2}$, assuming temperature $T$ is expressed in energy units ($k_B=1$), and assuming that the radius of curvature of the tube is about the knot size $R$, which is natural for an “ideal” knot. We further assume that the confinement entropy is the same as that for a straight tube for which case it was computed by Odijk [@Odijk_deflection_length] and turns out to be about unity per every so-called deflection length $\lambda\sim(D^{2}\ell)^{1/3}$; this results in the free energy contribution about $ TpD/\lambda\sim
Tp(D/\ell)^{1/3}$. Thus, using also (\[eq:densely\_packed\_tube\]), we obtain the following free energy estimate: $$\frac{\Delta F}{ T}\sim p^{1/3}\frac{\ell}{D}+p\left( \frac{D}{\ell
}\right) ^{1/3}\ .\label{eq:free_energy}$$ where $\Delta F$ is the free energy penalty for forming the knot (taking the reference free energy to be that of the unknotted chain). Obviously, it has a minimum at $$D^{\ast}\sim\ell p^{-1/2}\ ,\ \ \ \ \mathrm{or}\ \ \
R^{\ast}\sim\ell p^{-1/6}\ .\label{eq:optimal}$$ Notice that the resulting optimal $D^{\ast}$ meets the condition $D^{\ast
}<\ell$, so our estimate is self-consistent in this respect.
Of course, our result applies only as long as $D^{\ast}>d$, where $d$ is the thickness of the polymer itself. Furthermore, in the practically important case of a charged polymer, such as DNA, we should also require that tube diameter exceeds Debye screening length, $D^{\ast}>r_{s}$. In general, we can write roughly $p<(\ell/(d+r_{s} ))^{2}$. More complex knots get so tight that their further collapse is stopped by either the excluded volume or electrostatic repulsion. As an example, for the dsDNA, the ratio $\ell/(d+r_{s})\approx 20$ under physiological conditions [@DNA_effective_diameter], while all knots with 7 or fewer crossings on the projection have $p$ about $30$ or less [@Pieranski]. Therefore, in practice the condition on knot complexity $p<(\ell/(d+r_{s} ))^{2}$ is not very restrictive.
Let us now try to understand the physical meaning of our result (\[eq:optimal\]), because at the first glance it might seem counterintuitive. Indeed, the optimal $D^{\ast}$ results from the competition of two factors, each of which, as it seems, disfavors tightening! One, chain bending energy, obviously favors more loose states of the knot, or increasing $D$. The other one, however, related to the confinement entropy, favors tube widening *if* the chain length in the tube is fixed. In our case, when a knot tightens, it reduces the length of the chain, $pD$, confined in the knot. In other words, the part of the chain that remains in the tube suffers more when $D$ decreases, but the other part of the chain gets completely free of restriction, and that factor wins the whole game. That means, what really tightens the knot is the entropy gain of the chain tails outside of the knot. Alternatively, we can say that the knot gets compressed by the pressure of Rouse modes (or bending phonons) of the outside chain tails. Similar force can be observed during the translocation of a long polymer through a narrow channel across a membrane; while two long polymer ends are outside the membrane, their entropic favorability results in a force stretching the polymer portion inside the channel, and, accordingly, compressing the membrane. Notice that while in this planar geometry this force increases logarithmically with the length of the polymer tails, in our case of two long polymer ends sticking out of a compact knot, the force on the knot is independent of the length of these ends.
How tight should be the knot in the first place in order for our mechanism to take over and to bring the knot to its metastable size $R^{\ast}$? In other words, how wide is the basin of attraction of our metastable free energy minimum? We argue that the knot should be initially tightened to the state in which tube diameter is smaller or about chain persistence length $\ell$. Indeed, when knot tube is wider, $D>\ell$, the chain inside the tube is roughly Gaussian, with blobs of size $D$. Each blob contains about $D^{2} /\ell$ of polymer contour length, and the entire tube contains polymer length $pD^{2}/\ell$. At the same time, confinement entropy is about unity per blob, which results in overall confinement entropy of about $p$, independent of $D$. Thus, our entropic knot tightening effect does not work if the tube is wider than persistence length, and it comes into play only when the knot is prepared in a compact enough state such that $D<\ell$. To achieve this, the knot should be initially prepared by the pulling the ends with force $f>f^{\ast}$, where $f^{\ast}\sim T/\ell$.
Let us discuss now the dynamics aspect of the situation. Imagine once again that a sufficiently tight knot was initially tied in the polymer, similar to how it was done in the experimental work [@Quake_Knot_experiment] (see also [@Quake_Knot_experiment_Erratum]) with DNA. Suppose now that the chain is released and is free to move. We predict then that the knot tightens spontaneously within a time which is roughly independent of the total chain length $L$. After that, the knot will diffuse along the chain in pretty much the same way as it was observed experimentally [@Quake_Knot_experiment; @Quake_Knot_experiment_Erratum] and numerically [@Vologodski_knot_diffusion] for the stretched chain (even though in the experiment of ref. [@Quake_Knot_experiment] the chain ends were held at fixed separation throughout the diffusion process). As regards the knot diffusion coefficient along the chain, it was shown in the work [@Quake_Knot_experiment] that it can be quite accurately expressed in terms of the friction coefficient, $\zeta$, of a polymer with length $pD$ and diameter $d$, moving in the viscous solvent inside a tube of diameter $D$: $\zeta=\frac{2\pi\eta}{\ln(D/d)}pD$. The diffusion coefficient is then determined by Einstein’s relation as $ T/\zeta$, and the full relaxation time of knot diffusion to the chain end is about $L^{2}\zeta/ T$. In these formulae, $\eta$ is assumed to be the bulk viscosity of the solvent.
The above description of a soliton-like knot diffusing as a whole along the chain holds as long as diffusion time is shorter than the time of thermally-activated loosening of the knot. Indeed, we argued that the knot gets to its metastable size $R^{\ast}$ only when it is compact enough to begin with, such that $D < \ell$ and, therefore, a free energy barrier exists at $D \approx \ell$ (knots with $D>\ell$ will loosen up spontaneously). Using formula (\[eq:free\_energy\]), we can estimate the barrier height as $\left. \Delta F\right\vert
_{D/\ell=1}-\left. \Delta F\right\vert _{D=D^{\ast}}\sim
T\left( p-p^{5/6}\right) $. As one could have expected, this barrier is very high for complex knots, but even for $p \approx 30$, one gets a barrier of about $10 T$ which should be sufficient to keep a knot locked while diffusing over a distance of few hundred persistence lengths, typical of DNA manipulation experiments. For smaller values of $p$, the barrier may be too small to stabilize the tight knot and without the applied stretching it will spread due to thermal fluctuations. As a corollary to our result, let us mention the following rather unexpected prediction. Let us take the polymer with the knot tightened according to our mechanism, and now let us gently pull the chain ends by a weak force. The applied stretching will suppress transverse fluctuations of the open chain ends. But these fluctuations are exactly the reason why the knot was tightened in the first place. Therefore, when these fluctuations are suppressed by the applied force, the knot swells - instead of further tightening which one could have naively expected. Of course, at the larger forces the loosening of the knot stops and normal tightening takes over.
Let us support this physical argument by a little calculation. If the chain of contour length $L$ is stretched by a weak force $f$, it represents the succession of Pincus blobs, each involving the number of persistence lengths $g$ such that $f\ell g^{1/2}\sim T$; the stretching free energy is about $ T$ per blob, or $ TL/g\ell\sim
L\ell f^{2}/ T$ (see, e.g., [@deGennes_book] for further details). In our case, only the chain outside of the knot is subject to this stretching effect, which yields the overall free energy $$\frac{\Delta F}{ T}\sim p^{1/3}\frac{\ell}{D}+p\left( \frac{D}{\ell
}\right) ^{1/3}+\frac{L-pD}{\ell}\left( \frac{\ell f}{ T}\right)
^{2}\ , \label{eq:to_minimize_Pincus_regime}$$ subject to optimization with respect to $D$ (see Eq. (\[eq:result\_of\_optimization\]) below); here, the term $ \propto
L$ is large, but can be dropped as independent of $D$. Formula (\[eq:to\_minimize\_Pincus\_regime\]) remains valid as long as Pincus blob is larger than persistence length $f\ell/ T<1$. If we stretch the outside chain even further, beyond the Gaussian regime, it crosses-over to the so-called Marco-Siggia or Odijk regime in which stretching free energy is about $ T$ per Odijk deflection length. In this regime, $$\frac{\Delta F}{ T}\sim p^{1/3}\frac{\ell}{D}+p\left( \frac{D}{\ell
}\right) ^{1/3}+\frac{L-pD}{\ell}\left( \frac{\ell f}{ T}\right)
^{1/2}\ ; \label{eq:to_minimize_Odijk_regime}$$ once again, this has to be optimized with respect to $D$. Notice that in both equations (\[eq:to\_minimize\_Pincus\_regime\]) and (\[eq:to\_minimize\_Odijk\_regime\]) we have neglected the effect of force on the chain part inside of the knot. This is justified as long as the amount of transverse fluctuations of the outside chain, $D_{\mathrm{out}}(f)$, which is either the blob size in Pincus regime or the tube diameter in Odijk regime, remains smaller than the optimized tube diameter inside the knot, $D^{\ast}$. At still larger forces, the chain inside the knot is just as stretched as outside. Therefore, we can summarize all results of optimization in the following way: $$R^{\ast}\sim\left\{
\begin{array}
[c]{lcr} \ell p^{-1/6}+\ell p^{-1/2}\left( \frac{f\ell}{ T}\right)
^{2} &
\mathrm{for} & \frac{f\ell}{ T}<1\\
\ell p^{-1/6}+\ell p^{-1/2}\left( \frac{f\ell}{ T}\right) ^{1/2} &
\mathrm{for} & 1<\frac{f\ell}{ T}<p^{1/3}\\
\ell p^{1/3}\left( \frac{f\ell}{ T}\right) ^{-3/4} & \mathrm{for}
& p^{1/3}<\frac{f\ell}{ T}
\end{array}
\right. \label{eq:result_of_optimization}$$ These results are sketched in figure \[fig:R\_against\_force\]. In accordance with our qualitative argument, the application of weak force loosens the tight knot instead of tightening it further!
In practice, for the case of dsDNA, since $\ell\approx50 \ nm$ and $
T \approx4 \ pN \times nm$, and taking $p \approx30$, we get that both cross-overs $f \ell/ T \sim1$ and $f \ell/ T \sim p^{1/3}$ are within an experimentally feasible force range (about a few tenth and about a few piconewtons, respectively).
Speaking about experimental tests of our theory, we should of course consider first the experiment by Quake et al [@Quake_Knot_experiment; @Quake_Knot_experiment_Erratum] in which the knots were tied on DNA by the use of optical tweezers and then the knot motion along DNA was observed. Our theory indicates that the optimal tube diameter in the knot, $D^{\ast}$, does depend in a certain way on the knot complexity, on $p$. In the experiment, authors were unable to observe directly the knot and measure its size $R$, but they were able to detect the amount of fluorescence coming from the knot region in excess of the fluorescence from linear DNA. That means, in our notations, that authors measured quantity $pD^{\ast}-R^{\ast}$, because $pD^{\ast}$ is the length of DNA within the knot (proportional to the amount of fluorescence), while $R$ is the length of DNA that would have been in that place if there were no knot. Given the normalization condition (\[eq:densely\_packed\_tube\]), one could write $pD^{\ast}-R^{\ast}=\left( p-p^{1/3}\right) D^{\ast}$. Authors of the work [@Quake_Knot_experiment], assuming *a’priori* that $D$ is independent of $p$, plotted $pD^{\ast}-R^{\ast}$ against $p-p^{1/3}$, fitted the data to the linear function, and interpreted its slope as the tube diameter $D^{\ast}$. By contrast, our theory predicts that $D^{\ast}$ does depend on knot complexity $p$, such that $pD^{\ast}-R^{\ast}=\left( p^{1/2}-p^{-1/6}\right) \ell$. This is illustrated in the figure \[fig:D\_against\_p\], where both the linear fit and our theory results are plotted along with the data points reproduced from the work [@Quake_Knot_experiment]. At present, it is impossible to decide which theory is a better fit to the data.
To further test our predictions, one should prepare the knot and then consider what happens to it if the force is subsequently switched off or significantly reduced. Another test of our theory would be to see how the knot diffusion time, determined by the friction coefficient $\zeta$, depends on the knot type through $D$. Yet another possibility is to use granular macroscopic chain experiments, as described in [@Ben-Naim_Granular_Chain]. In the movie available on the site
http://cnls.lanl.gov/~ebn/research/pics/chain.mov
the knot does move along the chain like a soliton, it does not increase in size while moving, in accord with our theory. It would be interesting to make a detailed statistics on this subject, although we should also put in question the applicability of our arguments based on worm-like chain model to the granular chain examined in [@Ben-Naim_Granular_Chain].
To conclude, we have shown that a sufficiently complex knot in a worm-like polymer will attain a well-defined compact size which is smaller that the persistence length. This configuration is metastable and the knot will diffuse along the chain as a soliton, whether the ends of the chain are subjected to external force or are free to move. Notice that since the results depend crucially on the elastic properties of the polymer on length scales comparable to and smaller than the persistence length, the effects described in this work are quite sensitive to the choice of the polymer model and can not be captured by locally stiff (e.g., freely jointed or freely rotating) chain models. Likewise, solitary knots will not form in Gaussian chains for which the persistence length $\ell$ and the thickness of the chain $d$ coincide. Such knots are expected to form in semi-flexible polymers such as double stranded DNA for which (a) separation of length scales exists, $L \gg \ell \gg d$ and (b) elastic behavior on length scales smaller than $\ell$ was demonstrated in single molecule experiments [@Bustamante; @Bensimon].
We would like to acknowledge useful discussion with Eli Ben-Naim. We thank Xiaoyan Robert Bao for providing us with the data points used above in Figure \[fig:D\_against\_p\]. The work of AG was supported in part by the MRSEC Program of the National Science Foundation under Award Number DMR-0212302. This research was supported in part by a grant from the US-Israel Binational Science Foundation.
|
{
"pile_set_name": "ArXiv"
}
|
---
address: 'Theory Group, KEK, Tsukuba, Ibaraki 305-0801, Japan'
author:
- Toru Goto and Takeshi Nihei
title: 'New constraint on the minimal SUSY GUT model from proton decay[^1]'
---
=cmr8
1.5pt
\#1\#2\#3\#4[[\#1]{} [**\#2**]{}, \#3 (\#4)]{}
Introduction
============
Supersymmetric grand unified theory (SUSY GUT) [@SUSY_GUT] is strongly suggested by gauge coupling unification around $M_X \sim 2 \times 10^{16} {\,{\rm GeV}}$ [@Gauge_Coupling_Unification]. In this theory, the hierarchy between the weak scale and the GUT scale $M_X$ is protected against radiative corrections by supersymmetry. Also, this theory makes successful prediction for the charge quantization. Proton decay is one of the direct consequences of grand unification. The main decay mode $ p \rightarrow K^{+}\overline{\nu} $ [@DRW] in the minimal SU(5) SUSY GUT model [@SUGRA] has been searched for with underground experiments [@Kam; @IMB], and the previous results have already imposed severe constraints on this model. Recently new results of the proton decay search at Super-Kamiokande have been reported [@superK]. The bound on the partial lifetime of the $K^+\overline{\nu}$ mode is $\tau(p\rightarrow K^+ \overline{\nu})$ $>$ $5.5 \times 10^{32}$ years (90% C.L.)[^2], where three neutrinos are not distinguished.
There are many analyses on the nucleon decay in the minimal SU(5) SUSY GUT model [@dim5_op; @DRW; @NCA; @MATS+HMY; @HMTY; @GNA]. In the previous analyses, it was considered that the contribution from the dimension 5 operator with left-handed matter fields ($LLLL$ operator) was dominant for $ p \rightarrow K^{+} \overline{\nu} $ [@DRW]. In particular a Higgsino dressing diagram of the $RRRR$ operator has been estimated to be small or neglected in these analyses. It has been concluded that the main decay mode is $ p \rightarrow K^{+} \overline{\nu}_\mu $, and the decay rate of this mode can be suppressed sufficiently by adjusting relative phases between Yukawa couplings at colored Higgs interactions [@NCA].
In this talk, we present results of our analysis on the proton decay including the $RRRR$ operator in the minimal SU(5) SUSY GUT model [@GN][^3]. We calculate all the dressing diagrams [@NCA] (exchanging the charginos, the neutralinos and the gluino) of the $LLLL$ and $RRRR$ operators, taking account of various mixing effects among the SUSY particles, such as flavor mixing of quarks and squarks, left-right mixing of squarks and sleptons, and gaugino-Higgsino mixing of charginos and neutralinos. For this purpose we diagonalize mass matrices numerically to obtain the mixing factors at ‘ino’ vertices and the dimension 5 couplings. Comparing our calculation with the Super-Kamiokande limit, we derive constraints on the colored Higgs mass and the typical mass scale of squarks and sleptons. We find that these constraints are much stronger than those derived from the analysis neglecting the $RRRR$ effect.
Dimension 5 operators in the minimal SU(5) SUSY GUT model
=========================================================
Nucleon decay in the minimal SU(5) SUSY GUT model is dominantly caused by dimension 5 operators [@dim5_op], which are generated by the exchange of the colored Higgs multiplet. The dimension 5 operators relevant to the nucleon decay are described by the following superpotential: $$\begin{aligned}
W_5 & = & -\frac{1}{M_C} \left\{ \frac{1}{2}C_{5L}^{ijkl}
Q_k Q_l Q_i L_j + C_{5R}^{ijkl} U^c_i D^c_j U^c_k E^c_l \right\}.
\label{eqn:dim5_op}\end{aligned}$$ Here $Q$, $U^c$ and $E^c$ are chiral superfields which contain a left-handed quark doublet, a charge conjugation of a right-handed up-type quark, and a charge conjugation of a right-handed charged lepton, respectively, and are embedded in the 10 representation of SU(5). The chiral superfields $L$ and $D^c$ contain a left-handed lepton doublet and a charge conjugation of a right-handed down-type quark, respectively, and are embedded in the $\overline{5}$ representation. A mass of the colored Higgs superfields is denoted by $M_C$. The indices $ i,j,k,l = 1,2,3$ are generation labels. The first term in Eq. (\[eqn:dim5\_op\]) represents the $LLLL$ operator [@DRW] which contains only left-handed SU(2) doublets. The second term in Eq. (\[eqn:dim5\_op\]) represents the $RRRR$ operator which contains only right-handed SU(2) singlets. The coefficients $C_{5L}$ and $C_{5R}$ in Eq. (\[eqn:dim5\_op\]) at the GUT scale are determined by Yukawa coupling matrices as [@NCA] $$\begin{aligned}
C_{5L}^{ijkl} &=& (Y_D)_{ij} (V^T P Y_U V)_{kl}, \nonumber \\
C_{5R}^{ijkl} &=& (P^*V^*Y_D)_{ij}(Y_UV)_{kl},
\label{eqn:C5L&C5R}\end{aligned}$$ where $Y_U$ and $Y_D$ are diagonalized Yukawa coupling matrices for $10 \cdot 10 \cdot 5_H$ and $10 \cdot \overline{5} \cdot \overline{5}_H$ interactions, respectively. The unitary matrix $V$ is the Cabibbo-Kobayashi-Maskawa (CKM) matrix at the GUT scale. The matrix $P$ $=$ ${\rm diag}(P_1,P_2,P_3)$ is a diagonal unimodular phase matrix with $|P_i|=1$ and $ {\rm det}P=1$. We parametrize $P$ as $$\begin{aligned}
P_1/P_3 = e^{i\phi_{13}}, \hspace{3mm}
P_2/P_3 = e^{i\phi_{23}}.
\label{eqn:Phase_Matrix}\end{aligned}$$ The parameters $\phi_{13}$ and $\phi_{23}$ are relative phases between the Yukawa couplings at the colored Higgs interactions, and cannot be removed by field redefinitions [@Phase_Matrix]. The expressions for $C_{5L}$ and $C_{5R}$ in Eq. (\[eqn:C5L&C5R\]) are written in the flavor basis where the Yukawa coupling matrix for the $10 \cdot \overline{5} \cdot \overline{5}_H$ interaction is diagonalized at the GUT scale. Numerical values of $Y_U$, $Y_D$ and $V$ at the GUT scale are calculated from the quark masses and the CKM matrix at the weak scale using renormalization group equations (RGEs).
In this model, soft SUSY breaking parameters at the Planck scale are described by $m_0$, $M_{gX}$ and $A_X$ which denote universal scalar mass, universal gaugino mass, and universal coefficient of the trilinear scalar couplings, respectively. Low energy values of the soft breaking parameters are determined by solving the one-loop RGEs [@Radiative_Breaking]. The electroweak symmetry is broken radiatively due to the effect of a large Yukawa coupling of the top quark, and we require that the correct vacuum expectation values of the Higgs fields at the weak scale are reproduced. Thus we have all the values of the parameters at the weak scale. The masses and the mixings are obtained by diagonalizing the mass matrices numerically. We evaluate hadronic matrix elements using the chiral Lagrangian method [@Chiral_Lagrangian]. The parameters $\alpha_p$ and $\beta_p$ defined by $\langle 0| \epsilon^{abc} (d_R^a u_R^b) u_L^c |p
\rangle$ $=$ $\alpha_p N_L$ and $\langle 0| \epsilon^{abc} (d_L^a u_L^b) u_L^c |p
\rangle$ $=$ $\beta_p N_L$ ($N_L$ is a wave function of the left-handed proton) are evaluated as $0.003 \, {\rm GeV}^3$ $\leq$ $\beta_p$ $\leq$ $0.03 \, {\rm GeV}^3$ and $\alpha_p$ $=$ $-\beta_p$ by various methods [@beta_p]. We use the smallest value $\beta_p$ $=$ $-\alpha_p$ $=$ $0.003 \, {\rm GeV}^3$ in our analysis to obtain conservative bounds. For the details of the methods of our analysis, see Ref. [@GN].
$RRRR$ contribution to the proton decay
=======================================
The dimension 5 operators consist of two fermions and two bosons. Eliminating the two scalar bosons by gaugino or Higgsino exchange (dressing), we obtain the four-fermion interactions which cause the nucleon decay [@DRW; @NCA]. In the one-loop calculations of the dressing diagrams, we include all the dressing diagrams exchanging the charginos, the neutralinos and the gluino of the $LLLL$ and $RRRR$ dimension 5 operators. In addition to the contributions from the dimension 5 operators, we include the contributions from dimension 6 operators mediated by the heavy gauge boson and the colored Higgs boson. Though the effects of the dimension 6 operators ($\sim$ $1/M_X^2$) are negligibly small for $p\rightarrow K^+\overline{\nu}$, these could be important for other decay modes such as $p\rightarrow \pi^0 e^+$. The major contribution of the $LLLL$ operator comes from an ordinary diagram with wino dressing. The major contribution of the $RRRR$ operator arises from a Higgsino dressing diagram depicted in Fig. \[fig:diagram\].
The circle in this figure represents the complex conjugation of $C_{5R}^{ijkl}$ in Eq. (\[eqn:C5L&C5R\]) with $i$ $=$ $j$ $=1$ and $k$ $=$ $l$ $=3$. This diagram contains the Yukawa couplings of the top quark and the tau lepton. The importance of this diagram has been pointed out in Ref. [@RRRR] in the context of a SUSY SO(10) GUT model. The contribution of this diagram has been estimated to be negligible or simply ignored in previous analyses in the minimal SU(5) SUSY GUT [@dim5_op; @DRW; @NCA; @MATS+HMY; @HMTY; @GNA]. In particular, the authors of Ref. [@NCA] calculated the diagram in Fig. \[fig:diagram\]. However the effect was estimated to be small, because the authors assumed a relatively light top quark ($m_t$ $\sim$ 50 ${\,{\rm GeV}}$). We use an experimental value of the top quark mass, and show that this diagram indeed gives a significant contribution in the case of the minimal SU(5) SUSY GUT model also.
Before we present the results of our numerical calculations, we give a rough estimation for the decay amplitudes for a qualitative understanding of the results. In the actual calculations, however, we make full numerical analyses including contributions from all the dressing diagrams as well as those from dimension 6 operators. We also take account of various effects such as mixings between the SUSY particles.
Aside from the soft breaking parameter dependence arising from the loop calculations, relative magnitudes between various contributions can be roughly understood by the form of the dimension 5 operator in Eq. (\[eqn:C5L&C5R\]). Counting the CKM suppression factors and the Yukawa coupling factors, it is easily shown that the $RRRR$ contribution to the four-fermion operators $(u_R d_R)(s_L \nu_{\tau L})$ and $(u_R s_R)(d_L \nu_{\tau L})$ is dominated by a single Higgsino dressing diagram exchanging $ \tilde{t}_R$ (the right-handed scalar top quark) and $ \tilde{\tau}_R $ (the right-handed scalar tau lepton). For $K^+\overline{\nu}_\mu$ and $K^+\overline{\nu}_e$, the $RRRR$ contribution is negligible, since it is impossible to get a large Yukawa coupling of the third generation without small CKM suppression factors in this case. The main $LLLL$ contributions to $(u_L d_L)(s_L \nu_{i L})$ and $(u_L s_L)(d_L \nu_{i L})$ consist of two classes of wino dressing diagrams; they are $ \tilde{c}_L$ exchange diagrams and $ \tilde{t}_L$ exchange diagrams [@NCA]. Neglecting other subleading effects, we can write the amplitudes (the coefficients of the four-fermion operators) for $p\rightarrow K^+ \overline{\nu}_i$ as, $$\begin{aligned}
{\rm Amp.}(p\rightarrow K^+ \overline{\nu}_e) & \sim &
[ P_2 A_e(\tilde{c}_L) +
P_3 A_e(\tilde{t}_L) ]_{LLLL}, \nonumber \\
{\rm Amp.}(p\rightarrow K^+ \overline{\nu}_\mu) & \sim &
[ P_2 A_\mu(\tilde{c}_L) +
P_3 A_\mu(\tilde{t}_L) ]_{LLLL}, \nonumber \\
{\rm Amp.}(p\rightarrow K^+ \overline{\nu}_\tau) & \sim &
[ P_2 A_\tau(\tilde{c}_L) +
P_3 A_\tau(\tilde{t}_L) ]_{LLLL}
+ [ P_1 A_\tau(\tilde{t}_R) ]_{RRRR},
\label{eqn:rough_estimation}\end{aligned}$$ where the subscript $LLLL$ ($RRRR$) represents the contribution from the $LLLL$ ($RRRR$) operator. The $LLLL$ contributions for $A_\tau$ can be written in a rough approximation as $A_\tau(\tilde{c}_L)$ $\sim$ $g^2 Y_c Y_b
V_{ub}^* V_{cd}V_{cs}m_{\tilde{W}}$ $\!\! /(M_C m_{\tilde{f}}^2)$ and $A_\tau(\tilde{t}_L)$ $\sim$ $g^2 Y_t Y_b
V_{ub}^* V_{td}V_{ts}m_{\tilde{W}}$ $\!\! /(M_C m_{\tilde{f}}^2)$, where $g$ is the weak SU(2) gauge coupling, and $m_{\tilde{W}}$ is a mass of the wino $\tilde{W}$. A typical mass scale of the squarks and the sleptons is denoted by $m_{\tilde{f}}$. For $A_\mu$ and $A_e$, we just replace $Y_b V_{ub}^*$ in the expressions for $A_\tau$ by $Y_s V_{us}^*$ and $Y_d V_{ud}^*$, respectively. The $RRRR$ contribution is also evaluated as $A_\tau(\tilde{t}_R)$ $\sim$ $Y_d Y_t^2 Y_\tau V_{tb}^* V_{ud}V_{ts}\mu$ $\!\! /(M_C m_{\tilde{f}}^2)$, where $\mu$ is the supersymmetric Higgsino mass. The magnitude of $\mu$ is determined from the radiative electroweak symmetry breaking condition, and satisfies $|\mu|$ $>$ $|m_{\tilde{W}}|$ in the present scenario. Relative magnitudes between these contributions are evaluated as follows. The magnitude of the $\tilde{c}_L$ contribution is comparable with that of the $\tilde{t}_L$ contribution for each generation mode: $|A_i(\tilde{c}_L)|$ $\sim$ $|A_i(\tilde{t}_L)|$. Therefore, cancellations between the $LLLL$ contributions $P_2 A_i(\tilde{c}_L)$ and $P_3 A_i(\tilde{t}_L)$ can occur simultaneously for three modes $p\rightarrow K^+\overline{\nu}_i$ ($i$ $=$ $e$, $\mu$ and $\tau$) by adjusting the relative phase $\phi_{23}$ between $P_2$ and $P_3$ [@NCA]. The magnitudes of the $LLLL$ contributions satisfy $|P_2 A_\mu(\tilde{c}_L) + P_3 A_\mu(\tilde{t}_L)|$ $>$ $|P_2 A_\tau(\tilde{c}_L) + P_3 A_\tau(\tilde{t}_L)|$ $>$ $|P_2 A_e(\tilde{c}_L) + P_3 A_e(\tilde{t}_L)|$ independent of $\phi_{23}$. On the other hand, the magnitude of $A_\tau(\tilde{t}_R)$ is larger than those of $A_i(\tilde{c}_L)$ and $A_i(\tilde{t}_L)$, and the phase dependence of $P_1 A_\tau(\tilde{t}_R)$ is different from those of $P_2 A_i(\tilde{c}_L)$ and $P_3 A_i(\tilde{t}_L)$. Note that $A_i(\tilde{c}_L)$ and $A_i(\tilde{t}_L)$ are proportional to $\sim$ $1/(\sin \beta \cos \beta)$ $=$ $\tan \beta + 1/\tan \beta$, while $A_\tau(\tilde{t}_R)$ is proportional to $\sim$ $(\tan \beta + 1/\tan \beta)^2$, where $\tan \beta$ is the ratio of the vacuum expectation values of the two Higgs fields. Hence the $RRRR$ contribution is more enhanced than the $LLLL$ contributions for a large $\tan \beta$ [@RRRR].
Numerical results
=================
Now we present the results of our numerical calculations [@GN]. For the CKM matrix we fix the parameters as $V_{us}=0.2196$, $V_{cb}=0.0395$, $|V_{ub}/V_{cb}|=0.08$ and $\delta_{13}=90^\circ$ in the whole analysis, where $\delta_{13}$ is a complex phase in the CKM matrix in the convention of Ref. [@standard-parametrization]. The top quark mass is taken to be 175 ${\,{\rm GeV}}$ [@m_top]. The colored Higgs mass $M_C$ and the heavy gauge boson mass $M_V$ are assumed as $M_C$ $=$ $M_V$ $=$ $2 \times 10^{16} {\,{\rm GeV}}$. We require constraint on $ b \rightarrow s \gamma $ branching ratio from CLEO [@CLEO] and bounds on SUSY particle masses obtained from direct searches at LEP [@Neutralino_Bound], LEP II [@LEP2] and Tevatron [@CDF+D0].
Let us focus on the main decay mode $p\rightarrow K^+\overline{\nu}$. We first discuss the effects of the phases $\phi_{13}$ and $\phi_{23}$ parametrizing the matrix $P$ in Eq. (\[eqn:Phase\_Matrix\]). In Fig. \[fig:phi23\] we present the dependence of the decay rates $\Gamma(p\rightarrow K^+ \overline{\nu}_i)$ on the phase $\phi_{23}$.
=8.5cm
As an illustration we fix the other phase $\phi_{13}$ at $210^\circ$, and later we consider the whole parameter space of $\phi_{13}$ and $\phi_{23}$. The soft SUSY breaking parameters are also fixed as $m_0$ $=$ $1 {\,{\rm TeV}}$, $M_{gX}$ $=$ $125 {\,{\rm GeV}}$ and $A_X$ $=0$ here. The sign of the Higgsino mass $\mu$ is taken to be positive. With these parameters, all the masses of the scalar fermions other than the lighter $\tilde{t}$ are around 1 ${\,{\rm TeV}}$, and the mass of the lighter $\tilde{t}$ is about 400 ${\,{\rm GeV}}$. The lighter chargino is wino-like with a mass about 100 ${\,{\rm GeV}}$. This figure shows that there is no region for the total decay rate $\Gamma(p\rightarrow K^+ \overline{\nu})$ to be strongly suppressed, thus the whole region of $\phi_{23}$ in Fig. \[fig:phi23\] is excluded by the Super-Kamiokande limit. The phase dependence of $\Gamma(p\rightarrow K^+ \overline{\nu}_\tau)$ is quite different from those of $\Gamma(p\rightarrow K^+ \overline{\nu}_\mu)$ and $\Gamma(p\rightarrow K^+ \overline{\nu}_e)$. Though $\Gamma(p\rightarrow K^+ \overline{\nu}_\mu)$ and $\Gamma(p\rightarrow K^+ \overline{\nu}_e)$ are highly suppressed around $\phi_{23}$ $\sim$ $160^\circ$, $\Gamma(p\rightarrow K^+ \overline{\nu}_\tau)$ is not so in this region. There exists also the region $\phi_{23}$ $\sim$ $300^\circ$ where $\Gamma(p\rightarrow K^+ \overline{\nu}_\tau)$ is reduced. In this region, however, $\Gamma(p\rightarrow K^+ \overline{\nu}_\mu)$ and $\Gamma(p\rightarrow K^+ \overline{\nu}_e)$ are not suppressed in turn. Note also that the $K^+\overline{\nu}_\tau$ mode can give the largest contribution.
This behavior can be understood as follows. For $\overline{\nu}_\mu$ and $\overline{\nu}_e$, the effect of the $RRRR$ operator is negligible, and the cancellation between the $LLLL$ contributions directly leads to the suppression of the decay rates. This cancellation indeed occurs around $\phi_{23}$ $\sim$ $160^\circ$ for both $\overline{\nu}_\mu$ and $\overline{\nu}_e$ simultaneously in Fig. \[fig:phi23\]. For $\overline{\nu}_\tau$, the situation is quite different. The similar cancellation between $P_2 A_\tau(\tilde{c}_L)$ and $P_3 A_\tau(\tilde{t}_L)$ takes place around $\phi_{23}$ $\sim$ $160^\circ$ for $\overline{\nu}_\tau$ also. However, the $RRRR$ operator gives a significant contribution for $\overline{\nu}_\tau$. Therefore, $\Gamma(p\rightarrow K^+ \overline{\nu}_\tau)$ is not suppressed by the cancellation between the $LLLL$ contributions in the presence of the large $RRRR$ operator effect. Notice that it is possible to reduce $\Gamma(p\rightarrow K^+\overline{\nu}_\tau)$ by another cancellation between the $LLLL$ contributions and the $RRRR$ contribution. This reduction of $\Gamma(p\rightarrow K^+\overline{\nu}_\tau)$ indeed appears around $\phi_{23}$ $\sim$ $300^\circ$ in Fig. \[fig:phi23\]. The decay rate $\Gamma(p\rightarrow K^+\overline{\nu}_\mu)$ is rather large in this region. The reason is that $P_2 A_\tau(\tilde{c}_L)$ and $P_3 A_\tau(\tilde{t}_L)$ are constructive in this region in order to cooperate with each other to cancel the large $RRRR$ contribution $P_1 A_\tau(\tilde{t}_R)$, hence $P_2 A_\mu(\tilde{c}_L)$ and $P_3 A_\mu(\tilde{t}_L)$ are also constructive in this region. Thus we cannot reduce both $\Gamma(p\rightarrow K^+\overline{\nu}_\tau)$ and $\Gamma(p\rightarrow K^+\overline{\nu}_\mu)$ simultaneously. Consequently, there is no region for the total decay rate $\Gamma(p\rightarrow K^+ \overline{\nu})$ to be strongly suppressed. In the previous analysis [@HMTY] the region $\phi_{23}$ $\sim$ $160^\circ$ has been considered to be allowed by the Kamiokande limit $\tau(p\rightarrow K^+ \overline{\nu})$ $>$ $1.0 \times 10^{32}$ years (90% C.L.) [@Kam]. However the inclusion of the Higgsino dressing of the $RRRR$ operator excludes this region. We also examined the whole region of $\phi_{13}$ and $\phi_{23}$ with the same values for the other parameters as in Fig. \[fig:phi23\]. We found that we cannot reduce both $\Gamma(p\rightarrow K^+\overline{\nu}_\tau)$ and $\Gamma(p\rightarrow K^+\overline{\nu}_\mu)$ simultaneously, even if we adjust the two phases $\phi_{13}$ and $\phi_{23}$ anywhere.
Next we consider the case where we vary the parameters we have fixed so far. The relevant parameters are the colored Higgs mass $M_C$, the soft SUSY breaking parameters and $\tan \beta$. The partial lifetime $\tau(p\rightarrow K^+ \overline{\nu})$ is proportional to $M_C^2$ in a very good approximation, since this mode is dominated by the dimension 5 operators. Using this fact and the calculated value of $\tau(p\rightarrow K^+ \overline{\nu})$ for the fixed $M_C$ $=$ $2 \times 10^{16} {\,{\rm GeV}}$, we can obtain the lower bound on $M_C$ from the experimental lower limit on $\tau(p\rightarrow K^+ \overline{\nu})$. In Fig. \[fig:m\_sf\], we present the lower bound on $M_C$ obtained from the Super-Kamiokande limit as a function of the left-handed scalar up-quark mass $m_{\tilde{u}_L}$.
=8.5cm
Masses of the squarks other than the lighter $\tilde{t}$ are almost degenerate with $m_{\tilde{u}_L}$. The soft breaking parameters $m_0$, $M_{gX}$ and $A_X$ are scanned within the range of $0<m_0<3 {\,{\rm TeV}}$, $0<M_{gX}<1 {\,{\rm TeV}}$ and $ -5<A_X<5 $, and $\tan \beta$ is fixed at 2.5. Both signs of $\mu$ are considered. The whole parameter region of the two phases $\phi_{13}$ and $\phi_{23}$ is examined. The solid curve in this figure represents the result with all the $LLLL$ and $RRRR$ contributions. It is shown that the lower bound on $M_C$ decreases like $\sim$ $1/m_{\tilde{u}_L}$ as $m_{\tilde{u}_L}$ increases. This indicates that the $RRRR$ effect is indeed relevant, since the decay amplitude from the $RRRR$ operator is roughly proportional to $\mu/(M_C m_{\tilde{f}}^2)$ $\sim$ $1/(M_C m_{\tilde{f}})$, where we use the fact that the magnitude of $\mu$ is determined from the radiative electroweak symmetry breaking condition and scales as $\mu$ $\sim$ $m_{\tilde{f}}$. The dashed curve in Fig. \[fig:m\_sf\] represents the result in the case where we ignore the $RRRR$ effect. In this case the lower bound on $M_C$ behaves as $\sim$ $1/m_{\tilde{u}_L}^2$, since the $LLLL$ contribution is proportional to $m_{\tilde{W}}/(M_C m_{\tilde{f}}^2)$.
The solid curve in Fig. \[fig:m\_sf\] indicates that the colored Higgs mass $M_C$ must be larger than $6.5 \times 10^{16} {\,{\rm GeV}}$ for $\tan \beta$ $=$ 2.5 when the typical sfermion mass is less than $1 {\,{\rm TeV}}$. On the other hand, there is a theoretical upper bound of the colored Higgs mass $M_C$ ${
\raisebox{0.2em}{$<$} \hspace{-0.75em} \raisebox{-0.2em}{$\sim$} }$ $4 \times 10^{16} {\,{\rm GeV}}$ from an analysis of RGEs when we require the validity of the minimal SU(5) SUSY GUT model up to the Planck scale [^4]. Then it follows from Fig. \[fig:m\_sf\] that the minimal SU(5) SUSY GUT model with the sfermion masses less than $2 {\,{\rm TeV}}$ is excluded for $\tan \beta$ $=2.5$. The $RRRR$ effect plays an essential role here, since the lower bound on $m_{\tilde{f}}$ would be $600 {\,{\rm GeV}}$ if the $RRRR$ effect were ignored. We also find that the Kamiokande limit on the neutron partial lifetime $\tau (n\rightarrow K^0 \overline{\nu})$ $>$ $0.86 \times 10^{32}$ years (90% C.L.) [@Kam] already gives a comparable bound with that derived here from the Super-Kamiokande limit on $\tau (p\rightarrow K^+ \overline{\nu})$, as shown by the dash-dotted curve in Fig. \[fig:m\_sf\].
Fig. \[fig:tanB\] shows the $\tan \beta$ dependence of the lower bound on the colored Higgs mass $M_C$ obtained from the Super-Kamiokande limit.
=8.5cm
Here we vary $m_0$, $M_{gX}$ and $A_X$ as in Fig. \[fig:m\_sf\]. The phases $\phi_{13}$ and $\phi_{23}$ are fixed as $\phi_{13}$ $=210^\circ$ and $\phi_{23}$ $=150^\circ$. The result does not change much even if we take other values of $\phi_{13}$ and $\phi_{23}$. The region below the solid curve is excluded if $m_{\tilde{u}_L}$ is less than $1 {\,{\rm TeV}}$. The lower bound reduces to the dashed curve if we allow $m_{\tilde{u}_L}$ up to $3 {\,{\rm TeV}}$. It is shown that the lower bound on $M_C$ behaves as $\sim$ $\tan^2 \beta$ in a large $\tan \beta$ region, as expected from the fact that the amplitude of $p\rightarrow K^+ \overline{\nu}_\tau$ from the $RRRR$ operator is roughly proportional to $\tan^2 \beta /M_C$. On the other hand the $LLLL$ contribution is proportional to $\sim$ $\tan \beta /M_C$, as shown by the dotted curve in Fig. \[fig:tanB\]. Thus the $RRRR$ operator is dominant for large $\tan \beta$ [@RRRR]. Note that the lower bound on $M_C$ has the minimum at $\tan \beta$ $\approx$ 2.5. Thus we can conclude that for other value of $\tan \beta$ the constraints on $M_C$ and $m_{\tilde{f}}$ become severer than those shown in Fig. \[fig:m\_sf\]. In particular the lower bound on $m_{\tilde{f}}$ becomes larger than $\sim$ $2 {\,{\rm TeV}}$ for $\tan \beta$ $\neq$ 2.5.
The constraints obtained from the figures can be expressed as follows: $$\begin{aligned}
\left( \frac{M_C}{6.5 \times 10^{16} {\,{\rm GeV}}}\right) & {
\raisebox{0.2em}{$>$} \hspace{-0.75em} \raisebox{-0.2em}{$\sim$} }&
\left( \frac{\tau^{\rm exp}(p \rightarrow K^{+} \overline{\nu})}
{5.5 \times 10^{32} {\rm years}} \right)^{\frac{1}{2}}
\left( \frac{\beta_p}{0.003 {\,{\rm GeV}}^3} \right)
\left( \frac{1 {\,{\rm TeV}}}{m_{\tilde{f}}} \right)
\nonumber \\
& & \hspace{3cm} {\rm for} \ \tan \beta \approx 2.5, \nonumber \\
\left( \frac{M_C}{5.0 \times 10^{17} {\,{\rm GeV}}}\right) & {
\raisebox{0.2em}{$>$} \hspace{-0.75em} \raisebox{-0.2em}{$\sim$} }&
\left( \frac{\tau^{\rm exp}(p \rightarrow K^{+} \overline{\nu})}
{5.5 \times 10^{32} {\rm years}} \right)^{\frac{1}{2}}
\left( \frac{\beta_p}{0.003 {\,{\rm GeV}}^3} \right)
\left( \frac{1 {\,{\rm TeV}}}{m_{\tilde{f}}} \right)
\left( \frac{\tan \beta}{10} \right)^2
\nonumber \\
& & \hspace{3cm} {\rm for} \ \tan \beta \ {
\raisebox{0.2em}{$>$} \hspace{-0.75em} \raisebox{-0.2em}{$\sim$} }\ 5,
\label{eqn:constraint}\end{aligned}$$ where $\tau^{\rm exp}(p \rightarrow K^{+} \overline{\nu})$ is an experimental lower limit for the partial lifetime of the decay mode $p \rightarrow K^{+} \overline{\nu}$.
Conclusions
===========
We have reanalyzed the proton decay including the $RRRR$ dimension 5 operator in the minimal SU(5) SUSY GUT model. We have shown that the Higgsino dressing diagram of the $RRRR$ operator gives a dominant contribution for $p\rightarrow K^+\overline{\nu}_\tau$, and the decay rate of this mode can be comparable with that of $p\rightarrow K^+\overline{\nu}_\mu$. We have found that we cannot reduce both the decay rate of $p\rightarrow K^+\overline{\nu}_\tau$ and that of $p\rightarrow K^+\overline{\nu}_\mu$ simultaneously by adjusting the relative phases $\phi_{13}$ and $\phi_{23}$ between the Yukawa couplings at the colored Higgs interactions. We have obtained the bounds on the colored Higgs mass $M_C$ and the typical sfermion mass $m_{\tilde{f}}$ from the new limit on $\tau(p\rightarrow K^+ \overline{\nu})$ given by the Super-Kamiokande: $M_C$ $>$ $6.5 \times 10^{16} {\,{\rm GeV}}$ for $m_{\tilde{f}}$ $<$ $1 {\,{\rm TeV}}$. The minimal SU(5) SUSY GUT model with $m_{\tilde{f}}$ ${
\raisebox{0.2em}{$<$} \hspace{-0.75em} \raisebox{-0.2em}{$\sim$} }$ $2 {\,{\rm TeV}}$ is excluded if we require the validity of this model up to the Planck scale.
References {#references .unnumbered}
==========
[99]{} T. Goto and T. Nihei, . E. Witten, ; S. Dimopoulos, S. Raby and F. Wilczek, ; S. Dimopoulos and H. Georgi, ; N. Sakai, . C. Giunti, C.W. Kim and U.W. Lee, ; J. Ellis, S. Kelley and D.V. Nanopoulos, ; U. Amaldi, W. de Boer and H. Fürstenau, ; P. Langacker and M.-X. Luo, ; W.J. Marciano, [*Ann. Rev. Nucl. Part.*]{} [**41**]{}, 469 (1991). S. Dimopoulos, S. Raby and F. Wilczek, ; J. Ellis, D.V. Nanopoulos and S. Rudaz, . For reviews on the minimal SU(5) SUGRA GUT model, see for instance, H.P. Nilles, ; P. Nath, R. Arnowitt and A.H. Chamseddine, Applied $N=1$ Supergravity (World Scientific, Singapore, 1984). Kamiokande Collaboration, K.S. Hirata [*et al.*]{}, . IMB Collaboration, R. Becker-Szendy [*et al.*]{}, Proceedings of 23rd International Cosmic Ray Conference, Calgary 1993 [**Vol.4**]{} 589. M. Takita (Super-Kamiokande Collaboration), Talk presented in 29th International Conference on High Energy Physics, Vancouver, July 1998. Super-Kamiokande Collaboration, . N. Sakai and T. Yanagida, ; S. Weinberg, . P. Nath, A.H. Chamseddine and R. Arnowitt, . M. Matsumoto, J. Arafune, H. Tanaka and K. Shiraishi, ; J. Hisano, H. Murayama and T. Yanagida, . J. Hisano, T. Moroi, K. Tobe and T. Yanagida, . T. Goto, T. Nihei and J. Arafune, . V. Lucas and S. Raby, . J. Ellis, M.K. Gaillard and D.V. Nanopoulos, . K. Inoue, A. Kakuto, H. Komatsu and S. Takeshita, ; [*ibid.*]{} [**71**]{}, 413 (1984); L. Ibáñez and G.G. Ross, ; L. Alvarez-Gaumé, J. Polchinski and M.B. Wise, ; J. Ellis, J.S. Hagelin, D.V. Nanopoulos and K. Tamvakis, . A. Bouquet, J. Kaplan, and C.A. Savoy, ; . M. Claudson, M.B. Wise and L.J. Hall, ; S. Chadha and M. Daniel, . S.J. Brodsky, J. Ellis, J.S. Hagelin and C.T. Sachrajda, ; M.B. Gavela, S.F. King, C.T. Sachrajda, G. Martinelli, M.L. Paciello and B. Taglienti, . L.-L. Chau and W.-Y. Keung, ; H. Harari and M. Leurer, ; H. Fritzsch and J. Plankl, ; F.J. Botella and L.-L. Chao, . CDF Collaboration, F. Abe [*et al*]{}., ; ; D0 Collaboration, S. Abachi [*et al*]{}., . CLEO Collaboration, M.S. Alam, [*et al*]{}., . L3 Collaboration, M. Acciarri [*et al*]{}., . D. Treille, Talk presented in 29th International Conference on High Energy Physics, Vancouver, July 1998. CDF Collaboration, F. Abe [*et al*]{}., ; [*ibid.*]{} [**69**]{} (1992) 3439; D0 Collaboration, S. Abachi [*et al*]{}., . K.S. Babu and M.J. Strassler, hep-ph/9808447.
[^1]: Talk given by T. Nihei in International Symposium on Supersymmetry, Supergravity, and Superstring (SSS99), Seoul, Korea, June 23-27, 1999, based on the published work [@GN].
[^2]: After we finished our analysis, the latest limit of the Super-Kamiokande $\tau(p\rightarrow K^+ \overline{\nu})$ $>$ $6.7 \times 10^{32}$ years (90% C.L.) [@superK-new] was reported. An adaptation to the updated experimental limit is straightforward (See Eq.(\[eqn:constraint\])).
[^3]: See also Ref.[@Babu-Strassler].
[^4]: Also it has been pointed out that there exists an upper bound on $M_C$ given by $M_C$ $\leq$ $2.5 \times 10^{16} {\,{\rm GeV}}$ (90% C.L.) if we require the gauge coupling unification in the minimal contents of GUT superfields [@HMTY].
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'In this paper, we investigate the importance of a defense system’s learning rates to fight against the self-propagating class of malware such as worms and bots. To this end, we introduce a new propagation model based on the interactions between an adversary (and its agents) who wishes to construct a zombie army of a specific size, and a defender taking advantage of standard security tools and technologies such as honeypots (HPs) and intrusion detection and prevention systems (IDPSes) in the network environment. As time goes on, the defender can incrementally learn from the collected/observed attack samples (e.g., malware payloads), and therefore being able to generate attack signatures. The generated signatures then are used for filtering next attack traffic and thus containing the attacker’s progress in its malware propagation mission. Using simulation and numerical analysis, we evaluate the efficacy of signature generation algorithms and in general any learning-based scheme in bringing an adversary’s maneuvering in the environment to a halt as an adversarial containment strategy.'
author:
-
-
bibliography:
- 'sections/references.bib'
title: |
[On the Convergence Rates of Learning-based Signature\
Generation Schemes to Contain Self-propagating Malware]{}\
[^1]
---
Botnet, Malware propagation modeling, Self-replicating code, Worms, Intrusion detection and prevention system, Honeypots, Security games
Introduction {#sec:intro}
============
Background and Related Work {#sec:backg}
===========================
A Learning-based Propagation Model {#sec:model}
==================================
Numerical Analysis of the Learning-based Model
==============================================
\[sec:numAna\]
Simulation Results {#sec:sim}
==================
Concluding remarks and future directions {#sec:con}
========================================
\[appendix\]
[^1]: This work was funded by NSF grant CNS-1413996 “MACS: A Modular Approach to Cloud Security.”
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'The bosonic nature of light leads to counter-intuitive bunching effects. We describe an experimentally testable effect in which a single photon is induced through a highly reflecting beamsplitter by a large amplitude coherent state, with probability $1/e$ in the limit of large coherent state amplitude. We use this effect to construct a viable implementation of the bare raising operator on coherent states via conditional measurement, which succeeds with high probability and fidelity even in the high amplitude limit.'
author:
- 'Jennifer C. J.'
- 'Daniel K. L.'
- John
title: Photonic Quantum Operations via the Quantum Carburettor Effect
---
Introduction
============
Operations on quantum states of light have a wide range of applications in quantum information processing and communication [@Braunstein2005], as well as being of fundamental interest [@Kraus1983]. However, the operations that are easily implemented are limited to the class of Gaussian or linear operations [@Braunstein2005]. Conditional evolution overcomes this limitation and provides a richer set of operations useful for manipulating continuous variable systems [@Ban1996; @Filip2005; @Vitelli2010; @Dakna1998]. Here, we show how to mimic the application of the bare raising operator to coherent state inputs using a beamsplitter, a single photon source, and a detector. In doing so we exploit a process that we call the quantum carburettor effect, whereby a strong coherent beam entrains the passage of a single photon (from an independent source) through a highly reflective beamsplitter with high probability, thereby elegantly highlighting the bosonic nature of light. This effect could also be used as a method to the characterize highly reflective beamsplitters and high mean photon number states.
Conditional measurement-based evolution is a useful tool in discrete variable systems, such as the scheme by Knill, Laflamme and Milburn for efficient linear optical computing [@Knill2001]. Discrete systems have been extensively studied [@Nielsen], but have limitations particularly apparent in communications, where loss may be significant. Continuous variable schemes show greater promise here [@Braunstein2005], but while Gaussian states such as coherent or squeezed states are relatively well understood, non-Gaussian states and operations have been less well-studied [@Adesso2014]. Non-Gaussian operations are required for tasks such as entanglement distillation and error correction, essential to the use of continuous variables in information processing protocols. In the continuous variable regime conditional evolution allows non-Gaussian operations [@Braunstein2005]. There has significant interest in this approach [@Wenger2004; @Walker1986; @Parigi2007; @Andersson2006; @Pegg1998; @Dakna1997; @Ourjoumtsev2006], as the alternative of using nonlinear optics typically succeeds with low probability due to the weakness of nonlinear susceptibilities. Operations based on conditional measurement can be concatenated and this can allow operations which would not be possible deterministically, such as probabilistic state amplification [@Ralph2009; @Ferreyrol2010; @Marek2010; @Xiang2010; @Usuga2010; @Eleftheriadou2013; @Donaldson2015]. Here we look at a different way to increase the energy of a quantum state, by implementing the ladder or bare raising operator [@Susskind1964].
We first introduce the bare raising operator, then describe the basic setup for photon addition using a beamsplitter. In contrast to previous work [@Clausen1999; @Escher2005], our method works on an easily-generated state with Poisson photon number statistics such as the coherent state. The reflectivity of the beamsplitter is chosen to shape the photon number basis amplitudes of the output state to best match a photon-shifted coherent state. Our approach is not limited to coherent states as it does not rely on coherence between the input and the single-photon ancilla, and it will also provide a reasonably close implementation of the required operator on any state with similar support to a coherent state. We show that the probability of successful implementation of the operation remains high even for large coherent state amplitudes – the beamsplitter reflectivity in this case is close to unity, but the photon is still transmitted with high probability – a process that we dub the quantum carburettor effect. We examine the effect of imperfect detector efficiency and show that the operations and quantum carburettor effect persist for experimentally feasible values. We also consider a multi stage scheme.
The bare raising operator
=========================
The bare raising and lowering operators, sometimes known as the Susskind-Glogower operators [@Susskind1964], act on the space of harmonic oscillator energy eigenstates to shift the amplitudes of a state of a system up or down the ladder by exactly one quantum without modifying their relative amplitudes. They are $$\hat{E}^+ = \sum_{n=0}^\infty |n+1\rangle \langle n|,\label{addexact}\qquad
\hat{E}^- = \sum_{n=1}^\infty |n-1\rangle \langle n|,$$ with $\add=(\sub)^\dagger$. Note that while $\sub\add = \ident$, $\add \sub = \ident - |0 \rangle \langle 0|$, because acting with $\hat{E}^-$ on the ground state has no support. Unlike the more usual creation and annihilation operators $\cre = \sum_{n=0}^\infty \sqrt{n+1}|n+1\rangle \langle n|$ and $\ann= \sum_{n=1}^\infty \sqrt{n} |n-1\rangle \langle n|$ the bare operators do not introduce $\sqrt{n}$ bosonic enhancement factors, as shown in Fig. \[pncompare\]. The corresponding operations are identical in their actions on a Fock state, but for a superposition or mixture of Fock states the difference between these two operations can be clearly seen. $\add$ only shifts the Fock basis amplitudes up to a higher photon number, whereas after normalization $\cre$ increases the amplitudes of larger Fock states relative to lower ones. Therefore the bare raising and lowering operators can be used to shift the Fock state amplitudes of a quantum state up or down whilst preserving coherence, with the obvious exception of the ground state information being lost when the state is lowered.
These operators have long been of theoretical interest [@Susskind1964], as they can be used in applications such as generation and manipulation of nonclassical states [@Steinhoff2014] or their characterization [@Zou2006]. Also, the bare raising operator $\add$ is a Fock-space equivalent of the first Hilbert Hotel type operation [@Potocek2015], which demonstrates the mathematical concept of infinity by an apparent paradox: a fully occupied hotel with infinite rooms can accommodate one more guest by moving everyone up by one room.
(BS1) at (0,0) ; (BS1) – +(45:0.75cm) node\[anchor=south\] [$t_1$]{} – +(225:0.75cm) ; (BS2) at (1.5,0) ; (BS2) – +(45:0.75cm) node\[anchor=south\] [$t_2$]{} – +(225:0.75cm); (BS3) at (3,0) ; (BS3) – +(45:0.75cm) node\[anchor=south\] [$t_3$]{} – +(225:0.75cm); (BSN) at (6,0) ; (BSN) – +(45:0.75cm) node\[anchor=south\] [$t_{n-1}$]{} – +(225:0.75cm);
(D1) \[above=of BS1\] ; ($(D1) - 0.5*(0.5,0)$) arc (180:0:0.25) – cycle; ($(D1) + 0.5*(0,0.5)$) – ($(D1) + (0,0.5)$) node\[anchor=south\] [0]{}; (D2) \[above=of BS2\] ; ($(D2) - 0.5*(0.5,0)$) arc (180:0:0.25) – cycle; ($(D2) + 0.5*(0,0.5)$) – ($(D2) + (0,0.5)$)node\[anchor=south\] [0]{}; (D3) \[above=of BS3\] ; ($(D3) - 0.5*(0.5,0)$) arc (180:0:0.25) – cycle; ($(D3) + 0.5*(0,0.5)$) – ($(D3) + (0,0.5)$)node\[anchor=south\] [0]{}; (DN) \[above=of BSN\] ; ($(DN) - 0.5*(0.5,0)$) arc (180:0:0.25) – cycle; ($(DN) + 0.5*(0,0.5)$) – ($(DN) + (0,0.5)$)node\[anchor=south\] [0]{};
($(BS1) - (1,0)$) node\[anchor=east\] [$\1$]{} – (BS1); (BS1) – (BS2); (BS2) – (BS3); (BS3) – ($(BS3) + (1,0)$); ($(BS3) + (1.25,0)$) – ($(BSN) - (1.25,0)$); ($(BSN) - (1,0)$) – (BSN); (BSN) – ($(BSN)+(1,0)$) node\[anchor=west\] [$\ket{N}$]{} ;
($(BS1) - (0,1)$) node\[anchor=north\] [$\1$]{} – (BS1); ($(BS2) - (0,1)$) node\[anchor=north\] [$\1$]{} – (BS2); ($(BS3) - (0,1)$) node\[anchor=north\] [$\1$]{} – (BS3); ($(BSN) - (0,1)$) node\[anchor=north\] [$\1$]{} – (BSN); (BS1) – (D1); (BS2) – (D2); (BS3) – (D3); (BSN) – (DN);
($(D3) + (1.25,0.2)$) – ($(DN) - (1.25,-0.2)$); ($(BS3) + (1.25,-1.2)$) – ($(BSN) - (1.25,1.2)$);
A scheme to synthesize arbitrary Fock states using beamsplitters and conditional measurement with single photon inputs was considered in 2005 by Escher *et. al.* [@Escher2005] . Their system consisted of a cascade of beamsplitters, each combining a single photon Fock state with the output of the previous beamsplitter. This is depicted in Fig. \[escfig\]. The scheme requires $N$ single photons to make a Fock state $\ket{N}$, and succeeds when all $N-1$ perfectly efficient detectors do not fire. They found that the maximum probability of a given detector not firing to be $$P(0) = \left(\frac{n}{n+1}\right)^n \label{esceq}$$ for a Fock state $\ket{n}$ and single photon input, leading to a $\ket{n+1}$ output. This occurs with a beamsplitter of transmission coefficient $$t_n = \sqrt{\frac{n}{n+1}} \text{ .}$$
The standard creation operator $\cre$ can be implemented approximately, either using postselected spontaneous parametric down conversion [@Clausen2001], or using a beamsplitter, single photon and detector. We use the latter approach here, but we aim to implement the bare raising operator $\add \ket{n} = \ket{n+1}$ instead. We do this by an appropriate choice of the reflection coefficient. Recently, experimental implementations of $\add$ using cavity QED [@Oi2013] or circuit QED [@Govia2012; @Joo2016] have been proposed that may allow practical applications. Linear optical implementations of higher-order Hilbert-Hotel operations exist, but not in the Fock basis [@Potocek2015]. Here we replace the complexity of these proposals with linear optics in the Fock basis and postselection.
Implementation of $\add$ with a single beamsplitter
===================================================
Setup
-----
(4.7,2) – node\[anchor=north,align=center\] [**0**\
**counts**]{} (5.2,2);
The essential setup used throughout is shown in Fig. \[BS\], with coherent state and single photon inputs and measurement in one output mode [@Note1]. A coherent state of mean photon number $|\alpha|^2$ can be written as: $$\ket{\alpha} = e^{\frac{-|\alpha|^2}{2}} \sum_{n=0}^\infty \frac{\alpha^n}{\sqrt{n}} \ket{n} \text{ ,}$$ where $\{\ket{n}\}$ is the set of Fock states. The beamsplitter relations are $$\begin{pmatrix}
\cre_1 \\ \cre_2
\end{pmatrix} = \begin{pmatrix}
|t| e^{\imag \phi_T} & - |r| e^{-\imag \phi_R} \\ |r| e^{\imag \phi_R} & |t| e^{-\imag\phi_T}
\end{pmatrix} \begin{pmatrix}
\cre_3 \\ \cre_4 \end{pmatrix} \text{ ,}\label{BSeq}$$ where $|t|^2 + |r|^2 =1$. The operation implemented by this setup will be denoted $\addap$ to distinguish it from the ideal $\add$ operation. So that $\addap$ does not change the relative phase of the photon number states, we choose the convention $\phi_R = \pi$ and $\phi_T = 0$.
The joint input state to the beamsplitter is $$\ket{\alpha_1}\ket{1_2} = e^{\frac{-|\alpha|^2}{2}} \sum_{n=0}^\infty \frac{\alpha^n}{\sqrt{n}} \ket{n}_1\ket{1}_2 \equiv \sum_{n=0}^\infty q_n \ket{n}_1\ket{1}_2 \text{ ,}$$ where $q_n = e^{\frac{-|\alpha|^2}{2}} \frac{\alpha^n}{\sqrt{n!}}$. This can be written in terms of the input mode creation operators acting on the joint vacuum state: $$\ket{\Psi_\text{in}} = \sum_{n=0}^\infty q_n \frac{\cre_1 \,^n}{\sqrt{n!}} \cre_2 \ket{0 0}_{12} \text{ .}$$ Application of the beamsplitter transformation in Eq. gives the joint output as:
[rCl]{} &=& \_[n=0]{}\^ (|t|e\^[\_T]{} \_3 - |r|e\^[-\_R]{} \_4)\^n\
& & ( |r| e\^[\_R]{} \_3 + |t| e\^[-\_T]{} \_4) \_[34]{}
which can be expanded to give: $$\begin{aligned}
\hspace{-2ex}\ket{\Psi_\text{out}} & = & \sum_{n=0}^\infty \sum_{k=0}^n \frac{q_n \sqrt{n!} (-1)^k}{k! (n-k)!} \nonumber \\ & & \times \left[ A_1 \sqrt{(n-k+1)!k!} \ket{n-k+1}_3 \ket{k}_4 \right.\nonumber \\
& & \left. + A_2 \sqrt{(n-k)!(k+1)!} \ket{n-k}_3 \ket{k+1}_4 \right] \text{ ,}\label{cohout}\end{aligned}$$ where $$\begin{aligned}
A_1 = A_1(n,k) & = & |t|^{n-k}|r|^{k+1} e^{\imag(n-k)\phi_T -\imag (k-1)\phi_R} \\
A_2 = A_2(n,k) & = & |t|^{n-k+1}|r|^k e^{\imag(n-k-1)\phi_T -\imag k \phi_R} \text{ .}\end{aligned}$$
After conditioning on no counts at the detector in mode 3 (*i.e.* $k=n$, 2nd term only), we find the normalized output is $$\addap \ket{\alpha} = \frac{1}{\sqrt{P(0)}}\sum_{n=0}^\infty q_n |t| |r|^n \sqrt{n+1} \ket{n+1}_4 \text{ ,} \label{addapout}$$ where the success probability $P(0)$ is given by $$P(0) \sum_{n=0}^\infty |q_n|^2 |t|^2 |r|^{2n} (n+1) \text{ .} \label{p0}$$
The output state from the beamsplitter can be tuned by adjusting the value of $|r|$. We adjust $|r|$ to match the desired state $\add \ket{\alpha}$ more closely. We do this by optimizing $F_1 = |\bra{\alpha}(\add)^\dag \addap \ket{\alpha}|^2$, which is the fidelity of the beamsplitter-implemented operation with the exact operation.
Results
-------
### Perfect Detectors
The most striking result is the behavior of the success probability for the $\addap$ operation in the high $n$ or $\alpha$ limit. The limit of Eq. as $n \rightarrow \infty$ is $P(0) = 1/e$ for high-$n$ Fock states. This result also holds for $\alpha \rightarrow \infty$, as shown numerically in Fig. \[nondetgraph\_prob\]. Thus a single photon is, with high probability, transmitted through a highly-reflecting mirror by an intense light beam. We call this the quantum carburettor effect. Note that this is not due to any coherence between the photon and the coherent state, as would be the case in a Mach-Zender interferometer for instance.
The quantum carburettor effect can clearly be seen by the probability limit in Fig. \[nondetgraph\_prob\]. With a beamsplitter of the optimum reflection probability $|r_\text{opt}|^2$, the probability for the detection of zero photons with a perfect detector tends towards the limit $1/e \approx 0.37$. The optimum reflection probability tends towards a perfectly reflecting mirror, but even in this limit a single photon can be transmitted with probability $1/e$ if the appropriate amplitude coherent state is also incident on the beamsplitter.
We show the fidelity of the conditional output state after normalization with the ideal $\add \ket{\alpha}$ in Fig. \[nondetgraph\_fid\], and also for comparison the fidelity of the state $\cre\ket{\alpha}$ with the ideal $\add \ket{\alpha}$. The different effect of $\add$ and $\cre$ is clearest around $\alpha\approx1$, and the conditional output state has a much higher fidelity with $\add\ket{\alpha}$ than the state $\cre\ket{\alpha}$.
Numerical results indicate that the reflection probability $|r_\text{opt}|^2$ for implementing the $\add$ operation with the highest fidelity coincides closely with that for the highest success probability for large values of $|\alpha|$ ($\alpha>\approx 2.5$). To calculate the reflection probability corresponding to the maximum success probability, we differentiate Eq. w.r.t. $|r|^2$ and set equal to zero. Taking the positive solution for $|r|^2$ leads to $$|r|^2 = \frac{|\alpha|^2 - 3 + \sqrt{|\alpha|^4 + 2|\alpha|^2 +5}}{2|\alpha|^2} \label{cohoptprob} \text{ ,}$$ which is valid above $|\alpha|^2 = 0.5$. Hence in the high $|\alpha|$ limit Eq. is a good approximation of the optimal reflection probability.
### Inefficient Detectors
A major consideration for an experimental implementation would be the robustness to detector inefficiency. This can be accounted for by using the normally ordered measurement operator $:\exp(-\eta \cre \ann):$ in place of a simple projection [@Loudon2000 6.10], where $\eta$ is the detector efficiency (the probability that a photon incident on the detector will be counted). Numerical calculations shown in Fig. \[alphahumpgiveneta\] indicate that the model is not affected severely by the presence of moderate inefficiency. Numerical work was done in python with the aid of the QuTiP package [@qutip].
Whilst moderate detector inefficiency does not prevent implementation of $\add$, for $\eta < 1-1/e$ it introduces an upper bound on possible input coherent state amplitudes $\alpha$. This is due to the switch between two regimes: for low $\alpha$, the beamsplitter-implemented operation gives a better fidelity, while for higher $|\alpha|$ the loss of fidelity due to the inefficiency means that simply reflecting the input coherent state gives a better fidelity. In that case the success probability is the same as the detector inefficiency, although the operation cannot be said to be implemented. The boundary between these two regimes for various values of $\eta$ is depicted in Fig. \[basefid\] as the point at which the ‘do nothing’ fidelity exceeds the achievable fidelity with the current set up.
Cascaded Operation
==================
With a more elaborate approach, it may be possible to improve on the scheme in the previous section. Here we consider a straightforward extension of the implementation of $\add$, with extra components to correct a failed operation.
Setup
-----
The operation fails with 1 or more photocounts in mode 3. For the case of 1 count, we attempt the operation again on the failed output. This requires feedforward from the first detector, and an additional single photon, beamsplitter and detector as in Fig. \[2BSSchema\]. Figure \[tree\] shows the possible measurement results at each detector, with their associated outcomes and probabilities. This may improve the success probability of the operation.
(D1) at (5.2,2) ; (4.9,2) – node\[anchor=south\] [**D1**]{} (5.2,2); at (5.3,2) [**$\pmb{x=0, 1}$ or $\pmb{>1}$**\
**count(s)**]{}; at (3.2,3.2) [**BS1**]{}; (1,6) – node (centerii) (3,8) node\[anchor=south west\] [**BS2**]{}; (0,7) node\[anchor=east\] [$\pmb{\ket{1}}$]{} – node\[anchor=north\] [6]{} (centerii); (2,4.5) node\[anchor=north,draw\] (delay) [Switch]{} – node\[anchor=west\] [**5**]{} (centerii); (delay) – node\[anchor=south\] [If $x>1$]{} ++(-2,0) node\[anchor=east\] [**Fail**]{}; at ($(delay)+(0,0.7)$) [If $x=1$]{}; (centerii) – node\[anchor=south\] [**8**]{} (4,7) node\[anchor=west\] [$\pmb{\ket{\psi_2}}$]{}; (centerii) – node\[anchor=east\] [**7**]{} (2,9) node\[anchor=south\] ; (2,9.7) – node\[anchor=west\] [**0 counts**]{} node\[anchor=east\] [**D2**]{} (2,10.2); (D1.center) .. controls ($(D1.west) + (2,0)$) and ($(delay.south east) + (1,0)$) .. node\[anchor=west,align=right\] [Feedforward\
$x$]{} (delay.south east); (delay.east) – node\[anchor=south\] [If $x=0$]{} (4,2 |- delay.east) node\[anchor=west\] [$\pmb{\ket{\psi_1}}$]{};
(D1) at (1,0) [**D1 Result**]{}; (F1) at (0,-0.8) \[align=center\] [**Accept**\
**Fidelity** $\pmb{F_1}$]{}; (fail1) at (1,-1.2) [**Fail**]{}; (retry) at (2.2,-0.8) [**Retry**]{}; (2.2,-2) node (D2) [**D2 Result**]{} +(-1,-1) node \[align=center\] (F2) [**Accept**\
**Fidelity** $\pmb{F_2}$]{} +(1,-1) node (fail2) [**Fail**]{}; (D1) – node\[anchor=south east\] [$\pmb{P_1(0)}$]{}(F1); (D1) – node\[anchor=north west\] [$\pmb{P_1(>1)}$]{}(fail1); (D1) – node\[anchor=south west\] [$\pmb{P_1(1)}$]{} (retry); (retry) – (D2); (D2) – node \[anchor=south east\] [$\pmb{P_2(0)}$]{} (F2); (D2) – node\[anchor=south west\] [$\pmb{P_2(>0)}$]{} (fail2);
If there is no count at detector 1, then the first output state $\ket{\psi_1}$ is accepted as before. If there is a count, a correction is attempted. When there is no count at detector 2 the correction is accepted and the unnormalized output state $\ket{\psi_2}$ in mode 8 is $$\sum_{n=0}^\infty q_n |r_1|^{n-1} (n |t_1|^2 - |r_1|^2) |t_2||r_2|^n \sqrt{n+1} \ket{n+1}_8 \text{ ,} \label{2BSstate}$$ where subscript 1 refers to the first beamsplitter parameters and 2 to the second. The normalization gives the probability of an initial failure (1 count at detector 1) and then an accepted correction (no counts at detector 2).
The fidelity measure used is the mean fidelity of accepted output states: $$F_{mean} = \frac{P_1(0)F_1 + P_1(1) P_2(0) F_2}{P_1(0)+P_1(1) P_2(0)} \text{ ,}$$ where $P_j(0)$ indicates zero counts at the $j$th detector and therefore an accepted output state at the relevant beamsplitter, $P_j(1)$ indicates one photocount with the possibility to correct the state, and $F_j$ is the fidelity of that state. The total success probability is simply $P_1(0)+P_1(1)P_2(0)$.
Results
-------
Figure \[doubleBSgraph\] shows, for various input $\ket{\alpha}$, the possible mean fidelities against their probabilities. Points cover the full range of beamsplitter pairs, whilst the solid line indicates use of beamsplitter 1 alone. It can be seen that the best fidelity is obtained using a single beamsplitter. For lower $\alpha$ an increase in probability is possible, but this results in a large loss of fidelity. This indicates that the strategy delivers only marginal improvement at best. We explore the reason for this later.
Discussion and conclusions
==========================
From the basic scheme it is clear that $\add$ is an experimentally-viable, high probability non-Gaussian operation with potential applications in continuous variable tasks such as entanglement distillation. The application of $\add$ makes coherent states nonclassical. This is clearly seen by noting that $\add \ket{\alpha}$ has vanishing amplitude in the zero photon component, a sign of nonclassicality [@Mandel1995].
In the presented implementation of $\add$, the maximum success probability and optimal fidelity are achievable near simultaneously; there is no need to compromise one to improve the other as is the case with the implementation of $\cre$. More elaborate optical schemes may improve the success probability, as there is in principle no limit to the coherent state amplitude on which $\add$ may be implemented with high probability. This was demonstrated theoretically in [@Oi2013] in a cavity QED setup.
To observe the quantum carburettor effect experimentally, the main components required are a single photon source and laser suitable for interference, a beamsplitter and a photodetector. As the measurement is conditioned on zero photocounts, a non-photon-number-resolving detector should be sufficient. This would allow the investigation of the $1/e$ probability limit for the transmission of a single photon with matched beamsplitter $|r_\text{opt}|^2$ and $\alpha$.
An alternative experimental application for the quantum carburettor effect is shown in Fig. \[characbs\]. The quantum carburettor effect and the existence of an optimal reflection probability dependent on $\alpha$ leads to a peak in probability just above $1/e$ when $\alpha$ is adjusted. This effect can be used to characterize highly reflecting beamsplitters, or alternatively with a variable beamsplitter to characterize high $\alpha$ coherent states.
The poor performance of attempts to correct the failed operation with a second beamsplitter can be linked to the two photon interference and the production of a ‘hole’ in the photon number distribution of a coherent state, an effect described by Escher et al [@Escher2004] and depicted in Fig. \[pnfail\]. As $\alpha$ increases, the optimal beamsplitter reflection coefficient $|r_\text{opt}|$ for the $\add$ operation on a coherent state tends towards the reflection coefficient required to create a hole around $n=|\alpha|^2$ found by Escher et al. Figure \[pnfail\] shows this effect for $\alpha=2$, with the optimal beamsplitter for the $\add$ operation. This lack of amplitude at $|\alpha|^2$ severely impacts on the fidelity of any attempted recovery; while any divergence from this $|r_\text{opt}|$ in the first beamsplitter reduces the probability and fidelity of an initially successful operation. Hence, the attempt at correcting a failed operation is not particularly successful. Future work could consider other schemes to improve the success probability for $\add$, or look at ways to implement other non-Gaussian operations.
A scheme to implement the bare photon raising operator has been presented, using linear optics and conditioning on a measurement outcome. Through this the quantum carburettor effect (interference between a high amplitude single photon and coherent state) has been introduced and various applications considered, including characterizing beamsplitters and large coherent state amplitudes. A possible extension of the scheme was considered and found to give little improvement in success probability. These operators are experimentally realizable and of relevance in quantum information processing in continuous variable systems.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors acknowledge useful discussions with Brian Gerardot and Adetunmise Dada. J. C. J. Radtke acknowledges support from the EPSRC Doctoral Training Grant, University of Strathclyde.
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|
{
"pile_set_name": "ArXiv"
}
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a4 library.tex
=2
[**<span style="font-variant:small-caps;">Generalizations of Guillera-Sondow’s double integral formulas</span>**]{}\
[**<span style="font-variant:small-caps;">Sergey Zlobin</span>**]{}
[ We evaluate certain multidimensional integrals in terms of the Lerch transcendent function $\Phi$, generalizing Guillera-Sondow’s formulas. As an application, we get new representations of classical constants like Euler’s constant $\gamma$ and $\ln(4/\pi)$. ]{}
The [*Lerch transcendent*]{} $\Phi$ is defined as the analytic continuation of the series $$\Phi(z,s,u)=\frac{1}{u^s}+\frac{z}{(u+1)^s}+\frac{z^2}{(u+2)^s}+\cdots,$$ which converges for any complex number $u$ with $\Re u>0$ if $z$ and $s$ are any complex numbers with either $|z|<1$, or $|z|=1$ and $\Re s > 1$ (we suppose $\zeta^s=\exp(s \log \zeta)$, where $\log \zeta$ is the principal branch of the logarithm). The function $\Phi$ is holomorphic in $z$ and $s$, for $z \in \C \backslash [1,\infty]$ and all complex $s$ (see [@Bateman section 1.11] or [@GuilleraSondow section 2]). From the definition it follows that $$\label{LerchEq}
\Phi(z,s,u+1) = \frac{1}{z} \left( \Phi(z,s,u)- \frac{1}{u^s} \right),$$ $$\label{LerchEq2}
\Phi(z,s+1,u) = -\frac{1}{s} \frac{\partial \Phi}{\partial u}(z,s,u).$$
In the paper [@GuilleraSondow] J. Sondow and J. Guillera proved the following two theorems.
\[SondowGuilleraTh1\] Suppose $u>0$, $v>0$, $u \ne v$, either $z \in \C \backslash [1,\infty)$ and $\Re s > -2$, or $z=1$ and $\Re s > -1$. Then $$\begin{aligned}
\int_{\cube{2}} \frac{x_1^{u-1} x_2^{v-1}}{1-z x_1 x_2}
(-\ln x_1 x_2)^{s} dx_1 dx_2 &= \Gamma(s+1)
\frac{\Phi(z,s+1,v)-\Phi(z,s+1,u)}{u-v}, \\
\int_{\cube{2}} \frac{(x_1 x_2)^{u-1}}{1-z x_1 x_2}
(-\ln x_1 x_2)^{s} dx_1 dx_2 &= {\Gamma(s+2)} \Phi(z,s+2,u).\end{aligned}$$
\[SondowGuilleraTh2\] Suppose $u>0$, either $z \in \C \backslash [1, \infty)$ and $\Re s > -3$, or $z=1$ and $\Re s > -2$. Then $$\begin{gathered}
\int_{\cube{2}} \frac{1-x_1}{1-z x_1 x_2} (x_1 x_2)^{u-1}
(-\ln x_1 x_2)^{s} dx_1 dx_2
\\ =\Gamma(s+2) \left[ \Phi(z,s+2,u)+\frac{(1-z)
\Phi(z,s+1,u)-u^{-s-1}}{z (s+1)} \right].\end{gathered}$$
The purpose of this paper is to prove the following $m$-dimensional analogs of Theorems \[SondowGuilleraTh1\] and \[SondowGuilleraTh2\] (in what follows $d \overline x$ means $dx_1 dx_2 \cdots dx_m$, where $m$ is the dimension of an integral).
\[MainTh1\] Suppose $m$ is a positive integer, $\Re u>0$, $\Re v>0$, $u \ne v$, either $z \in \C \backslash [1,\infty)$ and $\Re s > -m$, or $z=1$ and $ \Re s > 1-m$. For the case $m>1$ define the function $$\begin{gathered}
F_{m,u,v}(x_1,x_2,\dots,x_m) =
(x_1 x_2 \cdots x_m)^{v-1} (x_1^{u-v} + (x_1 x_2)^{u-v} + \cdots +
(x_1 x_2 \cdots x_{m-1})^{u-v} ).
% x_1^{u-1} x_2^{v-1} \cdots x_{m-1}^{v-1} x_m^{v-1} \\
% + x_1^{u-1} x_2^{u-1} \cdots x_{m-1}^{v-1} x_m^{v-1}
% + \cdots +
% x_1^{u-1} x_2^{u-1} \cdots x_{m-1}^{u-1} x_m^{v-1}.\end{gathered}$$ Then $$\begin{gathered}
\int_{\cube{m}} \frac{F_{m,u,v}(x_1,x_2,\dots,x_m)}
{1-z x_1 x_2 \cdots x_m}
(-\ln x_1 x_2 \cdots x_m)^{s} d \overline x \\
=\frac{\Gamma(s+m-1)}{(m-2)!} \cdot
\frac{\Phi(z,s+m-1,v)-\Phi(z,s+m-1,u)}{u-v} \quad \mbox{for } m>1
\label{MainTh1Eq1},\end{gathered}$$ $$\label{MainTh1Eq2}
\int_{\cube{m}} \frac{(x_1 x_2 \cdots x_m)^{u-1}}{1-z x_1 x_2 \cdots x_m}
(-\ln x_1 x_2 \cdots s_m)^{s} d \overline x =
\frac{\Gamma(s+m)}{(m-1)!} \Phi(z,s+m,u) \quad \mbox{for } m \ge 1.$$
\[MainTh2\] Suppose $m$ is an integer $>1$, $\Re u>0$, either $z \in \C \backslash [1, \infty)$ and $\Re s > -m-1$, or $z=1$ and $\Re s > -m$. Then $$\begin{gathered}
\int_{\cube{m}} \frac{m-1 - x_1-x_1 x_2 - \cdots - x_1 x_2 \cdots x_{m-1}}
{1-z x_1 x_2 \cdots x_m} (x_1 x_2 \cdots x_m)^{u-1}
(-\ln x_1 x_2 \cdots x_m)^{s} \, d \overline x
\\ = \frac{\Gamma(s+m)}{(m-2)!} \left[ \Phi(z,s+m,u)+\frac{(1-z)
\Phi(z,s+m-1,u)-u^{-s-m+1}}{z (s+m-1)} \right]
\label{MainTh2Eq}.\end{gathered}$$
In the case $m=2$ Theorems \[MainTh1\] and \[MainTh2\] give Theorems \[SondowGuilleraTh1\] and \[SondowGuilleraTh2\]. As an example, we give also the case $m=3$.
a\) If $\Re u>0$, $\Re v>0$, $u \ne v$, either $z \in \C \backslash [1,\infty)$ and $\Re s > -3$, or $z=1$ and $ \Re s > -2$, then $$\int_{\cube{3}}
\frac{x_1^{u-1} x_2^{v-1} x_3^{v-1} + x_1^{u-1} x_2^{u-1} x_3^{v-1}}
{1-z x_1 x_2 x_3}
(-\ln x_1 x_2 x_3)^{s} d \overline x \\
=\Gamma(s+2)
\frac{\Phi(z,s+2,v)-\Phi(z,s+2,u)}{u-v}.$$ b) If $\Re u>0$, either $z \in \C \backslash [1,\infty)$ and $\Re s > -3$, or $z=1$ and $ \Re s > -2$, then $$\int_{\cube{3}}
\frac{(x_1 x_2 x_3)^{u-1}}
{1-z x_1 x_2 x_3}
(-\ln x_1 x_2 x_3)^{s} d \overline x \\
=\frac{\Gamma(s+3)}{2}
\Phi(z,s+3,u).$$ c) If $\Re u>0$, either $z \in \C \backslash [1,\infty)$ and $\Re s > -4$, or $z=1$ and $ \Re s > -3$, then $$\begin{gathered}
\int_{\cube{3}} \frac{2 - x_1-x_1 x_2}
{1-z x_1 x_2 x_3} (x_1 x_2 x_3)^{u-1}
(-\ln x_1 x_2 x_3)^{s} \, d \overline x
\\ = \Gamma(s+3) \left[ \Phi(z,s+3,u)+\frac{(1-z)
\Phi(z,s+2,u)-u^{-s-2}}{z (s+2)} \right].\end{gathered}$$
In [@GuilleraSondow] many interesting applications of Theorems \[SondowGuilleraTh1\] and \[SondowGuilleraTh2\] are given. All of them can be generalized by Theorems \[MainTh1\] and \[MainTh2\]; indeed, by these four theorems we have $$\begin{gathered}
\int_{\cube{2}} \frac{x_1^{u-1} x_2^{v-1}}{1-z x_1 x_2}
(-\ln x_1 x_2)^{s} dx_1 dx_2 \\
= (m-2)!
\int_{\cube{m}} \frac{F_{m,u,v}(x_1,x_2,\dots,x_m)}
{1-z x_1 x_2 \cdots x_m}
(-\ln x_1 x_2 \cdots x_m)^{s-m+2} d \overline x
\quad \mbox{for } m>1,\end{gathered}$$ $$\begin{gathered}
\int_{\cube{2}} \frac{(x_1 x_2)^{u-1}}{1-z x_1 x_2}
(-\ln x_1 x_2)^{s} dx_1 dx_2 \\
= (m-1)!
\int_{\cube{m}} \frac{(x_1 x_2 \cdots x_m)^{u-1}}
{1-z x_1 x_2 \cdots x_m}
(-\ln x_1 x_2 \cdots x_m)^{s-m+2} d \overline x
\quad \mbox{for } m \ge 1,\end{gathered}$$ $$\begin{gathered}
\int_{\cube{2}} \frac{1-x_1}{1-z x_1 x_2} (x_1 x_2)^{u-1}
(-\ln x_1 x_2)^{s} dx_1 dx_2 \\
=(m-2)!
\int_{\cube{m}} \frac{m-1 - x_1-x_1 x_2 - \cdots - x_1 x_2 \cdots x_{m-1}}
{1-z x_1 x_2 \cdots x_m} (x_1 x_2 \cdots x_m)^{u-1} \\
\times (-\ln x_1 x_2 \cdots x_m)^{s-m+2} \, d \overline x
\quad \mbox{for } m>1.\end{gathered}$$
We give here only two examples.
Let m be an integer $>1$, and $\gamma=\lim_{n \to \infty}
\left( 1+\frac{1}{2}+\cdots+\frac{1}{n} - \ln n \right)$ be Euler’s constant. Then $$\gamma=(m-2)! \int_{\cube{m}}
\frac{m-1 - x_1-x_1 x_2 - \cdots - x_1 x_2 \cdots x_{m-1}}
{(1-x_1 x_2 \cdots x_m) (-\ln x_1 x_2 \cdots x_m)^{m-1}}
d \overline x.$$
For an integer $m>1$ the following identity holds: $$\ln \frac{4}{\pi} = (m-2)! \int_{\cube{m}}
\frac{m-1 - x_1-x_1 x_2 - \cdots - x_1 x_2 \cdots x_{m-1}}
{(1+x_1 x_2 \cdots x_m) (-\ln x_1 x_2 \cdots x_m)^{m-1}}
d \overline x.$$
We omit details of proofs of these examples and refer to the case $m=2$, which was considered by J. Sondow [@SondowMonthly].
To prove Theorem \[MainTh1\], we will require two lemmas. The first is the identity (\[MainTh1Eq2\]) for $m=1$, and is classical.
\[Lemma1\] Suppose $\Re u>0$, either $z \in \C \backslash [1,\infty)$ and $\Re s > -1$, or $z=1$ and $ \Re s > 0$. Then $$\int_0^1 \frac{x^{u-1}}{1-z x} (-\ln x)^{s} dx =
\Gamma(s+1) \Phi(z,s+1,u).$$
The integral, call it $I$, defines a holomorphic function of $z$ and $s$ under the conditions stated. We can prove the statement for $|z|<1$ and $\Re s > 0$ and then use analytic continuation. Expand $1/(1-zx)$ into a geometric series and then integrate: $$I=\sum_{n=0}^{\infty} z^n \int_0^1 x^{u+n-1} (-\ln x)^{s} dx.$$ Making the substitution $x=e^{-y}$, we obtain $$I
=\sum_{n=0}^{\infty} z^n \int_0^{\infty} e^{-(u+n)y} y^{s} dy
=\sum_{n=0}^{\infty} \frac{\Gamma(s+1) z^n}{(u+n)^{s+1}}
=\Gamma(s+1) \Phi(z,s+1,u),$$ and the lemma follows.
Let $\alpha \ne 0$ and $x \in (0,1]$. Then the following identities hold for $k \ge 1$:\
a) $$\label{SimplexEq1}
\int_{1 \ge t_1 \ge t_2 \ge \cdots \ge t_k \ge x}
\frac{1}{t_1 t_2 \cdots t_k} dt_1 dt_2 \cdots dt_k = \frac{(-\ln x)^k}{k!},$$ b) $$\label{SimplexEq2}
\int_{1 \ge t_1 \ge t_2 \ge \cdots \ge t_k \ge x}
\frac{t_1^{\alpha} + t_2^{\alpha} + \cdots + t_k^{\alpha}}
{t_1 t_2 \cdots t_k} dt_1 dt_2 \cdots dt_k
= \frac{(-\ln x)^{k-1}}{(k-1)!} \cdot \frac{1-x^\alpha}{\alpha}.$$
The identity (\[SimplexEq1\]) is easily proved using induction and the equality $$\int_{1 \ge t_1 \ge t_2 \ge \cdots \ge t_k \ge x}
\frac{dt_1 dt_2 \cdots dt_k}{t_1 t_2 \cdots t_k} =
\int_x^1 \left(
\int_{1 \ge t_1 \ge t_2 \ge \cdots \ge t_{k-1} \ge t_k}
\frac{dt_1 dt_2 \cdots dt_{k-1}}{t_1 t_2 \cdots t_{k-1}}
\right) \frac{dt_k}{t_k}.$$ Denote the integral in (\[SimplexEq2\]) by $I_{k}(x)$. We prove by induction; the case $k=1$ is true. Suppose $k>1$ and the statement is true for $k-1$. We have $$I_{k}(x)=
\int_x^1 I_{k-1}(t_{k}) \frac{dt_k}{t_k} +
\int_x^1 \left(
\int_{1 \ge t_1 \ge t_2 \ge \cdots \ge t_{k-1} \ge t_k}
\frac{dt_1 dt_2 \cdots dt_{k-1}}{t_1 t_2 \cdots t_{k-1}}
\right) t_k^{\alpha-1} dt_k.$$ Apply (\[SimplexEq1\]) to the integral in parentheses: $$I_{k}(x)=
\int_x^1 I_{k-1}(t_{k}) \frac{dt_k}{t_k} +
\int_x^1 \frac{(-\ln t_k)^{k-1}}{(k-1)!} t_k^{\alpha-1} dt_k.$$ Using the induction hypothesis, we obtain $$\begin{aligned}
I_{k}(x)=&
\int_x^1
\left(
\frac{(-\ln t_k)^{k-2}}{(k-2)!} \cdot \frac{1-t_k^\alpha}{\alpha t_k}
+ \frac{(-\ln t_k)^{k-1}}{(k-1)!} t_k^{\alpha-1}
\right) dt_k \\
=& \left. \frac{(-\ln t_k)^{k-1}}{(k-1)!} \cdot \frac{t_k^{\alpha}-1}{\alpha}
\right|_x^1
=\frac{(-\ln x)^{k-1}}{(k-1)!} \cdot \frac{1-x^{\alpha}}{\alpha}.\end{aligned}$$ Now the lemma is completely proved.
[**Proof of Theorem \[MainTh1\].**]{} The integrals $J_1$ and $J_2$ in (\[MainTh1Eq1\]) and (\[MainTh1Eq2\]) define holomorphic functions of $s$ under the conditions stated. We can prove the statement for $\Re s > 0$ and then use analytic continuation.
First we prove (\[MainTh1Eq2\]). Make the substitution $$\label{Subst}
x_1=t_1, \quad x_2=t_2/t_1, \quad x_3=t_3/t_2,
\quad \dots, \quad x_m=t_m/t_{m-1}$$ in $J_2$. We obtain $$\begin{aligned}
J_2= & \int_{1 \ge t_1 \ge t_2 \ge \cdots \ge t_m \ge 0}
\frac{t_m^{u-1}}{1-z t_m} (-\ln t_m)^s \frac{1}{t_1 t_2 \cdots t_{m-1}}
dt_1 dt_2 \cdots dt_m \\
= & \int_{0}^1 \frac{t_m^{u-1}}{1-z t_m} (-\ln t_m)^s
\left(
\int_{1 \ge t_1 \ge t_2 \ge \cdots \ge t_{m-1} \ge t_m}
\frac{1}{t_1 t_2 \cdots t_{m-1}}
dt_1 dt_2 \cdots dt_{m-1} \right)
dt_m.\end{aligned}$$ Applying (\[SimplexEq1\]) with $x=t_m$ and $k=m-1$, we get $$J_2= \frac{1}{(m-1)!}
\int_{0}^1 \frac{t_m^{u-1}}{1-z t_m} (-\ln t_m)^{s+m-1} dt_m.$$ It remains to apply Lemma \[Lemma1\].
Now we prove (\[MainTh1Eq1\]). Denote $\alpha=u-v$, then $$J_1=\int_{\cube{m}} \frac{(x_1 x_2 \cdots x_m)^{v-1}}{1-z x_1 x_2 \cdots x_m}
(x_1^{\alpha} + (x_1 x_2)^{\alpha} + \cdots + (x_1 x_2 \cdots x_{m-1})^{\alpha})
(-\ln x_1 x_2 \cdots s_m)^{s} d \overline x.$$ Make the substitution (\[Subst\]) $$J_1=
\int_{0}^1 \frac{t_m^{v-1}}{1-z t_m} (-\ln t_m)^s
\left(
\int_{1 \ge t_1 \ge t_2 \ge \cdots \ge t_{m-1} \ge t_m}
\frac{t_1^{\alpha} + t_2^{\alpha} + \cdots + t_{m-1}^{\alpha}}
{t_1 t_2 \cdots t_{m-1}}
dt_1 dt_2 \cdots dt_{m-1} \right)
dt_m$$ and apply (\[SimplexEq1\]) $$J_1=\frac{1}{(m-2)! \alpha} \left(
\int_{0}^1 \frac{t_m^{v-1}}{1-z t_m} (-\ln t_m)^{s+m-2} dt_m -
\int_{0}^1 \frac{t_m^{v+\alpha-1}}{1-z t_m} (-\ln t_m)^{s+m-2} dt_m
\right).$$ It remains to apply Lemma \[Lemma1\] to both integrals and get back to $u$ from $\alpha$. The theorem is proved.\
[**Remark.**]{} The formula (\[MainTh1Eq2\]) can be also obtained by letting $v \to u$ in (\[MainTh1Eq1\]) and using the identity (\[LerchEq2\]).
[**Proof of Theorem \[MainTh2\].**]{} The integral $J$ in (\[MainTh2Eq\]) defines a function which is holomorphic in $s$, when $\Re s > -m-1$ if $z \in \C \backslash [1,\infty]$, and when $\Re s > -m $ if $z=1$. We prove the statement for $\Re s > 0$ and then use analytic continuation. We have $$\begin{gathered}
J=(m-1)
\int_{\cube{m}} \frac{(x_1 x_2 \cdots x_m)^{u-1}}
{1-z x_1 x_2 \cdots x_m}
(-\ln x_1 x_2 \cdots x_m)^{s} d \overline x \\
-
\int_{\cube{m}} \frac{F_{m,u+1,u}(x_1, x_2, \dots, x_m)}
{1-z x_1 x_2 \cdots x_m} (-\ln x_1 x_2 \cdots x_m)^{s} \, d \overline x.\end{gathered}$$ Apply Theorem \[MainTh1\] to both integrals: $$\begin{aligned}
J= & (m-1) \frac{\Gamma(s+m)}{(m-1)!} \Phi(z,s+m,u) \\
& -
\frac{\Gamma(s+m-1)}{(m-2)!} (\Phi(z,s+m-1,u)-\Phi(z,s+m-1,u+1)) \\
= & \frac{\Gamma(s+m)}{(m-2)!} \left[
\Phi(z,s+m,u) + \frac{\Phi(z,s+m-1,u+1)-\Phi(z,s+m-1,u)}{(s+m-1)}
\right].\end{aligned}$$ Use (\[LerchEq\]) and the theorem follows.
The way which we prove Theorem \[MainTh1\] can be applied to any integral $$\int_{\cube{m}} \frac{x_1^{u_1} x_2^{u_2} \cdots x_m^{u_m}}
{1-z x_1 x_2 \cdots x_m}
(-\ln x_1 x_2 \cdots x_m)^{s} d \overline x.$$ We give the formula for the case when all $u_i$ are different.
\[MainTh3\] Suppose $m \ge 1$ and $\Re u_1>0$ , $\Re u_2>0$, …, $\Re u_m>0$, $u_i \ne u_j$ whenever $i \ne j$, and either $z \in \C \backslash [1,\infty)$ and $\Re s > -1$, or $z=1$ and $ \Re s > 0$. Then the following identity holds: $$\label{MainTh3Eq}
\int_{\cube{m}} \frac{x_1^{u_1-1} x_2^{u_2-1} \cdots x_m^{u_m-1}}
{1-z x_1 x_2 \cdots x_m}
(-\ln x_1 x_2 \cdots x_m)^{s} d \overline x=
\Gamma(s+1) \sum_{i=1}^m \frac{\Phi(z,s+1,u_i)}{\prod_{j \ne i} (u_j-u_i)}.$$
To prove Theorem \[MainTh3\] we require the following
Let $k \ge 1$ and $u_1$, $u_2$, …, $u_{k+1}$ be arbitrary numbers with $u_i \ne u_j$ whenever $i \ne j$, and $x \in (0,1]$. Then the following identity hold:\
$$\label{SimplexEq3}
\int_{1 \ge t_1 \ge t_2 \ge \cdots \ge t_k \ge x}
t_1^{u_1-u_2-1} t_2^{u_2-u_3-1} \cdots t_k^{u_k-u_{k+1}-1}
dt_1 dt_2 \cdots dt_k =
\sum_{i=1}^{k+1} \frac{x^{u_i-u_{k+1}}}{\prod_{j=1, j \ne i}^{k+1} (u_j-u_i)}.$$
Denote the integral in (\[SimplexEq3\]) by $I(u_1,u_2,\dots,u_{k+1};x)$. We prove by induction; the case $k=1$ is true. Suppose $k>1$ and the statement is true for $k-1$, then $$\begin{aligned}
I(u_1,u_2,\dots,u_{k+1};x)=&
\int_x^1 I(u_1,u_2,\dots,u_{k};t_k) t_k^{u_k-u_{k+1}-1} dt_k \\
=&\int_x^1 \sum_{i=1}^{k} \frac{t_k^{u_i-u_{k}}}{\prod_{j=1, j \ne i}^k (u_j-u_i)}
t_k^{u_k-u_{k+1}-1} dt_k \\
=& \sum_{i=1}^{k} \frac{1}{\prod_{j=1, j \ne i}^k (u_j-u_i)}
\int_x^1 t_k^{u_i-u_{k+1}-1} dt_k \\
= & \sum_{i=1}^{k} \frac{1}{\prod_{j=1, j \ne i}^k (u_j-u_i)} \cdot
\frac{ 1 - x^{u_{i}-u_{k+1}} }{u_i-u_{k+1}} \\
= & \sum_{i=1}^{k} \frac{x^{u_i-u_{k+1}}}{\prod_{j=1, j \ne i}^{k+1} (u_j-u_i)}
+ x^{u_{k+1}-u_{k+1}} \cdot
\sum_{i=1}^{k} \frac{1}{(u_i-u_{k+1}) \prod_{j=1, j \ne i}^k (u_j-u_i)}.\end{aligned}$$ Thus the statement of the lemma is equivalent to the identity $$\label{LagrangeEq}
\sum_{i=1}^{k} \frac{1}{(u_i-u_{k+1}) \prod_{j=1, j \ne i}^k (u_j-u_i)}=
\frac{1}{\prod_{j=1}^k (u_j-u_{k+1})}.$$ To prove it, consider the polynomial $$P(x) = \sum_{i=1}^k \frac{\prod_{j=1, j \ne i}^k (u_j-x)}
{\prod_{j=1, j \ne i}^k (u_j-u_i)}.$$ of degree $k-1$. We have $P(u_i)=1$ for any $i \in \{ 1,2, \dots, k \}$. Hence $P(x) \equiv 1$. The equality $P(u_{k+1})=1$ yields (\[LagrangeEq\]) and the lemma follows.
[**Proof of Theorem \[MainTh3\].**]{} In the case $m=1$ the theorem is equivalent to Lemma 1. Now let $m>1$. Make the substitution (\[Subst\]) in the integral $J$ in (\[MainTh3Eq\]) $$J= \int_{1 \ge t_1 \ge t_2 \ge \cdots \ge t_m \ge 0}
\frac{t_m^{u_m-1}}{1-z t_m} (-\ln t_m)^s
t_1^{u_1-u_2-1} t_2^{u_2-u_3-1} \cdots t_{m-1}^{u_{m-1}-u_m-1}
dt_1 dt_2 \cdots dt_m.$$ Applying (\[SimplexEq3\]) for $k=m-1$ and $x=t_m$, we obtain $$J = \sum_{i=1}^{m} \frac{1}{\prod_{j=1, j \ne i}^{k+1} (u_j-u_i)}
\int_0^1 \frac{t_m^{u_i-1}}{1-z t_m} (-\ln t_m)^s dt_m.$$ Use Lemma \[Lemma1\] and the theorem follows.\
[**Remark.**]{} Theorem \[MainTh3\] is another generalization of the first equality in Theorem \[SondowGuilleraTh1\].
The author wishes to thank J. Sondow for reading a preliminary version of the paper and for some useful suggestions.
[99]{}
// Vol. 1, McGraw-Hill, New York, 1953.
// E-print math.NT/0506319, to appear in Ramanujan J.
// Amer. Math. Monthly 112 (2005) P. 61–65.
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'The latent space of normalizing flows must be of the same dimensionality as their output space. This constraint presents a problem if we want to learn low-dimensional, semantically meaningful representations. Recent work has provided compact representations by fitting flows constrained to manifolds, but hasn’t defined a density off that manifold. In this work we consider flows with full support in data space, but with ordered latent variables. Like in PCA, the leading latent dimensions define a sequence of manifolds that lie close to the data. We note a trade-off between the flow likelihood and the quality of the ordering, depending on the parameterization of the flow.'
bibliography:
- 'bibliography.bib'
---
Introduction
============
Normalizing flows provide a way to parameterize complex distributions in high-dimensional spaces via a sequence of invertible transformations of a simple base distribution. Unlike most other popular flexible generative models [e.g. @kingma2013; @goodfellow2014a_gan; @du2019], flows have a tractable likelihood, which simplifies training and model comparison. It is also easier to use tractable representations of distributions in other models and algorithms, such as for variational autoencoder (VAE) priors and posteriors.
However, generative models are not just used for estimating probabilities; we also fit these models to learn representations for down-stream tasks [@bengio2012]. The latent spaces learned by normalizing flows can be semantically meaningful [@dinh2016_realnvp; @kingma2018_glow] and can be useful for down-stream applications, such as classification [@nalisnick2019].
For standard flows the latent dimensionality must match the input dimensionality, in order to satisfy the invertibility constraint. This constraint presents a limitation for representation learning, where we are often interested in learning lower-dimensional representations (manifolds) that capture high-level semantic concepts [@bengio2012].
When data is constrained to a known manifold, it is possible to build a valid tractable flow on that space [@gemici2016; @rezende2020]. For data without strict constraints, @kumar2020_injective_flow and @brehmer2020_mfmf fit an invertible flow defined on a low-dimensional manifold, and bring the manifold close to the data by minimizing square error. Neither of these methods provide a tractable probabilistic model in the data space.
In this work we maintain a full-dimensional flow, but aim to order the latent variables, such that taking the first $K$ of them will reconstruct data with small square error. For a linear flow, our approach implements Principal Components Analysis (PCA). We encode this principle with an additional loss based on *nested dropout* [ND, @rippel2014_nested_dropout]. We show that our models successfully learn low dimensional representations, while obtaining similar likelihoods to standard flows. However, we find that the parameterization of the flow influences the representations, and we must make a trade-off between the compactness of representations and the likelihood of the flow. We hope that these findings will motivate research into flows that more naturally represent low-dimensional structure.
Nested dropout
==============
Nested dropout was introduced by @rippel2014_nested_dropout in the context of autoencoders, with a goal of imposing an ordering on representation dimensions. For a linear autoencoder, nested dropout fits PCA. In a non-linear case, the learned ordering was shown to be useful for efficient retrieval and adaptive compression.
As in dropout [@srivastava2014]: nested dropout randomly “drops” (zeros out) representation dimensions during training. However, rather than dropping features independently, we sample an index $k \sim p_k\br{\cdot}$, and drop dimensions $k+1 \dots K$, all dimensions above that index, assuming $K$ is the latent dimensionality.
We define $\vz_{\downarrow k}$ as a representation $\vz$ where all the dimensions above $k$-th have been dropped, as described above. We then define a low-dimensional reconstruction of a given datapoint $\vx$ for a given index $k$ as $$\label{eq:reconstruction}
\hat{\vx}_{\downarrow k} = g_\theta\br{f_\theta\br{\vx}_{\downarrow k}},$$ where $f_\theta$ and $g_\theta$ are the encoder and the decoder respectively, with parameters $\theta$.
To train the autoencoder for a given dataset $\set{\vx_n}_{n=1}^N$, we minimize the following objective: $$\label{eq:reconstruction_loss}
\mathcal{L}\br{\vect{\theta}} = \frac{1}{N}\sum_{n=1}^N \expect{d\br{\vx_n, \hat{\vx}_{n\downarrow k}}}{k\sim{p_k}},$$ where $d$ is the chosen distance metric, typically $L_2$ distance. This set-up encourages the auto-encoder to put more information into dimensions that correspond to smaller $k$ indices.
@rippel2014_nested_dropout used a geometric distribution for $p_k$, traditionally parameterized as $p_k\br{k} \!=\! \br{1 - p}^{k-1}p$ for a given Bernoulli probability $p$. We follow their choice in this work.
Normalizing flows with nested dropout {#sec:nested_dropout_flows}
-------------------------------------
We start with the standard training objective of a normalizing flow. For a given parametric invertible function $f_{\vect{\theta}}$ and a base density $\pi$, the density of a given datapoint $\vx$ is: $$\label{eq:flow_density}
p_\vect{\theta}\br{\vx} = \pi\br{f_{\vect{\theta}}\br{\vx}}\abs{\det\br{\deriv{f_{\vect{\theta}}}{\vx}}}.$$ Given a dataset $\set{\vx_n}_{n=1}^N$, we fit the flow parameters $\vect{\theta}$ by minimizing the negative log likelihood objective: $$\label{eq:likelihood_loss}
\mathcal{L}\br{\vect{\theta}} = -\frac{1}{N} \sum_{n=1}^N \log p_\vect{\theta}\br{\vect{x}_n}.$$ We can redefine a lower-dimensional reconstruction, \[eq:reconstruction\], for a normalizing flow by treating the invertible function $f_\theta$ as the “encoder”, and its inverse $f^{-1}_\theta$ as the “decoder”: $$\label{eq:flow_reconstruction}
\tilde{\vx}_{\downarrow k} = f_\theta^{-1}\br{f_\theta\br{\vx}_{\downarrow k}}.$$ We combine \[eq:reconstruction\_loss,eq:flow\_reconstruction,eq:likelihood\_loss\] into a single objective, which specifies that the flow should have high likelihood, while also giving good reconstructions from low-dimensional parts of its representation: $$\textstyle
\mathcal{L}\br{\vect{\theta}} = {\frac{1}{N}}\!\sum_{n=1}^N\bigl(
-\log p_\vect{\theta}\br{{\vect{x}_n}}
+ \lambda\,\expect{d\br{\vx_n, \tilde{\vx}_{n\downarrow k}}}{k\sim{p_k}}\bigl),$$ where $\lambda$ is a hyper-parameter that balances the two losses. In line with @rippel2014_nested_dropout we estimate the expectation with a single Monte-Carlo sample.
The loss is similar to the one used by @brehmer2020_mfmf [Eq.21], with two distinctions. First, $k$ is sampled from $p_k$ for each datapoint, rather than pre-set according to the desired manifold dimensionality. Second, while @brehmer2020_mfmf use two separate flows, one for each part of the objective, we train a single flow for both.
Experiments
===========
Synthetic dataset
-----------------
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LL MSE(2) MSE(1)
-------- ----------------------------------------------- -------------------------------- -------------------------------
PCA $\hphantom{-0.00}$ – $\hphantom{{}\pm 0.000}$ $0.003 \hphantom{{}\pm 0.000}$ $0.037\hphantom{{}\pm 0.000}$
QR $-0.804 \pm 0.000$ $0.240 \pm 0.053$ $0.321 \pm 0.025$
QR$^*$ $-0.804 \pm 0.000$ $0.003 \pm 0.000$ $0.037 \pm 0.000$
LU $-0.804 \pm 0.000$ $0.051 \pm 0.042$ $0.257 \pm 0.130$
LU$^*$ $-0.805 \pm 0.001$ $0.008 \pm 0.007$ $0.037 \pm 0.001$
: Average test log-likelihood (in nats) and reconstruction MSE when projecting down to 1 or 2 dimensions for the synthetic dataset. Mean $\pm$ 2 standard deviations over 10 different initializations. Superscript$^*$ for models trained with nested dropout.[]{data-label="tab:synthetic_results"}
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As a simple testcase, we sample data from a centered 3-dimensional normal distribution that is scaled along the axis and then rotated. The eigenvalues of the covariance are $1$, $0.1$, and $0.01$, so the target distribution looks like a fuzzy elliptical disc embedded into 3 dimensions. We sample $10^4$ points for training, and another $10^4$ points for evaluation.
We use a simple flow with a standard normal base distribution and a single invertible linear transformation. Such a flow can express the target distribution by learning to scale/rotate the base distribution as necessary. However, due to the spherical symmetry of the base distribution, the likelihood is invariant to permuting the dimensions (or any orthogonal transformation) of the base distribution.
![2D projections $\vz$ of the 3D synthetic dataset , colored by the value of the first principal component of PCA. Superscript$^*$ for models trained with nested dropout.[]{data-label="fig:synthetic_proj"}](fig/synthetic_proj){width="0.8\columnwidth"}
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The invertible linear transformation is parameterized by either an LU decomposition with a random, fixed permutation matrix $\mat{P}$ [@kingma2018_glow], or a QR decomposition [@hoogeboom2019] with the orthogonal matrix parameterized by 3 Householder transformations [@tomczak2016_householder]. We train the flow using the Adam optimizer [@kingma2014] for $30{\times}10^3$ iterations with a batch size of 500. We set the reconstruction coefficient to $\lambda=20$ for nested dropout runs, with $p_k$ set to a geometric distribution with $p=0.33$.
We use PCA as a baseline for reconstruction results, where we project onto 1 or 2 principal components. Likelihood and reconstruction results are summarized in \[tab:synthetic\_results\]. We visualise some of the projections in \[fig:synthetic\_proj\].
For both parameterizations the additional loss improves reconstructions from 1 or 2 dimensions. For the QR parameterization, matching PCA’s optimal representation isn’t a restriction, and the test likelihood is maintained. The results for the LU parameterization have higher variance with nested dropout. The permutation matrix $\mat{P}$ is not learned, and its initialization matters.
Images
------
To evaluate the method on high-dimensional data, we fit a normalizing flow with nested dropout to Fashion-MNIST images [@xiao2017_fashion_mnist]. To simplify the model architecture, we pad the images by 2 pixels on each side, giving $32{\times}32$ or 1024-dimensional images. We use the provided test set, but split the provided training set into $50\,000$ training images and $10\,000$ validation images. We dequantize the images by adding uniform noise $\mat{U} \in [0,1)^{32{\times}32}$.
We use a RQ-NSF (C) image flow [@durkan2019_nsf] containing 3 multi-scale levels with 4 transformation steps per level, where each step consists of an activation normalization layer, a 1x1 convolution, and a rational-quadratic coupling transform. Residual convolutional networks with 3 blocks and 128 hidden channels parameterize the 4-bin rational-quadratic splines in the coupling transforms. We train all models for $100{\times}10^3$ iterations, with a batch size of 256. We anneal the learning rate of the Adam optimizer from $5{\times}10^{-4}$ down to zero according to a cosine schedule [@loshchilov2016 Eq. 5].
![Mean squared error of Fashion-MNIST reconstructions for RQ-NSF (C) flow against the number of retained dimensions, varying the order in which the dimensions are dropped.[]{data-label="fig:nd_order"}](fig/nd_order){width="\columnwidth"}
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![Mean squared error of Fashion-MNIST reconstructions for RQ-NSF (C) flow against the number of retained dimensions. Comparing a flow trained with nested dropout to the baseline without the additional loss. *bpd* is the negative test log-likelihood in bits per dimension (lower is better).[]{data-label="fig:baseline_vs_nd"}](fig/baseline_vs_nd){width="\columnwidth"}
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-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![**Top:** samples from the flow. **Bottom:** reconstructions for an input image in the top left, decaying the latent dimension from 510 to 1 (left-to-right, top-to-bottom). **Left**: RQ-NSF (C) baseline. **Right**: RQ-NSF (C) with nested dropout.[]{data-label="fig:samples_reconstructions"}](fig/baseline_samples "fig:"){width=".49\columnwidth"} ![**Top:** samples from the flow. **Bottom:** reconstructions for an input image in the top left, decaying the latent dimension from 510 to 1 (left-to-right, top-to-bottom). **Left**: RQ-NSF (C) baseline. **Right**: RQ-NSF (C) with nested dropout.[]{data-label="fig:samples_reconstructions"}](fig/nd_samples "fig:"){width=".49\columnwidth"}
![**Top:** samples from the flow. **Bottom:** reconstructions for an input image in the top left, decaying the latent dimension from 510 to 1 (left-to-right, top-to-bottom). **Left**: RQ-NSF (C) baseline. **Right**: RQ-NSF (C) with nested dropout.[]{data-label="fig:samples_reconstructions"}](fig/baseline_reconstructions "fig:"){width=".49\columnwidth"} ![**Top:** samples from the flow. **Bottom:** reconstructions for an input image in the top left, decaying the latent dimension from 510 to 1 (left-to-right, top-to-bottom). **Left**: RQ-NSF (C) baseline. **Right**: RQ-NSF (C) with nested dropout.[]{data-label="fig:samples_reconstructions"}](fig/nd_reconstructions "fig:"){width=".49\columnwidth"}
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
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### Ordering in a multi-scale architecture
The multi-scale architecture with variable splitting, introduced by @dinh2016_realnvp and used in RQ-NSF (C), [already]{} induces a partial ordering on variables, even without the additional nested dropout objective. Some variables undergo more transformations than others, and those undergoing more transformations are eventually transformed at “coarser scale”, by transformations conditioned on variables that are further away spatially. @dinh2016_realnvp [Appendix D] demonstrate that such variables contain higher-level semantic information.
\[fig:nd\_order\] demonstrates the effect of this ordering. There is a sudden drop in error after including ${\approx}250$ dimensions from the reversed order. In our implementation of the multi-scale transform, it is this last quarter of the variables that are transformed at all 3 multi-scale levels. We use the reversed order in all further experiments, both during training with nested dropout, and during evaluation.
### Training with nested dropout
We now try to understand whether we can use nested dropout to improve upon the baseline RQ-NSF (C) model results in terms of reconstruction accuracy. We set the reconstruction coefficient $\lambda = 10^{-3}$, and we set $p = 10^{-3}$ for the geometric distribution $p_k$. Reconstruction and likelihood results are shown in \[fig:baseline\_vs\_nd\].
We note that the reconstruction curve is drastically improved, in particular for ${<}\,250$ dimension projections. However, we also note that the test log-likelihood of the model suffers, being reduced by $\approx\,$4% as a result of applying nested dropout. We hypothesize that this trade-off is controlled by the values of $\lambda$ and $p$, which we explore further in \[sec:hyperparams\].
We show samples in the ambient space and reconstructions for a baseline flow and a nested dropout flow in \[fig:samples\_reconstructions\]. Even though the likelihoods of the two models differ, their samples are of comparable perceptual quality. It has been noted that the relationship between the perceptual quality of samples and likelihood is complex [@theis2015]. The reconstructions demonstrate that nested dropout indeed guides the flow towards ordering the representations.
For the nested dropout flow, \[fig:manifold\_samples\] shows samples from the flow varying the number of dropped dimensions.[^1] Using 2 dimensions gives a generic, blurry item, which becomes sharper and more detailed given more latent features.
![Samples from RQ-NSF (C) trained with nested dropout. Each row shows samples with a different number of latent dimensions retained: $2,4,8,16,32,64$ for each row in order.[]{data-label="fig:manifold_samples"}](fig/manifold_samples){width="0.99\columnwidth"}
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### Hyper-parameters {#sec:hyperparams}
We perform a limited grid-search for the $\lambda$ and $p$ hyper-parameters. We perturb each hyper-parameter independently, starting with the baseline values used in the previous section. Results are shown in \[fig:hyperparams\].
-0.07in
![Mean squared error of Fashion-MNIST reconstructions for RQ-NSF (C) flow trained with nested dropout against the number of retained dimensions, varying hyper-parameters $\lambda$ and $p$. The dotted line for the baseline nested dropout model. *bpd* is the negative test log-likelihood in bits per dimension (lower is better).[]{data-label="fig:hyperparams"}](fig/hyperparams){width="\columnwidth"}
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The value of $p$ has a limited effect on the reconstruction curve. A lower value (which causes more dimensions to be dropped on average during the run) marginally improves the MSE numbers for $<200$ latent dimensions.
The effect of changing $\lambda$ is more pronounced. Larger values cause a noticeable improvement in reconstruction results, while lower values have the opposite effect.
All 4 models improve upon the baseline in terms of test log-likelihood, which is surprising. There could be an interaction between $\lambda$ and $p$ hyper-parameters, or large variation across runs. Unfortunately, all but one result, including the one with the best reconstruction results, have slightly lower likelihoods than the standard flow.
Discussion
==========
Nested dropout is a simple way to encourage a flow to represent data as closely as possible in the top, *principal* elements of its latent representation. Given the large redundancy in the way flows parameterize distributions, we can choose flows with similar likelihoods that provide much better low dimensional representations.
The small reduction in likelihood suggests that existing flows have some inductive bias *against* interpretable latent spaces. We also can’t always apply nested dropout, because the flow must be evaluated in both directions on every iteration. For architectures with one-pass sampling, training is typically slower by a factor of two. Flows without a cheap inverse, auto-regressive flows, would be impractical.
If data really lies on a low-dimensional manifold, densities in data space are not well defined, and other work is more appropriate [@kumar2020_injective_flow; @brehmer2020_mfmf]. However, PCA is often applied when data aren’t really restricted to a subspace of fixed dimensionality. We hope that nested dropout flows will be useful in similar circumstances.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported in part by the EPSRC Centre for Doctoral Training in Data Science, funded by the UK Engineering and Physical Sciences Research Council (grant EP/L016427/1) and the University of Edinburgh.
[^1]: As noted by @brehmer2020_mfmf, sampling in this fashion does not correspond to sampling from densities on corresponding manifolds.
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'J.-L. Waldspurger'
date: 13 septembre 2012
title: La formule des traces locale tordue
---
[**Introduction**]{}
On se propose de généraliser au cas tordu les résultats d’Arthur contenus dans les articles \[A1\] et \[A7\]. Soient $F$ un corps local, $G$ un groupe réductif connexe défini sur $F$ et $\tilde{G}$ un espace tordu sur $G$, au sens de Labesse (cf. 2.1). Nous imposons une condition à $\tilde{G}$ (2.1(2)) qui revient à dire qu’il existe un groupe algébrique non connexe $G^+$ défini sur $F$, de composante neutre $G$, tel que $\tilde{G}$ soit une composante connexe de $G^+$. Mais la structure de groupe sur $G^+$ ne joue aucun rôle, seules importent les actions à droite et à gauche de $G$ sur $\tilde{G}$. Notons $Z_{G}$ le centre de $G$ et $Z_{G}(F)^{\theta}$ le sous-groupe des $z\in Z_{G}(F)$ tels que $z\gamma=\gamma z$ pour tout $\gamma\in \tilde{G}$. On fixe un caractère unitaire $\omega$ de $G(F)$ dont la restriction à $Z_{G}(F)^{\theta}$ est triviale. On s’intéresse aux “distributions” $\omega$-équivariantes sur $\tilde{G}(F)$. Ce sont des formes linéaires $l:C_{c}^{\infty}(\tilde{G}(F))\to {\mathbb C}$ telles que, pour tout $f\in C_{c}^{\infty}(\tilde{G}(F))$ et tout $g\in G(F)$, on ait l’égalité $l(^gf)=\omega(g)^{-1}l(f)$, où $^gf$ est la fonction $^gf(\gamma)=f(g^{-1}\gamma g)$. Il y a deux types basiques de telles distributions. D’abord les intégrales orbitales. On fixe $\gamma\in \tilde{G}(F)$, disons fortement régulier. On note $Z_{G}(\gamma)$ son commutant dans $G$ et on munit le quotient $Z_{G}(\gamma,F)\backslash G(F)$ d’une mesure invariante à droite. Pour $f\in C_{c}^{\infty}(\tilde{G}(F))$, l’intégrale orbitale de $f$ au point $\gamma$ est $$I_{\tilde{G}}(\gamma,\omega,f)=D^{\tilde{G}}(\gamma)^{1/2}\int_{Z_{G}(\gamma,F)\backslash G(F)}\omega(x)f(x^{-1} \gamma x)\,dx.$$ La fonction $D^{\tilde{G}}(\gamma)$ est la variante tordue de la fonction habituelle. Il y a aussi les caractères de représentations. Soit $\pi$ une représentation admissible et irréductible de $G(F)$. Pour $\gamma\in \tilde{G}(F)$, notons $ad_{\gamma}$ l’automorphisme de $G$ tel que $\gamma g=ad_{\gamma}(g)\gamma$ pour tout $g\in G$. La classe d’équivalence de la représentation $\pi\circ ad_{\gamma}$ ne dépend pas de $\gamma$. Supposons que $\pi\circ ad_{\gamma}$ soit isomorphe à $\omega\otimes \pi$. On peut alors prolonger $\pi$ en une “$\omega$-représentation” $\tilde{\pi}$ de $\tilde{G}(F)$, c’est-à-dire une application $\tilde{\pi}$ de $\tilde{G}(F)$ dans le groupe des automorphismes de l’espace de $\pi$ qui vérifie la condition $\tilde{\pi}(g \gamma g')=\pi(g)\tilde{\pi}(\gamma)\pi(g')\omega(g')$ pour tous $\gamma\in \tilde{G}(F)$ et $g,g'\in G(F)$. Pour $f\in \tilde{G}(F)$, on définit l’opérateur $\tilde{\pi}(f)=\int_{\tilde{G}(F)}f(\gamma)\tilde{\pi}(\gamma)\,d\gamma$ (une mesure de Haar étant fixée sur $G(F)$ et transportée à $\tilde{G}(F)$), puis le caractère $$I_{\tilde{G}}(\tilde{\pi},f)=trace(\tilde{\pi}(f)).$$
La formule des traces locale tordue établit une égalité entre deux expressions, l’une contenant des intégrales orbitales, l’autre des caractères de $\omega$-représentations tempérées. Précisons un tout petit peu. Soient $f_{1},f_{2}\in C_{c}^{\infty}(\tilde{G}(F))$. On définit une expression $$J^{\tilde{G}}_{g\acute{e}om}(\omega,f_{1},f_{2})=\sum_{\tilde{M}\in {\cal P}(\tilde{M}_{0})}\vert \tilde{W}^M\vert \vert \tilde{W}^G\vert ^{-1}J_{\tilde{M},g\acute{e}om}^{\tilde{G}}(\omega,f_{1},f_{2}).$$ L’ensemble ${\cal P}(\tilde{M}_{0})$ est celui des “ensembles de Levi” $\tilde{M}$ de $\tilde{G}$ contenant un ensemble de Levi minimal fixé $\tilde{M}_{0}$. Si $\tilde{M}=\tilde{G}$, $J_{\tilde{G},g\acute{e}om}^{\tilde{G}}(\omega,f_{1},f_{2})$ est une certaine intégrale d’expressions $$\overline{I_{\tilde{G}}(\gamma,\omega,f_{1})}I_{\tilde{G}}(\gamma,\omega,f_{2})$$ en des points $\gamma\in \tilde{G}(F)$ qui sont fortement réguliers et elliptiques. Si $\tilde{M}\not=\tilde{G}$, l’expression est plus compliquée: elle fait intervenir des intégrales orbitales pondérées, qui généralisent les intégrales orbitales définies ci-dessus, mais ne sont plus $\omega$-équivariantes. On définit aussi une expression $$J^{\tilde{G}}_{spec}(\omega,f_{1},f_{2})=\sum_{\tilde{M}\in {\cal P}(\tilde{M}_{0})}\vert \tilde{W}^M\vert \vert \tilde{W}^G\vert ^{-1}J_{\tilde{M},spec}^{\tilde{G}}(\omega,f_{1},f_{2}).$$ Dans le cas où $\tilde{M}=\tilde{G}$ et où $Z_{G}(F)^{\theta}$ est compact, le terme $J_{\tilde{G},spec}^{\tilde{G}}(\omega,f_{1},f_{2})$ est une somme de produits $$\overline{I_{\tilde{G}}(\tilde{\pi},f_{1})}I_{\tilde{G}}(\tilde{\pi},f_{2}),$$ où $\tilde{\pi}$ décrit un certain ensemble de $\omega$-représentations de $\tilde{G}(F)$. Même dans ce cas simple, la définition de ces caractères doit être un peu généralisée, car les représentations sous-jacentes aux $\tilde{\pi}$ ne sont pas irréductibles mais seulement de longueur finie. Dans le cas où $Z_{G}(F)^{\theta}$ n’est plus compact, il faut intégrer les produits ci-dessus selon des paramètres réminiscents de l’existence de ce centre. Dans le cas où $\tilde{M}\not=\tilde{G}$, l’expression fait intervenir des généralisations des caractères, à savoir les caractères pondérés qui, eux non plus, ne sont pas $\omega$-équivariants.
La formule des traces locale tordue (théorème 5.1) affirme l’égalité $$J^{\tilde{G}}_{g\acute{e}om}(\omega,f_{1},f_{2})=J^{\tilde{G}}_{spec}(\omega,f_{1},f_{2}).$$
La première conséquence en est le “théorème 0” de Kazhdan: pour $f\in C_{c}^{\infty}(\tilde{G}(F))$, si $f$ vérifie $I_{\tilde{G}}(\tilde{\pi},f)=0$ pour toute $\omega$-représentation tempérée $\tilde{\pi}$ de $\tilde{G}(F)$, alors $I_{\tilde{G}}(\gamma,\omega,f)=0$ pour tout $\gamma$ fortement régulier (théorème 5.5).
Pour aller plus loin, on doit transformer la formule précédente en une formule invariante, comme dans \[A7\]. Pour cela, on doit utiliser le théorème de Paley-Wiener tordu. Celui-ci a été démontré par Rogawski (\[R\]) dans le cas où $F$ est non-archimédien et $\omega=1$. Il est démontré toujours dans le cas non-archimédien, mais pour tout $\omega$, dans le travail en cours de Henniart et Lemaire (\[HL\]). Dans le cas où $F$ est archimédien, il est démontré par Delorme et Mezo pour $\omega=1$ (\[DM\]). Nous montrons en 6.3 que leur résultat s’étend aisément au cas $\omega$ quelconque. A partir de ce point, les fonctions $f_{1}$ et $f_{2}$ sont supposées $K$-finies quand $F$ est archimédien. En utilisant le théorème de Paley-Wiener tordu, on transforme la formule en l’égalité suivante (théorème 6.6): $$I^{\tilde{G}}_{g\acute{e}om}(\omega,f_{1},f_{2})=I^{\tilde{G}}_{disc}(\omega,f_{1},f_{2}).$$ Le terme de gauche est de la forme $$\sum_{\tilde{M}\in {\cal P}(\tilde{M}_{0})}\vert \tilde{W}^M\vert \vert \tilde{W}^G\vert ^{-1}I_{\tilde{M},g\acute{e}om}^{\tilde{G}}(\omega,f_{1},f_{2}).$$ Il ne contient que des distributions $\omega$-équivariantes (intégrales orbitales pondérées invariantes). Mais le terme principal est le même que précédemment: $$I_{\tilde{G},g\acute{e}om}^{\tilde{G}}(\omega,f_{1},f_{2})=J_{\tilde{G},g\acute{e}om}^{\tilde{G}}(\omega,f_{1},f_{2}).$$ Du côté spectral, on a simplement $$I^{\tilde{G}}_{disc}(\omega,f_{1},f_{2})=J^{\tilde{G}}_{\tilde{G},spec}(\omega,f_{1},f_{2}).$$ Il ne contient que d’honnêtes caractères de $\omega$-représentations tempérées.
Supposons pour simplifier que $Z_{G}(F)^{\theta}$ soit compact. L’ensemble des $\omega$-représentations qui interviennent ici contient le sous-ensemble des représentations elliptiques au sens d’Arthur. Notons-le ici $E_{ell}(\tilde{G},\omega)$ (cette notation changera de sens dans le corps de l’article). On sait que le caractère d’une $\omega$-représentation $\tilde{\pi}$ de longueur finie est localement intégrable, donc donné par une fonction $\gamma\mapsto \Theta(\tilde{\pi},\gamma)$. D’après le théorème de Paley-Wiener, on peut associer à tout élément $\tilde{\pi}\in E_{ell}(\tilde{G},\omega)$ un “pseudo-coefficient” $f_{\tilde{\pi}}$. La formule ci-dessus permet d’exprimer $\Theta(\tilde{\pi},\gamma)$ au moyen des intégrales orbitales pondérées invariantes de $f_{\tilde{\pi}}$ (théorème 7.2). Notons $\tilde{G}(F)_{ell}$ l’ensemble des éléments fortement réguliers et elliptiques de $\tilde{G}(F)$. On peut munir l’espace des fonctions $\omega$-équivariantes sur $\tilde{G}(F)_{ell}$ d’un produit hermitien raisonnable. Notons $I_{cusp}(\tilde{G}(F),\omega)$ l’espace des fonctions sur $\tilde{G}(F)_{ell}$ de la forme $\gamma\mapsto I_{\tilde{G}}(\gamma,\omega,f)$, où $f$ est une fonction cuspidale sur $\tilde{G}(F)$ (c’est-à-dire que $I_{\tilde{G}}(\gamma,\omega,f)=0$ si $\gamma$ est fortement régulier et non elliptique). On montre que la famille des restrictions à $\tilde{G}(F)_{ell}$ des caractères $\Theta(\tilde{\pi},\gamma)$ forme une base orthonormale de $I_{cusp}(\tilde{G}(F),\omega)$ et on calcule la norme de chaque élément de base (théorème 7.3).
Une bonne partie de l’article n’est qu’un décalque des travaux d’Arthur. On s’est autorisé à passer rapidement sur les points dont les démonstrations sont similaires à celles du cas non tordu. Le point qui mérite attention est l’établissement de la partie spectrale de version non invariante de la formule des traces locale tordue (paragraphe 3). Comme dans le cas de la formule des traces d’Arthur-Selberg, l’utilisation dans le cas tordu de la combinatoire du cas non tordu ne conduit à rien de fructueux. On doit profondément modifier cette combinatoire. Heureusement pour nous, la bonne combinatoire a été mise au point par les rédacteurs du Morning Seminar, principalement dans la lecture 15 de Langlands. Ces notes ont été rédigées récemment (cf. \[LW\]). On utilise ici exactement la même méthode, dont la mise en oeuvre est évidemment beaucoup plus simple que dans le cas global.
Généralités
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Notations
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Soit $F$ un corps local de caractéristique nulle. On note $\vert .\vert _{F}$ sa valeur absolue usuelle. Soit $G$ un groupe réductif connexe défini sur $F$. On note $Z_{G}$ le centre de $G$ et $A_{G}$ le plus grand tore contenu dans $Z_{G}$ et déployé sur $F$. On pose ${\cal A}_{G}=X_{*}(A_{G})\otimes_{{\mathbb Z}}{\mathbb R}$, où $X_{*}(A_{G})$ désigne selon l’usage le groupe des sous-groupes à un paramètre de $A_{G}$. On note $a_{G}$ la dimension de ${\cal A}_{G}$. Notons $X_{F}^*(G)$ le groupe des caractères algébriques définis sur $F$ de $G$. Par restriction à $A_{G}$, il s’envoie dans le groupe $X^*(A_{G})$ des caractères de $A_{G}$, donc dans le dual ${\cal A}_{G}^*$ de ${\cal A}_{G}$. De façon générale, on note $<.,.>$ l’accouplement naturel entre un espace vectoriel et son dual. On définit l’homomorphisme $H_{G}:G(F)\to {\cal A}_{G}$ par la condition: $e^{<\chi,H_{G}(g)>}=\vert \chi(g)\vert_{F}$ pour tout $g\in G(F)$ et tout $\chi\in X_{F}^*(G)$. On note ${\cal A}_{G,F}$ l’image de $G(F)$ par cet homomorphisme et ${\cal A}_{A_{G},F}$ l’image de $A_{G}(F)$. Si $F$ est archimédien, ${\cal A}_{G,F}={\cal A}_{A_{G},F}={\cal A}_{G}$. Si $F$ est non archimédien, ${\cal A}_{G,F}$ et ${\cal A}_{A_{G},F}$ sont des réseaux dans ${\cal A}_{G}$. Plus précisément, ce sont des réseaux dans $log(q)X_{*}(A_{G})\otimes_{{\mathbb Z}}{\mathbb Q}$, où $q$ est le nombre d’éléments du corps résiduel de $F$. Pour tout sous-groupe fermé ${\cal L}$ dans ${\cal A}_{G}$, on note ${\cal L}^{\vee}$ le groupe des $\lambda\in {\cal A}_{G}^*$ tels que $<\lambda,H>\in 2\pi {\mathbb Z}$ pour tout $H\in{\cal L}$. Ainsi, via l’exponentielle, $i{\cal A}_{G}^*/i{\cal L}^{\vee}$ s’identifie au groupe dual de ${\cal L}$. On pose ${\cal A}_{G,F}^*={\cal A}_{G}^*/{\cal A}_{G,F}^{\vee}$.
Sauf mention expresse du contraire, tous les sous-groupes algébriques de $G$ que l’on considérera seront supposés définis sur $F$. Un Levi de $G$ est une composante de Levi d’un sous-groupe parabolique de $G$. Une paire parabolique est un couple $(P,M)$ où $P$ est un sous-groupe parabolique et $M$ est une composante de Levi de $P$. L’expression “soit $P=MU_{P}$ un sous-groupe parabolique” signifie que $M$ est une composante de Levi de $P$ et que $U_{P}$ est le radical unipotent de $P$. Plus précisément, si une paire parabolique minimale $(P_{0},M_{0})$ est fixée, l’expression “soit $P=MU_{P}$ un sous-groupe parabolique semi-standard (ou standard)” signifie que $P$ contient $M_{0}$ (ou $P_{0}$), que $M$ est la composante de Levi de $P$ qui contient $M_{0}$ et que $U_{P}$ est le radical unipotent de $P$. On utilise les notations habituelles d’Arthur. Par exemple, pour un Levi $M$, on note ${\cal P}(M)$, resp. ${\cal F}(M)$, l’ensemble des sous-groupes paraboliques $P$ dont $M$ est une composante de Levi, resp. qui contiennent $M$. Pour un sous-groupe parabolique $P=MU_{P}$, on note $\delta_{P}$ le module usuel, qui est une fonction sur $P(F)$.
Fixons une paire parabolique minimale $(P_{0},M_{0})$ et un sous-groupe compact maximal $K$ de $G(F)$. On suppose:
- si $F$ est non-archimédien, $K$ est le fixateur d’un point spécial dans l’appartement attaché à $A_{M_{0}}$ de l’immeuble de $G$;
- si $F$ est archimédien, les algèbres de Lie de $K$ et de $A_{M_{0}}$ sont orthogonales pour la forme de Killing.
On simplifie les notations en notant par un simple indice $0$ les objets relatifs à $M_{0}$ ou $P_{0}$. Par exemple $A_{0}=A_{M_{0}}$, ${\cal A}_{0}={\cal A}_{M_{0}}$, $H_{0}=H_{M_{0}}$, $\delta_{0}=\delta_{P_{0}}$. On note $\Delta_{0}$ l’ensemble des racines simples de $A_{0}$ associé à $P_{0}$. On note ${\cal A}_{0}^{\geq}$ l’ensemble des $H\in {\cal A}_{0}$ tels que $<\alpha,H>\geq 0$ pour tout $\alpha\in \Delta_{0}$. On note $M_{0}(F)^{\geq}$ l’ensemble des $m\in M_{0}(F)$ tels que $H_{0}(m)\in {\cal A}_{0}^{\geq}$. On a l’égalité $G(F)=KM_{0}(F)^{\geq}K$. Plus précisément, pour $g\in G(F)$, si l’on écrit $g=kmk'$, avec $k,k'\in K$ et $m\in M_{0}(F)^{\geq}$, l’élément $H_{0}(m)$ est uniquement déterminé par $g$, on le note $h_{0}(g)$. On note $W^G$ le groupe de Weyl de $G$ relatif au tore $A_{0}$, c’est-à-dire $W^G=Norm_{G(F)}(A_{0})/M_{0}(F)$ (de façon générale, si un groupe $H$ opère sur un ensemble $X$ et si $Y\subset X$, on note $Norm_{H}(Y)$ le sous-groupe des $h\in H$ tels que $h(Y)=Y$). On fixe une forme quadratique $(.,.)$ sur ${\cal A}_{0}$ définie positive et invariante par $W^G$. On définit la norme $\vert H\vert =(H,H)^{1/2}$ pour tout $H\in {\cal A}_{0}$. Par dualité, ${\cal A}_{0}^*$ est aussi muni d’une norme.
Quand on remplace le groupe $G$ par un autre groupe réductif, par exemple un Levi $M$, on ajoute des exposants $M$ dans les notations. Toutefois, si on remplace $G$ par une composante de Levi $M$ d’un sous-groupe parabolique $P$ fixé, il est parfois plus commode d’ajouter un exposant $P$ au lieu de $M$, ou de remplacer un indice $M$ par $P$. Par exemple, on pourra noter ${\cal A}_{P}$ au lieu de ${\cal A}_{M}$. Ou encore, si $P$ est standard, on notera $\Delta_{0}^M$ ou $\Delta_{0}^P$ l’ensemble des racines simples de $A_{0}$ dans $M$ associé au parabolique minimal $P_{0}\cap M$. Soit $M$ un Levi de $G$. Choissons un élément $x\in G(F)$ tel que $M'=x^{-1}Mx$ contienne $M_{0}$. On munit le sous-espace ${\cal A}_{M'}$ de ${\cal A}_{0}$ de la restriction de la forme quadratique fixée plus haut. De la conjugaison $ad_{x}$ se déduit fonctoriellement un isomorphisme noté simplement $H\mapsto xH$ de ${\cal A}_{M'}$ sur ${\cal A}_{M}$, grâce auquel on transporte à ${\cal A}_{M}$ la forme quadratique sur ${\cal A}_{M'}$. La forme obtenue ne dépend pas du choix de $x$.
Si $M'\subset M$ sont deux Levi, l’espace ${\cal A}_{M}$ est un sous-espace de ${\cal A}_{M'}$ dont on note ${\cal A}^M_{M'}$ l’orthogonal. Pour $H\in {\cal A}_{M'}$, on note $H_{M}$ et $H^M$ ses projections sur chacun de ces sous-espaces. Remarquons que
\(1) l’application $H\mapsto H_{M}$ envoie surjectivement ${\cal A}_{M',F}$ sur ${\cal A}_{M,F}$.
Preuve. En conjuguant $M'$, on peut supposer $M_{0}\subset M'$. Si $H=H_{M'}(x)$, pour $x\in M'(F)$, on a $H_{M}=H_{M}(x)$. Inversement, si $H=H_{M}(y)$, pour $y\in M(F)$, on fixe $P'\in {\cal P}^M(M')$ et on écrit $y=xuk$, avec $x\in M'(F)$, $u\in U_{P'}(F)$ et $k\in K\cap M(F)$. Alors $H_{M}(y)=H_{M}(x)$ et $H_{M}=(H_{M'}(x))_{M}$. D’où (1). $\square$
Dualement à (1), on a
\(2) l’injection ${\cal A}_{M}^*\to {\cal A}_{M'}^*$ se quotiente en une injection ${\cal A}_{M,F}^*\to {\cal A}_{M',F}^*$.
Soit $P=MU_{P}$ est un sous-groupe parabolique semi-standard. En utilisant l’égalité $G(F)=P(F)K$, on prolonge la fonction $H_{M}$ en une fonction $H_{P}:G(F)\to {\cal A}_{M}$ par $H_{P}(muk)=H_{M}(m)$ pour tous $m\in M(F)$, $u\in U_{P}(F)$, $k\in K$. Si de plus $P$ est standard, on note $M_{0}(F)^{\geq,M}$ ou $M_{0}(F)^{\geq,P}$ l’ensemble des $m\in M_{0}(F)$ tels que $<\alpha,H_{0}(m)>\geq 0$ pour tout $\alpha\in \Delta_{0}^P$.
Si $X(x_{1},x_{2},...)$ et $Y(x_{1},x_{2},...)$ sont deux expressions réelles positives ou nulles dépendant de variables $x_{1}$, $x_{2}$,..., on dit que $X(x_{1},x_{2},...)$ est essentiellement majorée par $Y(x_{1},x_{2},...)$ et on écrit $X(x_{1},x_{2},...)<<Y(x_{1},x_{2},...)$ s’il existe $c>0$ tel que, pour tous $x_{1},x_{2},...$, on ait l’inégalité $X(x_{1},x_{2},...)\leq cY(x_{1},x_{2},...)$. Cette terminologie est quelque peu imprécise (la question étant en pratique de savoir quelles données sont “variables”) mais évite d’introduire des kyrielles de constantes $c$ superflues.
On note $C_{c}^{\infty}(G(F))$ l’espace des fonctions $f:G(F)\to {\mathbb C}$ qui sont lisses et à support compact. Lisse signifie localement constante si $F$ est non-archimédien, $C^{\infty}$ si $F$ est archimédien.
Mesures
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On munit $G(F)$ d’une mesure de Haar. Pour tout Levi $M$, on munit l’espace ${\cal A}_{M}$ d’une mesure de Haar. On suppose que, si $x\in G(F)$, l’isomorphisme $H\mapsto xH$ de ${\cal A}_{M}$ sur ${\cal A}_{xMx^{-1}}$ identifie les mesures sur ces espaces. L’espace $i{\cal A}_{M}^*$ s’identifie via l’exponentielle au groupe dual de ${\cal A}_{M}$ et on le munit de la mesure duale. Cela entraîne que, si ${\cal L}$ est un réseau de ${\cal A}_{M}$, on a l’égalité $$mes({\cal A}_{M}/{\cal L})mes(i{\cal A}_{M}^*/i{\cal L}^{\vee})=1,$$ les mesures étant les quotients des mesures que l’on vient de fixer par les mesures de comptage sur les réseaux. On pose $A_{M}(F)_{c}=Ker(H_{M})\cap A_{M}(F)$. Dans le cas où $F$ est non-archimédien, on note $mes(i{\cal A}_{M,F}^*)$ la masse totale de $i{\cal A}_{M,F}^*$. Le groupe $ A_{M}(F)_{c}$ est un sous-groupe ouvert compact de $A_{M}(F)$. On munit $A_{M}(F)$ de la mesure de Haar pour laquelle $$mes( A_{M}(F)_{c})=mes({\cal A}_{M}/{\cal A}_{A_{M},F}).$$ Si $F$ est archimédien, on pose $mes(i{\cal A}_{M,F}^*)=1$. On a la suite exacte $$1\to A_{M}(F)_{c}\to A_{M}(F)\to {\cal A}_{M}\to 0.$$ On munit le groupe compact $ A_{M}(F)_{c}$ de la mesure de Haar de masse totale $1$ et le groupe $A_{M}(F)$ de la mesure de Haar compatible avec la suite ci-dessus. En tout cas, soit $f$ une fonction de Schwartz sur $i{\cal A}_{M,F}^*$, définissons une fonction $\hat{f}$ sur ${\cal A}_{M,F}$ par $$\hat{f}(H)=mes(i{\cal A}_{M,F}^*)^{-1}\int_{i{\cal A}_{M,F}^*}f(\Lambda)e^{-<\Lambda,H>}\,d\Lambda.$$ On a la formule d’inversion $$f(\Lambda)=\int_{{\cal A}_{M,F}}\hat{f}(H)e^{<\Lambda,H>}\,dH.$$ Remarquons qu’une intégrale sur ${\cal A}_{M,F}$ est en fait une série si $F$ est non-archimédien.
Pour deux Levi $M'\subset M$, on munit ${\cal A}_{M'}^M$ de la mesure compatible aux mesures sur ${\cal A}_{M}$ et ${\cal A}_{M'}$ et à l’isomorphisme ${\cal A}_{M'}={\cal A}_{M}\oplus {\cal A}_{M'}^M$. On munit $i{\cal A}_{M'}^{M,*}$ de la mesure duale comme ci-dessus. Dans le cas non archimédien, la masse totale de $i{\cal A}_{M'}^{M,*}/\left((i{\cal A}_{M',F}^{\vee}+i{\cal A}_{M}^*)\cap i{\cal A}_{M'}^{M,*}\right)$ est $mes(i{\cal A}_{M',F}^*)mes(i{\cal A}_{M,F}^*)^{-1}$.
Soit $\alpha$ une racine réduite de $A_{0}$ dans $G$. Comme on le sait, il est attaché à $\alpha$ un Levi $M_{\alpha}$ qui est minimal parmi les éléments de ${\cal L}(M_{0})-\{M_{0}\}$ et un sous-groupe parabolique $P_{\alpha}=M_{0}U_{\alpha}$ de $M_{\alpha}$. On munit $U_{\alpha}(F)$ d’une mesure de Haar. On suppose que si $\alpha$ et $\beta$ sont deux telles racines réduites et si $k\in K$ vérifie $kM_{0}k^{-1}=M_{0}$, $kP_{\alpha}k^{-1}=P_{\beta}$, alors la conjugaison par $k$ identifie les mesures sur $U_{\alpha}(F)$ et $U_{\beta}(F)$. Considérons un sous-groupe unipotent $U$ de $G$ qui est produit de tels groupes $U_{\alpha_{i}}$, pour $i=1,...,n$. On munit $U(F)$ de la mesure pour laquelle le produit $$\prod_{i=1,...,n}U_{\alpha_{i}}(F)\to U(F)$$ préserve les mesures. Cela ne dépend pas de l’ordre du produit. L’hypothèse sur $U$ est vérifiée si $U$ est le radical unipotent d’un sous-groupe parabolique semi-standard de $G$, ou si $U$ est le radical unipotent d’un sous-groupe parabolique semi-standard d’un Levi semi-standard de $G$, ou si $U$ est une intersection de tels groupes.
Pour tout Levi $M\in {\cal L}(M_{0})$, on munit $M(F)$ d’une mesure. On impose les conditions suivantes, dont on vérifie aisément qu’elles sont loisibles. D’abord, si $M$ et $M'$ sont deux éléments de ${\cal L}(M_{0})$ et si $x\in G(F)$ vérifie $xMx^{-1}=M'$, on suppose que la conjugaison par $x$ identifie les mesures sur $M(F)$ et $M'(F)$. Ensuite, si $M\subset L$ sont deux éléments de ${\cal L}(M_{0})$ et si $P=MU_{P}\in {\cal P}^L(M)$, on impose que l’on ait l’égalité $$\int_{L(F)}f(l)\,dl\,=\int_{U_{\bar{P}}(F)}\int_{M(F)}\int_{U_{P}(F)}f(\bar{u}mu)\delta_{P}(m)\,du\,dm\,d\bar{u}$$ pour toute $f\in C_{c}^{\infty}(L(F))$, où $\bar{P}=MU_{\bar{P}}$ est le parabolique opposé à $P$.
On munit le groupe $K$ de la mesure de Haar de masse totale $1$. Remarquons que, dans le cas où $F$ est non-archimédien, on n’impose pas que la mesure sur $G(F)$ se restreigne en cette mesure sur $K$. Plus généralement, pour tout Levi semi-standard $M$ de $G$, on pose $K^M=K\cap M(F)$ et on munit ce groupe de la mesure de Haar de masse totale $1$. On vérifie que, pour deux Lévi semi-standard $M\subset L$, il existe un réel $\gamma(L\vert M)>0$ tel que, pour tout $P=MU_{P}\in {\cal P}^L(M)$, on ait l’égalité $$\int_{L(F)}f(l)\,dl\,=\gamma(L\vert M)\int_{M(F)}\int_{U_{P}(F)} \int_{K^L}f(muk)\,dk\,du\,dm$$ pour toute $f\in C_{c}^{\infty}(L(F))$. On a l’égalité $\gamma(G\vert M)=\gamma(G\vert L)\gamma(L\vert M)$.
On introduit la fonction $D_{0}$ sur $M_{0}(F)^{\geq}$, à valeurs réelles positives, telle que l’on ait l’égalité $$\int_{G(F)}f(g)\,dg=\int_{K\times K}\int_{M_{0}(F)^{\geq}}D_{0}(m)f(kmk')\,dm\,dk\,dk'$$ pour toute $f\in C_{c}^{\infty}(G(F))$. Elle vérifie la majoration
\(1) $D_{0}(m)<<\delta_{0}(m)^{1/2}$ pour tout $m\in M_{0}(F)^{\geq}$.
Cf. \[A1\] corollaire 1.2.
[**Remarque.**]{} Nos mesures ne sont pas normalisées comme en \[A1\]: pour $M\in {\cal L}(M_{0})$, Arthur prend pour mesure sur $M(F)$ celle que l’on a définie multipliée par $\gamma(G\vert M)^2$. Cela entraîne que notre fonction $D_{0}$ est égale à celle d’Arthur multipliée par $\gamma(G\vert M_{0})^{2}$.
Définitions combinatoires
-------------------------
Les propriétés énoncées dans ce paragraphe sont aujourd’hui bien connues. La plupart sont dues à Arthur, Langlands et Labesse. On donne pour référence l’article récent \[LW\] qui les a rassemblées.
Soit $P=MU_{P}$ un sous-groupe parabolique semi-standard de $G$. On note $\Delta_{P}$ l’ensemble des racines simples de $A_{M}$ associé à $P$. Il s’agit d’une base de ${\cal A}_{M}^{G,*}$. On note $\{\check{\varpi}_{\alpha}; \alpha\in \Delta_{P}\}$ sa base duale. A toute racine $\alpha\in \Delta_{P}$, on peut associer une coracine $\check{\alpha}\in {\cal A}_{M}^G$. Dans le cas où $P=P_{0}$ et $M=M_{0}$, l’ensemble des racines de $A_{0}$ est un vrai système de racines et on definit les coracines selon l’usage. Dans le cas général, quitte à conjuguer $P$, on peut supposer $P$ standard. Alors $\Delta_{P}$ est l’ensemble des projections non nulles sur ${\cal A}_{M}$ d’éléments de $\Delta_{0}$ et on définit les coracines comme les projections non nulles des coracines associées aux éléments de $\Delta_{0}$. Cela ne dépend pas de la conjugaison. On introduit la base $\{\varpi_{\alpha};\alpha \in \Delta_{P}\}$ de ${\cal A}_{M}^{G,*}$ duale de celle des coracines. Une propriété essentielle est que $(\alpha,\beta)\leq0$ et $(\varpi_{\alpha},\varpi_{\beta})\geq0$ pour tous $\alpha\not=\beta \in \Delta_{P}$.
Soient $P=MU_{P}\subset Q=LU_{Q}$ deux sous-groupes paraboliques semi-standard. On note $\Delta_{P}^Q\subset \Delta_{P}$ l’ensemble des racines simples de $A_{M}$ dans l’algèbre de Lie de $L\cap U_{P}$. On définit les fonctions suivantes sur ${\cal A}_{M}$:
$\tau_{P}^Q$ fonction caractéristique de l’ensemble des $H\in {\cal A}_{M}$ tels que $<\alpha,H>>0$ pour tout $\alpha\in \Delta_{P}^Q$;
$\hat{\tau}_{P}^Q$ fonction caractéristique de l’ensemble des $H\in {\cal A}_{M}$ tels que $<\varpi_{\alpha}^L,H>>0$ pour tout $\alpha\in \Delta_{P}^Q$;
$\phi_{P}^Q$ fonction caractéristique de l’ensemble des $H\in {\cal A}_{M}$ tels que $<\varpi_{\alpha}^L,H>\leq 0$ pour tout $\alpha\in \Delta_{P}^Q$;
$\delta_{M}^Q$ fonction caractéristique du sous-ensemble ${\cal A}_{L}$ de ${\cal A}_{M}$.
Si $P'=M'U_{P'}$ est un sous-groupe parabolique semi-standard contenu dans $P$, ces fonctions peuvent être considérées comme des fonctions sur ${\cal A}_{M'}$, en les identifiant avec leurs composées avec la projection orthogonale de ${\cal A}_{M'}$ sur ${\cal A}_{M}$. On supprimera souvent les indices $P$ quand $P=P_{0}$.
On a $$(1) \qquad \sum_{R; M\subset R\subset Q}\delta_{M}^R(H)\tau_{R}^Q(H)=1.$$ Cela traduit la décomposition de ${\cal A}_{M}$ en chambres positives relatives aux paraboliques $R$ indiqués. On a l’égalité (qui est un lemme de Langlands):
$$(2) \qquad \sum_{R; P\subset R\subset Q}\phi_{P}^R(H)\tau_{R}^Q(H)=1.$$
On définit la fonction $\Gamma_{P}^Q$ sur ${\cal A}_{M}\times {\cal A}_{M}$ par $$\Gamma_{P}^Q(H,X)=\sum_{R; P\subset R\subset Q}(-1)^{a_{R}-a_{Q}}\tau_{P}^R(H)\hat{\tau}_{R}^Q(H-X).$$ On rappelle que $a_{R}$, resp. $a_{Q}$, est la dimension de ${\cal A}_{R}$, resp. ${\cal A}_{Q}$. Sur le support de cette fonction, on a une majoration $\vert H_{M}^L\vert <<\vert X_{M}^L\vert $. On a $$(3) \qquad \sum_{R; P\subset R\subset Q}\Gamma_{R}^Q(H,X)\phi_{P}^R(H)=\phi_{P}^Q(H-X).$$
Preuve. Il résulte de la formule du binôme que la définition de $\phi_{P}^Q(H)$ peut s’écrire $$(4) \qquad \phi_{P}^Q(H)=\sum_{S; P\subset S\subset Q}(-1)^{a_{S}-a_{Q}}\hat{\tau}_{S}^Q(H).$$ On applique cette définition en remplaçant $Q$ par $R$ ainsi que la définition de $\Gamma_{R}^Q(H,X)$. Le membre de gauche de (3) devient $$\sum_{S,R,S'; P\subset S\subset R\subset S'\subset Q}(-1)^{a_{S}-a_{R}+a_{S'}-a_{Q}}\hat{\tau}_{S}^R(H)\tau_{R}^{S'}(H)\hat{\tau}_{S'}^Q(H-X).$$ Pour $S,S'$ fixés, la somme $$\sum_{R; S\subset R\subset S'}(-1)^{a_{S}-a_{R}}\hat{\tau}_{S}^R(H)\tau_{R}^{S'}(H)$$ vaut $1$ si $S=S'$ et $0$ sinon (cf. \[LW\] proposition 1.7.2). L’expression précédente se simplifie en $$\sum_{S; P\subset S\subset Q}(-1)^{a_{S}-a_{Q}}\hat{\tau}_{S}^Q(H-X).$$ En appliquant de nouveau (4), c’est $\phi_{P}^Q(H-X)$. Cela prouve (3). $\square$
On a $$(5)\qquad \sum_{R; P\subset R\subset Q}\Gamma_{P}^R(H,X)\tau_{R}^Q(H-X)=\tau_{P}^Q(H).$$ La preuve est similaire à celle de (3).
Soit $M$ un Levi semi-standard. Considérons une famille ${\cal Y}=(Y[P])_{P\in {\cal P}(M)}$ où, pour tout $P$, $Y[P]$ est un élément de ${\cal A}_{M}$. On dit qu’elle est $(G,M)$-orthogonale si elle vérifie les conditions suivantes. Soient $P,P'\in {\cal P}(M)$ deux éléments adjacents. Soit $\alpha$ l’unique racine réduite de $A_{M}$ qui est positive pour $P$ et négative pour $P'$. Alors $Y[P]-Y[P']$ appartient à la droite portée par $\check{\alpha}$. On dit que ${\cal Y}$ est positive si $Y[P]-Y[P']$ appartient à la demi-droite portée par $\check{\alpha}$. Si $F$ est non-archimédien, on dit que ${\cal Y}$ est $p$-adique si $Y[P]\in {\cal A}_{M,F}\otimes_{{\mathbb Z}}{\mathbb Q}$ pour tout $P\in {\cal P}(M)$. Pour une famille $(G,M)$-orthogonale ${\cal Y}$ et pour $Q=LU_{Q}\in {\cal F}(M)$, on pose $Y[Q]=Y[P]_{L}$, où $P$ est un élément de ${\cal P}(M)$ tel que $P\subset Q$. Cela ne dépend pas du choix de $P$. Si on fixe $L\in {\cal L}(M)$, la famille $(Y[Q])_{Q\in {\cal P}(L)}$ est encore $(G,L)$-orthogonale et est positive si la famille de départ l’est. Dans le cas particulier où $M=M_{0}$, on associe à tout élément $T\in {\cal A}_{0}$ une famille $(G,M_{0})$-orthogonale ${\cal T}=(T[P])_{P\in {\cal P}(M_{0})}$ de la façon suivante. Pour $P\in {\cal P}(M_{0})$, soit $w\in W^G$ tel que $w(P_{0})=P$. On pose $T[P]=wT$. Cette famille est positive si et seulement si $T\in {\cal A}_{0}^{\geq}$.
Pour une famille $(G,M)$-orthogonale ${\cal Y}=(Y[P])_{P\in {\cal P}(M)}$ et pour $Q=LU_{Q}\in {\cal F}(M)$, on définit une fonction $\Gamma_{M}^Q(.,{\cal Y})$ sur ${\cal A}_{M}$ par $$\Gamma_{M}^Q(H,{\cal Y})=\sum_{R\in {\cal F}(M); R\subset Q}\delta_{M}^R(H)\Gamma_{R}^Q(H,Y[R]).$$ Sur le support de cette fonction, on a une majoration $\vert H^L\vert <<sup_{P\in {\cal P}(M); P\subset Q}\vert Y(P)^L\vert $. Dans le cas où la famille est positive, c’est la fonction caractéristique de l’ensemble des $H\in {\cal A}_{M}$ tels que $H^L$ appartient à l’enveloppe convexe de la famille des points $Y[P]^L$ pour $P\in {\cal P}(M)$, $P\subset Q$. On a l’égalité $$(6) \qquad \sum_{Q\in {\cal F}(M)}\Gamma_{M}^Q(H,{\cal Y})\tau_{Q}^G(H-Y[Q])=1.$$ Cf. \[LW\] lemme 1.8.4(3).
Fixons $P_{1}\in {\cal P}(M)$, ce qui définit un ordre sur l’ensemble des racines de $A_{M}$, noté $\alpha>_{P_{1}}0$. Pour $P\in {\cal P}(M)$, notons $\phi_{P,P_{1}}^G$ la fonction caractéristique de l’ensemble des $H\in {\cal A}_{M}$ tels que
- pour $\alpha\in \Delta_{P}$ tel que $ \alpha>_{P_{1}}0$, $<\varpi_{\alpha},H>\leq 0$;
- pour $\alpha\in \Delta_{P}$ tel que $ \alpha<_{P_{1}}0$, $<\varpi_{\alpha},H>> 0$.
Notons $s(P,P_{1})$ le nombre de $\alpha\in \Delta_{P}$ tel que $ \alpha<_{P_{1}}0$. On a l’égalité
$$(7)\qquad \Gamma_{M}^G(H,{\cal Y})=\sum_{P\in {\cal P}(M)}(-1)^{s(P,P_{1})}\phi_{P,P_{1}}^G(H-Y[P]).$$ Cf. \[LW\] proposition 1.8.7(2).
Pour $H\in {\cal A}_{0}$, on note ${\cal C}^G(H)$ l’enveloppe convexe des $wH$ pour $w\in W^G$. Supposons $H\in {\cal A}_{0}^{\geq}$. Alors, pour tout $H'\in {\cal A}_{0}$, $H'$ appartient à ${\cal C}^G(H)$ si et seulement si on a $\phi^G_{w(P_{0})}(H'-wH)=1$ pour tout $w\in W^G$. Pour $H'\in {\cal A}_{0}^{\geq}$, $H'$ appartient à ${\cal C}^G(H)$ si et seulement si $\phi_{P_{0}}^G(H'-H)=1$.
[0.3cm[**[Lemme]{}**]{}. [ *[Pour tout $m\in M_{0}(F)$, tout $k\in K$ et tout $P\in {\cal P}(M_{0})$, $H_{P}(km)$ appartient à ${\cal C}^G(H_{0}(m))$.]{}*]{}0.3cm]{}
Preuve. On utilise les deux ingrédients suivants.
\(8) Pour tous $g\in G(F)$ et tous $P,P'\in {\cal P}(M_{0})$, on a $\phi_{P}^G(H_{P}(g)-H_{P'}(g))=1$.
Autrement dit, la famille $(-H_{P}(g))_{P\in {\cal P}(M_{0})}$ est $(G,M_{0})$-orthogonale et positive, cf. \[A2\] p. 40.
\(9) Supposons $m\in M_{0}(F)^{\geq}$; alors, pour tout $k\in K$, $\phi_{P_{0}}^G(H_{\bar{P}_{0}}(km)-H_{0}(m))=1$.
Ceci est un ingrédient de la théorie de la transformée de Satake, cf. \[HC\] lemme 35 et \[BT\] corollaire 4.3.17.
Venons-en à la preuve du lemme. Conjuguer $m$ par un élément de $Norm_{K}(M_{0})$ ne change pas le problème. On peut donc supposer $m\in M_{0}(F)^{\geq}$. Soit $w\in W^G$. On doit prouver que $\phi^G_{w(P_{0})}(H_{P}(km)-wH_{0}(m))=1$. Ou encore que $\phi^G_{P_{0}}(w^{-1}H_{P}(km)-H_{0}(m))=1$. Or $w^{-1}H_{P}(km)=H_{w^{-1}(P)}(\dot{w}^{-1}km)$, où $\dot{w}$ est un représentant de $w$ dans $K$. Quitte à remplacer $k$ par $\dot{w}k$ et $P$ par $w(P)$, on est ramené à prouver $\phi_{P_{0}}^G(H_{P}(km)-H_{0}(m))=1$ pour tous $k$, $P$. D’après (8), on a $\phi_{\bar{P}_{0}}^G(H_{\bar{P}_{0}}(km)-H_{P}(km))=1$. C’est équivalent à $\phi_{P_{0}}^G(H_{P}(km)-H_{\bar{P}_{0}}(km))=1$. En utilisant (9) et le fait que $\phi_{P_{0}}^G$ est la fonction caractéristique d’un cône (qui est stable par addition), on en déduit la relation $\phi_{P_{0}}^G(H_{P}(km)-H_{0}(m))=1$ cherchée. $\square$
$(G,M)$-familles
----------------
Soit $M$ un Levi de $G$. D’après Arthur, une $(G,M)$-famille est une famille $(\varphi(\Lambda,P))_{P\in {\cal P}(M)}$ vérifiant les conditions suivantes:
- pour tout $P$, $\Lambda\mapsto \varphi(\Lambda,P)$ est une fonction $C^{\infty}$ sur $i{\cal A}_{M}^*$;
- soient $P,P'\in {\cal P}(M)$ deux paraboliques adjacents, soit ${\cal H}\subset {\cal A}_{M}^*$ le mur séparant les chambres positives associées à $P$ et $P'$; alors $\varphi(\Lambda,P)=\varphi(\Lambda,P')$ pour $\Lambda\in i{\cal H}$.
Pour $P\in {\cal P}(M)$, le réseau engendré par les coracines associées aux éléments de $\Delta_{P}$ ne dépend pas de $P$. On le note ${\mathbb Z}(\check{\Delta}_{M})$. On définit la fonction méromorphe $\epsilon_{P}^G$ sur ${\cal A}_{M,{\mathbb C}}^*={\cal A}_{M}^*\otimes_{{\mathbb R}}{\mathbb C}$ par $$\epsilon_{P}^G(\Lambda)=mes({\cal A}_{M}^G/{\mathbb Z}[\check{\Delta}_{M}])\prod_{\alpha\in \Delta_{P}}<\Lambda,\check{\alpha}>^{-1}.$$
[**Remarque.**]{} Arthur note cette fonction $\theta_{P}^G(\Lambda)^{-1}$. On réserve la lettre $\theta$ pour un autre usage.
Pour une $(G,M)$-famille comme ci-dessus, on définit la fonction $\varphi_{M}^G$ sur $i{\cal A}_{M}^*$ par $$\varphi_{M}^G(\Lambda)=\sum_{P\in {\cal P}(M)}\varphi(\Lambda,P)\epsilon_{P}^G(\Lambda).$$ Le point essentiel est qu’il s’agit d’une fonction $C^{\infty}$ (les singularités des fonctions $\epsilon_{P}^G$ disparaissent). Dans le cas où $F$ est archimédien, on a aussi:
[0.3cm[**[Lemme]{}**]{}. [ *[Supposons $F$ archimédien. Si les fonctions $\varphi(\Lambda,P)$ sont de Schwartz, $\varphi_{M}^G$ l’est aussi. Si les fonctions $\varphi(\Lambda,P)$ ainsi que leurs dérivées sont à croissance modérée, $\varphi_{M}^G$ et ses dérivées le sont aussi.]{}*]{}0.3cm]{}
Cela résulte par exemple du lemme 13.2.2 de \[LW\].
Variantes des fonctions $\epsilon_{P}^G$
-----------------------------------------
Soient $M$ un Levi semi-standard et $Y\in {\cal A}_{M}$. Si $F$ est non-archimédien, on suppose plus précisément que $Y\in {\cal A}_{M,F}\otimes_{{\mathbb Z}}{\mathbb Q}$. Soit $X\in {\cal A}_{G,F}$. On pose $${\cal A}_{M,F}^G(X)=\{H\in {\cal A}_{M,F}; H_{G}=X\}.$$ Si $F$ est non-archimédien, on munit cet ensemble de la mesure de comptage. Si $F$ est archimédien, cet ensemble est égal à $X+{\cal A}_{M}^G$ et on le munit de la mesure déduite de celle sur ${\cal A}_{M}^G$.
Pour $P\in {\cal P}(M)$ et $\Lambda\in {\cal A}_{M,{\mathbb C}}^*$, considérons l’intégrale $$\epsilon_{P}^{G,Y}(X;\Lambda)=\int_{{\cal A}_{M,F}^G(X)}\phi_{P}^G(H-Y)e^{<\Lambda,H>}\,dH.$$ Elle est absolument convergente si $<\Lambda,\check{\alpha}>>0$ pour tout $\alpha\in \Delta_{P}$. Dans ce domaine, elle ne dépend que de la projection de $\Lambda$ dans ${\cal A}_{M,{\mathbb C}}^{*}/i{\cal A}_{M,F}^{\vee}$. Si $F$ est archimédien, on calcule $$\epsilon_{P}^{G,Y}(X;\Lambda)=e^{<\Lambda,X+Y^G>}\epsilon_{P}^G(\Lambda).$$ Supposons $F$ non archimédien. On peut fixer $X'\in {\cal A}_{M,F}$ tel que $X'_{G}=X$ (cf. 1.1(1)). Notons ${\cal A}_{M,F}^G={\cal A}_{M,F}^G(0)={\cal A}_{M,F}\cap {\cal A}_{M}^G$. Alors ${\cal A}_{M,F}^G(X)=X'+{\cal A}_{M,F}^G$. Par le changement de variables $H\mapsto H+X'$, $$\epsilon_{P}^{G,Y}(X;\Lambda)=e^{<\Lambda,X'>}\epsilon_{P}^{G,Y-X'}(0;\Lambda).$$ Fixons un entier $k\geq1$, posons ${\cal L}_{k}=\frac{1}{k}log(q){\mathbb Z}[\check{\Delta}_{M}]$ où $q$ est le nombre d’éléments du corps résiduel de $F$. Si $k$ est assez grand, ce réseau contient ${\cal A}_{M,F}^G$ et $Y^G-(X')^G$. On fixe $k$ de sorte qu’il en soit ainsi. Par inversion de Fourier, on a $$\epsilon_{P}^{G,Y}(X;\Lambda)= [{\cal L}_{k}:{\cal A}_{M,F}^G]^{-1}e^{<\Lambda,X'>}\sum_{\nu\in i{\cal A}_{M,F}^{G,\vee}/i{\cal L}_{k}^{\vee}}\sum_{H\in {\cal L}_{k}}\phi_{P}^G(H+X'-Y)e^{<\Lambda+\nu,H>}$$ $$=[{\cal L}_{k}:{\cal A}_{M,F}^G]^{-1}e^{<\Lambda,X'>}\sum_{\nu\in i{\cal A}_{M,F}^{G,\vee}/i{\cal L}_{k}^{\vee}}\sum_{H\in {\cal L}_{k}}\phi_{P}^G(H)e^{<\Lambda+\nu,H+Y^G-(X')^G>}.$$ Les ensembles ${\cal A}_{M,F}^{G,\vee}$ et ${\cal L}_{k}^{\vee}$ sont des réseaux dans ${\cal A}_{M}^{G,*}$. L’ensemble des $H\in {\cal L}_{k}$ tels que $\phi_{P}^G(H)=1$ est celui des $\sum_{\alpha\in \Delta_{P}}\kappa n_{\alpha}\check{\alpha}$ pour des entiers $n_{\alpha}\leq0$, où $\kappa=\frac{1}{k}log(q)$. La série $$\sum_{H\in {\cal L}_{k}}\phi_{P}^G(H)e^{<\Lambda,H>}$$ se calcule. Elle vaut $\epsilon_{P,k}^G(\Lambda)$, où $$\epsilon^G_{P,k}(\Lambda)=\prod_{\alpha\in \Delta_{P}}(1-e^{-<\Lambda,\kappa\check{\alpha}>})^{-1}.$$ On obtient $$(1) \qquad \epsilon_{P}^{G,Y}(X;\Lambda)=[{\cal L}_{k}:{\cal A}_{M,F}^G]^{-1}e^{<\Lambda,X'>}\sum_{\nu\in i{\cal A}_{M,F}^{G,\vee}/i{\cal L}_{k}^{\vee}}e^{<\Lambda+\nu,Y^G-(X')^G>} \epsilon^G_{P,k}(\Lambda+\nu).$$ Cette expression se prolonge méromorphiquement à tout $\Lambda\in {\cal A}_{M,{\mathbb C}}^*$.
Fixons $P_{1}\in {\cal P}(M)$. Pour $P\in {\cal P}(M)$ et $\Lambda\in {\cal A}_{M,{\mathbb C}}^*$, posons $$\epsilon_{P,P_{1}}^{G,Y}(X;\Lambda)=\int_{{\cal A}_{M,F}^G(X)}\phi_{P,P_{1}}^G(H-Y)e^{<\Lambda,H>}\,dH.$$ Elle est absolument convergente si $<\Lambda,\check{\alpha}>>0$ pour tout $\alpha\in \Delta_{P_{1}}$. Montrons que
\(2) cette fonction se prolonge méromorphiquement à tout $\Lambda\in {\cal A}_{M,{\mathbb C}}^*$; on a l’égalité $\epsilon_{P,P_{1}}^{G,Y}(X;\Lambda)=(-1)^{s(P,P_{1})}\epsilon_{P}^{G,Y}(X;\Lambda)$.
On traite le cas où $F$ est non-archimédien, le cas archimédien étant plus facile. Le même calcul qui a conduit à l’égalité (1) conduit à une égalité similaire exprimant $\epsilon_{P,P_{1}}^{G,Y}(X;\Lambda)$. La fonction $\epsilon_{P,k}^G(\Lambda)$ y est remplacée par une fonction $\epsilon_{P,P_{1},k}^G(\Lambda)$. Supposons $<\Lambda,\check{\alpha}>>0$ pour tout $\alpha\in \Delta_{P_{1}}$. Alors cette fonction est définie par $$\epsilon_{P,P_{1},k}^G(\Lambda)=\sum_{H\in {\cal L}_{k}}\phi_{P,P_{1}}^G(H)e^{<\Lambda,H>}.$$ L’ensemble des $H\in {\cal L}_{k}$ tels que $\phi_{P,P_{1}}^G(H)=1$ est celui des $\sum_{\alpha\in \Delta_{P}}\kappa n_{\alpha}\check{\alpha}$, où $\kappa$ est comme ci-dessus et les entiers $n_{\alpha}$ vérifient
- $n_{\alpha}\leq0$ si $<\Lambda,\check{\alpha}>>0$;
- $n_{\alpha}>0$ si $<\Lambda,\check{\alpha}><0$.
Pour un réel $t\not=0$, on a les égalités élémentaires
$\sum_{n>0}e^{nt}=-(1-e^{-t})^{-1}$, si $t<0$;
$\sum_{n\leq0}e^{nt}=(1-e^{-t})^{-1}$, si $t>0$.
On calcule alors $\epsilon_{P,P_{1},k}^G(\Lambda)=(-1)^{s(P,P_{1})}\epsilon_{P,k}^G(\Lambda)$. Cette égalité se prolonge à tout $\Lambda$. Cela entraîne (2).
Variantes des fonctions $\varphi_{M}^G$
---------------------------------------
Soient $M$ un Levi semi-standard, $X\in {\cal A}_{G,F}$, ${\cal Y}=(Y[P])_{P\in {\cal P}(M)}$ une famille $(G,M)$-orthogonale et $(\varphi(\Lambda,P))_{P\in {\cal P}(M)}$ une $(G,M)$-famille.
Supposons d’abord $F$ archimédien. Définissons la fonction $$(1) \qquad \varphi_{M}^{G,{\cal Y}}(X;\Lambda)=\sum_{P\in {\cal P}(M)}\varphi(\Lambda,P)\epsilon_{P}^{G,Y[P]}(X;\Lambda).$$ Si l’on pose $\varphi({\cal Y};\Lambda,P)=\varphi(\Lambda,P)e^{<\Lambda,Y[P]^G>}$, la famille $(\varphi({\cal Y};\Lambda,P))_{P\in {\cal P}(M)}$ est une $(G,M)$-famille et on a l’égalité $\varphi_{M}^{G,{\cal Y}}(X;\Lambda)=e^{<\Lambda,X>}\varphi_{M}^G({\cal Y};\Lambda)$. Cette fonction est donc $C^{\infty}$.
Supposons maintenant $F$ non-archimédien. On suppose que la famille ${\cal Y}$ est $p$-adique, cf. 1.3. On suppose aussi que la famille $(\varphi(\Lambda,P))_{P\in {\cal P}(M)}$ est $p$-adique, notion que l’on définit de la façon suivante: on suppose que
- pour tout $P$, la fonction $\Lambda\mapsto \varphi(\Lambda,P)$ est invariante par $ i{\cal A}_{M,F}^{\vee}$, autrement dit se descend en une fonction sur $i{\cal A}_{M,F}^*$.
On définit la fonction $\varphi_{M}^{G,{\cal Y}}(X;\Lambda)$ par l’égalité (1).
[0.3cm[**[Lemme]{}**]{}. [ *[On suppose que $F$ est non-archimédien, que ${\cal Y}$ est une famille $(G,M)$-orthogonale $p$-adique et que $(\varphi(\Lambda,P))_{P\in {\cal P}(M)}$ est une $(G,M)$-famille $p$-adique. Alors la fonction $\varphi_{M}^{G,{\cal Y}}(X;\Lambda)$ est $C^{\infty}$ sur $i{\cal A}_{M}^*$ et invariante par $i{\cal A}_{M,F}^{\vee}$.]{}*]{}0.3cm]{}
Preuve. L’invariance est évidente puisque toutes nos fonctions le sont. Reprenons le réseau ${\cal L}_{k}$ de 1.5. On utilise le lemme 10.2 de \[A1\]. Il affirme l’existence d’une $(G,M)$-famille $(u_{k}(\Lambda,P))_{P\in {\cal P}(M)}$, où $\Lambda\mapsto u_{k}(\Lambda,P)$ appartient à $ C_{c}^{\infty}(i{\cal A}_{M}^*/i{\cal A}_{G}^*)$, vérifiant les propriétés suivantes:
- pour tout $P$ et tout $\Lambda$, $\sum_{\mu\in i{\cal L}_{k}^{\vee}}u_{k}(\Lambda+\mu,P)=1$;
- pour tout $P$, la fonction $v_{k}(\Lambda,P)=u_{k}(\Lambda,P)\epsilon_{P,k}^G(\Lambda)\epsilon_{P}^G(\Lambda)^{-1}$ est lisse sur $i{\cal A}_{M}^*/i{\cal A}_{G}^*$.
On peut glisser la somme $$\sum_{\mu\in i{\cal L}_{k}^{\vee}}u_{k}(\Lambda+\nu+\mu,P)$$ dans la formule 1.5(1). En regroupant les deux sommes de la formule obtenue, on obtient $$\epsilon_{P}^{G,Y}(X;\Lambda)= [{\cal L}_{k}:{\cal A}_{M,F}^G]^{-1}e^{<\Lambda,X'>}\sum_{\nu\in i{\cal A}_{M,F}^{G,\vee}}e^{<\Lambda+\nu,Y^G-(X')^G>}\epsilon_{P,k}^G(\Lambda+\nu)u_{k}(\Lambda+\nu,P)$$ $$= [{\cal L}_{k}:{\cal A}_{M,F}^G]^{-1}e^{<\Lambda,X'>}\sum_{\nu\in i{\cal A}_{M,F}^{G,\vee}}e^{<\Lambda+\nu,Y^G-(X')^G>}v_{k}(\Lambda+\nu,P)\epsilon_{P}^G(\Lambda+\nu).$$ Notons que la somme est localement finie d’après la compacité du support de $u_{k}(\Lambda,P)$. On en déduit $$(2) \qquad \varphi_{M}^{G,{\cal Y}}(X;\Lambda)=[{\cal L}_{k}:{\cal A}_{M,F}^G]^{-1}e^{<\Lambda,X>}\sum_{\nu\in i{\cal A}_{M,F}^{G,\vee}}e^{-<\nu,(X')^G>}$$ $$\sum_{P\in {\cal P}(M)}\varphi(\Lambda,P)e^{<\Lambda+\nu,Y[P]^G>}v_{k}(\Lambda+\nu,P)\epsilon_{P}^G(\Lambda+\nu).$$ Posons $$\varphi(\nu,{\cal Y};\Lambda,P)=\varphi(\Lambda-\nu,P)e^{<\Lambda,Y[P]^G>}v_{k}(\Lambda,P).$$ Montrons que
\(3) $(\varphi(\nu,{\cal Y};\Lambda,P))_{P\in {\cal P}(M)}$ est une $(G,M)$-famille.
La famille $(e^{<\Lambda,Y[P]^G>})_{P\in {\cal P}(M)}$ est une $(G,M)$-famillle. La famille $(v_{k}(\Lambda,P))_{P\in {\cal P}(M)}$ aussi: cela résulte aisément du fait que $(u_{k}(\Lambda,P))_{P\in {\cal P}(M)}$ en est une. Il reste à montrer que $(\varphi(\Lambda-\nu,P))_{P\in {\cal P}(M)}$ en est une. Or $i{\cal A}_{M,F}^{G,\vee}=(i{\cal A}_{M,F}^{\vee}+i{\cal A}_{G}^*)\cap i{\cal A}_{M}^{G,*}$. Il suffit de montrer que $(\varphi(\Lambda-\nu,P))_{P\in {\cal P}(M)}$ est une $(G,M)$-famille pour $\nu\in i{\cal A}_{M,F}^{\vee}+i{\cal A}_{G}^*$. Par hypothèse, nos fonctions sont invariantes par $i{\cal A}_{M,F}^{\vee}$. On peut donc se limiter à $\nu\in i{\cal A}_{G}^*$. Mais il est clair que les conditions définissant une $(G,M)$-famille sont invariantes par translation par $i{\cal A}_{G}^*$. D’où (3).
Alors (2) se récrit $$\varphi_{M}^{G,{\cal Y}}(X;\Lambda)=[{\cal L}_{k}:{\cal A}_{M,F}^G]^{-1}e^{<\Lambda,X>}\sum_{\nu\in i{\cal A}_{M,F}^{G,\vee}}e^{-<\nu,(X')^G>}\varphi_{M}^G(\nu,{\cal Y};\Lambda+\nu)$$ et ces fonctions sont $C^{\infty}$ d’après les propriétés des $(G,M)$-familles habituelles. $\square$
[**Remarque**]{} Pour $Z\in {\cal A}_{A_{G},F}$, on a l’égalité $$\varphi_{M}^{G,{\cal Y}}(X+Z;\Lambda)=e^{<\Lambda,Z>}\varphi_{M}^{G,{\cal Y}}(X;\Lambda).$$ En particulier, si l’on restreint la variable $\Lambda$ à l’ensemble $(i{\cal A}_{M}^{G,*}+i{\cal A}_{M,F}^{\vee})/i{\cal A}_{M,F}^{\vee}$, la fonction $X\mapsto \varphi_{M}^{G,{\cal Y}}(X;\Lambda).$ devient invariante par ${\cal A}_{A_{G},F}$.
Développement en fonction d’un paramètre $T$
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Si $F$ est archimédien, notons $PolExp$ l’ensemble des fonctions $f:{\cal A}_{0}\to {\mathbb C}$ pour lesquelles il existe une famille $(p_{\Lambda})_{\Lambda\in i{\cal A}_{0}^*}$ de polynômes sur ${\cal A}_{0}$ de sorte que
- l’ensemble des $\Lambda$ tels que $p_{\Lambda}\not=0$ est fini;
- pour tout $T$, on a l’égalité $$f(T)=\sum_{\Lambda\in i{\cal A}_{0}^*}e^{<\Lambda,T>}p_{\Lambda}(T).$$
La famille $(p_{\Lambda})_{\Lambda\in i{\cal A}_{0}^*}$ est uniquement déterminée. Plus précisément, la connaissance de $f$ dans un ouvert non vide de ${\cal A}_{0}$ suffit à déterminer cette famille. En particulier, le polynôme $p_{0}$ est bien déterminé. On pose $c_{0}(f)=p_{0}(0)$.
Si $\Xi\subset i{\cal A}_{0}^*$ est un ensemble fini et $N$ est un entier naturel, on note plus précisément $PolExp_{\Xi,N}$ l’ensemble des $f\in PolExp$ tels que, avec les notations ci-dessus, le degré des $p_{\Lambda}$ soit inférieur ou égal à $N$ et $p_{\Lambda}$ soit nul si $\Lambda\not\in \Xi$.
Si $F$ est non-archimédien, notons $PolExp$ l’ensemble des fonctions $f:{\cal A}_{M_{0},F}\otimes_{{\mathbb Z}}{\mathbb Q}\to {\mathbb C}$ qui vérifient la condition suivante. Pour tout réseau ${\cal R}\subset {\cal A}_{M_{0},F}\otimes_{{\mathbb Z}}{\mathbb Q}$, il existe une famille $(p_{{\cal R},\Lambda})_{\Lambda\in i{\cal A}_{0}^*/i{\cal R}^{\vee}}$ telle que
- l’ensemble des $\Lambda$ tels que $p_{{\cal R},\Lambda}\not=0$ est fini;
- pour tout $T\in {\cal R}$, on a l’égalité $$f(T)=\sum_{\Lambda\in i{\cal A}_{0}^*/i{\cal R}^{\vee}}e^{<\Lambda,T>}p_{{\cal R},\Lambda}(T).$$
De nouveau, la famille $(p_{{\cal R},\Lambda})_{\Lambda\in i{\cal A}_{0}^*/i{\cal R}^{\vee}}$ est uniquement déterminée. Plus précisément, la connaissance de $f$ dans l’intersection de ${\cal A}_{M_{0},F}\otimes_{{\mathbb Z}}{\mathbb Q} $ et d’un cône ouvert non vide de ${\cal A}_{0}$ suffit à déterminer cette famille. On pose $c_{{\cal R},0}(f)=p_{{\cal R},0}(0)$.
Soit $\boldsymbol{\Xi}=(\Xi_{{\cal R}})_{{\cal R}}$ une famille indexée par les réseaux dans ${\cal A}_{M_{0},F}\otimes_{{\mathbb Z}}{\mathbb Q}$, où $\Xi_{{\cal R}}$ est un sous-ensemble fini de $ i{\cal A}_{0}^*/i{\cal R}^{\vee}$. Soit $N$ un entier naturel. On note plus précisément $PolExp_{\boldsymbol{\Xi},N}$ l’ensemble des $f\in PolExp$ tels que, avec les notations ci-dessus et pour tout ${\cal R}$, le degré des $p_{{\cal R},\Lambda}$ soit inférieur ou égal à $N$ et que l’on ait $p_{{\cal R},\Lambda}=0$ si $\Lambda\not\in \Xi_{{\cal R}}$.
Soient $M$ un Levi semi-standard, ${\cal Y}=(Y[P])_{P\in {\cal P}(M)}$ une famille $(G,M)$-orthogonale, $(\varphi(\Lambda,P))_{P\in {\cal P}(M)}$ une $(G,M)$-famille, $T\in {\cal A}_{0}$ et $X\in {\cal A}_{G,F}$. Dans le cas où $F$ est non-archimédien, on suppose que les deux familles sont $p$-adiques et que $T\in {\cal A}_{M_{0},F}\otimes_{{\mathbb Z}}{\mathbb Q}$. On déduit de $T$ une famille $(G,M)$-orthogonale $(T[P])_{P\in {\cal P}(M)}$, cf. 1.3. On définit la famille ${\cal Y}(T)=(Y[P]+T[P])_{P\in {\cal P}(M)}$. Elle est encore $(G,M)$-orthogonale, $p$-adique si $F$ est non-archimédien. On a défini dans le paragraphe précédent un terme $\varphi_{M}^{G,{\cal Y}(T)}(X;\Lambda)$. D’autre part, pour $P\in {\cal P}(M)$, posons $\varphi({\cal Y};\Lambda,P)=\varphi(\Lambda,P)e^{<\Lambda,Y[P]^G>}$. La famille $(\varphi({\cal Y};\Lambda,P))_{P\in {\cal P}(M)}$ est une $(G,M)$-famille. On a défini en 1.4 la fonction $\varphi_{M}^G({\cal Y};\Lambda)$.
[**Remarque.**]{} On appliquera souvent ces constructions à la famille ${\cal Y}$ nulle, c’est-à-dire $Y[P]=0$ pour tout $P$. Dans ce cas on a $\varphi_{M}^G({\cal Y};\Lambda)= \varphi_{M}^G(\Lambda)$ et on note simplement $\varphi_{M}^{G,T}(X;\Lambda)=\varphi_{M}^{G,{\cal Y}(T)}(X;\Lambda)$.
Supposons $F$ non-archimédien. Fixons une base de l’espace des opérateurs différentiels à coefficients constants sur $i{\cal A}_{M}^*$, de degré borné par $a_{M}-a_{G}$. Appelons norme de la $(G,M)$-famille $(\varphi(\Lambda,P))_{P\in {\cal P}(M)}$ le sup des $\vert D\varphi(\Lambda,P)\vert $, quand $\Lambda$ parcourt $i{\cal A}_{M,F}^*$, $P$ parcourt ${\cal P}(M)$ et $D$ parcourt la base fixée.
0.3cm[**[Lemme]{}**]{}.
**
\(i) Supposons $F$ archimédien. Pour tout $\Lambda_{0}\in i{\cal A}_{M}^*$, la fonction $f:T\mapsto \varphi^{G,{\cal Y}(T)}_{M}(X;\Lambda_{0})$ appartient à $PolExp$. Plus précisément, il existe un entier $N$ et un sous-ensemble fini $\Xi\subset i{\cal A}_{0}^*$ ne dépendant que de $\Lambda_{0}$ tels que $f\in PolExp_{\Xi,N}$. On a $$c_{0}(f)=\left\lbrace\begin{array}{cc}e^{<\Lambda_{0},X>}\varphi_{M}^G({\cal Y};\Lambda_{0}),&\text{ si }\Lambda_{0}\in i{\cal A}_{G}^*,\\ 0,&\text{ sinon.}\\ \end{array}\right.$$
\(ii) Supposons $F$ non-archimédien. Pour tout $\Lambda_{0}\in i{\cal A}_{M}^*$, la fonction $f:T\mapsto \varphi^{G,{\cal Y}(T)}_{M}(X;\Lambda_{0})$ appartient à $PolExp$. Plus précisément, il existe un entier $N$ et une famille d’ensembles finis $\boldsymbol{\Xi}$ ne dépendant que de $\Lambda_{0}$ tels que $f\in PolExp_{\boldsymbol{\Xi},N}$. Soit ${\cal R}\subset {\cal A}_{M_0,F}\otimes_{{\mathbb Z}}{\mathbb Q}$ un réseau. Si $\Lambda_{0}\not\in i{\cal A}_{M,F}^{\vee}+i{\cal A}_{G}^*$, il existe un entier $k_{1}$ ne dépendant que de ${\cal R}$ tel que $c_{\frac{1}{k}{\cal R},0}(f)=0$ pour tout entier $k\geq k_{1}$. Supposons maintenant $\Lambda_{0}\in \Lambda_{1}+i{\cal A}_{M,F}^{\vee}$, où $\Lambda_{1}\in i{\cal A}_{G}^*$. Alors, il existe un réel $c>0$ ne dépendant que de ${\cal R}$ tel que, pour tout entier $k\geq1$, on ait la majoration $$\vert c_{\frac{1}{k}{\cal R},0}(f)- mes(i{\cal A}_{M,F}^*)mes(i{\cal A}_{G,F}^*)^{-1}e^{<\Lambda_{1},X>}\varphi^G_{M}({\cal Y};\Lambda_{1})\vert \leq cNk^{-1},$$ où $N$ est la norme de la $(G,M)$-famille $(\varphi({\cal Y};\Lambda,P))_{P\in {\cal P}(M)}$.
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Preuve. Rappelons comment on calcule un terme tel que $\varphi_{M}^G(\Lambda_{0})$. Pour tout $P\in {\cal P}(M)$, notons $\Delta_{P}(\Lambda_{0})$ l’ensemble des $\alpha\in \Delta_{P}$ tels que $<\Lambda_{0},\check{\alpha}>=0$. Notons $n_{P}(\Lambda_{0})$ le nombre d’éléments de cet ensemble. On fixe un élément $\xi\in i{\cal A}_{M}^{G,*}$ en position générale. Pour $t\in {\mathbb R}$, posons $$\varphi(t,\Lambda_{0},P)=\varphi(\Lambda_{0}+t\xi,P)\left(\prod_{\alpha\in \Delta_{P}-\Delta_{P}(\Lambda_{0})}<\Lambda_{0}+t\xi,\check{\alpha}>^{-1}\right)\left(\prod_{\alpha\in \Delta_{P}(\Lambda_{0})}<\xi,\check{\alpha}>^{-1}\right).$$ Cette fonction est $C^{\infty}$ en $t=0$. Alors $$(1)\qquad \varphi_{M}^G(\Lambda_{0})=mes({\cal A}_{M}^G/{\mathbb Z}[\check{\Delta}_{M}])\sum_{P\in {\cal P}(M)}(n_{P}(\Lambda_{0})!)^{-1}\left(\frac{d^{n_{P}(\Lambda_{0})}}{dt^{n_{P}(\Lambda_{0})}}\varphi(t,\Lambda_{0},P)\right) _{t=0}.$$ Cela résulte simplement de l’égalité $$\varphi_{M}^G(\Lambda_{0})=lim_{t\to 0}\sum_{P\in {\cal P}(M)}\varphi(\Lambda_{0}+t\xi,P)\epsilon_{P}^G(\Lambda_{0}+t\xi).$$
Preuve de (i). On a dit en 1.6 que l’on avait l’égalité $ f(T)=e^{<\Lambda,X>}\varphi_{M}^G({\cal Y}(T);\Lambda_{0})$. La formule (1) montre que $\varphi_{M}^G({\cal Y}(T);\Lambda_{0})$ est combinaison linéaire de termes $D\varphi({\cal Y}(T);\Lambda_{0},P)$ où $P$ parcourt ${\cal P}(M)$ et $D$ parcourt les opérateurs différentiels sur $i{\cal A}_{M}^*$, à coefficients constants et de degré borné par $a_{M}-a_{G}$. Comme fonction de $T$, un tel terme est de la forme $e^{<\Lambda_{0},T[P]^G>}p(T)$, où $p$ est un polynôme de degré inférieur ou égal à $a_{M}-a_{G}$. Cette fonction appartient à $PolExp$, donc $f$ aussi. Plus précisément, $f\in PolExp_{\Xi,a_{M}-a_{G}}$ où $\Xi=\{w\Lambda_{0}^G; w\in W^G\}$. Le coefficient $c_{0}(f)$ ne voit que les termes dont la partie exponentielle est triviale. Il n’y en a que si que si $\Lambda_{0}^G=0$. Supposons cette condition vérifiée. Alors toutes les exponentielles sont triviales et $c_{0}(f)$ est simplement $f(0)$. Mais on a $f(0)=e^{<\Lambda_{0},X>}\varphi_{M}^G({\cal Y};\Lambda_{0})$. D’où (i).
Preuve de (ii). Soit ${\cal R}$ un réseau dans ${\cal A}_{M_0,F}\otimes_{{\mathbb Z}}{\mathbb Q}$. On a défini en 1.5 des réseaux ${\cal L}_{k}$ pour tout entier $k\geq1$. On peut fixer $k_{0}$ de sorte que $T[P]^G\in {\cal L}_{k_{0}}$ pour tout $T\in {\cal R}$. On a alors l’égalité 1.6(2) que l’on récrit $$f(T) =[{\cal L}_{k_{0}}:{\cal A}_{M,F}^G]^{-1}e^{<\Lambda_{0},X>}\sum_{\nu\in i{\cal A}_{M,F}^{G,\vee}}e^{-<\nu,(X')^G>}\varphi_{M}^G(\nu,{\cal Y}(T);\Lambda_{0}+\nu),$$ où, pour tout $P\in {\cal P}(M)$, $$\varphi(\nu,{\cal Y}(T);\Lambda,P)=\varphi(\Lambda-\nu,P)e^{<\Lambda,Y[P]^G+T[P]^G>}v_{k_{0}}(\Lambda,P).$$ La somme en $\nu$ est finie. En appliquant (1), on obtient $$(2) \qquad f(T)=mes({\cal A}_{M}^G/{\mathbb Z}[\check{\Delta}_{M}])[{\cal L}_{k_{0}}:{\cal A}_{M,F}^G]^{-1}e^{<\Lambda_{0},X>}\sum_{\nu\in i{\cal A}_{M,F}^{G,\vee}}e^{-<\nu,(X')^G>}$$ $$\sum_{P\in {\cal P}(M)}(n_{P}(\Lambda_{0}+\nu)!)^{-1}\left(\frac{d^{n_{P}(\Lambda_{0}+\nu)}}{dt^{n_{P}(\Lambda_{0}+\nu)}}\varphi(\nu,{\cal Y}(T);t,\Lambda_{0}+\nu,P)\right) _{t=0}.$$ Comme dans le cas archimédien une fonction $$T\mapsto \left(\frac{d^{n_{P}(\Lambda_{0}+\nu)}}{dt^{n_{P}(\Lambda_{0}+\nu)}}\varphi(\nu,{\cal Y}(T);t,\Lambda_{0}+\nu,P)\right) _{t=0}$$ est produit de $e^{<\Lambda_{0}+\nu,T[P]^G>}$ et d’un polynôme en $T$ de degré inférieur ou égal à $a_{M}-a_{G}$. Une telle fonction appartient à $PolExp$, donc la fonction $f$ appartient aussi à cet espace. Plus précisément, $f\in PolExp_{\boldsymbol{\Xi},a_{M}-a_{G}}$, où $\boldsymbol{\Xi}=(\Xi_{{\cal R}})_{{\cal R}}$ est la famille telle que $\Xi_{{\cal R}}$ soit l’ensemble des projections dans $i{\cal A}_{0}^*/i{\cal R}^{\vee}$ des $w(\Lambda_{0}^G+\nu)$ pour $w\in W^G$ et $\nu\in i{\cal A}_{M,F}^{G,\vee}$. Pour tout $P\in {\cal P}(M)$, notons ${\cal S}(P)$ l’image de ${\cal R}$ par l’application $T\mapsto T[P]^G$. C’est un réseau de ${\cal A}_{M}^G$. Le coefficient $c_{{\cal R},0}(f)$ sélectionne les termes de la formule (2) pour lesquels les exponentielles sont triviales pour tout $T\in {\cal R}$, c’est-à-dire les couples $(\nu,P)$ tels que $\Lambda_{0}^G+\nu\in i{\cal S}(P)^{\vee}$. Supposons $\Lambda_{0}\not\in i{\cal A}_{M,F}^{\vee}+i{\cal A}_{G}^*= i{\cal A}_{M,F}^{G,\vee}\oplus i{\cal A}_{G}^*$. Si ${\cal R}$ est assez grand, on a ${\cal S}(P)^{\vee}\subset {\cal A}_{M,F}^{G,\vee}$ pour tout $P$ et aucun $\nu\in i{\cal A}_{M,F}^{G,\vee}$ ne contribue. Donc $c_{{\cal R},0}(f)=0$ ce qui prouve la première assertion du (ii). Supposons maintenant $\Lambda_{0}\in\Lambda_{1}+i{\cal A}_{M,F}^{\vee}$, où $\Lambda_{1}\in i{\cal A}_{G}^*$. On a les égalités $$\varphi_{M}^{G,{\cal Y}(T)}(X;\Lambda_{0})=\varphi_{M}^{G,{\cal Y}(T)}(X;\Lambda_{1})=e^{<\Lambda_{1},X>}(\varphi')_{M}^{G,{\cal Y}(T)}(X;0),$$ où $(\varphi'(\Lambda,P))_{P\in {\cal P}(M)}$ est la $(G,M)$-famille déduite de la famille initiale par translation par $\Lambda_{1}$. Cela nous ramène au cas où $\Lambda_{0}=0$, ce que l’on suppose désormais. Le calcul précédent conduit à l’égalité $$c_{{\cal R},0}(f)=mes({\cal A}_{M}^G/{\mathbb Z}[\check{\Delta}_{M}])[{\cal L}_{k_{0}}:{\cal A}_{M,F}^G]^{-1}\sum_{P\in {\cal P}(M)}$$ $$\sum_{\nu\in i{\cal S}(P)^{\vee}}e^{-<\nu,(X')^G>}(n_{P}(\nu)!)^{-1}\left(\frac{d^{n_{P}(\nu)}}{dt^{n_{P}(\nu)}}\varphi(\nu,{\cal Y};t,\nu,P)\right) _{t=0}.$$ Remplaçons ${\cal R}$ par $\frac{1}{k}{\cal R}$. Cela remplace ${\cal S}(P)^{\vee}$ par $k{\cal S}(P)^{\vee}$. On remarque que $\Delta_{P}(k\nu)=\Delta_{P}(\nu)$ et $n_{P}(k\nu)=n_{P}(\nu)$. On peut remplacer l’entier $k_{0}$ par $kk_{0}$. On peut choisir $u_{kk_{0}}(\Lambda,P)=u_{k_{0}}(\frac{\Lambda}{k},P)$. Alors $v_{kk_{0}}(\Lambda,P)=k^{a_{M}-a_{G}}v_{k_{0}}(\frac{\Lambda}{k},P)$. Puisqu’on a aussi $$[{\cal L}_{kk_{0}}:{\cal A}_{M,F}^G]=k^{a_{M}-a_{G}}[{\cal L}_{k_{0}}:{\cal A}_{M,F}^G],$$ l’égalité ci-dessus devient $$(3) \qquad c_{\frac{1}{k}{\cal R},0}(f)=mes({\cal A}_{M}^G/{\mathbb Z}[\check{\Delta}_{M}])[{\cal L}_{k_{0}}:{\cal A}_{M,F}^G]^{-1}\sum_{P\in {\cal P}(M)}$$ $$\sum_{\nu\in i{\cal S}(P)^{\vee}}e^{-<k\nu,(X')^G>}(n_{P}(\nu)!)^{-1}\left(\frac{d^{n_{P}(\nu)}}{dt^{n_{P}(\nu)}} f_{k}(t,\nu,P)\right) _{t=0},$$ où $$f_{k}(t,\nu,P)=\varphi(t\xi,P)e^{<k\nu+t\xi,Y[P]^G>}v_{k_{0}}(\nu+\frac{t}{k}\xi,P)\left(\prod_{\alpha\in \Delta_{P}-\Delta_{P}(\nu)}<k\nu+t\xi,\check{\alpha}>^{-1}\right)$$ $$\left(\prod_{\alpha\in \Delta_{P}(\nu)}<\xi,\check{\alpha}>^{-1}\right).$$ Notons que puisque $v_{k_{0}}(\Lambda,P)$ est à support compact, la somme est limitée à un ensemble fini indépendant de $k$. Fixons $\nu$ et $P$. Si $k$ est assez grand, on a $e^{-<k\nu,(X')^G>}=e^{<k\nu,Y[P]^G>}=1$ et ces termes disparaissent. Dans la définition de $f_{k}(t,\nu,P)$, le produit $\varphi(t\xi,P)e^{<k\nu+t\xi,Y[P]^G>}$ devient $\varphi({\cal Y};t\xi,P)$. Le terme $\left(\frac{d^{n_{P}(\nu)}}{dt^{n_{P}(\nu)}} f_{k}(t,\nu,P)\right) _{t=0}$ est combinaison linéaire de produits $$\left(\frac{d^{a}}{dt^{a}}\varphi({\cal Y};t\xi,P)\right)_{t=0}\left(\frac{d^b}{dt^b}v_{k_{0}}(\nu+\frac{t}{k}\xi,P)\right)_{t=0}\left(\frac{d^{c}}{dt^c}\prod_{\alpha\in \Delta_{P}-\Delta_{P}(\nu)}<k\nu+t\xi,\check{\alpha}>^{-1}\right)_{t=0}$$ pour des entiers naturels $a,b,c$ tels que $a+b+c=n_{P}(\nu)$. On voit qu’un tel produit est borné par $CNk^{-m}$, où $N$ est la norme de la $(G,M)$-famille $(\varphi({\cal Y};\Lambda,P))_{P\in {\cal P}(M)}$, $C$ ne dépend que de $m$ et $v_{k_{0}}$, et $m=b+c+\vert \Delta_{P}-\Delta_{P}(\nu)\vert $. Les termes pour lesquels $m$ est strictement positif sont négligeables pour la conclusion du lemme. Si $\nu\not=0$, on a $\Delta_{P}\not=\Delta_{P}(\nu)$ et tous les termes sont donc négligeables. Pour $\nu=0$, on ne conserve que les termes où les dérivations ne s’appliquent qu’à la fonction $\varphi({\cal Y};t\xi,P)$. Si on élimine les termes négligeables, le membre de droite de (3) devient $$mes({\cal A}_{M}^G/{\mathbb Z}[\check{\Delta}_{M}])[{\cal L}_{k_{0}}:{\cal A}_{M,F}^G]^{-1}\sum_{P\in {\cal P}(M)}v_{k_{0}}(0,P)n!^{-1}\left(\frac{d^n}{dt^n}\varphi({\cal Y};t\xi,P)\right)_{t=0}\prod_{\alpha\in \Delta_{P}}<\xi,\check{\alpha}>^{-1},$$ où $n=a_{M}-a_{G}$. On va montrer que
\(4) on a l’égalité $ [{\cal L}_{k_{0}}:{\cal A}_{M,F}^G]^{-1}v_{k_{0}}(0,P)= mes(i{\cal A}_{M,F}^*)mes(i{\cal A}_{G,F}^*)^{-1}$.
En admettant cette relation et en utilisant (1), l’expression ci-dessus n’est autre que $ \varphi_{M}^G({\cal Y};0)$ et on obtient la conclusion de (ii).
Il reste à prouver (4). Pour $\nu\in i{\cal L}_{k_{0}}^{\vee}$, $\nu\not=0$, la fonction $t\mapsto \epsilon_{P,{\cal L}_{k_{0}}}^G(\nu+t\xi)$ a un pôle d’ordre $n=a_{M}-a_{G}$ en $t=0$. La fonction $t\mapsto \epsilon_{P}^G(\nu+t\xi)^{-1}$ a un zéro d’ordre strictement inférieur. Puisque $v_{k_{0}}(\nu+t\xi,P)=u_{k_{0}}(\nu+t\xi,P)\epsilon_{P,{\cal L}_{k_{0}}}^G(\nu+t\xi)\epsilon_{P}^G(\nu+t\xi)^{-1}$ est régulière, on a nécessairement $u_{k_{0}}(\nu,P)=0$. Puisque $\sum_{\nu\in i{\cal L}_{k_{0}}^{\vee}}u_{k_{0}}(\nu,P)=1$, on en déduit $u_{k_{0}}(0,P)=1$. On calcule alors $$v_{k_{0}}(0,P)=mes({\cal A}_{M}^G/{\mathbb Z}[\check{\Delta}_{M}])^{-1}\lim_{\Lambda\to 0}\prod_{\alpha\in \Delta_{P}}\frac{<\Lambda,\check{\alpha}>}{1-e^{-<\Lambda,\kappa\check{\alpha}>}}=mes({\cal A}_{M}^G/{\mathbb Z}[\check{\Delta}_{M}])^{-1}\kappa^{-n},$$ où $\kappa=\frac{1}{k_{0}}log(q)$. Or ${\cal L}_{k_{0}}=\kappa {\mathbb Z}[\check{\Delta}_{M}]$, d’où $$\kappa^n mes({\cal A}_{M}^G/{\mathbb Z}[\check{\Delta}_{M}])=mes({\cal A}_{M}^G/\kappa{\mathbb Z}[\check{\Delta}_{M}])=mes({\cal A}_{M}^G/{\cal L}_{k_{0}}).$$ D’après les relations de compatibilité de nos mesures, on a $$mes({\cal A}_{M}^G/{\cal A}_{M,F}^G)=mes(i{\cal A}_{M,F}^*)^{-1}mes(i{\cal A}_{G,F}^*),$$ d’où $mes({\cal A}_{M}^G/{\cal L}_{k_{0}})=[{\cal L}_{k_{0}}:{\cal A}_{M,F}^G]^{-1}mes(i{\cal A}_{M,F}^*)^{-1}mes(i{\cal A}_{G,F}^*)$. En assemblant ces calculs, on obtient (4). $\square$
$(G,M)$-familles et inversion de Fourier
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Soient $M$ un Levi semi-standard et $X$ un élément de ${\cal A}_{G,F}$. Appliquons les constructions du paragraphe 1.6 à la famile ${\cal Y}=(Y[P])_{P\in {\cal P}(M)}$ nulle, c’est-à-dire $Y[P]=0$ pour tout $P$, et à la famille $(\varphi(\Lambda,P))_{P\in {\cal P}(M)}$ formée des fonctions constantes égales à $1$. On en déduit une fonction que l’on note $$\epsilon_{M}^{G,0}(X;\Lambda)=\sum_{P\in {\cal P}(M)}\epsilon_{P}^{G,0}(X;\Lambda).$$ On a l’égalité $$(1)\qquad \epsilon_{M}^{G,0}(X;\Lambda)=\left\lbrace\begin{array}{cc}e^{<\Lambda,X>},& \text{ si }F \text{ est non-archim\'edien et }X\in {\cal A}_{M,F}\cap {\cal A}_{G},\\&\text{ ou si }F\text{ est archim\'edien et }M=G,\\ 0,&\text{ sinon.}\end{array}\right.$$.
Preuve. On peut fixer un élément $P_{1}\in {\cal P}(M)$ et supposer $<\Lambda,\check{\alpha}>>0$ pour tout $\alpha\in \Delta_{P_{1}}$. En utilisant 1.5(2), on a $$\epsilon_{M}^{G,0}(X;\Lambda)=\sum_{P\in {\cal P}(M)}(-1)^{s(P,P_{1})}\epsilon_{P,P_{1}}^{G,0}(X;\Lambda)$$ $$=\sum_{P\in {\cal P}(M)}(-1)^{s(P,P_{1})}\int_{{\cal A}_{M,F}^G(X)}\phi_{P,P_{1}}^G(H)e^{<\Lambda,H>}\,dH$$ $$=\int_{{\cal A}_{M,F}^G(X)}e^{<\Lambda,H>}\sum_{P\in {\cal P}(M)}(-1)^{s(P,P_{1})}\phi_{P,P_{1}}^G(H) dH.$$ D’après 1.3(7), la somme intérieure est égale à $\Gamma_{M}^G(H,{\cal Y})$, où ${\cal Y}$ est la famille nulle. Ce n’est autre que $\delta_{M}^G(H)$. Si $F$ est archimédien et $M\not=G$, l’intégrale est nulle. Si $F$ est non archimédien, elle vaut $e^{<\Lambda,X>}$ s’il existe $H\in {\cal A}_{M,F}^G(X)$ avec $H^G=0$ et elle vaut $0$ sinon. Cette condition équivaut à $X\in {\cal A}_{M,F}\cap {\cal A}_{G}$. $\square$
Soit $T$ un élément de ${\cal A}_{0}$ et soit $(\varphi(\Lambda,P))_{P\in {\cal P}(M)}$ une $(G,M)$-famille. Dans le cas où $F$ est non-archimédien, on suppose que $T\in {\cal A}_{M_{0},F}\otimes_{{\mathbb Z}}{\mathbb Q}$ et que la $(G,M)$-famille est $p$-adique. Dans le cas archimédien, on suppose que les fonctions $\Lambda\mapsto \varphi(\Lambda,P)$ sont de Schwartz. Pour un sous-groupe parabolique $Q=LU_{Q}\in {\cal F}(M)$, on définit $\varphi(\Lambda,Q)$ pour $\Lambda\in i{\cal A}_{L,F}^*$ comme la restriction à $ i{\cal A}_{L,F}^*$ de $\varphi(\Lambda,P)$ pour n’importe quel $P\in {\cal P}(M)$ tel que $P\subset Q$. On définit sa transformée de Fourier $\hat{\varphi}(H,Q)$, qui est une fonction de $H\in {\cal A}_{L,F}$, par $$\hat{\varphi}(H,Q)=mes(i{\cal A}_{L,F}^*)^{-1}\int_{i{\cal A}_{L,F}^*}\varphi(\Lambda,Q)e^{-<\Lambda,H>}\,d\Lambda.$$ C’est une fonction de Schwartz sur ${\cal A}_{L,F}$.
[0.3cm[**[Lemme]{}**]{}. [ *[L’expression $$\sum_{Q=LU_{Q}\in {\cal F}(M)}\int_{ {\cal A}_{L,F}}\int_{{\cal A}_{M,F}^G(X+H_{G})}\delta_{M}^Q(H')\Gamma_{Q}^G(H',H+T[Q])\hat{\varphi}(H,Q) e^{<\Lambda,H'>}\,dH'\,dH$$ est convergente et égale à $\varphi_{M}^{G,T}(X;\Lambda)$.]{}*]{}0.3cm]{}
Preuve. La condition $\Gamma_{Q}^G(H',H+T[Q])\not=0$ implique une majoration $\vert (H')^G_{L}\vert <<1+\vert H^G\vert $ ($T$ est ici considéré comme une constante). Jointe aux conditions $\delta_{M}^Q(H')=1$ et $H_{G}'-H_{G}=X$, cela implique $\vert H'\vert <<1+\vert H\vert $. L’intégrale en $H'$ est donc essentiellement bornée par $(1+\vert H\vert )^D$ pour un entier $D$ convenable. Puisque $H\mapsto \hat{\varphi}(H,Q)$ est de Schwartz, l’intégrale en $H$ est convergente. Cela prouve la convergence énoncée.
Rappelons que, pour $\Lambda$ en position générale, $$\varphi_{M}^{G,T}(X;\Lambda)=\sum_{P\in {\cal P}(M)}\varphi(\Lambda,P)\epsilon^{G,T[P]}_{P}(X;\Lambda).$$ Fixons $P$. Par inversion de Fourier $$\varphi(\Lambda,P)=\int_{{\cal A}_{M,F}}\hat{\varphi}(H,P)e^{<\Lambda,H>}\,dH.$$ On va prouver l’égalité $$(2) \qquad \epsilon^{G,T[P]}_{P}(X;\Lambda)e^{<\Lambda,H>}=\sum_{Q=LU_{Q}; P\subset Q}\int_{{\cal A}_{L,F}^G(X+H_{G})}\Gamma_{Q}^G(H',H+T[Q])\epsilon_{P}^{Q,0}(H';\Lambda)\,dH'.$$ En admettant cela, on obtient $$\varphi_{M}^{G,T}(X;\Lambda)=\sum_{P\in {\cal P}(M)} \int_{{\cal A}_{M,F}}\hat{\varphi}(H,P)\epsilon^{G,T[P]}_{P}(X;\Lambda)e^{<\Lambda,H>}\,dH$$ $$=\sum_{P\in {\cal P}(M)} \int_{{\cal A}_{M,F}}\hat{\varphi}(H,P)\sum_{Q=LU_{Q}; P\subset Q}\int_{{\cal A}_{L,F}^G(X+H_{G})}\Gamma_{Q}^G(H',H+T[Q])\epsilon_{P}^{Q,0}(H';\Lambda)\,dH'\,dH.$$ Cette expression est absolument convergente pour $\Lambda$ en position générale pour les mêmes raisons que ci-dessus, la fonction $H'\mapsto \epsilon_{P}^{Q,0}(H';\Lambda)$ étant bornée. On sort la somme en $Q$ des intégrales et on la permute avec celle en $P$. On obtient $$\varphi_{M}^{G,T}(X;\Lambda)=\sum_{Q=LU_{Q}\in {\cal F}(M)}\sum_{P\in {\cal P}(M); P\subset Q} \int_{{\cal A}_{M,F}}\hat{\varphi}(H,P)$$ $$\int_{{\cal A}_{L,F}^G(X+H_{G})}\Gamma_{Q}^G(H',H+T[Q])\epsilon_{P}^{Q,0}(H';\Lambda)\,dH'\,dH.$$ On remarque que $H$ n’intervient que par sa projection $H_{L}$ dans l’intégrale intérieure. Décomposons la première intégrale en une intégrale sur $H\in {\cal A}_{L,F}$ et une intégrale sur ${\cal A}_{M,F}^L(H)$. On obtient $$(3) \qquad \varphi_{M}^{G,T}(X;\Lambda)=\sum_{Q=LU_{Q}\in {\cal F}(M)}\sum_{P\in {\cal P}(M); P\subset Q} \int_{{\cal A}_{L,F}}\int_{{\cal A}_{M,F}^L(H)}\hat{\varphi}(H'',P)$$ $$\int_{{\cal A}_{L,F}^G(X+H_{G})}\Gamma_{Q}^G(H',H+T[Q])\epsilon_{P}^{Q,0}(H';\Lambda)\,dH'\,dH''\,dH.$$ On voit apparaître l’intégrale $$\int_{{\cal A}_{M,F}^L(H)}\hat{\varphi}(H'',P)\,dH''.$$ Par inversion de Fourier, ceci n’est autre que $$mes(i{\cal A}_{L,F}^*)^{-1}\int_{i{\cal A}_{L,F}^*}\varphi(\Lambda,P)e^{-<\Lambda,H>}\,d\Lambda.$$ Puisque le domaine d’intégration est restreint à $i{\cal A}_{L,F}^*$, on peut aussi bien remplacer $\varphi(\Lambda,P)$ par $\varphi(\Lambda,Q)$. L’intégrale ci-dessus devient $\hat{\varphi}(H,Q)$, en particulier est indépendante de $P$. Dans la formule (3), la somme en $P$ ne porte plus que sur les termes $\epsilon_{P}^{Q,0}(H';\Lambda)$. Leur somme vaut $\epsilon_{M}^{Q,0}(H';\Lambda)$. Supposons $F$ non archimédien. L’égalité (1) dit que $\epsilon_{M}^{Q,0}(H';\Lambda)$ vaut $e^{<\Lambda,X>}$ si $H'\in {\cal A}_{M,F}$, $0$ sinon. La formule (3) devient $$\varphi_{M}^{G,T}(X;\Lambda)=\sum_{Q=LU_{Q}\in {\cal F}(M)}\int_{ {\cal A}_{L,F}}\int_{{\cal A}_{L,F}^G(X+H_{G})\cap {\cal A}_{M,F}}\Gamma_{Q}^G(H',H+T[Q])\hat{\varphi}(H,Q) e^{<\Lambda,H'>}\,dH'\,dH.$$ Mais ${\cal A}_{L,F}^G(X+H_{G})\cap {\cal A}_{M,F}$ est l’ensemble des $H'\in {\cal A}_{M,F}^G(X+H_{G})$ tels que $\delta_{M}^Q(H')=1$. La formule ci-dessus est donc équivalente à celle de l’énoncé. Supposons maintenant $F$ archimédien. L’égalité (1) transforme directement la formule (3) en celle de l’énoncé, à ceci près que l’on ne somme que sur les $Q\in {\cal P}(M)$. Mais si $Q\not\in {\cal P}(M)$, le support de la fonction $\delta_{M}^Q$ est de mesure nulle et la contribution de $Q$ à la formule de l’énoncé est nulle.
Prouvons (2). Puisqu’il s’agit de fonctions méromorphes en $\Lambda$, on peut supposer $Re<\Lambda,\check{\alpha}>>0$ pour tout $\alpha\in \Delta_{P}$. Alors $$\epsilon^{G,T}_{P}(X;\Lambda)e^{<\Lambda,H>}=\int_{ {\cal A}_{M,F}^G(X)}\phi_{P}^G(H'-T)e^{<\Lambda,H'+H>}\,dH'$$ $$=\int_{ {\cal A}_{M,F}^G(X+H_{G})}\phi_{P}^G(H'-H-T)e^{<\Lambda,H'>}\,dH'.$$ On utilise 1.3(3), d’où $$\epsilon^{G,T}_{P}(X;\Lambda)=\int_{ {\cal A}_{M,F}^G(X+H_{G})}\sum_{Q=LU_{Q}; P\subset Q}\Gamma_{Q}^G(H',H+T)\phi_{P}^Q(H')e^{<\Lambda,H'>}\,dH'.$$ On sort la somme en $Q$ de l’intégrale et on décompose celle-ci en une intégrale sur $H'\in {\cal A}_{L,F}^{G}(X+H_{G})$ et une intégrale sur $H''\in {\cal A}_{M,F}^{L}(H')$. On obtient $$\epsilon^{G,T}_{P}(X;\Lambda)=\sum_{Q=LU_{Q}; P\subset Q}\int_{{\cal A}_{L,F}^{G}(X+H_{G})}\Gamma_{Q}^G(H',H+T)\int_{{\cal A}_{M,F}^L(H')}\phi_{P}^Q(H'')e^{<\Lambda,H''>}\,dH''\,dH'.$$ La dernière intégrale n’est autre que $\epsilon_{P}^{Q,0}(H';\Lambda)$ et on obtient la formule (2). $\square$
Représentations
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Si $\pi$ est une représentation admissible de $G(F)$ et si $F$ est non-archimédien, on note $V_{\pi}$ un espace dans lequel elle se réalise. Dans le cas où $F$ est archimédien, pour être correct, il faudrait introduire deux espaces: un espace dans lequel $\pi$ se réalise et le sous-espace des vecteurs $K$-finis, qui est celui qui nous intéresse mais qui n’est pas invariant par l’action de $G(F)$. Pour simplifier, on n’introduit que ce dernier, que l’on note $V_{\pi}$, et on convient qu’il est plongé dans un espace implicite plus gros où $\pi$ se réalise. Si $\pi$ est irréductible et unitaire, on fixe un produit hermitien $(.,.)$ sur $V_{\pi}$ invariant par $\pi(g)$ pour tout $g\in G(F)$ (avec la convention ci-dessus dans le cas archimédien). Pour nous, un tel produit est linéaire en la seconde variable et antilinéaire en la première. On note $\Pi_{disc}(G(F))$ l’ensemble des classes de représentations irréductibles de la série discrète de $G(F)$. Pour $\sigma\in \Pi_{disc}(G(F))$, son degré formel $d(\sigma)$ est défini par l’égalité $$\int_{A_{G}(F)\backslash G(F)}(e_{1},\sigma(g)e_{2})(\sigma(g)e'_{1},e'_{2})\,dg\,=d(\sigma)^{-1}(e_{1},e'_{2})(e'_{1},e_{2})$$ pour tous $e_{1},e_{2},e'_{1},e'_{2}\in V_{\sigma}$.
Le groupe ${\cal A}_{G,{\mathbb C}}^*$ agit sur l’ensemble des classes d’isomorphisme de représentations admissibles de $G(F)$: à une représentation $\pi$ et à $\lambda\in {\cal A}_{G,{\mathbb C}}^* $, on associe la représentation $\pi_{\lambda}$ définie par $\pi_{\lambda}(g)=e^{<\lambda,H_{G}(g)>}\pi(g)$. Cette action se quotiente en une action de ${\cal A}_{G,{\mathbb C}}^*/i{\cal A}_{G,F}^{\vee}$. Pour deux représentations $\pi$, $\pi'$, considérons l’ensemble des $\lambda\in i{\cal A}_{G}^*$ tels que $\pi\simeq \pi'_{-\lambda}$. Il est stable par $i{\cal A}_{G,F}^{\vee}$ agissant par addition et on note $[\pi,\pi']$ son quotient par cette action. Si $\pi=\pi'$, c’est un groupe et il est plus suggestif de le noter $Stab(i{\cal A}_{G,F}^*,\pi)$. Ce groupe ne dépend que de l’orbite de $\pi$ pour l’action de $i{\cal A}_{G,F}^*$. L’action du groupe $i{\cal A}_{G,F}^*$ conserve l’ensemble $\Pi_{disc}(G(F))$. On note $\Pi_{disc}(G(F))/i{\cal A}_{G,F}^*$ l’ensemble des orbites.
Soit $P=MU_{P}$ un sous-groupe parabolique semi-standard de $G$ et soit $\sigma$ une représentation admissible de $M(F)$. Pour $\lambda\in {\cal A}_{M,{\mathbb C}}^*/i{\cal A}_{M,F}^{\vee}$, introduisons la représentation induite $\pi_{\lambda}=Ind_{P}^G(\sigma_{\lambda})$ de $G(F)$. Elle se réalise naturellement dans un espace $V_{\pi_{\lambda}}$ de fonctions $e:G(F)\to V_{\sigma}$ se transformant à gauche par $P(F)$ selon la formule usuelle. On définit l’espace $V_{\sigma,P}$ des fonctions $e:K\to V_{\sigma}$ telles que
- $e(muk)=\sigma(m)e(k)$ pour tous $m\in M(F)\cap K$, $u\in U(F)\cap K$, $k\in K$;
- si $F$ est non archimédien, $e$ est localement constante;
- si $F$ est archimédien, $e$ est lisse et $K$-finie.
Par l’homomorphisme $V_{\pi_{\lambda}}\to V_{\sigma,P}$ de restriction à $K$, $\pi_{\lambda}$ se réalise dans $V_{\sigma,P}$ pour tout $\lambda$. Supposons $\sigma$ irréductible et unitaire. Si $\lambda\in i{\cal A}_{M,F}^*$, $\pi_{\lambda}$ est unitaire. Plus précisément, $\pi_{\lambda}$ conserve le produit hermitien sur $V_{\sigma,P}$ défini par $$(e_{1},e_{2})=\int_{K}(e_{1}(k),e_{2}(k))\,dk.$$
Revenons à $\sigma$ admissible et $\lambda\in {\cal A}_{M,{\mathbb C}}^*/i{\cal A}_{M,F}^{\vee}$. Soit $P'=MU_{P'}\in {\cal P}(M)$. Posons $\pi'_{\lambda}=Ind_{P'}^G(\sigma_{\lambda})$. On définit l’opérateur d’entrelacement $$J_{P'\vert P}(\sigma_{\lambda}):V_{\sigma,P}\to V_{\sigma,P'}.$$ Modulo les isomorphismes $$\begin{array}{ccccccc}V_{\sigma,P}&\to&V_{\pi_{\lambda}}&&V_{\sigma,P'}&\to&V_{\pi'_{\lambda}}\\e&\mapsto &e_{\lambda}&&e'&\mapsto&e'_{\lambda},\\ \end{array}$$ il existe un réel $c$ tel que, pour $Re(<\lambda,\check{\alpha}>)>c$ pour tout $\alpha\in \Delta_{P}$, cet opérateur soit donné par la formule $$<\epsilon,(J_{P'\vert P}(\sigma_{\lambda})(e))_{\lambda}(g)>=\int_{(U_{P}(F)\cap U_{P'}(F))\backslash U_{P'}(F)}<\epsilon,e_{\lambda}(ug)>\,du$$ pour tout $e\in V_{\sigma,P}$ et tout $\epsilon\in V_{\check{\sigma}}$, où $\check{\sigma}$ est la contragrédiente de $\sigma$. Il se prolonge méromorphiquement à tout ${\cal A}_{M,{\mathbb C}}^*/i{\cal A}_{M,F}^{\vee}$. Les opérateurs d’entrelacement vérifient de multiples propriétés, cf. par exemple \[A3\] paragraphe 1.
Soit $x\in G(F)$. Posons $M'=xMx^{-1}$, $P'=xPx^{-1}$ et supposons $M'$ semi-standard. On note $x\sigma$ la représentation $m'\mapsto \sigma(x^{-1}m'x)$ de $M'(F)$ dans $V_{\sigma}$. On note $\lambda\mapsto x\lambda$ l’application de ${\cal A}_{M,{\mathbb C}}^*$ dans ${\cal A}_{M',{\mathbb C}}^*$ déduite par fonctorialité de la conjugaison par $x$. Posons $\pi'_{x\lambda}=Ind_{P'}^G((x\sigma)_{x\lambda})$. On réalise $\pi_{\lambda}$ et $\pi'_{x\lambda}$ dans leurs modèles $V_{\pi_{\lambda}}$ et $V_{\pi'_{x\lambda}}$. Notons $\partial_{P}(x)$ le jacobien de l’application $ad_{x}:U_{P}\to U_{xPx^{-1}}$ (en fait, on peut écrire $x=m'k'=km$, avec $m'\in M'(F)$, $m\in M(F)$, $k,k'\in K$; on a $\partial_{P}(x)=\delta_{P}(m)=\delta_{P'}(m')$). Définissons l’opérateur $$\gamma(x):V_{\pi_{\lambda}}\to V_{\pi'_{x\lambda}}$$ par $(\gamma(x)e)(g)=\partial_{P}(x)^{1/2}e(x^{-1}g)$. Il vérifie la relation $\gamma(x)\circ\pi_{\lambda} (g)=\pi'_{x\lambda}(g)\circ\gamma(x)$.
Supposons $\sigma$ irréductible et unitaire et limitons-nous à $\lambda\in i{\cal A}_{M,F}^*$. En identifiant $V_{\pi_{\lambda}}$ à $V_{\sigma,P}$ et $V_{\pi'_{x\lambda}}$ à $V_{x\sigma,P'}$, ces espaces sont munis de produits hermitiens définis positifs pour lesquels les représentations $\pi_{\lambda}$ et $\pi'_{x\lambda}$ sont unitaires. Alors l’application $\gamma(x)$ est unitaire.
On appliquera ces constructions à des éléments du groupe de Weyl $W^G$. Soit $w\in W^G$. On choisit un représentant $\dot{w}$ de $w$ dans $K$. Le Levi $M'=w(M)$ est semi-standard. Par abus de notation, on pose simplement $w\sigma=\dot{w}\sigma$ et $\gamma(w)=\gamma(\dot{w})$. Notons que $\gamma(w)$, vu comme un opérateur de $V_{\sigma,P}$ dans $V_{w\sigma,w(P)}$, ne dépend pas de $\lambda$. On pourra aussi appliquer cette construction à des quotients tels que $W^G/W^M$. On pourra vérifier que nos formules ne dépendent pas des choix des relèvements $\dot{w}$.
Opérateurs normalisés
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Soient $M$ un Levi semi-standard et $\sigma\in \Pi_{disc}(M(F))$. Pour $P\in {\cal P}(M)$, l’opérateur $J_{P\vert \bar{P}}(\sigma_{\lambda})\circ J_{\bar{P}\vert P}(\sigma_{\lambda})$ est une homothétie qui ne dépend pas de $P$. On définit $m^G(\sigma_{\lambda})$ comme le produit du degré formel $d(\sigma)$ et de l’inverse du rapport de cette homothétie. Remarquons qu’en vertu des relations de compatibilité de nos mesures, $m^G(\sigma_{\lambda})$ ne dépend que des mesures sur $G(F)$ et $A_{M}(F)$.
On introduit des opérateurs normalisés $R_{P'\vert P}(\sigma_{\lambda})$ et des fonctions de normalisation $r_{P'\vert P}(\sigma_{\lambda})$, pour $P,P'\in {\cal P}(M)$, comme en \[A3\] paragraphe 2. On a l’égalité $J_{P'\vert P}(\sigma_{\lambda})=r_{P'\vert P}(\sigma_{\lambda})R_{P'\vert P}(\sigma_{\lambda})$. Pour $\lambda\in i{\cal A}_{M,F}^*$, $R_{P'\vert P}(\sigma_{\lambda})$ est unitaire. On sait que, pour toute racine réduite $\alpha$ de $A_{M}$, on peut définir une fonction $r_{\alpha}(\sigma_{\lambda})$ de sorte que $$(1) \qquad r_{P'\vert P}(\sigma_{\lambda})=\prod_{\alpha>_{P}0,\alpha<_{P'}0}r_{\alpha}(\sigma_{\lambda}),$$ la notation signifiant que $\alpha$ parcourt les racines réduites qui sont positives pour $P$ et négatives pour $P'$.
Il est utile de rappeler quelle est la forme des fonctions $r_{\alpha}(\sigma_{\lambda})$ dans le cas où $F={\mathbb R}$. Fixons un sous-tore fondamental $S$ de $M$. Son algèbre de Lie $\mathfrak{s}$ se décompose de façon unique en une somme directe $\mathfrak{a}_{M}\oplus \mathfrak{s}^{M}$ stable par l’action galoisienne. On pose $\mathfrak{h}_{M}=\mathfrak{a}_{M}({\mathbb R})$, $\mathfrak{h}^M=i\mathfrak{s}^M({\mathbb R})$, on définit l’algèbre de Lie réelle $\mathfrak{h}= \mathfrak{h}_{M}\oplus \mathfrak{h}^M$ et sa complexifiée $\mathfrak{h}_{{\mathbb C}}$. Remarquons que ${\cal A}_{M}$ s’identifie à $\mathfrak{h}_{M}$. A la série discrète $\sigma$ est associé un caractère infinitésimal qui est une orbite dans $i\mathfrak{h}^*$ pour l’action du groupe de Weyl absolu de $S$. Choisissons $\mu_{\sigma}$ dans cette orbite. On sait que la projection $\mu_{\sigma}^M$ de $\mu_{\sigma}$ sur $i\mathfrak{h}^{M,*}$ appartient à un réseau fixe de cet espace, plus précisément à un réseau de $(X^*(S)\otimes_{{\mathbb Z}}{\mathbb Q})\cap i\mathfrak{h}^{M,*}$. Soit $\alpha$ une racine réduite de $A_{M}$. D’après \[A3\] paragraphe 3, pour un choix convenable du point $\mu_{\sigma}$, la fonction $r_{\alpha}(\sigma_{\lambda})$ est produit d’une constante uniformément bornée (c’est-à-dire bornée indépendamment de $\sigma$ et $\lambda$), d’un nombre fini de fonctions $$\frac{1}{<\mu_{\sigma}+\lambda,\check{\beta}>},$$ où $\check{\beta}$ est une coracine de $S$ se restreignant à $\mathfrak{h}_{M}$ en un multiple positif et non nul de $\check{\alpha}$, et, éventuellement, d’une fonction $$\Gamma(\frac{<\mu_{\sigma}+\lambda,\check{\beta}>+N}{2})\Gamma(\frac{<\mu_{\sigma}+\lambda,\check{\beta}>+N+1}{2})^{-1}$$ où $\check{\beta}$ est une coracine de $S$ égale à un multiple positif et non nul de $\check{\alpha}$ et $N\in \{0,1\}$.
On utilisera les propriétés suivantes, pour $\lambda\in i{\cal A}_{M,F}^*$.
\(2) La fonction $\lambda\mapsto m^G(\lambda)$ est régulière, à croissance modérée et à valeurs positives ou nulles.
Les première et troisième propriétés sont dues à Harish-Chandra. La deuxième (qui ne concerne que le cas où $F$ est archimédien) résulte de la formule explicite \[A3\] proposition 3.1.
\(3) On a l’égalité $r_{P\vert P'}(\sigma_{\lambda})=\overline{r_{P'\vert P}(\sigma_{\lambda})}$.
Cela résulte des relations d’adjonction vérifiées tant par les opérateurs d’entrelacement que par les opérateurs normalisés.
\(4) On a l’égalité $\vert r_{\bar{P}\vert P}(\sigma_{\lambda})\vert ^2=d(\sigma)m^G(\sigma_{\lambda})^{-1}$.
Cela résulte de (3) et de la définition de $m^G(\sigma_{\lambda})$.
\(5) Pour $P,P'\in {\cal P}(M)$, la fonction $$\lambda\mapsto r_{P\vert \bar{P}}(\sigma_{\lambda})r_{P'\vert \bar{P}'}(\sigma_{\lambda})^{-1}$$ est régulière de valeur absolue $1$. Si $F$ est archimédien, ses dérivées sont à croissance modérée.
La première assertion résulte de (4). Pour la seconde, on peut supposer $F={\mathbb R}$ car dans le cas où $F={\mathbb C}$, les fonctions de normalisation sont les mêmes que pour le groupe sur ${\mathbb R}$ déduit de $G$ par restriction des scalaires. La décomposition (1) nous ramène à considérer une fonction $r_{\alpha}(\sigma_{\lambda})r_{-\alpha}(\sigma_{\lambda})^{-1}$, ou encore, d’après (3), $r_{\alpha}(\sigma_{\lambda})\overline{r_{\alpha}(\sigma_{\lambda})}^{-1}$. D’après ce que l’on a dit ci-dessus, une telle fonction est produit fini de termes $$\frac{<\mu^M,\check{\beta}>-<\mu_{M}+\lambda,\check{\beta}>} {<\mu^M,\check{\beta}>+<\mu_{M}+\lambda,\check{\beta}>}$$ et éventuellement d’un terme $$\frac{\Gamma(\frac{<\mu_{M}+\lambda,\check{\beta}>}{2})\Gamma(\frac{-<\mu_{M}+\lambda,\check{\beta}>+1}{2})}{\Gamma(\frac{-<\mu_{M}+\lambda,\check{\beta}>}{2})\Gamma(\frac{<\mu_{M}+\lambda,\check{\beta}>+1}{2})}.$$ C’est un simple exercice de montrer que les dérivées de telles fonctions sont à croissance modérée en $\lambda$.
\(6) La fonction $\lambda\mapsto r_{P'\vert P}(\sigma_{\lambda})^{-1}$ est régulière et à croissance modérée.
Par la décomposition (1), on peut supposer $P'$ et $P$ adjacents. Il y a alors un parabolique $Q=LU_{Q}\in {\cal F}(M)$, qui est minimal parmi les paraboliques contenant strictement $M$, de sorte que $P,P'\subset Q$ et $P'\cap L=\overline{P\cap L}$. Par les propriétés habituelles, $r_{P'\vert P}(\sigma_{\lambda})=r^L_{\overline{P\cap L}\vert P\cap L}(\sigma_{\lambda})$. Alors la propriété cherchée résulte de (2) et (4) appliquées au groupe $L$.
\(7) Pour quatre éléments $P_{1},P_{2},P_{3},P_{4}\in {\cal P}(M)$, la fonction $$\lambda\mapsto m^G(\sigma_{\lambda})r_{P_{1}\vert P_{2}}(\sigma_{\lambda})r_{P_{3}\vert P_{4}}(\sigma_{\lambda})$$ est régulière et à croissance modérée.
On peut l’écrire comme produit de $$r_{\bar{P}_{2}\vert P_{1}}(\sigma_{\lambda})^{-1}r_{\bar{P}_{4}\vert P_{3}}(\sigma_{\lambda})^{-1}$$ et de $$m^G(\sigma_{\lambda})r_{\bar{P}_{2}\vert P_{2}}(\sigma_{\lambda})r_{\bar{P}_{4}\vert P_{4}}(\sigma_{\lambda}).$$ La première fonction vérifie les propriétés requises d’après (6). La seconde est de valeur absolue constante d’après (4).
$R$-groupes
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Soient $M$ un Levi semi-standard et $\sigma$ une représentation irréductible de $M(F)$ de la série discrète. On note ${\cal N}^G(\sigma)$ l’ensemble des couples $(A,n)$ où $A$ est un automorphisme unitaire de $V_{\sigma}$ et $n\in Norm_{G(F)}(M)$ (le normalisateur de $M$ dans $G(F)$) qui vérifient la condition $$A\circ n\sigma(x)=\sigma(x)\circ A$$ pour tout $x\in M(F)$. C’est un groupe pour le produit $(A,n)(A',n')=(AA',nn')$. Il y a un homomorphisme naturel $M(F)\to {\cal N}^G(\sigma)$ qui, à $x\in M(F)$, associe $(\sigma(m),m)$. L’image est un sous-groupe distingué de ${\cal N}^G(\sigma)$. On note ${\cal W}^G(\sigma)$ le quotient. On a une suite exacte $$(1) \qquad 1\to {\mathbb U}\to {\cal W}^G(\sigma)\to W^G(\sigma)\to 1,$$ où ${\mathbb U}$ est le groupe des nombres complexes de module $1$ et $$W^G(\sigma)=\{w\in W^G; w(M)=M, w\sigma\simeq \sigma\}/W^M.$$
Soit $P\in {\cal P}(M)$, posons $\pi=Ind_{P}^G(\sigma)$ que l’on réalise dans l’espace naturel $V_{\pi}$. Pour $(A,n)\in {\cal N}^G(\sigma)$, on définit l’automorphisme $r_{P}(A,n)$ de $V_{\pi}$ par $r_{P}(A,n)=R_{P\vert nPn^{-1}}(\sigma)\circ\gamma(n)\circ A$. Expliquons que $A$ désigne ici l’opérateur déduit de $A$ par fonctorialité, c’est-à-dire celui qui, à une fonction $e:G(F)\to V_{\sigma}$ associe la fonction $g\mapsto Ae(g)$. On vérifie que $r_{P}(A,n)$ est un entrelacement, c’est-à-dire que $$r_{P}(A,n)\circ \pi(g)=\pi(g)\circ r_{P}(A,n)$$ pour tout $g\in G(F)$. On vérifie que l’application $r_{P}$ ainsi définie est une représentation unitaire de ${\cal N}^G(\sigma)$. Si on remplace $P$ par un autre $P'\in {\cal P}(M)$, les représentations $r_{P}$ et $r_{P'}$ sont équivalentes: on a $r_{P}(A,n)\circ R_{P\vert P'}(\sigma)=R_{P\vert P'}(\sigma)\circ r_{P'}(A,n)$. Le sous-groupe des $(A,n)\in {\cal N}^G(\sigma)$ tels que $r_{P}(A,n)$ soit l’identité de $V_{\pi}$ est donc indépendant de $P$. Il contient $M(F)$ (identifié comme ci-dessus à un sous-groupe de ${\cal N}^G(\sigma)$). On note $W^G_{0}(\sigma)$ le quotient de ce sous-groupe par $M(F)$. On pose $${\cal R}^G(\sigma)={\cal W}^G(\sigma)/W^G_{0}(\sigma).$$
Par la suite (1), $W^G_{0}(\sigma)$ s’identifie à un sous-groupe distingué de $W^G(\sigma)$. On pose $R^G(\sigma)=W^G(\sigma)/W^G_{0}(\sigma)$. C’est le $R$-groupe de $\sigma$. On a une suite exacte $$1\to {\mathbb U}\to {\cal R}^G(\sigma)\to R^G(\sigma)\to 1.$$ La représentation $r_{P}$ se quotiente en une représentation de ${\cal R}^G(\sigma)$ telle que $z\in {\mathbb U}$ agisse par l’homothétie de rapport $z$.
[**Dans la suite, toutes les représentations de ${\cal R}^G(\sigma)$ seront supposées vérifier cette condition.**]{}
On note $Irr({\cal R}^G(\sigma))$ l’ensemble des classes d’isomorphisme de représentations irréductibles de ${\cal R}^G(\sigma)$. Le résultat principal de la théorie du $R$-groupe est qu’il existe une bijection $\rho\mapsto \pi_{\rho}$ de $Irr({\cal R}^G(\sigma))$ sur l’ensemble des classes de représentations irréductibles de $G(F)$ qui sont des sous-représentations de $\pi$, de sorte que la représentation $r_{P}\otimes \pi$ de ${\cal R}^G(\sigma)\times G(F)$ dans $V_{\pi}$ se décompose en $$(2) \qquad \oplus_{\rho\in Irr({\cal R}^G(\sigma))}\rho\otimes \pi_{\rho}.$$ Notons que la correspondance $\rho\mapsto \pi_{\rho}$ ne dépend pas de $P$: si on remplace $P$ par $P'$, l’opérateur $R_{P\vert P'}(\sigma)$ entrelace les représentations de ${\cal R}^G(\sigma)\times G(F)$ dans $V_{\pi}$ et dans $V_{\pi'}$, où $\pi'=Ind_{P'}^G(\sigma)$.
Harish-Chandra a décrit les groupes $W^G_{0}(\sigma)$ et $R^G(\sigma)$. Soit $\alpha$ une racine réduite de $A_{M}$ dans $G$. Il lui est associé un groupe de Levi $M_{\alpha}\in {\cal L}(M)$ qui est minimal dans ${\cal L}(M)-\{M\}$. Suposons que la mesure de Plancherel $m^{M_{\alpha}}(\sigma)$ soit nulle. On montre qu’alors $W^{M_{\alpha}}(M)$ a deux éléments: l’identité et une symétrie $s_{\alpha}$. L’ensemble des $\alpha$ vérifiant les conditions ci-dessus forme un système de racines et $W^G_{0}(\sigma)$ est le groupe de Weyl de ce système. En particulier, $W^G_{0}(\sigma)$ est engendré par ces symétries $s_{\alpha}$. Fixons un ensemble de racines positives dans ce système de racines. On peut identifier $R^G(\sigma)$ au sous-ensemble des $w\in W^G(\sigma)$ qui conservent l’ensemble fixé de racines positives. On a alors la décomposition $$W^G(\sigma)=W^{G}_{0}(\sigma)\rtimes R^G(\sigma).$$
[**Remarque.**]{} Les définitions ci-dessus dépendent de choix d’opérateurs d’entrelacement normalisés. Si on change de choix, on obtient une nouvelle représentation, notons-la $\underline{r}_{P}$. On vérifie qu’il existe un caractère unitaire $\chi$ de $W^G(\sigma)$, que l’on remonte en un caractère de ${\cal N}^G(\sigma)$, de sorte que $\underline{r}_{P}(A,n)=\chi(A,n)r_{P}(A,n)$. On peut introduire l’automorphisme $\boldsymbol{\chi}$ de ${\cal N}^G(\sigma)$ qui à $(A,n)$ associe $(\chi(A,n)A,n)$. On a alors $\underline{r}_{P}=r_{P}\circ\boldsymbol{\chi}$. A l’automorphisme $\boldsymbol{\chi}$ près, nos définitions sont donc indépendantes des choix d’opérateurs normalisés. En tout cas, dans la suite, ces choix sont fixés.
Soit $\lambda\in i{\cal A}_{G,F}^*$. L’application $$\begin{array}{ccc}{\cal N}^G(\sigma)&\to &{\cal N}^G(\sigma_{\lambda})\\ (A,n)&\mapsto& (Ae^{<\lambda,H_{G}(n)>},n)\\ \end{array}$$ est un isomorphisme. A cause de la torsion par $e^{<\lambda,H_{G}(n)>}$, cet isomorphisme est compatible avec les plongements de $M(F)$ dans les deux groupes ${\cal N}^G(\sigma)$ et ${\cal N}^G(\sigma_{\lambda})$. Il se quotiente en un isomorphisme ${\cal R}(\sigma)\simeq {\cal R}(\sigma_{\lambda})$ compatible avec l’identité $R(\sigma)= R(\sigma_{\lambda})$.
Coefficients
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Rappelons qu’Harish-Chandra a défini une fonction $\Xi$ sur $G(F)$. Elle vérifie les propriétés:
\(1) la fonction $\Xi$ est biinvariante par $K$.
\(2) il existe $d\in {\mathbb N}$ tel que l’on ait les majorations $$\delta_{0}(m)^{-1/2}<<\Xi(m)<<\delta_{0}(m)^{-1/2}(1+\vert H_{0}(m)\vert )^d$$ pour tout $m\in M_{0}(F)^{\geq}$.
Soient $P=MU_{P}$ un sous-groupe parabolique semi-standard et $\sigma\in \Pi_{disc}(M(F))$. Pour $\lambda\in i{\cal A}_{M,F}^*$, posons $\pi_{\lambda}=Ind_{P}^G(\sigma_{\lambda})$, que l’on réalise dans l’espace $V_{\sigma,P}$. Pour $u,v\in V_{\sigma,P}$, on définit une fonction coefficient sur $G(F)$ qui, à $g\in G(F)$, associe le produit scalaire $(u,\pi_{\lambda}(g)v)$. On a
\(3) $\vert (u,\pi_{\lambda}(g)v)\vert <<\Xi(g)$ pour tout $g\in G(F)$ et tout $\lambda\in i{\cal A}_{M,F}^*$;
plus précisément
\(4) $\int_{K}\vert (u(k),(\pi_{\lambda}(g)v)(k))\vert \,dk\,<<\Xi(g)$ pour tout $g\in G(F)$ et tout $\lambda\in i{\cal A}_{M,F}^*$.
Cela résulte de la preuve du lemme VI.2.2 de \[W\].
Soit $Q=LU_{Q}$ un sous-groupe parabolique standard. Notons $W^G(L\vert P)=\{s\in W^G; s(M)\subset L, P_{0}\cap L\subset s(P)\cap L\}/W^M$. Rappelons que pour tout $s$ dans cet ensemble, on fixe un relèvement de $s$ dans $K$, que l’on note encore $s$. On pose $Q_{s}=(s(P)\cap L)U_{Q}$ et $\underline{Q}_{s}=(s(P)\cap L)U_{\bar{Q}}$. On introduit la représentation $\pi^L_{s\lambda}=Ind_{s(P)\cap L}^L((s\sigma)_{s\lambda})$, que l’on réalise dans l’espace $V^L_{s\sigma,s(P)\cap L}$. Le terme $$J_{Q_{s}\vert s(P)}((s\sigma)_{s\lambda})\circ\gamma(s)u$$ est une fonction sur $K$ appartenant à l’espace de la représentation $Ind_{Q_{s}}^G((s\sigma)_{s\lambda})$. Sa restriction à $K\cap L(F)$ appartient à $V^L_{s\sigma,s(P)\cap L}$. On note $u_{s}(\lambda)$ cette restriction. De même, on note $v_{s}(\lambda)$ la restriction de $$J_{\underline{Q}_{s}\vert s(P)}((s\sigma)_{s\lambda})\circ\gamma(s)v$$ à $K\cap L(F)$. Pour $l\in L(F)$, posons $$E_{\flat}(l,\lambda)=\gamma(G\vert L)^{-1}\sum_{s\in W^G(L\vert P)}(u_{s}(\lambda),\pi^L_{s\lambda}(l)v_{s}(\lambda)).$$ On appelle cette fonction le terme constant faible du coefficient $(u,\pi_{\lambda}(g)v)$.
[0.3cm[**[Proposition]{}**]{}. [ *[Soit $\nu>0$ un réel. Il existe $c>0$ et une fonction $C$ sur $i{\cal A}_{M,F}^*$, lisse, à croissance modérée et à valeurs positives de sorte que l’on ait la majoration $$\vert (u,\pi_{\lambda}(m)v)-\delta_{Q}(m)^{-1/2}E_{\flat}(m,\lambda)\vert \leq C(\lambda)\delta_{Q}(m)^{-1/2}\Xi^L(m)e^{-c\vert H_{0}(m)\vert }$$ pour tout $\lambda\in i{\cal A}_{M,F}^*$ et tout $m\in M_{0}(F)^{\geq}$ vérifiant la condition $$<\alpha,H_{0}(m)>\geq \nu\vert H_{0}(m)\vert$$ pour tout $\alpha\in \Delta_{0}-\Delta_{0}^Q$.]{}*]{}0.3cm]{}
Cf. \[A1\] lemme 7.1 et \[W\] lemme VI.2.3 dans le cas non-archimédien.
La formule de Plancherel
------------------------
Cette formule exprime toute fonction $K$-finie $f\in C_{c}^{\infty}(G(F))$ à l’aide de ses actions dans les représentations induites unitaires de représentations de la série discrète des Levi de $G$ (la condition que $f$ est $K$-finie n’est une restriction que si $F$ est archimédien). Dans la formule ci-dessous, pour tout $M_{disc}\in {\cal L}(M_{0})$, on fixe un parabolique $S\in {\cal P}(M_{disc})$ et, pour tout élément de $\Pi_{disc}(M_{disc}(F))/i{\cal A}_{M_{disc},F}^*$, on fixe un point-base $\sigma$ dans cette orbite. Alors, pour toute fonction $K$-finie $f\in C_{c}^{\infty}(G(F))$ et tout $g\in G(F)$, on a l’égalité $$f(g)=\sum_{M_{disc}\in {\cal L}(M_{0})}\vert W^{M_{disc}}\vert \vert W^G\vert ^{-1}\sum_{\sigma\in \Pi_{disc}(M_{disc}(F))/i{\cal A}_{M_{disc},F}^*} \vert Stab(i{\cal A}_{M_{disc},F}^*,\sigma)\vert ^{-1}$$ $$\int_{i{\cal A}_{M_{disc},F}^*}m^G(\sigma_{\lambda})trace(Ind_{S}^G(\sigma_{\lambda},g^{-1})Ind_{S}^G(\sigma_{\lambda},f))\,d\lambda.$$
La formule du produit scalaire d’Arthur
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Pour exprimer cette formule sous une forme suffisamment générale, on fixe un sous-tore $D\subset A_{G}$ et un caractère unitaire $\omega$ de $G(F)$ dont la restriction à $D(F)$ est triviale. On adapte nos notations en notant par exemple ${\cal A}_{0}^D$ l’orthogonal du sous-espace ${\cal A}_{D}\subset {\cal A}_{0}$ et $H_{D}$, $H^D$ les projections d’un élément $H\in {\cal A}_{0}$ sur ${\cal A}_{D}$ et ${\cal A}^D$. On considère un élément $T\in {\cal A}_{0}$ que l’on traite comme une variable. Mais on suppose qu’il reste dans un sous-ensemble fixe de ${\cal A}_{0}$ défini par des relations
$<\alpha,T>>0$ pour tout $\alpha\in \Delta_{0}$;
$<\alpha,T>> c_{\star}\vert T\vert $ pour tout $\alpha\in \Delta_{0}$, où $c_{\star}>0$ est un réel fixé;
$T\in {\cal A}_{M_{0},F}\otimes_{{\mathbb Z}}{\mathbb Q}$ si $F$ est non-archimédien.
Remarquons que, dans un tel domaine, les fonctions $\vert T\vert $ et $<\alpha,T>$, pour $\alpha\in \Delta_{0}$, sont équivalentes. On note $\kappa^T$ la fonction caractéristique de l’ensemble des $g\in G(F)$ tels que $\phi^G(h_{0}(g)-T)=1$, cf. 1.1 pour la définition de $h_{0}(g)$.
Soient $P=MU_{P}$ et $P'=M'U_{P'}$ deux sous-groupes paraboliques semi-standard et soient $\sigma\in \Pi_{disc}(M(F))$, $\sigma'\in \Pi_{disc}(M'(F))$. On suppose que:
- les restrictions à $D(F)$ des caractères centraux de $\sigma$ et $\sigma'$ coïncident.
Fixons $u,v\in V_{\sigma,P}$ et $u',v'\in V_{\sigma',P'}$. Pour $\lambda\in i{\cal A}_{M,F}^*$ et $\lambda'\in i{\cal A}_{M',F}^*$ tels que $\lambda_{D}-\lambda'_{D}\in i{\cal A}_{D,F}^{\vee}$, posons $\pi_{\lambda}=Ind_{P}^G(\sigma_{\lambda})$ et $\pi'_{\lambda'}=Ind_{P'}^G(\sigma'_{\lambda'})$. Pour $X\in {\cal A}_{G,F}$, notons $G(F;X)$ l’ensemble des $g\in G(F)$ tels que $H_{G}(g)= X$. On munit cet ensemble d’une mesure de sorte que l’on ait l’égalité $$\int_{ G(F)}f(g)\,dg\,=\int_{ {\cal A}_{G,F}}\int_{G(F;X)}f(g)\,dg\,dX$$ pour toute $f\in C_{c}^{\infty}( G(F))$. On note $D(F)_{c}=Ker(H_{G})\cap D(F)$. On munit ce groupe d’une mesure de Haar, et l’ensemble $D(F)_{c}\backslash G(F;X)$ de la mesure quotient. Remarquons que $D(F)_{c}\backslash G(F;X)=D(F)\backslash D(F)G(F;X)$, ce dernier quotient est donc muni d’une mesure.
Posons $$\omega^T(X;\lambda,\lambda')=\int_{D(F)\backslash D(F)G(F;X)}(v,\pi_{\lambda}(g)u)(\pi'_{\lambda'}(g)v',u')\omega(g)\kappa^T(g)\,dg.$$ Cette expression ne dépend que de la classe $X+{\cal A}_{D,F}$.
La formule d’Arthur calcule une valeur approchée de $\omega^T(X;\lambda,\lambda')$. Rappelons ce résultat.
Soit $S=M_{S}U_{S}$ un sous-groupe parabolique standard. Notons $W^G(M_{S}\vert M)=\{s\in W^G; s(M)=M_{S}\}/W^{M}$. On définit de même $W^G(M_{S}\vert M')$. Soient $s\in W^G(M_{S}\vert M)$, $s'\in W^G(M_{S}\vert M')$, $k,k'\in K$. Définissons les éléments $u_{s}(k',\lambda), v_{s}(k,\lambda)\in V_{\sigma}$ et $u'_{s'}(k,\lambda'),v'_{s'}(k',\lambda')\in V_{\sigma'}$ par les formules suivantes $$u_{s}(k',\lambda)=(J_{\bar{S}\vert s(P)}((s\sigma)_{s\lambda})\circ\gamma(s)u)(k'),$$ $$v_{s}(k,\lambda)=(J_{S\vert s(P)}((s\sigma)_{s\lambda})\circ\gamma(s)v)(k),$$ $$u'_{s'}(k,\lambda')=(J_{S\vert s'(P')}((s'\sigma')_{s'\lambda'})\circ\gamma(s')u')(k),$$ $$v'_{s'}(k',\lambda')=(J_{\bar{S}\vert s'(P')}((s'\sigma')_{s'\lambda'})\circ\gamma(s')v')(k').$$ On rappelle que les termes entre parenthèses sont des fonctions sur $K$; cela a donc un sens de les évaluer en $k$ ou $k'$. Pour des éléments $\epsilon,\eta\in V_{\sigma}$ et $\epsilon',\eta'\in V_{\sigma'}$, et pour $\Lambda\in {\cal A}_{M_{S},{\mathbb C}}^*$ tel que $\Lambda_{D}\in i{\cal A}_{D,F}^{\vee}$, posons $$r^T_{S,s,s'}(X;\epsilon,\eta,\epsilon',\eta';\Lambda)= \int_{D(F)\backslash D(F) M_{S}^G(F;X)}(\eta,s\sigma(x)\epsilon)(s'\sigma'(x)\eta',\epsilon')$$ $$\phi_{S}^G(H_{0}(x)-T)e^{<\Lambda,H_{0}(x)>}\omega(x)\,dx,$$ où $M_{S}^G(F;X)=M_{S}(F)\cap G(F;X)$ et la mesure est définie de façon analogue à celle sur $D(F)\backslash D(F)G(F;X)$. Cette intégrale est absolument convergente si $Re<\Lambda,\check{\alpha}>>0$ pour tout $\alpha\in \Delta_{S}$ et se prolonge en une fonction méromorphe définie pour tout $\Lambda\in{\cal A}_{M_{S},{\mathbb C}}^*$ tel que $\Lambda_{D}\in i{\cal A}_{D,F}^{\vee}$. Posons $$r^T_{S,s,s'}(X;\lambda,\lambda';\Lambda)=\int_{K\times K}\omega(k'k^{-1})$$ $$r^T_{S,s,s'}(X,u_{s}(k',\lambda),v_{s}(k,\lambda),u'_{s'}(k,\lambda'),v'_{s'}(k',\lambda');\Lambda)\,dk\,dk'.$$ Pour des points $\lambda$ et $\lambda'$ où les opérateurs d’entrelacement intervenant ci-dessus n’ont pas de pôles, cette expression est holomorphe au point $\Lambda=s\lambda-s'\lambda'$. On pose
$$r^T(X;\lambda,\lambda')=\sum_{S; P_{0}\subset S}\sum_{s\in W^G(M_{S}\vert M),s'\in W^G(M_{S}\vert M')}r^T_{S,s,s'}(X;\lambda,\lambda'; s\lambda-s'\lambda').$$
On obtient ainsi une fonction méromorphe en $\lambda$ et $\lambda'$.
[0.3cm[**[Théorème (Arthur)]{}**]{}. [ *[Il existe $c>0$ et une fonction lisse et à croissance modérée $C$ sur $i{\cal A}_{M,F}^*\times i{\cal A}_{M',F}^*$, à valeurs positives, de sorte que, pour tous $\lambda\in i{\cal A}_{M,F}^*$, $\lambda'\in i{\cal A}_{M',F}^*$ pour tous $X$, $T$, $u,v,u',v'$, on ait la majoration $$\vert \omega^T(X;\lambda,\lambda')-r^T(X;\lambda,\lambda')\vert \leq m^G(\sigma_{\lambda})^{-1/2}m^G(\sigma_{\lambda'})^{-1/2}C(\lambda,\lambda')e^{-c\vert T\vert }\vert u\vert \vert v\vert \vert u'\vert \vert v'\vert .$$]{}*]{}0.3cm]{}
On peut exprimer plus explicitement le terme $r^T_{S,s,t}(X;\lambda,\lambda';\Lambda)$. Commençons par calculer $r^T_{S,s,s'}(X;\epsilon,\eta,\epsilon',\eta';\Lambda)$. Pour simplifier les notations, on fixe $\epsilon,\eta,\epsilon',\eta'$. Pour $Y\in {\cal A}_{M_{S},F}$, posons $$r_{S,s,s'}(Y )=\int_{D(F)\backslash D(F)M_{S}(F;Y)}(\eta,s\sigma(x)\epsilon)(s'\sigma'(x)\eta',\epsilon')\omega(x)\,dx,$$ où $M_{S}(F;Y)=\{x\in M_{S}(F); H_{M_{S}}(x)= Y\}$. Posons comme en 1.6 $${\cal A}_{M_{S},F}^{G}(X)=\{Y\in {\cal A}_{M_{S},F}; Y_{G}= X\} .$$ On a les égalités $$r^T_{S,s,s'}(X;\epsilon,\eta,\epsilon',\eta';\Lambda)=\int_{({\cal A}_{M_{S},F}^{G}(X)+{\cal A}_{D,F})/{\cal A}_{D,F}}\phi_{S}^G(Y-T)e^{<\Lambda,Y>}r_{S,s,s'}(Y)\,dY$$ $$=\int_{{\cal A}_{M_{S},F}^{G}(X)}\phi_{S}^G(Y-T)e^{<\Lambda,Y>}r_{S,s,s'}(Y)\,dY.$$ Notons $\omega_{\sigma}$ et $\omega_{\sigma'}$ les caractères centraux de $\sigma$ et $\sigma'$. Il est clair que, pour $Y\in {\cal A}_{M_{S},F}$, $r_{S,s,s'}(Y)=0$ si la restriction de $s(\omega_{\sigma})s'(\omega_{\sigma'})^{-1}\omega$ à $A_{M_{S}}(F)_{c}$ est non triviale. Supposons cette condition vérifiée. Il existe alors un élément $\Lambda_{s,s'}\in i{\cal A}_{M_{S}}^*$ tel que $$(s(\omega_{\sigma})s'(\omega_{\sigma'})^{-1}\omega)(a)=e^{<\Lambda_{s,s'},H_{M_{S}}(a)>}$$ pour tout $a\in A_{M_{S}}(F)$. Cet élément est uniquement déterminé modulo $i{\cal A}_{A_{M_{S}},F}^{\vee}$ et vérifie $(\Lambda_{s,s'})_{D}\in i{\cal A}_{D,F}^{\vee}$. La fonction $$Y\mapsto e^{-<\Lambda_{s,s'},Y>}r_{S,s,s'}( Y)$$ ne dépend que de la classe $Y+{\cal A}_{A_{M_{S}},F}$. L’ensemble $A_{M_{S}}(F)M_{S}(F;Y)$ est ouvert dans $M_{S}(F)$ et donc muni d’une mesure. On vérifie que, pour une fonction $f$ sur le quotient $A_{M_{S}}(F)\backslash A_{M_{S}}(F)M_{S}(F;Y)$, on a la formule d’intégration $$\int_{D(F)\backslash D(F)M_{S}(F;Y)}f(x)\,dx\,=C\int_{A_{M_{S}}(F)\backslash A_{M_{S}}(F)M_{S}(F;Y)}f(x)\,dx,$$ où $$C=mes(D(F)_{c})^{-1}mes(i{\cal A}_{M_{S},F}^*)^{-1}[{\cal A}_{M_{S},F}:{\cal A}_{A_{M_{S}},F}].$$ De ces considérations résulte l’égalité
$e^{-<\Lambda_{s,s'},Y>}r_{S,s,s'}( Y)=$ $$C\int_{A_{M_{S}}(F)\backslash A_{M_{S}}(F)M_{S}(F;Y)}(\eta,s\sigma(x)\epsilon)(s'\sigma'(x)\eta',\epsilon')\omega(x)e^{-<\Lambda_{s,s'},H_{M_{S}}(x)>}\,dx.$$
Pour $\mu\in i{\cal A}_{A_{M_{S}},F}^{\vee}/i{\cal A}_{M_{S},F}^{\vee}$, posons $$r^T_{S,s,s'}(\mu)=\sum_{Y\in{\cal A}_{M_{S},F}/{\cal A}_{A_{M_{S}},F}}e^{-<\Lambda_{s,s'}+\mu,Y>}r^T_{S,s,s'}(Y).$$ Par inversion de Fourier, on a l’égalité $$(1) \qquad r_{S,s,s'}( Y)= [{\cal A}_{M_{S},F}:{\cal A}_{A_{M_{S}},F}]^{-1}\sum_{\mu\in i{\cal A}_{A_{M_{S}},F}^{\vee}/i{\cal A}_{M_{S},F}^{\vee}}e^{<\Lambda_{s,s'}+\mu,Y>}r^T_{S,s,s'}(\mu ).$$ L’égalité $s(M)=M_{S}$ entraîne $mes(i{\cal A}_{M_{S},F}^*)=mes(i{\cal A}_{M,F}^*)$. D’autre part, on a $$r^T_{S,s,s'}(\mu )=C \int_{A_{M_{S}}(F)\backslash M_{S}(F)}(\eta,s\sigma(x)\epsilon)(s'\sigma'(x)\eta',\epsilon')e^{-<\Lambda_{s,s'}+\mu,H_{M_{S}}(x)>}\omega(x)\,dx.$$ Cette dernière expression est le produit scalaire de deux coefficients de représentations de la série discrète. Il est nul si la représentation $s'\sigma'$ n’est pas équivalente à la représentation $(\omega s\sigma)_{-\Lambda_{s,s'}-\mu}$, autrement dit si $\Lambda_{s,s'}+\mu\not\in [s'\sigma',\omega s\sigma]$. Soit $\nu\in [s'\sigma',\omega s\sigma]$. Introduisons un isomorphisme unitaire $A_{\nu}:V_{\sigma}\to V_{\sigma'}$ tel que $ (s'\sigma')(x)\circ A_{\nu}=\omega(x)A_{\nu}\circ(s\sigma)_{-\nu}(x) $ pour tout $x\in M'(F)$. Alors $$r^T_{S,s,s'}(\nu-\Lambda_{s,s'} )= Cd(\sigma)^{-1}(A_{\nu}\eta,\epsilon')(\eta',A_{\nu}\epsilon).$$ La formule (1) se récrit $$r_{S,s,s'}(Y)= mes(D(F)_{c})^{-1}mes(i{\cal A}_{M,F}^*)^{-1}d(\sigma)^{-1}\sum_{\nu\in [s'\sigma',\omega s\sigma]}e^{<\nu,Y>}(A_{\nu}\eta,\epsilon')(\eta',A_{\nu}\epsilon).$$ On a défini $\epsilon_{S}^{G,T}(X;\Lambda)$ en 1.6. On obtient finalement $$r^T_{S,s,s'}(X;\epsilon,\eta,\epsilon',\eta';\Lambda)=mes(D(F)_{c})^{-1}mes(i{\cal A}_{M,F}^*)^{-1} d(\sigma)^{-1}$$ $$\sum_{\nu\in [s'\sigma',\omega s\sigma]}(A_{\nu}\eta,\epsilon')(\eta',A_{\nu}\epsilon)\epsilon_{S}^{G,T}(X;\Lambda+\nu).$$
Pour calculer $r^T_{S,s,s'}(X;\lambda,\lambda';\Lambda)$, on doit remplacer $\epsilon,\eta,\epsilon',\eta'$ par $u_{s}(k',\lambda)$, $v_{s}(k,\lambda)$, $u'_{s'}(k,\lambda')$, $v'_{s'}(k',\lambda')$, multiplier par $\omega(k'k^{-1})$ puis intégrer en $k,k'\in K$. Il apparaît des intégrales $$\int_{K}(v'_{s'}(k',\lambda'),A_{\nu}u_{s}(k',\lambda) )\omega(k')\,dk',$$ $$\int_{K}(A_{\nu}v_{s}(k,\lambda),u'_{s'}(k,\lambda'))\omega(k)^{-1}\,dk.$$ Considérons la première, la seconde étant analogue. L’opérateur $A_{\nu}$ définit par fonctorialité un opérateur de $V_{ s\sigma,\bar{S}}$ dans $V_{\omega^{-1} s'\sigma',\bar{S}}$, que nous notons encore $A_{\nu}$. Notons $\underline{\omega}$ l’opérateur qui, à une fonction $f$ sur $K$, associe la fonction $k\mapsto \omega(k)f(k)$. Alors $\underline{\omega}\circ A_{\nu}$ envoie $V_{s\sigma,\bar{S}}$ dans $V_{s'\sigma',\bar{S}}$. L’intégrale ci-dessus se récrit $$\int_{K}((J_{\bar{S}\vert s'(P')}((s'\sigma')_{s'\lambda'})\circ\gamma(s')v')(k'), (\underline{\omega}\circ A_{\nu}\circ J_{\bar{S}\vert s(P)}((s\sigma)_{s\lambda})\circ \gamma(s)u)(k'))\,dk'.$$ Par définition du produit scalaire dans $V_{s'\sigma',\bar{S}}$, ce n’est autre que $$(J_{\bar{S}\vert s'(P')}((s'\sigma')_{s'\lambda'})\circ\gamma(s')v', \underline{\omega}\circ A_{\nu}\circ J_{\bar{S}\vert s(P)}((s\sigma)_{s\lambda})\circ \gamma(s)u).$$ On obtient alors $$(2) \qquad r^T_{S,s,s'}(X;\lambda,\lambda';\Lambda)=mes(D(F)_{c})^{-1}mes(i{\cal A}_{M,F}^*)^{-1}d(\sigma)^{-1}$$ $$\sum_{\nu\in [s'\sigma',\omega s\sigma]}(J_{\bar{S}\vert s'(P')}((s'\sigma')_{s'\lambda'})\circ\gamma(s')v', \underline{\omega}\circ A_{\nu}\circ J_{\bar{S}\vert s(P)}((s\sigma)_{s\lambda})\circ \gamma(s)u)$$ $$(\underline{\omega}\circ A_{\nu}\circ J_{S\vert s(P)}((s\sigma)_{s\lambda})\circ\gamma(s)v,J_{S\vert s'(P')}((s'\sigma')_{s'\lambda'})\circ\gamma(s')u')\epsilon_{S}^{G,T}(X;\Lambda+\nu).$$
Espaces tordus
==============
Notations
---------
Soit $\tilde{G}$ un espace tordu sous $G$. Rappelons qu’un tel espace est une variété algébrique sur $F$ munie d’actions algébriques de $G$ à droite et à gauche $$\begin{array}{ccc}G\times \tilde{G}\times G&\to&\tilde{G}\\ (g,\gamma,g')&\mapsto &g\gamma g'\\ \end{array}$$ pour chacune desquelles $\tilde{G}$ est un espace principal homogène. On impose la condition
\(1) $\tilde{G}(F)\not=\emptyset$.
Pour $\gamma\in \tilde{G}$, on note $ad_{\gamma}$ l’automorphisme de $G$ tel que $\gamma g=ad_{\gamma}(g)\gamma$ pour tout $g\in G$. Il arrive que $ad_{\gamma}$ induise sur certains objets des automorphismes indépendants de $\gamma$. On note alors $\theta$ ces automorphismes. Par exemple, $\tilde{G}$ détermine un automorphisme $\theta$ de $Z_{G}$ ou de $A_{G}$. On impose la condition
\(2) l’automorphisme $\theta$ de $Z_{G}$ est d’ordre fini.
[**Remarque.**]{} Les hypothèses (1) et (2) impliquent qu’il existe un groupe algébrique non connexe $G^+$ défini sur $F$, de composante neutre $G$, tel que $\tilde{G}$ s’identifie à une composante connexe de $G^+$, cf. \[L\] paragraphe 3.4. Cela nous permet d’utiliser pour notre espace $\tilde{G}$ les résultats démontrés dans la littérature pour les groupes non connexes. Il est néanmoins très probable que ces résultats restent vrais sous une hypothèse plus faible que (2).
On note $A_{\tilde{G}}$ le sous-tore de $A_{G}$ tel que $X_{*}(A_{\tilde{G}})=X_{*}(A_{G})^{\theta}$, l’exposant signifiant selon l’usage l’ensemble des invariants par $\theta$. On pose ${\cal A}_{\tilde{G}}=X_{*}(A_{\tilde{G}})\otimes_{{\mathbb Z}}{\mathbb R}={\cal A}_{G}^{\theta}$. On note $a_{\tilde{G}}$ la dimension de ${\cal A}_{\tilde{G}}$. On note $H_{\tilde{G}}:G(F)\to {\cal A}_{\tilde{G}}$ la composée de $H_{G}$ et de la projection orthogonale de ${\cal A}_{G}$ sur ${\cal A}_{\tilde{G}}$. On note ${\cal A}_{\tilde{G},F}$ l’image de $G(F)$ par l’application $H_{\tilde{G}}$.
Soit $(P,M)$ une paire parabolique de $G$. Notons $\tilde{P}$ le normalisateur de $P$ dans $\tilde{G}$, c’est-à-dire l’ensemble des $\gamma\in \tilde{G}$ tels que $ad_{\gamma}$ conserve $P$. Notons $\tilde{M}$ le normalisateur commun de $P$ et $M$ dans $\tilde{G}$. Si $\tilde{P}$ n’est pas vide, les ensembles $\tilde{M}$, $\tilde{P}(F)$, $\tilde{M}(F)$ ne le sont pas non plus. On dit alors que $\tilde{P}$ est un espace parabolique et $\tilde{M}$ est un espace de Levi. Dans le cas de la paire minimale $(P_{0},M_{0})$, il est toujours vrai que $\tilde{P}_{0}$ n’est pas vide. Soit $\gamma_{0}\in \tilde{M}_{0}(F)$. Un sous-groupe parabolique standard $P$ donne naissance à un espace parabolique $\tilde{P}$ si et seulement si $ad_{\gamma_{0}}(P)=P$. On peut supposer et on suppose que la forme quadratique fixée sur ${\cal A}_{0}$ est invariante par l’automorphisme déduit de $ad_{\gamma_{0}}$ par fonctorialité.
On adopte pour les espaces de Levi et les espaces paraboliques des notations similaires à celles introduites pour les groupes. Par exemple, on note ${\cal P}(\tilde{M})$ l’ensemble des espaces paraboliques de composante de Levi $\tilde{M}$. Ou bien, pour $H\in {\cal A}_{M}$, on note $H_{\tilde{M}}$ et $H^{\tilde{M}}$ ses projections sur l’espace ${\cal A}_{\tilde{M}}$, resp. sur son orthogonal ${\cal A}_{M}^{\tilde{M}}$ dans ${\cal A}_{M}$. On munit les espaces ${\cal A}_{\tilde{M}}$, $i{\cal A}_{\tilde{M}}^*$ et le groupe $A_{\tilde{M}}(F)$ de mesures de Haar vérifiant des conditions similaires à celles de 1.2.
Soient $L$ un Levi de $G$ et $\gamma\in \tilde{G}(F)$ tel que $ad_{\gamma}(L)=L$. Notons $\theta$ l’automorphisme de ${\cal A}_{L}$ déduit fonctoriellement de $ad_{\gamma}$. Posons ${\cal A}_{L}^{\theta}=\{H\in {\cal A}_{L}; \theta H=H\}$. Notons $M$ le plus grand Levi contenant $L$ tel que ${\cal A}_{L}^{\theta}\subset {\cal A}_{M}$. Posons $\tilde{M}=M\gamma$. On a
\(3) l’ensemble $\tilde{M}$ est un ensemble de Levi;
\(4) on a l’égalité ${\cal A}_{L}^{\theta}={\cal A}_{\tilde{M}}$;
\(5) l’ensemble des espaces paraboliques $\tilde{P}$ tels que $L\subset P$ et $ad_{\gamma}(P)=P$ est égal à ${\cal F}(\tilde{M})$.
Preuve. Notons $T$ le sous-tore de $A_{L}$ tel que $X_{*}(T)=X_{*}(A_{L})\cap {\cal A}_{L}^{\theta}$. Le Levi $M$ est le commutant de ce tore dans $G$. Par définition de $T$ et $\theta$, $ad_{\gamma}$ fixe tout point de $T$. Donc $\tilde{M}$ est le commutant de $T$ dans $\tilde{G}$ (c’est-à-dire l’ensemble des $\gamma'\in \tilde{G}$ tels que $ad_{\gamma'}$ fixe tout point de $T$). On sait qu’un tel commutant est un espace de Levi, pourvu qu’il ne soit pas vide. Cette dernière condition est vérifiée. D’où (1). Evidemment $T$ est inclus dans $A_{\tilde{M}}$, donc ${\cal A}_{L}^{\theta}\subset {\cal A}_{\tilde{M}}$. Inversement, un élément $H\in {\cal A}_{\tilde{M}}$ appartient à ${\cal A}_{M}$ donc aussi à ${\cal A}_{L}$. Puisque $\gamma\in \tilde{M}$, on a aussi $\theta H=H$, donc $H\in {\cal A}_{L}^{\theta}$. Cela prouve (2). Soit $\tilde{P}=\tilde{M}_{P}U_{P}$ un espace parabolique tel que $L\subset M_{P}$ et $ad_{\gamma}(P)=P$. Puisque $ad_{\gamma}$ conserve $L$ et que $M_{P}$ est l’unique composante de Levi de $P$ contenant $L$, on a aussi $ad_{\gamma}(M_{P})=M_{P}$. Donc $\tilde{M}_{P}=M_{P}\gamma$. Alors ${\cal A}_{\tilde{M}_{P}}$ est un sous-espace de ${\cal A}_{L}$ sur lequel $\theta$ agit trivialement. D’où ${\cal A}_{\tilde{M}_{P}}\subset {\cal A}_{L}^{\theta}={\cal A}_{\tilde{M}}$. D’où $\tilde{M}\subset \tilde{M}_{P}$, ce qui équivaut à $\tilde{P}\in {\cal F}(\tilde{M})$. La réciproque est claire. $\square$
On a fixé une paire parabolique minimale $(P_{0},M_{0})$ et un groupe compact maximal $K$. On a
\(6) si $F$ est archimédien, il existe $\gamma_{0}\in \tilde{M}_{0}(F)$ tel que $ad_{\gamma_{0}}$ conserve la composante neutre de $K$.
Cf. \[DM\] lemme 1.
Définitions combinatoires
-------------------------
Soit $\tilde{P}=\tilde{M}U$ un espace parabolique tel que $M_{0}\subset M$. Pour $\alpha\in \Delta_{P}$, on note $\tilde{\alpha}$ la restriction de $\alpha$ à $A_{\tilde{M}}$. On note $\Delta_{\tilde{P}}$ l’ensemble de ces $\tilde{\alpha}$ pour $\alpha\in \Delta_{P}$. Si on fixe $\gamma\in \tilde{M}$, l’automorphisme $ad_{\gamma}$ définit par fonctorialité une permutation de $\Delta_{P}$ indépendante de $\gamma$, que l’on peut noter $\theta$. Alors $\Delta_{\tilde{P}}$ s’identifie à l’ensemble des orbites dans $\Delta_{P}$ pour l’action du groupe de permutations engendré par $\theta$. Pour $\alpha\in \Delta_{P}$, on note $\varpi_{\tilde{\alpha}}$ la restriction de $\varpi_{\alpha}$ à $A_{\tilde{M}}$. Cette restriction ne dépend en effet que de la restriction $\tilde{\alpha}$ de $\alpha$. Les ensembles $\Delta_{\tilde{P}}$ et $\{\varpi_{\tilde{\alpha}};\tilde{\alpha}\in \Delta_{\tilde{P}}\}$ sont des bases de ${\cal A}_{\tilde{M}}^{\tilde{G},*}$.
Soient $\tilde{P}=\tilde{M}U_{P}\subset \tilde{Q}=\tilde{L}U_{Q}$ deux espaces paraboliques tels que $M_{0}\subset M$. On définit naturellement le sous-ensemble $\Delta_{\tilde{P}}^{\tilde{Q}}\subset \Delta_{\tilde{P}}$. On définit les fonctions $\tau_{\tilde{P}}^{\tilde{Q}}$, $\hat{\tau}_{\tilde{P}}^{\tilde{Q}}$, $\phi_{\tilde{P}}^{\tilde{Q}}$, $\delta_{\tilde{P}}^{\tilde{Q}}$ sur ${\cal A}_{\tilde{M}}$ et $\Gamma_{\tilde{P}}^{\tilde{Q}}$ sur ${\cal A}_{\tilde{M}}\times {\cal A}_{\tilde{M}}$ en remplaçant $\alpha$ par $\tilde{\alpha}$ et $\varpi_{\alpha}$ par $\varpi_{\tilde{\alpha}}$ dans les définitions de 1.3. On définit aussi la notion de famille de points $(\tilde{G},\tilde{M})$-orthogonale. Pour une telle famille ${\cal Y}$, on définit la fonction $\Gamma_{\tilde{M}}^{\tilde{Q}}(.,{\cal Y})$. Toutes les relations énoncées dans le paragraphe 1.3 restent valables pour ces nouvelles fonctions. Evidemment, toutes ces fonctions peuvent être considérées comme des fonctions sur ${\cal A}_{M}$ ou même sur ${\cal A}_{0}$, par composition avec la projection orthogonale sur ${\cal A}_{\tilde{M}}$.
Soient maintenant $Q=LU_{Q}\subset R$ deux sous-groupes paraboliques tels que $M_{0}\subset L$ et soit $\tilde{P}=\tilde{M}U_{P}$ un espace parabolique tel que $Q\subset P\subset R$. Notons $\tilde{\sigma}_{Q}^R$ la fonction caractéristique du sous-ensemble des $H\in {\cal A}_{L}$ qui vérifient les conditions suivantes:
$ <\alpha,H>>0$ pour tout $\alpha\in \Delta_{Q}^R$;
$<\alpha,H>\leq0$ pour tout $\alpha\in \Delta_{Q}-\Delta_{Q}^R$;
$<\varpi_{\tilde{\alpha}},H_{\tilde{M}}>>0$ pour tout $\tilde{\alpha}\in \Delta_{\tilde{P}}$.
On démontre que cette fonction ne dépend pas de $\tilde{P}$ vérifiant les conditions ci-dessus, cf. \[LW\] lemme 2.10.3. Soulignons que cette fonction n’est définie que s’il existe au moins un $\tilde{P}$ vérifiant ces conditions. Pour un sous-groupe parabolique $Q=LU_{Q}$ et un espace parabolique $\tilde{P}$ tel que $Q\subset P$, on a l’égalité
\(1) $\tau_{Q}^P(H)\hat{\tau}_{\tilde{P}}(H)=\sum_{R; P\subset R}\tilde{\sigma}_{Q}^R(H)$ pour tout $H\in {\cal A}_{L}$,
cf. \[LW\] lemme 2.10.5.
$(\tilde{G},\tilde{M})$-familles
--------------------------------
Les définitions et résultats des paragraphes 1.4 à 1.8 s’adaptent aux espaces tordus. Pour un espace de Levi $\tilde{M}$, on définit une notion de $(\tilde{G},\tilde{M})$-famille: c’est une famille $(\varphi(\Lambda,\tilde{P}))_{\tilde{P}\in {\cal P}(\tilde{M})}$ où $\Lambda\mapsto\varphi(\Lambda,\tilde{P})$ est une fonction $C^{\infty}$ sur $i{\cal A}_{\tilde{M}}^*$; pour deux espaces paraboliques adjacents $\tilde{P}$ et $\tilde{P}'$, les fonctions $\varphi(\Lambda,\tilde{P})$ et $\varphi(\Lambda,\tilde{P}')$ se recollent sur le mur séparant les chambres positives associées aux deux espaces paraboliques. Pour une telle famille, on définit la fonction $\varphi_{\tilde{M}}^{\tilde{G}}(\Lambda)$. Pour $X\in {\cal A}_{\tilde{G},F}$ et pour une famille $(\tilde{G},\tilde{M})$-orthogonale ${\cal Y}$, on définit la fonction $\varphi_{\tilde{M}}^{\tilde{G},{\cal Y}}(X;\Lambda)$. Ces fonctions vérifient les mêmes propriétés que dans le cas non tordu.
Considérons un Levi $L$ contenu dans $\tilde{M}$. Soit $(\varphi(\Lambda,Q))_{Q\in {\cal P}(L)}$ une $(G,L)$-famille. Pour $\tilde{P}\in {\cal P}(\tilde{M})$, choisissons $Q\in {\cal P}(L)$ tel que $Q\subset P$. La restriction de $\Lambda\mapsto \varphi(\Lambda,Q)$ à $i{\cal A}_{\tilde{M}}^*$ ne dépend pas du choix de $Q$. On la note $\varphi(\Lambda,\tilde{P})$. Alors la famille $(\varphi(\Lambda,\tilde{P}))_{\tilde{P}\in {\cal P}(\tilde{M})}$ est une $(\tilde{G},\tilde{M})$-famille.
Une construction auxiliaire
---------------------------
Pour un caractère $\mu$ de $G(F)$ et pour $\gamma\in \tilde{G}(F)$, le caractère $\mu\circ ad_{\gamma}$ ne dépend pas de $\gamma$, on le note $\mu\circ \theta$. On note $\mu\circ(1-\theta)$ le caractère $\mu(\mu\circ\theta)^{-1}$. Dans la suite, on considérera un caractère unitaire $\omega$ de $G(F)$ dont la restriction à $Z_{G}(F)^{\theta}$ est triviale. Certains aspects de la théorie se simplifient quand ce caractère est de la forme $\mu\circ(1-\theta)$. Ce n’est pas toujours le cas mais nous allons voir que l’on peut se ramener à ce cas en effectuant des extensions.
Soient $(G',\tilde{G}')$ un couple vérifiant les mêmes hypothèses que notre couple $(G,\tilde{G})$. Soit $p:G'\to G$ un homomorphisme de groupes algébriques et $\tilde{p}:\tilde{G}'\to \tilde{G}$ un homomorphisme de variétés algébriques (tous deux définis sur $F$). On dit que $(p,\tilde{p})$ est un homomorphisme d’espaces tordus si et seulement si on a l’égalité $$\tilde{p}(x'\gamma'y')=p(x')\tilde{p}(\gamma')p(y')$$ pour tous $\gamma'\in \tilde{G}'$ et $x',y'\in G'$.
0.3cm[**[Proposition]{}**]{}.
**
Soit $\omega$ un caractère unitaire de $G(F)$ dont la restriction à $Z_{G}(F)^{\theta}$ est triviale. Il existe un espace tordu $(G',\tilde{G}')$ vérifiant les hypothèses (1) et (2) de 2.1, et un homomorphisme d’espaces tordus $(p,\tilde{p}):(G',\tilde{G}')\to (G,\tilde{G})$ vérifiant les conditions suivantes:
\(i) l’homomorphisme $p$ s’insère dans une suite exacte $$1\to C\to G'\stackrel{p}{\to}G\to 1$$ où $C$ est un sous-tore central de $G'$;
\(ii) pour toute extension $F'$ de $F$, l’homomorphisme $p:G'(F')\to G(F')$ est surjectif;
\(iii) il existe un caractère unitaire $\mu'$ de $G'(F)$ tel que $\omega\circ p=\mu'\circ(1-\theta')$ (en notant $\theta'$ l’analogue de $\theta$ pour $(G',\tilde{G}')$).
0.3cm
Preuve. On commence par montrer
\(1) il existe $(G',\tilde{G}')$ vérifiant (i) et (ii) et tel que le groupe dérivé de $G'$ soit simplement connexe.
C’est une simple adaptation de la théorie des $z$-extensions. Fixons $\gamma\in \tilde{G}(F)$, posons $\theta=ad_{\gamma}$. Notons $G_{SC}$ le revêtement simplement connexe du groupe dérivé de $G$. On sait que l’on a un isomorphisme $$(G_{SC}\times Z_{G}^0)/\Xi\simeq G,$$ où $\Xi$ est un sous-goupe abélien fini de $G_{SC}\times Z_{G}^0$. L’automorphisme $\theta$ se relève en un automorphisme de $G_{SC}$, donc aussi en un automorphisme de $G_{SC}\times Z_{G}^0$ qui préserve $\Xi$. On note tous ces automorphismes $\theta$. Fixons une clôture algébrique $\bar{F}$ de $F$, posons $\Gamma_{F}=Gal(\bar{F}/F)$. Fixons une extension galoisienne finie $F'$ de $F$ telle que $\Gamma_{F'}$ agisse trivialement sur $\Xi$. Notons $\hat{\Xi}$ le groupe $Hom(\Xi,\bar{F}^{\times})$ des caractères algébriques de $\Xi$ et ${\mathbb Z}[\hat{\Xi}]$ le ${\mathbb Z}$-module libre de base $\hat{\Xi}$. Pour plus de clarté, si $\mu\in \hat{\Xi}$, on note $\underline{\mu}$ l’élément de base de ${\mathbb Z}[\hat{\Xi}]$. Le module ${\mathbb Z}[\hat{\Xi}]$ est muni d’une action de $\Gamma_{F}$ triviale sur $\Gamma_{F'}$. Notons $\Lambda$ l’ensemble des fonctions $f:\Gamma_{F'}\backslash \Gamma_{F}\to{\mathbb Z}[\hat{\Xi}]$. On le munit de l’action galoisienne par translations à droite. On note $C$ le tore sur $F$ tel que $X_{*}(C)=\Lambda$, muni de cette action galoisienne. Par ailleurs $\theta$ se transporte en une permutation de $\hat{\Xi}$, puis en un automorphisme de ${\mathbb Z}[\hat{\Xi}]$, puis en un automorphisme de $\Lambda$ (par action sur l’espace d’arrivée), enfin en un automorphisme $\theta_{C}$ de $C$. Il est clair que $\theta_{C}$ est défini sur $F$. On définit deux applications $$\Xi\to {\mathbb Z}[\hat{\Xi}]\otimes_{{\mathbb Z}}\bar{F}^{\times}$$ $${\mathbb Z}[\hat{\Xi}]\otimes_{{\mathbb Z}}\bar{F}^{\times}\to C.$$ La première envoie $\xi\in \Xi$ sur $\sum_{\mu\in \hat{\Xi}}\underline{\mu}\otimes \mu(\xi)$. La seconde est déduite de l’application ${\mathbb Z}[\hat{\Xi}]\to \Lambda$ qui à $x\in {\mathbb Z}[\hat{\Xi}]$ associe la fonction $f$ définie par $f(\sigma)=\sigma x$. Notons $\iota:\Xi\to C$ la composée des deux applications. Elle est injective, équivariante pour les actions galoisiennes et vérifie $\iota\circ \theta=\theta_{C}\circ\iota$. Posons $$G'=(G_{SC}\times Z_{G}^0\times C)/\{(\xi,\iota(\xi)); \xi\in \Xi\}.$$ On a bien la suite exacte $$1\to C\to G'\stackrel{p}{\to} G\to 1.$$ La condition (ii) est vérifiée parce que $C$ est un tore induit, donc $H^1(\Gamma_{F'},C)=0$ pour toute extension $F'$ de $F$. Enfin, le groupe dérivé de $G'$ est simplement connexe parce que l’homomorphisme naturel $G_{SC}\to G'$ est injectif. Le produit des automorphismes $\theta$ de $G_{SC}\times Z_{G}^0$ et $\theta_{C}$ de $C$ se descend en un automorphisme $\theta'$ de $G'$. Sa restriction à $Z_{G'}$ est d’ordre fini. On introduit une variété $\tilde{G}'$ muni d’un isomorphisme défini sur $F$ de $G'$ sur $\tilde{G}'$. On note $g\mapsto g\boldsymbol{\theta}'$ cet isomorphisme. On définit les actions à gauche et à droite de $G'$ sur $\tilde{G}'$ par $(x,g\boldsymbol{\theta},y)\mapsto xg\theta'(y)\boldsymbol{\theta}'$. On définit $\tilde{p}:\tilde{G}'\to \tilde{G}$ par $\tilde{p}(g\boldsymbol{\theta}')=g\gamma$. Le couple $(p,\tilde{p})$ est un homomorphisme d’espaces tordus. Cela prouve (1).
On montre ensuite
\(2) les assertions de la proposition sont vérifiées si le groupe dérivé de $G$ est simplement connexe.
On fixe de nouveau $\gamma\in \tilde{G}(F)$ et on en déduit un automorphisme $\theta$ de $G$. On introduit le tore $D=Z_{G}^0/(Z_{G_{SC}}\cap Z_{G}^0)$. On a la suite exacte $$1\to G_{SC} \to G\stackrel{\underline{d}}{\to}D\to 1,$$ d’où $$1\to G_{SC}(F)\to G(F)\stackrel{\underline{d}}{\to} D(F).$$ De $\theta$ se déduit un automorphisme encore noté $\theta$ de $D$. Tout caractère de $G_{SC}(F)$ étant trivial, $\omega$ se factorise en un caractère du quotient $G_{SC}(F)\backslash G(F)$, que l’on peut prolonger en un caractère de $D(F)$. On note $\omega_{D}$ ce prolongement. Notons $Z_{G}^{\theta,0}$ et $D^{\theta,0}$ les composantes neutres des sous-groupes de points fixes par $\theta$ dans $Z_{G}$ et $D$. L’homomorphisme $Z_{G}^{\theta,0}\to D^{\theta,0}$ est surjectif, donc $\underline{d}(Z_{G}^{\theta,0}(F))$ est d’indice fini dans $D^{\theta,0}(F)$. Par hypothèse, $\omega$ est trivial sur $Z_{G}(F)^{\theta}$, a fortiori sur $Z_{G}^{\theta,0}(F)$. Donc la restriction de $\omega_{D}$ à $D^{\theta,0}(F)$ est d’ordre fini. Notons $n_{1}$ l’ordre de $\omega_{D}$, $n_{2}$ celui de l’automorphisme $\theta$ de $D$ et posons $N=n_{1}n_{2}$. Si $N=1$, on a $D=D^{\theta,0}$ puis $\omega=1$. On prend $G'=G$, $\tilde{G}'=\tilde{G}$ et c’est terminé. Supposons $N>1$. Posons $$C=\{(x_{1},...,x_{N})\in D^N; \prod_{i=1,...,N}\theta_{D}^{i}x_{i}=1\}.$$ C’est un tore isomorphe à $D^{N-1}$. Posons $G'=G\times C$. L’extension $$1\to C\to G'\stackrel{p}{\to }G\to 1$$ vérifie les conditions (i) et (ii). Définissons un automorphisme $\theta'$ de $G'$ par $$\theta'(g,x_{1},...,x_{N})=(\theta(g),\underline{d}(g)x_{N},x_{1},...,x_{N-2},\underline{d}\circ\theta(g^{-1})x_{N-1}).$$ On vérifie que $$(\theta')^N(g,x_{1},...,x_{N})=(\theta^N(g),x_{1},...,x_{N}).$$ Il en résulte que la restriction de $\theta'$ à $Z_{G'}$ est d’ordre fini. On définit l’espace tordu $\tilde{G}'$ et l’application $\tilde{p}$ comme dans la preuve de (1). Définissons un caractère $\mu'$ de $G'(F)$ par $$\mu'(g,x_{1},...,x_{N})=\prod_{i=1,...,N-1}\prod_{j=0,...,i-1}\omega_{D}^{-1}\circ\theta^j(x_{i}).$$ On calcule $$\mu'\circ(1-\theta')(g,x_{1},...,x_{N})=\omega_{D}^{-1}(x_{1}x_{N}^{-1}\underline{d}(g)^{-1})\prod_{i=2,...,N-1}\prod_{j=0,...,i-1}\omega_{D}^{-1}\circ\theta^j(x_{i}x_{i-1}^{-1})$$ $$=\omega_{D}\circ\underline{d}(g)\omega_{D}(x_{1}^{-1}x_{N})\left(\prod_{i=2,...,N-1}\prod_{j=0,...,i-1}\omega_{D}^{-1}\circ\theta^j(x_{i})\right)\left(\prod_{i=1,...,N-2}\prod_{j=0,...,i}\omega_{D}^{-1}\circ\theta^j(x_{i}^{-1})\right)$$ $$=\omega(g)\omega_{D}(x_{N})\left(\prod_{i=1,...,N-2}\omega_{D}\circ\theta^{i}(x_{i})\right)\prod_{j=0,...,N-2}\omega_{D}^{-1}\circ\theta^j(x_{N-1}).$$ On utilise la relation $$x_{N}=\prod_{i=1,...,N-1}\theta^{i}(x_{i}^{-1})$$ et on obtient $$\mu'\circ(1-\theta')(g,x_{1},...,x_{N})=\omega(g)\prod_{j=0,...,N-1}\omega_{D}^{-1}\circ\theta^{j}(x_{N-1}).$$ L’application $x\mapsto \prod_{j=0,...,n_{2}-1}\theta^j(x)$ envoie $D$ dans $D^{\theta,0}$, donc $D(F)$ dans $D^{\theta,0}(F)$. Donc, pour tout $x\in D(F)$, $\prod_{j=0,...,N-1}\theta^j(x)$ est la puissance $n_{1}$-ième d’un élément de $D^{\theta,0}(F)$. Il en résulte que $$\prod_{j=0,...,N-1}\omega_{D}\circ\theta^j(x)=1$$ pour tout $x\in D(F)$. L’égalité ci-dessus devient $$\mu'\circ(1-\theta')(g,x_{1},...,x_{N})=\omega(g),$$ comme on le voulait. Cela prouve (2).
Dans le cas général, on commence par construire un couple vérifiant (1). On affecte les objets relatifs à ce couple d’un indice $1$: $G'_{1}$, $\tilde{G}'_{1}$, $p_{1}$, $\tilde{p}_{1}$. On applique (2) à ce couple et au caractère $\omega\circ p_{1}$. On obtient des objets $G'$, $\tilde{G}'$, $\mu'$ et des applications $p_{2}:G'\to G'_{1}$, $\tilde{p}_{2}:\tilde{G}'\to \tilde{G}'_{1}$. On pose $p=p_{1}\circ p_{2}$, $\tilde{p}=\tilde{p}_{1}\circ\tilde{p}_{2}$ et on note $C$ l’image réciproque de $C_{1}$ dans $G'$. Il est clair que les objets $G'$, $\tilde{G}'$, $C$, $p$ et $\tilde{p}$ vérifient les conditions de l’énoncé. $\square$
Représentations
---------------
Soit $\omega$ un caractère unitaire de $G(F)$. On suppose que sa restriction à $Z_{G}(F)^{\theta}$ est triviale. On appelle $\omega$-représentation lisse de $\tilde{G}(F)$ un couple $(\pi,\tilde{\pi})$, où $\pi$ est une représentation lisse de $G(F)$ dans un espace $V_{\pi}$ et $\tilde{\pi} $ est une application de $\tilde{G}(F)$ dans le groupe des automorphismes de $V_{\pi}$ telle que $\tilde{\pi}(g\gamma g')=\pi(g)\tilde{\pi}(\gamma)\pi(g')\omega(g')$ pour tous $g,g'\in G(F)$ et $\gamma\in \tilde{G}(F)$.
[**Remarque.**]{} Dans le cas où $F$ est archimédien, il faudrait distinguer, comme on l’a dit en 1.9, l’espace de la représentation et son sous-espace $V_{\pi}$ des vecteurs $K$-finis.
En pratique, on parlera plutôt de la $\omega$-représentation $\tilde{\pi}$, en occultant la représentation $\pi$ sous-jacente. Pour une $\omega$-représentation lisse $\tilde{\pi}$ et pour $z\in {\mathbb C}^{\times}$, on définit $z\tilde{\pi}$ par $(z\tilde{\pi})(\gamma)=z\tilde{\pi}(\gamma)$.
[**Variante.**]{} On dira que $\tilde{\pi}$ est unitaire s’il existe un produit hermitien défini positif sur $V_{\pi}$ tel que $\tilde{\pi}(\gamma)$ soit un opérateur unitaire pour tout $\gamma\in \tilde{G}(F)$. Notons ${\mathbb U}$ le groupe des nombres complexes de module $1$. Pour une $\omega$-représentation lisse unitaire $\tilde{\pi}$ et pour $z\in {\mathbb U}$, $z\tilde{\pi}$ est encore unitaire.
On dit que $\tilde{\pi}$ est admissible si et seulement si $\pi$ l’est. On dit que $\tilde{\pi}$ est tempérée si et seulement si $\tilde{\pi}$ est unitaire et $\pi$ est tempérée. Nos hypothèses que $\omega$ unitaire et que l’automorphisme $\theta$ de $A_{G}$ est d’ordre fini impliquent que $\tilde{\pi}$ est de longueur finie si et seulement si $\pi$ l’est. Il y a une notion naturelle d’irréductibilité et une notion plus fine de $G$-irréductibilité: $\tilde{\pi}$ est $G$-irréductible si et seulement si $\pi$ est irréductible. Notons $Irr(G(F);\theta,\omega)$ l’ensemble des classes de représentations lisses irréductibles $\pi$ de $G(F)$ telles que $\omega\otimes \pi \simeq \pi\circ \theta$, où $\pi\circ\theta$ est la classe de $\pi\circ ad_{\gamma}$ pour un quelconque $\gamma\in \tilde{G}(F)$. L’application $\tilde{\pi}\mapsto \pi$ est une surjection de l’ensemble des $\omega$-représentations lisses $G$-irréductibles de $\tilde{G}(F)$ sur $Irr(G(F);\theta,\omega)$. Les fibres sont isomorphes à ${\mathbb C}^{\times}$.
La mesure de Haar fixée sur $G(F)$ détermine une mesure sur $\tilde{G}(F)$. Pour $f\in C_{c}^{\infty}(\tilde{G}(F))$, on définit l’opérateur $$\tilde{\pi}(f)=\int_{\tilde{G}(F)}f(\gamma)\tilde{\pi}(\gamma)d\gamma.$$ Supposons $\tilde{\pi}$ admissible. Alors la trace de cet opérateur est bien définie. A $\tilde{\pi}$ est associé son caractère, qui est la distribution sur $C_{c}^{\infty}(\tilde{G}(F))$ définie par $$I_{\tilde{G}}(\tilde{\pi},f)=trace(\tilde{\pi}(f)).$$ Si $g\in G(F)$ et si l’on note $^gf$ la fonction définie par $^gf(\gamma)=f(g^{-1}\gamma g)$, on a $$I_{\tilde{G}}(\tilde{\pi},{^gf})=\omega(g)^{-1}I_{\tilde{G}}(\tilde{\pi},f).$$ On note $D_{spec}(\tilde{G}(F);\omega)$ l’espace de distributions engendré par les caractères de $\omega$-représentations admissibles de longueur finie. Il est évidemment engendré par les caractères de $\omega$-représentations irréductibles. Une $\omega$-représentation irréductible a un caractère non nul si et seulement si elle est $G$-irréductible, cf. \[L\] proposition A.4.1. Tout élément $\pi\in Irr(G(F);\theta,\omega)$ détermine une droite $D_{\pi}$ dans $D_{spec}(\tilde{G}(F);\omega)$, à savoir la droite portée par le caractère de $\tilde{\pi}$, où $\tilde{\pi}$ est un prolongement quelconque de $\pi$ en une représentation de $\tilde{G}(F)$. On a $$(1)\qquad D_{spec}(\tilde{G}(F);\omega)=\oplus_{\pi\in Irr(G(F);\theta,\omega)}D_{\pi};$$ cf. \[L\] proposition A.4.1 dans le cas non archimédien; la preuve est similaire dans le cas archimédien;
\(2) l’espace $D_{spec}(\tilde{G}(F);\omega)$ est formé de distributions localement intégrables sur $\tilde{G}(F)$ et lisses sur les éléments fortement réguliers.
Preuve. Notons ${\bf 1}$ le caractère trivial de $G(F)$. Si $\omega={\bf 1}$, l’assertion est due à Bouaziz dans le cas archimédien (\[B\] théorème 2.1.1) et à Clozel dans le cas non-archimédien (\[C\]). Supposons qu’il existe un caractère unitaire $\mu$ de $G(F)$ tel que $\omega=\mu\circ(1-\theta)$ (cf. 2.4). Soit $\tilde{\pi}$ une $\omega$-représentation de $\tilde{G}(F)$ de représentation sous-jacente $\pi$. Posons $\pi_{1}=\pi\otimes \mu$. Fixons $\gamma_{0}\in \tilde{G}(F)$ et définissons $\tilde{\pi}_{1}$ par $\tilde{\pi}_{1}(g\gamma_{0})=\mu(g)\tilde{\pi}(g\gamma_{0})$. On vérifie que $\tilde{\pi}_{1}$ est une ${\bf 1}$-représentation de $\tilde{G}(F)$, de représentation sous-jacente $\pi_{1}$. Son caractère est localement intégrable, donc associé à une fonction localement intégrable $\gamma\mapsto\Theta(\tilde{\pi}_{1},\gamma)$ sur $\tilde{G}(F)$. Le caractère de $\tilde{\pi}$ est alors associé à la fonction $g\gamma_{0}\mapsto \mu(g)^{-1}\Theta(\tilde{\pi}_{1},g\gamma_{0})$, qui est elle-aussi localement intégrable. Dans le cas général, on introduit des objets $G'$, $\tilde{G}'$, $C$, $p$, $\tilde{p}$ vérifiant la proposition 2.4. On pose $\pi'=\pi\circ p$, $\tilde{\pi}'=\tilde{\pi}\circ\tilde{p}$, $\omega'=\omega\circ p$. Le terme $\tilde{\pi}'$ est une $\omega'$-représentation de $\tilde{G}'(F)$. L’hypothèse du cas précédent est vérifiée: $\omega'$ est de la forme $\mu'\circ(1-\theta')$. Donc le caractère de $\tilde{\pi}'$ est localement intégrable, associé à une fonction localement intégrable $\gamma'\mapsto \Theta(\tilde{\pi}',\gamma')$ sur $\tilde{G}'(F)$. Le caractère central de la représentation $\pi'$ est trivial sur $C(F)$ par construction. Il en résulte que la fonction ci-dessus est invariante par translations (à droite ou à gauche) par $C(F)$. Il existe donc une fonction $\gamma\mapsto \Theta(\tilde{\pi},\gamma)$ sur $\tilde{G}(F)$ telle que l’on ait l’égalité $\Theta(\tilde{\pi}',\gamma')=\Theta(\tilde{\pi},\tilde{p}(\gamma'))$ pour tout $\gamma'\in \tilde{G}'(F)$. On vérifie aisément que cette fonction est elle-aussi localement intégrable et que sa distribution associée est le caractère de $\tilde{\pi}$. $\square$
Soit $\tilde{M}$ un espace de Levi de $\tilde{G}$ tel que $M_{0}\subset M$ et soit $\tilde{\sigma}$ une $\omega$-représentation lisse de $\tilde{M}(F)$. Pour $\tilde{P}\in {\cal P}(\tilde{M})$, on définit l’induite habituelle $Ind_{P}^G(\sigma)$. On définit une représentation $\tilde{\pi}=Ind_{\tilde{P}}^{\tilde{G}}(\tilde{\sigma})$ de $\tilde{G}(F)$ dans l’espace de cette induite par la formule $$(3) \qquad (\tilde{\pi}(\gamma)f)(g)=\omega(g')\delta_{P}(\gamma')^{1/2}\tilde{\sigma}(\gamma')f(g'),$$ où:
- on a écrit $g\gamma =\gamma'g'$, avec $\gamma'\in \tilde{M}(F)$;
- $\delta_{P}(\gamma')$ est la valeur absolue du déterminant de l’action de $ad_{\gamma'}$ dans l’algèbre de Lie du radical unipotent de $P$.
[**Remarque.**]{} Il peut arriver que $\tilde{\sigma}$ soit $M$-irréductible mais que $Ind_{\tilde{P}}^{\tilde{G}}(\tilde{\sigma})$ ait un caractère nul. Cela se produit quand $Ind_{P}^G(\sigma)$ n’est pas irréductible et que l’action de $\tilde{G}(F)$ permute sans point fixe ses composantes irréductibles. Donnons un exemple. Supposons $F$ non-archimédien et $G=SL_{2}$, prenons pour $M$ le tore diagonal et pour $P$ le Borel supérieur. Soient $\chi$ un caractère quadratique non trivial de $F^{\times}$ et $D\in F^{\times}$ tel que $\chi(D)=-1$. Supposons $\tilde{G}=\{x\in GL_{2}; det(x)=D\}$. Prenons pour $\sigma$ le caractère $$\left(\begin{array}{cc}a&0\\0&a^{-1}\\ \end{array}\right)\mapsto \chi(a)$$ de $M(F)$ et pour $\tilde{\sigma}$ la représentation $$\left(\begin{array}{cc}a&0\\0&d\\ \end{array}\right)\mapsto \chi(a)$$ de $\tilde{M}(F)$. On vérifie que le caractère de $Ind_{\tilde{P}}^{\tilde{G}}(\tilde{\sigma})$ coïncide avec la restriction à $\tilde{G}(F)$ du caractère de la représentation $\pi$ de $GL_{2}(F)$ induite du caractère $\chi\times 1$ du tore diagonal. Or $\pi\simeq (\chi\circ det)\pi$, donc le caractère de $\pi$ est nul sur les éléments de $GL_{2}(F)$ de déterminant $D$.
Torsion par un caractère
------------------------
Notons $G(F)^1$ le noyau de l’homomorphisme $H_{\tilde{G}}$. Il est invariant par $ad_{\gamma}$ pour tout $\gamma\in \tilde{G}(F)$. Posons $\tilde{{\cal A}}_{\tilde{G},F}=G(F)^1\backslash \tilde{G}(F)=\tilde{G}(F)/G(F)^1$ et notons $\tilde{H}_{\tilde{G}}:\tilde{G}(F)\to \tilde{{\cal A}}_{\tilde{G},F}$ la projection naturelle. L’ensemble $\tilde{{\cal A}}_{\tilde{G},F}$ est un espace principal homogène sous l’action de ${\cal A}_{\tilde{G},F}$ (les actions à gauche et à droite coïncident). On pose $\tilde{{\cal A}}_{\tilde{G}}={\cal A}_{\tilde{G}}\otimes_{{\cal A}_{\tilde{G},F}}\tilde{{\cal A}}_{\tilde{G},F}$. C’est un espace affine sous ${\cal A}_{\tilde{G}}$. Par abus de notation, on note $\tilde{{\cal A}}_{\tilde{G}}^*$ l’espace des fonctions affines sur $\tilde{{\cal A}}_{\tilde{G}}$, c’est-à-dire les fonctions $$\begin{array}{cccc}\tilde{\lambda}:&\tilde{{\cal A}}_{\tilde{G}}&\to& {\mathbb R}\\ &\tilde{H}&\mapsto &<\tilde{\lambda},\tilde{H}>\\ \end{array}$$ telles qu’il existe $\lambda\in {\cal A}_{\tilde{G}}^*$ de sorte que $<\tilde{\lambda},H+\tilde{H}>=<\lambda,H>+<\tilde{\lambda},\tilde{H}>$ pour tout $H\in {\cal A}_{\tilde{G}}$ et $\tilde{H}\in \tilde{{\cal A}}_{\tilde{G}}$. On définit de façon évidente le complexifié $\tilde{{\cal A}}_{\tilde{G},{\mathbb C}}^*$ et son sous-espace imaginaire $i\tilde{{\cal A}}_{\tilde{G}}^*$. On note $\tilde{{\cal A}}_{\tilde{G},F}^{\vee}$ le sous-groupe des $\tilde{\lambda}\in \tilde{{\cal A}}_{\tilde{G}}^*$ tels que $\tilde{\lambda}(\tilde{{\cal A}}_{\tilde{G},F})\subset 2\pi{\mathbb Z}$. On pose $i\tilde{{\cal A}}_{\tilde{G},F}^*=(i\tilde{{\cal A}}_{\tilde{G}}^*)/(i\tilde{{\cal A}}_{\tilde{G},F}^{\vee})$. On a des suites exactes $$0\to {\mathbb R}\to \tilde{{\cal A}}_{\tilde{G}}^*\to {\cal A}_{\tilde{G}}^*\to 0,$$ $$0\to {\mathbb C}\to \tilde{{\cal A}}_{\tilde{G},{\mathbb C}}^*\to {\cal A}_{\tilde{G},{\mathbb C}}^*\to 0,$$ $$0\to i{\mathbb R}/2\pi i{\mathbb Z}\to i\tilde{{\cal A}}_{\tilde{G},F}^*\to i{\cal A}_{\tilde{G},F}^*\to 0.$$ Le choix d’un point-base $\gamma\in \tilde{G}(F)$ scinde ses suites: on identifie par exemple $\tilde{\cal A}_{\tilde{G}}^*$ au sous-espace des fonctions affines qui valent $0$ au point $\tilde{H}_{\tilde{G}}(\gamma)$. Pour $\tilde{\lambda}\in \tilde{{\cal A}}_{\tilde{G}}^*$ (ou $\tilde{\lambda}\in i\tilde{{\cal A}}_{\tilde{G},F}^*$), on note sans plus de commentaire $\lambda$ son image dans ${\cal A}_{\tilde{G}}^*$ (ou $i{\cal A}_{\tilde{G},F}^*$).
Soit $\tilde{\pi}$ une $\omega$-représentation lisse de $\tilde{G}(F)$. Pour $\tilde{\lambda}\in \tilde{{\cal A}}_{\tilde{G},{\mathbb C}}^*$, on définit la représentation $\tilde{\pi}_{\tilde{\lambda}}$ par $\tilde{\pi}_{\tilde{\lambda}}(\gamma)=e^{<\tilde{\lambda},\tilde{H}_{\tilde{G}}(\gamma)>}\tilde{\pi}(\gamma)$. Sa représentation sous-jacente de $G(F)$ est $\pi_{\lambda}$. Evidemment, $\tilde{\pi}_{\tilde{\lambda}}$ ne dépend que de l’image de $\tilde{\lambda}$ dans $\tilde{{\cal A}}_{\tilde{G},{\mathbb C}}^*/i\tilde{{\cal A}}_{\tilde{G},F}^{\vee}$. Remarquons que:
\(1) si $\pi$ est irréductible, le stabilisateur de $\tilde{\pi}$ dans $i\tilde{{\cal A}}_{\tilde{G},F}^*$ s’identifie au stabilisateur $Stab(i{\cal A}_{\tilde{G},F}^*,\pi)$ de $\pi$ dans $i{\cal A}_{\tilde{G},F}^*$.
Preuve. Il est clair que le premier groupe se projette injectivement dans le second. Inversement, soit $\lambda\in Stab(i{\cal A}_{\tilde{G},F}^*,\pi)$ et fixons arbitrairement $\tilde{\lambda}\in i\tilde{{\cal A}}_{\tilde{G},F}^*$ au-dessus de $\lambda$. Alors $\tilde{\pi}_{\tilde{\lambda}}$ est isomorphe à une représentation prolongeant $\pi$, donc à $z\tilde{\pi}$ pour un certain $z\in {\mathbb C}^{\times}$. On en déduit $\tilde{\pi}_{n\tilde{\lambda}}\simeq z^n\tilde{\pi}$ pour tout $n\in {\mathbb N}$. On choisit $n$ tel que $n\lambda=0$ dans $i{\cal A}_{\tilde{G},F}^*$. Alors $n\tilde{\lambda}\in i{\mathbb R}/2\pi i{\mathbb Z}$ et $\tilde{\pi}_{n\tilde{\lambda}}=e^{n\tilde{\lambda}}\tilde{\pi}$, donc $z^n=e^{n\tilde{\lambda}}$ et $z$ est de module $1$. En écrivant $z=e^{x}$, avec $x\in i{\mathbb R}/2\pi i{\mathbb Z}$, l’élément $\tilde{\lambda}-x$ appartient au stabilisateur de $\tilde{\pi}$. $\square$
On devra considérer des fonctions à valeurs complexes $\varphi$ définies sur $\tilde{{\cal A}}_{\tilde{G},{\mathbb C}}$, resp. $i\tilde{{\cal A}}_{\tilde{G},F}^*$. La plupart vérifieront la condition
\(2) pour tout $z\in {\mathbb C}\subset \tilde{{\cal A}}_{\tilde{G},{\mathbb C}}^*$ et tout $\tilde{\lambda}\in \tilde{{\cal A}}_{\tilde{G},{\mathbb C}}^*$, $\varphi(z+\tilde{\lambda})=e^{z}\varphi(\tilde{\lambda})$
resp. une condition similaire. Modulo le choix d’un point-base permettant de scinder les suites exactes ci-dessus, une telle fonction s’identifie à une fonction sur ${\cal A}_{\tilde{G},{\mathbb C}}^*$, resp. $i{\cal A}_{\tilde{G},F}^*$.
Une fonction $\varphi:i\tilde{{\cal A}}_{\tilde{G},F}^*\to {\mathbb C}$ est dite de Paley-Wiener si et seulement s’il existe une fonction $b:\tilde{{\cal A}}_{\tilde{G},F}\to {\mathbb C}$, lisse et à support compact, de sorte que $$\varphi(\tilde{\lambda})=\int_{\tilde{{\cal A}}_{\tilde{G},F}}b(X)e^{<\tilde{\lambda},X>}\,dX.$$ Il revient au même de demander que $\varphi$ vérifie (2) et que la fonction sur $i{\cal A}_{\tilde{G},F}^*$ déduite de $\varphi$ comme ci-dessus soit de Paley-Wiener.
Caractères pondérés
-------------------
Soit $\tilde{M}$ un espace de Levi de $\tilde{G}$ tel que $M_{0}\subset M$. Soit $\tilde{\pi}$ une $\omega$-représentation admissible et de longueur finie de $\tilde{M}(F)$. Fixons $\tilde{P}\in {\cal P}(\tilde{M})$, introduisons la représentation $Ind_{\tilde{P}}^{\tilde{G}}(\tilde{\pi})$, que l’on réalise dans l’espace $V_{\pi,P}$. Supposons dans un premier temps que $\tilde{\pi}$ est en position générale, en ce sens que les opérateurs d’entrelacement intervenant ci-dessous sont bien définis. Pour $\tilde{Q}\in {\cal P}(\tilde{M})$, l’opérateur $J_{P\vert Q}(\pi)J_{Q\vert P}(\pi)$ est un automorphisme de $V_{\pi,P}$. Notons $\mu_{Q\vert P}(\pi)$ son inverse. Pour $\Lambda\in i{\cal A}_{\tilde{M}}^*$, posons $${\cal M}(\pi;\Lambda,\tilde{Q})=\mu_{Q\vert P}(\pi)^{-1}\mu_{Q\vert P}(\pi_{\Lambda/2})J_{Q\vert P}(\pi)^{-1}J_{Q\vert P}(\pi_{\Lambda}).$$ On souligne la présence d’un indice $\Lambda/2$ dans cette formule. La famille $({\cal M}(\pi;\Lambda,\tilde{Q}))_{\tilde{Q}\in {\cal P}(\tilde{M})}$ est une $(\tilde{G},\tilde{M})$-famille à valeurs opérateurs. On en déduit un opérateur ${\cal M}_{\tilde{M}}^{\tilde{G}}(\pi;\Lambda)$. On pose ${\cal M}_{\tilde{M}}^{\tilde{G}}(\pi)={\cal M}_{\tilde{M}}^{\tilde{G}}(\pi;0)$. Si $F$ est archimédien, notons $C_{c}^{\infty}(\tilde{G}(F),K)$ le sous-espace des éléments de $C_{c}^{\infty}(\tilde{G}(F))$ qui sont $K$-finis à droite et à gauche. Pour unifier les notations, posons $C_{c}^{\infty}(\tilde{G}(F),K)=C_{c}^{\infty}(\tilde{G}(F))$ dans le cas où $F$ est non-archimédien. Le caractère pondéré de $\tilde{\pi}$ est la distribution sur $C_{c}^{\infty}(\tilde{G}(F),K)$ définie par $$J_{\tilde{M}}^{\tilde{G}}(\tilde{\pi},f)=trace({\cal M}_{\tilde{M}}^{\tilde{G}}(\pi)Ind_{\tilde{P}}^{\tilde{G}}(\tilde{\pi},f)).$$ On a choisi un parabolique $\tilde{P}$, mais on vérifie que cette trace ne dépend pas de ce choix.
[**Remarque.**]{} On verra en 5.4 que cette définition s’étend à tout $C_{c}^{\infty}(\tilde{G}(F))$ au moins si $\tilde{\pi}$ est tempérée.
Levons l’hypothèse que $\tilde{\pi}$ est en position générale. Pour $\tilde{\pi}$ quelconque et pour $\tilde{\lambda}\in \tilde{{\cal A}}_{\tilde{M},{\mathbb C}}^*$, $\tilde{\pi}_{\tilde{\lambda}}$ est en position générale pour $\tilde{\lambda}$ hors d’un sous-ensemble fermé de mesure nulle. On peut donc définir l’opérateur ${\cal M}_{\tilde{M}}^{\tilde{G}}(\pi_{\lambda})$ et la distribution $f\mapsto J_{\tilde{M}}^{\tilde{G}}(\tilde{\pi}_{\tilde{\lambda}},f)$ pour $\tilde{\lambda}$ dans un ouvert dense. Il est clair que ces termes vérifient la condition (2) de 2.6 et s’étendent en des fonctions méromorphes de $\tilde{\lambda}$ définies pour tout $\tilde{\lambda}$. Si ces fonctions sont régulières en $\tilde{\lambda}=0$, on définit ${\cal M}_{\tilde{M}}^{\tilde{G}}(\pi)$ et $J_{\tilde{M}}^{\tilde{G}}(\tilde{\pi},f)$ comme leurs valeurs en ce point $\tilde{\lambda}=0$.
[0.3cm[**[Proposition]{}**]{}. [ *[Si $\tilde{\pi}$ est unitaire, les fonctions ci-dessus sont régulières en $\tilde{\lambda}=0$.]{}*]{}0.3cm]{}
Cf. \[A4\] proposition 2.3. Arthur traite le cas d’une représentation irréductible d’un groupe connexe mais sa preuve s’étend à notre situation.
Remarquons que, dans notre construction, on n’a pas imposé à $\tilde{\pi}$ d’être irréductible. On vérifie que l’opérateur ${\cal M}_{\tilde{M}}^{\tilde{G}}(\pi)$ dépend fonctoriellement de $\tilde{\pi}$, pourvu qu’il soit défini, et que, si l’on a une suite exacte $$1\to \tilde{\pi}_{1}\to \tilde{\pi}_{2}\to \tilde{\pi}_{3}\to 1,$$ on a l’égalité $$(1) \qquad J_{\tilde{M}}^{\tilde{G}}(\tilde{\pi}_{2},f)=J_{\tilde{M}}^{\tilde{G}}(\tilde{\pi}_{1},f)+J_{\tilde{M}}^{\tilde{G}}(\tilde{\pi}_{3},f),$$ pourvu que tous les termes soient définis.
On a:
\(2) supposons que $\tilde{\pi}$ soit irréductible mais pas $M$-irréductible; alors $J_{\tilde{M}}^{\tilde{G}}(\tilde{\pi},f)=0$ pourvu que ce terme soit défini.
Preuve. Puisque $\tilde{\pi}_{\tilde{\lambda}}$ vérifie les mêmes hypothèses que $\tilde{\pi}$, il suffit de considérer le cas où $\tilde{\pi}$ est en position générale. La décomposition de $\pi$ en composantes irréductibles pour $M(F)$ induit une décomposition de $Ind_{P}^G(\pi)$. L’opérateur ${\cal M}_{\tilde{M}}^{\tilde{G}}(\pi)$ préserve chaque composante tandis que $Ind_{\tilde{P}}^{\tilde{G}}(\tilde{\pi},f)$ les permute sans point fixe. $\square$
Plus généralement,
\(3) supposons que le caractère de $\tilde{\pi}$ soit nul; alors $J_{\tilde{M}}^{\tilde{G}}(\tilde{\pi},f)=0$ pourvu que ce terme soit défini.
Preuve. On peut encore supposer $\tilde{\pi}$ en position générale. Grâce à (1), on peut supposer que $\tilde{\pi}$ est somme d’irréductibles. On peut supprimer les irréductibles non $M$-irréductibles: cela ne modifie pas l’hypothèse puisque leur caractère est nul, ni la conclusion d’après (2). Notons $\rho_{1},...,\rho_{k}$ les différentes représentations irréductibles de $M(F)$ intervenant dans $\pi$. Pour chaque $\rho_{i}$, fixons un prolongement $\tilde{\rho}_{i}$ de $\rho_{i}$ à $\tilde{M}(F)$. Alors on a une égalité $$\tilde{\pi}=\oplus_{i=1,...,k}\oplus_{j=1,...,l_{i}}z_{i,j}\tilde{\rho}_{i},$$ pour des familles $(z_{i,j})_{j=1,...,l_{i}}$ de nombres complexes. On a alors $$J_{\tilde{M}}^{\tilde{G}}(\tilde{\pi},f)=\sum_{i=1,...,k}(\sum_{j=1,...,l_{j}}z_{i,j})J_{\tilde{M}}^{\tilde{G}}(\tilde{\rho}_{i},f).$$ D’après 2.5(1), l’hypothèse implique que $\sum_{j=1,...,l_{j}}z_{i,j}=0$ pour tout $i$. D’où la conclusion. $\square$
[**Remarque.**]{} Le terme $J_{\tilde{M}}^{\tilde{G}}(\tilde{\sigma},f)$ dépend de la mesure sur $G(F)$ (nécessaire pour définir $Ind_{\tilde{P}}^{\tilde{G}}(\tilde{\sigma},f)$) et de la mesure sur ${\cal A}_{\tilde{M}}^{\tilde{G}}$ (nécessaire pour définir le terme ${\cal M}_{\tilde{M}}^{\tilde{G}}(\sigma)$). Il ne dépend d’aucune autre mesure.
$R$-groupes tordus
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Soient $M$ un Levi semi-standard de $G$ et $\sigma$ une représentation irréductible et de la série discrète de $M(F)$. On note ${\cal N}^{\tilde{G}}(\sigma)$ l’ensemble des couples $(A,\gamma)$ où $A$ est un automorphisme unitaire de $V_{\sigma}$ et $\gamma\in Norm_{\tilde{G}(F)}(M)$ (c’est-à-dire $\gamma\in \tilde{G}(F)$ et $ad_{\gamma}(M)=M$) qui vérifient la condition $$\sigma( ad_{\gamma}(x))\circ A=\omega(x)A\circ \sigma(x)$$ pour tout $x\in M(F)$. Le groupe ${\cal N}^G(\sigma)$ agit à gauche et à droite sur ${\cal N}^{\tilde{G}}(\sigma)$ par $$\begin{array}{ccc}{\cal N}^G(\sigma)\times {\cal N}^{\tilde{G}}(\sigma)\times {\cal N}^G(\sigma)&\to&{\cal N}^{\tilde{G}}(\sigma)\\ ((A',n'),(A,\gamma),(A'',n''))&\mapsto &(A'AA''\omega(n''),n'\gamma n'').\\ \end{array}$$ L’ensemble ${\cal N}^{\tilde{G}}(\sigma)$ peut être vide. Supposons qu’il ne l’est pas. C’est alors un espace principal homogène sous ${\cal N}^G(\sigma)$, pour l’une ou l’autre des deux actions.
Soit $P\in {\cal P}(M)$, posons $\pi=Ind_{P}^G(\sigma)$. On définit une application $\tilde{\nabla}_{P}$ de ${\cal N}^{\tilde{G}}(\sigma)$ dans le groupe des opérateurs unitaires de $V_{\pi}$ de la façon suivante. Soit $(A,\gamma)\in {\cal N}^{\tilde{G}}(\sigma)$. Alors $\tilde{\nabla}_{P}(A,\gamma)$ est la composée des opérateurs suivants:
- l’opérateur $e\mapsto A\circ e$ de $V_{\pi}$ dans $V_{\pi_{1}}$, où $\pi_{1}=Ind_{P}^G(\omega^{-1}(\sigma\circ ad_{\gamma}))$;
- l’opérateur $\boldsymbol{\omega}$ de $V_{\pi_{1}}$ dans $V_{\pi_{2}}$, où $\pi_{2}=Ind_{P}^G(\sigma\circ ad_{\gamma})$, qui à $e\in V_{\pi_{1}}$, associe la fonction $g\mapsto \omega(g)e(g)$;
- l’opérateur $e\mapsto \partial_{P}(\gamma)^{1/2}e\circ ad_{\gamma}^{-1}$ de $V_{\pi_{2}}$ dans $V_{\pi'}$, où $\pi'=Ind_{ad_{\gamma}(P)}^G(\sigma)$ et $\partial_{P}(\gamma)$ est le jacobien de l’application $ad_{\gamma}:U_{P}(F)\to U_{ad_{\gamma}(P)}(F)$;
- l’opérateur $R_{P\vert ad{\gamma}(P)}(\sigma):V_{\pi'}\to V_{\pi}$.
On vérifie les relations
$(1) \qquad \pi(ad_{\gamma}(g))\circ \tilde{\nabla}_{P}(A,\gamma)=\omega(g)\tilde{\nabla}_{P}(A,\gamma)\circ \pi(g)$ pour tout $g\in G(F)$;
$(2) \qquad \pi(n')r_{P}(A',n')\tilde{\nabla}_{P}(A,\gamma)r_{P}(A'',n'')\pi(n'')\omega(n'')=\tilde{\nabla}_{P}(A'AA''\omega(n''),n'\gamma n'')$ pour tous $(A',n'), (A'',n'')\in {\cal N}^G(\sigma)$ et $(A,\gamma)\in {\cal N}^{\tilde{G}}(\sigma)$.
Il s’en déduit que si $(A',n')(A,\gamma)=(A,\gamma)(A'',n'')$, on a l’égalité $$(3) \qquad r_{P}(A',n')\tilde{\nabla}_{P}(A,\gamma)=\tilde{\nabla}_{P}(A,\gamma)r_{P}(A'',n'').$$
On vérifie directement que les orbites dans ${\cal N}^{\tilde{G}}(\sigma)$ pour l’action du sous-groupe $M(F)\subset {\cal N}^G(\sigma)$ sont les mêmes, que l’on considère l’action à gauche ou l’action à droite. Notons ${\cal W}^{\tilde{G}}(\sigma)$ l’ensemble de ces orbites. Grâce à (3), les orbites dans ${\cal W}^{\tilde{G}}(\sigma)$ pour l’action de $W^G_{0}(\sigma)$ sont les mêmes, que l’on considère l’action à gauche ou l’action à droite. Notons ${\cal R}^{\tilde{G}}(\sigma)$ l’ensemble de ces orbites. L’ensemble ${\cal R}^{\tilde{G}}(\sigma)$ est un espace principal homogène sous l’action à droite ou à gauche de ${\cal R}^G(\sigma)$. Posons $$W^{\tilde{G}}(\sigma)=\{\gamma\in \tilde{G}(F); ad_{\gamma}(M)=M, \sigma\circ ad_{\gamma}\simeq \sigma\otimes \omega\}/M(F),$$ et $R^{\tilde{G}}(\sigma)=W^{\tilde{G}}(\sigma)/W_{0}^G(\sigma)$. On a une application surjective $${\cal R}^{\tilde{G}}(\sigma)\to R^{\tilde{G}}(\sigma)$$ dont les fibres sont isomorphes à ${\mathbb U}$.
Soit $\tilde{\lambda}\in i\tilde{{\cal A}}_{\tilde{G},F}^*$. L’application $$\begin{array}{ccc}{\cal N}^{\tilde{G}}(\sigma)&\to &{\cal N}^{\tilde{G}}(\sigma_{\lambda})\\ (A,\gamma)&\mapsto&(Ae^{<\tilde{\lambda},\tilde{H}_{\tilde{G}}(\gamma)>},\gamma)\\ \end{array}$$ est bijective. Il s’en déduit une bijection ${\cal R}^{\tilde{G}}(\sigma)\simeq {\cal R}^{\tilde{G}}(\sigma_{\lambda})$ compatible avec l’identité $R^{\tilde{G}}(\sigma)= R^{\tilde{G}}(\sigma_{\lambda})$. Ces bijections sont aussi compatibles avec les isomorphismes ${\cal N}^G(\sigma)\simeq {\cal N}^G(\sigma_{\lambda})$ et ${\cal R}^G(\sigma)\simeq {\cal R}^G(\sigma_{\lambda})$.
Soit $g\in G(F)$ tel que $M'=gMg^{-1}$ soit semi-standard. Posons $\sigma'=g\sigma=\sigma\circ ad_{g}^{-1}$. On définit une application $$\begin{array}{ccc}{\cal N}^{\tilde{G}}(\sigma)&\to&{\cal N}^{\tilde{G}}(\sigma')\\ (A,\gamma)&\mapsto& (A\omega(g),g\gamma g^{-1}).\\ \end{array}$$ Elle est bijective et se quotiente en des bijections ${\cal R}^{\tilde{G}}(\sigma)\simeq {\cal R}^{\tilde{G}}(\sigma')$, $R^{\tilde{G}}(\sigma)\simeq R^{\tilde{G}}(\sigma')$. On note toutes ces applications $ad_{g}$. Ce sont ces applications qui servent à définir les notions de conjugaison par $G(F)$ utilisées dans la suite. Par exemple, pour $\boldsymbol{\tilde{r}}\in {\cal R}^{\tilde{G}}(\sigma)$, on dit que les triplets $(M,\sigma,\boldsymbol{\tilde{r}})$ et $(M',\sigma',ad_{g}(\boldsymbol{\tilde{r}}))$ sont conjugués.
Appelons représentation de ${\cal R}^{\tilde{G}}(\sigma)$ un couple $(\rho,\tilde{\rho})$, où $\rho$ est une représentation (disons unitaire et de dimension finie) de ${\cal R}^G(\sigma)$ et $\tilde{\rho}$ est un homomorphisme de ${\cal R}^{\tilde{G}}(\sigma)$ dans le groupe des automorphismes de $V_{\rho}$ vérifiant la condition $$\rho({\bf r}')\tilde{\rho}(\boldsymbol{\tilde{r}})\rho({\bf r}'')=\tilde{\rho}({\bf r}'\boldsymbol{\tilde{r}}{\bf r}'')$$ pour tous ${\bf r}',{\bf r}''\in {\cal R}^G(\sigma)$, $\boldsymbol{\tilde{r}}\in {\cal R}^{\tilde{G}}(\sigma)$. On dit que $\tilde{\rho}$ est ${\cal R}^G(\sigma)$-irréductible si $\rho$ est irréductible. Le groupe ${\mathbb U}$ agit par multiplication sur ces représentations. Par ailleurs, tout élément $\boldsymbol{\tilde{r}}\in {\cal R}^{\tilde{G}}(\sigma)$ détermine un automorphisme $\theta_{\boldsymbol{\tilde{r}}}$ de ${\cal R}^G(\sigma)$ par l’égalité $\boldsymbol{\tilde{r}}{\bf r}=\theta_{\boldsymbol{\tilde{r}}}({\bf r})\boldsymbol{\tilde{r}}$. Notons $\theta_{{\cal R}}$ sa classe modulo automorphismes intérieurs, qui ne dépend pas du choix de $\boldsymbol{\tilde{r}}$. Alors les orbites de ${\mathbb U}$ dans l’ensemble des classes de représentations ${\cal R}^G(\sigma)$-irréductibles de ${\cal R}^{\tilde{G}}(\sigma)$ sont en bijection avec l’ensemble $Irr({\cal R}^G(\sigma);\theta_{{\cal R}})$ des représentations irréductibles $\rho$ de ${\cal R}^G(\sigma)$ telles que $\rho\circ\theta_{{\cal R}}\simeq \rho$.
Considérons la décomposition 1.11(2) de l’espace $V_{\pi}$. Soit $\rho\in Irr({\cal R}^G(\sigma))$. La relation (3) implique que, pour $(A,\gamma)\in {\cal N}^{\tilde{G}}(\sigma)$, $\tilde{\nabla}_{P}(A,\gamma)$ envoie la composante $\rho$-isotypique $\rho\otimes \pi_{\rho}$ sur la composante isotypique $(\rho\circ\theta_{{\cal R}})\otimes \pi_{\rho\circ \theta_{{\cal R}}}$. La relation (1) entraîne alors que $\pi_{\rho\circ\theta_{{\cal R}}}\circ\theta\simeq \omega\pi_{\rho}$. En particulier, $\pi_{\rho}\circ\theta\simeq \omega\pi_{\rho}$ si et seulement si $\rho\in Irr({\cal R}^G(\sigma),\theta_{{\cal R}})$. Supposons cette relation vérifiée et choisissons une représentation $\tilde{\rho}$ de ${\cal R}^{\tilde{G}}(\sigma)$ prolongeant $\rho$. Alors il existe une unique représentation $\tilde{\pi}_{\tilde{\rho}}$ de $\tilde{G}(F)$ prolongeant $\pi_{\rho}$ de sorte que, pour tout $(A,\gamma)\in {\cal N}^{\tilde{G}}(\sigma)$, la restriction de $\tilde{\nabla}_{P}(A,\gamma)$ à $\rho\otimes \pi_{\rho}$ soit égale à $\tilde{\rho}(\boldsymbol{\tilde{r}})\otimes \tilde{\pi}_{\tilde{\rho}}(\gamma)$, où $\boldsymbol{\tilde{r}}$ est l’image de $(A,\gamma)$ dans ${\cal R}^{\tilde{G}}(\sigma)$.
Ainsi, à un triplet $(M,\sigma,\rho)$, où $M$ est un Levi semi-standard, $\sigma$ est une représentation irréductible de la série discrète de $M(F)$ telle que ${\cal N}^{\tilde{G}}(\sigma)\not=\emptyset$ et $\rho\in Irr({\cal R}^G(\sigma);\theta_{{\cal R}})$, on a associé une représentation $G$-irréductible et tempérée $\tilde{\pi}_{\tilde{\rho}}$ de $\tilde{G}(F)$, uniquement définie à multiplication près par ${\mathbb U}$. Comme en 1.11, on voit que cette correspondance ne dépend pas du sous-groupe parabolique $P$ choisi. Sur l’ensemble des triplets $(M,\sigma,\rho)$, on définit de façon évidente la relation de conjugaison par $G(F)$. La construction ci-dessus se quotiente en une bijection entre l’ensemble des classes de conjugaison par $G(F)$ de triplets $(M,\sigma,\rho)$ et l’ensemble des orbites de ${\mathbb U}$ dans l’ensemble des classes d’isomorphisme de représentations $G$-irréductibles et tempérées de $\tilde{G}(F)$.
L’ensemble $E(\tilde{G},\omega)$
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Soient $M$ et $\sigma$ comme dans le paragraphe précédent. On suppose ${\cal N}^{\tilde{G}}(\sigma)\not=\emptyset$. On fixe $P\in {\cal P}(M)$ et on note $\pi=Ind_{P}^G(\sigma)$. Soit $(A,\gamma)\in {\cal N}^{\tilde{G}}(\sigma)$. Grâce à 2.7(1), on peut définir une $\omega$-représentation $\tilde{\pi}$ de $\tilde{G}(F)$ par la formule $$\tilde{\pi}(g\gamma)=\pi(g)\tilde{\nabla}_{P}(A,\gamma)$$ pour tout $g\in G(F)$. Notons que cette représentation n’est pas irréductible en général. La relation 2.7(2) montre que $\tilde{\pi}$ ne dépend que de l’image $\boldsymbol{\tilde{r}}$ de $(A,\gamma)$ dans ${\cal R}^{\tilde{G}}(\sigma)$. Elle ne dépend donc que du triplet $\boldsymbol{\tau}=(M,\sigma,\boldsymbol{\tilde{r}})$ et on peut la noter $\tilde{\pi}_{\boldsymbol{\tau}}$. On voit aussi que sa classe ne dépend pas de $P$. On vérifie que la classe de $\tilde{\pi}_{\boldsymbol{\tau}}$ ne dépend que de la classe de conjugaison par $G(F)$ du triplet $ \boldsymbol{\tau}$ (cf. 2.8 pour cette notion de conjugaison). Il est tout aussi clair que la correspondance est équivariante pour les actions de ${\mathbb U}$: pour $z\in {\mathbb U}$, la représentation correspondant à $(M,\sigma,z\boldsymbol{\tilde{r}})$ est $z\tilde{\pi}_{\boldsymbol{\tau}}$. Le groupe ${\cal R}^G(\sigma)$ agissant à gauche et à droite sur ${\cal R}^{\tilde{G}}(\sigma)$, il agit aussi par conjugaison. Pour $\boldsymbol{\tilde{r}}\in {\cal R}^{\tilde{G}}(\sigma)$, il peut exister $z\in {\mathbb U}$, $z\not=1$, tel que $z\boldsymbol{\tilde{r}}$ soit conjugué à $\boldsymbol{\tilde{r}}$. Dans ce cas, $z\tilde{\pi}\simeq \tilde{\pi}$ donc le caractère de $\tilde{\pi}$ est nul. On dit que le triplet $(M,\sigma,\boldsymbol{\tilde{r}})$ est essentiel si la classe de conjugaison de $\boldsymbol{\tilde{r}}$ par ${\cal R}^G(\sigma)$ ne coupe ${\mathbb U}\boldsymbol{\tilde{r}}$ qu’en le point $\boldsymbol{\tilde{r}}$. On note ${\cal E}(\tilde{G},\omega)$ l’ensemble des triplets $(M,\sigma,\boldsymbol{\tilde{r}})$ qui sont essentiels.
Soit $\boldsymbol{\tau}=(M,\sigma,\boldsymbol{\tilde{r}})$ un triplet comme ci-dessus et soit $\tilde{\lambda}\in i\tilde{{\cal A}}_{\tilde{G},F}^*$. A l’aide de $\tilde{\lambda}$, on a défini en 2.8 un isomorphisme ${\cal R}^{\tilde{G}}(\sigma)\simeq {\cal R}^{\tilde{G}}(\sigma_{\lambda})$. En identifiant ces deux ensembles, le triplet $\boldsymbol{\tau}_{\tilde{\lambda}}=(M,\sigma_{\lambda},\boldsymbol{\tilde{r}})$ vérifie encore les conditions requises. On vérifie que $\boldsymbol{\tau}$ est essentiel si et seulement si $\boldsymbol{\tau}_{\tilde{\lambda}}$ l’est et qu’on a l’égalité $\tilde{\pi}_{\boldsymbol{\tau}_{\tilde{\lambda}}}=(\tilde{\pi}_{\boldsymbol{\tau}})_{\tilde{\lambda}}$.
On note $E(\tilde{G},\omega)$ l’ensemble des triplets $(M,\sigma,\tilde{r})$, où $(M,\sigma)$ est comme ci-dessus, $\tilde{r}\in R^{\tilde{G}}(\sigma)$ et il existe $\boldsymbol{\tilde{r}}\in {\cal R}^{\tilde{G}}(\sigma)$ relevant $\tilde{r}$ de sorte que le triplet $(M,\sigma,\boldsymbol{\tilde{r}})$ soit essentiel. Autrement dit, $E(\tilde{G},\omega)$ est le quotient de ${\cal E}(\tilde{G},\omega)$ par l’action naturelle de ${\mathbb U}$. On note $E(\tilde{G},\omega)/conj$ l’ensemble des classes de conjugaison par $G(F)$ dans l’ensemble $E(\tilde{G},\omega)$. Pour $\boldsymbol{\tau}\in {\cal E}(\tilde{G},\omega)$, l’espace porté par le caractère de $\tilde{\pi}_{\boldsymbol{\tau}}$, c’est-à-dire $\{0\}$ si ce caractère est nul et une droite sinon, ne dépend que de l’image $\tau$ de $\boldsymbol{\tau}$ dans $E(\tilde{G},\omega)$. On le note $D_{\tau}$. Il ne dépend aussi que de la classe de conjugaison de $\tau$. On a défini l’espace $D_{spec}(\tilde{G}(F);\omega)$ en 2.5. On note $D_{temp}(\tilde{G}(F);\omega)$ le sous-espace engendré par les caractères de $\omega$-représentations tempérées.
0.3cm[**[Proposition]{}**]{}.
**
\(i) Pour tout $\boldsymbol{\tau}\in {\cal E}(\tilde{G},\omega)$, le caractère de $\tilde{\pi}_{\boldsymbol{\tau}}$ est non nul.
\(ii) On a l’égalité $$D_{temp}(\tilde{G}(F);\omega)=\oplus_{\tau\in E(\tilde{G},\omega)/conj}D_{\tau}.$$
0.3cm
Preuve. Pour tout couple $(M,\sigma)$ comme en 2.8 et pour tout $\rho\in Irr({\cal R}^G(\sigma),\theta_{{\cal R}})$, on fixe une extension $\tilde{\rho}$ de $\rho$ à ${\cal R}^{\tilde{G}}(\sigma)$. Pour tout $\tilde{r}\in R^{\tilde{G}}(\sigma)$, on fixe aussi un relèvement $\boldsymbol{\tilde{r}}\in {\cal R}^{\tilde{G}}(\sigma)$. Pour tout triplet $\beta=(M,\sigma,\rho)$ comme en 2.8, on pose $\tilde{\pi}_{\beta}=\tilde{\pi}_{\tilde{\rho}}$ et, pour tout $\tau=(M,\sigma,\tilde{r})\in E(\tilde{G},\omega)$, on pose $\tilde{\pi}_{\tau}=\tilde{\pi}_{\boldsymbol{\tau}}$, où $\boldsymbol{\tau}=(M,\sigma,\boldsymbol{\tilde{r}})$. On a décrit les $\omega$-représentations tempérées et $G$-irréductibles en 2.8, à l’aide de triplets $\beta=(M,\sigma,\rho)$. L’espace $D_{temp}(\tilde{G}(F);\omega)$ est somme directe des $D_{\tilde{\pi}_{\beta}}$ quand $\beta$ décrit les classes de conjugaison de triplets $\beta$. Il nous suffit de prouver l’assertion suivante. Fixons un couple $(M,\sigma)$. Notons $(\tau_{i})_{i=1,...,n}$ les différents éléments de $E(\tilde{G},\omega)$ de la forme $(M,\sigma,\tilde{r}_{i})$ et notons $(\beta_{j})_{j=1,...,m}$ les differents triplets comme ci-dessus de la forme $(M,\sigma,\rho_{j})$. Notons $X$ la matrice colonne à $n$ lignes dont les coefficients sont les caractères $trace(\tilde{\pi}_{\tau_{i}})$ des $\tilde{\pi}_{\tau_{i}}$ et notons $Y$ la matrice colonne à $m$ lignes dont les coefficients sont les caractères $trace(\tilde{\pi}_{\beta_{j}})$ des $\tilde{\pi}_{\beta_{j}}$. Alors
\(1) il existe une matrice à coefficients complexes $M$ telle que $X=MY$ et $M$ est inversible.
Soit $(A_{i},\gamma_{i})\in {\cal N}^{\tilde{G}}(\sigma)$ se projetant sur $\boldsymbol{\tilde{r}}_{i}$. On a vu en 2.8 que l’opérateur $\tilde{\nabla}_{P}(A_{i},\gamma_{i})$ se restreint à une composante $\rho_{j}\otimes \pi_{\rho_{j}}$ en l’opérateur $\tilde{\rho}_{j}(\boldsymbol{\tilde{r}}_{i})\otimes \tilde{\pi}_{\beta_{j}}(\gamma)$ et qu’il permute sans point fixe les composantes $\rho\otimes \pi_{\rho}$ pour $\rho\not\in Irr({\cal R}^G(\sigma),\theta_{{\cal R}})$. Il résulte alors de la définition de $\tilde{\pi}_{\tau_{i}}$ que l’on a l’égalité $$trace(\tilde{\pi}_{\tau_{i}})=\sum_{j=1,...,m}trace(\tilde{\rho}_{j}(\boldsymbol{\tilde{r}}_{i}))trace(\tilde{\pi}_{\beta_{j}}).$$ Autrement dit, la matrice $M=(trace(\tilde{\rho}_{j}(\boldsymbol{\tilde{r}}_{i})))_{i=1,...,n;j=1,...,m}$ vérifie $X=MY$. On va montrer que $M$ est inversible. Le groupe ${\cal R}^G(\sigma)$ est une extension de $R^G(\sigma)$ par ${\mathbb U}$. Le groupe $R^G(\sigma)$ étant fini, l’extension provient d’une extension par un sous-groupe fini $Z\subset {\mathbb U}$. Substituons ce groupe $Z$ à ${\mathbb U}$ dans toutes les constructions. Fixons $\boldsymbol{\tilde{r}}\in {\cal R}^{\tilde{G}}(\sigma)$. On en déduit un automorphisme $\theta_{\boldsymbol{\tilde{r}}}$ de ${\cal R}^G(\sigma)$. Fixons un entier $N\geq2$ tel que l’ordre de $\theta_{\boldsymbol{\tilde{r}}}$ divise $N$. Introduisons le produit semi-direct $H={\cal R}^G(\sigma)\rtimes({\mathbb Z}/N{\mathbb Z})$ où $k\in {\mathbb Z}/N{\mathbb Z}$ agit par $(\theta_{\boldsymbol{\tilde{r}}})^k$. On identifie ${\cal R}^{\tilde{G}}(\sigma)$ à la composante ${\cal R}^G(\sigma)\times\{1\}$ en envoyant ${\bf r}\boldsymbol{\tilde{r}}$ sur $({\bf r},1)$, pour tout ${\bf r}\in {\cal R}^G(\sigma)$. On est ramené à un problème concernant les représentations irréductibles du groupe fini $H$. Sa solution est bien connue, indiquons simplement le résultat. Pour tout $\tilde{r}\in R^{\tilde{G}}(\sigma)$, notons $Stab(R^G(\sigma),\tilde{r})$ le stabilisateur de $\tilde{r}$ dans $R^G(\sigma)$ (agissant par conjugaison dans $R^{\tilde{G}}(\sigma)$). Introduisons la matrice $$M'=(\vert Stab(R^G(\sigma),\tilde{r}_{i})\vert ^{-1}\overline{trace(\tilde{\rho}_{j}(\boldsymbol{\tilde{r}}_{i}))})_{i=1,...,n; j=1,...,m}.$$ Alors la transposée de $M'$ est l’inverse de $M$. $\square$
Le groupe $i{\cal A}_{\tilde{G},F}^*$ agit naturellement sur $E(\tilde{G},\omega)$: pour $\lambda\in i{\cal A}_{\tilde{G},F}^*$ et pour $\tau=(M,\sigma,\tilde{ r})$, on pose $\tau_{\lambda}=(M,\sigma_{\lambda},\tilde{r})$. Ainsi, on dispose d’une action de $W^G\times i{\cal A}_{\tilde{G},F}^*$ sur $E(\tilde{G},\omega)$. Pour $\tau\in E(\tilde{G},\omega)$, on note $Stab(W^G\times i{\cal A}_{\tilde{G},F}^*,\tau)$ son stabilisateur dans $W^G\times i{\cal A}_{\tilde{G},F}^*$. C’est un groupe fini qui contient le stabilisateur $ Stab(i{\cal A}_{\tilde{G},F}^*,\sigma)$ de $\sigma$ dans $i{\cal A}_{\tilde{G},F}^*$. Il contient aussi le groupe $W^M$ comme sous-groupe distingué et son quotient $Stab(W^G\times i{\cal A}_{\tilde{G},F}^*,\tau)/W^M$ contient $W_{0}^G(\sigma)$ comme sous-groupe distingué. On pose $${\bf Stab}(W^G\times i{\cal A}_{\tilde{G},F}^*,\tau)=(Stab(W^G\times i{\cal A}_{\tilde{G},F}^*,\tau)/W^M)/W_{0}^G(\sigma).$$ Remarquons que ${\bf Stab}(W^G\times i{\cal A}_{\tilde{G},F}^*,\tau_{\lambda})={\bf Stab}(W^G\times i{\cal A}_{\tilde{G},F}^*,\tau)$ pour $\lambda\in i{\cal A}_{\tilde{G},F}^*$ et que $\vert {\bf Stab}(W^G\times i{\cal A}_{\tilde{G},F}^*,\tau)\vert $ ne dépend que de l’image de $\tau$ dans $E(\tilde{G},\omega)/conj$.
[**Remarque.**]{} Le groupe $Stab(W^G\times i{\cal A}_{\tilde{G},F}^*,\tau)$ peut être plus gros que le produit des stabilisateurs de $\tau$ dans chacun des groupes $W^G$ et $i{\cal A}_{\tilde{G},F}^*$: on peut avoir une relation $w\sigma\simeq \sigma_{\lambda}\not\simeq \sigma$.
Triplets et induction
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Soit $\tilde{L}$ un espace de Levi de $\tilde{G}$ contenant $\tilde{M}_{0}$, soit $M$ un Levi semi-standard de $L$ et soit $\sigma$ une représentation irréductible et de la série discrète de $M(F)$. Le groupe ${\cal N}^L(\sigma)$ est contenu dans ${\cal N}^G(\sigma)$ et l’ensemble ${\cal N}^{\tilde{L}}(\sigma)$ est contenu dans l’ensemble ${\cal N}^{\tilde{G}}(\sigma)$. Fixons $\tilde{Q}\in {\cal P}(\tilde{L})$ et $P\in {\cal P}(M)$ tel que $P\subset Q$. Posons $\Pi=Ind_{P}^G(\sigma)$, $\pi=Ind_{P\cap L}^L(\sigma)$. On sait que $Ind_{Q}^G(\pi)\simeq \Pi$. Dans les modèles naturels, l’isomorphisme envoie une fonction $e$ du premier espace sur la fonction $g\mapsto (e(g))(1)$. Soit $(A,n)\in {\cal N}^L(\sigma)$. On dispose de l’entrelacement $r_{P\cap L}^L(A,n)$ de $\pi$ et de l’entrelacement $r_{P}(A,n)$ de $\Pi$. On a
\(1) $r_{P}(A,n)$ s’identifie à l’opérateur déduit par fonctorialité de $r^L_{P\cap L}(A,n)$.
C’est un simple calcul utilisant le fait que l’opérateur $R_{P\vert ad_{n}(P)}(\sigma)$ se déduit par fonctorialité de $R^L_{P\vert ad_{n}(P)}(\sigma)$.
Puisque $W^G_{0}(\sigma)$ est par définition le noyau de $(A,n)\mapsto r_{P}(A,n)$ dans ${\cal N}^G(\sigma)/M(F)$ et de même pour $W^L_{0}(\sigma)$ , on déduit de (1) que $W^L_{0}(\sigma)=W^G_{0}(\sigma)\cap ({\cal N}^L(\sigma)/M(F))$. Alors, d’après les définitions, les plongements ${\cal N}^L(\sigma)\to {\cal N}^G(\sigma)$ et ${\cal N}^{\tilde{L}}(\sigma)\to {\cal N}^{\tilde{G}}(\sigma)$ se quotientent en des plongements ${\cal R}^L(\sigma)\to {\cal R}^G(\sigma)$ et ${\cal R}^{\tilde{L}}(\sigma)\to {\cal R}^{\tilde{G}}(\sigma)$. Evidemment, on a aussi des plongements $R^L(\sigma)\to R^G(\sigma)$ et $R^{\tilde{L}}(\sigma)\to R^{\tilde{G}}(\sigma)$. Supposons ${\cal R}^{\tilde{L}}(\sigma)\not=\emptyset$.
[**Remarque.**]{} Cette hypothèse entraîne que la restriction de $\omega$ à $Z_{L}(F)^{\theta}$ est triviale.
Soit $\boldsymbol{\tilde{r}}\in {\cal R}^{\tilde{L}}(\sigma)$, posons $\boldsymbol{\tau}=(M,\sigma,\boldsymbol{\tilde{r}})$. On peut considérer ce triplet relativement à chacun des espaces ambiants $\tilde{L}$ ou $\tilde{G}$. Remarquons que la notion de triplet essentiel dépend de l’espace ambiant. Si le triplet est essentiel relativement à $\tilde{G}$, il l’est relativement à $\tilde{L}$, mais la réciproque semble fausse en général (je dis semble car je n’ai pas d’exemple). Notons qu’associer une $\omega$-représentation à un triplet ne nécessite pas que le triplet soit essentiel (s’il ne l’est pas, le caractère de cette représentation est nul). Ainsi, on associe à $\boldsymbol{\tau}$ une représentation $\tilde{\pi}_{\boldsymbol{\tau}}$ de $\tilde{L}(F)$, resp. $\tilde{\Pi}_{\boldsymbol{\tau}}$ de $\tilde{G}(F)$. [0.3cm[**[Lemme]{}**]{}. [ *[La représentation $\tilde{\Pi}_{\boldsymbol{\tau}}$ est isomorphe à l’induite $Ind_{\tilde{Q}}^{\tilde{G}}(\tilde{\pi}_{\boldsymbol{\tau}})$.]{}*]{}0.3cm]{}
Preuve. Fixons $(A,\gamma)\in {\cal N}^{\tilde{G}}(\sigma)$ se projetant sur $\boldsymbol{\tilde{r}}$. On construit les représentations $\tilde{\Pi}_{\boldsymbol{\tau}}$ et $\tilde{\pi}_{\boldsymbol{\tau}}$ à l’aide de ces éléments comme en 2.8. Leurs représentations sous-jacentes de $G(F)$ sont respectivement $ \Pi$ et $\pi$. Comme on l’a dit ci-dessus, $\Pi\simeq Ind_{Q}^G(\pi)$. L’action de $\gamma$ sur $Ind_{\tilde{Q}}^{\tilde{G}}(\tilde{\pi}_{\boldsymbol{\tau}})$ se déduit de $\tilde{\nabla}_{P}^{\tilde{L}}(A,\gamma)$ par la formule 2.5(3). Il suffit de vérifier que, par l’isomorphisme précédent, cette action coïncide avec $\tilde{\Pi}_{\boldsymbol{\tau}}(\gamma)=\tilde{\nabla}_{P}(A,\gamma)$. Comme pour (1), c’est un simple calcul reposant sur le fait que l’opérateur $R_{P\vert ad_{\gamma}(P)}(\sigma)$ se déduit par fonctorialité de $R^L_{P\vert ad_{\gamma}(P)}(\sigma)$. $\square$
Les ensembles $E_{disc}(\tilde{G},\omega)$ et $E_{ell}(\tilde{G},\omega)$
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Soient $M$ un Levi semi-standard de $G$ et $\sigma$ une représentation irréductible et de la série discrète de $M(F)$. Un élément $\tilde{w}\in W^{\tilde{G}}(\sigma)$ agit naturellement sur ${\cal A}_{M}$. Notons ${\cal A}_{M}^{\tilde{w}}$ le sous-espace des points fixes par cette action. Il contient ${\cal A}_{\tilde{G}}$. On note $W^{\tilde{G}}_{reg}(\sigma)$ l’ensemble des $\tilde{w}\in W^{\tilde{G}}(\sigma)$ tels que ${\cal A}_{M}^{\tilde{w}}={\cal A}_{\tilde{G}}$. Rappelons que $R^{\tilde{G}}(\sigma)=W^G_{0}(\sigma)\backslash W^{\tilde{G}}(\sigma)$, en particulier $R^{\tilde{G}}(\sigma)=W^{\tilde{G}}(\sigma)$ si $W^G_{0}(\sigma)=\{1\}$.
0.3cm[**[Lemme]{}**]{}.
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Soit $\tilde{r}\in R^{\tilde{G}}(\sigma)$. Les conditions suivantes sont équivalentes:
\(a) il n’existe pas d’espace de Levi $\tilde{L}$ tel que $M\subset L\subsetneq G$ et $\tilde{r}\in R^{\tilde{L}}(\sigma)$;
\(b) $W^G_{0}(\sigma)=\{1\}$ et $\tilde{r}\in W^{\tilde{G}}_{reg}(\sigma)$.
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Preuve. Fixons $\tilde{w}\in W^{\tilde{G}}(\sigma)$ d’image $\tilde{r}$. Pour un espace de Levi $\tilde{L}$ tel que $M\subset L\subsetneq G$, l’image de $R^{\tilde{L}}(\sigma)$ dans $R^{\tilde{G}}(\sigma)$ est $W^G_{0}(\sigma)\backslash(W^G_{0}(\sigma)W^{\tilde{L}}(\sigma))$. La condition (a) équivaut donc à
\(1) pour tout $w'\in W^{G}_{0}(\sigma)$, il n’existe pas de $\tilde{L}$ comme ci-dessus tel que $w'\tilde{w}\in W^{\tilde{L}}(\sigma)$.
D’après 2.1(4), la condition $w'\tilde{w}\in W^{\tilde{L}}(\sigma)$ équivaut à ${\cal A}_{\tilde{L}}\subset{\cal A}_{M}^{w'\tilde{w}}$. Donc (1) est équivalent à
\(2) pour tout $w'\in W^{G}_{0}(\sigma)$, $w'\tilde{w}\in W^{\tilde{G}}_{reg}(\sigma)$.
Si $W^G_{0}(\sigma)=\{1\}$, (2) est équivalente à (b). Si $W^G_{0}(\sigma)\not=\{1\}$, (b) n’est pas vérifiée et on doit prouver que (2) ne l’est pas non plus. Comme on l’a dit en 1.11, $W^G_{0}(\sigma)$ est le groupe de Weyl d’un système de racines inclus dans l’ensemble des racines de $A_{M}$ dans $G$. Il résulte des définitions que l’action de $\tilde{w}$ conserve ce système de racines. Fixons une base de ce système. On peut alors trouver $w'\in W^G_{0}(\sigma)$ tel que $w'\tilde{w}$ conserve cette base (en permutant ses éléments). La somme des coracines associées aux éléments de cette base est alors un élément de ${\cal A}_{M}^{w'\tilde{w}}$ qui n’appartient pas à ${\cal A}_{\tilde{G}}$. Donc $w'\tilde{w}\not\in W^{\tilde{G}}_{reg}(\sigma)$ et (2) n’est pas vérifiée. $\square$
Soit $\tilde{w}\in W^{\tilde{G}}(\sigma)$. Comme on vient de le dire, l’action de $\tilde{w}$ conserve l’ensemble de racines associé à $W_{0}^G(\sigma)$, cf.1.11. En fixant un sous-ensemble positif, on pose $\epsilon_{\sigma}(\tilde{w})=(-1)^{n(\tilde{w})}$, où $n(\tilde{w})$ est le nombre de racines positives $\alpha$ telles que $\tilde{w}(\alpha)$ soit négative. Ce signe ne dépend pas de l’ensemble positif choisi. Soit $\tau=(M,\sigma,\tilde{r})\in E(\tilde{G},\omega)$. On choisit $\tilde{w}\in {\cal W}^{\tilde{G}}(\sigma)$ se projetant sur $\tilde{r}$ et on pose $$\iota(\tau)=\vert W^G_{0}(\sigma)\vert ^{-1}\sum_{\tilde{w}'\in W^G_{0}(\sigma)\tilde{w}\cap W^{\tilde{G}}_{reg}(\sigma)}\epsilon_{\sigma}(\tilde{w}')\vert det((1-\tilde{w}')_{\vert {\cal A}_{M}^{\tilde{G}}})\vert ^{-1}.$$
On dit que le triplet $\tau\in E(\tilde{G},\omega) $ est discret si $W^G_{0}(\sigma)\tilde{w}\cap W^{\tilde{G}}_{reg}(\sigma)\not=\emptyset$. On dit que $\tau$ est elliptique s’il est discret et de plus $W^G_{0}(\sigma)=\{1\}$. On note $E_{disc}(\tilde{G},\omega)$, resp. $E_{ell}(\tilde{G},\omega)$, l’ensemble des triplets $(M,\sigma,r)$ qui sont discrets, resp. elliptiques. On note $E_{disc}(\tilde{G},\omega)/conj$, resp. $E_{ell}(\tilde{G},\omega)/conj$, l’ensemble des classes de conjugaison par $G(F)$ dans $E_{disc}(\tilde{G},\omega)$, resp. $E_{ell}(\tilde{G},\omega)$.
Représentations elliptiques
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Posons, avec les notations de 2.9, $$D_{ell}(\tilde{G}(F);\omega)=\oplus_{\tau\in E_{ell}(\tilde{G},\omega)/conj}D_{\tau}.$$ Pour tout ensemble de Levi $\tilde{L}\in {\cal L}(\tilde{M}_{0})$, posons $W^G(\tilde{L})=Norm_{G(F)}(\tilde{L})/L(F)$. Supposons que $\omega$ soit trivial sur $Z_{L}(F)^{\theta}$. Le groupe $Norm_{G(F)}(\tilde{L})$ agit sur $D_{temp}(\tilde{L}(F);\omega)$: à $n\in Norm_{G(F)}(\tilde{L})$ et $d\in D_{temp}(\tilde{L}(F);\omega)$, on associe la distribution $f\mapsto \omega(n)^{-1}d(f^n)$, où $f^n(\gamma)=f(n\gamma n^{-1})$. Cette action se descend en une action de $W^G(\tilde{L})$ sur $D_{temp}(\tilde{L}(F);\omega)$. Cette action conserve le sous-espace $D_{ell}(\tilde{L}(F);\omega)$. Selon l’usage, on note $D_{ell}(\tilde{L}(F);\omega)^{W^G(\tilde{L})}$ le sous-espace des invariants.
Définissons une action du groupe $Norm_{G(F)}(\tilde{L})$ sur l’ensemble ${\cal E}(\tilde{L})/conj$ des classes de conjugaison par $L(F)$ dans ${\cal E}(\tilde{L})$. Soient $n\in Norm_{G(F)}(\tilde{L})$ et $\boldsymbol{\tau}=(M,\sigma,\boldsymbol{\tilde{r}})\in {\cal E}(\tilde{L})$. Posons $M'=nMn^{-1}$, $\sigma'=n\sigma$. Quitte à multiplier $n$ à gauche par un élément de $L(F)$, on peut supposer $M'$ semi-standard. On définit une bijection $ad_{n}:{\cal N}^{\tilde{L}}(\sigma)\simeq {\cal N}^{\tilde{L}}(\sigma')$ comme en 2.8: à $(A,\gamma)$, on associe $(A\omega(n),n\gamma n^{-1})$. Cette bijection se descend en une bijection $ad_{n}:{\cal R}^{\tilde{G}}(\sigma)\to {\cal R}^{\tilde{G}}(\sigma')$. La classe de conjugaison par $L(F)$ du triplet $\boldsymbol{\tau}'=(M',\sigma',ad_{n}(\boldsymbol{\tilde{r}}))$ ne dépend pas de la modification de $n$ faite ci-dessus et ne dépend que de la classe de conjugaison par $L(F)$ de $\boldsymbol{\tau}$. L’action cherchée est celle qui associe à la classe de $\boldsymbol{\tau}$ celle de $\boldsymbol{\tau}'$. Cette action se quotiente en une action de $W^G(\tilde{L})$.
Notons ${\cal L}(\tilde{M}_{0};\omega)$ l’ensemble des $\tilde{L}\in {\cal L}(\tilde{M}_{0})$ tels que $\omega$ soit trivial sur $Z_{L}(F)^{\theta}$. Soit $\tilde{L}\in {\cal L}(\tilde{M}_{0};\omega)$. Pour $\tilde{Q}\in {\cal P}(\tilde{M})$, l’opération d’induction $Ind_{\tilde{Q}}^{\tilde{G}}$ définit une application de $D_{temp}(\tilde{L}(F);\omega)$ dans $ D_{temp}(\tilde{G}(F);\omega)$ qui ne dépend pas du choix de $\tilde{Q}$. On note $$Ind_{\tilde{L}}^{\tilde{G}}:D_{temp}(\tilde{L}(F);\omega)\to D_{temp}(\tilde{G}(F);\omega)$$ cette application. Posons simplement $\tilde{W}^G=W^G(\tilde{M}_{0})$. Ce groupe agit sur ${\cal L}(\tilde{M}_{0})$ et conserve ${\cal L}(\tilde{M}_{0},\omega)$.
0.3cm[**[Proposition]{}**]{}.
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\(i) Pour tout $\tilde{L}\in {\cal L}(\tilde{M}_{0},\omega)$, l’application $Ind_{\tilde{L}}^{\tilde{G}}$ est injective sur $D_{ell}(\tilde{L}(F);\omega)^{W^G(\tilde{L})}$.
(ii)On a l’égalité $$D_{temp}(\tilde{G}(F),\omega)=\oplus_{\tilde{L}\in {\cal L}(\tilde{M}_{0},\omega)/\tilde{W}^G}Ind_{\tilde{L}}^{\tilde{G}}(D_{ell}(\tilde{L}(F);\omega)^{W^G(\tilde{L})}).$$
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Preuve. D’après la proposition 2.9, l’espace $D_{temp}(\tilde{G}(F);\omega)$ a une base paramétrée par $E(\tilde{G},\omega)/conj$. On note ici $(e^{\tilde{G}}(\tau))_{\tau\in E(\tilde{G},\omega)/conj}$ une telle base. Alors $(e^{\tilde{G}}(\tau))_{\tau\in E_{ell}(\tilde{G},\omega)/conj}$ est une base de $D_{ell}(\tilde{G}(F);\omega)$. Soit $\tilde{L}\in {\cal L}(\tilde{M}_{0},\omega)$. Le groupe $W^G(\tilde{L})$ agit tant sur $D_{ell}(\tilde{L}(F);\omega)$ que sur $E_{ell}(\tilde{L})/conj$. Pour $w\in W^G(\tilde{L})$ et $\tau\in E_{ell}(\tilde{L})/conj$, on a une égalité $w(e^{\tilde{L}}(\tau))=z(w,\tau)e^{\tilde{L}}(w\tau)$, avec $z(w,\tau)\in {\mathbb C}^{\times}$. Posons $\underline{e}^{\tilde{L}}(\tau)=\sum_{w\in W^G(\tilde{L})}w(e^{\tilde{L}}(\tau))$. Cet élément peut être nul: la restriction de l’application $w\mapsto z(w,\tau)$ au stabilisateur de $\tau$ est un caractère de ce groupe; $\underline{e}^{\tilde{L}}(\tau)$ est nul si et seulement si ce caractère est non trivial. Notons $\underline{E}_{ell}(\tilde{L})$ l’ensemble des $\tau\in E_{ell}(\tilde{L})$ dont la classe vérifie $\underline{e}^{\tilde{L}}(\tau)\not=0$. L’action de $W^G(\tilde{L})$ respecte cet ensemble. Identifions l’ensemble quotient $(\underline{E}_{ell}(\tilde{L})/conj)/W^G(\tilde{L})$ à un ensemble de représentants (on fera d’autres identifications similaires dans la suite). Alors $(\underline{e}^{\tilde{L}}(\tau))_{\tau\in (\underline{E}_{ell}(\tilde{L})/conj)/W^G(\tilde{L})}$ est une base de $D_{ell}(\tilde{L}(F);\omega)^{W^G(\tilde{L})}$.
Pour ce qui est des triplets $(M,\sigma,\tilde{r})$, il y a une correspondance naturelle entre triplets pour $\tilde{L}$ et triplets pour $\tilde{G}$: c’est l’identité, modulo l’identification de ${\cal R}^{\tilde{L}}(\sigma)$ à un sous-ensemble de ${\cal R}^{\tilde{G}}(\sigma)$. On a déjà dit qu’un triplet pouvait être essentiel pour $\tilde{L}$ mais pas pour $\tilde{G}$. Notons $E^{\tilde{G}}(\tilde{L})$ l’ensemble des triplets pour $\tilde{L}$ qui sont essentiels dans $\tilde{G}$. De la correspondance précédente se déduit une application $\iota_{\tilde{L}}^{\tilde{G}}:E^{\tilde{G}}(\tilde{L})/conj\to E(\tilde{G},\omega)/conj$. Celle-ci est invariante par l’action de $W^G(\tilde{L})$ sur l’espace de départ. D’après le lemme 2.10, l’application $Ind_{\tilde{L}}^{\tilde{G}}$ se décrit ainsi: pour $\tau\in E(\tilde{L},\omega)/conj$, elle envoie $e^{\tilde{L}}(\tau)$ sur $0$ si $\tau\not\in E^{\tilde{G}}(\tilde{L})/conj$ et sur $z_{\tilde{L}}^{\tilde{G}}(\tau)e^{\tilde{G}}(\iota_{\tilde{L}}^{\tilde{G}}(\tau))$ si $\tau\in E^{\tilde{G}}(\tilde{L})/conj$, où $z_{\tilde{L}}^{\tilde{G}}(\tau)\in {\mathbb C}^{\times}$. Soit $\tau\in E^{\tilde{G}}_{ell}(\tilde{L})/conj$ ($=(E^{\tilde{G}}(\tilde{L})/conj)\cap (E_{ell}(\tilde{L})/conj)$). Puisque l’induction est insensible à l’action de $W^G(\tilde{L})$, $Ind_{\tilde{L}}^{\tilde{G}}$ envoie $\underline{e}^{\tilde{L}}(\tau)$ sur $\vert W^G(\tilde{L})\vert z_{\tilde{L}}^{\tilde{G}}(\tau) e^{\tilde{G}}(\iota_{\tilde{L}}^{\tilde{G}}(\tau))$. A fortiori $\underline{e}^{\tilde{L}}(\tau)\not=0$, ce qui démontre l’inclusion $E^{\tilde{G}}_{ell}(\tilde{L})\subset \underline{E}_{ell}(\tilde{L})$.
On a décrit des bases de chacun de nos espaces et les matrices des applications d’induction dans ces bases. Le lemme résulte alors des deux assertions suivantes:
\(1) pour tout $\tilde{L}\in {\cal L}(\tilde{M}_{0},\omega)$, on a l’égalité $E^{\tilde{G}}_{ell}(\tilde{L})= \underline{E}_{ell}(\tilde{L})$;
\(2) notons $$\iota:\bigsqcup_{\tilde{L}\in {\cal L}(\tilde{M}_{0},\omega)/\tilde{W}^G}(E^{\tilde{G}}_{ell}(\tilde{L})/conj)/W^G(\tilde{L})\to E(\tilde{G},\omega)/conj$$ l’application qui coïncide avec $\iota_{\tilde{L}}^{\tilde{G}}$ sur le sous-ensemble indexé par $\tilde{L}$ de l’ensemble de départ; alors $\iota$ est bijective.
Prouvons (2). Soit $\tau=(M,\sigma,\tilde{r})\in E(\tilde{G},\omega)$. Parmi les espaces de Levi $\tilde{L}$ tels que $M\subset L$ et $\tilde{r}\in {\cal R}^{\tilde{L}}(\sigma)$, considérons un élément minimal $\tilde{L}$. Quitte à conjuguer $\tau$ par un élément de $G(F)$, on peut supposer $\tilde{L}\in {\cal L}(\tilde{M}_{0},\omega)$. Notons $\tau^{\tilde{L}}$ le même triplet vu comme un élément de $E(\tilde{L},\omega)$. On a $\tau=\iota_{\tilde{L}}^{\tilde{G}}(\tau^{\tilde{L}})$. Le lemme 2.11 et la minimalité de $\tilde{L}$ assurent que $\tau^{\tilde{L}}\in E_{ell}(\tilde{L})$ (et forcément $\tau^{\tilde{L}}\in E^{\tilde{G}}_{ell}(\tilde{L})$). Cela prouve la surjectivité de $\iota$. Démontrons l’injectivité. On identifie $ {\cal L}(\tilde{M}_{0},\omega)/\tilde{W}^G$ à un ensemble de représentants dans $ {\cal L}(\tilde{M}_{0},\omega)$. Soient $\tilde{L},\tilde{L}' $ dans cet ensemble, $\tau=(M,\sigma,\tilde{r})\in E^{\tilde{G}}_{ell}(\tilde{L})$, $\tau'=(M',\sigma',\tilde{r}')\in E^{\tilde{G}}_{ell}(\tilde{L}')$, supposons les triplets $(M,\sigma,\tilde{r})$ et $(M',\sigma',\tilde{r}')$ conjugués par un élément de $G(F)$. On doit prouver qu’alors $\tilde{L}=\tilde{L}'$ et que $\tau$ et $\tau'$ sont conjugués par l’action du normalisateur $Norm_{G(F)}(\tilde{L})$. Soit $g\in G(F)$ qui conjugue le premier triplet en le second. On a $gMg^{-1}=M'$, $g\sigma=\sigma'$. Notons que, puisque $\tau$ est elliptique, $\tilde{r}$ est au départ un élément de $W^{\tilde{L}}_{reg}(\sigma)$. Mais, après passage à $\tilde{G}$, le triplet n’est plus elliptique et $\tilde{r}$ devient une classe modulo $W^{G}_{0}(\sigma)$. De même pour $\tilde{r}'$. En continuant à considérer $\tilde{r}$ et $\tilde{r}'$ comme des éléments de $W^{\tilde{L}}_{reg}(\sigma)$ et $W^{\tilde{L}'}_{reg}(\sigma')$, la condition de conjugaison est $ad_{g}(W^G_{0}(\sigma)\tilde{r})=W_{0}^G(\sigma')\tilde{r}'$. On a dit que $W^G_{0}(\sigma)$ est le groupe de Weyl d’un sous-système de racines $\Sigma$ de l’ensemble des racines de $A_{M}$ dans $G$. Puisque $W^L_{0}(\sigma)=\{1\}$ (condition d’ellipticité), aucune de ces racines n’intervient dans $L$. On peut donc trouver un élément $H\in {\cal A}_{\tilde{L}}$ qui n’annule aucune de ces racines. Fixons un tel élément. Il détermine un sous-ensemble positif $\Sigma_{+}\subset\Sigma$: l’ensemble des $\alpha\in \Sigma$ tels que $<\alpha,H>>0$. Puisque $\tilde{r}\in W^{\tilde{L}}$ fixe $H$, l’action de $\tilde{r}$ conserve $\Sigma_{+}$. On a des objets analogues pour $\tau'$, que l’on affecte d’un $'$. La conjugaison par $g$ envoie $\Sigma$ sur $\Sigma'$. Quitte à multiplier $g$ à gauche par un élément de $W^G_{0}(\sigma')$, on peut supposer qu’elle envoie $\Sigma_{+} $ sur $\Sigma'_{+}$. Ecrivons $ad_{g}(\tilde{r})=w\tilde{r}'$, avec $w\in W_{0}^G(\sigma)$. Puisque $\tilde{r}$ conserve $\Sigma_{+}$, $ad_{g}(\tilde{r})$ conserve $\Sigma'_{+}$. Il en est de même de $\tilde{r}'$. Donc aussi de $w$. Un élément du groupe de Weyl qui conserve un ensemble positif est l’identité. D’où $w=1$. Puisque $\tilde{r}\in W^{\tilde{L}}(\sigma)$, $\tilde{L}$ est le plus petit espace de Levi contenant à la fois $M$ et $\tilde{r}$. De même pour $\tilde{L}'$. Alors $g$ conjugue $\tilde{L}$ en $\tilde{L}'$. Deux éléments de ${\cal L}(\tilde{M}_{0})$ qui sont conjugués par un élément de $G(F)$ le sont par un élément de $\tilde{W}^G$. Puisque les deux espaces de Levi $\tilde{L}$ et $\tilde{L}'$ appartiennent à notre ensemble de représentants, on a $\tilde{L}=\tilde{L}'$ et $g$ appartient à $Norm_{G(F)}(\tilde{L})$. C’est ce que l’on voulait prouver.
Prouvons (1). Soient $\tilde{L}\in {\cal L}(\tilde{M}_{0},\omega)$ et $\tau\in E_{ell}(\tilde{L})-E^{\tilde{G}}_{ell}(\tilde{L})$. On veut prouver que $\tau\not\in \underline{E}_{ell}(\tilde{L})$, autrement dit que la fonction $w\mapsto z(w,\tau)$ n’est pas constante sur le stabilisateur $Stab(W^G(\tilde{L}),\tau)$ de $\tau$ dans $W^G(\tilde{L})$ (en notant encore $\tau$ la classe de conjugaison de $\tau$ par $L(F)$). Relevons $\tau$ en un élément $\boldsymbol{\tau}=(M,\sigma,\boldsymbol{\tilde{r}})\in {\cal E}(\tilde{L})$. Il résulte des définitions que, pour $w\in Stab(W^G(\tilde{L}),\tau)$, $w(\boldsymbol{\tau})$ est égal à $(M,\sigma,z(w,\tau)\boldsymbol{\tilde{r}})$, à conjugaison près par un élément de $L(F)$. Il s’agit donc de trouver un élément $g\in Norm_{G(F)}(\tilde{L})$ tel que $ad_{g}(\boldsymbol{\tau})=(M,\sigma,z\boldsymbol{\tilde{r}})$, avec $z\not=1$. Puisque $\tau\not\in E^{\tilde{G}}_{ell}(\tilde{L})$, il existe en tout cas un élément $g\in G(F)$ qui vérifie cette dernière relation. La preuve de l’injectivité de $\iota$ montre que, quitte à modifier $g$ par un élément de $W^G_{0}(\sigma)$ (ce qui ne change par $ad_{g}(\boldsymbol{\tau})$), on peut supposer $g\in Norm_{G(F)}(\tilde{L})$. C’est ce qu’on voulait. $\square$
Le calcul spectral
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Position du problème
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On considère un élément $T\in {\cal A}_{0}$, qui intervient dans ce qui suit comme un paramètre. On lui impose de vérifier les propriétés suivantes:
$\theta(T)=T$;
$<\alpha,T>>0$ pour tout $\alpha\in \Delta_{0}$;
$<\alpha,T>\geq c_{\star}\vert T\vert $, où $c_{\star}>0$ est un réel fixé;
si $F$ est non-archimédien, $T\in {\cal A}_{M_0,F}\otimes_{\mathbb Z}{\mathbb Q}$.
On note ${\tilde{\kappa}}^T$ la fonction caractéristique du sous-ensemble des $g\in G(F)$ tels que $\phi^{\tilde{G}}(h_{0}(g)-T)=1$. Ce sous-ensemble est invariant par $A_{G}(F)$ et sa projection dans $A_{G}(F)\backslash G(F)$ est compact.
On définit le sous-espace $C_{c}^{\infty}(\tilde{G}(F);K)$ de $C_{c}^{\infty}(\tilde{G}(F))$ comme en 2.7. Soient $f_{1},f_{2}\in C_{c}^{\infty}(\tilde{G}(F),K)$. On pose $$J^T(\omega,f_{1},f_{2})=\int_{A_{\tilde{G}}(F)\backslash G(F)}\int_{\tilde{G}(F)} \bar{f}_{1}(\gamma)f_{2}(g^{-1}\gamma g) \omega(g){\tilde{\kappa}}^T(g)\,d\gamma\,dg.$$ C’est une intégrale à support compact. En effet, l’intégration en $\gamma$ est à support compact puisque $f_{1}$ l’est. A cause de la fonction ${\tilde{\kappa}}^T$, on peut écrire $g=ay$, où $y$ reste dans un compact et $a\in A_{G}(F)$. La condition $f_{2}(g^{-1}\gamma g)\not=0$ impose alors que $a^{-1}\theta(a)$ reste dans un compact ce qui entraîne que l’image de $a$ dans $A_{\tilde{G}}(F)\backslash A_{G}(F)$ reste dans un compact.
On se propose dans cette section de montrer que la fonction $T\mapsto J^T(\omega,f_{1},f_{2})$ est asymptote à un élément de $PolExp$ et de calculer une expression “spectrale” du terme constant de cet élément.
Utilisation de la formule de Plancherel
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On fixe un élément $\gamma_{0}\in \tilde{M}_{0}(F)$, vérifiant la condition 2.1(6) quand $F$ est archimédien. On note simplement $\theta=ad_{\gamma_{0}}$.
On définit les fonctions $\varphi_{1}$ et $\varphi_{2}$ sur $G(F)$ par $\varphi_{i}(x)=f_{i}(x\gamma_{0})$ pour $i=1,2$. Alors $$J^T(\omega,f_{1},f_{2})=\int_{A_{\tilde{G}}(F)\backslash G(F)}\int_{G(F)} \bar{\varphi}_{1}(x)\varphi_{2}(g^{-1}x\theta(g)) \omega(g)\tilde{\kappa}^T(g)\,dx\,dg.$$ On utilise la formule de Plancherel-Harish-Chandra pour exprimer $\varphi_{2}$. C’est-à-dire $$\varphi_{2}(g)=\sum_{M_{disc}\in {\cal L}(M_{0})}\vert W^{M_{disc}}\vert \vert W^G\vert ^{-1}\sum_{\sigma\in \Pi_{disc}(M_{disc}(F))/i{\cal A}_{M_{disc},F}^*} \vert Stab(i{\cal A}_{M_{disc},F}^*,\sigma)\vert ^{-1}$$ $$\int_{i{\cal A}_{M_{disc},F}^*}m^G(\sigma_{\lambda})trace(Ind_{S}^G(\sigma_{\lambda},g^{-1})Ind_{S}^G(\sigma_{\lambda},\varphi_{2}))\,d\lambda.$$
Pour tous $M$, $\sigma$, l’expression $$\int_{A_{\tilde{G}}(F)\backslash G(F)}\int_{G(F)} \vert \varphi_{1}(x)\vert$$ $$\vert \int_{i{\cal A}_{M_{disc},F}^*}m^G(\sigma_{\lambda})trace(Ind_{S}^G(\sigma_{\lambda},\theta(^{-1}x^{-1}g)Ind_{S}^G(\sigma_{\lambda},\varphi_{2}))\,d\lambda\vert {\tilde{\kappa}}^T(g)\,dx\,dg$$ est convergente. En effet, l’intégrale en $x$ est à support compact puisque $\varphi_{1}$ l’est. L’intégrale intérieure définit une fonction de Schwartz-Harish-Chandra en $\theta(g)^{-1}x^{-1}g$. Comme dans le paragraphe précédent, on peut écrire $g=ay$ où $y$ reste dans un compact. L’intégrale restante en $a$ est celle d’une fonction essentiellement bornée par $(1+\vert \theta(H_0(a))-H_0(a)\vert )^{-r}$ pour tout réel $r$. Une telle intégrale est convergente. On pose $$J^T_{M_{disc},\sigma}(\omega,f_{1},f_{2})= \int_{A_{\tilde{G}}(F)\backslash G(F)}\int_{G(F)} \bar{\varphi}_{1}(x)$$ $$\int_{i{\cal A}_{M_{disc},F}^*}m^G(\sigma_{\lambda})trace(Ind_{S}^G(\sigma_{\lambda},\theta(g)^{-1}x^{-1}g)Ind_{S}^G(\sigma_{\lambda},\varphi_{2}))\,d\lambda \omega(g){\tilde{\kappa}}^T(g)\,dx\,dg$$ et on a $$(1) \qquad J^T(\omega,f_{1},f_{2})=\sum_{M_{disc}\in {\cal L}(M_{0})}\vert W^{M_{disc}}\vert \vert W^G\vert ^{-1}$$ $$\sum_{\sigma\in \Pi_{disc}(M_{disc}(F))/i{\cal A}_{M_{disc},F}^*}\vert Stab(i{\cal A}_{M_{disc},F}^*,\sigma)\vert ^{-1}J^T_{M_{disc},\sigma}(\omega,f_{1},f_{2}).$$
Apparition d’intégrales de coefficients
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Nous fixons maintenant un Levi semi-standard $M_{disc}$ et $\sigma\in \Pi_{disc}(M_{disc}(F))$. On fixe aussi $S\in {\cal P}(M_{disc})$. On pose $\pi_{\lambda}=Ind_{S}^G(\sigma_{\lambda})$ que l’on réalise dans l’espace $V_{\sigma,S}$. On pose $S'=\theta^{-1}(S)$, $\sigma'=\sigma\circ \theta$, $\lambda'=\theta^{-1}\lambda$, $\pi'_{\lambda'}=Ind_{S'}^G(\sigma'_{\lambda'})$, que l’on réalise dans l’espace $V_{\sigma',S'}$. On fixe des bases orthonormées ${\cal B}$ de $V_{\sigma,S}$ et ${\cal B}'$ de $V_{\sigma',S'}$, réunions de bases des différents $K$-types intervenant. On introduit l’opérateur $V_{\pi'_{\lambda'}}\to V_{\pi_{\lambda}}$ qui à $e\in V_{\pi'_{\lambda'}}$ associe la fonction $g\mapsto e(\theta^{-1}(g))$. Par restriction à $K$, on obtient un opérateur unitaire $U_{\theta,\sigma_{\lambda}}:V_{\sigma',S'}\to V_{\sigma,S}$, qui vérifie $U_{\theta,\sigma_{\lambda}}\pi'_{\lambda'}(g)=\pi_{\lambda}(\theta(g))U_{\theta,\sigma_{\lambda}}$. Si $K$ est stable par $\theta$, il est indépendant de $\lambda$. Mais on n’a pas posé cette hypothèse sur $K$ et l’opérateur peut dépendre de $\lambda$.
Pour $x,g\in G(F)$, on a $$trace(Ind_{S}^G(\sigma_{\lambda},\theta(g)^{-1}x^{-1}g)Ind_{S}^G(\sigma_{\lambda},\varphi_{2}))=trace(U_{\theta,\sigma_{\lambda}}^{-1}Ind_{S}^G(\sigma_{\lambda},\theta(g)^{-1}x^{-1}g)Ind_{S}^G(\sigma_{\lambda},\varphi_{2})U_{\theta,\sigma_{\lambda}})$$ $$=\sum_{v'\in{\cal B}'}(v',U_{\theta,\sigma_{\lambda}}^{-1}\pi_{\lambda}(\theta(g)^{-1}x^{-1}g)\pi_{\lambda}(\varphi_{2})U_{\theta,\sigma_{\lambda}}v')$$ $$=\sum_{v'\in {\cal B}'}(\pi_{\lambda}(x\theta(g))U_{\theta,\sigma_{\lambda}}v',\pi_{\lambda}(g)\pi_{\lambda}(\varphi_{2})U_{\theta,\sigma_{\lambda}}v').$$ Cette somme est finie: $\varphi_{2}$ est $K$-finie à droite donc $\pi_{\lambda}(\varphi_{2})U_{\theta,\sigma_{\lambda}}$ annule presque tout $v'\in {\cal B}'$. L’expression $$\int_{G(F)} \bar{\varphi}_{1}(x) \int_{i{\cal A}_{M_{disc},F}^*}m^G(\sigma_{\lambda})\sum_{v'\in {\cal B}'}(\pi_{\lambda}(x\theta(g))U_{\theta,\sigma_{\lambda}}v',\pi_{\lambda}(g)\pi_{\lambda}(\varphi_{2})U_{\theta,\sigma_{\lambda}}v')\,d\lambda\,dx$$ est absolument convergente. En permutant les intégrales, on voit qu’elle vaut $$\int_{i{\cal A}_{M_{disc},F}^*}m^G(\sigma_{\lambda})\sum_{v'\in {\cal B}'}(\pi_{\lambda}(\varphi_{1})\pi_{\lambda}(\theta(g))U_{\theta,\sigma_{\lambda}}v',\pi_{\lambda}(g)\pi_{\lambda}(\varphi_{2})U_{\theta,\sigma_{\lambda}}v')\,d\lambda.$$ On a l’égalité $$(\pi_{\lambda}(\varphi_{1})\pi_{\lambda}(\theta(g))U_{\theta,\sigma_{\lambda}}v',\pi_{\lambda}(g)\pi_{\lambda}(\varphi_{2})U_{\theta,\sigma_{\lambda}}v')=( \pi_{\lambda}(\varphi_{1})U_{\theta,\sigma_{\lambda}}\pi'_{\lambda'}(g)v',\pi_{\lambda}(g) \pi_{\lambda}(\varphi_{2})U_{\theta,\sigma_{\lambda}}v').$$ On exprime matriciellement tous les opérateurs intervenant. L’expression ci-dessus devient $$\sum_{u,v\in {\cal B},u'\in {\cal B}'}( \pi_{\lambda}(\varphi_{1})U_{\theta,\sigma_{\lambda}}u',v)(u,\pi_{\lambda}(\varphi_{2})U_{\theta,\sigma_{\lambda}}v')(v,\pi_{\lambda}(g)u)(\pi'_{\lambda'}(g)v',u').$$ Cette somme est en fait finie d’après les propriétés de $K$-finitude des fonctions $\varphi_{1}$ et $\varphi_{2}$ et de l’opérateur $U_{\theta,\sigma_{\lambda}}$. Posons $$B_{u,v,u',v'}(\lambda)= ( \pi_{\lambda}(\varphi_{1})U_{\theta,\sigma_{\lambda}}u',v)(u,\pi_{\lambda}(\varphi_{2})U_{\theta,\sigma_{\lambda}}v').$$ C’est une fonction de Schwartz sur $i{\cal A}_{M_{disc},F}^*$. On obtient l’égalité $$(1) \qquad J^T_{M_{disc},\sigma}(\omega,f_{1},f_{2})=\sum_{u,v\in {\cal B},u',v'\in {\cal B}'} \int_{A_{\tilde{G}}(F)\backslash G(F)}$$ $$\int_{i{\cal A}_{M_{disc},F}^*}m^G(\sigma_{\lambda})B_{u,v,u',v'}(\lambda)(v,\pi_{\lambda}(g)u)(\pi'_{\lambda'}(g)v',u')\,d\lambda\,dx\,\omega(g){\tilde{\kappa}}^T(g)\,dg.$$
Une première approximation d’une intégrale de coefficients
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Fixons $u,v\in {\cal B}$, $u',v'\in {\cal B}'$ et une fonction de Schwartz $B$ sur $i{\cal A}_{M_{disc},F}^*$. On pose $$j^T=
\int_{A_{\tilde{G}}(F)\backslash G(F)}\int_{i{\cal A}_{M_{disc},F}^*}m^G(\sigma_{\lambda})B(\lambda) (v,\pi_{\lambda}(g)u)(\pi'_{\lambda'}(g)v',u')\,d\lambda\,\omega(g)\tilde{\kappa}^T(g)\,dg,$$ où on rappelle que $\lambda'=\theta^{-1}\lambda$. Cette expression est convergente dans l’ordre indiqué pour les mêmes raisons que précédemment. La formule (1) du paragraphe précédent exprime $J^T_{M_{disc},\sigma}(\omega,f_{1},f_{2})$ comme combinaison linéaire de telles expressions $j^T$.
On définit une fonction $\Omega_{G}$ sur $M_{0}(F)$ par $$\Omega_{G}(m)=\int_{K\times K}\int_{i{\cal A}_{M_{disc},F}^*}m^G(\sigma_{\lambda})B(\lambda) (v,\pi_{\lambda}(kmk')u)(\pi'_{\lambda'}(kmk')v',u')\,d\lambda\,\omega(kmk')\,dk\,dk'.$$ Alors $$j^T=\int_{A_{\tilde{G}}(F)\backslash M_{0}(F)^{\geq}}\Omega_{G}(m)D_{0}(m)\tilde{\kappa}^T(m)\,dm.$$
Soit $Q=LU_{Q}$ un sous-groupe parablique standard de $G$. Comme en 1.12, notons $W^G(L\vert S)$ l’ensemble des $w\in W^G/W^{M_{disc}}$ tels que $w(M_{disc})\subset L$, $w(S)\cap L\supset P_{0}\cap L$. On identifiera souvent cet ensemble à un sous-ensemble de $W^G$ formé d’éléments $w$ de longueur minimale dans leur classe $W^Lw$. Pour un élément $w$ de cet ensemble, on note $Q_{w}=(w(S)\cap L)U_{Q}$, $\underline{Q}_{w}=(w(S)\cap L)U_{\bar{Q}}$. Notons $\pi_{w}=Ind_{Q_{w}}^G(w\sigma)$ et $\underline{\pi}_{w}=Ind_{\underline{Q}_{w}}^G(w\sigma)$, réalisées dans leurs espaces habituels $V_{w\sigma,Q_{w}}$ et $V_{w\sigma,\underline{Q}_{w}}$. Pour $v\in V_{w\sigma,Q_{w}}$ et $\underline{v}\in V_{w\sigma,\underline{Q}_{w}}$, on pose $$(v,\underline{v})^L=\int_{K\cap L(F)}(v(k),\underline{v}(k))\,dk.$$ Le produit intérieur est le produit hermitien sur $V_{w\sigma}$. On pose des définitions analogues en remplaçant $S$ par $S'=\theta^{-1}(S)$. On prendra garde que les définitions de $Q_{w}$ et $\underline{Q}_{w}$ changent: pour $w\in W^G(L\vert S')$, on a $Q_{w}=(w(S')\cap L)U_{Q}$.
Soient $w\in W^G(L\vert S)$ et $w'\in W^G(L\vert S')$. On définit une fonction $\omega_{Q,w,w'}$ sur $G(F)\times K\times K\times {\cal A}_{M_{disc},{\mathbb C}}^*$ par $$\omega_{Q,w,w'}(g,k,k',\lambda)=(J_{Q_{w}\vert w(S)}((w\sigma)_{w\lambda})\circ \gamma(w)\circ \pi_{\lambda}(k)v,J_{\underline{Q}_{w}\vert w(S)}((w\sigma)_{w\lambda})\circ\gamma(w)\circ\pi_{\lambda}(gk')u)^L$$ $$( J_{\underline{Q}_{w'}\vert w'(S')}((w'\sigma')_{w'\lambda'})\circ \gamma(w')\circ \pi'_{\lambda'}(gk')v',J_{Q_{w'}\vert w'(S')}((w'\sigma')_{w'\lambda'})\circ\gamma(w')\circ\pi'_{\lambda'}(k)u')^L.$$ Cette fonction est méromorphe en $\lambda$. Pour comprendre les questions de régularité et de croissance en $\lambda$, il est commode de récrire cette définition en utilisant les opérateurs normalisés et les facteurs de normalisation. Pour $\lambda\in i{\cal A}_{M_{disc},F}^*$, on a l’égalité $$(1) \qquad \omega_{Q,w,w'}(g,k,k',\lambda)={\bf r}_{w,w'}(\sigma_{\lambda})$$ $$( R_{Q_{w}\vert w(S)}((w\sigma)_{w\lambda})\circ \gamma(w)\circ \pi_{\lambda}(k)v,R_{\underline{Q}_{w}\vert w(S)}((w\sigma)_{w\lambda})\circ\gamma(w)\circ\pi_{\lambda}(gk')u,)^L$$ $$( R_{\underline{Q}_{w'}\vert w'(S')}((w'\sigma')_{w'\lambda'})\circ \gamma(w')\circ \pi'_{\lambda'}(gk')v',R_{Q_{w'}\vert w'(S')}((w'\sigma')_{w'\lambda'})\circ\gamma(w')\circ\pi'_{\lambda'}(k)u')^L,$$ où $${\bf r}_{w,w'}(\sigma_{\lambda})=r_{ \underline{Q}_{w}\vert Q_{w}}((w\sigma)_{w\lambda})r_{Q_{w'}\vert \underline{Q}_{w'} }((w'\sigma')_{w'\lambda'}).$$ Les opérateurs normalisés sont holomorphes et unitaires sur $i{\cal A}_{M_{disc},F}^*$. Les produits scalaires de l’expression ci-dessus sont donc holomorphes et bornés en $\lambda$. On a
\(2) le produit $m^G(\sigma_{\lambda}){\bf r}_{w,w'}(\sigma_{\lambda})$ est holomorphe et à croissance modérée sur $i{\cal A}_{M_{disc},F}^*$.
En effet, par transport de structure, on a aussi $${\bf r}_{w,w'}(\sigma_{\lambda})=r_{S_{1}\vert S_{2}}(\sigma_{\lambda})r_{S_{3}\vert S_{4}}(\sigma_{\lambda}),$$ où $S_{1}=w^{-1}(\underline{Q}_{w})$, $S_{2}=w^{-1}(Q_{w})$, $S_{3}=\theta((w')^{-1}(Q_{w'}))$, $S_{4}=\theta((w')^{-1}(\underline{Q}_{w'}))$. Ces quatre paraboliques appartiennent à ${\cal P}(M_{disc})$ et il reste à appliquer 1.10(7).
On pose $$\omega_{Q,w,w'}(g)=\int_{K\times K}\int_{i{\cal A}_{M_{disc},F}^*}\omega_{Q,w,w'}(g,k,k',\lambda)m^G(\sigma_{\lambda})B(\lambda)\omega(k^{-1}gk')\,d\lambda\,dk\,dk'.$$
Pour tout sous-groupe parabolique standard $Q'=L'U_{Q'}\supset Q$, on note ${\cal W}_{Q}^{Q'}$ l’ensemble des $(w,w')\in W^G(L\vert S)\times W^G(L\vert S')$ tels que $\theta(W^{L'}w')\cap W^{L'}w\not=\emptyset$. On pose simplement ${\cal W}_{Q}={\cal W}_{Q}^Q$.
Soit $R$ un parabolique standard contenant $Q$. Pour $w\in W^G(L\vert S)$ et $w'\in W^G(L\vert S')$, notons $s_{Q}^R(w,w')$ la somme des $(-1)^{a_{\tilde{P}}-a_{\tilde{G}}}$ sur les ensembles paraboliques $\tilde{P}$ tels que $Q\subset P\subset R$ et $(w,w')\in {\cal W}_{Q}^P$. Remarquons que, si cet ensemble d’ensembles paraboliques n’est pas vide, il existe $\tilde{P}_{-}\subset \tilde{P}_{+}$ de sorte que cet ensemble soit simplement celui des $\tilde{P}$ tels que $\tilde{P}_{-}\subset \tilde{P}\subset \tilde{P}_{+}$. Donc $s_{Q}^R(w,w')$ est nul si l’ensemble est vide ou si $\tilde{P}_{-}\not=\tilde{P}_{+}$ et est égal à $(-1)^{a_{\tilde{P}_{+}}-a_{\tilde{G}}}$ si l’ensemble est non vide et $\tilde{P}_{-}=\tilde{P}_{+}$. Posons $$j^T_{\star}=\sum_{Q=LU_{Q},R; P_{0}\subset Q\subset R}\sum_{w\in W^G(L\vert S),w'\in W^G(L\vert S')}s_{Q}^R(w,w')$$ $$\int_{A_{\tilde{G}}(F)\backslash M_{0}(F)^{\geq,Q}}\delta_{Q}(m)D_{0}^L(m)\tilde{\sigma}_{Q}^R(H_0(m)-T)\phi^Q(H_0(m)-T)\omega_{Q,w,w'}(m)\,dm.$$
[0.3cm[**[Proposition]{}**]{}. [ *[L’expression ci-dessus est absolument convergente. Pour tout réel $r$, on a la majoration $$\vert j^T-j^T_{\star}\vert <<\vert T\vert ^{-r}$$ pour tout $T$.]{}*]{}0.3cm]{}
La preuve de cette proposition sera donnée en 3.15.
Un lemme de majoration
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Soit $Q=LU_{Q}$ un parabolique standard. Pour $H\in {\cal A}_{0}$, on note ${\cal C}^L(H)$ l’enveloppe convexe des $sH$ pour $s\in W^L$. On pose $$N^L(H)=1+inf\{\vert\theta H'-H''\vert ; H',H''\in {\cal C}^L(H)\}.$$ Soient $w\in W^G(L\vert S)$ et $w'\in W^G(L\vert S')$, que l’on relève en des éléments de $W^G$. On pose $$N^L_{w,w'}(H)=1+inf\{\vert w\circ\theta\circ (w')^{-1} H'-H''\vert ; H',H''\in {\cal C}^L(H)\}.$$ Ce terme ne dépend pas des relèvements choisis: changer de relèvement multiplie $w$ et $w'$ à gauche par des éléments de $W^L$ dont l’action conserve ${\cal C}^L(H)$. Remarquons que, dans le cas où $W^Lw\cap \theta(W^Lw')\not=\emptyset$, on peut pour la même raison remplacer $w'$ et $w$ par des éléments tels que $w=\theta(w')$ et on obtient $N^L_{w,w'}(H)=N^L(H)$.
0.3cm[**[Lemme]{}**]{}.
**
Soit $Q=LU_{Q}$ un sous-groupe parabolique standard.
\(i) Soient $w\in W^G(L\vert S)$ et $w'\in W^G(L\vert S')$ et soit $\tilde{P}=\tilde{M}U_{P}$ le plus petit espace parabolique standard tel que $Q\subset P$ et $(w,w')\in {\cal W}_{Q}^P$. Alors on a une majoration $$\vert (H-T)_{L}^M\vert <<N_{w,w'}^L(H)$$ pour tout $T$ et tout $H\in {\cal A}_{0}$ tel que $\phi^Q(H-T)\tau_{Q}^P(H-T)=1$ et $<\alpha,H>\geq0$ pour tout $\alpha\in \Delta_{0}^Q$.
\(ii) Soit $\tilde{P}=\tilde{M}U_{P}$ le plus petit espace parabolique standard tel que $Q\subset P$. Alors on a une majoration $$\vert (H-T)_{L}^M\vert <<N^L(H)$$ pour tout $T$ et tout $H\in {\cal A}_{0}$ tel que $\phi^Q(H-T)\tau_{Q}^P(H-T)=1$ et $<\alpha,H>\geq0$ pour tout $\alpha\in \Delta_{0}^Q$.
\(iii) Soient $w\in W^G(L\vert S)$ et $w'\in W^G(L\vert S')$ et soit $\tilde{P}=\tilde{M}U_{P}$ le plus petit espace parabolique standard tel que $Q\subset P$ et $(w,w')\in {\cal W}_{Q}^P$. Supposons $(w,w')\not\in {\cal W}_{Q}$. Alors on a une majoration $$\vert T\vert +\vert (H-T)^M_{L}\vert <<N^L_{w,w'}(H)$$ pour tout $T$ et tout $H\in {\cal A}_{0}$ tel que $\phi^Q(H-T)\tau_{Q}^P(H-T)=1$ et $<\alpha,H>\geq0$ pour tout $\alpha\in \Delta_{0}^Q$.
\(iv) Soient $w\in W^G(L\vert S)$ et $w'\in W^G(L\vert S')$ et soit $R$ un sous-groupe parabolique contenant $Q$. Supposons $s_{Q}^R(w,w')\not=0$. Alors on a une majoration $$\vert (H-T)_{L}^{\tilde{G}}\vert <<N^L_{w,w'}(H)$$ pour tout $T$ et tout $H\in {\cal A}_{0}$ tel que $\phi^Q(H-T)\tilde{\sigma}_{Q}^R(H-T)=1$ et $<\alpha,H>\geq0$ pour tout $\alpha\in \Delta_{0}^Q$.
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Preuve de (i). Les hypothèses $\phi^Q(H-T)=1$ et $<\alpha,H>\geq0$ pour tout $\alpha\in \Delta_{0}^Q$ entraînent que $H^L$ appartient à ${\cal C}^L(T^L)$. Donc ${\cal C}^L(H)\subset {\cal C}^L(H_{L}+T^L)$. Posons $H_{\star}=H_{L}-T_{L}$. On obtient ${\cal C}^L(H)\subset {\cal C}^L(T+H_{\star})$. Soient $H',H''\in {\cal C}^L(T+H_{\star})$. Posons $s=w\theta(w')^{-1}$. On veut minorer $\vert H'-s\theta H''\vert $. On a $$(1) \qquad \vert H'-s\theta H''\vert \geq \vert X\vert ,$$ où $X=(H'-s\theta H'')_{L}^M$. On a $H_{L}^{_{'}M}=T_{L}^M+H_{\star}^M$ parce que $H'\in {\cal C}^L(T+H_{\star})$, d’où $X= T_{L}^M+H_{\star}^M-(s\theta H'')_{L}^M $. L’hypothèse que $H''\in {\cal C}^L(T+H_{\star})$ signifie que l’on peut écrire $H''=H_{\star}+\sum_{u\in W^L}y_{u}uT$, avec des $y_{u}\geq0$ tels que $\sum_{u\in W^L}y_{u}=1$. Donc $$X=T_{L}^M+H^M_{\star}-(s\theta H_{\star})_{L}^M-\sum_{u\in W^L}y_{u}(s\theta uT)_{L}^M.$$ Posons $Y_{u}=T_{L}^M-(s\theta uT)_{L}^M$ et $Y=\sum_{u\in W^L}y_{u}Y_{u}$. Alors $ X=H_{\star}^M-(s\theta H_{\star})_{L}^M+Y$. Pour tout $u\in W^L$, on a $Y_{u}=(T-s\theta uT)_{L}^M=(T-u' T)_{L}^M$, où $u'=s\theta(u)$, puisque $\theta T=T$. L’hypothèse $(w,w')\in {\cal W}_{Q}^P$ entraîne que $s\in W^M$ donc aussi $u'\in W^M$. Puisque $T$ est dominant, $T-u'T$ est combinaison linéaire à coefficients positifs ou nuls de $\check{\alpha}$ pour $\alpha\in \Delta_{0}^P$. Donc $Y_{u}$ est combinaison linéaire à coefficients positifs ou nuls de $\check{\alpha}_{L}$ pour $\alpha\in \Delta_{0}^P-\Delta_{0}^Q$. L’élément $Y$ vérifie la même propriété. Par ailleurs, l’hypothèse $\tau_{Q}^P(H-T)=1$ signifie que $H_{\star}^M$ est combinaison linéaire à coefficients strictement positifs de $\check{\varpi}_{\alpha}^M$ pour $\alpha\in \Delta_{0}^P-\Delta_{0}^Q$. Soit $U=\sum_{\alpha\in \Delta_{0}^P-\Delta_{0}^Q}u_{\alpha}\check{\varpi}_{\alpha}^M$ et $V=\sum_{\alpha\in \Delta_{0}^P-\Delta_{0}^Q}v_{\alpha}\check{\alpha}_{L}$ avec des coefficients $u_{\alpha}$ et $v_{\alpha}$ positifs ou nuls. Montrons que l’on a une majoration $$(2)\qquad \vert U\vert +\vert V\vert << \vert U-(s\theta U)_{L}+V\vert .$$ Il suffit de prouver que le cône engendré par les $\check{\varpi}_{\alpha}^M-(s\theta\check{\varpi}_{\alpha}^M)_{L}$ et les $\check{\alpha}_{L}$ est un “vrai” cône, c’est-à-dire ne contient pas d’espace vectoriel non nul. Il revient au même de prouver que, pour $U$ et $V$ comme ci-dessus, l’égalité $U-(s\theta U)_{L}+V=0$ entraîne $U=0$, $V=0$. Or le produit scalaire $(U,V)$ est positif ou nul. Par produit scalaire avec $U$, l’égalité $U-(s\theta U)_{L}+V=0$ entraîne donc $(U,U-(s\theta U)_{L})\leq0$, d’où $(U,U)\leq (U,(s\theta U)_{L})$. Par Cauchy-Schwartz, cela implique $U=(s\theta U)_{L}
$ puis $U=s\theta U$. Si $U\not=0$, notons $\Delta_{0}(U)$ l’ensemble des $\alpha\in \Delta_{0}^P$ tels que $<\alpha,U>>0$. Cet ensemble est non vide et inclus dans $\Delta_{0}^P-\Delta_{0}^Q$. Introduisons le sous-groupe parabolique standard $P'=M'U_{P'}$ tel que $\Delta_{0}^{M'}=\Delta_{0}^P-\Delta_{0}(U)$.On a $Q\subset P'\subsetneq P$. L’élément $U$ appartient à la chambre positive relative au sous-groupe parabolique $M\cap P'$ de $M$. L’égalité $U=s\theta U$ entraîne $P'=s\theta(P')$. Puisque $P'$ et $s\theta(P')$ sont standard, cela implique $P'=\theta(P') $ et $s\in W^{M'}$. Mais alors $(w,w')\in {\cal W}_{Q}^{P'}$, ce qui contredit l’hypothèse de minimalité de $\tilde{P}$. Cette contradiction prouve que $U=0$. L’égalité $U-(s\theta U)_{L}+V=0$ entraîne alors que $V=0$, ce qui prouve (2).
On applique (2) à $U=H_{\star}^M$ et $V=Y$. En abandonnant le terme $\vert Y\vert $, on obtient la majoration $\vert H_{\star}^M\vert <<\vert X\vert $. Grâce à (1), cela entraîne la majoration du (i) de l’énoncé.
La preuve du (ii) est similaire, il suffit de supprimer le terme $s$ des calculs.
Preuve de (iii). On reprend la preuve de (i). A la fin de cette preuve, on avait abandonné le terme $\vert Y\vert $. Rétablissons-le. Pour obtenir le (iii) de l’énoncé, il suffit de prouver que, pour tout $u\in W^L$, on a une majoration $$(3)\qquad \vert T\vert <<\vert Y_{u}\vert .$$ Notons $\Sigma^M$ l’ensemble des racines de $A_{0}$ dans l’algèbre de Lie de $M$, muni de la positivité définie par $P_{0}\cap M$. On sait plus précisément que $T-u'T$ est combinaison linéaire des $\check{\alpha}$ pour toutes les racines $\alpha\in \Sigma^M$ telles que $\alpha>0$ et $(u')^{-1}\alpha<0$. Les coefficients sont de la forme $<\beta,T>$ pour des $\beta\in \Sigma^M$, $\beta>0$, donc sont essentiellement minorés par $\vert T\vert $. L’assertion (3) s’en déduit pourvu qu’il y ait au moins une racine $\alpha$ telle que $\alpha>0$, $(u')^{-1}\alpha<0$ et $\check{\alpha}_{L}\not=0$. S’il n’en est pas ainsi, on a $u'\in W^L$. L’égalité $u'=w\theta(w')^{-1}\theta(u)$, jointe au fait que $u\in W^L$, entraîne alors que $W^Lw\cap \theta(W^Lw')\not=\emptyset$, contrairement à l’hypothèse. Cela prouve (3) et achève la preuve du (iii) de l’énoncé.
Preuve de (iv). L’hypothèse $s_{Q}^R(w,w')\not=0$ signifie qu’il existe un unique espace parabolique $\tilde{P}=\tilde{M}U_{P}$ tel que $Q\subset P\subset R$ et $(w,w')\in {\cal W}_{Q}^P$. On reprend la preuve du (i). On a encore $s\in W^M$. Pour $H',H''\in {\cal C}^L(T+H_{\star})$, on a $$\vert H'-s\theta H''\vert >>\vert (H'-s\theta H'')_{L}^M\vert +\vert (H'-s\theta H'')_{M}\vert .$$ La preuve de (i) s’applique au premier terme: on a une majoration $$(4) \qquad \vert (H'-s\theta H'')_{L}^M\vert >> \vert H_{\star}^M\vert ,$$ où $H_{\star}=(H-T)_{L}$. On a $(H'-s\theta H'')_{M}=H_{\star,M}-\theta(H_{\star,M})$ puisque $s\in W^M$. Ecrivons $H_{\star, M}=H_{1}+H_{2}$ où $H_{1}\in {\cal A}_{\tilde{M}}$ et $H_{2}$ appartient à l’orthogonal ${\cal A}_{M}^{\tilde{M}}$ de ce sous-espace dans ${\cal A}_{M}$. On a $H_{\star,M}-\theta(H_{\star,M})=H_{2}-\theta H_{2}$ et $1-\theta$ est injective sur ${\cal A}_{M}^{\tilde{M}}$, d’où une majoration $$(5)\qquad \vert H_{2}\vert <<\vert H_{\star,M}-\theta( H_{\star,M})\vert.$$ La condition $\tilde{\sigma}_{Q}^R(H_{\star})=1$ entraîne que
\(6) $<\varpi_{\alpha},H_{1}>>0$ pour tout $\alpha\in \Delta_{0}-\Delta_{0}^P$.
Elle entraîne aussi que $ <\alpha,H_{\star}>\leq 0$ pour tout $\alpha\in \Delta_{0}-\Delta_{0}^R$. On remarque que, pour tout $\alpha\in \Delta_{0}-\Delta_{0}^P$, il existe $i\geq1$ tel que $\theta^{i}\alpha\not\in \Delta_{0}^R$: sinon l’espace parabolique standard $\tilde{P}'$ associé à la réunion de $\tilde{\Delta}_{0}^P$ et de $\{\tilde{\alpha}\}$ vérifierait $P\subsetneq P'\subset R$, ce qui contredirait l’hypothèse d’unicité de $\tilde{P}$. Pour $\alpha\in \Delta_{0}-\Delta_{0}^P$ et $i\geq1$ comme ci-dessus, on a
$$(7) \qquad<\alpha,H_{1}>=<\theta^{i}\alpha,H_{1}>=<\theta^{i}\alpha,H_{\star}-H_{2}-H_{\star}^M>\leq -<\theta^{i}\alpha,H_{2}+H_{\star}^M>.$$ Les conditions (6) et (7) bornent $\vert H^{\tilde{G}}_{1}\vert $: on a une majoration $$\vert H_{1}^{\tilde{G}}\vert <<\vert H_{2}\vert +\vert H_{\star}^M\vert .$$ Grâce à (5), on en déduit $$\vert H_{\star,M}^{\tilde{G}}\vert <<\vert H_{\star,M}-\theta (H_{\star,M})\vert+\vert H_{\star}^M\vert .$$ D’où, grâce à (4), $$\vert H_{\star}^{\tilde{G}}\vert << \vert H_{\star,M}-\theta (H_{\star,M})\vert+ \vert (H'-s\theta H'')_{L}^M\vert <<\vert H'-s\theta H''\vert .$$ Cela prouve l’assertion (iv) de l’énoncé. $\square$
Majoration de coefficients
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[0.3cm[**[Lemme]{}**]{}. [ *[Soient $Q=LU_{Q}$ un parabolique standard, $w\in W^G(L\vert S)$ et $w'\in W^G(L\vert S')$. Quel que soit le réel $r$, il existe $c>0$ tel que l’on ait la majoration $$\vert \omega_{Q,w,w'}(m)\vert\leq c\delta_{Q}(m)^{-1}\Xi^L(m)^2N^L_{w,w'}(H_0(m))^{-r}$$ pour tout $m\in M_{0}(F)$. ]{}*]{}0.3cm]{}
Preuve. Notons $\rho_{w\lambda}^G=Ind_{\underline{Q}_{w}}^G((w\sigma)_{w\lambda})$, $\rho^{_{'}G}_{w'\lambda'}=Ind_{\underline{Q}_{w'}}((w'\sigma')_{w'\lambda'})$, $\rho_{w\lambda}=Ind_{w(S)\cap L}^L((w\sigma)_{w\lambda})$, $\rho'_{w'\lambda'}=Ind_{w'(S')\cap L}^L((w'\sigma')_{w'\lambda'})$ que l’on réalise dans leurs espaces habituels $V_{w\sigma,\underline{Q}_{w}}$, $V_{w'\sigma',\underline{Q}_{w'}}$, $V^L_{w\sigma,w(S)\cap L}$, $V^L_{w'\sigma',w'(S')\cap L}$. D’après 3.4(1) et (2), $\omega_{Q,w,w'}(m,k,k',\lambda)m^G(\sigma_{\lambda})$ est combinaison linéaire de termes $$(v_{0},\rho^G_{w\lambda}(m)u_{0})^L(\rho^{_{'}G}_{w'\lambda'}(m)v'_{0},u'_{0})^L$$ pour des éléments $u_{0},v_{0}\in V_{w\sigma,\underline{Q}_{w}}$ et $u'_{0},v'_{0}\in V_{w'\sigma',\underline{Q}_{w'}}$. Les coefficients sont des fonctions $C^{\infty}$ de $\lambda$, $k$ et $k'$ et sont à croissance modérée en $\lambda$. On dispose d’applications $$V_{w\sigma,\underline{Q}_{w}}\to V^L_{w\sigma,w(S)\cap L},\,\,V_{w'\sigma',\underline{Q}_{w'}}\to V^L_{w'\sigma',w'(S')\cap L}$$ qui sont les restrictions à $K\cap L(F)$. En notant $e,f,e',f'$ les images de $u_{0},v_{0},u'_{0}, v'_{0}$ par ces applications, on a les égalités $$(v_{0,}\rho^G_{w\lambda}(m)u_{0})^L=\delta_{Q}(m)^{-1/2}(f,\rho_{w\lambda}(m)e),$$ $$(\rho^{_{'}G}_{w'\lambda'}(m)v'_{0},u'_{0})^L=\delta_{Q}(m)^{-1/2}(\rho'_{w'\lambda'}(m)f',e').$$ Quitte à remplacer la fonction de Schwartz $B$ par son produit avec une fonction à croissance modérée, on est ramené à évaluer $$(1) \qquad \delta_{Q}(m)^{-1}\int_{i{\cal A}_{M_{disc},F}^*} (f,\rho_{w\lambda}(m)e)(\rho'_{w'\lambda'}(m)f',e')B(\lambda)\,d\lambda.$$ On a $$(f,\rho_{w\lambda}(m)e)=\int_{K\cap L(F)}(f(k),(\rho_{w\lambda}(m)e)(k))\,dk,$$ où le produit intérieur est le produit hermitien sur $V_{w\sigma}$. On a $$(\rho_{w\lambda}(m)e)(k)=e^{<w\lambda,H_{\underline{Q}_{w}}(km)>}(\rho_{w}(m)e)(k),$$ où on a noté $\rho_{w}$ la représentation $\rho_{w\lambda}$ pour $\lambda=0$. En utilisant des formules similaires pour le terme $(\rho'_{w'\lambda'}(m)f',e')$, on obtient que (1) est égal à $$\delta_{Q}(m)^{-1}\int_{(K\cap L(F))\times (K\cap L(F))}(f(k),(\rho_{w}(m)e)(k))((\rho'_{w'}(m)f')(k'),e'(k'))\beta(m,k,k') \,dk\,dk',$$ où $$\beta(m,k,k')=\int_{i{\cal A}_{M_{disc},F}^*}B(\lambda)e^{<w\lambda,H_{\underline{Q}_{w}}(km)>-<w'\lambda',H_{\underline{Q}_{w'}}(k'm)>}\,d\lambda.$$ On va prouver que, pour tout réel $r$, il existe $c>0$ tel que $$(2)\qquad \vert \beta(m,k,k')\vert \leq cN^L_{w,w'}(H_0(m))^{-r}.$$ Admettons cela et finissons la démonstration. De (2) résulte que (1) est essentiellement majoré par $$\delta_{Q}(m)^{-1}N^L_{w,w'}(H_0(m))^{-r}$$ $$\int_{(K\cap L(F))\times (K\cap L(F))}\vert (f(k),(\rho_{w}(m)e)(k))((\rho'_{w'}(m)f')(k'),e'(k'))\vert \,dk\,dk'.$$ En utilisant 1.12(4), la double intégrale est essentiellement majorée par $\Xi^L(m)^2$ et on obtient la majoration de l’énoncé.
Démontrons (2). On commence par prouver
\(3) pour tout parabolique standard $Q'=L'U_{Q'}\subset Q$ et tout $k\in K\cap L(F)$, $H_{Q'}(km)$ appartient à ${\cal C}^L(H_0(m))$.
Quitte à conjuguer $m$ par un élément de $K\cap Norm_{L(F)}(M_{0})$, ce qui ne change pas notre problème, on peut supposer $<\alpha,H_0(m)>\geq0$ pour tout $\alpha\in \Delta_{0}^Q$. D’après le lemme 1.3, $H_{0}(km)$ appartient à ${\cal C}^L(H_0(m))$. L’élément $H_{Q'}(km)$ est la projection orthogonale de $H_{0}(km)$ sur ${\cal A}_{L'}$. On est ramené à montrer que, pour $H\in {\cal C}^L(H_0(m))$, alors la projection $H_{L'}$ de $H$ sur ${\cal A}_{L'}$ appartient aussi à ${\cal C}^L(H_0(m))$. L’espace ${\cal A}_{L'}^L$ est réunion des chambres positives associées aux paraboliques $Q''\in {\cal P}^L(L')$. On fixe un tel $Q''$ tel que $H_{L'}^L$ appartienne à la chambre associée à $Q''$ et on fixe $s\in W^L$ tel que $s(Q'')$ soit standard. Alors $H_{L'}=s^{-1}((sH)_{s(L')})$. Puisque ${\cal C}^L(H_{0}(m))$ est invariant par l’action de $s$, on peut aussi bien remplacer $H$ par $sH$ et $L'$ par $s(L')$. En oubliant cette construction, on est ramené au cas où $<\alpha,H_{L'}>\geq 0$ pour tout $\alpha\in \Delta_{0}^Q$. Dans ce cas, l’appartenance à ${\cal C}^L(H_0(m))$ équivaut à ce que $ H_{0}(m)-H_{L'}$ soit une combinaison linéaire à coefficients positifs ou nuls de $\check{\alpha}$ pour $\alpha\in\Delta_{0}^Q$. On a $$H_{0}(m)-H_{L'}=H_{0}(m)-H_{0}(m)_{L'}+H_{0}(m)_{L'}-H_{L'}.$$ La première différence est égale à $H_{0}(m)^{L'}$ qui appartient à la chambre positive fermée de ${\cal A}_{0}^{L'}$, a fortiori est combinaison linéaire à coefficients positifs ou nuls de $\check{\alpha}$ pour $\alpha\in \Delta_{0}^{Q'}$. La deuxième différence est la projection de $H_{0}(m)-H$. Puisque $H\in {\cal C}^L(H_{0}(m))$, $H_{0}(m)-H$ est une combinaison linéaire à coefficients positifs ou nuls de $\check{\alpha}$ pour $\alpha\in\Delta_{0}^Q$. Donc $H_{0}(m)_{L'}-H_{L'}$ est une telle combinaison linéaire de $\check{\alpha}_{L'}$, pour $\alpha\in \Delta_{0}^Q$. Il suffit de voir qu’une telle projection est de la forme voulue. C’est clair si $\alpha\in \Delta_{0}^{Q'}$ puisqu’alors $\check{\alpha}_{L'}=0$. Soit $\alpha\in \Delta_{0}^Q-\Delta_{0}^{Q'}$. Alors $\check{\alpha}_{L'}=\check{\alpha}-\check{\alpha}^{L'}$. Parce que $\{\check{\alpha}; \alpha\in\Delta_{0}^Q\}$ est une base obtuse de ${\cal A}_{0}^L$, $-\check{\alpha}^{L'}$ appartient à la chambre positive fermée de ${\cal A}_{0}^{L'}$. A fortiori, $-\check{\alpha}^{L'}$ est combinaison linéaire à coefficients positifs ou nuls de $\check{\beta}$ pour $\beta\in\Delta_{0}^{Q'}$. Cela prouve (3).
Le terme $\beta(m,k,k')$ est la transformée de Fourier de $B$ évaluée en $\theta((w')^{-1}H') -w^{-1} H$, où $H=H_{\underline{Q}_{w}}(km)$ et $H'=H_{\underline{Q}_{w'}}(k'm)$. Il est donc essentiellement majoré par $(1+\vert \theta((w')^{-1}H') -w^{-1} H\vert )^{-r}$ pour tout réel $r$, ou encore par $ (1+\vert (w\circ\theta\circ(w')^{-1})H'- H\vert )^{-r}$. Puisque $k'm$ et $km$ appartiennent à $L(F)$, on a les égalités $H=H_{Q_{w}}(km)$ et $H'=H_{Q_{w'}}(k'm)$. Les paraboliques $Q_{w}$ et $Q_{w'}$ sont standard. Grâce à (3), $H$ et $H'$ appartiennent à ${\cal C}^L(H_{0}(m))$. Donc $1+\vert \theta((w')^{-1}H') -w^{-1} H\vert \geq N^L_{w,w'}(H_{0}(m))$ et la majoration (2) s’ensuit. $\square$
Pour tout parabolique standard $Q=LU_{Q}$ et pour $m\in M_{0}(F)$, posons $$\Omega_{Q}(m)=\sum_{(w,w')\in {\cal W}_{Q}}\omega_{Q,w,w'}(m).$$ Remarquons que, dans le cas où $Q=G$, on retrouve la fonction $\Omega_{G}$ déjà définie.
[0.3cm[**[Corollaire]{}**]{}. [ *[Pour tout parabolique standard $Q=LU_{Q}$ et pour tout réel $r$, il existe $c>0$ tel que l’on ait la majoration $$\vert \Omega_{Q}(m)\vert \leq c\delta_{Q}(m)^{-1}\Xi^L(m)^2N^L(H_{0}(m))^{-r}$$ pour tout $m\in M_{0}(F)$.]{}*]{}0.3cm]{}
Un lemme d’équivalence
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Soient $x^T(m)$ et $y^T(m)$ deux fonctions des variables $T$ et $m\in M_{0}(F)^{\geq}$. On dit que ces fonctions sont équivalentes si et seulement si elles vérifient la condition suivante: pour tout réel $\nu>0$ et pour tout réel $r$, il existe $c>0$ de sorte que l’on ait l’inégalité $$\vert x^T(m)-y^T(m)\vert \leq c \vert T\vert ^{-r}$$ pour tout $T$ et tout $m\in M_{0}(F)^{\geq}$ tel que $\vert H_0(m)\vert \leq \nu\vert T\vert $.
0.3cm[**[Lemme]{}**]{}.
**
Soit $\tilde{P}=\tilde{M}U_{P}$ un espace parabolique contenant $\tilde{P}_{0}$, soit $Q=LU_{Q}$ un sous-groupe parabolique tel que $P_{0}\subset Q\subset P$ et soit $\epsilon>0$.
\(i) Les fonctions $\phi^Q(H_{0}(m)-\epsilon T)\tau_{Q}^P(H_{0}(m)-\epsilon T)\Omega_{P}(m)\delta_{P_{0}}(m)$ et $\gamma(M\vert L)^{2}\phi^Q(H_{0}(m)-\epsilon T)\tau_{Q}^P(H_{0}(m)-\epsilon T)\Omega_{Q}(m)\delta_{P_{0}}(m)$ sont équivalentes.
\(ii) Soient $w\in W^G(L\vert S)$ et $w'\in W^G(L\vert S')$. Supposons $(w,w')\in {\cal W}_{Q}^P-{\cal W}_{Q}$. Alors la fonction $\phi^Q(H_{0}(m)-\epsilon T)\tau_{Q}^P(H_{0}(m)-\epsilon T)\omega_{Q,w,w'}(m)\delta_{P_{0}}(m)$ est équivalente à $0$.
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Preuve. On fixe $\nu>0$. Les $m$ à considérer vérifient
- $m\in M_{0}(F)^{\geq}$;
- $\phi^Q(H_{0}(m)-\epsilon T)\tau_{Q}^P(H_{0}(m)-\epsilon T)=1$, ce qui entraîne $<\alpha,H_{0}(m)>><\alpha,\epsilon T>$ pour tout $\alpha\in \Delta_{0}^P-\Delta_{0}^Q$;
- $\vert H_{0}(m)\vert\leq \nu\vert T\vert $.
En conséquence
\(1) il existe un réel $\nu'$ ne dépendant que de $\nu$ tel que l’on ait $<\alpha,H_{0}(m)>>\nu'\vert H_{0}(m)\vert $ pour $\alpha\in \Delta_{0}^P-\Delta_{0}^Q$.
On peut donc approximer les produits scalaires intervenant dans $\Omega_{P}(m)$ par leurs termes constants faibles. Précisément, soient $(w,w')\in {\cal W}_{P}$. Considérons le terme $$X_{w}(m,k,k',\lambda)=(R_{P_{w}\vert w(S)}((w\sigma)_{w\lambda})\circ \gamma(w)\circ \sigma_{\lambda}(k)v,R_{\underline{P}_{w}\vert w(S)}((w\sigma)_{w\lambda})\circ\gamma(w)\circ\sigma_{\lambda}(mk')u)^M$$ qui intervient dans $\omega_{P,w,w'}(m,k,k',\lambda)$. On a vu dans la preuve précédente que l’on pouvait aussi l’écrire $$\delta_{P}(m)^{-1/2}(v_{w}(k,\lambda),\rho_{P,w\lambda}(m)u_{w}(k',\lambda)),$$ où $\rho_{P,w\lambda}=Ind_{w(S)\cap M}^M((w\sigma)_{w\lambda})$, $u_{w}(k',\lambda)$ est la restriction à $K\cap M(F)$ de $$R_{\underline{P}_{w}\vert w(S)}((w\sigma)_{w\lambda})\circ\gamma(w)\circ\sigma_{\lambda}(k')u,$$ et $v_{w}(k,\lambda)$ est la restriction à $K\cap M(F)$ de $$R_{P_{w}\vert w(S)}((w\sigma)_{w\lambda})\circ \gamma(w)\circ \sigma_{\lambda}(k)v.$$ Notons $Y_{Q,w}(m,k,k',\lambda)$ le terme constant faible relatif à $Q$ de $$(v_{w}(k,\lambda),\rho_{P,w\lambda}(m)u_{w}(k',\lambda)),$$ multiplié par $\delta_{Q}(m)^{-1/2}$. Remarquons que les éléments $u_{w}(k',\lambda)$ et $v_{w}(k,\lambda)$ restent dans des espaces de dimension finie et, quand on les écrit dans une base de ces espaces, leurs coefficients sont des fonctions bornées de $k,k'$ et de $\lambda$. D’après (1), on peut appliquer la proposition 1.12 où l’on remplace $G$ par $M$. Il existe donc $c>0$ et une fonction $C_{1}$ sur $i{\cal A}_{M_{disc},F}^*$, lisse, à croissance modérée et à valeurs positives, de sorte que l’on ait la majoration $$\vert X_{w}(m,k,k',\lambda)-Y_{Q,w}(m,k,k',\lambda)\vert \leq C_{1}(\lambda)\delta_{Q}(m)^{-1/2}\Xi^{L}(m)e^{-c\vert H_{0}(m)\vert }$$ pour tous $k$, $k'$, $\lambda$ et tout $m$ dans le domaine décrit ci-dessus. Notons que la condition $\tau_{Q}^P(H_{0}(m)-\epsilon T)=1$ entraîne une minoration $\vert H_{0}(m)\vert >> \vert T\vert $ (sauf dans le cas où $P=Q$, mais alors le lemme est tautologique). Pour $m\in M_{0}(F)^{\geq}$, on a aussi $$\Xi^L(m)<<(1+\vert H_{0}(m)\vert )^{D_{1}}\delta_{P_{0}}^Q(m)^{-1/2}<<\vert T\vert ^{\nu D_{1}}\delta_{P_{0}}^Q(m)^{-1/2}$$ pour un entier $D_{1}$ convenable. Quitte à réduire $c$, on a donc $$(2)\qquad \vert X_{w}(m,k,k',\lambda)-Y_{Q,w}(m,k,k',\lambda)\vert \leq C_{1}(\lambda) \delta_{P_{0}}(m)^{-1/2}e^{-c\vert T\vert }.$$ D’après 1.2(2) et 1.12(3), on a aussi $$(3)\qquad \vert X_{w}(m,k,k',\lambda)\vert<<(1+\vert H_{0}(m)\vert )^{D_{2}}\delta_{P_{0}}(m)^{-1/2}<<\vert T\vert ^{\nu D_{2}}\delta_{P_{0}}(m)^{-1/2} ,$$ pour un autre entier $D_{2}$. Par différence, on en déduit $$(4) \qquad \vert Y_{Q,w}(m,k,k',\lambda)\vert <<C_{2}(\lambda)\vert T \vert ^{\nu D_{2}}\delta_{P_{0}}(m)^{-1/2},$$ pour une autre fonction $C_{2}$.
On traite de la même façon le terme $$X_{w'}(m,k,k',\lambda)=$$ $$(R_{\underline{P}_{w'}\vert w'(S')}((w'\sigma')_{w'\lambda'})\circ\gamma(w')\circ \pi'_{\lambda'}(mk')v',R_{P_{w'}\vert w'(S')}((w'\sigma')_{w'\lambda'})\circ\gamma(w')\circ \pi'_{\lambda'}(k)u')^M$$ qui intervient dans $\omega_{P,w,w'}(m,k,k',\lambda)$. On définit $Y_{Q,w'}(m,k,k',\lambda)$, qui vérifie des majorations analogues à celles ci-dessus.
Rappelons l’égalité $$\omega_{P,w,w'}(m)=\int_{K\times K}\int_{i{\cal A}_{M_{disc},F}^*}m^G(\sigma_{\lambda})B(\lambda){\bf r}_{w,w'}(\sigma_{\lambda})X_{w}(m,k,k',\lambda)$$ $$X_{w'}(m,k,k',\lambda)\omega(k^{-1}mk')\,d\lambda\,dk\,dk'.$$ Posons $$\omega_{Q,w,w'}(m)=\gamma(M\vert L)^2\int_{K\times K}\int_{i{\cal A}_{M_{disc},F}^*}m^G(\sigma_{\lambda})B(\lambda){\bf r}_{w,w'}(\sigma_{\lambda})Y_{Q,w}(m,k,k',\lambda)$$ $$Y_{Q,w'}(m,k,k',\lambda)\omega(k^{-1}mk')\,d\lambda\,dk\,dk'.$$ En utilisant les majorations précédentes, on obtient $$\vert \omega_{P,w,w'}(m)-\gamma(M\vert L)^{-2}\omega_{Q,w,w'}(m)\vert \delta_{P_{0}}(m)\leq e^{-c\vert T\vert }\int_{i{\cal A}_{M_{disc},F}^*}\vert C(\lambda)B(\lambda)m^G(\sigma_{\lambda}){\bf r}_{w,w'}(\sigma_{\lambda})\vert \,d\lambda$$ pour une certaine fonction $C$ à croissance modérée. L’intégrale ci-dessus est convergente, donc les fonctions $\phi^Q(H_{0}(m)-\epsilon T)\tau_{Q}^P(H_{0}(m)-\epsilon T)\omega_{P,w,w'}(m)\delta_{P_{0}}(m)$ et $\gamma(M\vert L)^{-2}\phi^Q(H_{0}(m)-\epsilon T)\tau_{Q}^P(H_{0}(m)-\epsilon T)\omega_{Q,w,w'}(m)\delta_{P_{0}}(m)$ sont équivalentes.
On calcule $Y_{Q,w}(m,k,k',\lambda)$ en utilisant les définitions de 1.12. On doit y remplacer $G$ par $M$, $P$ par $w(S)\cap M$, $Q$ par $Q\cap M$ et $\sigma_{\lambda}$ par $(w\sigma)_{w\lambda}$. Notons que, pour $s\in W^M(L\vert w(S)\cap M)$, les opérateurs $J^M_{(Q\cap M)_{s}\vert sw(S)\cap M}((sw\sigma)_{sw\lambda})$ et $J^M_{\underline{(Q\cap M)}_{s}\vert sw(S)\cap M}((sw\sigma)_{sw\lambda})$ qui interviennent en 1.12 définissent par induction les opérateurs $J_{Q_{sw}\vert s(P_{w})}((sw\sigma)_{sw\lambda})$ et $J_{\underline{Q}_{sw}\vert s(\underline{P}_{w})}((sw\sigma)_{sw\lambda})$, avec les mêmes définitions qu’en 3.4. En rétablissant les opérateurs d’entrelacement non normalisés dans la définition de $X_{w}(m,k,k',\lambda)$ et en utilisant les propriétés habituelles de ces opérateurs, on obtient l’égalité $$Y_{Q,w}(m,k,k',\lambda)=\gamma(M\vert L)^{-1}r_{ \underline{P}_{w}\vert w(S)}((w\sigma)_{w\lambda})^{-1}r_{w(S)\vert P_{w}}((w\sigma)_{w\lambda})^{-1}$$ $$\sum_{s\in W^M(L\vert w(S)\cap M)}( J_{Q_{sw}\vert sw(S)}((sw\sigma)_{sw\lambda})\circ\gamma(sw)\circ\pi_{\lambda}(k)v,J_{\underline{Q}_{sw}\vert sw(S)}((sw\sigma)_{sw\lambda})\circ\gamma(sw)\circ\pi_{\lambda}(mk')u)^L.$$ Remarquons que l’appication $s\mapsto sw$ est une bijection de $W^M(L\vert w(S)\cap M)$ sur l’ensemble des $s\in W^G(L\vert S)$ tels que $s\in W^Mw$. On calcule de même $Y_{Q,w'}(m,k,k',\lambda)$.
Notons ${\cal W}_{Q,w,w'}^P$ l’ensemble des $(s,s')\in W^G(L\vert S)\times W^G(L\vert S')$ tels que $s\in W^Mw$ et $s'\in W^Mw'$. Les calculs précédents conduisent à l’égalité $$\omega_{Q,w,w'}(m)=\sum_{(s,s')\in {\cal W}_{Q,w,w'}^P}\omega_{Q,s,s'}(m).$$ Le terme $\Omega_{P}(m)$ est la somme des $\omega_{P,w,w'}(m)$ sur les $(w,w')\in {\cal W}_{P}$. On a $\cup_{(w,w')\in {\cal W}_{P}}{\cal W}_{Q,w,w'}={\cal W}_{Q}^P$. Posons $$\Omega_{Q}^P(m)=\sum_{(s,s')\in {\cal W}_{Q}^P}\omega_{Q,s,s'}(m).$$ On obtient que les fonctions $\phi^Q(H_{0}(m)-\epsilon T)\tau_{Q}^P(H_{0}(m)-\epsilon T)\Omega_{P}(m)\delta_{P_{0}}(m)$ et $\gamma(M\vert L)^{-2}\phi^Q(H_{0}(m)-\epsilon T)\tau_{Q}^P(H_{0}(m)-\epsilon T)\Omega_{Q}^P(m)\delta_{P_{0}}(m)$ sont équivalentes. Pour obtenir le (i) de l’énoncé, il reste à prouver que la dernière fonction est équivalente à $\gamma(M\vert L)^{-2}\phi^Q(H_{0}(m)-\epsilon T)\tau_{Q}^P(H_{0}(m)-\epsilon T)\Omega_{Q}(m)\delta_{P_{0}}(m)$. Or il suffit pour cela de prouver le (ii) de l’énoncé.
Soient maintenant $w$, $w'$ comme en (ii). En utilisant le lemme 3.6 et la majoration $\Xi^L(m)^2<<(1+\vert H_{0}(m)\vert )^D\delta_{P_{0}}^Q(m)^{-1}$ pour $m\in M_{0}(F)^{\geq,Q}$, cf. 1.2(2), il suffit pour prouver (ii) que, pour $m$ dans le domaine qui nous intéresse, on ait une minoration $$N_{w,w'}^L(H_{0}(m))>>\vert T\vert .$$ Mais soit $\tilde{P}'=\tilde{M}'U_{P'}$ le plus petit espace parabolique standard tel que $Q\subset P'$ et $(w,w')\in {\cal W}_{Q}^{P'}$. On a $P'\subset P$. La condition $\tau_{Q}^P(H_{0}(m)-\epsilon T)=1$ entraîne $\tau_{Q}^{P'}(H_{0}(m)-\epsilon T)=1$. On peut appliquer le lemme 3.5(iii) qui nous fournit la minoration $$N_{w,w'}^L(H_{0}(m))>>\epsilon\vert T\vert+\vert (H_{0}(m)-T)_{L}^{M'}\vert$$ plus forte que celle dont on a besoin. $\square$
Un deuxième lemme d’équivalence
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0.3cm[**[Lemme]{}**]{}.
**
Soit $\tilde{P}=\tilde{M}U_{P}$ un espace parabolique contenant $\tilde{P}_{0}$, soit $Q=LU_{Q}$ un sous-groupe parabolique tel que $P_{0}\subset Q\subset P$ et soit $\epsilon>0$.
\(i) Les fonctions $\phi^Q(H_{0}(m)-\epsilon T)\tau_{Q}^P(H_{0}(m)-\epsilon T)\Omega_{P}(m)\delta_{P}(m)D_{0}^M(m)$ et $\phi^Q(H_{0}(m)-\epsilon T)\tau_{Q}^P(H_{0}(m)-\epsilon T)\Omega_{Q}(m)\delta_{Q}(m)D_{0}^L(m)$ sont équivalentes.
\(ii) Soient $w\in W^G(L\vert S)$ et $w'\in W^G(L\vert S')$. Supposons $(w,w')\in {\cal W}_{Q}^P-{\cal W}_{Q}$. Alors la fonction $\phi^Q(H_{0}(m)-\epsilon T)\tau_{Q}^P(H_{0}(m)-\epsilon T)\omega_{Q,w,w'}(m)\delta_{Q}(m)D_{0}^L(m)$ est équivalente à $0$.
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Preuve. On a une majoration $\delta_{P}(m)D_{0}^M(m)<<\delta_{P_{0}}(m)$ pour tout $m\in M_{0}(F)^{\geq}$, cf. 1.2(1). On peut donc multiplier les deux fonctions du (i) l’énoncé précédent par la fonction bornée $\delta_{P_{0}}(m)^{-1}\delta_{P}(m)D_{0}^M(m)$, elles restent équivalentes. On obtient que la première fonction du (i) du présent énoncé est équivalente à la fonction $$(1) \qquad \gamma(M\vert L)^{-2} \phi^Q(H_{0}(m)-\epsilon T)\tau_{Q}^P(H_{0}(m)-\epsilon T)\Omega_{Q}(m)\delta_{P}(m)D_{0}^M(m).$$ Comme on l’a vu dans la preuve précédente, une fois fixé un réel $\nu$, on a une minoration $<\alpha,H_{0}(m)>>>\nu'\vert H_{0}(m)\vert $ pour les $m$ intervenant. D’après \[A1\] lemme 1.1 et en tenant compte de la remarque 1.2, il existe $c>0$ tel que l’on ait une majoration $$\vert \gamma(M\vert L)^{-2}\delta_{P}(m)D_{0}^M(m)-\delta_{Q}(m)D_{0}^L(m)\vert <<\delta_{P_{0}}(m)e^{-c\vert H_{0}(m)\vert }.$$ De nouveau, on a vu dans la preuve précédente que l’on avait une minoration $\vert H_{0}(m)\vert >>\vert T\vert$, sauf si $P=Q$ auquel cas le lemme est tautologique. On peut remplacer la majoration ci-dessus par $$\vert\gamma(M\vert L)^{-2} \delta_{P}(m)D_{0}^M(m)-\delta_{Q}(m)D_{0}^L(m)\vert <<\delta_{P_{0}}(m)e^{-c\vert T\vert }$$ pour un autre $c>0$. Il résulte du corollaire 3.6 et de 1.2(2) que l’on a une majoration $$\vert \Omega_{Q}(m)\vert<< (1+\vert H_{0}(m)\vert )^D\delta_{P_{0}}(m)^{-1}$$ pour un entier $D$ convenable. De ces deux dernières majorations résulte que la fonction (1) est équivalente à la deuxième fonction du (i) de l’énoncé. Cela démontre ce (i). Le (ii) résulte du (ii) du lemme précédent par multiplication par la fonction bornée $\delta_{P_{0}}(m)^{-1}\delta_{P}(m)D_{0}^M(m)$. $\square$
Un troisième lemme d’équivalence
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[0.3cm[**[Lemme]{}**]{}. [ *[Soient $Q=LU_{Q}$ un sous-groupe parabolique standard, $\tilde{P}=\tilde{M}U_{P}$ et $\tilde{P}'=\tilde{M}'U_{P'}$ deux espaces paraboliques standard et soit $\epsilon>0$. Notons $\tilde{P}_{-}=\tilde{M}_{-}U_{P_{-}}$ le plus grand espace parabolique standard tel que $P_{-}\subset Q$. On suppose $Q\subset P$ et $P_{-}\subset P'\subset P$. Alors les fonctions $$\phi^Q(H_{0}(m-\epsilon T)\tau_{Q}^P(H_{0}(m)-\epsilon T)\Omega_{P'}(m)\delta_{P'}(m)D_{0}^{M'}(m)$$ et $$\phi^Q(H_{0}(m-\epsilon T)\tau_{Q}^P(H_{0}(m)-\epsilon T)\Omega_{P_{-}}(m)\delta_{P_{-}}(m)D_{0}^{M_{-}}(m)$$ sont équivalentes.]{}*]{}0.3cm]{}
Preuve. Fixons $\epsilon'>0$ que l’on précisera plus tard. On utilise l’égalité $$\sum_{Q'; P_{0}\subset Q'\subset P'}\phi^{Q'}(H_{0}(m)-\epsilon' T)\tau_{Q'}^{P'}(H_{0}(m)-\epsilon'T)=1$$ cf. 1.3(2). On peut fixer $Q'=L'U_{Q'}$ et prouver que les produits de nos fonctions avec $\phi^{Q'}(H_{0}(m)-\epsilon' T)\tau_{Q'}^{P'}(H_{0}(m)-\epsilon'T)$ sont équivalentes. D’après le lemme précédent, les fonctions $$\phi^{Q'}(H_{0}(m)-\epsilon' T)\tau_{Q'}^{P'}(H_{0}(m)-\epsilon'T)\Omega_{P'}(m)\delta_{P'}(m)D_{0}^{M'}(m)$$ et $$\phi^{Q'}(H_{0}(m)-\epsilon' T)\tau_{Q'}^{P'}(H_{0}(m)-\epsilon'T)\Omega_{Q'}(m)\delta_{Q'}(m)D_{0}^{L'}(m)$$ sont équivalentes. La première fonction de l’énoncé, multipliée par $\phi^{Q'}(H_{0}(m)-\epsilon' T)\tau_{Q'}^{P'}(H_{0}(m)-\epsilon'T)$, est alors équivalente à $$(1) \qquad \phi^Q(H_{0}(m-\epsilon T)\tau_{Q}^P(H_{0}(m)-\epsilon T) \phi^{Q'}(H_{0}(m)-\epsilon' T)$$ $$\tau_{Q'}^{P'}(H_{0}(m)-\epsilon'T)\Omega_{Q'}(m)\delta_{Q'}(m)D_{0}^{L'}(m).$$ Supposons que $Q'\subset P_{-}$. Alors l’égalité $\phi^{Q'}(H_{0}(m)-\epsilon' T)\tau_{Q'}^{P'}(H_{0}(m)-\epsilon'T)=1$ entraîne $\phi^{Q'}(H_{0}(m)-\epsilon' T)\tau_{Q'}^{P_{-}}(H_{0}(m)-\epsilon'T)=1$. On peut appliquer le raisonnement ci-dessus en remplaçant $P'$ par $P_{-}$. On obtient que la deuxième fonction de l’énoncé, multipliée par $\phi^{Q'}(H_{0}(m)-\epsilon' T)\tau_{Q'}^{P'}(H_{0}(m)-\epsilon'T)$, est équivalente à (1), d’où la conclusion cherchée.
Supposons maintenant que $Q'\not\subset P_{-}$. Il nous suffit de prouver que la fonction (1) est équivalente à $0$ et que la deuxième fonction de l’énoncé, multipliée par $\phi^{Q'}(H_{0}(m)-\epsilon' T)\tau_{Q'}^{P'}(H_{0}(m)-\epsilon'T)$, est elle-aussi équivalente à $0$. En utilisant le corollaire 3.6 et les majorations maintenant familières portant sur les fonctions $D_{0}$ et $\Xi$, on voit qu’il suffit de prouver que, pour $m\in M_{0}(F)^{\geq}$ tel que $$\phi^Q(H_{0}(m-\epsilon T)\tau_{Q}^P(H_{0}(m)-\epsilon T) \phi^{Q'}(H_{0}(m)-\epsilon' T)\tau_{Q'}^{P'}(H_{0}(m)-\epsilon'T)=1,$$ on a des minorations $$(2)\qquad N^{L'}(H_{0}(m))>>\vert T\vert ,\,\,N^{M_{-}}(H_{0}(m))>>\vert T\vert .$$
Remarquons que la condition $\phi^{Q'}(H_{0}(m)-\epsilon' T)=1$ entraîne $<\alpha,H_{0}(m)>\leq c_{1}\epsilon'\vert T\vert $ pour un certain $c_{1}>0$ et pour tout $\alpha\in \Delta_{0}^{Q'}$, tandis que la condition $\phi^Q(H_{0}(m)-\epsilon T)\tau_{Q}^P(H_{0}(m)-\epsilon T)=1$ entraîne $<\alpha,H_{0}(m)>> c_{2}\vert T\vert $ pour un certain $c_{2}>0$ et pour tout $\alpha\in \Delta_{0}^P-\Delta_{0}^{Q}$. En supposant $c_{1}\epsilon'<c_{2}$, nos fonctions sont nulles sauf si $Q'\subset Q$. On suppose donc $Q'\subset Q$. L’hypothèse $Q'\not\subset P_{-}$ signifie qu’il existe $\alpha\in \Delta_{0}^{Q'}$ tel que $\alpha\not\in \Delta_{0}^{P_{-}}$. Fixons un tel $\alpha$, qui appartient à $\Delta_{0}^Q$ par l’hypothèse $Q'\subset Q$ que l’on vient de poser. Par définition de $P_{-}$, $\Delta_{0}^{P_{-}}$ est l’ensemble des éléments de $\Delta_{0}$ dont les images par les puissances de $\theta$ appartiennent toutes à $\Delta_{0}^Q$. Puisque $\alpha\not\in \Delta_{0}^{P_{-}}$, on peut fixer un entier $k>0$ tel que $\theta^{-k}\alpha\not\in \Delta_{0}^Q$.
Notons $\tilde{P}''=\tilde{M}''U_{P''}$ le plus petit espace parabolique tel que $Q'\subset P''$. Il est inclus dans $\tilde{P}'$ et l’hypothèse $\phi^{Q'}(H_{0}(m)-\epsilon' T)\tau_{Q'}^{P'}(H_{0}(m)-\epsilon'T)=1$ entraîne $\phi^{Q'}(H_{0}(m)-\epsilon' T)\tau_{Q'}^{P''}(H_{0}(m)-\epsilon'T)=1$. Appliquons le lemme 3.5(i). On obtient $$(3)\qquad N^{L'}(H_{0}(m))>>\vert (H_{0}(m)-\epsilon'T)_{L'}^{M''}\vert .$$ Des hypothèses $m\in M_{0}(F)^{\geq}$ et $ \phi^{Q'}(H_{0}(m)-\epsilon' T)=1$ résulte une majoration $$\vert (H_{0}(m)-\epsilon'T)_{L'}^{M''}\vert\geq \vert H_{0}(m)^{M''}\vert -c_{3}\epsilon'\vert T\vert$$ pour une constante $c_{3}>0$ convenable. Puisque $\alpha\in \Delta_{0}^{Q'}$, on a $\theta^{-k}\alpha\in \Delta_{0}^{P''}$. Donc $$<\theta^{-k}\alpha,H_{0}(m)^{M''}>=<\theta^{-k}\alpha,H_{0}(m)>.$$ Puisque $\theta^{-k}\alpha\in \Delta_{0}^P-\Delta_{0}^Q$, l’hypothèse $\tau_{Q}^P(H_{0}(m)-\epsilon T)=1$ entraîne $$<\theta^{-k}\alpha,H_{0}(m)>> <\theta^{-k}\alpha, \epsilon T>.$$ On en déduit une majoration $\vert H_{0}(m)^{M''}\vert \geq c_{4}\epsilon \vert T\vert $ pour une constante $c_{4}>0$ convenable. On obtient $$\vert (H_{0}(m)-\epsilon'T)_{L'}^{M''}\vert\geq (c_{4}\epsilon-c_{3}\epsilon')\vert T\vert .$$ En supposant $c_{3}\epsilon'<c_{4}\epsilon$, on en déduit $$\vert (H_{0}(m)-\epsilon'T)_{L'}^{M''}\vert>> \vert T\vert .$$ Jointe à (3), cette minoration entraîne la première relation de (2).
Soient maintenant $H',H''\in {\cal C}^{M_{-}}(H_{0}(m))$. Parce que $P_{-}$ est stable par $\theta$, on a $(\theta H')_{M_{-}}=\theta(H'_{M_{-}})$. On a aussi $H'_{M_{-}}=H''_{M_{-}}=H_{M_{-}}(m)$. D’où $$\vert \theta H'-H''\vert \geq\vert (\theta H')_{M_{-}}-H''_{M_{-}}\vert =\vert \theta H_{M_{-}}(m)-H_{M_{-}}(m)\vert\geq\vert \theta(H_{M_{-}}(m)^M)-H_{M_{-}}(m)^M\vert .$$ Parce que $\phi^Q(H_{0}(m-\epsilon T)\tau_{Q}^P(H_{0}(m)-\epsilon T)=1$, on peut écrire $H_{0}(m)^M=\epsilon T^M+X-Y$, avec $X=\sum_{\beta\in \Delta_{0}^P-\Delta_{0}^{Q}}x_{\beta}\check{\varpi}_{\beta}^M$, $Y=\sum_{\beta\in \Delta_{0}^Q}y_{\beta}\check{\beta}$, les coefficients $x_{\beta}$ et $y_{\beta}$ étant positifs ou nuls. Donc $H_{M_{-}}(m)^M=\epsilon T_{M_{-}}^M+X-Y_{M_{-}}$, avec $Y_{M_{-}}=\sum_{\beta\in \Delta_{0}^L-\Delta_{0}^M}y_{\beta}\check{\beta}_{M}$. Remarquons que $T_{M_{-}}^M$ est fixe par $\theta$ puisque $T$ l’est. D’où $$\vert \theta H_{M_{-}}(m)^M-H_{M_{-}}(m)^M\vert^2=\vert \theta X-X+Y_{M_{-}}-\theta(Y_{M_{_{}}})\vert ^2$$ $$=\vert \theta X-X\vert ^2+\vert\theta(Y_{M_{-}})-Y_{M_{-}}\vert ^2+2(\theta X-X,Y_{M_{-}}-\theta(Y_{M_{-}})).$$ Les éléments $X$ et $Y_{M_{-}}$ sont orthogonaux, et les éléments $\theta X$ et $\theta(Y_{M_{-}})$ aussi. L’élément $\theta X$ est combinaison linéaire à coefficients positifs ou nuls de $\check{\varpi}_{\beta}$ pour $\beta\in \Delta_{0}^P-\Delta_{0}^{P_{-}}$. Donc $(\theta X,Y_{M_{-}})\geq0$. Pour une raison analogue, $(X,\theta(Y_{M_{-}}))\geq0$. On obtient alors $$\vert \theta H_{M_{-}}(m)-H_{M_{-}}(m)\vert \geq\vert \theta(Y_{M_{-}})-Y_{M_{-}}\vert .$$ Puisque $\alpha\in \Delta_{0}^Q$, on a $$<\alpha,H_{0}(m)>=\epsilon<\alpha,T>-2y_{\alpha}-\sum_{\beta\in \Delta_{0}^Q, \beta\not=\alpha}y_{\beta}<\alpha,\check{\beta}>\geq c_{5}\epsilon\vert T\vert -2y_{\alpha}$$ pour un certain $c_{5}>0$, puisque les $<\alpha,\check{\beta}>$ sont négatifs ou nuls. Puisque $\alpha\in \Delta_{0}^{Q'}$, on a aussi $<\alpha,H_{0}(m)>\leq c_{1}\epsilon'\vert T\vert $. D’où $$y_{\alpha}\geq \frac{1}{2}(c_{5}\epsilon-c_{1}\epsilon')\vert T\vert.$$ En supposant $c_{1}\epsilon' <c_{5}\epsilon$, on obtient une minoration $y_{\alpha}>>\vert T\vert $. Puisque $\theta^{-k}\alpha\not\in \Delta_{0}^Q$, on a $<\theta^{-k}\varpi_{\alpha},Y_{M_{-}}>=0$, d’où $<\varpi_{\alpha},\theta^k(Y_{M_{-}})>=0$. Alors $<\varpi_{\alpha},Y_{M_{-}}-\theta^k(Y_{M_{-}})>=y_{\alpha}>>\vert T\vert $, d’où $\vert Y_{M_{-}}-\theta^k(Y_{M_{-}})\vert >>\vert T\vert $. L’égalité $1-\theta^k=(1-\theta)(1+\theta+...+\theta^{k-1})$ et le fait que $\theta$ soit une isométrie entraîne $\vert Y_{M_{-}}-\theta^k(Y_{M_{-}})\vert\leq k\vert Y_{M_{-}}-\theta(Y_{M_{-}})\vert $. D’où $\vert Y_{M_{-}}-\theta(Y_{M_{-}})\vert >>\vert T\vert $. En rassemblant nos calculs, on obtient la minoration $$\vert \theta H'-H''\vert >>\vert T\vert ,$$ qui démontre la seconde assertion de (2) et le lemme. $\square$
Une fonction ${\cal E}^T$
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Pour $m\in M_{0}(F)$, posons $${\cal E}^T(m)=\sum_{\tilde{P}=\tilde{M}U_{P}; \tilde{P}_{0}\subset \tilde{P}}(-1)^{dim(\mathfrak{a}_{\tilde{M}})-dim(\mathfrak{a}_{\tilde{G}})}\Omega_{P}(m)\hat{\tau}_{\tilde{P}}(H_{0}(m)-T)\delta_{P}(m)D_{0}^M(m).$$
[0.3cm[**[Proposition]{}**]{}. [ *[Les fonctions ${\cal E}^T(m)$ et $\Omega_{G}(m)\phi^{\tilde{G}}(H_{0}(m)-T)D_{0}^G(m)$ sont équivalentes.]{}*]{}0.3cm]{}
Preuve. Fixons $\epsilon>0$ que l’on précisera plus tard. Pour $m\in M_{0}(F)^{\geq}$, on a l’égalité $$\sum_{Q; P_{0}\subset Q}\phi^Q(H_{0}(m)-\epsilon T)\tau_{Q}(H_{0}(m)-\epsilon T)=1,$$ cf. 1.3(2). On peut donc fixer $Q$ et montrer que les fonctions $\phi^Q(H_{0}(m)-\epsilon T)\tau_{Q}(H_{0}(m)-\epsilon T){\cal E}^T(m)$ et $\phi^Q(H_{0}(m)-\epsilon T)\tau_{Q}(H_{0}(m)-\epsilon T)\phi^{\tilde{ G}}(H_{0}(m)-T)\Omega_{G}(m)D_{0}^G(m)$ sont équivalentes. Soit $\tilde{P}_{-}=\tilde{M}_{-}U_{P_{-}}$ le plus grand espace parabolique tel que $P_{0}\subset P \subset Q$. On introduit la fonction $\phi_{\tilde{P}_{-}}^{\tilde{G}}$, cf. 2.2. On décompose encore le problème en deux. On va prouver:
\(1) les fonctions $\phi_{\tilde{P}_{-}}^{\tilde{G}}(H_{0}(m)-T)\phi^Q(H_{0}(m)-\epsilon T)\tau_{Q}(H_{0}(m)-\epsilon T){\cal E}^T(m)$ et $\phi_{\tilde{P}_{-}}^{\tilde{G}}(H_{0}(m)-T)\phi^Q(H_{0}(m)-\epsilon T)\tau_{Q}(H_{0}(m)-\epsilon T)\phi^{\tilde{ G}}(H_{0}(m)-T)\Omega_{G}(m)D_{0}^G(m)$ sont égales sur $M_{0}(F)^{\geq}$;
\(2) les fonctions $(1-\phi_{\tilde{P}_{-}}^{\tilde{G}}(H_{0}(m)-T))\phi^Q(H_{0}(m)-\epsilon T)\tau_{Q}(H_{0}(m)-\epsilon T){\cal E}^T(m)$ et $(1-\phi_{\tilde{P}_{-}}^{\tilde{G}}(H_{0}(m)-T))\phi^Q(H_{0}(m)-\epsilon T)\tau_{Q}(H_{0}(m)-\epsilon T)\phi^{\tilde{ G}}(H_{0}(m)-T)\Omega_{G}(m)D_{0}^G(m)$ sont toutes deux équivalentes à $0$.
Traitons (1). On se limite à considérer des $m\in M_{0}(F)^{\geq}$ tels que $\phi_{\tilde{P}_{-}}^{\tilde{G}}(H_{0}(m)-T)\phi^Q(H_{0}(m)-\epsilon T)\tau_{Q}(H_{0}(m)-\epsilon T)=1$. Pour $\alpha\in \Delta^{P_{-}}_{0}$, on a une égalité $$\varpi_{\alpha}=\varpi_{\alpha}^{M_{-}}+\sum_{\beta\in \Delta_{0}^G-\Delta_{0}^{P_{-}}}x_{\beta}\varpi_{\beta}.$$ Parce que les poids fondamentaux forment une base aigu" e de ${\cal A}_{0}^*$, les coefficients $x_{\beta}$ sont positifs ou nuls. On en déduit une formule analogue $$\varpi_{\tilde{\alpha}}=\varpi_{\tilde{\alpha}}^{M_{-}}+\sum_{\tilde{\beta}\in \Delta_{0}^{\tilde{G}}-\Delta_{0}^{\tilde{P}_{-}}}x_{\tilde{\beta}}\varpi_{\tilde{\beta}}.$$ Puisque $\phi_{\tilde{P}_{-}}^{\tilde{G}}(H_{0}(m)-T)=1$, on a $<\varpi_{\tilde{\beta}},H_{0}(m)-T>\leq 0$ pour tous les $\tilde{\beta}$ intervenant, donc $$(3) \qquad <\varpi_{\tilde{\alpha}},H_{0}(m)-T>\leq <\varpi_{\tilde{\alpha}}^{M_{-}},H_{0}(m)-T>.$$ Les hypothèses $m\in M_{0}(F)^{\geq}$ et $\phi^Q(H_{0}(m)-\epsilon T)=1$ entraînent une majoration $\vert H_{0}(m)^L\vert\leq c\epsilon\vert T\vert $ pour un certain $c>0$, a fortiori $\vert H_{0}(m)^{M_{-}}\vert \leq c\epsilon\vert T\vert $. D’où $\vert <\varpi_{\tilde{\alpha}}^{M_{-}},H_{0}(m)>\vert \leq c_{\tilde{\alpha}} \epsilon \vert T\vert$ pour un certain $c_{\tilde{\alpha}}>0$. On précise $\epsilon$ en imposant que pour tout $T$, on ait l’inégalité $c_{\tilde{\alpha}}\epsilon\vert T\vert \leq <\varpi_{\tilde{\alpha}}^{M_{-}},T>$ ( pour tout $\tilde{\alpha}$). Grâce à (3), cela entraîne $$<\varpi_{\tilde{\alpha}},H_{0}(m)-T>\leq 0.$$ On avait supposé $\tilde{\alpha}\in \Delta_{0}^{\tilde{P}_{-}}$, mais cette relation est aussi vraie pour $\tilde{\alpha}\in \Delta_{0}^{\tilde{G}}-\Delta_{0}^{\tilde{P}_{-}}$ grâce à l’hypothèse $\phi_{\tilde{P}_{-}}^{\tilde{G}}(H_{0}(m)-T)=1$. Mais alors on a $\hat{\tau}_{\tilde{P}}(H_{0}(m)-T)=0$ pour tout $\tilde{P}$ propre. Dans la somme définissant ${\cal E}^T(m)$, il ne reste que la contribution de $\tilde{P}=\tilde{G}$, qui est simplement $\Omega_{G}(m)D_{0}^G(m)$. Comme on vient de le voir, on a $\phi^{\tilde{ G}}(H_{0}(m)-T)=1$ donc $\Omega_{G}(m)=\phi^{\tilde{ G}}(H_{0}(m)-T)\Omega_{G}(m)$ et la conclusion de (1) s’ensuit.
Traitons (2). On peut supposer $P_{-}\not=G$, sinon $1-\phi_{\tilde{P}_{-}}^{\tilde{G}}(H_{0}(m)-T)=0$. Pour la deuxième fonction, l’assertion est évidente car $(1-\phi_{\tilde{P}_{-}}^{\tilde{G}}(H_{0}(m)-T))\phi^{\tilde{ G}}(H_{0}(m)-T)=0$. Pour tout $\tilde{\alpha}\in \Delta_{0}^{\tilde{G}}-\Delta_{0}^{\tilde{P}_{-}}$, notons simplement $\hat{\tau}_{\tilde{\alpha}}$ la fonction caractéristique de l’ensemble des $H\in {\cal A}_{0}$ tels que $\varpi_{\tilde{\alpha}}(H)>0$. Le support de $1-\phi_{\tilde{P}_{-}}^{\tilde{G}}(H_{0}(m)-T)$ est contenu dans la réunion des supports des $\hat{\tau}_{\tilde{\alpha}}(H_{0}(m)-T)$ quand $\tilde{\alpha}$ décrit $ \Delta_{0}^{\tilde{G}}-\Delta_{0}^{\tilde{P}_{-}}$. On peut fixer $\tilde{\alpha}$ dans cet ensemble et se contenter de prouver que la fonction $$\hat{\tau}_{\tilde{\alpha}}(H_{0}(m)-T)\phi^Q(H_{0}(m)-\epsilon T)\tau_{Q}(H_{0}(m)-\epsilon T){\cal E}^T(m)$$ est équivalente à $0$. Dans la somme définissant ${\cal E}^T(m)$, on regroupe les $\tilde{P}$ en paires $(\tilde{P}',\tilde{P})$ de sorte que $\tilde{\Delta}_{0}^{P}=\tilde{\Delta}_{0}^{P'}\sqcup \{\tilde{\alpha}\}$. Il nous suffit de prouver que, pour une telle paire,
\(4) la fonction $$\hat{\tau}_{\tilde{\alpha}}(H_{0}(m)-T)\phi^Q(H_{0}(m)-\epsilon T)\tau_{Q}(H_{0}(m)-\epsilon T)$$ $$\left(\Omega_{P}(m)\hat{\tau}_{\tilde{P}}(H_{0}(m)-T)\delta_{P}(m)D_{0}^{M}(m)-\Omega_{P'}(m)\hat{\tau}_{\tilde{P}'}(H_{0}(m)-T)\delta_{P'}(m)D_{0}^{M'}(m)\right)$$ est équivalente à $0$.
On fixe $\epsilon'>0$ que l’on précisera par la suite. En utilisant le découpage déjà maintes fois utilisé, on peut fixer un sous-groupe parabolique standard $Q'\subset P$ et montrer que la fonction (4) multipliée par $\phi^{Q'}(H_{0}(m)-\epsilon' T)\tau_{Q'}^P(H_{0}(m)-\epsilon' T)$ est équivalente à $0$. Par hypothèse, $\tilde{\alpha}\not\in \Delta_{0}^{\tilde{P}_{-}}$. Par définition de $\tilde{P}_{-}$, il existe donc une racine $\alpha\in \Delta_{0}-\Delta_{0}^Q$ telle que sa restriction à ${\cal A}_{\tilde{M}_{0}}$ soit $\tilde{\alpha}$. Fixons une telle racine. L’hypothèse $\phi^Q(H_{0}(m)-\epsilon T)\tau_{Q}(H_{0}(m)-\epsilon T)=1$ entraîne une minoration $<\alpha,H_{0}(m)>>> \epsilon\vert T\vert $. Si $\alpha\in \Delta_{0}^{Q'}$, l’hypothèse $\phi^{Q'}(H_{0}(m)-\epsilon' T)=1$ entraîne une majoration $<\alpha,H_{0}(m)><< \epsilon'\vert T\vert $. On impose que $\epsilon'$ soit assez petit pour que ces deux inégalités soient contradictoires. Alors $\alpha\not\in \Delta_{0}^{Q'}$. Notons $\tilde{P}'_{-}=\tilde{M}'_{-}U_{P'_{-}}$ le plus grand espace parabolique standard tel que $P'_{-}\subset Q'$. Alors $\tilde{\alpha}\not\in \Delta^{\tilde{P}'_{-}}$. Cette relation, jointe aux inclusions $P'_{-}\subset Q'\subset P$ et à l’égalité $\Delta^{\tilde{P}}=\Delta^{\tilde{P}'}\sqcup \{\tilde{\alpha}\}$, entraîne que $P'_{-}\subset P'$. On peut alors appliquer le lemme 3.9: les fonctions $$\phi^{Q'}(H_{0}(m)-\epsilon' T)\tau_{Q'}^P(H_{0}(m)-\epsilon' T)\Omega_{P}(m)\delta_{P}(m)D_{0}^M(m)$$ et $$\phi^{Q'}(H_{0}(m)-\epsilon' T)\tau_{Q'}^P(H_{0}(m)-\epsilon' T)\Omega_{P'}(m)\delta_{P'}(m)D_{0}^{M'}(m)$$ sont toutes deux équivalentes à $$\phi^{Q'}(H_{0}(m)-\epsilon' T)\tau_{Q'}^P(H_{0}(m)-\epsilon' T)\Omega_{P'_{-}}(m)\delta_{P'_{-}}(m)D_{0}^{M'_{-}}(m).$$ Leur différence est donc équivalente à $0$. Par ailleurs, on a l’égalité $$\hat{\tau}_{\tilde{\alpha}}(H_{0}(m)-T)\hat{\tau}_{\tilde{P}}(H_{0}(m)-T)=\hat{\tau}_{\tilde{\alpha}}(H_{0}(m)-T)\hat{\tau}_{\tilde{P}'}(H_{0}(m)-T)$$ d’après l’hypothèse sur $P$ et $P'$. Donc la fonction (4), multipliée par $\phi^{Q'}(H_{0}(m)-\epsilon' T)\tau_{Q'}^P(H_{0}(m)-\epsilon' T)$, est égale à la différence des deux fonctions ci-dessus, multipiée par une fonction bornée. Cela prouve (4) et la proposition. $\square$
Contrôle de la partie centrale
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On sait que ${\cal A}_{A_{G},F}$ est un sous-groupe d’indice fini dans ${\cal A}_{G,F}$ (ces groupes sont tous deux égaux à ${\cal A}_{G}$ si $F$ est archimédien). Fixons un ensemble de représentants ${\bf b}$ du groupe quotient. Pour tout sous-ensemble ${\cal X}\subset G(F)$, notons ${\cal X}^{{\bf b}}$ l’ensemble des $g\in X$ tels que $H_{G}(g)\in {\bf b}$. Notons $H_{G}^{\tilde{G}}$ la composée de $H_{G}$ et de la projection sur l’orthogonal de ${\cal A}_{\tilde{G}}$.
[0.3cm[**[Lemme]{}**]{}. [ *[Soient $Q$ un sous-groupe parabolique standard, $w\in W^G(L\vert S)$ et $w'\in W^G(L\vert S')$. Il existe un entier $D$ et, quel que soit le réel $r$, il existe $c>0$ tel que l’on ait la majoration $$\vert \omega_{Q,w,w'}(am)\vert \delta_{Q}(am)D^L(am)\leq c(1+\vert H_{0}(m)\vert)^D(1+\vert H_{G}^{\tilde{G}} (a)\vert )^{-r}$$ pour tout $m\in M_{0}(F)^{\geq,{\bf b}}$ et tout $a\in A_{\tilde{G}}(F)\backslash A_{G}(F)$.]{}*]{}0.3cm]{}
Preuve. Il résulte des définitions que, pour tout $\lambda\in i{\cal A}_{M_{disc},F}^*$ et tous $k,k'\in K$, on a l’égalité $$\omega_{Q,w,w'}(am,k,k',\lambda)=e^{-<\lambda',H_{G}(a)>+<\lambda,H_{G}(a)>}\omega_{Q,w,w'}(m,k,k',\lambda)$$ $$=e^{<\lambda,H_{G}(a)-\theta H_{G}(a)>}\omega_{Q,w,w'}(m,k,k',\lambda).$$ Alors $$\omega_{Q,w,w'}(am)=\int_{K\times K}\int_{i{\cal A}_{M_{disc}}^*/i({\cal A}_{G}^*+{\cal A}_{M_{disc},F}^{\vee})}\omega_{Q,w,w'}(m,k,k',\lambda)m^G(\sigma_{\lambda})$$ $$B'(\lambda,a)\omega(k^{-1}amk')\,d\lambda\,dk\,dk',$$ où $$B'(\lambda,a)=\int_{i{\cal A}_{G,F}^*}B(\lambda+\mu)e^{<\lambda+\mu,H_{G}(a)-\theta H_{G}(a)>}\,d\mu.$$ On peut remplacer l’ensemble d’intégration en $\lambda$ par un domaine fondamental ${\cal X}$ dans $i{\cal A}_{M_{disc}}^*$. Alors $B'(\lambda,a)$ est de Schwartz en les deux variables. On remarque que les fonctions $\vert H_{G}(a)-\theta H_{G}(a)\vert $ et $\vert H_{G}^{\tilde{G}}(a)\vert $ sont équivalentes. On peut alors trouver une fonction $B''$ de Schwartz sur ${\cal X}$, à valeurs positives, de sorte que l’on ait l’inégalité $\vert B'(\lambda,a)\vert \leq B''(\lambda)(1+\vert H_{G}^{\tilde{G}} (a)\vert )^{-r}$. Alors $\omega_{Q,w,w'}(am)$ est majorée par le produit de $(1+\vert H_{G}^{\tilde{G}} (a)\vert )^{-r}$ et d’une fonction qui a essentiellement la même forme que $\omega_{Q,w,w'}(m)$, sauf que l’on a supprimé l’intégration centrale. Cette dernière fonction est essentiellement majorée par $\delta_{Q}(m)^{-1}\Xi^L(m)^2$ par le même argument qu’au lemme 3.6. On en déduit l’énoncé. $\square$
Un lemme de convergence
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Soient $Q=LU_{Q}\subset R$ deux sous-groupes paraboliques standard et soient $w\in W^G(L\vert S)$ et $w'\in W^G(L\vert S')$. On suppose $s_{Q}^R(w,w')\not=0$. Il existe donc un unique espace parabolique $\tilde{P}=\tilde{M}U_{P}$ tel que $Q\subset P\subset R$ et $ (w,w')\in {\cal W}_{Q}^P$. Pour $m\in M_{0}(F)$, on pose $$E^T_{Q,R,w,w'}(m)=\phi^Q(H_{0}(m)-T)\tilde{\sigma}_{Q}^R(H_{0}(m)-T)\delta_{Q}(m)D_{0}^L(m)\omega_{Q,w,w'}(m).$$
[0.3cm[**[Lemme]{}**]{}. [ *[Sous ces hypothèses, l’intégrale $$\int_{A_{\tilde{G}}(F)\backslash M_{0}(F)^{\geq,Q}}\vert E^T_{Q,R,w,w'}(m)\vert \,dm$$ est convergente.]{}*]{}0.3cm]{}
Preuve. Le lemme 3.5(iv) nous fournit une majoration $$\vert (H_{0}(m)-T)^{\tilde{G}}_{L}\vert << N^L_{w,w'}(H_{0}(m))$$ pour tout $T$ et tout $m\in M_{0}(F)^{\geq,Q}$ tel que $\phi^Q(H_{0}(m)-T)\tilde{\sigma}_{Q}^R(H_{0}(m)-T)=1$. On a $(H_{0}(m)-T)^{\tilde{G}}_{L}=H_{0}^{\tilde{G}}(m)-H_{0}^L(m)-T_{L}^{\tilde{G}}$. Les hypothèses $\phi^Q(H_{0}(m)-T)=1$ et $m\in M_{0}(F)^{\geq,Q}$ entraînent une majoration $\vert H_{0}^L(m)\vert <<\vert T\vert $. D’où $$\vert H_{0}^{\tilde{G}}(m)\vert <<\vert T\vert+\vert (H_{0}(m)-T)^{\tilde{G}}_{L}\vert,$$ puis $$(1) \qquad \vert H_{0}^{\tilde{G}}(m)\vert <<\vert T\vert +N^L_{w,w'}(H_{0}(m)).$$ Remarquons que, si l’on considère $T$ comme fixé (comme on le peut ici), la majoration (1) entraîne plus simplement $$\vert H_{0}^{\tilde{G}}(m)\vert <<N^L_{w,w'}(H_{0}(m))$$ dans le domaine considéré. En utilisant cette majoration, le lemme 3.6 et les majorations familières concernant les fonctions $D_{0}^L$ et $\Xi^L$, on voit que l’intégrale de l’énoncé est essentiellement majorée par $$\int_{A_{\tilde{G}}F)\backslash M_{0}(F)^{\geq,Q}}(1+\vert H_{0}^{\tilde{G}}(m)\vert )^{-r}\,dm$$ pour tout réel $r$. Cette intégrale est convergente, d’où le lemme. $\square$
Comparaison de deux intégrales
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On conserve les hypothèses du paragraphe précédent. Soit $\nu$ un réel strictement positif. Notons ${\bf 1}_{\nu T}$ la fonction caractéristique de l’ensemble des $m\in M_{0}(F)$ tels que $\vert H_{0}^{\tilde{G}}(m)\vert\leq \nu\vert T\vert $.
[0.3cm[**[Lemme]{}**]{}. [ *[Sous ces hypothèses, si $\nu$ est assez grand, la différence entre les deux intégrales $$\int_{A_{\tilde{G}}(F)\backslash M_{0}(F)^{\geq,Q}} E^T_{Q,R,w,w'}(m)\,dm$$ et $$\int_{A_{\tilde{G}}(F)\backslash M_{0}(F)^{\geq}}E^T_{Q,R,w,w'}(m){\bf 1}_{\nu T}(m) \,dm$$ est essentiellement majorée par $\vert T\vert ^{-r}$ pour tout réel $r$.]{}*]{}0.3cm]{}
Preuve. On peut considérer que le domaine d’intégration de la seconde intégrale est l’ensemble des $m\in A_{\tilde{G}}(F)\backslash M_{0}(F)^{\geq}$ tels que ${\bf 1}_{\nu T}(m)=1$. Ce domaine est contenu dans le domaine d’intégration $A_{\tilde{G}}(F)\backslash M_{0}(F)^{\geq,Q}$ de la première intégrale. Notons ${\cal D}^T$ le complémentaire du premier domaine dans le second. On va prouver plus précisément:
\(1) pour $\nu$ assez grand, on a pour tout réel $r$ une majoration $$\int_{{\cal D}^T}\vert E^T_{Q,R,w,w'}(m)\vert \,dm\,<<\vert T\vert ^{-r}.$$
L’ensemble ${\cal D}^T$ est réunion disjointe des deux sous-ensembles
${\cal D}^T_{1}=A_{\tilde{G}}(F)\backslash \{m\in M_{0}(F)^{\geq,Q}; {\bf 1}_{\nu T}(m)=0\}$;
${\cal D}^T_{2}=A_{\tilde{G}}(F)\backslash \{m\in M_{0}(F)^{\geq,Q}-M_{0}(F)^{\geq}; {\bf 1}_{\nu T}(m)=1\}$.
Compte tenu de la définition de $E^T_{Q,R,w,w'}(m)$, on peut ajouter la condition $\phi^Q(H_{0}(m)-T)\tilde{\sigma}_{Q}^R(H_{0}(m)-T)=1$. Sur ${\cal D}^T_{1}$, on utilise la relation (1) du paragraphe précédent, que l’on écrit plus précisément $$\vert H_{0}^{\tilde{G}}(m)\vert \leq c_{1}(\vert T\vert +N^L_{w,w'}(H_{0}(m))) .$$ Jointe à la condition ${\bf 1}_{\nu T}(m)=0$, elle nous dit que $$(\nu-c_{1})\vert T\vert \leq c_{1}N^L_{w,w'}(H_{0}(m))$$ et $$(1-\frac{c_{1}}{\nu})\vert H_{0}^{\tilde{G}}(m)\vert \leq c_{1}N^L_{w,w'}(H_{0}(m)).$$ On suppose $\nu>c_{1}$. On en déduit une majoration $$\vert H_{0}^{\tilde{G}}(m)\vert +\vert T\vert <<N^L_{w,w'}(H_{0}(m)).$$ En utilisant le lemme 3.6 et les majorations habituelles des fonctions $D_{0}^L$ et $\Xi^L$, on voit que l’intégrale $$\int_{{\cal D}_{1}^T}\vert E^T_{Q,R,w,w'}(m)\vert \,dm$$ est essentiellement majorée par $$\vert T\vert ^{-r}\int_{A_{\tilde{G}}(F)\backslash M_{0}(F)}(1+\vert H_{0}^{\tilde{G}}(m)\vert )^{-r}\,dm$$ pour tout réel $r$. Cette dernière intégrale est convergente, d’où la majoration $$\int_{{\cal D}_{1}^T}\vert E^T_{Q,R,w,w'}(m)\vert \,dm<<\vert T\vert ^{-r}.$$
Traitons maintenant l’intégrale sur ${\cal D}^T_{2}$. Le lemme 3.5(iv) nous fournit la majoration $$\vert (H_{0}(m)-T)_{L}^{\tilde{G}}\vert <<N^L_{w,w'}(H_{0}(m))$$ pour tout $T$ et tout $m\in M_{0}(F)^{\geq,Q}$ tel que $\phi^Q(H_{0}(m)-T)\tilde{\sigma}_{Q}^R(H_{0}(m)-T)=1$. Supposons de plus $m\not\in M_{0}(F)^{\geq}$. Il existe donc $\alpha\in \Delta_{0}$ tel que $<\alpha,H_{0}(m)>\leq0$. Nécessairement, $\alpha\not\in \Delta_{0}^Q$ (puisque $m\in M_{0}(F)^{\geq,Q}$). On a $$<\alpha,(H_{0}(m)-T)_{L}^{\tilde{G}}>=<\alpha,H_{0}(m)-T>+<\alpha,(T-H_{0}(m))^L> .$$ Le premier terme est inférieur ou égal à $-<\alpha,T>$. Le second est négatif ou nul: l’hypothèse $\phi^Q(H_{0}(m)-T)=1$ entraîne que $(T-H_{0}(m))^L$ est combinaison linéaire à coefficients positifs ou nuls de $\check{\beta}$ pour $\beta\in \Delta_{0}^Q$ et on a $<\alpha,\check{\beta}>\leq0$ pour tout tel $\beta$. Donc $$<\alpha,(H_{0}(m)-T)_{L}^{\tilde{G}}>\leq -<\alpha,T>.$$ A fortiori $\vert (H_{0}(m)-T)_{L}^{\tilde{G}}\vert >>\vert T\vert$, ce qui prouve la majoration $$\vert T\vert <<N^L_{w,w'}(H_{0}(m))$$ pour tout $m\in M_{0}(F)^{\geq,Q}-M_{0}(F)^{\geq}$ tel que $\phi^Q(H_{0}(m)-T)\tilde{\sigma}_{Q}^R(H_{0}(m)-T)=1$. En utilisant encore le lemme 3.6 et les majorations habituelles des fonctions $D_{0}^L$ et $\Xi^L$, on voit que l’intégrale $$\int_{{\cal D}_{2}^T}\vert E^T_{Q,R,w,w'}(m)\vert \,dm$$ est essentiellement majorée par $$\vert T\vert ^{-r}\int_{A_{\tilde{G}}(F)\backslash M_{0}(F)}(1+\vert H_{0}^{\tilde{G}}(m)\vert )^D{\bf 1}_{\nu T}(m)\,dm$$ pour un certain entier $D$ et pour tout réel $r$. La dernière intégrale est convergente et essentiellement bornée par $\vert T\vert ^{D'}$ pour un certain entier $D'$. On en déduit $$\int_{{\cal D}_{2}^T}\vert E^T_{Q,R,w,w'}(m)\vert \,dm<<\vert T\vert ^{-r}$$ pour tout réel $r$. Cela prouve (1) et le lemme. $\square$
Un lemme d’équivalence
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Posons $$E^T(m)=\sum_{Q,R; P_{0}\subset Q\subset R}\sum_{w\in W^G(L\vert S),w'\in W^G(L\vert S')}s_{Q}^R(w,w')E^T_{Q,R,w,w'}(m)$$ $$=\sum_{Q,R; P_{0}\subset Q\subset R}\sum_{w\in W^G(L\vert S),w'\in W^G(L\vert S')}s_{Q}^R(w,w')\phi^Q(H_{0}(m)-T)\tilde{\sigma}_{Q}^R(H_{0}(m)-T)$$ $$\delta_{Q}(m)D_{0}^L(m)\omega_{Q,w,w'}(m).$$
[0.3cm[**[Lemme]{}**]{}. [ *[Les fonctions $E^T(m)$ et $\Omega_{G}(m)\phi^{\tilde{G}}(H_{0}(m)-T)D_{0}^G(m)$ sont équivalentes.]{}*]{}0.3cm]{}
Preuve. Dans la somme définissant ${\cal E}^T(m)$, on peut glisser une sous-somme $$\sum_{Q; P_{0}\subset Q\subset P}\phi^Q(H_{0}(m)-T)\tau_{Q}^P(H_{0}(m)-T)$$ puisque celle-ci vaut $1$. En utilisant le lemme 3.7 (i) et (ii), on peut alors, à équivalence près, remplacer la fonction $\Omega_{P}(m)\delta_{P}(m)D_{0}^M(m)$ par $$\sum_{(w,w')\in {\cal W}_{Q}^P}\omega_{Q,w,w'}(m)\delta_{Q}(m)D_{0}^L(m).$$ On a aussi l’égalité $$\tau_{Q}^P(H_{0}(m)-T)\hat{\tau}_{\tilde{P}}(H_{0}(m)-T)=\sum_{R; P\subset R}\tilde{\sigma}_{Q}^R(H_{0}(m)-T),$$ cf. 2.2(1). A ce point, on a montré que ${\cal E}^T(m)$ est équivalente à $$\sum_{Q,R; P_{0}\subset Q\subset R}\sum_{w\in W^G(L\vert S),w'\in W^G(L\vert S')}\phi^Q(H_{0}(m)-T)\tilde{\sigma}_{Q}^R(H_{0}(m)-T)$$ $$\delta_{Q}(m)D_{0}^L(m)\omega_{Q,w,w'}(m)\sum_{\tilde{P}; Q\subset P\subset R, (w,w')\in {\cal W}_{Q}^P}(-1)^{dim(\mathfrak{a}_{\tilde{P}})-dim(\mathfrak{a}_{\tilde{G}})}.$$ La dernière somme est égale par définition à $s_{Q}^R(w,w')$, donc l’expression ci-dessus est égale à $E^T(m)$. Donc $E^T(m)$ est équivalente à $ {\cal E}^T(m)$. La proposition 3.10 nous dit que ${\cal E}^T(m)$ est équivalente à la deuxième fonction de l’énoncé. $\square$
Preuve de la proposition 3.4
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Que l’expression définissant $j_{\star}^T$ soit absolument convergente résulte du lemme 3.12. Fixons un réel $\nu>0$ que l’on précisera plus tard. Introduisons l’expression $$j_{1}^T= \int_{A_{\tilde{G}}(F)\backslash M_{0}(F)^{\geq}}E^T(m){\bf 1}_{\nu T}(m)\,dm.$$ Elle aussi est absolument convergente. Le lemme 3.13 nous dit que $$lim_{T\to \infty}j_{\star}^T-j_{1}^T=0.$$ Pour $m\in M_{0}(F)^{\geq}$, posons $$\psi(m)=\Omega_{G}(m)D_{0}^G(m)\tilde{\kappa}^T(m)-E^T(m){\bf 1}_{\nu T}(m).$$ Remarquons que toute intégrale sur $A_{\tilde{G}}(F)\backslash M_{0}(F)^{\geq}$ se décompose en produit d’une intégrale sur $M_{0}(F)^{\geq,{\bf b}}$ et d’une intégrale sur $A_{\tilde{G}}(F)\backslash A_{G}(F)$, cf. 3.11 pour la définition de $M_{0}(F)^{\geq,{\bf b}}$. On décompose $j^T-j_{1}^T$ en $j_{2}^T+j_{3}^T$, où $$j_{2}^T= \int_{M_{0}(F)^{\geq,{\bf b}}}\int_{a\in A_{\tilde{G}}(F)\backslash A_{G}(F); \vert H_{G}^{\tilde{G}}(a)\vert >\vert T\vert } \psi(am)\,da\,dm,$$ $$j_{3}^T= \int_{M_{0}(F)^{\geq,{\bf b}}}\int_{a\in A_{\tilde{G}}(F)\backslash A_{G}(F); \vert H_{G}^{\tilde{G}}(a)\vert \leq\vert T\vert } \psi(am)\,da\,dm.$$ Dans la première, on utilise le lemme 3.11. On peut l’appliquer à chaque fonction intervenant dans la définition de $E^T(m)$, ainsi qu’à la fonction $\Omega_{G}(m)D_{0}^G(m)$ qui en est un cas particulier. Ce lemme nous fournit une majoration $$\vert \psi(am)\vert <<(1+\vert H_{0}(m)\vert)^D(1+\vert H_{G}^{\tilde{G}}(a)\vert )^{-r}$$ pour un certain entier $D$ et pour tout réel $r$. L’intégrale en $a$ est essentiellement majorée par $\vert T\vert ^{-r}$ pour tout réel $r$. L’intégrale en $m$ porte sur un domaine où $\vert H_{0}(m)\vert <<\vert T\vert $ car une telle inégalité est vérifiée sur l’intersection de $M_{0}(F)^{\geq,{\bf b}}$ et des supports des fonctions $\tilde{\kappa}^T(am)$ ou ${\bf 1}_{\nu T}(am)$. L’intégrale en $m$ est donc essentiellement bornée par $\vert T\vert ^{D'}$ pour un certain entier $D'$. Il en résulte que $lim_{T\to \infty}j_{2}^T=0$. Considérons $j_{3}^T$. Si $\nu$ est assez grand, les conditions $m\in M_{0}(F)^{\geq,{\bf b}}$, $\vert H_{G}^{\tilde{G}}(a)\vert \leq \vert T\vert $ et $\tilde{\kappa}^T(am)=1$ impliquent $\vert H_{0}(am)^{\tilde{G}}\vert \leq \nu\vert T\vert $. On ne change alors rien en multipliant la fonction $\Omega_{G}(am)D_{0}^G(am)\tilde{\kappa}^T(am)$ par ${\bf 1}_{\nu T}(am)$. Pour $m\in M_{0}(F)^{\geq}$, on a $\tilde{\kappa}^T(m)=\phi^{\tilde{ G}}(H_{0}(m)-T)$. Le lemme 3.14 nous fournit pour tout $r$ une majoration $$\vert \psi(am)\vert<< \vert T\vert ^{-r}$$ sur le domaine d’intégration de $j_{3}^T$, pour un certain entier $D$ et pour tout $r$. De nouveau, l’intégrale de cette fonction sur le domaine ${\bf 1}_{\nu T}(am)=1$ est majorée par $\vert T\vert ^{-r}$ pour tout $r$. Donc $lim_{T\to \infty}j_{3}^T=0$. Cela achève la preuve. $\square$
Définition de $(G,M)$-familles
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Posons $M'_{disc}=\theta^{-1}(M_{disc})$. Soit $t\in W^G(M_{disc}\vert M'_{disc})$, c’est-à-dire que $t\in W^G/W^{M'_{disc}}$ et $t(M'_{disc})=M_{disc}$. Soit $\nu\in [t\sigma',\omega \sigma]$. On fixe un automorphisme unitaire $A_{\nu}$ de $V_{\sigma}$ tel que $(t\sigma')(x)\circ A_{\nu}=\omega(x)A_{\nu}\circ \sigma_{-\nu}(x)$ pour tout $x\in M_{disc}(F)$. Par fonctorialité, il définit des homomorphismes entre différentes représentations induites. Notons $\underline{\omega}$ l’opérateur qui, à une fonction $\varphi$ sur $G(F)$, associe la fonction $g\mapsto \omega(g)\varphi(g)$. Lui-aussi définit des homomorphismes entre certaines représentations induites. Pour $\Lambda\in i{\cal A}_{M_{disc},F}^*$, introduisons l’opérateur $$A(t,\nu;\Lambda):\pi'_{t^{-1}(\Lambda+\nu)}=Ind_{S'}^G(\sigma'_{t^{-1}(\lambda+\nu)})\to \pi_{\Lambda}=Ind_{S}^G(\sigma_{\Lambda})$$ défini par $$A(t,\nu;\Lambda)=R_{S\vert t(S')}(\sigma_{\Lambda})\circ\gamma(t)\circ A_{\nu}^{-1}\underline{\omega}^{-1}.$$ Il vérifie la relation d’entrelacement $\omega(g)\pi_{\Lambda}(g)\circ A(t,\nu;\Lambda)=A(t,\nu;\Lambda)\circ \pi'_{t^{-1}(\Lambda+\nu)}(g)$.
Pour $S''\in {\cal P}(M_{disc})$, on définit la fonction $(\lambda,\Lambda)\mapsto \phi(t,\nu;\lambda,\Lambda,S'')$ des deux variables $\lambda,\Lambda\in i{\cal A}_{M_{disc},F}^*$ par $$\phi(t,\nu;\lambda,\Lambda,S'')=(A(t,\nu;t\lambda'-\nu)v',J_{\bar{S}''\vert S}(\sigma_{t\lambda'-\nu})^{-1}\circ J_{\bar{S}''\vert S}(\sigma_{t\lambda'-\nu+\Lambda})u )$$ $$( J_{S''\vert S}(\sigma_{t\lambda'-\nu})^{-1}\circ J_{S''\vert S}(\sigma_{t\lambda'-\nu+\Lambda})v,A(t,\nu;t\lambda'-\nu)u').$$ Pour $\lambda$ fixé, la famille $(\phi(t,\nu;\lambda,\Lambda,S''))_{S''\in {\cal P}(M_{disc})}$ de fonctions en $\Lambda$ est presque une $(G,M_{disc})$-famille et même presque une $(G,M_{disc})$-famille $p$-adique dans le cas où $F$ est non-archimédien. “Presque” parce que les fonctions ne sont pas forcément $C^{\infty}$: il peut y avoir des singularités. Etudions cette question de régularité. En convertissant les opérateurs d’entrelacement en opérateurs normalisés, on peut récrire la définition de $\phi(t,\nu;\lambda,\Lambda,S'')$ sous la forme $$\phi(t,\nu;\lambda,\Lambda,S'')=r_{\bar{S}''\vert S''}(\sigma_{t\lambda'-\nu})^{-1}r_{ \bar{S}''\vert S''}(\sigma_{t\lambda'-\nu+\Lambda})$$ $$(A(t,\nu;t\lambda'-\nu)v',R_{ \bar{S}''\vert S}(\sigma_{t\lambda'-\nu})^{-1}\circ R_{\bar{S}''\vert S}(\sigma_{t\lambda'-\nu+\Lambda})u)$$ $$( R_{ S''\vert S}(\sigma_{t\lambda'-\nu})^{-1}\circ R_{S''\vert S}(\sigma_{t\lambda'-\nu+\Lambda})v,A(t,\nu;t\lambda'-\nu)u').$$ Posons $$r_{S'',reg}(\sigma_{\lambda})=r_{ \bar{S}''\vert S''}(\sigma_{\lambda})r_{ \bar{S}\vert S}(\sigma_{\lambda})^{-1}.$$ On peut écrire $$\phi(t,\nu;\lambda,\Lambda,S'')=r_{ \bar{S}\vert S}(\sigma_{t\lambda'-\nu})^{-1}r_{ \bar{S}\vert S}(\sigma_{t\lambda'-\nu+\Lambda})\phi_{reg}(t,\nu;\lambda,\Lambda,S''),$$ où $$\phi_{reg}(t,\nu;\lambda,\Lambda,S'')=r_{S'',reg}(\sigma_{t\lambda'-\nu})^{-1}r_{S'',reg}(\sigma_{t\lambda'-\nu+\Lambda})$$ $$(A(t,\nu;t\lambda'-\nu)v',R_{ \bar{S}''\vert S}(\sigma_{t\lambda'-\nu})^{-1}\circ R_{\bar{S}''\vert S}(\sigma_{t\lambda'-\nu+\Lambda})u)$$ $$( R_{ S''\vert S}(\sigma_{t\lambda'-\nu})^{-1}\circ R_{S''\vert S}(\sigma_{t\lambda'-\nu+\Lambda})v,A(t,\nu;t\lambda'-\nu)u').$$
Grâce à 1.10(5), la fonction $\phi_{reg}(t,\nu;\lambda,\Lambda,S'')$ est régulière en $\lambda$ et $\Lambda$. La famille $(\phi_{reg}(t,\nu;\lambda,\Lambda,S''))_{S''\in {\cal P}(M_{disc})}$ est vraiment une $(G,M)$-famille, $p$-adique dans le cas où $F$ est non-archimédien. Toutes les singularités de la famille de départ se concentrent dans la fonction $r_{ \bar{S}\vert S}(\sigma_{t\lambda'-\nu})^{-1}r_{ \bar{S}\vert S}(\sigma_{t\lambda'-\nu+\Lambda})$ en facteur.
Soit $Q=LU_{Q}\in {\cal F}(M_{disc})$. On déduit de la famille $(\phi(t,\nu;\lambda,\Lambda,S''))_{S''\in {\cal P}(M_{disc})}$ une famille $(\phi(t,\nu;\lambda,\Lambda,S^{_{''}L}))_{S^{_{''}L}\in {\cal P}^L(M_{disc})}$, qui n’est autre que $(\phi(t,\nu;\lambda,\Lambda,S''))_{S''\in {\cal P}(M_{disc}),S''\subset Q}$.
Pour $X\in {\cal A}_{L,F}$ et $S''\in {\cal P}(M_{disc})$ tel que $S''\subset Q$, on a défini en 1.6 la fonction $$\epsilon_{S''}^{Q,T[S'']}(X;\Lambda)=\int_{{\cal A}_{M_{disc},F}^{L}(X)}\phi_{S''}^L(Y-T[S''])e^{<\Lambda,Y>}\,dY$$ ($\phi_{S''}^L$ est la fonction combinatoire de 1.3, qui n’a rien à voir avec les fonctions de la $(G,M_{disc})$-famille). On pose $$\phi_{M_{disc}}^{Q,T}(t,\nu,X;\lambda,\Lambda)=\sum_{S''\in {\cal P}(M_{disc}),S''\subset Q}\phi(t,\nu;\lambda,\Lambda,S'')\epsilon_{S''}^{Q,T[S'']}(X;\Lambda).$$ On définit de même $\phi_{reg,M_{disc}}^{Q,T}(t,\nu,X;\lambda,\Lambda)$. Rappelons que l’on a posé $\lambda'=\theta^{-1}\lambda$. On pose $$\epsilon(t,\nu;\lambda)=r_{\bar{S}\vert S}(\sigma_{\lambda})r_{\bar{S}\vert S}(\sigma_{t\lambda'-\nu})^{-1} .$$
0.3cm[**[Lemme]{}**]{}.
**
\(i) Pour $\lambda$ en position générale, la fonction $\Lambda\mapsto \phi_{M_{disc}}^{Q,T}(t,\nu,X;\lambda,\Lambda)$ est régulière en $\Lambda=\lambda-t\lambda'+\nu$.
\(ii) Les fonctions $$\lambda\mapsto \phi_{M_{disc}}^{Q,T}(t,\nu,X;\lambda, \lambda-t\lambda'+\nu),$$ $$\lambda\mapsto \phi_{reg,M_{disc}}^{Q,T}(t,\nu,X;\lambda, \lambda-t\lambda'+\nu),$$ $$\lambda\mapsto \epsilon(t,\nu;\lambda)$$ sont $C^{\infty}$ sur $i{\cal A}_{M,F}^*$. Si $F$ est archimédien, toutes leurs dérivées sont à croissance lente.
\(iii) On a l’égalité $$\phi_{M_{disc}}^{Q,T}(t,\nu,X;\lambda, \lambda-t\lambda'+\nu)= \epsilon(t,\nu;\lambda) \phi_{reg,M_{disc}}^{Q,T}(t,\nu,X;\lambda, \lambda-t\lambda'+\nu) .$$
\(iv) Comme fonction de $X$, $ \phi_{M_{disc}}^{Q,T}(t,\nu,X;\lambda,\lambda-t\lambda'+\nu)$ ne dépend que de la classe $X+{\cal A}_{A_{\tilde{G}},F}$.
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Preuve. Supposons d’abord $F$ non-archimédien. La discussion ci-dessus montre que l’on a l’égalité $$\phi_{M_{disc}}^{Q,T}(t,\nu,X;\lambda,\Lambda)=r_{ \bar{S}\vert S}(\sigma_{t\lambda'-\nu+\Lambda})r_{ \bar{S}\vert S}(\sigma_{t\lambda'-\nu})^{-1}\phi_{reg,M_{disc}}^{Q,T}(t,\nu,X;\lambda,\Lambda).$$ La dernière fonction est $C^{\infty}$ en $\lambda$ et $\Lambda$ d’après le lemme 1.6. Le facteur $r_{ \bar{S}\vert S}(\sigma_{t\lambda'-\nu+\Lambda})r_{ \bar{S}\vert S}(\sigma_{t\lambda'-\nu})^{-1}$ est évidemment régulier en $\Lambda=\lambda-t\lambda'+\nu$ pour $\lambda$ en position générale. Il vaut alors $r_{ \bar{S}\vert S}(\sigma_{ \lambda})r_{ \bar{S}\vert S}(\sigma_{t\lambda'-\nu})^{-1}$. Pour prouver (ii) et (iii), il reste à montrer que ce dernier terme est régulier en $\lambda$. Par l’isomorphisme $\omega\sigma_{-\nu}\simeq t\sigma'$, on a $\sigma_{t\lambda'-\nu}\simeq \omega^{-1}t(\sigma_{\lambda}\circ\theta) $. Par transport de structure et parce que les facteurs de normalisation sont insensibles à la torsion par le caractère $\omega$, on a $r_{ \bar{S}\vert S}(\sigma_{t\lambda'-\nu})=r_{\theta t^{-1}(\bar{S})\vert \theta t^{-1}(S)}(\sigma_{\lambda})$. Mais alors, le quotient $$r_{\bar{S}\vert S}(\sigma_{\lambda})r_{ \bar{S}\vert S}(\sigma_{t\lambda'-\nu})^{-1}=r_{\bar{S}\vert S}(\sigma_{\lambda})r_{\theta t^{-1}(\bar{S})\vert \theta t^{-1}(S)}(\sigma_{\lambda})^{-1}$$ est régulier d’après 1.10(5).
Supposons maintenant $F$ archimédien. Le même raisonnement prouve les assertions de régularité. Pour montrer l’assertion de croissance lente, le lemme 1.4 nous ramène à prouver que que
- les dérivées des fonctions intervenant dans la définitions des termes $\phi_{reg}(t,\nu;\lambda,\Lambda,S'')$ sont à croissance lente;
- les dérivées de la fonction $r_{ \bar{S}\vert S}(\sigma_{\lambda})r_{ \bar{S}\vert S}(\sigma_{t\lambda'-\nu})^{-1}$ sont à croissance lente.
Les opérateurs d’entrelacement normalisés ont des dérivées à croissance lente. Les autres fonctions intervenant sont toutes de la forme $r_{ \bar{S}_{1}\vert S_{1}}(\sigma_{\mu})r_{\bar{S}_{2}\vert S_{2}}(\sigma_{\mu})^{-1}$ où $S_{1},S_{2}\in {\cal P}(M_{disc})$ et $\mu$ dépend linéairement de $\lambda$. L’assertion résulte de 1.10(5).
Preuve de (iv). Pour $Y\in {\cal A}_{A_{\tilde{G}},F}$, l’égalité suivante résulte des définitions $$\phi_{M_{disc}}^{Q,T}(t,\nu,X+Y;\lambda,\lambda-t\lambda'+\nu)=e^{<\lambda-t\lambda'+\nu,Y>} \phi_{M_{disc}}^{Q,T}(t,\nu,X;\lambda,\lambda-t\lambda'+\nu).$$ Evidemment $e^{<\lambda-t\lambda',Y>}=1$. En comparant les restrictions à $A_{\tilde{G}}(F)$ des caractères centraux des représentations $\omega\sigma$ et $ t\sigma'$, l’isomorphisme $\omega\sigma_{-\nu}\simeq t\sigma'$ et le fait que $\omega$ soit trivial sur $A_{\tilde{G}}(F)$ entraîne que $\nu_{\tilde{G}}\in i{\cal A}_{A_{\tilde{G}},F}^{\vee}$. Donc $e^{<\nu,Y>}=1$. Cela prouve (iii). $\square$
Définition d’une nouvelle intégrale
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Pour deux sous-groupes paraboliques $Q=LU_{Q},R\in {\cal F}(M_{disc})$ tels que $Q\subset R$ et pour $t\in W^G(M_{disc}\vert M'_{disc})$, notons $s_{Q}^{R}(t)$ la somme des $$(-1)^{dim(\mathfrak{a}_{\tilde{M}})-dim(\mathfrak{a}_{\tilde{G}})}$$ sur les espaces paraboliques $\tilde{P}=\tilde{M}U_{P}$ tels que $Q\subset P\subset R$ et $t\theta^{-1}(P)=P$. Cette dernière condition équivaut à $\gamma_{0}t^{-1}\in \tilde{P}(F)$ (où on identifie $t$ à un relèvement dans $K$). La condition $s_{Q}^{R}(t)\not=0$ équivaut à ce qu’il existe un et un seul $\tilde{P}$ vérifiant ces conditions. Soient $t\in W^G(M_{disc}\vert M'_{disc})$ et $\nu\in [t\sigma',\omega\sigma]$. Posons $${\bf E}^T_{Q,R,t,\nu}=mes(i{\cal A}_{M_{disc},F}^*)^{-1}\int_{{\cal A}_{L,F}/{\cal A}_{A_{\tilde{G}},F}}\tilde{\sigma}_{Q}^{R}(X-T[Q])$$ $$\int_{i{\cal A}_{M_{disc},F}^*}B(\lambda) \phi_{M_{disc}}^{Q,T}(t,\nu,X;\lambda,\lambda-t\lambda'+\nu)\,d\lambda\,dX.$$
[0.3cm[**[Lemme]{}**]{}. [ *[Si $s_{Q}^R(t)\not=0$, l’expression ${\bf E}^T_{Q,R,t,\nu}$ est convergente dans l’ordre indiqué.]{}*]{}0.3cm]{}
Preuve. La convergence absolue de l’intégrale intérieure résulte du (ii) du lemme précédent et du fait que $B$ est de Schwartz. Puisque ${\cal A}_{A_{L},F}$ est d’indice fini dans ${\cal A}_{L,F}$, on peut fixer $X_{0}\in {\cal A}_{L,F}$ et prouver que l’expression $$\int_{{\cal A}_{A_{L},F}/{\cal A}_{A_{\tilde{G}},F}}\tilde{\sigma}_{Q}^{R}(X+X_{0}-T[Q])\vert \int_{i{\cal A}_{M_{disc},F}^*}B(\lambda) \phi_{M_{disc}}^{Q,T}(t,\nu,X+X_{0};\lambda,\lambda-t\lambda'+\nu)\,d\lambda\vert \,dX$$ est convergente. Il résulte des définitions que $$\phi_{M_{disc}}^{Q,T}(t,\nu,X+X_{0};\lambda,\lambda-t\lambda'+\nu)=e^{<\lambda-t\lambda'+\nu,X>} \phi_{M_{disc}}^{Q,T}(t,\nu,X_{0};\lambda,\lambda-t\lambda'+\nu).$$ Ainsi l’expression ci-dessus est de la forme $$\int_{{\cal A}_{A_{L},F}/{\cal A}_{A_{\tilde{G}},F}}\tilde{\sigma}_{Q}^{R}(X+X_{0}-T[Q])\vert \int_{i{\cal A}_{M_{disc},F}^*}B'(\lambda)e^{<\lambda-t\lambda',X>}\,d\lambda\vert \,dX,$$ où $B'$ est une fonction de Schwartz. En utilisant l’égalité $$<\lambda-t\lambda',X>=<\lambda,(1-\theta t^{-1})X>,$$ on voit que l’intégrale intérieure est essentiellement majorée par $(1+\vert ( 1-\theta t^{-1})X\vert )^{-r}$ pour tout réel $r$. Pour démontrer la convergence cherchée, et puisque $T$ peut être ici considéré comme une constante, il suffit de prouver que l’on a une majoration $$(1)\qquad \vert X^{\tilde{G}}\vert <<1+\vert T\vert +\vert ( 1-\theta t^{-1})X\vert$$ pour tout $T$ et tout $X\in {\cal A}_{A_{L},F}$ tel que $\tilde{\sigma}_{Q}^R(X+X_{0}-T[Q])=1$. Puisque $X_{0}$ est fixé, cela résulte par translation par $-X_{0}$ d’une majoration $$\vert (X-T[Q])^{\tilde{G}}_{L}\vert <<1+\vert ( 1-\theta t^{-1})X\vert$$ pour tout $X\in {\cal A}_{L}$ tel que $\tilde{\sigma}_{Q}^R(X-T[Q])=1$. Pour mieux comprendre la situation, introduisons un élément $s\in W^G$ tel que $s(Q)$ soit standard. Posons $Q'=s(Q)=L'U_{Q'}$, $R'=s(R)$. La majoration précédente résulte de la majoration
\(2) $ \vert (X-T)^{\tilde{G}}_{L'}\vert <<1+\vert (s\theta t^{-1}s^{-1}-1)X\vert $ pour tout $X\in {\cal A}_{L'}$ tel que $\tilde{\sigma}_{Q'}^{R'}(X)=1$.
La condition $s_{Q}^R(t)\not=0$ se traduit ainsi: il existe un unique espace parabolique $\tilde{P}=\tilde{M}U_{P}$ tel que $Q'\subset P\subset R'$ et $ \theta(st)s^{-1}\in W^M$, autrement dit $s_{Q'}^{R'}(s,st)\not=0$. On applique le lemme 3.5(iv) en y remplaçant les termes $Q,R,S,w,w',H$ par $Q',R',P_{0},s,st,X$ (ces données vérifient les conditions de ce lemme). Ce lemme implique $$\vert (X-T)^{\tilde{G}}_{L'}\vert <<N_{s,st}^{L'}(X).$$ Mais, par définition de ce dernier terme, on a $$N_{s,st}^{L'}(X)<<1+\vert (s\theta t^{-1}s^{-1}-1)X\vert .$$ Cela prouve (2) et le lemme. $\square$
Apparition des $(G,M)$-familles
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Soient $Q=LU_{Q}\subset R$ deux sous-groupes paraboliques standard et soient $w\in W^G(L\vert S)$, $w'\in W^G(L\vert S')$ deux éléments tels que $s_{Q}^R(w,w')\not=0$. On note $\tilde{P}=\tilde{M}U_{P}$ l’unique espace parabolique tel que $Q\subset P\subset R$ et $(w,w')\in {\cal W}_{Q}^P$. On pose $$E^T_{Q,R,w,w'}=\int_{A_{\tilde{G}}(F)\backslash M_{0}(F)^{\geq,Q}}E^T_{Q,R,w,w'}(m)\,dm$$ $$=\int_{A_{\tilde{G}}(F)\backslash M_{0}(F)^{\geq,Q}}\tilde{\sigma}_{Q}^R(H_{0}(m)-T)\phi^Q(H_{0}(m)-T)\delta_{Q}(m)D_{0}^L(m)\omega_{Q,w,w'}(m)\,dm.$$ Le but du paragraphe est de définir une approximation plus explicite de $E^T_{Q,R,w,w'}$. On pose $$E^T_{\star;Q,R,w,w'}= \sum_{t\in W^G(M_{disc}\vert M'_{disc})\cap w^{-1}W^Lw'}\sum_{\nu\in[t\sigma',\omega \sigma]}{\bf E}^T_{w^{-1}(Q),w^{-1}(R),t,\nu}.$$ On identifie comme toujours un élément $t\in W^G(M_{disc}\vert M'_{disc})$ à un relèvement dans $W^G$. La condition que $t$ appartienne à $w^{-1}W^Lw'$ ne dépend pas du relèvement choisi. D’autre part, on voit que, pour $t$ dans l’ensemble de sommation, on a $s_{w^{-1}(Q)}^{w^{-1}(R)}(t)=s_{Q}^R(w,w')$. Notre hypothèse est que ce nombre est non nul. Les termes ${\bf E}^T_{w^{-1}(Q),w^{-1}(R),t,\nu}$ sont donc bien définis d’après le lemme précédent.
[0.3cm[**[Lemme]{}**]{}. [ *[On a une majoration $$\vert E^T_{Q,R,w,w'}-mes(A_{\tilde{G}}(F)_{c})^{-1}E^T_{\star;Q,R,w,w'}\vert <<\vert T\vert ^{-r}$$ pour tout réel $r$.]{}*]{}0.3cm]{}
Preuve. Fixons un réel $\zeta>0$. Notons ici ${\bf 1}_{\zeta T}$ la fonction caractéristique de l’ensemble des $H\in {\cal A}_{0}$ tels que $\vert H\vert \leq \zeta\vert T\vert $ (ce n’est pas la même fonction qu’en 3.13). Notons $E^{T,\zeta}_{Q,R,w,w'}$ la variante de l’expression $E^T_{Q,R,w,w'}$ où on glisse la fonction ${\bf 1}_{\zeta T}(H_{L}^{\tilde{G}}(m))$ dans l’intégrale. Notons $E^{T,\zeta}_{\star;Q,R,w,w'}$ la variante de $E^T_{\star;Q,R,w,w'}$ où on glisse la fonction ${\bf 1}_{\zeta T}(X^{\tilde{G}})$ dans les intégrales définissant chaque ${\bf E}^T_{w^{-1}(Q),w^{-1}(R),t,\nu}$. Il résulte de 3.13(1) que, si $\zeta$ est assez grand, on a la majoration $$\vert E^T_{Q,R,w,w'}-E^{T,\zeta}_{Q,R,w,w'}\vert<<\vert T\vert ^{-r}$$ pour tout réel $r$. De même, en reprenant la preuve du lemme 3.17, il résulte de 3.17(1) que, si $\zeta$ est assez grand, on a la majoration $$\vert E^T_{\star;Q,R,w,w'}-E^{T,\zeta}_{\star;Q,R,w,w'}\vert<<\vert T\vert ^{-r}$$ pour tout réel $r$. On fixe $\zeta$ tel qu’il en soit ainsi. Il nous suffit alors de majorer $$\vert E^{T,\zeta}_{Q,R,w,w'}-mes(A_{\tilde{G}}(F)_{c})^{-1}E^{T,\zeta}_{\star;Q,R,w,w'}\vert .$$
Pour $m\in M_{0}(F)^{\geq,Q}$, on a l’égalité $\phi^Q(H_{0}(m)-T)=\kappa^{L,T}(m)$, ce dernier terme étant l’analogue de $\kappa^T(m)$ quand on remplace $G$ par $L$, cf. 1.14. On a aussi trivialement $\tilde{\sigma}_{Q}^R(H_{0}(m)-T)=\tilde{\sigma}_{Q}^R(H_{L}(m)-T)$. Alors $E^{T,\zeta}_{Q,R,w,w'}$ est l’intégrale sur $A_{\tilde{G}}(F)\backslash M_{0}(F)^{\geq,Q}$ du produit de $D_{0}^L(m)$ et de $${\bf 1}_{\zeta T}(H_{L}^{\tilde{G}}(m))\tilde{\sigma}_{Q}^R(H_{L}(m)-T)\kappa^{L,T}(m)\delta_{Q}(m)\omega_{Q,w,w'}(m) .$$ Cette dernière fonction est définie non seulement pour $m\in M_{0}(F)$, mais en tout point de $L(F)$. Elle est biinvariante par $K\cap L(F)$ (cela résulte de la présence d’une intégrale sur $K\times K$ dans la définition de $\omega_{Q,w,w'}(g)$, cf. 3.4). En se rappelant la définition de la fonction $D_{0}^L(m)$, cf. 1.2, on voit que $$E^{T,\zeta}_{Q,R,w,w'}=\int_{A_{\tilde{G}}(F)\backslash L(F)}{\bf 1}_{\zeta T}(H_{L}^{\tilde{G}}(l))\tilde{\sigma}_{Q}^R(H_{L}(l)-T)\kappa^{L,T}(l)\delta_{Q}(l)\omega_{Q,w,w'}(l) \,dl.$$ L’intégrale sur $A_{\tilde{G}}(F)\backslash L(F)$ se décompose en une intégrale sur $ X\in{\cal A}_{L,F}/{\cal A}_{A_{\tilde{G}},F}$ d’intégrales portant sur $A_{\tilde{G}}(F)\backslash A_{\tilde{G}}(F)L(F;X)$ (on rappelle que $L(F;X)$ est l’ensemble des $l\in L(F)$ tels que $H_{L}(l)= X $). Précisément $$E^{T,\zeta}_{Q,R,w,w'}=\int_{{\cal A}_{L,F}/{\cal A}_{A_{\tilde{G}},F}}{\bf 1}_{\zeta T}(X^{\tilde{G}})\tilde{\sigma}_{Q}^R(X-T) e^T_{Q,w,w'}(X)\,dX,$$ où $$e^T_{Q,w,w'}(X)=\int_{ A_{\tilde{G}}(F)\backslash A_{\tilde{G}}(F)L(F;X)}\delta_{Q}(l)\kappa^{L,T}(l)\omega_{Q,w,w'}(l)\,dl.$$
Fixons $X$. Revenons à la définition de $\omega_{Q,w,w'}(l)$, cf. 3.4. C’est une intégrale portant sur $K\times K\times i{\cal A}_{M_{disc},F}^*$. Ces intégrales commutent entre elles et commutent à l’intégrale ci-dessus sur $A_{\tilde{G}}(F)\backslash A_{\tilde{G}}(F) L(F;X)$ car celle-ci est à support compact à cause de la fonction $\kappa^{L,T}(l)$. On obtient $$e^T_{Q,w,w'}(X)=\int_{i{\cal A}_{M_{disc},F}^*}B(\lambda)m^G(\sigma_{\lambda})\omega^T_{Q,w,w'}(X,\lambda)\,d\lambda,$$ où $$\omega^T_{Q,w,w'}(X,\lambda)=\int_{K\times K}\int_{ A_{\tilde{G}}(F)\backslash A_{\tilde{G}}(F) L(F;X)}\omega_{Q,w,w'}(l,k,k',\lambda)\omega(l)\delta_{Q}(l)\kappa^{L,T}(l) \,dl\,\omega(k^{-1}k')\,dk\,dk'.$$
Fixons $\lambda$ que l’on suppose provisoirement en position générale. Introduisons les représentations $\rho_{w\lambda}=Ind_{w(S)\cap L}^L((w\sigma)_{w\lambda})$ et $\rho'_{w' \lambda'}=Ind_{w'(S')\cap L}^L((w'\sigma')_{w'\lambda'})$ de $L(F)$, que l’on réalise dans les espaces $V^L_{w\sigma,w(S)\cap L}$ et $V^L_{w'\sigma',w'(S')\cap L}$. Pour $k,k'\in K$, introduisons des éléments $u_{w}(k',\lambda),v_{w}(k,\lambda)\in V^L_{w\sigma,w(S)\cap L}$ et $u'_{w'}(k,\lambda'),v'_{w'}(k',\lambda')\in V^L_{w'\sigma',w'(S')\cap L}$ définis par $$u_{w}(k',\lambda)=(R_{\underline{Q}_{w}\vert w(S)}((w\sigma)_{w\lambda})\circ\gamma(w)\circ\pi_{\lambda}(k')u)_{K\cap L(F)},$$ $$v_{w}(k,\lambda)=(R_{Q_{w}\vert w(S)}((w\sigma)_{w\lambda})\circ\gamma(w)\circ\pi_{\lambda}(k)v)_{K\cap L(F)},$$ $$u'_{w'}(k,\lambda')=(R_{Q_{w'}\vert w'(S')}((w'\sigma')_{w'\lambda'})\circ\gamma(w')\circ\pi'_{\lambda'}(k)u')_{K\cap L(F)},$$ $$v'_{w'}(k',\lambda')=(R_{\underline{Q}_{w'}\vert w'(S')}((w'\sigma')_{w'\lambda'})\circ\gamma(w')\circ\pi'_{\lambda'}(k')v')_{K\cap L(F)}.$$ Les termes entre parenthèses de ces expressions sont des éléments de $V_{w\sigma,\underline{Q}_{w}}$, resp. $V_{w\sigma,Q_{w}}$, $V_{w'\sigma',Q_{w'}}$, $V_{w'\sigma',\underline{Q}_{w'}}$. Par définition de ces espaces, ce sont des fonctions sur $K$. L’indice final $K\cap L(F)$ signifie que l’on prend leurs restrictions à $K\cap L(F)$. On obtient des éléments des espaces indiqués. Avec ces notations, la définition de $\omega^T_{Q,w,w'}(X,\lambda)$ se récrit $$\omega^T_{Q,w,w'}(X,\lambda)={\bf r}_{w,w'}(\sigma_{\lambda})\int_{K\times K}\int_{A_{\tilde{G}}(F)\backslash A_{\tilde{G}}(F)L(F;X)}(v_{w}(k,\lambda),\rho_{w\lambda}(l)u_{w}(k',\lambda))$$ $$( \rho'_{w'\lambda'}(l)v'_{w'}(k',\lambda'),u'_{w'}(k,\lambda'))\omega(l)\kappa^{L,T}(l)\,dl\,\omega(k'k^{-1})\,dk\,dk'.$$ Notons que le $\delta_{Q}(l)$ disparaît dans la transition entre induites pour $G(F)$ et induites pour $L(F)$. L’intégrale intérieure est de la forme de celles considérées en 1.14, le groupe ambiant $G$ de ce paragraphe étant remplacé par $L$ et le tore $D$ étant $A_{\tilde{G}}$. On déduit des constructions de 1.14 une valeur approchée de $\omega^T_{Q,w,w'}(X,\lambda)$, notons-la $r^T_{Q,w,w'}(X,\lambda)$. On l’étudiera plus loin. Le théorème 1.14 entraîne l’existence d’un réel $c>0$ et d’une fonction lisse et à croissance modérée $C$ sur $i{\cal A}_{M_{disc},F}^*$ de sorte que l’on ait la majoration $$\vert \omega^T_{Q,w,w'}(X,\lambda)-r^T_{Q,w,w'}(X,\lambda)\vert \leq {\bf r}_{w,w'}(\sigma_{\lambda})m^L((w\sigma)_{w\lambda})^{-1/2}m^L((w'\sigma')_{w'\lambda'})^{-1/2}C(\lambda)e^{-c\vert T\vert }.$$ Définissons au moins formellement $$r^T_{Q,w,w'}(X)=\int_{i{\cal A}_{M_{disc},F}^*}B(\lambda)m^G(\sigma_{\lambda})r^T_{Q,w,w'}(X,\lambda)\,d\lambda,$$ $$R^{T,\zeta}_{Q,R,w,w'}=\int_{{\cal A}_{L,F}/{\cal A}_{A_{\tilde{G}},F}}{\bf 1}_{\zeta T}(X^{\tilde{G}})\tilde{\sigma}_{Q}^R(X-T) r^T_{Q,w,w'}(X)\,dX.$$ On obtient $$\vert E^{T,\zeta}_{Q,R,w,w'}-R^{T,\zeta}_{Q,R,w,w'}\vert \leq e^{-c\vert T\vert }\int_{{\cal A}_{L,F}/{\cal A}_{A_{\tilde{G}},F}}{\bf 1}_{\zeta T}(X^{\tilde{G}})\tilde{\sigma}_{Q}^R(X-T)\,dX$$ $$\int_{i{\cal A}_{M_{disc},F}^*}\vert B(\lambda)m^G(\sigma_{\lambda}){\bf r}_{w,w'}(\sigma_{\lambda})m^L((w\sigma)_{w\lambda})^{-1/2}m^L((w'\sigma')_{w'\lambda'})^{-1/2}C(\lambda)\vert \,d\lambda.$$ Ces calculs sont justifiés par le résultat suivant:
\(1) le membre de droite de l’expression ci-dessus est convergent; il est essentiellement majoré par $\vert T\vert ^{-r}$ pour tout réel $r$.
A cause de la fonction ${\bf 1}_{\zeta T}(X^{\tilde{G}})$, l’intégrale en $X$ est convergente et essentiellement majorée par $\vert T\vert ^D$ pour un certain entier $D$. Il suffit donc de prouver la convergence de l’intégrale en $\lambda$. Puisque $B$ est de Schwartz, celle-ci résulte de:
\(2) $\vert m^G(\sigma_{\lambda}){\bf r}_{w,w'}(\sigma_{\lambda})m^L((w\sigma)_{w\lambda})^{-1/2}m^L((w'\sigma')_{w'\lambda'})^{-1/2}\vert =1$.
On rappelle que $${\bf r}_{w,w'}(\sigma_{\lambda})=r_{ \underline{Q}_{w}\vert Q_{w}}((w\sigma)_{w\lambda})r_{Q_{w'}\vert \underline{Q}_{w'}}((w'\sigma')_{w'\lambda'}).$$ On a l’égalité $$r_{ \overline{Q_{w}}\vert Q_{w}}((w\sigma)_{w\lambda})=r_{\overline{Q_{w}}\vert \underline{Q}_{w}}((w\sigma)_{w\lambda})r_{\underline{Q}_{w}\vert Q_{w}}((w\sigma)_{w\lambda}).$$ D’après la définition de $\underline{Q}_{w}$ et les propriétés usuelles des facteurs de normalisation, on a $$r_{\overline{Q_{w}}\vert \underline{Q}_{w}}((w\sigma)_{w\lambda})=r^L_{ (\overline{w(S)\cap L})\vert (w(S)\cap L)}((w\sigma)_{w\lambda}).$$ On utilise 1.10(4) dans $G$ et dans $L$. Parce que $d(\sigma)=d(w\sigma)$ et $m^G(\sigma_{\lambda})=m^G((w\sigma)_{w\lambda})$, on en déduit $$\vert r_{ \underline{Q}_{w}\vert Q_{w}}((w\sigma)_{w\lambda})\vert =m^G(\sigma_{\lambda})^{-1/2}m^L((w\sigma)_{w\lambda})^{1/2}.$$ De même, on a $$\vert r_{Q_{w'}\vert \underline{Q}_{w'}}((w'\sigma')_{w'\lambda'})\vert =m^G(\sigma_{\lambda})^{-1/2}m^L((w'\sigma')_{w'\lambda'})^{1/2}.$$ L’assertion (2) en résulte, puis (1).
Pour démontrer le lemme, il nous reste à prouver l’égalité $$(3) \qquad R^{T,\zeta}_{Q,R,w,w'}=mes(A_{\tilde{G}}(F)_{c})^{-1}{\bf E}^{T,\zeta}_{Q,R,w,w'}.$$ Pour cela, il faut revenir à la définition du membre de gauche, et d’abord du terme $r^T_{Q,w,w'}(X,\lambda)$. On suppose $\lambda$ en position générale et on dédouble la variable $\lambda$. Plus précisément, on conserve inchangés les termes où intervient la variable $\lambda'=\theta^{-1}\lambda$ et, dans les autres termes, on remplace $\lambda$ par $\mu\in i{\cal A}_{M_{disc},F}^*$. En particulier, on effectue cette substitution dans la définition de ${\bf r}_{w,w'}(\sigma_{\lambda})$, qui devient un terme dépendant de $\lambda'$ et $\mu$, notons-le ${\bf r}_{w,w'}(\lambda',\mu)$. D’après les définitions ci-dessus, d’après celles de 1.14 et la formule (2) de ce paragraphe, $r^T_{Q,w,w'}(X,\lambda)$ est la valeur en $\mu=\lambda$ d’une expression composée de:
- une intégrale sur $(k,k')\in K\times K$;
- une somme sur les sous-groupes paraboliques $S_{1}=M_{1}U_{1}$ tels que $P_{0}\subset S_{1}\subset Q$;
- une somme sur $s\in W^L(M_{1}\vert w(M_{disc}))$, $s'\in W^L(M_{1}\vert w'(M'_{disc}))$;
- une somme sur $\nu\in [s'w'\sigma',\omega sw\sigma]$;
le terme que l’on somme est
$$(4)\qquad C {\bf r}_{w,w'}(\lambda',\mu)\omega(k'k^{-1}) d(\sigma)^{-1}\epsilon_{S_{1}}^{Q,T}(X;sw\mu-s'w'\lambda'+\nu)$$ $$(J^L_{(\bar{S}_{1}\cap L)\vert (s'w'(S')\cap L)}((s'w'\sigma')_{s'w'\lambda'})\circ \gamma(s')v'_{w'}(k',\lambda'),\underline{\omega}\circ A_{\nu}\circ J^L_{(\bar{S}_{1}\cap L)\vert (sw(S)\cap L)}((sw\sigma)_{sw\mu})\circ\gamma(s)u_{w}(k',\mu) )$$ $$( \underline{\omega}\circ A_{\nu}\circ J^L_{(S_{1}\cap L)\vert (sw(S)\cap L)}((sw\sigma)_{sw\mu})\circ\gamma(s)v_{w}(k,\mu), J^L_{(S_{1}\cap L)\vert (s'w'(S')\cap L)}((s'w'\sigma')_{s'w'\lambda'})\circ\gamma(s')u'_{w'}(k,\lambda')) ,$$ où $$C=mes(i{\cal A}_{M_{disc},F}^*)^{-1}mes(A_{\tilde{G}}(F)_{c})^{-1}.$$
On fait disparaître le terme ${\bf r}_{w,w'}( \lambda',\mu)$ en rétablissant les opérateurs d’entrelacement non normalisés dans les définitions des éléments $u_{w}(k',\mu)$, etc... Notons ${\bf u}_{w}(k',\mu)$ etc... les éléments définis à l’aide des opérateurs non normalisés. La propriété usuelle de compatibilité des opérateurs d’entrelacement à l’induction conduit à l’égalité $$J^L_{(\bar{S}_{1}\cap L)\vert (sw(S)\cap L)}((sw\sigma)_{sw\mu})\circ\gamma(s){\bf u}_{w}(k',\mu)=$$ $$(J_{\bar{S}_{1}\vert s(\underline{Q}_{w})}((sw\sigma)_{sw\mu})\circ\gamma(s) \circ J_{\underline{Q}_{w}\vert w(S)}((w\sigma)_{w\mu})\circ\gamma(w)\circ\pi_{\mu}(k')u)_{K\cap L(F)}.$$ D’après les définitions, la distance entre les deux sous-groupes paraboliques $\bar{S}_{1}$ et $sw(S)$ est la somme des distances entre $\bar{S}_{1}$ et $s(\underline{Q}_{w})$ et entre $s(\underline{Q}_{w})$ et $sw(S)$. Par composition des opérateurs d’entrelacement, on obtient $$J^L_{(\bar{S}_{1}\cap L)\vert (sw(S)\cap L)}((sw\sigma)_{sw\mu})\circ\gamma(s){\bf u}_{w}(k',\mu)=(J_{\bar{S}_{1}\vert sw(S)}((sw\sigma)_{sw\mu})\circ\gamma(sw)\circ\pi_{\mu}(k')u)_{K\cap L(F)},$$ puis
$\underline{\omega}\circ A_{\nu}\circ J^L_{(\bar{S}_{1}\cap L)\vert (sw(S)\cap L)}((sw\sigma)_{sw\mu})\circ\gamma(s){\bf u}_{w}(k',\mu)=$ $$(\underline{\omega}\circ A_{\nu}\circ J_{\bar{S}_{1}\vert sw(S)}((sw\sigma)_{sw \mu})\circ\gamma(sw)\circ\pi_{\mu}(k')u)_{K\cap L(F)}.$$ Il y a là un abus d’écriture: le premier $\underline{\omega}$ porte sur des fonctions sur $K\cap L(F)$, le second sur des fonctions sur $K$. De même $$J^L_{(\bar{S}_{1}\cap L)\vert (s'w'(S')\cap L)}((s'w'\sigma')_{s'w'\lambda'})\circ \gamma(s'){\bf v}'_{w'}(k',\lambda')=$$ $$( J_{\bar{S}_{1}\vert s'w'(S')}((s'w'\sigma')_{s'w'\lambda'})\circ\gamma(s'w')\circ\pi'_{\lambda'}(k')v')_{K\cap L(F)}$$ Le premier produit scalaire de l’expression (4) (transformé comme on l’a dit ci-dessus) s’écrit donc $$\int_{K\cap L(F)}(( J_{\bar{S}_{1}\vert s'w'(S')}((s'w'\sigma')_{s'w'\lambda'})\circ\gamma(s'w')\circ\pi'_{\lambda'}(k')v')(h),$$ $$(\underline{\omega}\circ A_{\nu}\circ J_{\bar{S}_{1}\vert sw(S)}((sw\sigma)_{sw\mu})\circ\gamma(sw)\circ\pi_{\mu}(k')u)(h))\,dh.$$ On peut commuter l’opérateur $\pi'_{\lambda'}(k')$ aux opérateurs qui le précèdent et remplacer le premier terme par $$( J_{\bar{S}_{1}\vert s'w'(S')}((s'w'\sigma')_{s'w'\lambda'})\circ\gamma(s'w')v')(hk').$$ On peut commuter l’opérateur $\pi_{\mu}(k')$ aux opérateurs qui le précèdent, mais, d’après la définition de $\underline{\omega}$, la commutation à cet opérateur fait sortir un terme $\omega(k')^{-1}$. On peut donc remplacer le deuxième terme par $$\omega(k')^{-1}(\underline{\omega}\circ A_{\nu}\circ J_{\bar{S}_{1}\vert sw(S)}((sw\sigma)_{sw\mu})\circ\gamma(sw)u)(hk').$$ On se rappelle que dans (4) figure une multiplication par $\omega(k')$ et que l’on doit intégrer tout cela en $k'\in K$. Après ces opérations, le premier produit scalaire de (4) devient plus simplement le produit scalaire $$(J_{\bar{S}_{1}\vert s'w'(S')}((s'w'\sigma')_{s'w'\lambda'})\circ\gamma(s'w')v' ,\underline{\omega}\circ A_{\nu}\circ J_{\bar{S}'\vert sw(S)}((sw\sigma)_{sw\mu})\circ\gamma(sw)u) ,$$ ou encore $$(A_{\nu}^{-1}\circ \underline{\omega}^{-1}\circ J_{\bar{S}_{1}\vert s'w'(S')}((s'w'\sigma')_{s'w'\lambda'})\circ\gamma(s'w')v' , J_{\bar{S}'\vert sw(S)}((sw\sigma)_{sw\mu})\circ\gamma(sw)u) .$$
On traite de même le second produit scalaire. On obtient que $r^T_{Q,w,w'}(X,\lambda)$ est la valeur en $\mu=\lambda$ d’une expression composée de sommes sur les $S_{1}$, $s$, $s'$, $\nu$ de termes $$(5)\qquad Cd(\sigma)^{-1}\epsilon_{S_{1}}^{Q,T}(X;sw\mu-s'w'\lambda'+\nu)$$ $$(A_{\nu}^{-1}\circ\underline{\omega}^{-1}\circ J_{\bar{S}_{1}\vert s'w'(S')}((s'w'\sigma')_{s'w'\lambda'})\circ\gamma(s'w')v' , J_{\bar{S}'\vert sw(S)}((sw\sigma)_{sw\mu})\circ\gamma(sw)u)$$
$$( J_{S_{1}\vert sw(S)}((sw\sigma)_{sw\mu})\circ\gamma(sw)v,A_{\nu}^{-1}\circ\underline{\omega}^{-1}\circ J_{S_{1}\vert s'w'(S')}((s'w'\sigma')_{s'w'\lambda'})\circ\gamma(s'w')u').$$ L’ensemble de sommation $[s'w'\sigma';\omega sw\sigma]$ est celui des $\nu\in i{\cal A}_{M_{1},F}^{*}$ tels que $s'w'\sigma'\simeq \omega(sw\sigma)_{-\nu}$. Posons $t=w^{-1}s^{-1}s'w'$. C’est un élément de $W^G(M_{disc}\vert M'_{disc})$. L’application $$\begin{array}{ccc}[t\sigma';\omega \sigma]&\to&[s'w'\sigma';\omega sw\sigma]\\ \nu&\mapsto &sw\nu\\ \end{array}$$ est bijective. Plus précisément, remarquons que toutes les représentations $\sigma$, $t\sigma'$, $sw\sigma$ et $s'w'\sigma'$ se réalisent naturellement dans le même espace $V_{\sigma}$. Soit $\nu\in [t \sigma';\omega \sigma]$, introduisons un automorphisme unitaire $A_{\nu}$ de $V_{\sigma}$ tel que $(t \sigma')(x)\circ A_{\nu}=\omega(x)A_{\nu}\circ \sigma_{-\nu}(x)$ pour tout $x\in M_{disc}(F)$. Alors $A_{\nu}$ vérifie aussi $ s'w'\sigma'(x)\circ A_{\nu}=\omega(x) A_{\nu}\circ(sw\sigma)_{-sw\nu}(x)$ pour tout $x\in M_{1}(F)$. On peut donc remplacer l’ensemble de sommation $[s'w'\sigma';\omega sw\sigma]$ par $[t \sigma';\omega \sigma]$, en remplaçant $\nu$ par $sw\nu$, tout en prenant $A_{sw\nu}=A_{\nu}$. Rappelons que les opérateurs intervenant dans (5) sont en fait déduits par fonctorialité de ces opérateurs $A_{\nu}$. On vérifie l’égalité $$A_{\nu}^{-1}\circ\underline{\omega}^{-1}\circ J_{\bar{S}_{1}\vert s'w'(S')}((s'w'\sigma')_{s'w'\lambda'})\circ\gamma(s'w')=J_{\bar{S}_{1}\vert s'w'(S')}((sw\sigma)_{s'w'\lambda'-sw\nu})\circ\gamma(s'w')\circ A_{\nu}^{-1}\circ\underline{\omega}^{-1}.$$
Comme en 3.16, on introduit l’opérateur $$A(t,\nu;\Lambda)=R_{S\vert t(S')}(\sigma_{\Lambda})\circ\gamma(t)\circ A_{\nu}^{-1}\circ\underline{\omega}^{-1}:Ind_{S'}^G(\sigma'_{t^{-1}(\Lambda+\nu)})\to Ind_{S}^G(\sigma_{\Lambda}).$$ On calcule $$J_{\bar{S}_{1}\vert s'w'(S')}((sw\sigma)_{s'w'\lambda'-sw\nu})\circ\gamma(s'w')\circ A_{\nu}^{-1}\circ\underline{\omega}^{-1}=r_{\bar{S}_{1}\vert s'w'(S')}((sw\sigma)_{s'w'\lambda'-sw\nu})$$ $$R_{\bar{S}_{1}\vert s'w'(S')}((sw\sigma)_{s'w'\lambda'-sw\nu})\circ\gamma(s'w')\circ A_{\nu}^{-1}\circ\underline{\omega}^{-1}$$ $$=r_{\bar{S}_{1}\vert s'w'(S')}((sw\sigma)_{s'w'\lambda'-sw\nu})R_{\bar{S}_{1}\vert sw(S)}((sw\sigma)_{s'w'\lambda'-sw\nu})\circ \gamma(sw)\circ A(t,\nu;t\lambda'-\nu)$$ $$=r_{\bar{S}_{1}\vert s'w'(S')}((sw\sigma)_{s'w'\lambda'-sw\nu})R_{ sw(S)\vert \bar{S}_{1}}((sw\sigma)_{s'w'\lambda'-sw\nu})^{-1}\circ \gamma(sw)\circ A(t,\nu;t\lambda'-\nu)$$ $$=r_{\bar{S}_{1}\vert s'w'(S')}((sw\sigma)_{s'w'\lambda''-sw\nu})r_{ sw(S)\vert \bar{S}_{1}}((sw\sigma)_{s'w'\lambda'-sw\nu})$$ $$J_{ sw(S)\vert \bar{S}_{1}}((sw\sigma)_{s'w'\lambda'-sw\nu})^{-1}\circ \gamma(sw)\circ A(t,\nu;t\lambda'-\nu).$$ La deuxième égalité nécessite que nos relèvements vérifient $t=w^{-1}s^{-1}s'w$, ce que l’on peut supposer. Un calcul analogue vaut si l’on remplace $\bar{S}'$ par $S'$. En utilisant ensuite les propriétés d’adjonction des opérateurs d’entrelacement, on voit que $r^T_{Q,w,w'}(X,\lambda)$ est la valeur en $\mu=\lambda$ d’une expression composée de sommes sur les $S_{1}$, $s$, $s'$ comme précédemment et sur $\nu\in [t\sigma',\omega \sigma]$ de termes $$(6)\qquad C\epsilon_{S_{1}}^{Q,T}(X;sw\mu-s'w'\lambda'+sw\nu)r(S_{1},s,s';s'w'\lambda'-sw\nu)$$ $$(A(t,\nu;t\lambda'-\nu)v',\gamma(w^{-1}s^{-1})\circ J_{ \bar{S}_{1}\vert sw(S)}((sw\sigma)_{s'w'\lambda'-sw\nu})^{-1}\circ J_{\bar{S}_{1}\vert sw(S)}((sw\sigma)_{sw\mu})\circ\gamma(sw)u )$$ $$( \gamma(w^{-1}s^{-1})\circ J_{S_{1}\vert sw(S)}((sw\sigma)_{s'w'\lambda'-sw\nu})^{-1}\circ J_{S'\vert sw(S)}((sw\sigma)_{sw\mu})\circ\gamma(sw)v,A(t,\nu,t\lambda'-\nu)u'),$$ où, pour $\xi\in i{\cal A}_{M_{S'},F}^*$, on a posé $$r(S_{1},s,s';\xi)=d(\sigma)^{-1}r_{S_{1}\vert s'w'S'}((sw\sigma)_{\xi})r_{ sw(S)\vert S_{1}}((sw\sigma)_{\xi})\overline{r_{\bar{S}_{1}\vert s'w'S'}((sw\sigma)_{\xi})}\overline{r_{sw(S)\vert \bar{S}_{1}}((sw\sigma)_{\xi})}.$$ En utilisant 1.10(3), on obtient $$r(S_{1},s,s';\xi)=d(\sigma)^{-1}r_{S_{1}\vert \bar{S}_{1}}((sw\sigma)_{\xi})r_{\bar{S}_{1}\vert S_{1}}((sw\sigma)_{\xi}).$$ Puisque $d(\sigma)=d(sw\sigma)$, ceci n’est autre que $m^G((sw\sigma)_{\xi})^{-1}$, lui-même égal à $m^G(\sigma_{w^{-1}s^{-1}\xi})$. Pour $\xi=s'w'\lambda'-sw\nu$, on a $\sigma_{w^{-1}s^{-1}\mu}=\sigma_{t\lambda'-\nu}\simeq \omega^{-1}(t\sigma')_{t\lambda'}=\omega^{-1} t((\sigma_{\lambda})\circ\theta)$. La mesure de Plancherel est invariante par automorphisme et on vérifie facilement qu’elle est aussi invariante par torsion par un caractère unitaire. On obtient $r(S_{1},s,s';s'w'\lambda'-sw\nu)=m^G(\sigma_{\lambda})^{-1}$.
On peut remplacer la sommation sur $s'$ par une sommation sur $t=w^{-1}s^{-1}s'w'$. Cet élément décrit l’ensemble $W^G(M_{disc}\vert M'_{disc})\cap w^{-1}W^Lw'$, en identifiant pour simplifier le premier ensemble à un ensemble de représentants dans $W^G$. On peut remplacer la somme en $S_{1}$ et $s$ par une somme sur $S''=w^{-1}s^{-1}(S_{1})$. Ce parabolique décrit l’ensemble ${\cal P}^{w^{-1}(Q)}(M_{disc})$ des $S''\in {\cal P}(M_{disc})$ tels que $S''\subset w^{-1}(Q)$. On a les égalités $$\gamma(w^{-1}s^{-1})\circ J_{ \bar{S}_{1}\vert sw(S)}((sw\sigma)_{s'w'\lambda'-sw\nu})^{-1}\circ J_{\bar{S}_{1}\vert sw(S)}((sw\sigma)_{sw\mu})\circ\gamma(sw)=$$ $$J_{ \bar{S}''\vert S}(\sigma_{t\lambda'-\nu})^{-1}\circ J_{\bar{S}''\vert S}(\sigma_{\mu}),$$ $$\gamma(w^{-1}s^{-1})\circ J_{ S_{1}\vert sw(S)}((sw\sigma)_{s'w'\lambda'-sw\nu})^{-1}\circ J_{S_{1}\vert sw(S)}((sw\sigma)_{sw\mu})\circ\gamma(sw)=$$ $$J_{ S''\vert S}(\sigma_{t\lambda'-\nu})^{-1}\circ J_{S''\vert S}(\sigma_{\mu}),$$ $$\epsilon_{S_{1}}^{Q,T}(X,sw\mu-s'w'\lambda'+sw\nu)=\epsilon_{S''}^{w^{-1}(Q),T[S'']}(w^{-1}X,\mu-t\lambda'+\nu).$$ En utilisant les définitions de 3.16, l’expression (6) se récrit $$C m^G(\sigma_{\lambda})^{-1}\epsilon_{S''}^{w^{-1}(Q),T[S'']}(w^{-1}X,\mu-t\lambda'+\nu)\phi(t,\nu;\lambda,\mu-t\lambda'+\nu,S'').$$ En sommant sur $S''$, on obtient $$C m^G(\sigma_{\lambda})^{-1}\phi_{M_{disc}}^{w^{-1}(Q),T}(t,\nu,w^{-1}(X);\lambda,\mu-t\lambda'+\nu).$$ A ce point, le lemme 3.16 nous autorise à égaler $\mu$ à $\lambda$. On obtient finalement $$r^T_{Q,w,w'}(X,\lambda)=C m^G(\sigma_{\lambda})^{-1}\sum_{t\in W^G(M_{disc}\vert M'_{disc})\cap w^{-1}W^Lw'}$$ $$\sum_{\nu\in [t\sigma',\omega \sigma]}\phi_{M_{disc}}^{w^{-1}(Q),T}(t,\nu,w^{-1}(X);\lambda,\lambda-t\lambda'+\nu).$$ En appliquant les définitions, on obtient ensuite $$R^{T,\zeta}_{Q,R,w,w'}=C \sum_{t\in W^G(M_{disc}\vert M'_{disc})\cap w^{-1}W^Lw'}\sum_{\nu\in [t\sigma',\omega \sigma]}
\int_{{\cal A}_{L,F}/{\cal A}_{A_{\tilde{G}},F}}{\bf 1}_{\zeta T}(X^{\tilde{G}})\tilde{\sigma}_{Q}^R(X-T)$$ $$\int_{i{\cal A}_{M_{disc},F}^*}B(\lambda)\phi_{M_{disc}}^{w^{-1}(Q),T}(t,\nu,w^{-1}(X);\lambda,\lambda-t\lambda'+\nu)\,d\lambda\,dX.$$ On a $\tilde{\sigma}_{Q}^R(X-T)=\tilde{\sigma}_{w^{-1}(Q)}^{w^{-1}(R)}(w^{-1}X-T[w^{-1}(Q)])$. Par le changement de variables $X\mapsto wX$, la double intégrale multipliée par $C$ devient $mes(A_{\tilde{G}}(F)_{c})^{-1}{\bf E}^T_{w^{-1}(Q),w^{-1}(R),t,\nu}$ et on obtient l’égalité cherchée $$R^{T,\zeta}_{Q,R,w,w'}= mes(A_{\tilde{G}}(F)_{c})^{-1}E^T_{\star;Q,R,w,w'}.$$ Cela achève la preuve. $\square$
Une nouvelle expression approchant $j^T$
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Pour $Q=LU_{Q}\in {\cal F}(M_{disc})$ et $t\in W^G(M_{disc}\vert M'_{disc})$, définissons une fonction $H\mapsto S_{Q}(t;H)$ sur ${\cal A}_{0}$ par $$S_{Q}(t;H)=\sum_{R; Q\subset R}s_{Q}^R(t)\tilde{\sigma}_{Q}^R(H).$$ En dévissant la définition de $s_{Q}^R(t)$, on a aussi $$S_{Q}(t;H)=\sum_{\tilde{P}=\tilde{M}U_{P}; Q\subset P, t\theta^{-1}(P)=P}(-1)^{dim(\mathfrak{a}_{\tilde{M}})-dim(\mathfrak{a}_{\tilde{G}})}\sum_{R; P\subset R}\tilde{\sigma}_{Q}^R(H),$$ ou encore, en utilisant 2.2(1), $$S_{Q}(t;H)=\sum_{\tilde{P}=\tilde{M}U_{P}; Q\subset P,t\theta^{-1}(P)=P}(-1)^{dim(\mathfrak{a}_{\tilde{M}})-dim(\mathfrak{a}_{\tilde{G}})}\tau_{Q}^P(H)\hat{\tau}_{\tilde{P}} (H).$$
Pour $\nu\in [t\sigma',\omega \sigma]$, posons $${\bf E}_{t,\nu}^T=mes(i{\cal A}_{M_{disc},F}^*)^{-1}\sum_{Q=LU_{Q}\in {\cal F}(M_{disc})}\int_{{\cal A}_{L,F}/{\cal A}_{A_{\tilde{G}},F}}S_Q(t;X-T[Q])$$ $$\int_{i{\cal A}_{M_{disc},F}^*}B(\lambda)\phi_{M_{disc}}^{Q,T}(t,\nu,X;\lambda;\lambda-t\lambda'+\nu)\,d\lambda\,dX.$$ Ceci n’est autre que $$\sum_{Q,R\in {\cal F}(M_{disc}); Q\subset R}s_{Q}^R(t){\bf E}_{Q,R,t,\nu}^T$$ qui est une expression convergente d’après le lemme 3.18. Posons $${\bf E}^T= \sum_{t\in W(M_{disc}\vert M'_{disc})}\sum_{\nu\in [t\sigma',\omega \sigma]}{\bf E}_{t,\nu}^T.$$
[0.3cm[**[Lemme]{}**]{}. [ *[On a la majoration $$\vert j^T-mes(A_{\tilde{G}}(F)_{c})^{-1}{\bf E}^T\vert <<\vert T\vert ^{-r}$$ pour tout réel $r$.]{}*]{}0.3cm]{}
Preuve. La proposition 3.4 nous dit que l’on peut aussi bien remplacer $j^T$ par $j^T_{\star}$, c’est-à-dire par $$\sum_{Q=LU_{Q},R; P_{0}\subset Q\subset R}\sum_{w\in W^G(L\vert S),w'\in W^G(L\vert S')}s_{Q}^R(w,w')E^T_{Q,R,w,w'}.$$ Le lemme 3.18 nous dit que l’on peut aussi bien remplacer $E^T_{Q,R,w,w'}$ par $mes(A_{\tilde{G}}(F)_{c})^{-1}E^T_{\star;Q,R,w,w'}$. L’expression ci-dessus est alors remplacée par $$mes(A_{\tilde{G}}(F)_{c})^{-1}\sum_{Q=LU_{Q},R; P_{0}\subset Q\subset R}\sum_{w\in W^G(L\vert S),w'\in W^G(L\vert S')}s_{Q}^R(w,w')$$ $$\sum_{t\in W^G(M_{disc}\vert M'_{disc})\cap w^{-1}W^Lw'}\sum_{\nu\in [t\sigma',\omega \sigma]}{\bf E}^T_{w^{-1}(Q),w^{-1}(R),t,\nu}.$$ Comme on l’a remarqué en 3.18, pour $Q,...,t$ intervenant ci-dessus, on a l’égalité $s_{Q}^R(w,w')=s_{w^{-1}(Q)}^{w^{-1}(R)}(t)$. On peut donc récrire l’expression précédente sous la forme $$mes(A_{\tilde{G}}(F)_{c})^{-1}\sum_{t\in W(M_{disc}\vert M'_{disc})}\sum_{\nu\in [t\sigma',\omega \sigma]}E^T_{\star;t,\nu},$$ où $$E^T_{\star;t,\nu}=\sum_{Q=LU_{Q},R; P_{0}\subset Q\subset R}\sum_{w\in W^G(L\vert S),w'\in W^G(L\vert S'); t\in w^{-1}W^Lw'}s_{w^{-1}(Q)}^{w^{-1}(R)}(t){\bf E}^T_{w^{-1}(Q),w^{-1}(R),t,\nu}.$$ Fixons $t$ et $\nu$. Pour tous $Q,R$ et tout $w\in W^G(L\vert S)$ il y a un et un seul $w'\in W^G(L\vert S')$ tel que $t\in w^{-1}W^Lw'$: en identifiant tous ces éléments à des relèvements dans $W^G$, $w'$ est l’élément de longueur minimale dans la classe $W^Lwt$. On peut donc supprimer la somme en $w'$ et la condition $t\in w^{-1}W^Lw'$. L’application $$\begin{array}{ccc}\{Q=LU_{Q},R,w; P_{0}\subset Q\subset R, w\in W^G(L\vert S)\}&\to& \{Q',R'; M_{disc}\subset Q'\subset R'\}\\ (Q,R,w)&\mapsto& (w^{-1}(Q),w^{-1}(R))\\ \end{array}$$ est bijective. On voit alors que $E^T_{\star;t,\nu}={\bf E}^T_{t,\nu}$. Cela démontre le lemme. $\square$
Dans toutes nos expressions intervient la fonction $B$ fixée en 3.4. Il peut être utile de préciser la dépendance en $B$ de la majoration du lemme. Pour cela, notons pour quelques instants $j^T(B)$ et ${\bf E}^T(B)$ les termes notés précédemment $j^T$ et ${\bf E}^T$. Introduisons un ensemble ${\cal N}$ de semi-normes sur l’espace des fonctions de Schwartz sur $i{\cal A}_{M_{disc},F}^*$. Dans le cas où $F$ est non-archimédien, ces semi-normes sont $$B\mapsto sup\{ \vert (XB)(\lambda)\vert ;\lambda\in i{\cal A}_{M_{disc},F}^*\},$$ où $X$ parcourt les opérateurs différentiels à coefficients constants sur $i{\cal A}_{M_{disc}}^*$. Dans le cas où $F$ est archimédien, ce sont les $$B\mapsto sup\{\vert (XB)(\lambda)\vert (1+\vert \lambda\vert )^N; \lambda\in i{\cal A}_{M_{disc}}^*\},$$ où $X$ parcourt les mêmes opérateurs que ci-dessus et $N$ parcourt les entiers naturels. Le lemme se précise en
\(1) pour tout réel $r\geq1$, il existe $c_{r}>0$ et un sous-ensemble fini ${\cal N}_{r}\subset {\cal N}$ de sorte que l’on ait la majoration $$\vert j^T(B)-mes(A_{\tilde{G}}(F)_{c})^{-1}{\bf E}^T(B)\vert \leq c_{r}sup\{\underline{n}(B); \underline{n}\in {\cal N}_{r}\}\vert T\vert ^{-r}$$ pour tout $T$ et toute fonction de Schwartz $B$ sur $i{\cal A}_{M_{disc},F}^*$.
Il suffit de reprendre patiemment toutes nos majorations. A chaque fois que $B$ y intervient, c’est par l’intermédiaire d’une semi-norme appartenant à l’ensemble ${\cal N}$ et il est clair que, pour $r$ fixé, il n’y a qu’un nombre fini de telles semi-normes qui interviennent.
Un lemme d’inversion de Fourier
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Dans ce paragraphe et jusqu’en 3.23, on fixe un élement $t\in W^G(M_{disc}\vert M'_{disc})$ et un élément $\nu\in [t\sigma',\omega \sigma]$. On définit un Levi $M_{t}$: c’est le plus grand Levi contenant $M_{disc}$ tel que ${\cal A}_{M_{t}}$ contienne l’ensemble $$\{H\in {\cal A}_{M_{disc}}; t\theta^{-1}H=H\}.$$ On pose $\tilde{M}_{t}=M_{t}\gamma_{0}t^{-1}$, où on relève $t$ en un élément de $G$. D’après 2.1(3) et (5), $\tilde{M}_{t}$ est un ensemble de Levi et l’ensemble des espaces paraboliques $\tilde{P}$ tels que $M_{disc}\subset P$ et $t\theta^{-1}(P)=P$ n’est autre que ${\cal F}(\tilde{M}_{t})$.
Considérons l’ensemble des couples $(\lambda,\Lambda)\in i{\cal A}_{M_{disc}}^*\times i{\cal A}_{\tilde{M}_{t}}^*$ tels que $\Lambda-\lambda+t\lambda'-\nu\in i{\cal A}_{M_{disc},F}^{\vee}$. Il est invariant par translation par $(i{\cal A}_{M_{disc},F}^{\vee}+i{\cal A}_{\tilde{M}_{t}}^*)\times i{\cal A}_{\tilde{M}_{t},F}^{\vee}$. Considérons son quotient par l’action de ce groupe. Parce que $1-t\theta^{-1}$ se restreint en un automorphisme de $i{\cal A}_{M_{disc}}^{\tilde{M}_{t},*}$, on voit que ce quotient est fini, réduit à un élément si $F$ est archimédien. On voit aussi que la projection $(\lambda,\Lambda)\to \lambda$ est injective. Autrement dit, si on note $\{\nu\}_{t}$ l’image de cette projection, il existe une application $\lambda\mapsto \Lambda(\lambda)$ de $\{\nu\}_{t}$ dans $i{\cal A}_{\tilde{M}_{t},F}^*$ de sorte que le quotient ci-dessus soit égal à $\{(\lambda,\Lambda(\lambda)); \lambda\in \{\nu\}_{t}\}$.
[0.3cm[**[Lemme]{}**]{}. [ *[Soit $f$ une fonction de Schwartz sur $i{\cal A}_{M_{disc},F}^*$ et soit $X\in {\cal A}_{\tilde{M}_{t},F}$. L’intégrale $$mes(i{\cal A}_{M_{disc},F}^*)^{-1}\int_{{\cal A}_{M_{disc},F}^{\tilde{M}_{t}}(X)}\int_{i{\cal A}_{M_{disc},F}^*}e^{<\lambda-t\lambda'+\nu,Y>}f(\lambda)\,d\lambda\,dY$$ est convergente dans cet ordre. Elle est égale à $$mes(i{\cal A}_{\tilde{M}_{t},F}^*)^{-1}\vert det((1-t\theta^{-1})_{\vert {\cal A}_{M_{disc}}^{\tilde{M}_{t},*}})\vert ^{-1}\sum_{\lambda\in \{\nu\}_{t}}e^{<\Lambda(\lambda),X>}\int_{i{\cal A}_{\tilde{M}_{t},F}^*}f(\lambda+\xi)\,d\xi.$$]{}*]{}0.3cm]{}
Preuve. On suppose $F$ non-archimédien, le cas archimédien étant plus simple. On fixe un relèvement de $\nu$ dans $i{\cal A}_{M_{disc}}^*$, que l’on note encore $\nu$. On fixe $\mu\in i{\cal A}_{M_{disc}}^*$ tel que $\nu^{\tilde{M}_{t}}=t\theta^{-1}\mu-\mu$. On fixe $X'\in {\cal A}_{M_{disc},F}$ tel que $(X')^{\tilde{M}_{t}}=X$. On a ${\cal A}_{M_{disc},F}^{\tilde{M}_{t}}(X)=X'+{\cal A}_{M_{disc},F}^{\tilde{M}_{t}}$ (où ${\cal A}_{M_{disc},F}^{\tilde{M}_{t}}={\cal A}_{M_{disc},F}\cap {\cal A}_{M_{disc}}^{\tilde{M}_{t}}$) et l’intégrale de l’énoncé se récrit $$mes(i{\cal A}_{M_{disc},F}^*)^{-1}\sum_{Y\in{\cal A}_{M_{disc},F}^{\tilde{M}_{t}}}e^{<\nu_{\tilde{M}_{t}},X'>}\int_{i{\cal A}_{M_{disc},F}^*}e^{<(\lambda-\mu)-t\theta^{-1}(\lambda-\mu),Y+X'>}f(\lambda)\,d\lambda.$$ Toutes les intégrations se font sur des groupes compacts. Une intégrale sur $i{\cal A}_{M_{disc},F}^*$ se décompose en le produit d’une intégrale sur $
i{\cal A}_{M_{disc}}^*/(i{\cal A}_{M_{disc},F}^{\vee}+i{\cal A}_{\tilde{M}_{t}}^*)$ et d’une intégrale sur $i{\cal A}_{\tilde{M}_{t},F}^*$ (avec des pondérations provenant de nos choix de mesures). L’expression ci-dessus se transforme en $$(1) \qquad mes(i{\cal A}_{M_{disc},F}^*)^{-1}mes(i{\cal A}_{\tilde{M}_{t},F}^*)\sum_{Y\in {\cal A}_{M_{disc},F}^{\tilde{M}_{t}}}e^{<\nu_{\tilde{M}_{t}},X'>}$$ $$\int_{i{\cal A}_{M_{disc}}^*/(i{\cal A}_{M_{disc},F}^{\vee}+i{\cal A}_{\tilde{M}_{t}}^*)}e^{<(\lambda-\mu)-t\theta^{-1}(\lambda-\mu),Y+X'>} f'(\lambda)\,d\lambda,$$ où $$f'(\lambda)=mes(i{\cal A}_{\tilde{M}_{t},F}^*)^{-1}\int_{i{\cal A}_{\tilde{M}_{t},F}^*}f(\lambda+\xi)\,d\xi.$$ La fonction $f'$ est lisse sur $ i{\cal A}_{M_{disc}}^*/(i{\cal A}_{M_{disc},F}^{\vee}+i{\cal A}_{\tilde{M}_{t}}^*)$. L’accouplement $$\begin{array}{ccc}( i{\cal A}_{M_{disc}}^*/(i{\cal A}_{M_{disc},F}^{\vee}+i{\cal A}_{\tilde{M}_{t}}^*))\times {\cal A}_{M_{disc},F}^{\tilde{M}_{t}}&\to &{\mathbb C}^{\times}\\ (\lambda,Y)&\mapsto&e^{<\lambda-t\theta^{-1}\lambda,Y>}\\ \end{array}$$ n’est pas parfait. Son noyau ${\cal K}$ est l’ensemble des $\lambda\in i{\cal A}_{M_{disc}}^*/(i{\cal A}_{M_{disc},F}^{\vee}+i{\cal A}_{\tilde{M}_{t}}^*)$ tels que $\lambda-t\theta^{-1}\lambda\in i{\cal A}_{M_{disc},F}^{\vee}+i{\cal A}_{\tilde{M}_{t}}^*$. A ce défaut près, la formule (1) est une inversion de Fourier. Elle est donc convergente dans l’ordre indiqué. On la calcule selon la formule usuelle. Elle vaut $$\vert {\cal K}\vert ^{-1}e^{<\nu_{\tilde{M}_{t}},X'>}\sum_{\lambda\in \mu+{\cal K}}e^{<(\lambda-\mu)-t\theta^{-1}(\lambda-\mu),X'>}f'(\lambda).$$ Notons $p:i{\cal A}_{M_{disc}}^*\to i{\cal A}_{M_{disc}}^{\tilde{M}_{t},*}$ la projection orthogonale. Via cette projection, on vérifie que ${\cal K}$ s’identifie au noyau de $1-t\theta^{-1}$ agissant dans $i{\cal A}_{M_{disc}}^{\tilde{M}_{t},*}/p(i{\cal A}_{M_{disc},F}^{\vee})$. Donc $\vert {\cal K}\vert =\vert det((1-t\theta^{-1})_{\vert {\cal A}_{M_{disc}}^{\tilde{M}_{t},*}})\vert $. D’autre part, en comparant les définitions, on voit que $\mu+{\cal K}=\{\nu\}_{t}$. Notre expression vaut donc $$(2) \qquad \vert det((1-t\theta^{-1})_{\vert {\cal A}_{M_{disc}}^{\tilde{M}_{t},*}})\vert ^{-1}e^{<\nu_{\tilde{M}_{t}},X'>}\sum_{\lambda\in \{\nu\}_{t}}e^{<(\lambda-\mu)-t\theta^{-1}(\lambda-\mu),X'>}f'(\lambda).$$ Pour $\lambda\in \{\nu\}_{t}$, on introduit comme ci-dessus $\Lambda(\lambda)\in {\cal A}_{\tilde{M}_{t},F}^*$ tel que $\Lambda(\lambda)-\lambda+t\lambda'-\nu\in i{\cal A}_{M_{disc},F}^{\vee}$, ou encore $\Lambda(\lambda)-\lambda+\mu+t\theta^{-1}(\lambda-\mu)-\nu_{\tilde{M}_{t}}\in i{\cal A}_{M_{disc},F}^{\vee}$. Puisque $X'\in {\cal A}_{M_{disc},F}$, on a $$e^{<(\lambda-\mu)-t\theta^{-1}(\lambda-\mu),X'>}=e^{<\Lambda(\lambda)-\nu_{\tilde{M}_{t}},X'>},$$ d’où $$e^{<\nu_{\tilde{M}_{t}},X'>}e^{<(\lambda-\mu)-t\theta^{-1}(\lambda-\mu),X'>}=e^{<\Lambda(\lambda),X'>}=e^{<\Lambda(\lambda),X>}.$$ L’expression (2) se transforme en celle de l’énoncé. $\square$
Transformation de ${\bf E}^T_{t,\nu}$
-------------------------------------
Dans la définition de ${\bf E}^T_{t,\nu}$ intervient la $(G,M_{disc})$-famille $(\phi(t,\nu;\lambda,\Lambda,S''))_{S''\in {\cal P}(M_{disc})}$ de 3.16. Les fonctions qui composent celle-ci ont le défaut d’avoir des singularités et, quand $F$ est archimédien, de ne pas être de Schwartz en $\Lambda$: elles sont seulement à croissance modérée. Modifions cette famille. On fixe une fonction $C$ sur $i{\cal A}_{M_{disc},F}^*$, qui est $C^{\infty}$ et à support compact et telle que $C(0)=1$. On pose $$\phi_{\star}(t,\nu;\lambda,\Lambda,S'')=r_{\bar{S}\vert S}(\sigma_{t\lambda'-\nu+\Lambda})^{-1}r_{\bar{S}\vert S}(\sigma_{\lambda})C(\Lambda-\lambda'+t\lambda'-\nu)B(\lambda)\phi(t,\nu;\lambda,\Lambda,S'').$$ On a aussi $$\phi_{\star}(t,\nu;\lambda,\Lambda,S'')= \epsilon(t,\nu;\lambda)C(\Lambda-\lambda'+t\lambda'-\nu)B(\lambda)\phi_{reg}(t,\nu;\lambda,\Lambda,S'').$$ Les lemmes 1.6 et 3.16(ii) montrent que $\phi_{\star}(t,\nu;\lambda,\Lambda,S'')$ est $C^{\infty}$ en les deux variables $\lambda,\Lambda\in i{\cal A}_{M_{disc},F}^*$. Les propriétés de $B$ et $C$ assurent qu’elle est de Schwartz en ces deux variables. D’après le lemme 3.16(iii), on a l’égalité $$\phi_{\star,M_{disc}}^{Q,T}(t,\nu,X;\lambda,\lambda-t\lambda'+\nu)=B(\lambda)\phi_{M_{disc}}^{Q,T}(t,\nu,X;\lambda,\lambda-t\lambda'+\nu)$$ pour tout $Q=LU_{Q}\in {\cal F}(M_{disc})$ et tout $X\in {\cal A}_{L,F}$. La définition de ${\bf E}^T_{t,\nu}$ se récrit $${\bf E}^T_{t,\nu}=mes(i{\cal A}_{M_{disc},F}^*)^{-1}\sum_{Q=LU_{Q}\in {\cal F}(M_{disc})}\int_{{\cal A}_{L,F}\backslash {\cal A}_{A_{\tilde{G}},F}}S_Q(t;X-T[Q])$$ $$\int_{i{\cal A}_{M_{disc},F}^*}\phi_{\star,M_{disc}}^{Q,T}(t,\nu,X;\lambda,\lambda-t\lambda'+\nu)\,d\lambda\,dX.$$ En utilisant le lemme 1.8, on peut écrire $$\phi_{\star,M_{disc}}^{Q,T}(t,\nu,X;\lambda,\lambda-t\lambda'+\nu)=\sum_{Q'=L'U_{Q'}; M_{disc}\subset Q'\subset Q}\int_{{\cal A}_{L',F}}\int_{{\cal A}_{M_{disc},F}^L(X+H_{L}) }$$ $$\delta_{M_{disc}}^{Q'}(H')\Gamma_{Q'}^Q(H',H+T[Q'])\hat{\phi}_{\star}(t,\nu;\lambda,H,Q') e^{<\lambda-t\lambda'+\nu,H'>}\,dH'\,dH,$$ où $$\hat{\phi}_{\star}(t,\nu;\lambda,H,Q')=mes(i{\cal A}_{L',F}^*)^{-1}\int_{i{\cal A}_{L',F}^*}\phi_{\star}(t,\nu;\lambda,\Lambda,Q')e^{-<\Lambda,H>}\,d\Lambda.$$ Fixons $Q,Q'\in {\cal F}(M_{disc})$, avec $Q'\subset Q$.
0.3cm[**[Lemme]{}**]{}.
**
\(i) Pour $X\in {\cal A}_{L,F}$, l’intégrale $$\int_{i{\cal A}^*_{M_{disc},F}}\int_{{\cal A}_{L',F}}\int_{{\cal A}_{M_{disc},F}^L(X+H_{L})}$$ $$\vert\delta_{M_{disc}}^{Q'}(H') \Gamma_{Q'}^Q(H',H+T[Q'])\hat{\phi}_{\star}(t,\nu;\lambda,H,Q')\vert \,dH'\,dH\,d\lambda$$ est convergente.
\(ii) L’intégrale $$\int_{{\cal A}_{L,F}\backslash {\cal A}_{A_{\tilde{G}},F}} \int_{{\cal A}_{L',F}}\int_{{\cal A}_{M_{disc},F}^L(X+H_{L})}\left\vert \delta_{M_{disc}}^{Q'}(H')S_Q(t;X-T[Q])\Gamma_{Q'}^Q(H',H+T[Q'])\right.$$ $$\int_{i{\cal A}_{M_{disc},F}^*}\left.\hat{\phi}_{\star}(t,\nu;\lambda,H,Q') e^{<\lambda-t\lambda'+\nu,H'>}\,d\lambda\right\vert \,dH'\,dH\,dX$$ est convergente.
0.3cm
Preuve de (i). Les deux conditions $\delta_{M_{disc}}^{Q'}(H')\Gamma_{Q'}^Q(H',H+T[Q'])=1$ et $H'\in {\cal A}_{M_{disc},F}^L(X+H_{L})$ entraînent une majoration $\vert H'\vert <<1+\vert H\vert$, les éléments $T$ et $X$ étant ici considérés comme constants. L’intégrale en $H'$ est donc convergente et est essentiellement bornée par $(1+\vert H\vert )^D$ pour un certain entier $D$. Puisque $\phi_{\star}(t,\nu;\lambda,\Lambda,Q')$ est de Schwartz en les deux variables $\lambda$ et $\Lambda$, $\hat{\phi}_{\star}(t,\nu;\lambda,H,Q')$ est de Schwartz en les deux variables $\lambda$ et $H$. Alors l’intégrale $$\int_{i{\cal A}^*_{M_{disc},F}}\int_{{\cal A}_{L',F}}(1+\vert H\vert )^D\vert \hat{\phi}_{\star}(t,\nu;t\lambda'-\nu,H,Q')\vert\,dH\,d\lambda$$ est convergente, ce qui prouve (i).
Preuve de (ii). L’intégrale intérieure en $\lambda$ est le produit de $e^{<\nu,H'>}$, qui est de valeur absolue $1$, et de la transformée de Fourier en $\lambda$ de $\hat{\phi}_{\star}(t,\nu;\lambda,H,Q')$, évaluée au point $(\theta t^{-1}-1)H'$. Puisque $\hat{\phi}_{\star}(t,\nu;\lambda,H,Q')$ est de Schwartz en $\lambda$ et $H$, on peut fixer des fonctions de Schwartz $h$ sur ${\cal A}_{L',F}$ et $h'$ sur ${\cal A}_{M_{disc},F}$, à valeurs positives, de sorte que la valeur absolue de cette intégrale intérieure soit majorée par $h'((\theta t^{-1}-1)H')h(H)$. Les conditions $H'\in {\cal A}_{M_{disc},F}^L(X+H_{L})$ et $\delta_{M_{disc}}^{Q'}(H')=1$ entraînent que $H'=X+H_{L}+(H')^L_{L'}$. La condition $\Gamma_{Q'}^Q(H',H+T[Q'])$ implique une majoration $\vert (H')^L_{L'}\vert <<1+\vert H^L\vert $, l’élément $T$ étant considéré comme constant. On en déduit $$(\theta t^{-1}-1)H'=(\theta t^{-1}-1)X+Y(H'),$$ avec $\vert Y(H')\vert <<1+\vert H\vert $. D’autre part, on a une majoration $$1+\vert U+V\vert >>(1+\vert U\vert )(1+\vert V\vert )^{-1}$$ pour tous $U,V\in {\cal A}_{M_{disc}}$. Pour tout réel $r>0$, on a une majoration $$h'((\theta t^{-1}-1)H')<<(1+\vert (\theta t^{-1}-1)H'\vert )^{-r}.$$ En utilisant les majorations précédentes, on obtient $$h'((\theta t^{-1}-1)H')<<(1+\vert (\theta t^{-1}-1)X\vert )^{-r}(1+\vert H\vert )^r.$$ A ce point, on a montré que l’expression du (ii) de l’énoncé était majorée par $$\int_{{\cal A}_{L,F}/ {\cal A}_{A_{\tilde{G}},F}}\int_{{\cal A}_{L',F}}\int_{{\cal A}_{M_{disc},F}^L(X+H_{L})}\vert \delta_{M_{disc}}^{Q'}(H')S_Q(t;X-T[Q])\Gamma_{Q'}^Q(H',H+T[Q'])\vert$$ $$(1+\vert (\theta t^{-1}-1)X\vert )^{-r}(1+\vert H\vert )^rh(H)\,dH'\,dH\,dX.$$ Par des raisonnements déjà faits, l’intégrale en $H'$ est convergente et essentiellement majorée par $(1+\vert H\vert )^D$ pour un certain entier $D$. L’intégrale en $H$ est convergente quel que soit $r$ puisque $h$ est de Schwartz. L’expression ci-dessus est donc essentiellement majorée par $$\int_{{\cal A}_{L,F}/ {\cal A}_{A_{\tilde{G}},F}}\vert S_Q(t;X-T[Q])\vert (1+\vert (\theta t^{-1}-1)X\vert )^{-r} \,dX.$$ Il reste à montrer que l’on peut choisir $r$ tel que cette intégrale soit convergente. En revenant à la définition de $S_{Q}(t;X)$, on voit qu’il suffit de fixer un sous-groupe parabolique $R$ tel que $Q\subset R$ et $s_{Q}^R(t)\not=0$ et de prouver la même assertion pour l’intégrale $$\int_{{\cal A}_{L,F}/ {\cal A}_{A_{\tilde{G}},F}}\tilde{\sigma}_{Q}^R(X-T[Q])(1+\vert (\theta t^{-1}-1)X\vert )^{-r}\,dX.$$ Or il résulte de 3.17(1) que, pour $\tilde{\sigma}_{Q}^R(X-T[Q])=1$, on a une majoration $$1+\vert X^{\tilde{G}}\vert<<1+ \vert (\theta t^{-1}-1)X\vert .$$ L’intégrale ci-dessus est donc essentiellement majorée par $$\int_{{\cal A}_{L,F}/ {\cal A}_{A_{\tilde{G}},F}}(1+\vert X^{\tilde{G}}\vert )^{-r}\,dX.$$ Pour $r$ assez grand, ceci est convergent. $\square$
Ce lemme nous autorise à récrire ${\bf E}_{t,\nu}^T$ sous la forme $$(1) \qquad {\bf E}_{t,\nu}^T=mes(i{\cal A}_{M_{disc},F}^*)^{-1}\sum_{Q=LU_{Q},Q'=L'U_{Q'}\in {\cal F}(M_{disc}); Q'\subset Q}\int_{{\cal A}_{L,F}/{\cal A}_{A_{\tilde{G}},F}}$$ $$\int_{{\cal A}_{L',F}}\int_{{\cal A}_{M_{disc},F}^L(X+H_{L})}\delta_{M_{disc}}^{Q'}(H')S_Q(t;X-T[Q])\Gamma_{Q'}^Q(H',H+T[Q'])$$ $$\int_{i{\cal A}_{M_{disc},F}^*}e^{<\lambda-t\lambda'+\nu,H'>}\hat{\phi}_{\star}(t,\nu;\lambda,H,Q')\,d\lambda\,dH'\,dH\,dX,$$ où on peut permuter librement la somme en $Q$, $Q'$ et les trois premières intégrales.
Calcul de ${\bf E}_{t,\nu}^T$
-----------------------------
On a défini en 3.16 une $(G,M)$-famille $(\phi_{reg}(t,\nu;\lambda,\Lambda,S'))_{S'\in {\cal P}(M_{disc})}$. Il s’en déduit une $(\tilde{G},\tilde{M}_{t})$-famille $(\phi_{reg}(t,\nu;\lambda,\Lambda,\tilde{P}))_{\tilde{P}\in {\cal P}(\tilde{M}_{t})}$, cf. 2.3 . Les éléments $\lambda$ et $\Lambda$ appartiennent ici à $i{\cal A}_{M_{disc},F}^*$ et $i{\cal A}_{\tilde{M}_{t},F}^*$, $\lambda$ jouant un rôle de paramètre. Pour $X\in {\cal A}_{\tilde{G},F}$, on construit la fonction $$\phi^{\tilde{G},T}_{reg,\tilde{M}_{t}}(t,\nu,X;\lambda,\Lambda)=\sum_{\tilde{P}\in {\cal P}(\tilde{M}_{t})}\phi_{reg}(t,\nu;\lambda,\Lambda,\tilde{P})\epsilon_{\tilde{P}}^{\tilde{G},T[\tilde{P}]}(X;\Lambda).$$ C’est une fonction lisse en $\lambda$ et $\Lambda$. Si $F$ est archimédien, toutes ses dérivées sont à croissance modérée.
[0.3cm[**[Proposition]{}**]{}. [ *[On a l’égalité $${\bf E}_{t,\nu}^T=mes(i{\cal A}_{\tilde{M}_{t},F}^*)^{-1}\vert det((1-t\theta^{-1})_{\vert {\cal A}_{M_{disc}}^{\tilde{M}_{t},*}})\vert ^{-1}\sum_{\lambda\in \{\nu\}_{t}}\sum_{X\in {\cal A}_{\tilde{G},F}/{\cal A}_{A_{\tilde{G},F}}}$$ $$\int_{i{\cal A}_{\tilde{M}_{t},F}^*}B(\lambda+\xi)\epsilon(t,\nu;\lambda+\xi)\phi^{\tilde{G},T}_{reg,\tilde{M}_{t}}(t,\nu,X;\lambda+\xi,\Lambda(\lambda))\,d\xi.$$]{}*]{}0.3cm]{}
[**Remarque.**]{} On vérifie que, pour $\lambda$ intervenant ci-desus, on a $e^{<\Lambda(\lambda),Z>}=1$ pour tout $Z\in {\cal A}_{A_{\tilde{G}},F}$. La fonction que l’on somme en $X$ est donc bien invariante par ${\cal A}_{A_{\tilde{G}},F}$.
Preuve. On part de la formule (1) du paragraphe précédent. Permutons les intégrales en $X$ et $H$. Effectuons ensuite le changement de variables $X\mapsto X-H_{L}$. L’intégrale en $H'$ devient une intégrale sur ${\cal A}_{M_{disc},F}^L(X)$. La composition de l’intégrale en $X$ et de cette intégrale en $H'$ devient une unique intégrale en $X\in {\cal A}_{M_{disc},F}/{\cal A}_{A_{\tilde{G}},F}$. Cela conduit à l’égalité $${\bf E}_{t,\nu}^T=mes(i{\cal A}_{M_{disc},F}^*)^{-1}\sum_{Q=LU_{Q},Q'=L'U_{Q'}\in {\cal F}(M_{disc}); Q'\subset Q}\int_{{\cal A}_{L',F}}\int_{{\cal A}_{M_{disc},F}/{\cal A}_{A_{\tilde{G}},F}}\delta_{M_{disc}}^{Q'}(X)$$ $$S_Q(t;X-H_{L}-T[Q])\Gamma_{Q'}^Q(X,H+T[Q'])\int_{i{\cal A}_{M_{disc},F}^*}e^{<\lambda-t\lambda'+\nu,X>}\hat{\phi}_{\star}(t,\nu;\lambda,H,Q')\,d\lambda\,dX\,dH.$$ On peut toujours permuter librement les sommes en $Q$ et $Q'$ et les deux premières intégrales, d’où $${\bf E}_{t,\nu}^T=mes(i{\cal A}_{M_{disc},F}^*)^{-1}\int_{{\cal A}_{M_{disc},F}/{\cal A}_{A_{\tilde{G}},F}}\sum_{Q'=L'U_{Q'}\in {\cal F}(M_{disc})}\int_{{\cal A}_{L',F}}\delta_{M_{disc}}^{Q'}(X)\sum_{Q=LU_{Q}; Q'\subset Q}$$ $$S_Q(t;X-H_{L}-T[Q])\Gamma_{Q'}^Q(X,H+T[Q'])\int_{i{\cal A}_{M_{disc},F}^*}e^{<\lambda-t\lambda'+\nu,X>}\hat{\phi}_{\star}(t,\nu;\lambda,H,Q')\,d\lambda\,dH\,dX.$$ Fixons $Q'$. En se reportant à la définition de $S_{Q}(t,X)$ et en utilisant 3.20(3), on a l’égalité $$\sum_{Q=LU_{Q}; Q'\subset Q}S_Q(t;X-H_{L}-T[Q])\Gamma_{Q'}^Q(X,H+T[Q'])=$$ $$\sum_{\tilde{P}\in {\cal F}(\tilde{M}_{t}); Q'\subset P}(-1)^{a_{\tilde{P}}-a_{\tilde{G}}} \hat{\tau}_{\tilde{P}}(X-H-T[\tilde{P}])\sum_{Q=LU_{Q}; Q'\subset Q\subset P}\tau_{Q}^P(X-H-T[Q])\Gamma_{Q'}^Q(X,H+T[Q']).$$ D’après 1.3(4), la dernière somme en $Q$ vaut $\tau_{Q'}^P(X)$. D’où $${\bf E}_{t,\nu}^T=\int_{{\cal A}_{M_{disc},F}/{\cal A}_{A_{\tilde{G}},F}}{\bf E}_{t,\nu}^T(X)\,dX,$$ où $${\bf E}_{t,\nu}^T(X)=mes(i{\cal A}_{M_{disc},F}^*)^{-1}\sum_{Q'=L'U_{Q'}\in {\cal F}(M_{disc})}\int_{{\cal A}_{L',F}}\delta_{M_{disc}}^{Q'}(X)\sum_{\tilde{P}=\tilde{M}U_{P}\in {\cal F}(\tilde{M}_{t}); Q'\subset P}$$ $$(-1)^{a_{\tilde{P}}-a_{\tilde{G}}} \hat{\tau}_{\tilde{P}}(X-H-T[\tilde{P}])\tau_{Q'}^P(X)\int_{i{\cal A}_{M_{disc},F}^*}e^{<\lambda-t\lambda'+\nu,X>}\hat{\phi}_{\star}(t,\nu;\lambda,H,Q')\,d\lambda\,dH.$$ Fixons $X$. Puisque $\hat{\phi}_{\star}(t,\nu;\lambda,H,Q')$ est de Schwartz en $\lambda$ et $H$, l’expression ${\bf E}^T_{t,\nu}(X)$ est absolument convergente. On peut donc écrire $${\bf E}^T_{t,\nu}(X)=mes(i{\cal A}_{M_{disc},F}^*)^{-1}\int_{i{\cal A}_{M_{disc},F}^*}e^{<\lambda-t\lambda'+\nu,X>}{\bf E}_{t,\nu}^T(X,\lambda)\,d\lambda,$$ où $${\bf E}_{t,\nu}^T(X,\lambda)=\sum_{Q'=L'U_{Q'}\in {\cal F}(M_{disc})}\int_{{\cal A}_{L',F}}\delta_{M_{disc}}^{Q'}(X)\sum_{\tilde{P}=\tilde{M}U_{P}\in {\cal F}(\tilde{M}_{t}); Q'\subset P}(-1)^{a_{\tilde{P}}-a_{\tilde{G}}} \hat{\tau}_{\tilde{P}}(X-H-T[\tilde{P}])$$ $$\tau_{Q'}^P(X)\hat{\phi}_{\star}(t,\nu;\lambda,H,Q')\,dH.$$ Fixons $\lambda$. On peut permuter les sommes en $\tilde{P}$ et en $Q'$, puis décomposer l’intégrale en $H\in {\cal A}_{L',F}$ en une intégrale sur $H\in {\cal A}_{\tilde{M},F}$ et une intégrale en $H'\in {\cal A}_{L',F}^{\tilde{M}}(H)$ et on peut permuter la première avec la somme en $Q'$ On obtient $${\bf E}_{t,\nu}^T(X,\lambda) =\sum_{\tilde{P}=\tilde{M}U_{P}\in {\cal F}(\tilde{M}_{t})}(-1)^{a_{\tilde{P}}-a_{\tilde{G}}}\int_{{\cal A}_{\tilde{M},F}}\hat{\tau}_{\tilde{P}}(X-H-T[\tilde{P}])$$ $$\sum_{Q'=L'U_{Q'}; M_{disc}\subset Q'\subset P}\delta_{M_{disc}}^{Q'}(X)\tau_{Q'}^P(X)\int_{{\cal A}_{L',F}^{\tilde{M}}(H)}\hat{\phi}_{\star}(t,\nu;\lambda,H',Q')\,dH'\,dH.$$ En se rappelant la définition de $\hat{\phi}_{\star}(t,\nu;\lambda,H',Q')$ et par inversion de Fourier partielle, on a $$\int_{{\cal A}_{L',F}^{\tilde{M}}(H)}\hat{\phi}_{\star}(t,\nu;\lambda,H',Q')\,dH'\,=mes(i{\cal A}_{\tilde{M},F})^{-1}\int_{i{\cal A}_{\tilde{M},F}^*}\phi_{\star}(t,\nu;\lambda,\Lambda,Q')e^{-<\Lambda,H>}\,d\Lambda.$$ Puisque $\Lambda$ ne parcourt plus que $i{\cal A}_{\tilde{M},F}^*$, on peut aussi bien remplacer $\phi_{\star}(t,\nu;\lambda,\Lambda,Q')$ par $\phi_{\star}(t,\nu;\lambda,\Lambda,\tilde{P})$ et l’expression ci-dessus est égale à $\hat{\phi}_{\star}(t,\nu;\lambda,H,\tilde{P})$. Notons que $Q'$ a ici disparu donc, dans l’expression précédente de ${\bf E}_{\nu,t}^T$, la somme en $Q'$ n’est plus que $$\sum_{Q'; M_{disc}\subset Q'\subset P}\delta_{M_{disc}}^{Q'}(X)\tau_{Q'}^P(X).$$ Celle-ci est égale à $$\sum_{\tilde{P}'; \tilde{M}_{t}\subset \tilde{P}'\subset \tilde{P}}\delta_{\tilde{M}_{t}}^{\tilde{P}'}(X)\tau_{\tilde{P}'}^{\tilde{P}}(X).$$ En effet, les deux expressions sont égales à $1$ d’après 1.3(1) et sa variante tordue. On obtient $${\bf E}_{t,\nu}^T(X,\lambda)= \sum_{\tilde{P}=\tilde{M}U_{P}\in {\cal F}(\tilde{M}_{t})}(-1)^{a_{\tilde{P}}-a_{\tilde{G}}}\int_{{\cal A}_{\tilde{M},F}}\hat{\tau}_{\tilde{P}}(X-H-T[\tilde{P}])$$ $$\sum_{\tilde{P}'=\tilde{M}'U_{P'};\tilde{M}_{t}\subset\tilde{P}'\subset \tilde{P}}\delta_{\tilde{M}_{t}}^{\tilde{P}'}(X)\tau_{\tilde{P}'}^{\tilde{P}}(X) \hat{\phi}_{\star}(t,\nu;\lambda,H,\tilde{P})\,dH.$$ On remonte maintenant le calcul ci-dessus: on permute les sommes en $\tilde{P}$ et $\tilde{P}'$; on remplace $\hat{\phi}_{\star}(t,\nu;\lambda,H,\tilde{P})$ par l’expression égale $$\int_{{\cal A}_{\tilde{M}',F}^{\tilde{M}}(H)}\hat{\phi}_{\star}(t,\nu;\lambda,H',\tilde{P}')\,dH';$$ à partir des intégrales en $H\in {\cal A}_{\tilde{M},F}$ et $H'\in {\cal A}_{\tilde{M}',F}^{\tilde{M}}(H)$, on reconstitue une intégrale en $H\in {\cal A}_{\tilde{M}',F}$. On obtient $${\bf E}^T_{t,\nu}(X,\lambda)=\sum_{\tilde{P}'=\tilde{M}'U_{P'}\in {\cal F}(\tilde{M}_{t})}\int_{{\cal A}_{\tilde{M}',F}}\delta_{\tilde{M}_{t}}^{\tilde{P}'}(X)$$ $$\sum_{\tilde{P}; \tilde{P}'\subset \tilde{P}}(-1)^{a_{\tilde{P}}-a_{\tilde{G}}}\hat{\tau}_{\tilde{P}}(X-H-T[\tilde{P}'])\tau_{\tilde{P}'}^{\tilde{P}}(X)\hat{\phi}_{\star}(t,\nu;\lambda,H,\tilde{P}')\,dH.$$ Par définition, $$\sum_{\tilde{P}; \tilde{P}'\subset \tilde{P}}(-1)^{a_{\tilde{P}}-a_{\tilde{G}}}\hat{\tau}_{\tilde{P}}(X-H-T[\tilde{P}'])\tau_{\tilde{P}'}^{\tilde{P}}(X)=\Gamma_{\tilde{P}'}^{\tilde{G}}(X,H+T[P']).$$ Cela fait disparaître les $\tilde{P}$ de la formule ci-dessus et nous autorise à abandonner les $'$ des $\tilde{P}'$. D’où $${\bf E}^T_{t,\nu}(X,\lambda)=\sum_{\tilde{P}=\tilde{M}U_{P}\in {\cal F}(\tilde{M}_{t})}\int_{{\cal A}_{\tilde{M},F}}\delta_{\tilde{M}_{t}}^{\tilde{P}}(X)\Gamma_{\tilde{P}}^{\tilde{G}}(X,H+T[\tilde{P}])\hat{\phi}_{\star}(t,\nu;\lambda,H,\tilde{P})\,dH.$$
Revenons à ${\bf E}^T_{t,\nu}$ qui est l’intégrale en $\lambda$ puis $X$ de l’expression ci-dessus, multipliée par $mes(i{\cal A}_{M_{disc},F}^*)^{-1}e^{<\lambda-t\lambda'+\nu,X>}$. On peut décomposer l’intégrale en $X$ en une intégrale en $X\in {\cal A}_{\tilde{M}_{t},F}/{\cal A}_{A_{\tilde{G}},F}$ et une intégrale en $Y\in {\cal A}_{M_{disc},F}^{\tilde{M}_{t}}(X)$. Remarquons que pour $Y$ dans cet ensemble, on a ${\bf E}^T_{t,\nu}(Y,\lambda)={\bf E}^T_{t,\nu}(X,\lambda)$. On a alors $${\bf E}^T_{t,\nu}=mes(i{\cal A}_{M_{disc},F}^*)^{-1}\int_{{\cal A}_{\tilde{M}_{t},F}/{\cal A}_{A_{\tilde{G}},F}}\int_{{\cal A}_{M_{disc},F}^{\tilde{M}_{t}}(X)}\int_{i{\cal A}_{M_{disc},F}^*}e^{<\lambda-t\lambda'+\nu,Y>}{\bf E}^T_{t,\nu}(X,\lambda)\,d\lambda\,dY\,dX.$$ Pour tout $X$, la fonction $\lambda\mapsto {\bf E}^T_{t,\nu}(X,\lambda)$ est de Schwartz. La double intégrale intérieure est calculée par le lemme 3.20. On obtient $${\bf E}^T_{t,\nu}=mes(i{\cal A}_{\tilde{M}_{t},F}^*)^{-1}\vert det((1-t\theta^{-1})_{\vert {\cal A}_{M_{disc}}^{\tilde{M}_{t}*}}\vert ^{-1}\int_{{\cal A}_{\tilde{M}_{t},F}/{\cal A}_{A_{\tilde{G}},F}}\sum_{\lambda\in \{\nu\}_{t}}e^{<\Lambda(\lambda),X>}$$ $$\int_{i{\cal A}_{\tilde{M}_{t},F}^*}{\bf E}^T_{t,\nu}(X,\lambda+\xi)\,d\xi\,dX.$$ En développant le dernier terme, on obtient $${\bf E}^T_{t,\nu}=mes(i{\cal A}_{\tilde{M}_{t},F}^*)^{-1}\vert det((1-t\theta^{-1})_{\vert {\cal A}_{M_{disc}}^{\tilde{M}_{t}*}}\vert ^{-1}\int_{{\cal A}_{\tilde{M}_{t},F}/{\cal A}_{A_{\tilde{G}},F}}\sum_{\lambda\in \{\nu\}_{t}}e^{<\Lambda(\lambda),X>}\int_{i{\cal A}_{\tilde{M}_{t},F}^*}$$ $$\sum_{\tilde{P}=\tilde{M}U_{P}\in {\cal F}(\tilde{M}_{t})}\int_{{\cal A}_{\tilde{M},F}}\delta_{\tilde{M}_{t}}^{\tilde{P}}(X)\Gamma_{\tilde{P}}^{\tilde{G}}(X,H+T[\tilde{P}])\hat{\phi}_{\star}(t,\nu;\lambda+\xi,H,\tilde{P})\,dH\,d\xi\,dX.$$ Cette expression est absolument convergente. En effet, pour $\tilde{P}$ fixé, l’intégrale en $X$ est à support compact et est essentiellement bornée par $(1+\vert H\vert )^D$ pour un entier $D$ convenable. Les intégrales restantes en $H$ et $\xi$ sont convergentes puisque $\hat{\phi}_{\star}(t,\nu;\lambda+\xi,H,\tilde{P})$ est de Schwartz en $\xi$ et $H$. On peut donc commencer par intégrer en $\xi$, puis en $H$, puis en $X$. On décompose ensuite l’intégrale en $X$ en une somme sur $X\in {\cal A}_{\tilde{G},F}/{\cal A}_{A_{\tilde{G}},F}$ d’intégrales en $Y\in {\cal A}_{\tilde{M}_{t},F}^{\tilde{G}}(X+H_{\tilde{G}})$. Après encore quelques permutations, on obtient $${\bf E}^T_{t,\nu}=mes(i{\cal A}_{\tilde{M}_{t},F}^*)^{-1}\vert det((1-t\theta^{-1})_{\vert {\cal A}_{M_{disc}}^{\tilde{M}_{t}*}}\vert ^{-1}\sum_{\lambda\in \{\nu\}_{t}}\sum_{X\in {\cal A}_{\tilde{G},F}/{\cal A}_{A_{\tilde{G}},F}}\int_{i{\cal A}_{\tilde{M}_{t},F}^*}\sum_{\tilde{P}=\tilde{M}U_{P}\in {\cal F}(\tilde{M}_{t})}$$ $$\int_{{\cal A}_{\tilde{M},F}}\int_{{\cal A}_{\tilde{M}_{t},F}^{\tilde{G}}(X+H_{\tilde{G}})}e^{<\Lambda(\lambda),Y>}\delta_{\tilde{M}_{t}}^{\tilde{P}}(Y)\Gamma_{\tilde{P}}^{\tilde{G}}(Y,H+T[\tilde{P}])\hat{\phi}_{\star}(t,\nu;\lambda+\xi,H,\tilde{P})\,dY\,dH\,d\xi.$$ L’expression intérieure (somme en $\tilde{P}$ et intégrales en $Y$ et $H$) est calculée par la variante tordue du lemme 1.8. C’est $\phi_{\star,\tilde{M}_{t}}^{\tilde{G},T}(t,\nu,X;\lambda+\xi,\Lambda(\lambda))$. D’où $${\bf E}^T_{t,\nu}=mes(i{\cal A}_{\tilde{M}_{t},F}^*)^{-1}\vert det((1-t\theta^{-1})_{\vert {\cal A}_{M_{disc}}^{\tilde{M}_{t}*}}\vert ^{-1}\sum_{\lambda\in \{\nu\}_{t}}\sum_{X\in {\cal A}_{\tilde{G},F}/{\cal A}_{A_{\tilde{G}},F}}$$ $$\int_{i{\cal A}_{\tilde{M}_{t},F}^*}\phi_{\star,\tilde{M}_{t}}^{\tilde{G},T}(t,\nu,X;\lambda+\xi,\Lambda(\lambda))\,d\xi.$$ En revenant aux définitions des fonctions $\phi_{reg}(t,\nu;\lambda,\Lambda,S')$ et $\phi_{\star}(t,\nu;\lambda,\Lambda,S')$, on voit que la seconde est le produit de la première et de $\epsilon(t,\nu;\lambda)B(\lambda)C( \Lambda-\lambda+t\lambda'-\nu)$. Mais, pour $\lambda\in \{\nu\}_{t}$ et $\xi\in i{\cal A}_{\tilde{M}_{t},F}^*$, on a par définition $\Lambda(\lambda)-\lambda-\xi+t\theta^{-1}(\lambda+\xi)-\nu\in i{\cal A}_{M_{disc},F}^{\vee}$. Donc $C( \Lambda(\lambda)-\lambda-\xi+t\theta^{-1}(\lambda+\xi)-\nu)=1$ et $$\phi_{\star,\tilde{M}_{t}}^{\tilde{G},T}(t,\nu,X;\lambda+\xi,\Lambda(\lambda))=B(\lambda+\xi)\epsilon(t,\nu;\lambda+\xi)\phi_{reg,\tilde{M}_{t}}^{\tilde{G},T}(t,\nu,X;\lambda+\xi,\Lambda(\lambda)).$$ La formule ci-dessus devient celle de l’énoncé. $\square$
Le “terme constant” de ${\bf E}_{t,\nu}^T$
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Posons $$j_{spec,t,\nu}=\vert det((1-t\theta^{-1})_{\vert {\cal A}_{M_{disc}}^{\tilde{M}_{t},*}})\vert ^{-1} \sum_{\lambda\in\{\nu\}_{t}; \Lambda(\lambda)=0}\int_{i{\cal A}_{\tilde{M}_{t},F}^*}B(\lambda+\xi)\phi_{reg,\tilde{M}_{t}}^{\tilde{G}}(t,\nu;\lambda+\xi,0)\,d\xi.$$ Cette expression est absolument convergente, la fonction $\xi\mapsto \phi_{reg,\tilde{M}_{t}}^G(t,\nu;\lambda+\xi,0)$ étant lisse et à croissance modérée.
[0.3cm[**[Lemme]{}**]{}. [ *[Il existe une unique fonction $T\mapsto f(T)$ qui appartient à $PolExp$ et qui coïncide avec $mes({\cal A}_{\tilde{G}}(F)_{c})^{-1}{\bf E}_{t,\nu}^T$ dans le cône où celle-ci est définie. Si $F$ est archimédien, on a $c_{0}(f)=j_{spec,t,\nu}$. Si $F$ est non-archimédien, pour tout réseau ${\cal R}\subset {\cal A}_{M_{0},F}\otimes_{{\mathbb Z}}{\mathbb Q}$, on a l’égalité $$lim_{k\to \infty}c_{\frac{1}{k}{\cal R},0}(f)=j_{spec,t,\nu}.$$]{}*]{}0.3cm]{}
[**Remarque.**]{} Pour simplifier, on appellera “terme constant” de $f$ le terme $c_{0}(f)$ si $F$ est archimédien, $lim_{k\to \infty}c_{\frac{1}{k}{\cal R},0}(f)$ si $F$ est non-archimédien.
Preuve. On suppose $F$ non-archimédien, le cas archimédien étant similaire. Considérons la formule de la proposition précédente. Pour tous $\lambda$, $X$ et pour tout $\xi\in i{\cal A}_{\tilde{M}_{t},F}^*$, la fonction $T\mapsto f_{\lambda,X,\xi}(T)=\phi^T_{reg,\tilde{M}_{t}}(t,\nu,X;\lambda+\xi,\Lambda(\lambda))$ est définie pour $T\in {\cal A}_{M_{0},F}\otimes_{{\mathbb Z}}{\mathbb Q}$ et elle appartient à $PolExp$, cf. lemme 1.7. Plus précisément, elle appartient à un espace $PolExp_{\boldsymbol{\Xi},N}$, où $\boldsymbol{\Xi}$ et $N$ ne dépendent pas de $\xi$. Autrement dit, pour tout réseau ${\cal R}\subset {\cal A}_{M_{0},F}\otimes_{{\mathbb Z}}{\mathbb Q}$, on peut écrire $$f_{\lambda,X,\xi}(T)=\sum_{\mu\in {\cal X}_{{\cal R}}}e^{<\mu,T>}p_{{\cal R},\mu}(T)$$ pour $T\in {\cal R}$, avec un ensemble ${\cal X}_{{\cal R}}$ indépendant de $\xi$ et des polynômes $p_{{\cal R},\mu}$ de degré borné indépendamment de $\xi$. Les coefficients de ces polynômes se calculent par interpolation et vérifient donc les mêmes propriétés que la fonction $ f_{\lambda,X,\xi}$ elle-même. Ils sont donc $C^{\infty}$ en $\xi$. Il en résulte que le développement en $T$ commute à l’intégrale en $\xi$. Cela implique que, si on note $f(T)$ le membre de droite de l’égalité de la proposition 3.22 multiplié par $mes({\cal A}_{\tilde{G}}(F)_{c})^{-1}$, la fonction $T\mapsto f(T)$ appartient à $PolExp$ et que son coefficient $c_{{\cal R},0}(f)$ se calcule en remplaçant $ f_{\lambda,X,\xi}$ par son coefficient $c_{{\cal R},0}(f_{\lambda,X,\xi})$ dans la formule intégrale. Remarquons que la norme (au sens de 1.7) de la $(\tilde{G},\tilde{M}_{t})$-famille $(\phi_{reg}(t,\nu;\lambda+\xi,\Lambda,\tilde{P}))_{\tilde{P}\in {\cal P}(\tilde{M}_{t})}$ est bornée indépendamment de $\xi$. Il résulte donc du lemme 1.7 que $lim_{k\to \infty}c_{\frac{1}{k}{\cal R},0}(f)$ se calcule en remplaçant $f_{\lambda,X,\xi}$ par $lim_{k\to \infty}c_{\frac{1}{k}{\cal R},0}(f_{\lambda,X,\xi})$ dans la formule intégrale (multipliée par $mes({\cal A}_{\tilde{G}}(F)_{c})^{-1}$). Cette dernière limite est $0$ si $\Lambda(\lambda)\not\in ( i{\cal A}_{\tilde{M}_{t},F}^{\vee}+i{\cal A}_{\tilde{G}}^*)/ i{\cal A}_{\tilde{M}_{t},F}^{\vee}$. Si $\Lambda(\lambda)\in(\Lambda_{1}(\lambda)+i{\cal A}_{\tilde{M}_{t},F}^{\vee})/ i{\cal A}_{\tilde{M}_{t},F}^{\vee}$, avec $\Lambda_{1}(\lambda)\in i{\cal A}_{\tilde{G}}^*$, c’est $$mes({\cal A}_{\tilde{G}}(F)_{c})^{-1}mes(i{\cal A}_{\tilde{M}_{t},F}^*)mes(i{\cal A}_{\tilde{G},F}^*)^{-1}e^{<\Lambda_{1}(\lambda),X>}\phi_{reg,\tilde{M}_{t}}^{\tilde{G}}(t,\nu;\lambda+\xi,\Lambda_{1}(\lambda)).$$ La somme en $X$ devient simplement $$\sum_{X\in {\cal A}_{\tilde{G},F}/{\cal A}_{A_{\tilde{G}},F}}e^{<\Lambda_{1}(\lambda),X>}.$$ Cette somme vaut $ [{\cal A}_{\tilde{G},F}:{\cal A}_{A_{\tilde{G}},F}]=mes(i{\cal A}_{\tilde{G},F}^*)mes(A_{\tilde{G}}(F)_{c})$ si $\Lambda_{1}(\lambda)\in i{\cal A}_{\tilde{G},F}^{\vee}$, $0$ sinon. La condition $\Lambda_{1}(\lambda)\in i{\cal A}_{\tilde{G},F}^{\vee}$ équivaut à $\Lambda(\lambda)=0$. Ces calculs conduisent à l’égalité de l’énoncé. $\square$
Le terme constant de $j^T$
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Considérons l’ensemble des $\lambda\in i{\cal A}_{M_{disc}}^*$ tels que $t(\sigma_{\lambda}\circ\theta)\simeq \omega \sigma_{\lambda}$. Il est invariant par translations par $i{\cal A}_{M_{disc},F}^{\vee}+i{\cal A}_{\tilde{M}_{t}}^*$. On note $[\sigma]_{t}$ son quotient par l’action de ce groupe. Ce quotient est fini. On vérifie que l’application $\lambda\mapsto (t\lambda'-\lambda,\lambda)$ est une bijection de $[\sigma]_{t}$ sur l’ensemble des couples $(\nu,\lambda)$, où
- $\nu\in [t\sigma',\omega\sigma]$;
- $\lambda\in \{\nu\}_{t}$;
- $\Lambda(\lambda)=0$.
Posons $$j_{spec}= \sum_{t\in W^G(M_{disc}\vert M'_{disc})}\vert det((1-t\theta^{-1})_{\vert {\cal A}_{M_{disc}}^{\tilde{M}_{t},*}})\vert ^{-1} \sum_{\lambda\in [\sigma]_{t}}$$ $$\int_{i{\cal A}_{\tilde{M}_{t},F}^*}B(\lambda+\xi)\epsilon(t,t\lambda'-\lambda;\lambda+\xi)\phi_{reg,\tilde{M}_{t}}^{\tilde{G}}(t,t\lambda'-\lambda;\lambda+\xi,0)\,d\xi.$$ Ce terme est bien défini d’après les propriétés ci-dessus.
[0.3cm[**[Corollaire]{}**]{}. [ *[Il existe une unique fonction $T\mapsto f(T)$ qui appartient à $PolExp$ et qui vérifie la majoration $$\vert j^T-f(T)\vert <<\vert T\vert ^{-r}$$ pour tout réel $r$ et tout $T$ dans le cône où $j^T$ est définie. Si $F$ est archimédien, on a $c_{0}(f)=j_{spec}$. Si $F$ est non-archimédien, pour tout réseau ${\cal R}\subset {\cal A}_{M_{0},F}\otimes_{{\mathbb Z}}{\mathbb Q}$, on a l’égalité $$lim_{k\to \infty}c_{\frac{1}{k}{\cal R},0}(f)=j_{spec}.$$]{}*]{}0.3cm]{}
Preuve. L’existence de la fonction $f$ résulte du lemme 3.19, de la définition de ${\bf E}^T$ et du lemme précédent. L’unicité est claire: un élément de $PolExp$ est nul s’il est à décroissance rapide dans un cône. On calcule le terme constant en utilisant le lemme précédent. Ce calcul conduit à une formule similaire à $j_{spec}$ ci-dessus, à ceci près que la somme en $\lambda\in [\sigma]_{t}$ y est remplacée par une double somme sur $\nu\in [\sigma,\omega t\theta\sigma]$ et $\lambda\in \{\nu\}_{t}$ tel que $\Lambda(\lambda)=0$. Comme on l’a dit ci-dessus, cette double somme coïncide avec une somme en $\lambda\in [\sigma]_{t}$. $\square$
Définition d’une expression spectrale
-------------------------------------
Soit $\tilde{M}$ un espace de Levi de $\tilde{G}$. On suppose que $M_{0}\subset M$. Soit $\tau=(M_{disc},\sigma,r)$ un triplet formé d’un Levi semi-standard $M_{disc}\subset M$, d’une représentation $\sigma$ de $M_{disc}(F)$ irréductible et de la série discrète et d’un élément $\tilde{r}\in R^{\tilde{M}}(\sigma)$. On fixe un relèvement $\boldsymbol{\tilde{r}}\in {\cal R}^{\tilde{M}}(\sigma)$ et on pose $\boldsymbol{\tau}=(M_{disc},\sigma,\boldsymbol{\tilde{r}})$. Fixons aussi un élément $\tilde{P}\in {\cal P}(\tilde{M})$, puis un élément $S\in {\cal P}(M_{disc})$ tel que $S\subset P$. Posons $\pi_{\tau}=Ind_{S\cap M}^M(\sigma)$ et $\Pi_{\tau}=Ind_{S}^G(\sigma)\simeq Ind_{P}^G(\pi_{\tau})$. A l’aide de $ \boldsymbol{\tau}$, on a défini en 2.9 une représentation de $\tilde{M}(F)$ dans l’espace $V_{\sigma,S\cap P}$ de $\pi_{\tau}$ et une représentation de $\tilde{G}(F)$ dans l’espace $V_{\sigma,P}$ de $\Pi_{\tau}$. Notons-les respectivement $\tilde{\pi}_{\boldsymbol{\tau}}$ et $\tilde{\Pi}_{\boldsymbol{\tau}}$. On a $\tilde{\Pi}_{\boldsymbol{\tau}}=Ind_{\tilde{P}}^{\tilde{G}}(\tilde{\pi}_{\boldsymbol{\tau}})$. Soit $\tilde{\lambda}\in i\tilde{{\cal A}}_{\tilde{M},F}^*$ en position générale. On peut remplacer $\tau$ par $\tau_{\lambda}=(M,\sigma_{\lambda},\tilde{r})$ et $\boldsymbol{\tau}$ par $\boldsymbol{\tau}_{\tilde{\lambda}}=(M,\sigma_{\lambda},\boldsymbol{\tilde{r}})$ cf. 2.9. On définit comme en 2.7 la $(\tilde{G},\tilde{M})$-famille $({\cal M}(\pi_{\tau_{\lambda}};\Lambda,\tilde{Q}))_{\tilde{Q}\in {\cal P}(\tilde{M})}$ dont les fonctions prennent leurs valeurs dans l’espace des endomorphismes de $V_{\sigma,S\cap P}$. Rappelons que l’on a fixé deux fonctions $f_{1},f_{2}\in C_{c}^{\infty}(\tilde{G}(F))$. On définit une $(\tilde{G},\tilde{M})$-famille $({\cal J}(\pi_{\tau_{\lambda}},f_{1},f_{2};\Lambda,\tilde{Q}))_{\tilde{Q}\in {\cal P}(\tilde{M})}$ par $${\cal J}(\pi_{\tau_{\lambda}},f_{1},f_{2};\Lambda,\tilde{Q})=\overline{trace({\cal M}(\pi_{\tau_{\lambda}};\Lambda,\tilde{\bar{Q}}))\tilde{\Pi}_{\boldsymbol{\tau}_{\tilde{\lambda}}}(f_{1})}trace({\cal M}(\pi_{\tau_{\lambda}};\Lambda,\tilde{Q})\tilde{\Pi}_{\boldsymbol{\tau}_{\tilde{\lambda}}}(f_{2})).$$ On en déduit une fonction ${\cal J}_{\tilde{M}}^{\tilde{G}}(\pi_{\tau_{\lambda}},f_{1},f_{2};\Lambda)$. On pose $$J_{\tilde{M}}^{\tilde{G}}(\pi_{\tau_{\lambda}},f_{1},f_{2})={\cal J}_{\tilde{M}}^{\tilde{G}}(\pi_{\tau_{\lambda}},f_{1},f_{2};0).$$ [**Remarque.**]{} A cause de la double apparition de $\tilde{\pi}_{\boldsymbol{\tau}_{\tilde{\lambda}}}$ dans les définitions ci-dessus, les fonctions ainsi définies ne dépendent ni du relèvement $\boldsymbol{\tau}$ de $\tau$, ni du relèvement $\tilde{\lambda}$ de $\lambda$.
On rappelle que l’on a défini la notion de triplet essentiel, cf. 2.9. Cette notion dépend de l’espace ambiant. Ici, c’est l’espace $\tilde{M}$. On a
\(1) si le triplet $\tau$ n’est pas essentiel pour $\tilde{M}$, les fonctions ci-dessus sont nulles.
Le caractère de $\tilde{\pi}_{\boldsymbol{\tau}_{\tilde{\lambda}}}$ étant nul pour tout $\tilde{\lambda}$, la preuve est la même que celle de 2.7(3).
On suppose désormais $\tau$ essentiel, c’est-à-dire $\tau\in E(\tilde{M},\omega)$. Il est clair que la fonction $\lambda\mapsto J_{\tilde{M}}^{\tilde{G}}(\pi_{\tau_{\lambda}},f_{1},f_{2})$ est la restriction à $i{\cal A}_{\tilde{M},F}^*$ d’une fonction méromorphe.
[0.3cm[**[Lemme]{}**]{}. [ *[La fonction $\lambda\mapsto J_{\tilde{M}}^{\tilde{G}}(\pi_{\tau_{\lambda}},f_{1},f_{2})$ est régulière sur tout $i {\cal A}_{\tilde{M},F}^*$. Si $F$ est archimédien, c’est une fonction de Schwartz sur cet espace.]{}*]{}0.3cm]{}
Preuve. Convertissons tous les opérateurs d’entrelacement qui interviennent dans les définitions en produits d’opérateurs nomalisés et de facteurs de normalisation. On obtient une expression $${\cal J}(\pi_{\tau_{\lambda}},f_{1},f_{2};\Lambda,\tilde{Q})={\bf r}(\pi_{\tau_{\lambda}};\Lambda,\tilde{Q}){\cal J}_{reg}(\pi_{\tau_{\lambda}},f_{1},f_{2};\Lambda,\tilde{Q}),$$ où on a regroupé dans ${\bf r}(\pi_{\tau_{\lambda}};\Lambda,\tilde{Q})$ les facteurs de normalisation. On calcule ${\bf r}(\pi_{\tau_{\lambda}};\Lambda,\tilde{Q})$. C’est le produit de $$r_{P\vert Q}(\sigma_{\lambda})r_{Q\vert P}(\sigma_{\lambda+\Lambda})r_{P\vert Q}(\sigma_{\lambda+\Lambda/2})^{-1}r_{Q\vert P}(\sigma_{\lambda+\Lambda/2})^{-1}$$ et du conjugué de $$r_{P\vert \bar{Q}}(\sigma_{\lambda})r_{\bar{Q}\vert P}(\sigma_{\lambda+\Lambda})r_{P\vert \bar{Q}}(\sigma_{\lambda+\Lambda/2})^{-1}r_{\bar{Q}\vert P}(\sigma_{\lambda+\Lambda/2})^{-1}.$$ On obtient $${\bf r}(\pi_{\tau_{\lambda}};\Lambda,\tilde{Q})=r_{\bar{Q}\vert Q}(\sigma_{\lambda})r_{Q\vert\bar{ Q}}(\sigma_{\lambda+\Lambda})r_{Q\vert \bar{Q}}(\sigma_{\lambda+\Lambda/2})^{-1}r_{\bar{Q}\vert Q}(\sigma_{\lambda+\Lambda/2})^{-1}.$$ Le produit des deux derniers termes est indépendant de $\tilde{Q}$. On peut donc écrire $${\bf r}(\pi_{\tau_{\lambda}};\Lambda,\tilde{Q})=C(\lambda,\Lambda) {\bf r}_{reg}(\pi_{\tau_{\lambda}};\Lambda,\tilde{Q}),$$ où $$C(\lambda,\Lambda)=r_{\bar{P}\vert P}(\sigma_{\lambda})r_{P\vert\bar{ P}}(\sigma_{\lambda+\Lambda})r_{P\vert \bar{P}}(\sigma_{\lambda+\Lambda/2})^{-1}r_{\bar{P}\vert P}(\sigma_{\lambda+\Lambda/2})^{-1}$$ et $${\bf r}_{reg}(\pi_{\tau_{\lambda}};\Lambda,\tilde{Q})=r_{\bar{Q}\vert Q}(\sigma_{\lambda})r_{\bar{P}\vert P}(\sigma_{\lambda})^{-1}r_{Q\vert \bar{Q}}(\sigma_{\lambda+\Lambda})r_{P\vert \bar{P}}(\sigma_{\lambda+\Lambda})^{-1}.$$ En un point $\lambda$ général, $J_{\tilde{M}}^{\tilde{G}}(\rho_{\tau_{\lambda}},f_{1},f_{2})$ est donc le produit de $C(\lambda,0)$ et d’un terme issu de la $(\tilde{G},\tilde{M})$-famille $$({\bf r}_{reg}(\rho_{\tau_{\lambda}};\Lambda,\tilde{Q}){\cal J}_{reg}(\rho_{\tau_{\lambda}},f_{1},f_{2};\Lambda,\tilde{Q}))_{\tilde{Q}\in {\cal P}(\tilde{M})}.$$ Le terme $C(\lambda,0)$ vaut $1$. La $(\tilde{G},\tilde{M})$-famille ci-dessus est formée de fonctions régulières en $\lambda$ et $\Lambda$ (d’après 1.10(5)). Elle donne donc naissance à une fonction régulière en $\lambda$. Dans le cas où $F$ est archimédien, on décompose plus finement cette dernière $(\tilde{G},\tilde{M})$-famille en combinaison linéaire de produit de familles similaires et de coefficients matriciels des opérateurs $\tilde{\Pi}_{\boldsymbol{\tau}_{\tilde{\lambda}}}(f_{1})$ et $\tilde{\Pi}_{\boldsymbol{\tau}_{\tilde{\lambda}}}(f_{2})$. Ces coefficients sont de Schwartz tandis les autres termes donnent naissance à des fonctions à croissance modérée en $\lambda$ (lemme 1.4). D’où le lemme. $\square$
[**Remarque.**]{} Dans le cas où $\tilde{M}=\tilde{G}$, on a simplement l’égalité $$J_{\tilde{G}}^{\tilde{G}}(\pi_{\tau_{\lambda}},f_{1},f_{2})=\overline{trace(\tilde{\Pi}_{\boldsymbol{\tau}_{\tilde{\lambda}}}(f_{1}))}trace(\tilde{\Pi}_{\boldsymbol{\tau}_{\tilde{\lambda}}}(f_{2})).$$
D’après le lemme, le terme $J_{\tilde{M}}^{\tilde{G}}(\pi_{\tau_{\lambda}},f_{1},f_{2})$ est défini pour tout $\lambda$. On vérifie qu’il ne dépend pas de l’espace parabolique $\tilde{P}$ choisi et qu’il ne dépend que de la classe de conjugaison par $M(F)$ de l’élément $\tau_{\lambda}$. On pose $$J_{\tilde{M},spec}^{\tilde{G}}(\omega,f_{1},f_{2})=\sum_{\tau\in (E_{disc}(\tilde{M},\omega)/conj)/i{\cal A}_{\tilde{M},F}^*}\vert {\bf Stab}(W^M\times i{\cal A}_{\tilde{M},F}^*,\tau)\vert ^{-1} \iota(\tau)$$ $$\int_{i{\cal A}_{\tilde{M},F}^*}J_{\tilde{M}}^{\tilde{G}}(\pi_{\tau_{\lambda}},f_{1},f_{2})\,d\lambda.$$ Pour toute classe $\tau\in (E_{disc}(\tilde{M},\omega)/conj)/i{\cal A}_{\tilde{M},F}^*$, on a choisi un point-base que l’on a également noté $\tau$. Les termes ${\bf Stab}(W^M\times i{\cal A}_{\tilde{M},F}^*,\tau)$ et $\iota(\tau)$ ont été définis en 2.9 et 2.10. On vérifie que l’expression $J_{\tilde{M},spec}^{\tilde{G}}(\omega,f_{1},f_{2})$ ne dépend que de la classe de conjugaison par $\tilde{G}(F)$ de l’espace de Levi $\tilde{M}$.
La formule spectrale
--------------------
On pose $\tilde{W}^G=Norm_{G(F)}(\tilde{M}_{0})/M_{0}(F)$ (on ne confondra pas ce groupe avec le quotient par $M_{0}(F)$ du normalisateur de $M_{0}$ dans $\tilde{G}(F)$). Posons $$J^{\tilde{G}}_{spec}(\omega,f_{1},f_{2})= \sum_{\tilde{M}\in {\cal L}(\tilde{M}_{0})}\vert \tilde{W}^M\vert \vert \tilde{W}^G\vert ^{-1}(-1)^{a_{\tilde{M}}-a_{\tilde{G}}}J_{\tilde{M},spec}^{\tilde{G}}(\omega,f_{1},f_{2}).$$
[0.3cm[**[Proposition]{}**]{}. [ *[Il existe une unique fonction $T\mapsto \varphi(T)$ qui appartient à $PolExp$ et qui vérifie pour tout réel $r$ la majoration $$\vert J^T(\omega,f_{1},f_{2})-\varphi(T)\vert <<\vert T\vert ^{-r}$$ pour tout $T$ dans le cône où $J^T(\omega,f_{1},f_{2})$ est définie. Si $F$ est archimédien, on a l’égalité $$c_{0}(\varphi)=J^{\tilde{G}}_{spec}(\omega,f_{1},f_{2}).$$ Si $F$ est non-archimédien, pour tout réseau ${\cal R}\subset {\cal A}_{M_{0},F}\otimes_{{\mathbb Z}}{\mathbb Q}$, on a l’égalité $$lim_{k\to \infty}c_{\frac{1}{k}{\cal R},0}(\varphi)=J^{\tilde{G}}_{spec}(\omega,f_{1},f_{2}).$$]{}*]{}0.3cm]{}
Preuve. Il résulte des formules 3.2(1) et 3.3(1) que $J^T(\omega,f_{1},f_{2})$ est somme finie de termes $j^T$ tels qu’en 3.4. L’existence de la fonction $\varphi$ résulte donc du corollaire 3.24. Comme toujours, l’unicité de $\varphi$ est évidente. Le terme constant de $\varphi$ se calcule en sommant les termes constants des différents $j^T$ intervenant, lesquels sont calculés par le même corollaire 3.24. Commençons par calculer le terme constant de l’élément de $PolExp$ asymptote à l’expression $J^T_{M_{disc},\sigma}(\omega,f_{1},f_{2})$ de 3.3(1). Il est égal à $$(1) \qquad \sum_{u,v\in {\cal B}, u',v'\in {\cal B}'} \sum_{t\in W^G(M_{disc}\vert M'_{disc})}\vert det((1-t\theta^{-1})_{\vert {\cal A}_{M_{disc}}^{\tilde{M}_{t},*}})\vert ^{-1} \sum_{\lambda\in [\sigma]_{t}}$$ $$\int_{i{\cal A}_{\tilde{M}_{t},F}^*}B_{u,v,u',v'}(\lambda+\xi)\epsilon(t,t\lambda'-\lambda;\lambda+\xi)\phi_{reg,\tilde{M}_{t}}^{\tilde{G}}(u,v,u',v';t,t\lambda'-\lambda;\lambda+\xi,0)\,d\xi.$$ On a précisé la notation en rétablissant le quadruplet $(u,v,u',v')$ dans la fonction notée simplement $\phi_{reg,\tilde{M}_{t}}^{\tilde{G}}(t,t\lambda'-\lambda;\lambda+\xi,0)$ dans le corollaire 3.24. Les sommes en $u$, $v$, $u'$ et $v'$ sont finies, on peut les faire entrer sous l’intégrale. Fixons $t$, $\lambda$ et $\xi$, calculons $$\sum_{u,v\in {\cal B}, u',v'\in {\cal B}'}B_{u,v,u',v'}(\lambda+\xi)\phi_{reg,\tilde{M}_{t}}^{\tilde{G}}(u,v,u',v';t,t\lambda'-\lambda;\lambda+\xi,0).$$ C’est la valeur en $\Lambda=0$ de la fonction $h_{\tilde{M}_{t}}^{\tilde{G}}(\Lambda)$ associée à la $(\tilde{G},\tilde{M}_{t})$-famille $(h(\Lambda,\tilde{P}))_{\tilde{P}\in {\cal P}(\tilde{M}_{t})}$ définie par $$h(\Lambda,\tilde{P})=\sum_{u,v\in {\cal B}, u',v'\in {\cal B}'}B_{u,v,u',v'}(\lambda+\xi)\phi_{reg}(u,v,u',v';t,t\lambda'-\lambda;\lambda+\xi,\Lambda,\tilde{P}).$$ Fixons $S''\in {\cal P}(M_{disc})$ tel que $S''\subset P$. Posons pour simplifier $\nu=t\lambda'-\lambda$, $\underline{\sigma}=\sigma_{\lambda}$, $\underline{\pi}=\pi_{\lambda}$. En revenant aux définitions de 3.3 et 3.16, on a $$h(\Lambda,\tilde{P})=\sum_{u,v\in {\cal B}, u',v'\in {\cal B}'}(\underline{\pi}_{\xi}(\varphi_{1})U_{\theta,\underline{\sigma}_{\xi}}u',v)(u,\underline{\pi}_{\xi}(\varphi_{2})U_{\theta,\underline{\sigma}_{\xi}}v') r_{S'',reg}(\underline{\sigma}_{\xi})^{-1}r_{S'',reg}(\underline{\sigma}_{\xi+\Lambda})$$ $$(A(t, \nu;\lambda+\xi)v',R_{\bar{S}''\vert S}(\underline{\sigma}_{\xi})^{-1}\circ R_{\bar{S}''\vert S}(\underline{\sigma}_{\xi+\Lambda})u)$$ $$(R_{S''\vert S}(\underline{\sigma}_{\xi})^{-1}\circ R_{S''\vert S}(\underline{\sigma}_{\xi+\Lambda})v,A(t,\nu;\lambda+\xi)u').$$ Les sommes en $u$ et $v$ se simplifient en $$(2) \qquad h(\Lambda,\tilde{P})=\sum_{u',v'\in {\cal B}'} r_{S'',reg}(\underline{\sigma}_{\xi})^{-1}r_{S'',reg}(\underline{\sigma}_{\xi+\Lambda})$$ $$(A(t, \nu;\lambda+\xi)v',R_{\bar{S}''\vert S}(\underline{\sigma}_{\xi})^{-1}\circ R_{\bar{S}''\vert S}(\underline{\sigma}_{\xi+\Lambda})\underline{\pi}_{\xi}(\varphi_{2})U_{\theta,\underline{\sigma}_{\xi}}v')$$ $$(R_{S''\vert S}(\underline{\sigma}_{\xi})^{-1}\circ R_{S''\vert S}(\underline{\sigma}_{\xi+\Lambda})\underline{\pi}_{\xi}(\varphi_{1})U_{\theta,\underline{\sigma}_{\xi}}u',A(t,\nu;\lambda+\xi)u').$$ On a $$\sum_{v'\in {\cal B}'}(A(t, \nu;\lambda+\xi)v',R_{\bar{S}''\vert S}(\underline{\sigma}_{\xi})^{-1}\circ R_{\bar{S}''\vert S}(\underline{\sigma}_{\xi+\Lambda})\underline{\pi}_{\xi}(\varphi_{2})U_{\theta,\underline{\sigma}_{\xi}}v')$$ $$=\sum_{v'\in {\cal B}'}( v',A(t, \nu;\lambda+\xi)^{-1}R_{\bar{S}''\vert S}(\underline{\sigma}_{\xi})^{-1}\circ R_{\bar{S}''\vert S}(\underline{\sigma}_{\xi+\Lambda})\underline{\pi}_{\xi}(\varphi_{2})U_{\theta,\underline{\sigma}_{\xi}}v')$$ $$=trace(A(t, \nu;\lambda+\xi)^{-1}R_{\bar{S}''\vert S}(\underline{\sigma}_{\xi})^{-1}\circ R_{\bar{S}''\vert S}(\underline{\sigma}_{\xi+\Lambda})\underline{\pi}_{\xi}(\varphi_{2})U_{\theta,\underline{\sigma}_{\xi}})$$ $$(3) \qquad =trace( R_{\bar{S}''\vert S}(\underline{\sigma}_{\xi})^{-1}\circ R_{\bar{S}''\vert S}(\underline{\sigma}_{\xi+\Lambda})\underline{\pi}_{\xi}(\varphi_{2})U_{\theta,\underline{\sigma}_{\xi}}A(t, \nu;\lambda+\xi)^{-1}).$$ La condition $\lambda\in [\sigma]_{t}$ équivaut à l’équivalence $t\underline{\sigma}_{\xi}'\simeq\omega \underline{\sigma}_{\xi}$, ou encore $\underline{\sigma}_{\xi}\circ ad_{\gamma}\simeq \omega\underline{\sigma}_{\xi}$, où $\gamma=\gamma_{0}t^{-1}$ (en identifiant $t$ à un relèvement dans $G(F)$). On vérifie que le couple $(A_{\nu},\gamma)$ appartient à ${\cal N}^{\tilde{M_{t}}}(\underline{\sigma}_{\xi})\subset {\cal N}^{\tilde{G}}(\underline{\sigma}_{\xi})$. En considérant $(A_{\nu},\gamma)$ comme un élément de ce dernier ensemble, on a défini l’opérateur $\tilde{\nabla}_{S}(A_{\nu},\gamma)$ en 2.8. En comparant les définitions, on obtient l’égalité $$U_{\theta,\underline{\sigma}_{\xi}}A(t, \nu;\lambda+\xi)^{-1}=\tilde{\nabla}_{S}(A_{\nu},\gamma)\underline{\pi}_{\xi}(t).$$ Relevons $\xi$ en l’élément $\tilde{\xi}\in i\tilde{{\cal A}}_{\tilde{M}_{t},F}^*$ tel que $<\tilde{\xi},\tilde{H}_{\tilde{M}_{t}}(\gamma)>=0$. A l’aide de cet élément, on identifie ${\cal R}^{\tilde{M}_{t}}(\underline{\sigma})$ à ${\cal R}^{\tilde{M}_{t}}(\underline{\sigma}_{\xi})$. Notons $\boldsymbol{\tilde{r}}$ l’image de $(A_{\nu},\gamma)$ dans ${\cal R}^{\tilde{M_{t}}}(\underline{\sigma}_{\xi})={\cal R}^{\tilde{M_{t}}}(\underline{\sigma})$, posons $\boldsymbol{\tau}=(M_{disc},\underline{\sigma},\boldsymbol{\tilde{r}})$ et $\tau=(M_{disc},\underline{\sigma},\tilde{r})$, où $\tilde{r}$ est l’image de $\boldsymbol{\tilde{r}}$ dans $R^{\tilde{M}_{t}}(\underline{\sigma})$. Introduisons la représentation de $\tilde{G}(F)$ associée en 2.8 à $\boldsymbol{\tau}_{\tilde{\xi}}$, vu comme un triplet pour $\tilde{G}$, que l’on note ici simplement $\underline{\tilde{\Pi}}_{\xi}$. Alors $$\tilde{\nabla}_{S}(A_{\nu},\gamma)\underline{\pi}_{\xi}(t)=\omega(t)^{-1}\underline{\tilde{\Pi}}_{\xi}(\gamma_{0}).$$ En revenant à la définition de $\varphi_{2}$ en 3.2, on voit que $$\underline{\pi}_{\xi}(\varphi_{2})U_{\theta,\underline{\sigma}_{\xi}}A(t, \nu;\lambda+\xi)^{-1}=\omega(t)^{-1}\underline{\tilde{\Pi}}_{\xi}(f_{2}).$$ L’expression (3) devient simplement $$\omega(t)^{-1}trace( R_{\bar{S}''\vert S}(\underline{\sigma}_{\xi})^{-1}\circ R_{\bar{S}''\vert S}(\underline{\sigma}_{\xi+\Lambda})\underline{\tilde{\Pi}}_{\xi}(f_{2})).$$ On traite de même la somme en $u'$. Les opérateurs portant cette fois sur la première variable des produits scalaires, on obtient une expression conjuguée. En particulier, il sort un facteur $\overline{\omega(t)}^{-1}$, dont le produit avec $\omega(t)^{-1}$ ci-dessus vaut $1$. L’expression (2) devient $$(4) \qquad h(\Lambda,\tilde{P})= r_{S'',reg}(\underline{\sigma}_{\xi})^{-1}r_{S'',reg}(\underline{\sigma}_{\xi+\Lambda})\overline{trace( R_{S''\vert S}(\underline{\sigma}_{\xi})^{-1}\circ R_{S''\vert S}(\underline{\sigma}_{\xi+\Lambda})\underline{\tilde{\Pi}}_{\xi}(f_{1}))}$$ $$trace( R_{\bar{S}''\vert S}(\underline{\sigma}_{\xi})^{-1}\circ R_{\bar{S}''\vert S}(\underline{\sigma}_{\xi+\Lambda})\underline{\tilde{\Pi}}_{\xi}(f_{2})).$$ Tous les termes sont relatifs à un parabolique $S$ de référence, que l’on a fixé au début du calcul. Fixons un élément $\tilde{P}_{1}\in {\cal P}(\tilde{M}_{t})$ et un élément $S_{1}\in {\cal P}(M_{disc})$ tel que $S_{1}\subset P_{1}$. Remplaçons $S$ par $S_{1}$ dans les définitions des termes ci-dessus. On obtient une nouvelle expression $h_{1}(\Lambda,\tilde{P})$. On a
\(5) les valeurs en $\Lambda=0$ des fonctions $h_{\tilde{M}}^{\tilde{G}}(\Lambda)$ et $h_{1,\tilde{M}}^{\tilde{G}}(\Lambda)$ sont égales.
On a $$trace( R_{\bar{S}''\vert S}(\underline{\sigma}_{\xi})^{-1}\circ R_{\bar{S}''\vert S}(\underline{\sigma}_{\xi+\Lambda})\underline{\tilde{\Pi}}_{\xi}(f_{2}))=$$ $$trace( R_{S_{1}\vert S}(\underline{\sigma}_{\xi})^{-1}R_{\bar{S}''\vert S_{1}}(\underline{\sigma}_{\xi})^{-1}\circ R_{\bar{S}''\vert S_{1}}(\underline{\sigma}_{\xi+\Lambda})R_{S_{1}\vert S}(\underline{\sigma}_{\xi+\Lambda})\underline{\tilde{\Pi}}_{\xi}(f_{2}))$$ $$=trace( R_{\bar{S}''\vert S_{1}}(\underline{\sigma}_{\xi})^{-1}\circ R_{\bar{S}''\vert S_{1}}(\underline{\sigma}_{\xi+\Lambda})R_{S_{1}\vert S}(\underline{\sigma}_{\xi+\Lambda})\underline{\tilde{\Pi}}_{\xi}(f_{2})R_{S_{1}\vert S}(\underline{\sigma}_{\xi})^{-1}).$$ Le produit final $$R_{S_{1}\vert S}(\underline{\sigma}_{\xi+\Lambda})\underline{\tilde{\Pi}}_{\xi}(f_{2})R_{S_{1}\vert S}(\underline{\sigma}_{\xi})^{-1}$$ ne dépend pas de $S''$. Un principe général valable pour toute $(\tilde{G},\tilde{M})$-famille dit que l’on ne modifie pas la valeur $h_{\tilde{M}}^{\tilde{G}}(0)$ si on remplace un facteur indépendant de $S''$ par sa valeur en $\Lambda=0$. Mais, en $\Lambda=0$, le produit ci-dessus est égal à $ \underline{\tilde{\Pi}}_{1,\xi}(f_{2})$, où $ \underline{\tilde{\Pi}}_{1,\xi}$ est l’analogue de $ \underline{\tilde{\Pi}}_{\xi}$ relatif au parabolique $S_{1}$. Un même raisonnement s’applique aux autres termes et (5) s’ensuit.
En oubliant ce changement de $S$ en $S_{1}$, on suppose pour simplifier que $S\subset P_{1}$. Pour calculer $h(\Lambda,\tilde{P})$, on a choisi $S''$ avec $S''\subset P$. On peut lui imposer de plus $S''\cap M_{t}=S\cap M_{t}$. Introduisons l’élément $\underline{S}''\in {\cal P}(M_{disc})$ tel que $\underline{S}''\subset \bar{P}$ et $\underline{S}''\cap M_{t}=S\cap M_{t}$. On a $$R_{\bar{S}''\vert S}(\underline{\sigma}_{\xi})^{-1}\circ R_{\bar{S}''\vert S}(\underline{\sigma}_{\xi+\Lambda})=R_{\underline{S}''\vert S}(\underline{\sigma}_{\xi})^{-1}\circ R_{\bar{S}''\vert \underline{S}''}(\underline{\sigma}_{\xi})^{-1}\circ R_{\bar{S}''\vert \underline{S}''}(\underline{\sigma}_{\xi+\Lambda})\circ R_{\underline{S}''\vert S}(\underline{\sigma}_{\xi+\Lambda}).$$ Mais $\bar{S}''$ et $\underline{S}''$ sont tous deux contenus dans $\bar{P}$. L’opérateur $R_{\bar{S}''\vert \underline{S}''}(\underline{\sigma}_{\xi+\Lambda})$ est induit de l’opérateur $R_{\bar{S}''\cap M_{t}\vert \underline{S}''\cap M_{t}}^{M_{t}}(\underline{\sigma}_{\xi+\Lambda})$ et celui-ci ne dépend pas de $\Lambda$ puisque $\Lambda\in i{\cal A}_{\tilde{M}_{t},F}^*$. D’où $$R_{\bar{S}''\vert \underline{S}''}(\underline{\sigma}_{\xi})^{-1}\circ R_{\bar{S}''\vert \underline{S}''}(\underline{\sigma}_{\xi+\Lambda})=1,$$ puis $$R_{\bar{S}''\vert S}(\underline{\sigma}_{\xi})^{-1}\circ R_{\bar{S}''\vert S}(\underline{\sigma}_{\xi+\Lambda})=R_{\underline{S}''\vert S}(\underline{\sigma}_{\xi})^{-1}\circ R_{\underline{S}''\vert S}(\underline{\sigma}_{\xi+\Lambda}).$$ Rétablissons maintenant les opérateurs d’entrelacement non normalisés. La formule (4) se transforme en $$(6)\qquad h(\Lambda,\tilde{P})={\bf r}_{S''}(\underline{\sigma}_{\xi})^{-1}{\bf r}_{S''}(\underline{\sigma}_{\xi+\Lambda})\overline{trace(J_{S''\vert S}(\underline{\sigma}_{\xi})^{-1}\circ J_{S''\vert S}(\underline{\sigma}_{\xi+\Lambda})\underline{\tilde{\Pi}}_{\xi}(f_{1}))}$$ $$trace(J_{\underline{S}''\vert S}(\underline{\sigma}_{\xi})^{-1}\circ J_{\underline{S}''\vert S}(\underline{\sigma}_{\xi+\Lambda})\underline{\tilde{\Pi}}_{\xi}(f_{2})),$$ où $${\bf r}_{S''}(\underline{\sigma}_{\xi})=r_{S'',reg}(\underline{\sigma}_{\xi})r_{\underline{S}''\vert S}(\underline{\sigma}_{\xi})^{-1}r_{S\vert S''}(\underline{\sigma}_{\xi})^{-1}=r_{S'',reg}(\underline{\sigma}_{\xi})r_{\underline{S}''\vert S''}(\underline{\sigma}_{\xi})^{-1},$$ puisque les distances entre $\underline{S}''$ et $S$ et entre $S$ et $S''$ s’ajoutent. Si l’on suppose $\xi$ en position générale, tous les termes ci-dessus sont bien définis. Introduisons la représentation $\tilde{\pi}_{\boldsymbol{\tau}_{\tilde{\xi}}}$ de $\tilde{M}_{t}(F)$ associée à $\boldsymbol{\tau}_{\tilde{\xi}}$. On note $\pi_{\tau_{\xi}}$ la représentation sous-jacente de $M_{t}(F)$. On a simplement $\underline{\tilde{\Pi}}_{\xi}=\tilde{\Pi}_{\boldsymbol{\tau}_{\tilde{\xi}}}=Ind_{\tilde{P}_{1}}^{\tilde{G}}(\tilde{\pi}_{\boldsymbol{\tau}_{\tilde{\xi}}})$. En identifiant ces représentations, les propriétés d’induction des opérateurs d’entrelacement entraînent que $J_{\underline{S}''\vert S}(\underline{\sigma}_{\xi})^{-1}\circ J_{\underline{S}''\vert S}(\underline{\sigma}_{\xi+\Lambda})$ coïncide avec $J_{\bar{P}\vert P_{1}}(\pi_{\tau_{\xi}})^{-1}\circ J_{\bar{P}\vert P_{1}}(\pi_{\tau_{\xi+\Lambda}})$. En appliquant les définitions, la formule (6) se récrit $$(7) \qquad h(\Lambda,\tilde{P})=C(\underline{\sigma}_{\xi},\Lambda,\tilde{P})\overline{trace({\cal M}(\pi_{\tau_{\xi}};\Lambda,\tilde{P})\underline{\tilde{\Pi}}_{\xi}(f_{1}))}trace({\cal M}(\pi_{\tau_{\xi}};\Lambda,\tilde{\bar{P}})\underline{\tilde{\Pi}}_{\xi}(f_{2})),$$ où $$C(\underline{\sigma}_{\xi},\Lambda,\tilde{P})={\bf r}_{S''}(\underline{\sigma}_{\xi})^{-1}{\bf r}_{S''}(\underline{\sigma}_{\xi+\Lambda})\mu_{\bar{P}\vert P_{1}}(\pi_{\tau_{\xi}})\mu_{\bar{P}\vert P_{1}}( \pi_{\tau_{\xi+\Lambda/2}})^{-1}\overline{\mu_{P\vert P_{1}}(\pi_{\tau_{\xi}})\mu_{P\vert P_{1}}(\pi_{\tau_{\xi+\Lambda/2}})^{-1}}.$$
On a l’égalité $$\mu_{\bar{P}\vert P_{1}}(\pi_{\tau_{\xi}})\overline{\mu_{P\vert P_{1}}(\pi_{\tau_{\xi}})}=r_{\bar{P}\vert P}(\underline{\sigma}_{\xi})^{-1}r_{P\vert \bar{P}}(\underline{\sigma}_{\xi})^{-1}.$$ Ce terme est indépendant de $\tilde{P}$: c’est le produit des $r_{\alpha}(\underline{\sigma}_{\xi})^{-1}$ pour toutes les racines simples $\alpha$ de $A_{M_{disc}}$ intervenant dans $G$ et pas dans $M_{t}$. Pour la même raison, le produit $$\mu_{\bar{P}\vert P_{1}}( \pi_{\tau_{\xi+\Lambda/2}})^{-1}\overline{\mu_{P\vert P_{1}}(\pi_{\tau_{\xi+\Lambda/2}})^{-1}}$$ est indépendant de $\tilde{P}$. Comme dans la preuve de (5), on ne change pas le terme $h_{\tilde{M}_{t}}^{\tilde{G}}(0)$ en remplaçant ces termes par leurs valeurs en $\Lambda=0$. Mais alors le produit de ces deux termes vaut $1$. On peut définir ${\bf r}_{S''}(\underline{\sigma}_{\xi+\mu})$ pour tout $\mu\in i{\cal A}_{M_{disc},F}^*$. Par définition ${\bf r}_{S''}(\underline{\sigma}_{\xi})$ est la valeur de cette fonction en $\mu=0$. Pour $\mu$ en position générale, on a $${\bf r}_{S''}(\underline{\sigma}_{\xi+\mu})=r_{\bar{S}''\vert S''}(\underline{\sigma}_{\xi+\mu})r_{\underline{S}''\vert S''}(\underline{\sigma}_{\xi+\mu})^{-1}r_{\bar{S}\vert S}(\underline{\sigma}_{\xi+\mu})^{-1}.$$ On introduit l’élément $\underline{S}\in {\cal P}(M_{disc})$ tel que $\underline{S}\subset \bar{P}_{1}$ et $\underline{S}\cap M_{t}=S\cap M_{t}$. Les distances entre $\bar{S}''$ et $ \underline{S}''$ et entre $\underline{S}''$ et $S''$ s’ajoutent. De même, les distances entre $\bar{S}$ et $ \underline{S}$ et entre $\underline{S}$ et $S$ s’ajoutent. D’où $${\bf r}_{S''}(\underline{\sigma}_{\xi+\mu})=r_{\bar{S}''\vert \underline{S}''}(\underline{\sigma}_{\xi+\mu})r_{\bar{S}\vert \underline{S}}(\underline{\sigma}_{\xi+\mu})^{-1}r_{\underline{S}\vert S}(\underline{\sigma}_{\xi+\mu})^{-1}.$$ Par les propriétés habituelles d’induction, $$r_{\bar{S}''\vert \underline{S}''}(\underline{\sigma}_{\xi+\mu})=r^{M_{t}}_{(\bar{S}''\cap M_{t})\vert (\underline{S}''\cap M_{t})}(\underline{\sigma}_{\xi+\mu})=r^{M_{t}}_{(\bar{S}\cap M_{t})\vert (\underline{S}\cap M_{t})}(\underline{\sigma}_{\xi+\mu})=r_{\bar{S}\vert \underline{S}}(\underline{\sigma}_{\xi+\mu}).$$ D’où ${\bf r}_{S''}(\underline{\sigma}_{\xi+\mu})=r_{\underline{S}\vert S}(\underline{\sigma}_{\xi+\mu})^{-1}$. Ce terme est régulier en $\mu=0$ (pour $\xi$ en position générale), d’où ${\bf r}_{S''}(\underline{\sigma}_{\xi})=r_{\underline{S}\vert S}(\underline{\sigma}_{\xi})^{-1}$. On obtient $${\bf r}_{S''}(\underline{\sigma}_{\xi})^{-1}{\bf r}_{S''}(\underline{\sigma}_{\xi+\Lambda})=r_{\underline{S}\vert S}(\underline{\sigma}_{\xi})r_{\underline{S}\vert S}(\underline{\sigma}_{\xi+\Lambda})^{-1}.$$ Ce terme est indépendant de $S''$. Comme ci-dessus, on ne change pas le terme $h_{\tilde{M}_{t}}^{\tilde{G}}(0)$ en remplaçant l’expression précédente par sa valeur en $\Lambda=0$. Mais celle-ci est $1$. Cela prouve qu’on peut supprimer le terme $C(\underline{\sigma}_{\xi},\Lambda,\tilde{P})$ de la formule (7) sans changer la valeur de $h_{\tilde{M}_{t}}^{\tilde{G}}(0)$. Quand on supprime ce terme $C(\underline{\sigma}_{\xi},\Lambda,\tilde{P})$, le membre de droite de (6) devient ${\cal J}(\pi_{\tau_{\xi}},f_{1},f_{2},\tilde{\bar{P}})$. Si c’était plus simplement ${\cal J}(\pi_{\tau_{\xi}},f_{1},f_{2},\tilde{P})$, on en déduirait $h_{\tilde{M}_{t}}^{\tilde{G}}(0)=J_{\tilde{M}_{t}}^{\tilde{G}}(\pi_{\tau_{\xi}},f_{1},f_{2})$. A cause du changement de $P$ en $\bar{P}$ et parce que $\epsilon_{\tilde{\bar{P}}}^{\tilde{G}}(\Lambda)=(-1)^{a_{\tilde{M}_{t}}-a_{\tilde{G}}}\epsilon_{\tilde{P}}^{\tilde{G}}(\Lambda)$, on obtient $$h_{\tilde{M}_{t}}^{\tilde{G}}(0)=(-1)^{a_{\tilde{M}_{t}}-a_{\tilde{G}}}J_{\tilde{M}_{t}}^{\tilde{G}}(\pi_{\tau_{\xi}},f_{1},f_{2}).$$
On doit aussi calculer le terme $\epsilon(t,\nu;\lambda+\xi)$ qui intervient dans (1). D’après sa définition en 3.16, c’est la valeur en $\mu=\lambda+\xi$ de $r_{\bar{S}\vert S}(\sigma_{\mu})r_{\bar{S}\vert S}(\sigma_{t\mu'-\nu})^{-1}$. Ou encore la valeur en $\mu=0$ de $r_{\bar{S}\vert S}(\underline{\sigma}_{\xi+\mu})r_{\bar{S}\vert S}(\underline{\sigma}_{\xi+t\mu'})^{-1}$. Le calcul est fait par Arthur en \[A1\] p.87. Le rapport précédent est égal à $$(9)\qquad \prod_{\alpha>_{S}0}r_{\alpha}(\underline{\sigma}_{\xi+\mu})r_{\alpha}(\underline{\sigma}_{\xi+t\mu'})^{-1},$$ où $\alpha$ parcourt les racines simples de $A_{M_{disc}}$ dans $G$ qui sont positives pour $S$. Si $\alpha$ n’intervient pas dans $M_{t}$, $r_{\alpha}(\underline{\sigma}_{\xi+\mu})$ est régulière en $\mu=0$ puisqu’on a supposé $\xi$ en position générale. Le rapport $r_{\alpha}(\underline{\sigma}_{\xi+\mu})r_{\alpha}(\underline{\sigma}_{\xi+t\mu'})^{-1}$ vaut $1$ en $\mu=0$. Supposons que $\alpha$ intervient dans $M_{t}$. On définit un Levi $M_{\alpha}$ contenant $M_{disc}$ comme en 1.11. Si $m^{M_{\alpha}}(\underline{\sigma}_{\xi})\not=0$, $r_{\alpha}(\underline{\sigma}_{\xi+\mu})$ est encore régulière en $\mu=0$ et le rapport $r_{\alpha}(\underline{\sigma}_{\xi+\mu})r_{\alpha}(\underline{\sigma}_{\xi+t\mu'})^{-1}$ vaut encore $1$ en $\mu=0$. Supposons $m^{M_{\alpha}}(\underline{\sigma}_{\xi})=0$. On sait qu’alors $r_{\alpha}(\underline{\sigma}_{\xi+\mu})$ a un pôle d’ordre $1$ en $\mu=0$. Plus précisément, $r_{\alpha}(\underline{\sigma}_{\xi+\mu})$ est équivalente au voisinage de $\mu=0$ à $c_{\alpha}<\mu,\check{\alpha}>$, où $c_{\alpha}$ est un nombre réel non nul. Le rapport $r_{\alpha}(\underline{\sigma}_{\xi+\mu})r_{\alpha}(\underline{\sigma}_{\xi+t\mu'})^{-1}$ est donc équivalent à $<\mu,\check{\alpha}><t\mu',\check{\alpha}>^{-1}$, ou encore à $<\mu,\check{\alpha}><\mu,\theta t^{-1}\check{\alpha}>^{-1}$. En notant $\Sigma$ l’ensemble des racines $\alpha$ intervenant dans $M_{t}$ pour lesquelles $m^{M_{\alpha}}(\underline{\sigma}_{\xi})=0$, l’expression (1) est donc équivalente à $$\prod_{\alpha\in \Sigma; \alpha>_{S}0}<\mu,\check{\alpha}><\mu,\theta t^{-1}\check{\alpha}>^{-1}.$$ Ce produit est égal au signe $\epsilon_{\underline{\sigma}}(w)$ défini en 2.9, où $w$ est l’image de $\gamma=\gamma_{0}t^{-1}$ dans l’ensemble $W^{\tilde{M_{t}}}(\underline{\sigma}_{\xi})=W^{\tilde{M_{t}}}(\underline{\sigma})$. D’où l’égalité $$\epsilon(t,\nu;\lambda+\xi)=\epsilon_{\underline{\sigma}}(w).$$
A ce point, on a transformé la formule (1) en $$\sum_{t\in W^G(M_{disc}\vert M'_{disc})}(-1)^{a_{\tilde{M}_{t}}-a_{\tilde{G}}} \vert det((1-t\theta^{-1})_{\vert {\cal A}_{M_{disc}}^{\tilde{M}_{t}}})\vert ^{-1}$$ $$\sum_{\lambda\in [\sigma]_{t}}\epsilon_{\sigma_{\lambda}}(w)\int_{i{\cal A}_{\tilde{M}_{t},F}^*}J_{\tilde{M}_{t}}^{\tilde{G}}(\pi_{\tau_{\xi}},f_{1},f_{2})\,d\xi.$$ On rappelle que $w$ est l’image de $\gamma=\gamma_{0}t^{-1}$ dans $W^{\tilde{M_{t}}}(\sigma_{\lambda})$ et que $\tau=(M_{disc},\sigma_{\lambda},r)$, où $r$ est l’image de $ w$ dans $R^{\tilde{M}_{t}}(\sigma_{\lambda})$. On décompose la formule ci-dessus selon les espaces de Levi $\tilde{M}_{t}$. Elle devient $$(10)\qquad \sum_{\tilde{M};M_{disc}\subset M}X(\tilde{M},M_{disc},\sigma),$$ où $X(\tilde{M},M_{disc},\sigma)$ est l’expression obtenue à partir de l’expression ci-dessus en limitant la somme aux $t$ tels que $\tilde{M}_{t}=\tilde{M}$. Fixons $\tilde{M}$. On peut remplacer la double somme en $t\in W^G(M_{disc},M'_{disc})$ tels que $\tilde{M}_{t}=\tilde{M}$ et en $\lambda\in [\sigma]_{t}$ par une double somme en $\lambda\in i{\cal A}_{M_{disc}}^*/(i{\cal A}_{M_{disc},F}^{\vee}+i{\cal A}_{\tilde{M}}^*)$ et en $t\in W^G(M_{disc},M'_{disc})$ tels que $\tilde{M}_{t}=\tilde{M}$ et $\sigma_{\lambda}\circ ad_{\gamma_{0}t^{-1}}\simeq \omega\sigma_{\lambda}$. Fixons $\lambda$. L’application $t\mapsto w$ est alors une bijection entre l’ensemble de sommation en $t$ et l’ensemble $W^{\tilde{M}}_{reg}(\sigma_{\lambda})$. On peut encore décomposer cette somme en une somme sur $r\in R^{\tilde{M}}(\sigma_{\lambda})$, que l’on écrit $r=W_{0}^{M}(\sigma_{\lambda})w$, et une somme sur $w'\in W_{0}^{M}(\sigma_{\lambda})w\cap W_{reg}^{\tilde{M}}(\sigma_{\lambda})$. On voit que les seuls termes de l’expression qui dépendent de $w'$ sont $\epsilon_{\sigma_{\lambda}}(w')\vert det((1-w')_{\vert {\cal A}_{M_{disc}}^{\tilde{M}}}\vert ^{-1}$. Leur somme vaut $\vert W_{0}^M(\sigma_{\lambda})\vert \iota(\tau)$ où $\tau$ est comme ci-dessus. D’où $$X(\tilde{M},M_{disc},\sigma)= (-1)^{a_{\tilde{M}}-a_{\tilde{G}}}\sum_{\lambda\in i{\cal A}_{M_{disc}}^*/(i{\cal A}_{M_{disc},F}^{\vee}+i{\cal A}_{\tilde{M}}^*)}\vert W_{0}^M(\sigma_{\lambda})\vert$$ $$\sum_{r\in R^{\tilde{M}}(\sigma_{\lambda})} \iota(\tau)\int_{i{\cal A}_{\tilde{M},F}^*}J_{\tilde{M}}^{\tilde{G}}(\pi_{\tau_{\xi}},f_{1},f_{2})\,d\xi.$$ Notons $\Pi_{disc}(M_{disc}(F))/i{\cal A}_{\tilde{M},F}^*$ l’ensemble des orbites dans $\Pi_{disc}(M_{disc}(F))$ pour l’action de $i{\cal A}_{\tilde{M},F}^*$. Pour $\underline{\sigma}\in \Pi_{disc}(M_{disc}(F))$, notons $Stab(i{\cal A}_{\tilde{M},F}^*,\underline{\sigma})$ son stabilisateur dans $i{\cal A}_{\tilde{M},F}^*$. Il ne dépend que de l’image de $\underline{\sigma}$ dans l’ensemble d’orbites précédent. Notons ${\cal O}_{\sigma}$ l’orbite de $\sigma$ pour l’action de $i{\cal A}_{M_{disc},F}^*$. Elle se décompose en orbites pour l’action de $i{\cal A}_{\tilde{M},F}^*$, notons ${\cal O}_{\sigma}/i{\cal A}_{\tilde{M},F}^*$ l’ensemble de ces orbites. Notons que $Stab(i{\cal A}_{\tilde{M},F}^*,\underline{\sigma})$ est indépendant de $\underline{\sigma}$ dans cet ensemble. L’application qui, à $\lambda$, associe l’orbite de $\sigma_{\lambda}$ pour l’action de $i{\cal A}_{\tilde{M},F}^*$, a pour image ${\cal O}_{\sigma}/i{\cal A}_{\tilde{M},F}^*$ et a un noyau d’ordre $\vert Stab(i{\cal A}_{M_{disc},F}^*,\sigma)\vert \vert Stab(i{\cal A}_{\tilde{M},F}^*,\sigma)\vert ^{-1} $. Alors $$X(\tilde{M},M_{disc},\sigma)= (-1)^{a_{\tilde{M}}-a_{\tilde{G}}}\vert Stab(i{\cal A}_{M_{disc},F}^*,\sigma)\vert$$ $$\sum_{\underline{\sigma}\in {\cal O}_{\sigma}/i{\cal A}_{\tilde{M},F}^*}\vert Stab(i{\cal A}_{\tilde{M},F}^*,\underline{\sigma})\vert ^{-1}\vert W_{0}^M(\underline{\sigma})\vert \sum_{r\in R^{\tilde{M}}(\underline{\sigma})} \iota(\tau)\int_{i{\cal A}_{\tilde{M},F}^*}J_{\tilde{M}}^{\tilde{G}}(\pi_{\tau_{\xi}},f_{1},f_{2})\,d\xi,$$ où maintenant $\tau=(M_{disc},\underline{\sigma},\tilde{r})$.
Considérons la formule 3.2(1). Le terme constant de son membre de droite est la somme sur $M_{disc}\in {\cal L}(M_{0})$ et sur $\sigma\in \Pi_{disc}(M_{disc}(F))/i{\cal A}_{M_{disc},F}^*$ de l’expression (10), multipliée par $\vert W^{M_{disc}}\vert \vert W^G\vert ^{-1}\vert Stab(i{\cal A}_{M_{disc},F}^*,\sigma)\vert^{-1}$. On peut l’écrire $$(11) \qquad \sum_{\tilde{M}; M_{0}\subset M}\vert W^M\vert \vert W^G\vert ^{-1}X(\tilde{M}),$$ où $$X(\tilde{M})=\sum_{M_{disc};M_{0}\subset M_{disc}\subset M}\vert W^{M_{disc}}\vert \vert W^M\vert ^{-1}$$ $$\sum_{\sigma\in \Pi_{disc}(M_{disc}(F))/i{\cal A}_{M_{disc},F}^*}\vert Stab(i{\cal A}_{M_{disc},F}^*,\sigma)\vert^{-1} X(\tilde{M},M_{disc},\sigma).$$ Fixons $\tilde{M}$. La somme en $\sigma$ et celle en $\underline{\sigma}\in {\cal O}_{\sigma}/i{\cal A}_{\tilde{M},F}^*$ figurant dans $X(\tilde{M},M_{disc},\sigma)$ se simplifient en une somme sur l’ensemble $\Pi_{disc}(M_{disc}(F))/i{\cal A}_{\tilde{M},F}^*$. Considérons les triplets $\tau$ qui apparaissent. Ils ne sont pas forcément essentiels, mais on peut se limiter aux essentiels (pour $\tilde{M}$) puisque la contribution des autres est nulle d’après 3.25(1). Ils sont alors discrets, puisque sinon, le terme $\iota(\tau)$ est nul. On a donc $$(12)\qquad X(\tilde{M})= (-1)^{a_{\tilde{M}}-a_{\tilde{G}}}\sum_{\tau\in (E_{disc}(\tilde{M},\omega)/conj)/i{\cal A}_{\tilde{M},F}^*}C(\tau)\iota(\tau)\int_{i{\cal A}_{\tilde{M},F}^*}J_{\tilde{M}}^{\tilde{G}}(\pi_{\tau_{\xi}},f_{1},f_{2})\,d\xi,$$ où $C(\tau)$ est la somme de $$(13)\qquad \vert W^{M_{disc}}\vert \vert W^M\vert ^{-1}\vert Stab(i{\cal A}_{\tilde{M},F}^*,\underline{\sigma}) \vert ^{-1}\vert W_{0}^M(\underline{\sigma})\vert$$ sur les $M_{disc}$ tels que $M_{0}\subset M_{disc}\subset M$, $\underline{\sigma}\in \Pi_{disc}(M_{disc}(F))/i{\cal A}_{\tilde{M},F}^*$, $\tilde{r}\in R^{\tilde{G}}(\underline{\sigma})$ tels que $(M_{disc},\underline{\sigma},\tilde{r})$ ait pour image $\tau$ dans $(E_{disc}(\tilde{M},\omega)/conj)/i{\cal A}_{\tilde{M},F}^*$. Si on fixe un représentant $(M_{disc},\underline{\sigma},\tilde{r})$ de $\tau$, l’ensemble de sommation est celui des différents $(w(M_{disc}),w\underline{\sigma}, w(\tilde{r}))$ pour $w\in W^M$. Ou encore l’ensemble de ces triplets pour $w\in W^M/Stab(W^M,\tau/i{\cal A}_{\tilde{M},F}^*)$, où $Stab(W^M,\tau/i{\cal A}_{\tilde{M},F}^*)$ est le stabilisateur de $\tau\in E_{disc}(\tilde{M},\omega)/i{\cal A}_{\tilde{M},F}^*$ dans $W^M$. Le terme (13) est constant sur l’ensemble de sommation. D’où $$C(\tau)=\vert W^M/Stab(W^M,\tau/i{\cal A}_{\tilde{M},F}^*)\vert \vert W^{M_{disc}}\vert \vert W^M\vert ^{-1}\vert Stab(i{\cal A}_{\tilde{M},F}^*,\underline{\sigma})\vert ^{-1}\vert W_{0}^M(\underline{\sigma})\vert$$ $$=\vert Stab(W^M,\tau/i{\cal A}_{\tilde{M},F}^*)\vert ^{-1} \vert W^{M_{disc}}\vert\vert Stab(i{\cal A}_{\tilde{M},F}^*,\underline{\sigma})\vert ^{-1}\vert W_{0}^M(\underline{\sigma})\vert.$$ On a défini en 2.9 le groupe $Stab(W^M\times i{\cal A}_{\tilde{M},F}^*,\tau)$ des $(w,\lambda)\in W^M\times i{\cal A}_{\tilde{M},F}^*$ qui conservent $\tau$. Prenons garde que dans la définition de ce groupe, $\tau$ est considéré comme un élément de $E_{disc}(\tilde{M},\omega)$ tandis que, dans celle ci-dessus de $Stab(W^M,\tau/i{\cal A}_{\tilde{M},F}^*)$, $\tau$ est considéré comme un élément du quotient $E_{disc}(\tilde{M},\omega)/i{\cal A}_{\tilde{M},F}^*$. On voit qu’il y a une suite exacte $$1\to Stab(i{\cal A}_{\tilde{M},F}^*,\underline{\sigma})\to Stab(W^M\times i{\cal A}_{\tilde{M},F}^*,\tau)\to Stab(W^M,\tau/i{\cal A}_{\tilde{M},F}^*)\to 1$$ Donc $$\vert Stab(W^M,\tau/i{\cal A}_{\tilde{M},F}^*)\vert \vert Stab(i{\cal A}_{\tilde{M},F}^*,\underline{\sigma})\vert =\vert Stab(W^M\times i{\cal A}_{\tilde{M},F}^*,\tau)\vert .$$ En appliquant les définitions de 2.9, on a aussi $$\vert Stab(W^M\times i{\cal A}_{\tilde{M},F}^*,\tau)\vert =\vert W^{M_{disc}}\vert \vert W_{0}^M(\underline{\sigma})\vert \vert {\bf Stab}(W^M\times i{\cal A}_{\tilde{M},F}^*,\tau)\vert .$$ On conclut $$C(\tau)=\vert {\bf Stab}(W^M\times i{\cal A}_{\tilde{M},F}^*,\tau)\vert .$$ Alors (12) devient $$X(\tilde{M})=(-1)^{a_{\tilde{M}}-a_{\tilde{G}}}J_{\tilde{M},spec}^{\tilde{G}}(\omega,f_{1},f_{2}).$$ Ce terme ne dépendant que de la classe de conjugaison de $\tilde{M}$, on vérifie que la somme (11) est égale à $$(14) \qquad \sum_{\tilde{M}\in{\cal L}( \tilde{M}_{0})}\vert \tilde{W}^M\vert \vert \tilde{W}^G\vert ^{-1}X(\tilde{M})$$ En effet, les deux termes sont égaux à $$\sum_{\tilde{P}=\tilde{M}U_{P};P_{0}\subset P}\vert {\cal P}(\tilde{M})\vert ^{-1}X(\tilde{M}).$$ On se rappelle que (11) est le terme constant de l’élément de $PolExp$ asymptote à $J^T(\omega,f_{1},f_{2})$. D’après la formule ci-dessus calculant $X(\tilde{M})$, (14) est égal à $J^{\tilde{G}}_{spec}(\omega,f_{1},f_{2})$. Cela prouve la proposition. $\square$
Le calcul géométrique
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La formule de Weyl
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Pour quelques instants, nous allons considérer des sous-groupes algébriques de $G$ qui ne sont pas forcément définis sur $F$. Considérons un sous-groupe de Borel $B$ de $G$ et un sous-tore maximal $T$ de $B$. On suppose $T$ défini sur $F$ mais $B$ défini seulement sur la clôture algébrique $\bar{F}$ de $F$. On définit leur normalisateur commun $\tilde{T}=\{\gamma\in \tilde{G}; ad_{\gamma}(T)=T, ad_{\gamma}(B)=B\}$. On dit que $\tilde{T}$ est un tore tordu maximal de $\tilde{G}$ si et seulement si $\tilde{T}\cap \tilde{G}(F)\not=\emptyset$. Remarquons que cela entraîne que $\tilde{T}$ est défini sur $F$: si on choisit $\gamma\in \tilde{T}\cap \tilde{G}(F)$, on a $\tilde{T} =T\gamma=\gamma T$. Remarquons aussi qu’à tout tore tordu maximal $\tilde{T}$ est associé un sous-tore maximal $T$ de $G$. Pour $\gamma\in \tilde{T}(F)$, la restriction de $ad_{\gamma}$ à $\tilde{T}$ ne dépend pas de $\gamma$. On la note simplement $\theta$. On note $A_{T}$ le plus grand sous-tore déployé de $T$ et $A_{\tilde{T}}$ le plus grand sous-tore contenu dans le sous-groupe $A_{T}^{\theta}$ des points fixes par $\theta$ (autrement dit, $A_{\tilde{T}}$ est la composante neutre $A_{T}^{\theta,0}$ de $A_{T}^{\theta}$). On dit que $\tilde{T}$ est elliptique dans $\tilde{G}$ si $A_{\tilde{T}}=A_{\tilde{G}}$. En général, notons $M$ le commutant de $A_{\tilde{T}}$ dans $G$ et posons $\tilde{M}=M\tilde{T}$. Alors $\tilde{M}$ est un espace de Levi et $\tilde{T}$ est un tordu maximal et elliptique de $\tilde{M}$.
Pour tout tore tordu maximal $\tilde{T}$, on munit le groupe $T^{\theta}(F)$ d’une mesure de Haar. On suppose que, si deux tores tordus maximaux sont conjugués par un élément de $G(F)$, cette conjugaison est compatible aux mesures.
Soit $\tilde{T}$ un tore tordu maximal. Le groupe $T(F)$ agit sur $\tilde{T}(F)$ par conjugaison. L’ensemble des classes de conjugaison est le quotient $\tilde{T}(F)/(1-\theta)(T(F))=(1-\theta)(T(F))\backslash \tilde{T}(F)$, où $(1-\theta)(T(F))=\{t\theta(t^{-1}); t\in T(F)\}$. Pour $\gamma\in \tilde{T}(F)$, l’application naturelle $$\begin{array}{ccc}T^{\theta,0}(F)&\to& \tilde{T}(F)/(1-\theta)(T(F))\\ t&\mapsto& t\gamma\\ \end{array}$$ est un isomorphisme local. On munit $\tilde{T}(F)/(1-\theta)(T(F))$ de la mesure invariante par translation à droite ou à gauche par $T(F)$ et telle que l’application ci-dessus préserve localement les mesures. Il est immédiat que cette définition ne dépend pas du choix de $\gamma$. On pose $$W^{G}(\tilde{T})=Norm_{G(F)}(\tilde{T})/T(F).$$
L’ensemble des classes de conjugaison par $G(F)$ de tores tordus maximaux est fini. Fixons un ensemble de représentants $T(\tilde{G})$. On note $T_{ell}(\tilde{G})$ le sous-ensemble des éléments elliptiques de $T(\tilde{G})$.
Pour $\gamma\in \tilde{G}$, on note $Z_{G}(\gamma)$ son centralisateur dans $G$ (l’ensemble des $x\in G$ tels que $x\gamma x^{-1}=\gamma$) et $G_{\gamma}$ la composante neutre de $Z_{G}(\gamma)$. On dit que $\gamma$ est fortement régulier si et seulement si $G_{\gamma}$ est un tore et $Z_{G}(\gamma)$ est commutatif. Un tel élément est semi-simple. On note $G_{reg}$ l’ensemble des éléments fortement réguliers. Un élément $\gamma\in G_{reg}(F)$ appartient à un unique tore tordu maximal: c’est $\tilde{T}=T\gamma$, où $T$ est le commutant de $G_{\gamma}$ dans $G$. On définit une fonction $D^{\tilde{G}}$ sur $\tilde{G}(F)$ de la façon suivante. Si $\gamma\in \tilde{G}(F)$ est semi-simple, $D^{\tilde{G}}(\gamma)$ est la valeur absolue du déterminant de $1-ad_{\gamma}$ agissant sur $\mathfrak{g}/\mathfrak{g}_{\gamma}$, où on note par des lettres gothiques les algèbres de Lie. Si $\gamma$ est quelconque, $D^{\tilde{G}}(\gamma)=D^{\tilde{G}}(\gamma_{ss})$, où $\gamma_{ss}$ est la partie semi-simple de $\gamma$.
La formule d’intégration de Weyl prend l’une ou l’autre des deux formes suivantes, pour une fonction $f\in C_{c}^{\infty}(\tilde{G}(F))$:
$$\int_{\tilde{G}(F)}f(\gamma)\,d\gamma\,=\sum_{\tilde{T}\in T(\tilde{G})}\vert W^G(\tilde{T})\vert ^{-1} \int_{\tilde{T}(F)/(1-\theta)(T(F))}\int_{T^{\theta}(F)\backslash G(F)}f(x^{-1}\gamma x)\,dx\,D^{\tilde{G}}(\gamma)\,d\gamma$$
$=\sum_{\tilde{M}\in {\cal L}(\tilde{M}_{0})}\vert \tilde{W}^M\vert \vert \tilde{W}^G\vert ^{-1}\sum_{\tilde{T}\in T_{ell}(\tilde{M})}\vert W^M(\tilde{T})\vert ^{-1}$ $$\int_{\tilde{T}(F)/(1-\theta)(T(F))}\int_{T^{\theta}(F)\backslash G(F)}f(x^{-1}\gamma x)\,dx\,D^{\tilde{G}}(\gamma)\,d\gamma.$$
Quelques majorations
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Soient $f\in C_{c}^{\infty}(\tilde{G}(F))$ et $\gamma\in \tilde{G}_{reg}(F)$ tel que $\omega$ soit trivial sur $Z_{G}(\gamma,F)$. On pose $$I_{\tilde{G}}(\gamma,\omega,f)=D^{\tilde{G}}(\gamma)^{1/2}\int_{Z_{G}(\gamma,F)\backslash G(F)}\omega(x) f(x^{-1}\gamma x)\,dx.$$ Par définition, c’est l’intégrale orbitale de $f$ en $\gamma$. Si $\omega$ n’est pas trivial sur $Z_{G}(\gamma,F)$, on pose $I_{\tilde{G}}(\gamma,\omega,f)=0$.
Dans le cas où le caractère $\omega$ est trivial, on note simplement $I_{\tilde{G}}(\gamma,f)=I_{\tilde{G}}(\gamma,\omega,f)$.
Soit $\tilde{T}$ un tore tordu maximal de $\tilde{G}$. Supposons $\omega$ trivial sur $T^{\theta}(F)$. On a
\(1) la fonction $\gamma\mapsto I_{\tilde{G}}(\gamma,\omega,f)$ est bornée sur $\tilde{T}(F)$.
Preuve. Il existe une fonction $f'\in C_{c}^{\infty}(\tilde{G}(F))$, à valeurs positives ou nulles, de sorte que $\vert f(\gamma)\vert \leq f'(\gamma)$ pour tout $\gamma\in \tilde{G}(F)$. Alors $$\vert I_{\tilde{G}}(\gamma,\omega,f)\vert\leq I_{\tilde{G}}(\gamma,f').$$ Il suffit de majorer le membre de droite. En oubliant cette construction, on peut supposer $\omega=1$ et $f$ à valeurs positives ou nulles. La fonction $\gamma\mapsto I_{\tilde{G}}(\gamma,f)$ sur $\tilde{T}(F)$ se quotiente en une fonction sur $\tilde{T}(F)/(1-\theta)(T(F))$ qui est à support compact. Il suffit de vérifier qu’elle est localement bornée. Il suffit de fixer $\gamma\in \tilde{T}(F)$ et de trouver un voisinage $U$ de $0$ dans $\mathfrak{t}^{\theta}(F)$ tel que la fonction soit bornée sur $exp(U)\gamma$. Mais la théorie de la descente vaut dans le cas tordu. Elle entraîne que l’on peut trouver un tel voisinage $U$ et une fonction $\varphi\in C_{c}^{\infty}(\mathfrak{g}_{\gamma}(F))$ de sorte que $$I_{\tilde{G}}(exp(X)\gamma,f)=I_{G_{\gamma_{0}}}(X,\varphi)$$ pour tout $X\in U$ tel que $exp(X)\gamma\in \tilde{G}_{reg}(F)$, avec une définition évidente de l’intégrale de droite. Le membre de droite est borné d’après Harish-Chandra. $\square$
On a
\(2) il existe $\eta>0$ tel que la fonction $\gamma\mapsto D^{\tilde{G}}(\gamma)^{-\eta}$ soit localement intégrable sur $\tilde{T}(F)/(1-\theta)(T(F))$.
Preuve. Il suffit de prouver l’existence de $\eta>0$ tel que, pour tout $\gamma\in \tilde{T}(F)$, la fonction $X\mapsto D^{\tilde{G}}(exp(X)\gamma)^{-\eta}$ est intégrable au voisinage de $0$ dans $\mathfrak{t}^{\theta}(F)$. Pour $X$ proche de $0$, on a $D^{\tilde{G}}(exp(X)\gamma)=D^{\tilde{G}}(\gamma)D^{G_{\gamma}}(exp(X))$. D’après Harish-Chandra, il existe $\eta_{\gamma}>0$ tel que $X\mapsto D^{G_{\gamma}}(exp(X))^{-\eta_{\gamma}}$ soit intégrable au voisinage de $0$. Le réel $\eta_{\gamma}$ ne dépend en fait que du groupe $G_{\gamma}$. Or il n’y a qu’un nombre fini de tels commutants possibles. En prenant pour $\eta$ le plus petit des réels $\eta_{\gamma}$, on obtient (2). $\square$
On fixe une norme $\vert \vert .\vert \vert $ sur $G(F)$ selon la méthode habituelle. On suppose qu’elle est biinvariante par $K$ et que $\vert \vert g\vert \vert \geq1$ pour tout $g\in G(F)$. On rappelle que, pour tout $g\in G(F)$, on a défini $h_{0}(g)\in {\cal A}_{0}^{\geq}$. Les fonctions $1+log(\vert \vert g\vert \vert )$ et $1+\vert h_{0}(g)\vert $ sont équivalentes. Fixons $\gamma_{0}\in \tilde{G}(F)$. On définit une norme sur $\tilde{G}(F)$ par $\vert \vert \gamma\vert \vert =\vert \vert g\vert \vert $ si $\gamma=g\gamma_{0}$ avec $g\in G(F)$. Elle dépend du point-base $\gamma_{0}$ mais sa classe d’équivalence (en un sens plus ou moins clair) n’en dépend pas.
0.3cm[**[Lemme]{}**]{}.
**
Soit $\tilde{T}$ un tore tordu maximal de $\tilde{G}$. Il existe deux entiers $N,k>0$ et un réel $c>0$ tels que l’on ait les majorations
\(i) $inf_{t\in T^{\theta,0}(F)}\vert \vert tx\vert \vert \leq c (inf_{t\in T^{\theta,0}(F)}\vert \vert t\gamma\vert \vert)^N\vert \vert x^{-1}\gamma x\vert \vert ^ND^{\tilde{G}}(\gamma)^{-k}$
\(ii) $inf_{t\in T(F)}\vert \vert tx\vert \vert \leq c \vert \vert x^{-1}\gamma x\vert \vert ^ND^{\tilde{G}}(\gamma)^{-k}$
pour tout $x\in G(F)$ et tout $\gamma\in \tilde{T}(F)\cap \tilde{G}_{reg}(F)$.
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Preuve. La preuve reprend celle du lemme 4.2 de \[A1\]. Considérons une extension galoisienne finie $F'$ de $F$ que l’on précisera plus tard. On fixe comme précédemment des normes sur $G(F')$ et $\tilde{G}(F')$, notons-les pour un instant $\vert \vert .\vert \vert_{F'}$. Il existe un entier $N_{1}\geq1$ tel que l’on ait les majorations $$\vert \vert x\vert \vert <<\vert \vert x\vert \vert _{F'}^{N_{1}},\,\,\vert \vert x\vert \vert _{F'}<<\vert \vert x\vert \vert ^{N_{1}}$$ pour tout $x\in G(F)$ et on a des majorations similaires pour $\gamma\in \tilde{G}(F)$. On voit que l’énoncé est équivalent à celui obtenu en remplaçant $\vert \vert .\vert \vert $ par $\vert \vert .\vert \vert _{F'}$ dans les inégalités (i) et (ii). On peut donc travailler avec les seules normes $\vert \vert .\vert \vert _{F'}$ et simplifier la notation pour le reste de la preuve en abandonnant les indices $F'$.
Soit $S$ un sous-tore de $G$ défini sur $F$, pas forcément maximal. On montre d’abord
\(3) il existe un entier $N_{2}$ tel que l’on ait la majoration $$inf_{t\in S(F)}\vert \vert tx\vert \vert <<( inf_{t\in S(F')}\vert \vert tx\vert \vert)^{N_{2}}$$ pour tout $x\in G(F)$.
Notons $d$ le degré de l’extension $F'/F$ et $S(F')^d$ le sous-groupe des puissances $d$-ièmes dans $S(F')$. Le quotient $S(F')/S(F')^d$ est compact, d’où une majoration $$(4) \qquad inf_{t\in S(F')^d}\vert \vert tx\vert \vert<<inf_{t\in S(F')}\vert \vert tx\vert \vert$$ pour tout $x\in G(F')$. Fixons $x\in G(F)$, posons $X=inf_{t\in S(F')^d}\vert \vert tx\vert \vert$ et choisissons $u\in S(F')$ tel que $\vert \vert u^dx\vert \vert =X$ (il est à peu près clair que la borne inférieure est atteinte). Pour $\sigma\in Gal(F'/F)$, l’application $y\mapsto \vert \vert \sigma(y)\vert \vert $ est encore une norme sur $G(F')$, il y a donc un entier $N_{3}\geq1$ tel que $\vert \vert \sigma(y)\vert \vert <<\vert \vert y\vert \vert ^{N_{3}}$ pour tout $y\in G(F')$ et tout $\sigma\in Gal(F'/F)$. Donc $\vert \vert \sigma(u)^dx\vert \vert <<X^{N_{3}}$ pour tout $\sigma$. Parce que $\sigma(u)^du^{-d}=\sigma(u)^dx(u^dx)^{-1}$, on en déduit l’existence de $N_{4}\geq1$ tel que $\vert \vert \sigma(u)^du^{-d}\vert \vert <<X^{N_{4}}$. On vérifie qu’il existe $N_{5}\geq1$ tel que $\vert \vert v\vert \vert <<\vert \vert v^d\vert \vert ^{N_{5}}$ pour tout $v\in S(F')$. D’où $\vert \vert \sigma(u)u^{-1}\vert \vert<<X^{N_{4}N_{5}}$. Posons $v=\prod_{\sigma\in Gal(F'/F)}\sigma(u)$. On a $vx=(\prod_{\sigma\in Gal(F'/F)}\sigma(u)u^{-1}) u^dx$, d’où l’existence de $N_{6}\geq1$ tel que $\vert \vert vx\vert \vert <<X^{N_{6}}$. Mais $v\in S(F)$. Donc $inf_{t\in S(F)}\vert \vert tx\vert \vert <<X^{N_{6}}$. Jointe à (4), cette relation démontre (3).
On choisit $F'$ de sorte que $T$ soit déployé sur $F'$. En appliquant (3) à $S=T^{\theta,0}$ ou $S=T$, on voit que les assertions de l’énoncé sont équivalentes aux mêmes assertions où l’on remplace le corps de base $F$ par $F'$. En oubliant cette construction, on peut supposer que $T$ est déployé sur $F$. On fixe un sous-groupe de Borel $B=TU$ de $G$ contenant $T$. La décomposition d’Iwasawa montre que l’on peut se limiter à prouver l’énoncé pour $x\in B(F)$. Ecrivons $x=yu$ avec $y\in T(F)$ et $u\in U(F)$. Pour $S=T^{\theta,0}$ ou $S=T$, il existe $N_{7}\geq1$ tel que $$(5)\qquad inf_{t\in S(F)}\vert \vert tx\vert \vert<<(inf_{t\in S(F)}\vert \vert ty\vert \vert)^{N_{7}}\vert \vert u\vert \vert ^{N_{7}}.$$ On a $x^{-1}\gamma x=u'\gamma' $, où $\gamma'=y^{-1} \gamma y$ et $u'=u^{-1}ad_{\gamma'}(u)$. Il résulte de la filtration habituelle du groupe unipotent $U$ que les coefficients de $u$ sont des fractions rationnelles en les coefficients de $\gamma'$ et $u'$, avec pour dénominateurs des puissances de $det((ad_{\gamma'}-1)_{\vert \mathfrak{g}/\mathfrak{t}^{\theta}})$. Il y a donc des entiers $N_{8},k_{1}\geq1$ tels que $\vert \vert u\vert \vert <<\vert \vert x^{-1}\gamma x\vert \vert^{N_{8}}D^{\tilde{G}}(\gamma')^{-k_{1}}$. Puisque $D^{\tilde{G}}(\gamma')=D^{\tilde{G}}(\gamma)$, la relation (5) nous ramène à prouver l’existence d’un entier $N_{9}\geq1$ tel que $$(6) \qquad inf_{t\in S(F)}\vert \vert ty\vert \vert<<(inf_{t\in S(F)}\vert \vert t\gamma\vert \vert)^{N_{9}}\vert \vert y^{-1}\gamma y\vert \vert ^{N_{9}}.$$ Le cas où $S=T$ (qui concerne le (ii) de l’énoncé) est trivial puisque le terme de gauche vaut $1$. On suppose maintenant $S=T^{\theta,0}$. Parce que les deux groupes $(1-\theta)(T(F))$ et $T^{\theta,0}(F)$ sont d’intersection finie et que leur produit est un sous-groupe d’indice fini de $T(F)$, on vérifie qu’il existe $N_{10}\geq1$ tel que $$inf_{t\in T^{\theta,0}(F)}\vert \vert ty\vert \vert<<( inf_{t\in T^{\theta,0}(F)}\vert \vert ty^{-1}\theta(y)\vert \vert)^{N_{10}}.$$ Soit $t'\in T^{\theta,0}(F)$ tel que $\vert \vert (t')^{-1}\gamma\vert \vert $ soit minimal, posons $\gamma'=(t')^{-1}\gamma$. Alors $$inf_{t\in T^{\theta,0}(F)}\vert \vert ty\vert \vert<<\vert \vert t'y^{-1}\theta(y)\vert \vert ^{N_{10}}<<\vert \vert t'y^{-1}\theta(y)\gamma'\vert \vert ^{N_{11}}\vert \vert \gamma'\vert \vert ^{N_{11}}$$ pour un certain $N_{11}\geq1$. Mais $t'y^{-1}\theta(y)\gamma'=y^{-1}\gamma y$ et $\vert \vert \gamma'\vert \vert =inf_{t\in T^{\theta,0}(F)}\vert \vert t\gamma\vert \vert$. La relation précédente n’est autre que (6). Cela achève la preuve. $\square$
Le lemme entraîne évidemment
\(7) il existe un entier $k\geq0$ et, pour tous sous-ensembles compacts $\boldsymbol{\Omega}$ de $\tilde{G}(F)$ et $\Omega$ de $\tilde{T}(F)$, il existe $c>0$ tel que l’on ait la majoration $$inf_{t\in T^{\theta,0}(F)}\vert \vert tx\vert \vert \leq c D^{\tilde{G}}(\gamma)^{-k}$$ pour tout couple $(x,\gamma)$ tel que $x\in G(F)$, $\gamma\in \Omega\cap \tilde{G}_{reg}(F)$ et $x^{-1}\gamma x\in \boldsymbol{\Omega}$.
Application de la formule de Weyl
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On a défini l’intégrale $J^T(\omega,f_{1},f_{2})$ en 3.1. C’est une intégrale à support compact. On applique la formule de Weyl sous sa deuxième forme à l’intégrale intérieure en $\gamma$. On obtient $$(1) \qquad J^T(\omega,f_{1},f_{2})=\sum_{\tilde{M}\in {\cal L}(\tilde{M}_{0})}\vert \tilde{W}^M\vert \vert \tilde{W}^G\vert ^{-1}\sum_{\tilde{S}\in T_{ell}(\tilde{M})}J_{\tilde{M},\tilde{S}}^T(\omega,f_{1},f_{2}),$$ où $$J_{\tilde{M},\tilde{S}}^T(\omega,f_{1},f_{2})=\vert W^M(\tilde{S})\vert ^{-1} \int_{\tilde{S}(F)/(1-\theta)(S(F))}\int_{S^{\theta}(F)\backslash G(F)}$$ $$\int_{A_{\tilde{G}}(F)\backslash G(F)}\bar{f}_{1}(x^{-1}\gamma x)f_{2}(g^{-1}x^{-1}\gamma xg)\omega(g)\tilde{\kappa}^T(g)\,dg\,dx\,D^{\tilde{G}}(\gamma)\,d\gamma.$$ On a noté nos tores $\tilde{S}$ plutôt que $\tilde{T}$ pour éviter les confusions avec le paramètre $T$.
On fixe jusqu’en 4.8 un espace de Levi $\tilde{M}$ contenant $\tilde{M}_{0}$ et un tore tordu elliptique $\tilde{S}$ de $\tilde{M}$. Par changement de variables $g\mapsto x^{-1}y$, on a $$J_{\tilde{M},\tilde{S}}^T(\omega,f_{1},f_{2})=\vert W^M(\tilde{S})\vert ^{-1} \int_{\tilde{S}(F)/(1-\theta)(S(F))}\int_{S^{\theta}(F)\backslash G(F)}$$ $$\int_{A_{\tilde{G}}(F)\backslash G(F)}\bar{f}_{1}(x^{-1}\gamma x)f_{2}(y^{-1}\gamma y)\omega(x^{-1}y)\tilde{\kappa}^T(x^{-1}y)\,dy\,dx\,D^{\tilde{G}}(\gamma)\,d\gamma.$$ Pour tout $\gamma$, la fonction $y\mapsto f_{2}(y^{-1}\gamma y)$ est invariante par $S^{\theta}(F)$, a fortiori par $A_{\tilde{S}}(F)=A_{\tilde{M}}(F)$. D’où $$J_{\tilde{M},\tilde{S}}^T(\omega,f_{1},f_{2})=\vert W^M(\tilde{S})\vert ^{-1} \int_{\tilde{S}(F)/(1-\theta)(S(F))}\int_{S^{\theta}(F)\backslash G(F)}$$ $$\int_{A_{\tilde{M}}(F)\backslash G(F)}\bar{f}_{1}(x^{-1}\gamma x)f_{2}(y^{-1}\gamma y)\omega(x^{-1}y)u_{\tilde{M}}^T(x,y) \,dy\,dx\,D^{\tilde{G}}(\gamma)\,d\gamma,$$ où $$u_{\tilde{M}}^T(x,y)=\int_{A_{\tilde{G}}(F)\backslash A_{\tilde{M}}(F)}\omega(a)\tilde{\kappa}^T(x^{-1}ay)\,da.$$
Soient $x,y\in G(F)$. Pour $\tilde{P}\in {\cal F}(\tilde{M})$, on a défini $T[\tilde{P}]$ en 1.3. On pose $Y(x,y,T;\tilde{P})=T[\tilde{P}]+H_{\tilde{P}}(x)-H_{\tilde{\bar{P}}}(y)$. La famille ${\cal Y}(x,y,T)=(Y(x,y,T;\tilde{P}))_{\tilde{P}\in {\cal P}(\tilde{M})}$ est $(\tilde{G},\tilde{M})$-orthogonale. On introduit la fonction $H\mapsto \Gamma_{\tilde{M}}(H,{\cal Y}(x,y,T))$ sur ${\cal A}_{\tilde{M}}$, cf. 2.2. Rappelons que l’on note $A_{\tilde{M}}(F)_{c}$ le plus grand sous-groupe compact de $A_{\tilde{M}}(F)$, c’est-à-dire le sous-groupe des $a\in A_{\tilde{M}}(F)$ tels que $H_{\tilde{M}}(a)=0$. Si la restriction de $\omega$ à ce groupe est non triviale, on pose $$v_{\tilde{M}}^T(x,y)=0.$$ Supposons maintenant que la restriction de $\omega$ à $A_{\tilde{M}}(F)_{c}$ est triviale. Alors $\omega$ se factorise en un caractère de ${\cal A}_{A_{\tilde{M}},F}$ que l’on note encore $\omega$. On pose alors $$v_{\tilde{M}}^T(x,y)=C\int_{{\cal A}_{A_{\tilde{G}},F}\backslash {\cal A}_{A_{\tilde{M}},F}}\Gamma_{\tilde{M}}(H,{\cal Y}(x,y,T))\omega(H)\,dH,$$ où $$C=mes(A_{\tilde{M}}(F)_{c})mes(A_{\tilde{G}}(F)_{c})^{-1}.$$
Posons $$J_{v,\tilde{M},\tilde{S}}^T(\omega,f_{1},f_{2})=\vert W^M(\tilde{S})\vert ^{-1} \int_{\tilde{S}(F)/(1-\theta)(S(F))}\int_{S^{\theta}(F)\backslash G(F)}$$ $$\int_{A_{\tilde{M}}(F)\backslash G(F)}\bar{f}_{1}(x^{-1}\gamma x)f_{2}(y^{-1}\gamma y)\omega(x^{-1}y)v_{\tilde{M}}^T(x,y) \,dy\,dx\,D^{\tilde{G}}(\gamma)\,d\gamma.$$
[0.3cm[**[Proposition]{}**]{}. [ *[L’expression ci-dessus est absolument convergente. Pour tout réel $r$, on a la majoration $$\vert J_{\tilde{M},\tilde{S}}^T(\omega,f_{1},f_{2})-J_{v,\tilde{M},\tilde{S}}^T(\omega,f_{1},f_{2})\vert << \vert T\vert ^{-r}$$ pour tout $T$.]{}*]{}0.3cm]{}
Cette proposition sera prouvée en 4.6.
Dans les formules définissant $J_{\tilde{M},\tilde{S}}^T(\omega,f_{1},f_{2})$ et $J_{v,\tilde{M},\tilde{S}}^T(\omega,f_{1},f_{2})$, les variables d’intégration $\gamma$, $x$ et $y$ appartiennent à des ensembles quotients mais il convient de les relever dans $\tilde{S}(F)$, resp. $G(F)$. On fixe un sous-ensemble compact $\boldsymbol{\Omega}$ de $\tilde{G}(F)$ contenant les supports de $f_{1}$ et $f_{2}$. L’ensemble des éléments de $\tilde{S}(F)/(1-\theta)(S(F))$ dont la classe de conjugaison coupe $\boldsymbol{\Omega}$ est compact. On peut donc fixer un sous-ensemble compact $\Omega$ de $\tilde{S}(F)$ et supposer
\(2) $\gamma\in \Omega$.
Appliquons 4.2(7). Puisque $S^{\theta,0}(F)/A_{\tilde{M}}(F)$ est compact, on peut remplacer dans cette relation le groupe $S^{\theta,0}(F)$ par $A_{\tilde{M}}(F)$. Cela nous permet de supposer $$(3) \qquad \vert \vert x\vert \vert ,\vert \vert y\vert \vert \leq cD^{\tilde{G}}(\gamma)^{-k}.$$
Une majoration de $u_{\tilde{M}}^T(x,y)$ et $v_{\tilde{M}}^T(x,y)$
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Fixons $\epsilon>0$. Notons $\chi^T_{\epsilon}$ la fonction caractéristique des $\gamma\in\tilde{S}(F)/(1-\theta)(S(F))$ tels que $D^{\tilde{G}}(\gamma)\leq e^{-\epsilon\vert T\vert }$.
[0.3cm[**[Lemme]{}**]{}. [ *[Pour tout réel $r$, on a la majoration $$\int_{\tilde{S}(F)/(1-\theta)(S(F))}\int_{S^{\theta}(F)\backslash G(F)}\int_{A_{\tilde{M}}(F)\backslash G(F)}\vert f_{1}(x^{-1}\gamma x)f_{2}(y^{-1}\gamma y)\vert$$ $$(\vert u_{\tilde{M}}^T(x,y)\vert +\vert v_{\tilde{M}}^T(x,y)\vert )\,dy\,dx\,\chi_{\epsilon}^T(\gamma)D^{\tilde{G}}(\gamma)\,d\gamma<<\vert T\vert ^{-r}$$ pour tout $T$.]{}*]{}0.3cm]{}
Preuve. Il suffit de traiter le cas où $\omega=1$ et les fonctions $f_{1}$ et $f_{2}$ sont à valeurs positives ou nulles. Montrons qu’il existe un entier $D>0$ tel que, pour $x$ et $y$ vérifiant 4.3(3), on ait la majoration $$(1) \qquad \vert u_{\tilde{M}}^T(x,y)\vert <<(\vert T\vert +\vert log(D^{\tilde{G}}(\gamma))\vert)^D .$$ On a une majoration $$log(\vert \vert g\vert \vert )<<\vert T\vert +\vert H_{G}(g)\vert$$ pour tout $g\in G(F)$ tel que $\tilde{\kappa}^T(g)=1$. Le terme $u_{\tilde{M}}^T(x,y)$ est défini par une intégration sur $A_{\tilde{M}}(F)/A_{\tilde{G}}(F)$. Il existe un ensemble fini ${\bf b}\subset {\cal A}_{\tilde{G}}$ tel qu’en posant $A_{\tilde{M}}(F)^{{\bf b}}=\{a\in A_{\tilde{M}}(F); H_{\tilde{G}}(a)\in {\bf b}\}$, on ait $$\vert u_{\tilde{M}}^T(x,y)\vert<<\int_{A_{\tilde{M}}(F)^{{\bf b}}}\tilde{\kappa}^T(x^{-1}ay)\,da.$$ Pour $a$ dans l’ensemble d’intégration, on a $$log(\vert \vert a\vert \vert )<<\vert T\vert +log(\vert \vert x\vert \vert )+log(\vert \vert y\vert \vert ),$$ d’où $$log(\vert \vert a\vert \vert )<<\vert T\vert +\vert log(D^{\tilde{G}}(\gamma))\vert$$ d’après 4.3(3). L’intégrale est alors essentiellement bornée par la mesure des $H\in {\cal A}_{A_{\tilde{M}},F}$ tels que $\vert H\vert $ soit majoré par le membre de droite ci-dessus. On en déduit (1).
Une preuve similaire montre que la fonction $v_{\tilde{M}}^T(x,y)$ vérifie la même propriété.
D’après les explications de la fin du paragraphe précédent, l’intégrale de l’énoncé est donc majorée par $$\int_{\tilde{S}(F)/(1-\theta)(S(F))}\int_{S^{\theta}(F)\backslash G(F)}\int_{A_{\tilde{M}}(F)\backslash G(F)} f_{1}(x^{-1}\gamma x)f_{2}(y^{-1}\gamma y)$$ $$(\vert T\vert +\vert log(D^{\tilde{G}}(\gamma))\vert)^D \,dy\,dx\,\chi_{\epsilon}^T(\gamma)D^{\tilde{G}}(\gamma)\,d\gamma$$ $$<<\int_{\tilde{S}(F)/(1-\theta)(S(F))}I_{\tilde{G}}(\gamma,f_{1})I_{\tilde{G}}(\gamma,f_{2})(\vert T\vert +\vert log(D^{\tilde{G}}(\gamma))\vert)^D \chi_{\epsilon}^T(\gamma)\,d\gamma.$$ On choisit $\eta$ comme en 4.2(2). Dans le domaine où les fonctions à intégrer ne sont pas nulles, on a $$(\vert T\vert +\vert log(D^{\tilde{G}}(\gamma))\vert)^D <<\vert T\vert ^DD^{\tilde{G}}(\gamma)^{-\eta/2},$$ $$\chi_{\epsilon}^T(\gamma)\leq e^{-\epsilon\eta\vert T\vert /2}D^{\tilde{G}}(\gamma)^{-\eta/2}.$$ En utilisant aussi 4.2(1), on voit que l’intégrale ci-dessus est donc essentiellement majorée par $$\vert T\vert ^De^{-\epsilon\eta\vert T\vert /2}\int_{\Omega}D^{\tilde{G}}(\gamma)^{-\eta}\,d\gamma.$$ La dernière intégrale est convergente et le lemme s’ensuit. $\square$
Majoration de $u_{\tilde{M}}(x,y)-v_{\tilde{M}}(x,y)$
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[0.3cm[**[Lemme]{}**]{}. [ *[Il existe $\epsilon>0$ tel que, pour tout réel $r$, on ait la majoration $$\vert u^T_{\tilde{M}}(x,y)-v^T_{\tilde{M}}(x,y)\vert <<\vert T\vert ^{-r}$$ pour tout $T$ et tous $x,y\in G(F)$ tels que $\vert \vert x\vert \vert, \vert \vert y\vert \vert \leq e^{\epsilon\vert T\vert }$.]{}*]{}0.3cm]{}
Preuve. Fixons $\epsilon'>0$ que l’on précisera plus tard. On note $\epsilon'{\cal T}$ la famille de points $(\epsilon'T[\tilde{P}])_{\tilde{P}\in {\cal P}(\tilde{M})}$. Pour $\tilde{Q}=\tilde{L}U_{Q}\in {\cal F}(\tilde{M})$, posons $$u^T_{\tilde{M}}(x,y,\tilde{Q})=\int_{A_{\tilde{M}}(F)/A_{\tilde{G}}(F)}\omega(a)\tilde{\kappa}^T(x^{-1}ay)\Gamma_{\tilde{M}}^{\tilde{Q}}(H_{\tilde{M}}(a),\epsilon'{\cal T})\tau_{\tilde{Q}}^{\tilde{G}}(H_{\tilde{L}}(a)-\epsilon'T[\tilde{Q}])\,da.$$ Si $\omega$ est non trivial sur $A_{\tilde{M}}(F)_{c}$, posons $v^T_{\tilde{M}}(x,y,\tilde{Q})=0$. Sinon, posons $$v^T_{\tilde{M}}(x,y,\tilde{Q})=C\int_{{\cal A}_{A_{\tilde{M}},F}/{\cal A}_{A_{\tilde{G}},F}}\omega(H)\Gamma_{\tilde{M}}^{\tilde{G}}(H,{\cal Y}(x,y,T))\Gamma_{\tilde{M}}^{\tilde{Q}}(H,\epsilon'{\cal T})\tau_{\tilde{Q}}^{\tilde{G}}(H-\epsilon'T[\tilde{Q}])\,dH.$$ L’égalité 1.3(6) nous permet d’écrire $$u^T_{\tilde{M}}(x,y)=\sum_{\tilde{Q}\in {\cal F}(\tilde{M})}u^T_{\tilde{M}}(x,y;\tilde{Q}),$$ $$v^T_{\tilde{M}}(x,y)=\sum_{\tilde{Q}\in {\cal F}(\tilde{M})}v^T_{\tilde{M}}(x,y;\tilde{Q}).$$ On peut fixer $\tilde{Q}$ et majorer $\vert u^T_{\tilde{M}}(x,y,\tilde{Q})-v^T_{\tilde{M}}(x,y,\tilde{Q})\vert $. Ecrivons $x=ulk$, $y=\bar{u}'l'k'$, avec $l,l'\in L(F)$, $u\in U_{Q}(F)$, $\bar{u}'\in U_{\bar{Q}}(F)$, $k,k'\in K$. Soit $a\in A_{\tilde{M}}(F)$ tel que
\(1) $\Gamma_{\tilde{M}}^{\tilde{Q}}(H_{\tilde{M}}(a),\epsilon'{\cal T})\tau_{\tilde{Q}}^{\tilde{G}}(H_{\tilde{M}}(a)-\epsilon'T[\tilde{Q}])=1$.
On a $x^{-1}ay=k^{-1}hl^{-1}al'h'k'$, avec $h'=(a\bar{u}'l')^{-1}ua\bar{u}'l'$, $h=l^{-1}a\bar{u}'a^{-1}l$. Les éléments $u$, $\bar{u}'$, $h'$ et $h$ sont unipotents, écrivons $u=exp(Y)$, $\bar{u}'=exp(Y')$, $h=exp(X)$, $h'=exp(X')$. Munissons $\mathfrak{g}(F)$ d’une norme $\vert .\vert $. Parce qu’il s’agit d’éléments unipotents, on a des majorations $\vert Y\vert <<\vert \vert u\vert \vert ^D$, $ \vert Y'\vert <<\vert \vert \bar{u}'\vert \vert ^D$ pour un entier $D$ convenable. Fixons $\epsilon$ et supposons $\vert \vert x\vert \vert ,\vert \vert y\vert \vert \leq e^{\epsilon\vert T\vert }$. On a essentiellement la même majoration pour $l$, $u$, $l'$, $\bar{u}'$. On a donc aussi $\vert Y\vert ,\vert Y'\vert <<e^{D\epsilon\vert T\vert }$. L’élément $X$ se déduit de $Y'$ par conjugaison par $l^{-1}a$. Dans la suite interviendront des réels $c_{1}, c_{2}$ etc... que, pour simplifier, on n’introduira pas. A chaque fois, cela signifie qu’il existe un réel $c_{1}, c_{2}...>0$, indépendant des données, tel que la relation soit vérifiée. Ainsi, la relation (1) entraîne $$(2) \qquad \vert \alpha(a)\vert >e^{c_{1}\epsilon'\vert T\vert }$$ pour toute racine $\alpha$ de $A_{\tilde{M}}$ agissant dans $\mathfrak{u}_{Q} $. Ici $a$ agit dans $\mathfrak{u}_{\bar{Q}}$ et les racines y sont inférieures à $e^{-c_{1}\epsilon'\vert T\vert }$. La conjugaison par $l^{-1}$ est de norme bornée par $e^{D'\epsilon\vert T\vert }$ pour un entier convenable. On en déduit des majorations $$(3)(a)\qquad \vert X\vert <<e^{(c_{2}\epsilon-c_{3}\epsilon')\vert T\vert }.$$ De même, $$(3)(b)\qquad \vert Y\vert <<e^{(c_{2}\epsilon-c_{3}\epsilon')\vert T\vert }.$$
Supposons d’abord $F$ non archimédien. On suppose $c_{2}\epsilon-c_{3}\epsilon'<0$. On peut toujours supposer $\vert T\vert $ assez grand (la majoration de l’énoncé étant triviale pour $\vert T\vert $ borné). Alors les relations ci-dessus entraînent que $h,h'\in K$. On a donc $\tilde{\kappa}^T(x^{-1}ay)=\tilde{\kappa}^T(l^{-1}al')$. Fixons un élément $\tilde{P}\in {\cal P}(\tilde{M})$ tel que $\tilde{P}\subset \tilde{Q}$ et un élément $w\in W^{\tilde{G}}$, relevé en un élément de $K$, tel que $\tilde{P}_{0}\subset w(\tilde{P})$. Posons $\underline{\tilde{M}}=w(\tilde{M})$, $\underline{\tilde{L}}=w(\tilde{L})$, $\underline{\tilde{Q}}=w(\tilde{Q})$, $\underline{l}=wlw^{-1}$, $\underline{a}=waw^{-1}$, $\underline{l}'=wl'w^{-1}$. On a $\tilde{\kappa}^T(l^{-1}al')=\tilde{\kappa}^T(\underline{l}^{-1}\underline{a}\underline{l}')$. Fixons $u,u'\in K\cap \underline{L}(F)$ et $m\in M_{0}(F)^{\geq,\underline{Q}}$ tels que $\underline{l}^{-1}\underline{a}\underline{l}'=umu'$. On a
\(4) si $\epsilon<c_{4}\epsilon'$ et $\vert T\vert $ est assez grand, alors $m\in M_{0}(F)^{\geq}$.
Il s’agit de prouver que $\alpha(m)\geq1$ pour toute racine $\alpha$ de $A_{0}$ intervenant dans $\mathfrak{u}_{\underline{Q}}$. Il suffit de prouver que, pour tout $N\in \mathfrak{u}_{\underline{Q}}(F)$, on a $\vert ad_{m}(N)\vert \geq\vert N\vert $. En remplaçant $N$ par $ad_{u'(\underline{l}')^{-1}}(N)$, il suffit que, pour tout $N\in \mathfrak{u}_{\underline{Q}}(F)$, on ait $$\vert ad_{u^{-1}\underline{l}^{-1}\underline{a}}(N)\vert \geq \vert ad_{u'(\underline{l}')^{-1}}(N)\vert .$$ On a $$\vert ad_{u'(\underline{l}')^{-1}}(N)\vert<<\vert \vert l'\vert \vert^D \vert N\vert$$ pour un entier $D$ convenable, d’où $$\vert ad_{u'(\underline{l}')^{-1}}(N)\vert<<e^{c_{5}\epsilon\vert T\vert }\vert N\vert .$$ Posons $N'=ad_{u^{-1}\underline{l}^{-1}\underline{a}}(N)$. On a $ad_{\underline{a}}(N)=ad_{\underline{l}u}(N')$ d’où, par le même raisonnement $$\vert ad_{\underline{a}}(N)\vert <<e^{c_{6}\epsilon\vert T\vert }\vert N'\vert .$$ Mais pour $\alpha$ intervenant dans $\mathfrak{u}_{\underline{Q}}$, $\alpha(\underline{a})$ est minoré par la relation (2). On en déduit $$\vert ad_{\underline{a}}(N)\vert >>e^{c_{1}\epsilon'\vert T\vert }\vert N\vert .$$ Toutes ces relations entraînent $$\vert ad_{u'(\underline{l}')^{-1}}(N)\vert<<e^{((c_{5}+c_{6})\epsilon-c_{1}\epsilon')\vert T\vert }\vert N'\vert .$$ En supposant $(c_{5}+c_{6})\epsilon-c_{1}\epsilon'<0$, cela entraîne la majoration cherchée. Cela prouve (4).
On a
\(5) si $\epsilon<\epsilon'$, alors $$\vert H_{0}(m)^{\underline{\tilde{L}}}\vert \leq c_{7}\epsilon'\vert T\vert.$$
On fixe un ensemble fini ${\bf b}\subset {\cal A}_{\tilde{L},F}$ tel que ${\cal A}_{\tilde{L},F}\subset {\cal A}_{A_{\tilde{L}},F}+{\bf b}$. On peut choisir un élément $b\in A_{\tilde{L}}(F)$ tel que $H_{\tilde{L}}(\underline{a})-H_{\tilde{L}}(b)\in {\bf b}$. Posons $\underline{a}'=\underline{a}b^{-1}$. La relation (1) entraîne $\Gamma_{\underline{\tilde{M}}}^{\underline{\tilde{Q}}}(H_{\underline{\tilde{M}}}(\underline{a}'),\epsilon'{\cal T})=1$. Puisque de plus $H_{\underline{\tilde{L}}}(\underline{a}')\in {\bf b}$, on a $$\vert H_{\underline{\tilde{M}}}(\underline{a}')\vert \leq c_{8}\epsilon'\vert T\vert .$$ On en déduit une majoration $$\vert \vert a'\vert \vert <<e^{c_{9}\epsilon'\vert T\vert }.$$ Posons $m'=mb^{-1}$. On a $H_{0}(m')^{\underline{\tilde{L}}}=H_{0}(m)^{\underline{\tilde{L}}}$. D’où une majoration $$\vert H_{0}(m)^{\underline{\tilde{L}}}\vert \leq c_{10}log(\vert \vert m'\vert \vert ).$$ Mais $m'=\underline{l}^{-1}\underline{a}'\underline{l}'$. Par un calcul maintenant familier, on en déduit $$log(\vert \vert m'\vert \vert )\leq1+ (c_{9}\epsilon'+c_{11}\epsilon)\vert T\vert .$$ La relation (5) résulte des deux majorations précédentes.
On suppose $\epsilon$, $\epsilon'$ et $T$ tels que (4) et (5) soient vérifiés. Montrons que
\(6) si $\epsilon'$ est assez petit, on a l’égalité $\phi_{\tilde{P}_{0}}^{\tilde{G}}(H_{0}(m)-T)=\phi_{\underline{\tilde{Q}}}^{\tilde{G}}(H_{\underline{\tilde{L}}}(m)-T)$.
Notons simplement $H=H_{\tilde{P}_{0}}(m)$. L’égalité $\phi_{\tilde{P}_{0}}^{\tilde{G}}(H_{0}(m)-T)=1$ équivaut à
\(7) $<\varpi_{\tilde{\alpha}},H-T>\leq0$ pour tout $\tilde{\alpha}\in \Delta_{\tilde{P}_{0}}$.
L’égalité $\phi_{\underline{\tilde{Q}}}^{\tilde{G}}(H_{\underline{\tilde{L}}}(m)-T)=1$ équivaut à
\(8) $<\varpi_{\tilde{\alpha}},H-T>\leq0$ pour tout $\tilde{\alpha}\in \Delta_{\tilde{P}_{0}}-\Delta_{\tilde{P}_{0}}^{\underline{\tilde{Q}}}$.
Evidemment (7) entraîne (8). Inversement, supposons (8) vérifiée. Soit $\tilde{\alpha}\in \Delta_{\tilde{P}_{0}}^{\underline{\tilde{Q}}}$. On a $$<\varpi_{\tilde{\alpha}},H-T>=<\varpi_{\tilde{\alpha}},H^{\underline{\tilde{L}}}-T^{\underline{\tilde{L}}}>+<\varpi_{\tilde{\alpha}},H_{\underline{\tilde{L}}}-T_{\underline{\tilde{L}}}>.$$ On va prouver que ces deux termes sont négatifs pourvu que $\epsilon'$ soit assez petit. Cela prouvera que (8) entraîne (7) et achèvera la preuve de (6). Le terme $H^{\underline{\tilde{L}}}$ est une projection de $H_{0}(m)^{\underline{\tilde{L}}}$. D’après (5), on a donc une majoration $$\vert <\varpi_{\tilde{\alpha}},H^{\underline{\tilde{L}}}>\vert \leq c_{12}\epsilon'\vert T\vert .$$ On a aussi facilement $$<\varpi_{\tilde{\alpha}},T>\geq c_{13}\vert T\vert .$$ Il résulte de ces deux inégalités que $<\varpi_{\tilde{\alpha}},H^{\underline{\tilde{L}}}-T^{\underline{\tilde{L}}}>$ est négatif si $\epsilon'$ est assez petit. D’après (7), le terme $H_{\underline{\tilde{L}}}-T_{\underline{\tilde{L}}}$ est combinaison linéaire à coefficients négatifs ou nuls de termes $\check{\tilde{\beta}}_{\underline{\tilde{L}}}$ pour $\tilde{\beta}\in \Delta_{\tilde{P}_{0}}-\Delta_{\tilde{P}_{0}}^{\underline{\tilde{Q}}}$. Il reste à voir que, pour un tel $\tilde{\beta}$, on a $<\varpi_{\tilde{\alpha}},\check{\tilde{\beta}}_{\underline{\tilde{L}}}>\geq0$. On a $$<\varpi_{\tilde{\alpha}},\check{\tilde{\beta}}_{\underline{\tilde{L}}}>=<\varpi_{\tilde{\alpha}},\check{\tilde{\beta}}>-<\varpi_{\tilde{\alpha}},\check{\tilde{\beta}}^{\underline{\tilde{L}}}>.$$ Le premier terme est nul puisque $\tilde{\alpha}\in \Delta_{\tilde{P}_{0}}^{\underline{\tilde{Q}}}$ et $\tilde{\beta}\not\in \Delta_{\tilde{P}_{0}}^{\underline{\tilde{Q}}}$. Parce que $\Delta_{\tilde{P}_{0}}$ est une base aigu" e, $\check{\tilde{\beta}}^{\underline{\tilde{L}}}$ appartient à la chambre négative fermée associée à $\Delta_{\tilde{P}_{0}}^{\underline{\tilde{Q}}}$, a fortiori est combinaison linéaire à coefficients négatifs ou nuls de $\check{\tilde{\alpha}}'$ pour $\tilde{\alpha}'\in \Delta_{\tilde{P}_{0}}^{\underline{\tilde{Q}}}$. Donc $<\varpi_{\tilde{\alpha}},\check{\tilde{\beta}}^{\underline{\tilde{L}}}>\leq0$. Cela achève la preuve de (6).
On a
\(9) si $\epsilon'$ est assez petit, on a l’égalité $\tilde{\kappa}^T(x^{-1}ay)=\phi_{\tilde{Q}}^{\tilde{G}}(H_{\tilde{L}}(a)-H_{\tilde{Q}}(x)+H_{\tilde{\bar{Q}}}(y)-T[\tilde{Q}])$.
On a vu que $\tilde{\kappa}^T(x^{-1}ay)=\tilde{\kappa}^T(\underline{l}^{-1}\underline{a}\underline{l}')=\tilde{\kappa}^T(m)$. Puisque $m\in M_{0}(F)^{\geq}$ d’après (4), on a $\tilde{\kappa}^T(m)=\phi_{\tilde{P}_{0}}^{\tilde{G}}(H_{0}(m)-T)$. Grâce à (5), on a donc $\tilde{\kappa}^T(x^{-1}ay)=
\phi_{\underline{\tilde{Q}}}^{\tilde{G}}(H_{\underline{\tilde{L}}}(m)-T)$ pourvu que $\epsilon'$ soit assez petit. Par conjugaison par $w$, cela entraîne $\tilde{\kappa}^T(x^{-1}ay)=\phi_{\tilde{Q}}^{\tilde{G}}(w^{-1}H_{\underline{\tilde{L}}}(m)-T[\tilde{Q}])$. Par définition de $m$, on a l’égalité $$H_{\underline{\tilde{L}}}(m)=-H_{\underline{\tilde{L}}}(\underline{l})+H_{\underline{\tilde{L}}}(\underline{a})+H_{\underline{\tilde{L}}}(\underline{l}').$$ On a aussi $w^{-1}H_{\underline{\tilde{L}}}(\underline{l})=H_{\tilde{L}}(l)$ etc... D’où $$w^{-1}H_{\underline{\tilde{L}}}(m)=-H_{\tilde{L}}(l)+H_{\tilde{L}}(a)+H_{\tilde{L}}(l').$$ Il résulte des définitions que $H_{\tilde{L}}(l)=H_{\tilde{Q}}(x)$ et $H_{\tilde{L}}(l')=H_{\tilde{\bar{Q}}}(y)$. On obtient alors l’égalité de (9).
Supposons $\epsilon'$ assez petit pour que (9) soit vérifiée. Alors l’intégrale définissant $u^T_{\tilde{M}}(x,y;\tilde{Q})$ se factorise en $$(10)\qquad u^T_{\tilde{M}}(x,y;\tilde{Q})=C(\omega) \int_{{\cal A}_{A_{\tilde{M}},F}/{\cal A}_{A_{\tilde{G}},F}}\phi_{\tilde{Q}}^{\tilde{G}}(H-H_{\tilde{Q}}(x)+H_{\tilde{\bar{Q}}}(y)-T[\tilde{Q}])$$ $$\Gamma_{\tilde{M}}^{\tilde{Q}}(H,\epsilon'{\cal T})\tau_{\tilde{Q}}^{\tilde{G}}(H-\epsilon'T[\tilde{Q}])\,dH,$$ où $$C(\omega)=\int_{A_{\tilde{M}}(F)_{c}A_{\tilde{G}}(F)/A_{\tilde{G}}(F)}\omega(a)\,da.$$ Si $\omega$ n’est pas trivial sur $A_{\tilde{M}}(F)_{c}$, $C(\omega)$ est nul. Alors $u^T_{\tilde{M}}(x,y;\tilde{Q})=v_{\tilde{M}}^T(x,y;\tilde{Q})$ par définition de ce dernier terme. Supposons maintenant $\omega$ trivial sur $A_{\tilde{M}}(F)_{c}$. Alors $C(\omega)=C$. Fixons $H\in {\cal A}_{A_{\tilde{M}},F}$ tel que $$(11)\qquad \Gamma_{\tilde{M}}^{\tilde{Q}}(H,\epsilon'{\cal T})\tau_{\tilde{Q}}^{\tilde{G}}(H-\epsilon'T[\tilde{Q}])=1.$$ On va prouver
\(12) si $\epsilon'$ et $\epsilon<\epsilon'$ sont assez petits, on a l’égalité $$\Gamma_{\tilde{M}}^{\tilde{G}}(H,{\cal Y}(x,y,T))=\phi_{\tilde{Q}}^{\tilde{G}}(H-H_{\tilde{Q}}(x)+H_{\tilde{\bar{Q}}}(y)-T[\tilde{Q}]).$$
Fixons un sous-espace parabolique $\tilde{P}\in {\cal P}(\tilde{M})$ tel que $\tilde{P}\subset \tilde{Q}$ et que $H$ appartienne à la clôture de la chambre positive associée à l’espace parabolique $\tilde{P}\cap \tilde{L}$ de $\tilde{L}$. On a alors l’égalité $\Gamma_{\tilde{M}}^{\tilde{Q}}(H,\epsilon'{\cal T})=\phi_{\tilde{P}}^{\tilde{Q}}(H-\epsilon'T[\tilde{P}])$. L’égalité (11) entraîne alors que $H$ appartient à la clôture de la chambre positive associée à l’espace parabolique $\tilde{P}$. Par des arguments déjà vus, on a des majorations $$\vert H_{\tilde{P}'}(x)\vert, \vert H_{\tilde{P}'}(y)\vert \leq c_{14}\epsilon'\vert T\vert$$ pour tout $\tilde{P}'\in {\cal P}(\tilde{M})$. Si $\epsilon'$ est assez petit, il en résulte que ${\cal Y}(x,y,T)$ est une famille positive. Donc $H'\mapsto\Gamma_{\tilde{M}}^{\tilde{G}}(H',{\cal Y}(x,y,T))$ est la fonction caractéristique de l’ensemble des $H'$ tels que $(H')^{\tilde{G}}$ appartient à l’enveloppe convexe des $Y(x,y,T,\tilde{P}')^{\tilde{G}}$. Pour $H$ dans la clôture de la chambre positive associée à $\tilde{P}$, on a simplement $$\Gamma_{\tilde{M}}^{\tilde{G}}(H,{\cal Y}(x,y,T))=\phi_{\tilde{P}}^{\tilde{G}}(H-H_{\tilde{P}}(x)+H_{\tilde{\bar{P}}}(y)-T[\tilde{P}]).$$ Posons $X=H-H_{\tilde{P}}(x)+H_{\tilde{\bar{P}}}(y)$. De la même façon que l’on a prouvé (5) et (6), on montre que si $\epsilon<\epsilon'$, on a $\vert X^{\tilde{L}}\vert \leq c_{15}\epsilon'\vert T\vert $, puis que, si $\epsilon'$ est assez petit, on a l’égalité $$\phi_{\tilde{P}}^{\tilde{G}}(H-H_{\tilde{P}}(x)+H_{\tilde{\bar{P}}}(y)-T[\tilde{P}])=\phi_{\tilde{Q}}^{\tilde{G}}(H-H_{\tilde{Q}}(x)+H_{\tilde{\bar{Q}}}(y)-T[\tilde{Q}]).$$ Cela prouve (12).
L’égalité (10) jointe à (12) prouve l’égalité $u_{\tilde{M}}^T(x,y;\tilde{Q})=v_{\tilde{M}}^T(x,y;\tilde{Q})$. On a dû supposer $\epsilon'$ et $\epsilon/\epsilon'$ assez petits. Si $\epsilon$ lui-même est assez petit, on peut choisir le paramètre auxiliaire $\epsilon'$ satisfaisant ces conditions. Cela prouve le lemme. Cela prouve même plus, à savoir l’égalité $u_{\tilde{M}}^T(x,y;\tilde{Q})=v_{\tilde{M}}^T(x,y;\tilde{Q})$.
Rappelons que l’on avait supposé $F$ non archimédien. Remarquons que cette hypothèse n’a été utilisée que pour affirmer que $exp(X)$ et $exp(Y)$ appartenaient à $K$. Tout le reste est valable sans hypothèse sur $F$. Ainsi, supposons maintenant $F$ archimédien et définissons ${\bf u}^T_{\tilde{M}}(x,y,\tilde{Q})$ comme l’intégrale analogue à $u_{\tilde{M}}^T(x,y;\tilde{Q})$ où l’on remplace $\tilde{\kappa}^T(x^{-1}ay)$ par $\tilde{\kappa}^T(l^{-1}al')$. On a pour cette intégrale les mêmes résultats démontrés précédemment pour $u_{\tilde{M}}^T(x,y;\tilde{Q})$. Revenons à l’égalité $x^{-1}ay=k^{-1}exp(X)l^{-1}al'exp(Y)k'$, qui entraîne $\tilde{\kappa}^T(x^{-1}ay)=\tilde{\kappa}^T(exp(X)l^{-1}al'exp(Y))$. Fixons un élément $m'\in M_{0}(F)^{\geq}$ tel que $exp(X)l^{-1}al'exp(Y)\in Km'K$ et définissons $m$ comme dans le cas non archimédien. On a encore (4): $m\in M_{0}(F)^{\geq}$ (sous hypothèses sur $\epsilon$ et $\epsilon'$). Il résulte alors de (3)(a) et (3)(b) et du lemme 5.2 de \[A1\] que l’on a une majoration $$\vert H_{0}(m)-H_{0}(m')\vert \leq c_{15}exp^{-c_{16}\vert T\vert }.$$ Posons $\eta_{+}=1+exp^{-c_{16}\vert T\vert /2}$, $\eta_{-}=1-exp^{-c_{16}\vert T\vert /2}$. De l’inégalité ci-dessus résulte les majorations $$\tilde{\kappa}^{\eta_{-}T}(m)\leq\left\lbrace\begin{array}{c}\tilde{\kappa}^T(m)\\\tilde{\kappa}^T(m')\\ \end{array}\right\rbrace \leq\tilde{\kappa}^{\eta_{+}T}(m),$$ puis $$\vert \tilde{\kappa}^T(m)-\tilde{\kappa}^T(m')\vert \leq\tilde{\kappa}^{\eta_{+}T}(m)-\tilde{\kappa}^{\eta_{-}T}(m).$$ Puisque $\tilde{\kappa}^T(m')=\tilde{\kappa}^T(x^{-1}ay)$, on en déduit $$\vert {\bf u}_{\tilde{M}}^T(x,y;\tilde{Q})-u_{\tilde{M}}^T(x,y;\tilde{Q})\vert \leq\int_{A_{\tilde{M}}(F)/A_{\tilde{G},F}}(\tilde{\kappa}^{\eta_{+}T}(l^{-1}al')-\tilde{\kappa}^{\eta_{-}T}(l^{-1}al'))$$ $$\Gamma_{\tilde{M}}^{\tilde{Q}}(H_{\tilde{M}}(a),\epsilon'{\cal T})\tau_{\tilde{Q}}^{\tilde{G}}(H_{\tilde{L}}(a)-\epsilon'T[\tilde{Q}]\,da.$$ Le membre de droite est la différence entre une expression analogue à ${\bf u}_{\tilde{M}}^T(x,y;\tilde{Q})$ où l’on remplace $\omega$ par $1$, $T$ par $\eta_{+}T$ et $\epsilon'$ par $\eta_{+}^{-1}\epsilon'$, et l’expression similaire où $\eta_{-}$ remplace $\eta_{+}$. Ces expressions sont calculées par (10). On obtient $$\vert {\bf u}_{\tilde{M}}^T(x,y;\tilde{Q})-u_{\tilde{M}}^T(x,y;\tilde{Q})\vert \leq$$ $$\int_{{\cal A}_{A_{\tilde{M}},F}/{\cal A}_{A_{\tilde{G}},F}}(\phi_{\tilde{Q}}^{\tilde{G}}(H-H_{\tilde{Q}}(x)+H_{\tilde{\bar{Q}}}(y)-\eta_{+}T[\tilde{Q}])-\phi_{\tilde{Q}}^{\tilde{G}}(H-H_{\tilde{Q}}(x)+H_{\tilde{\bar{Q}}}(y)-\eta_{-}T[\tilde{Q}]))$$ $$\Gamma_{\tilde{M}}^{\tilde{Q}}(H,\epsilon'{\cal T})\tau_{\tilde{Q}}^{\tilde{G}}(H-\epsilon'T[\tilde{Q}])\,dH.$$ Le membre de droite est la différence entre les volumes de deux polyèdres dont l’un est inclus dans l’autre. Leurs sommets ont une norme essentiellement bornée par $\vert T\vert $ et tout point de la frontière du petit polyèdre est à distance au plus $c_{17}(\eta_{+}-\eta_{-})\vert T\vert $ de la frontière du plus grand. La différence de leur volume est donc essentiellement bornée par $(\eta_{+}-\eta_{-})\vert T\vert ^D$ pour un entier $D$ convenable. Donc par $e^{-c_{16}\vert T\vert /2}\vert T\vert ^D$. On conclut que, pour tout réel $r$, $$\vert {\bf u}_{\tilde{M}}^T(x,y;\tilde{Q})-u_{\tilde{M}}^T(x,y;\tilde{Q})\vert << \vert T\vert ^{-r}.$$ Comme on l’a dit, la preuve du cas non-archimédien montre que ${\bf u}_{\tilde{M}}^T(x,y;\tilde{Q})=v_{\tilde{M}}^T(x,y;\tilde{Q})$. Cela prouve le lemme. $\square$
Preuve de la proposition 4.3
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Pour prouver que $J_{v,\tilde{M},\tilde{S}}^T(\omega,f_{1},f_{2})$ est absolument intégrable, on considère $T$ comme fixé. L’expression $J_{v,\tilde{M},\tilde{S}}^T(\omega,f_{1},f_{2})$ est une intégrale en $\gamma$. Elle est à support compact. Au voisinage des éléments singuliers, elle est absolument intégrable d’après le lemme 4.4. En utilisant la relation 4.3(3), on vérifie que la fonction $v_{\tilde{M}}^T(x,y)$ reste bornée (pour $T$ fixé) si $\gamma$ reste hors d’un voisinage des éléments singuliers. Alors l’intégrale hors d’un tel voisinage est essentiellement bornée par $$(1) \qquad \int_{\tilde{S}(F)/(1-\theta)(S(F))}
\int_{S^{\theta}(F)\backslash G(F)}\int_{A_{\tilde{M}}(F)\backslash G(F)}\vert f_{1}(x^{-1}\gamma x)f_{2}(y^{-1}\gamma y)\vert \,dy\,dx\,D^{\tilde{G}}(\gamma)\,d\gamma,$$ ou encore par $$\int_{\tilde{S}(F)/(1-\theta)(S(F))}I_{\tilde{G}}(\gamma,\vert f_{1}\vert )I_{\tilde{G}}(\gamma,\vert f_{2}\vert )d\gamma,$$ qui est convergente d’après 4.2(1).
Démontrons maintenant la majoration de l’énoncé de la proposition 4.3. On fixe $\epsilon$ comme dans le lemme 4.5. On fixe ensuite $\epsilon'>0$ tel que la relation $D^{\tilde{G}}(\gamma)\geq e^{-\epsilon'\vert T\vert }$ et la relation 4.3(3): $$\vert \vert x\vert \vert ,\vert \vert y\vert \vert \leq cD^{\tilde{G}}(\gamma)^{-k}$$ entraînent $\vert \vert x\vert \vert ,\vert \vert y\vert \vert \leq e^{\epsilon\vert T\vert }$, du moins si $T$ est grand (on peut supposer $T$ grand pour démontrer notre majoration). On décompose $J_{\tilde{M},\tilde{S}}^T(\omega,f_{1},f_{2})-J_{v,\tilde{M},\tilde{S}}^T(\omega,f_{1},f_{2})$ en somme de deux intégrales. Dans la première, on glisse le terme $\chi_{\epsilon'}^T(\gamma)$. Dans la seconde, on glisse le terme $1-\chi_{\epsilon'}^T(\gamma)$. La première intégrale est majorée par le lemme 4.4: pour tout réel $r$, elle est essentiellement bornée par $\vert T\vert ^{-r}$. Dans la seconde, grâce à 4.3(3) et nos choix de $\epsilon'$ et $\epsilon$, on a la majoration du lemme 4.5. Pour tout $r$, cette intégrale est donc essentiellement majorée par le produit de $\vert T\vert ^{-r}$ et de l’intégrale (1). Celle-ci étant convergente, cela prouve la majoration cherchée. $\square$
Terme constant de $v_{\tilde{M}}^T(x,y)$
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Soient $x,y\in G(F)$. Pour $\tilde{P}\in {\cal P}(\tilde{M})$ et $\Lambda\in i{\cal A}_{\tilde{M}}^*$, posons $$v(x,y;\Lambda,\tilde{P})=e^{<\Lambda, H_{\tilde{\bar{P}}}(y)-H_{\tilde{P}}(x)>}.$$ La famille $(v(x,y;\Lambda,\tilde{P}))_{\tilde{P}\in {\cal P}(\tilde{M})}$ est une $(\tilde{G},\tilde{M})$-famille, dont on déduit une fonction $v^{\tilde{G}}_{\tilde{M}}(x,y;\Lambda)$. On pose $v^{\tilde{G}}_{\tilde{M}}(x,y)=v^{\tilde{G}}_{\tilde{M}}(x,y;0)$.
0.3cm[**[Lemme]{}**]{}.
**
\(i) La fonction $T\mapsto f(T)=v_{\tilde{M}}^T(x,y)$ appartient à $PolExp$. Plus précisément, elle appartient à un espace $PolExp_{\Xi,N}$ si $F$ est archimédien, resp. $PolExp_{\boldsymbol{\Xi},N}$ si $F$ est non-archimédien, où $\Xi$ et $N$, resp. $\boldsymbol{\Xi}$ et $N$, sont indépendants de $x$ et $y$.
\(ii) Supposons $F$ archimédien. On a $$c_{0}(f)=\left\lbrace\begin{array}{cc}(-1)^{a_{\tilde{M}}-a_{\tilde{G}}}v^{\tilde{G}}_{\tilde{M}}(x,y),&\text{ si }\omega\text{ est trivial sur }A_{\tilde{M}}(F),\\0, &\text{ sinon.}\\ \end{array}\right.$$
\(iii) Supposons $F$ non-archimédien. Soit ${\cal R}\subset {\cal A}_{M_{0},F}\otimes_{{\mathbb Z}}{\mathbb Q}$ un réseau. Alors il existe un entier $k_{1}>0$ et un réel $c>0$ ne dépendant tous deux que de ${\cal R}$ de sorte que, pour tout entier $k\geq1$, on ait la majoration $$\vert c_{\frac{1}{k}{\cal R},0}(f)\vert\leq ck^{-1}(log(\vert \vert x\vert \vert )+log(\vert \vert y\vert \vert )) ^{a_{\tilde{M}}-a_{\tilde{G}}}$$ si $\omega$ est non trivial sur $A_{\tilde{M}}(F)$, respectivement $$\vert c_{\frac{1}{k}{\cal R},0}(f)- (-1)^{a_{\tilde{M}}-a_{\tilde{G}}}v^{\tilde{G}}_{\tilde{M}}(x,y)\vert\leq ck^{-1}(log(\vert \vert x\vert \vert )+log(\vert \vert y\vert \vert )) ^{a_{\tilde{M}}-a_{\tilde{G}}}$$ si $\omega$ est trivial sur $A_{\tilde{M}}(F)$.
0.3cm
Preuve. On traite le cas où $F$ est non-archimédien, le cas archimédien étant plus simple. Si $\omega$ n’est pas trivial sur $A_{\tilde{M}}(F)_{c}$, $v_{\tilde{M}}^T(x,y)=0$ et les assertions sont évidentes. Supposons $\omega$ trivial sur $A_{\tilde{M}}(F)_{c}$. On a déduit de $\omega$ un caractère de ${\cal A}_{A_{\tilde{M}},F}$, encore noté $\omega$. Il y a un unique $\Lambda_{\omega}\in i{\cal A}_{\tilde{M}}^*/{\cal A}_{A_{\tilde{M}},F}^{\vee}$ tel que $\omega(H)=e^{<\Lambda_{\omega},H>}$ pour tout $H\in {\cal A}_{A_{\tilde{M}},F}$. Notons que, puisque $\omega$ est trivial sur $A_{\tilde{G}}(F)$, la projection $(\Lambda_{\omega})_{\tilde{G}}$ appartient à $i{\cal A}_{A_{\tilde{G}},F}^*$. On a $$v_{\tilde{M}}^T(x,y)=C\sum_{{\cal A}_{A_{\tilde{M}},F}/{\cal A}_{A_{\tilde{G}},F}}\Gamma_{\tilde{M}}^{\tilde{G}}(H,{\cal Y}(x,y,T))e^{<\Lambda_{\omega},H>}.$$ On peut décomposer la somme en une somme sur $X\in {\cal A}_{\tilde{G},F}/{\cal A}_{A_{\tilde{G}},F}$ suivie d’une somme sur $H\in {\cal A}_{A_{\tilde{M}},F}\cap {\cal A}_{\tilde{M},F}^{\tilde{G}}(X)$. Par inversion de Fourier, cette dernière somme peut être remplacée par la somme sur $H\in {\cal A}_{\tilde{M},F}^{\tilde{G}}(X)$, suivie de $$[{\cal A}_{\tilde{M},F}:{\cal A}_{A_{\tilde{M}},F}]^{-1}\sum_{\nu\in i{\cal A}_{A_{\tilde{M}},F}^{\vee}/i{\cal A}_{\tilde{M},F}^{\vee}}e^{<\nu,H>}.$$ Remarquons que $C[{\cal A}_{\tilde{M},F}:{\cal A}_{A_{\tilde{M}},F}]^{-1}=mes(A_{\tilde{G}}(F)_{c})^{-1}mes(i{\cal A}_{\tilde{M},F}^*)^{-1}$. Tout ceci est absolument convergent puisque la fonction $H\mapsto \Gamma_{\tilde{M}}^{\tilde{G}}(H,{\cal Y}(x,y,T))$ est à support compact. On obtient $$v_{\tilde{M}}^T(x,y)= mes(A_{\tilde{G}}(F)_{c})^{-1}mes(i{\cal A}_{\tilde{M},F}^*)^{-1} \sum_{\nu\in i{\cal A}_{A_{\tilde{M}},F}^{\vee}/i{\cal A}_{\tilde{M},F}^{\vee} }\sum_{X\in {\cal A}_{\tilde{G},F}/{\cal A}_{A_{\tilde{G}},F}}I^T(x,y,X;\Lambda_{\omega}+\nu),$$ où, pour $\Lambda\in {\cal A}_{\tilde{M},{\mathbb C}}^*$, on a posé $$I^T(x,y,X;\Lambda)=\sum_{H\in {\cal A}_{\tilde{M},F}^{\tilde{G}}(X)}\Gamma_{\tilde{M}}^{\tilde{G}}(H,{\cal Y}(x,y,T))e^{<\Lambda,H>}.$$ Cette dernière somme est finie donc définit une fonction holomorphe en $\Lambda$. Fixons $\tilde{P}_{1}\in {\cal P}(\tilde{M})$ et supposons $<\Lambda,\check{\tilde{\alpha}}>>0$ pour tout $\tilde{\alpha}\in \Delta_{\tilde{P}_{1}}$. La variante tordue de 1.3(7) nous dit que $$\Gamma_{\tilde{M}}^{\tilde{G}}(H,{\cal Y}(x,y,T))=\sum_{\tilde{P}\in {\cal P}(\tilde{M})}(-1)^{s(\tilde{P},\tilde{P}_{1})}\phi_{\tilde{P},\tilde{P}_{1}}^{\tilde{G}}(H-Y(x,y,T;\tilde{P}) ),$$ où $Y(x,y,T;\tilde{P})=T[\tilde{P}]-H_{\tilde{P}}(x)+H_{\tilde{\bar{P}}}(y)$. L’hypothèse sur $\Lambda$ nous permet de permuter les deux sommes: $$I^T(x,y,X;\Lambda)=\sum_{\tilde{P}\in {\cal P}(\tilde{M})}(-1)^{s(\tilde{P},\tilde{P}_{1})}\sum_{H\in {\cal A}_{\tilde{M},F}^{\tilde{G}}(X)}\phi_{\tilde{P},\tilde{P}_{1}}^{\tilde{G}}(H- Y(x,y,T;\tilde{P}))$$ $$=\sum_{\tilde{P}\in {\cal P}(\tilde{M})}(-1)^{s(\tilde{P},\tilde{P}_{1})}\epsilon_{\tilde{P},\tilde{P}_{1}}^{\tilde{G},Y(x,y,T;\tilde{P})}(X;\Lambda).$$ La variante tordue de 1.5(2) conduit à l’égalité $$I^T(x,y,X;\Lambda)=\sum_{\tilde{P}\in {\cal P}(\tilde{M})}\epsilon_{\tilde{P}}^{\tilde{G},Y(x,y,T;\tilde{P})}(X;\Lambda).$$ Introduisons la $(\tilde{G},\tilde{M})$-famille $(\varphi(\Lambda;\tilde{P}))_{\tilde{P}\in {\cal P}(\tilde{M})}$ formée des fonctions constantes valant $1$. Alors $$I^T(x,y,X;\Lambda)=\varphi_{\tilde{M}}^{\tilde{G},{\cal Y}(x,y,T)}(X;\Lambda).$$ On a prouvé cette égalité sous des hypothèses portant sur $\Lambda$ mais elle se prolonge à tout $\Lambda$, les deux membres étant méromorphes (et en fait holomorphes). Notons simplement ${\cal Y}_{x,y}={\cal Y}(x,y,0)$. Avec la définition de 1.7 (dans sa version tordue), on a ${\cal Y}(x,y,T)={\cal Y}_{x,y}(T)$. On obtient $$v_{\tilde{M}}^T(x,y)=mes(A_{\tilde{G}}(F)_{c})^{-1}mes(i{\cal A}_{\tilde{M},F}^*)^{-1} \sum_{\nu\in i{\cal A}_{A_{\tilde{M}},F}^{\vee}/i{\cal A}_{\tilde{M},F}^{\vee} }\sum_{X\in {\cal A}_{\tilde{G},F}/{\cal A}_{A_{\tilde{G}},F}}\varphi_{\tilde{M}}^{\tilde{G},{\cal Y}_{x,y}(T)}(X;\Lambda_{\omega}+\nu).$$ La variante tordue du lemme 1.7 entraîne l’assertion (i) de l’énoncé. Pour un réseau ${\cal R}$ et un entier $k$, le coefficient $c_{\frac{1}{k}{\cal R},0}(f)$ est le produit de $mes(A_{\tilde{G}}(F)_{c})^{-1}mes(i{\cal A}_{\tilde{M},F}^*)^{-1}$ et de la somme des coefficients analogues pour les fonctions $T\mapsto \varphi_{\tilde{M}}^{\tilde{G},{\cal Y}_{x,y}(T)}(X;\Lambda_{\omega}+\nu)$, lesquels sont calculés par le même lemme. Notons ${\cal N}$ l’ensemble des $\nu \in i{\cal A}_{A_{\tilde{M}},F}^{\vee}/i{\cal A}_{\tilde{M},F}^{\vee}$ tel que $\Lambda_{\omega}+\nu\in i{\cal A}_{\tilde{M},F}^{\vee}+i{\cal A}_{\tilde{G}}^*$. Pour $\nu\in {\cal N}$, fixons $\Lambda_{\omega,\nu}\in i{\cal A}_{\tilde{G}}^*$ tel que $\Lambda_{\omega}+\nu\in\Lambda_{\omega,\nu} +i{\cal A}_{\tilde{M},F}^{\vee}$. Posons $$(1) \qquad c(f)= mes(A_{\tilde{G}}(F)_{c})^{-1}mes(i{\cal A}_{\tilde{G},F}^*)^{-1} \sum_{\nu\in {\cal N}}\sum_{X\in {\cal A}_{\tilde{G},F}/{\cal A}_{A_{\tilde{G}},F}}e^{<\Lambda_{\omega,\nu},X>}\varphi_{\tilde{M}}^{\tilde{G}}({\cal Y}_{x,y};\Lambda_{\omega,\nu}).$$ Notons $N(x,y)$ la norme de la $(\tilde{G},\tilde{M})$-famille $(\varphi({\cal Y}_{x,y};\Lambda,\tilde{P}))_{\tilde{P}\in {\cal P}(\tilde{M})}$. La variante du lemme 1.7 implique qu’il existe un entier $k_{1}>0$ et un réel $c>0$, ne dépendant tous deux que de ${\cal R}$, de sorte que, si $k\geq k_{1}$, on ait la majoration $$(2) \qquad \vert c_{\frac{1}{k}{\cal R},0}(f)-c(f)\vert \leq cN(x,y)k^{-1}.$$ En comparant les définitions, on voit que $\varphi({\cal Y}_{x,y};\Lambda,\tilde{P})=v(x,y;-\Lambda,\tilde{P})$ pour tout $\tilde{P}$. Parce que $\epsilon_{\tilde{P}}^{\tilde{G}}(-\Lambda)=(-1)^{a_{\tilde{M}}-a_{\tilde{G}}}\epsilon_{\tilde{P}}^{\tilde{G}}(\Lambda)$, on en déduit $$\varphi_{\tilde{M}}^{\tilde{G}}({\cal Y}_{x,y};\Lambda_{\omega,\nu})=(-1)^{a_{\tilde{M}}-a_{\tilde{G}}}v^{\tilde{G}}_{\tilde{M}}(x,y;-\Lambda_{\omega,\nu}).$$ Dans la formule (1), la somme en $X$ vaut $[{\cal A}_{\tilde{G},F}:{\cal A}_{A_{\tilde{G}},F}] $ si $\Lambda_{\omega,\nu}\in i{\cal A}_{\tilde{G},F}^{\vee}$, $0$ sinon. Notons que si $\Lambda_{\omega,\nu}\in i{\cal A}_{\tilde{G},F}^{\vee}$, on a $\Lambda_{\omega}+\nu\in i{\cal A}_{\tilde{M},F}^{\vee}$, d’où $\Lambda_{\omega}\in i{\cal A}_{A_{\tilde{M}},F}^{\vee}$. Cela équivaut à ce que $\omega$ soit trivial sur $A_{\tilde{M}}(F)$. Donc, si $\omega$ n’est pas trivial sur $A_{\tilde{M}}(F)$, $c(f)=0$. Supposons $\omega$ trivial sur $A_{\tilde{M}}(F)$. On peut supposer $\Lambda_{\omega}=0$. La condition $\nu\in {\cal N}$ signifie que $\nu\in( i{\cal A}_{A_{\tilde{M}},F}^{\vee}\cap i{\cal A}_{\tilde{G}}^*)/i{\cal A}_{\tilde{G},F}^{\vee}$. La condition $\Lambda_{\omega,\nu}\in i{\cal A}_{\tilde{G},F}^{\vee}$ sélectionne le terme $\nu=0$. En remarquant que $mes(A_{\tilde{G}}(F)_{c})^{-1}mes(i{\cal A}_{\tilde{G},F}^*)^{-1}[{\cal A}_{\tilde{G},F}:{\cal A}_{A_{\tilde{G}},F}] =1$, on obtient $$c(f)= (-1)^{a_{\tilde{M}}-a_{\tilde{G}}}v_{\tilde{M}}(x,y).$$ La majoration $$N(x,y)<<sup_{\tilde{P}\in {\cal P}(\tilde{M})}(1+\vert H_{\tilde{P}}(x)\vert +\vert H_{\tilde{\bar{P}}}(y)\vert )^{a_{\tilde{M}}-a_{\tilde{G}}}$$ résulte des définitions. Il est clair que l’on a $$(3) \qquad 1+\vert H_{\tilde{P}}(x)\vert +\vert H_{\tilde{\bar{P}}}(y)\vert<<1+log(\vert \vert x\vert \vert )+log(\vert \vert y\vert \vert ) .$$ Alors les assertions de l’énoncé résultent de (2). $\square$
Terme constant de $J_{v,\tilde{M},\tilde{S}}^T(\omega,f_{1},f_{2})$
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Supposons d’abord $\omega$ trivial sur $S^{\theta}(F)$. Pour $\gamma\in \tilde{S}(F)$, posons $$J_{\tilde{M}}^{\tilde{G}}(\gamma,\omega,f_{1},f_{2})= D^{\tilde{G}}(\gamma)\int_{S^{\theta}(F)\backslash G(F)}\int_{S^{\theta}(F)\backslash G(F)}\bar{f}_{1}(x^{-1}\gamma x)f_{2}(y^{-1}\gamma y)\omega(x^{-1}y)v^{\tilde{G}}_{\tilde{M}}(x,y) \,dy\,dx.$$ Posons $$J^{\tilde{G}}_{\tilde{M},\tilde{S}}(\omega,f_{1},f_{2})= \vert W^M(\tilde{S})\vert ^{-1} mes(A_{\tilde{M}}(F)\backslash S^{\theta}(F))\int_{\tilde{S}(F)/(1-\theta)(S(F))}J_{\tilde{M}}^{\tilde{G}}(\gamma,\omega,f_{1},f_{2})\,d\gamma.$$ Si $\omega$ n’est pas trivial sur $S^{\theta}(F)$, on pose simplement $$J^{\tilde{G}}_{\tilde{M},\tilde{S}}(\omega,f_{1},f_{2})=0.$$
0.3cm[**[Lemme]{}**]{}.
**
\(i) Les expressions ci-dessus sont absolument convergentes.
\(ii) La fonction $T\mapsto f(T)= J^T_{v,\tilde{M},\tilde{S}}(\omega,f_{1},f_{2})$ appartient à $PolExp$. Si $F$ est archimédien, on a l’égalité $c_{0}(f)=(-1)^{a_{\tilde{M}}-a_{\tilde{G}}}J^{\tilde{G}}_{\tilde{M},\tilde{S}}(\omega,f_{1},f_{2})$. Si $F$ est non-archimédien, pour tout réseau ${\cal R}\subset {\cal A}_{M_{0},F}\otimes_{{\mathbb Z}}{\mathbb Q}$, on a l’égalité $$lim_{k\to \infty}c_{\frac{1}{k}{\cal R},0}(f)=(-1)^{a_{\tilde{M}}-a_{\tilde{G}}}J^{\tilde{G}}_{\tilde{M},\tilde{S}}(\omega,f_{1},f_{2}).$$
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[**Remarque.**]{} Comme en 3.23, on appellera “terme constant” de $f$ le terme $c_{0}(f)$ si $F$ est archimédien, $lim_{k\to \infty}c_{\frac{1}{k}{\cal R},0}(f)$ si $F$ est non-archimédien.
Preuve. Pour l’assertion (i), on peut supposer $\omega$ trivial sur $S^{\theta}(F)$. Il résulte de la variante tordue de 1.7(1) et de 4.7(3) que l’on a une majoration $$\vert v_{\tilde{M}}(x,y)\vert <<(1+log(\vert \vert x\vert \vert )+log(\vert \vert y\vert \vert ))^D$$ pour tous $x,y\in G(F)$, où $D=a_{\tilde{M}}-a_{\tilde{G}}$. En tenant compte de 4.3(3), il suffit de prouver la convergence de $$(1)\qquad \int_{S^{\theta}(F)\backslash G(F)}\int_{A_{\tilde{M}}(F)\backslash G(F)}\vert f_{1}(x^{-1}\gamma x)f_{2}(y^{-1}\gamma y) \vert (1+\vert log(D^{\tilde{G}}(\gamma))\vert )^D\,dy\,dx\,D^{\tilde{G}}(\gamma)\,d\gamma.$$ Grâce à 4.2(1), on est ramené à celle de l’intégrale $$\int_{\Omega}(1+\vert log(D^{\tilde{G}}(\gamma))\vert )^D\,d\gamma.$$ Puisqu’on a sur $\Omega$ une majoration $(1+\vert log(D^{\tilde{G}}(\gamma))\vert )^D<<D^{\tilde{G}}(\gamma)^{-\eta}$, cela résulte de 4.2(2).
Prouvons (ii), en supposant comme souvent $F$ non-archimédien. Le lemme précédent nous dit que chaque fonction $T\mapsto v_{\tilde{M}}^T(x,y)$ appartient à $PolExp$. Plus précisément, pour tout réseau ${\cal R}\subset {\cal A}_{M_{0},F}\otimes_{{\mathbb Z}}{\mathbb Q}$, on peut écrire $$v_{\tilde{M}}^T(x,y)=\sum_{\mu\in \chi_{{\cal R}}}e^{<\mu,T>}p_{{\cal R},\mu,x,y}(T),$$ où $\chi_{{\cal R}}$ est indépendant de $x$ et $y$ et les polynômes $p_{{\cal R},\mu,x,y}$ ont un degré borné indépendamment de $x$ et $y$. Comme en 3.23, les coefficients de ces polynômes se calculent par interpolation, donc ont les mêmes propriétés de croissance que les fonctions $v_{\tilde{M}}^T(x,y)$ elles-mêmes. Il en résulte que, dans la formule intégrale définissant $J^T_{v,\tilde{M},\tilde{S}}(\omega,f_{1},f_{2})$, les intégrales commutent au développement en $T$. Cela entraîne que la fonction $T\mapsto f(T)= J^T_{v,\tilde{M},\tilde{S}}(\omega,f_{1},f_{2})$ appartient à $PolExp$. Cela entraîne aussi que le terme $c_{\frac{1}{k}{\cal R},0}(f)$ se calcule en remplaçant dans l’intégrale les fonctions $v_{\tilde{M}}^T(x,y)$ par leurs termes similaires. Ceux-ci sont calculés par le lemme 4.7, à une erreur près. L’intégrale des termes d’erreurs est essentiellement majorée par le produit de $k^{-1}$ et de l’intégrale (1) ci-dessus. Puisque celle-ci est convergente, ce terme d’erreur tend vers $0$ quand $k$ tend vers l’infini. Le terme principal est nul si $\omega$ n’est pas trivial sur $A_{\tilde{M}}(F)$ et on obtient dans ce cas la formule cherchée. Supposons $\omega$ trivial sur $A_{\tilde{M}}(F)$. On obtient dans ce cas pour terme constant $$(2) \qquad (-1)^{a_{\tilde{M}}-a_{\tilde{G}}} \vert W^M(\tilde{S})\vert ^{-1} \int_{\tilde{S}(F)/(1-\theta)(S(F))}$$ $$\int_{S^{\theta}(F)\backslash G(F)}\int_{A_{\tilde{M}}(F)\backslash G(F)}\bar{f}_{1}(x^{-1}\gamma x)f_{2}(y^{-1}\gamma y)\omega(x^{-1}y)v^{\tilde{G}}_{\tilde{M}}(x,y) \,dy\,dx\,D^{\tilde{G}}(\gamma)\,d\gamma.$$ On remarque que la fonction $y\mapsto v_{\tilde{M}}(x,y)$ est invariante à gauche par $M(F)$, donc par $S(F)$. En factorisant l’intégrale en $y$ en une intégrale sur $S^{\theta}(F)\backslash G(F)$ et une intégrale en $A_{\tilde{M}}(F)\backslash S^{\theta}(F)$, on voit apparaître l’intégrale $$\int_{A_{\tilde{M}}(F)\backslash S^{\theta}(F)}\omega(y)\,dy.$$ Elle est nulle (et donc aussi le terme constant), si $\omega$ n’est pas trivial sur $S^{\theta}(F)$. Si $\omega$ est trivial sur $S^{\theta}(F)$, elle vaut $mes(A_{\tilde{M}}(F)\backslash S^{\theta}(F))$. La formule (2) devient $(-1)^{a_{\tilde{M}}-a_{\tilde{G}}}J^{\tilde{G}}_{\tilde{M},\tilde{S}}(\omega,f_{1},f_{2})$. $\square$
La formule géométrique
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Pour tout $\tilde{M}\in {\cal L}(\tilde{M}_{0})$, on note $T_{ell}(\tilde{M},\omega)$ l’ensemble des $\tilde{S}\in T_{ell}(\tilde{M})$ tels que $\omega$ soit trivial sur $S^{\theta}(F)$. On pose $$J_{g\acute{e}om,\tilde{M}}^{\tilde{G}}(\omega,f_{1},f_{2})=\sum_{\tilde{S}\in T_{ell}(\tilde{M},\omega)}\vert W^M(\tilde{S})\vert ^{-1}mes(A_{\tilde{M}}(F)\backslash S^{\theta}(F))$$ $$\int_{\tilde{S}(F)/(1-\theta)(S(F))}J_{\tilde{M}}^{\tilde{G}}(\gamma,\omega,f_{1},f_{2})\,d\gamma.$$ On pose $$J^{\tilde{G}}_{g\acute{e}om}(\omega,f_{1},f_{2})=\sum_{\tilde{M}\in {\cal L}(\tilde{M}_{0})}\vert \tilde{W}^M\vert \vert \tilde{W}^G\vert ^{-1}(-1)^{a_{\tilde{M}}-a_{\tilde{G}}}J_{g\acute{e}om,\tilde{M}}^{\tilde{G}}(\omega,f_{1},f_{2}).$$
[0.3cm[**[Proposition]{}**]{}. [ *[Il existe une unique fonction $T\mapsto \varphi(T)$ qui appartient à $PolExp$ et qui vérifie pour tout réel $r$ la majoration $$\vert J^T(\omega,f_{1},f_{2})-\varphi(T)\vert <<\vert T\vert ^{-r}$$ pour tout $T$ dans le cône où $J^T(\omega,f_{1},f_{2})$ est définie. Si $F$ est archimédien, on a l’égalité $$c_{0}(\varphi)=J_{g\acute{e}om}^{\tilde{G}}(\omega,f_{1},f_{2}).$$ Si $F$ est non-archimédien, pour tout réseau ${\cal R}\subset {\cal A}_{M_{0},F}\otimes_{{\mathbb Z}}{\mathbb Q}$, on a l’égalité $$lim_{k\to \infty}c_{\frac{1}{k}{\cal R},0}(\varphi)=J_{g\acute{e}om}^{\tilde{G}}(\omega,f_{1},f_{2}).$$]{}*]{}0.3cm]{}
Preuve. Cela résulte de 4.3(1) et du lemme précédent. $\square$
La formule des traces locale tordue, version non invariante
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Le théorème
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On a défini des expressions $J_{spec}^{\tilde{G}}(\omega,f_{1},f_{2})$ et $J_{g\acute{e}om}^{\tilde{G}}(\omega,f_{1},f_{2})$ en 3.26 et 4.9.
[0.3cm[**[Théorème]{}**]{}. [ *[On a l’égalité $J_{spec}^{\tilde{G}}(\omega,f_{1},f_{2})=J_{g\acute{e}om}^{\tilde{G}}(\omega,f_{1},f_{2})$.]{}*]{}0.3cm]{}
Il suffit de comparer les propositions 3.26 et 4.9. $\square$
[**Remarque.**]{} Nos mesures vérifiant les conditions de cohérence de 1.2, on voit que les expressions du théorème sont proportionnelles au carré de la mesure sur $G(F)$, en un sens évident, et inversement proportionnelles à la mesure sur $A_{\tilde{G}}(F)$. Elles ne dépendent d’aucune autre mesure.
Extension de la formule des traces locale tordue aux fonctions $C^{\infty}$ à support compact
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Dans ce paragraphe, $F$ est archimédien. Notons $\mathfrak{U}$ l’algèbre enveloppante de l’algèbre de Lie réelle de $G(F)$. Elle agit à droite et à gauche sur les fonctions $C^{\infty}$ sur $\tilde{G}(F)$. Pour $X,Y\in \mathfrak{U}$ et $f$ une telle fonction, on note $XfY$ l’image de $f$ par les actions à gauche de $X$ et à droite de $Y$. Soit $\Omega$ un sous-ensemble compact de $\tilde{G}(F)$. Notons $C_{c}^{\infty}(\Omega)$ l’espace des éléments de $C_{c}^{\infty}(\tilde{G}(F))$ à support dans $\Omega$. On munit $C_{c}^{\infty}(\Omega)$ de la topologie définie par les semi-normes $$f\mapsto sup_{\gamma\in \Omega}\vert XfY(\gamma)\vert ,$$ pour $X,Y\in \mathfrak{U}$. On a l’égalité $$C_{c}^{\infty}(\tilde{G}(F))=\bigcup_{\Omega}C_{c}^{\infty}(\Omega)$$ où $\Omega$ décrit les sous-ensembles compacts de $\tilde{G}(F)$. On munit $C_{c}^{\infty}(\tilde{G}(F))$ de la topologie limite inductive des topologies que l’on vient de définir sur les sous-espaces $C_{c}^{\infty}(\Omega)$. Autrement dit, une suite $(f_{n})_{n\in {\mathbb N}}$ d’éléments de $C_{c}^{\infty}(\tilde{G}(F))$ converge vers $f\in C_{c}^{\infty}(\tilde{G}(F))$ si et seulement si les deux conditions suivantes sont satisfaites:
- il existe un sous-ensemble compact $\Omega\subset \tilde{G}(F)$ tel que $f$ et chaque $f_{n}$ soient à support dans $\Omega$;
- pour tous $X,Y\in \mathfrak{U}$, on a $lim_{n\to \infty}sup_{\gamma\in \tilde{G}(F)}\vert XfY(\gamma)-Xf_{n}Y(\gamma)\vert =0$.
Comme dans le cas non tordu, le théorème de Peter-Weyl entraîne que $C_{c}^{\infty}(\tilde{G}(F),K)$ est dense dans $C_{c}^{\infty}(\tilde{G}(F))$.
0.3cm[**[Proposition]{}**]{}.
**
Soient $f_{1},f_{2}\in C_{c}^{\infty}(\tilde{G}(F))$.
\(i) Toutes les expressions intervenant dans la définition de $J^{\tilde{G}}_{spec}(\omega,f_{1},f_{2})$ et $J^{\tilde{G}}_{g\acute{e}om}(\omega,f_{1},f_{2})$ sont convergentes et ces expressions elles-mêmes le sont.
\(ii) On a l’égalité $J^{\tilde{G}}_{spec}(\omega,f_{1},f_{2})=J^{\tilde{G}}_{g\acute{e}om}(\omega,f_{1},f_{2})$.
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Preuve. Le côté géométrique est à peu près trivial. Pour $i=1,2$, on peut majorer $\vert f_{i}\vert$ par une fonction $f'_{i}\in C_{c}^{\infty}(\tilde{G}(F),K)$ à valeurs positives ou nulles. Chaque terme de $J^{\tilde{G}}_{g\acute{e}om}(\omega,f_{1},f_{2})$ est majoré par le même terme relatif aux fonctions $f'_{1}$ et $f'_{2}$ et au caractère $\omega$ trivial. La convergence de ces termes entraîne celle des termes initiaux. On voit de même que $J^{\tilde{G}}_{g\acute{e}om}(\omega,f_{1},f_{2})$ est continu en $f_{1}$ et $f_{2}$.
Passons aux termes spectraux. Dans le cas où $F={\mathbb C}$, on voit que notre problème est équivalent à celui concernant les objets déduits de $G$ et $\tilde{G}$ par restriction des scalaires de ${\mathbb C}$ à ${\mathbb R}$. On suppose donc $F={\mathbb R}$. On peut fixer un sous-ensemble compact $\Omega\subset \tilde{G}({\mathbb R})$ et supposer nos fonctions $f_{1}$ et $f_{2}$ à support dans $\Omega$. On a une formule de la forme $$(1) \qquad J_{spec}^{\tilde{G}}(\omega,f_{1},f_{2})=\sum_{\tilde{M}\in {\cal L}(\tilde{M}_{0})}\sum_{\tau\in (E_{disc}(\tilde{M},\omega)/conj)/i{\cal A}_{\tilde{M}}^*}c(\tau)\int_{i{\cal A}_{\tilde{M}}^*} J_{\tilde{M}}^{\tilde{G}}(\pi_{\tau_{\lambda}},f_{1},f_{2})\,d\lambda,$$ où les $c(\tau)$ sont des coefficients uniformément bornés. On fixe $\gamma_{0}\in \tilde{M}_{0}({\mathbb R})$ vérifiant la condition 2.1(4) et on définit des fonctions $\varphi_{i}$ sur $G({\mathbb R})$ pour $i=1,2$ par $\varphi_{i}(x)=f_{i}(x\gamma_{0})$. Fixons $\tilde{M}$ et $\tau$ intervenant dans la formule ci-dessus. On a une égalité $$J_{\tilde{M}}^{\tilde{G}}(\pi_{\tau_{\lambda}},f_{1},f_{2})=trace({\cal X}^{\tilde{G}}_{\tilde{M}}(\pi_{\tau_{\lambda}})(\overline{\tilde{\Pi}_{\boldsymbol{\tau}_{\tilde{\lambda}}}(f_{1})}\otimes \tilde{\Pi}_{\boldsymbol{\tau}_{\tilde{\lambda}}}(f_{2}))),$$ où ${\cal X}^{\tilde{G}}_{\tilde{M}}(\pi_{\tau_{\lambda}})$ est l’opérateur déduit de la $(\tilde{G},\tilde{M})$ famille $({\cal X}(\pi_{\tau_{\lambda}};\Lambda,\tilde{Q}))_{\tilde{Q}\in {\cal P}(\tilde{M})}$ définie par $${\cal X}(\pi_{\tau_{\lambda}};\Lambda,\tilde{Q})=\overline{{\cal M}(\pi_{\tau_{\lambda}};\Lambda,\tilde{\bar{Q}})}\otimes {\cal M}(\pi_{\tau_{\lambda}};\Lambda,\tilde{Q}).$$ On a aussi $$\overline{\tilde{\Pi}_{\boldsymbol{\tau}_{\tilde{\lambda}}}(f_{1})}\otimes \tilde{\Pi}_{\boldsymbol{\tau}_{\tilde{\lambda}}}(f_{2})=\left(\overline{\Pi_{\tau_{\lambda}}(\varphi_{1})}\otimes \Pi_{\tau_{\lambda}}(\varphi_{2})\right) U_{\tau_{\lambda}},$$ où $$U_{\tau_{\lambda}}=\overline{\tilde{\Pi}_{\boldsymbol{\tau}_{\tilde{\lambda}}}(\gamma_{0})}\otimes \tilde{\Pi}_{\boldsymbol{\tau}_{\tilde{\lambda}}}(\gamma_{0}).$$ On a donc $$J_{\tilde{M}}^{\tilde{G}}(\pi_{\tau_{\lambda}},f_{1},f_{2})=trace(j_{\tilde{M}}^{\tilde{G}}(\pi_{\tau_{\lambda}},\varphi_{1},\varphi_{2})),$$ où $$j_{\tilde{M}}^{\tilde{G}}(\pi_{\tau_{\lambda}},\varphi_{1},\varphi_{2})={\cal X}^{\tilde{G}}_{\tilde{M}}(\pi_{\tau_{\lambda}})\left(\overline{\Pi_{\tau_{\lambda}}(\varphi_{1})}\otimes \Pi_{\tau_{\lambda}}(\varphi_{2})\right) U_{\tau_{\lambda}}.$$
On peut bien sûr fixer $\tilde{M}$ et ne considérer que la sous-somme de l’expression (1) indexée par $\tilde{M}$. Un élément $\tau\in E_{disc}(\tilde{M},\omega)$ est de la forme $(M_{disc},\sigma,\tilde{r})$, où $M_{disc}\subset M$ et $\sigma$ est de la série discrète de $M_{disc}({\mathbb R})$. On peut encore fixer $M_{disc}$, que l’on suppose semi-standard, et ne considérer que la sous-somme des $\tau$ issus de ce Levi. On peut identifier la composante neutre de $A_{M_{disc}}({\mathbb R})$ à ${\cal A}_{M_{disc}}$. Puisque $\tau $ est discret, l’ensemble $W^{\tilde{M}}_{reg}(\sigma)$ n’est pas vide. La restriction du caractère central de $\sigma$ à $ {\cal A}_{M_{disc}}$ est naturellement paramétrée par un élément de $i{\cal A}_{M_{disc}}^*$. Cette restriction étant fixée par tout élément de $W^{\tilde{M}}_{reg}(\sigma)$ et l’ensemble des points fixes de l’action d’un tel élément dans $i{\cal A}_{M_{disc}}^*$ étant égal à $i{\cal A}_{\tilde{M}}^*$, le paramètre en question appartient à $i{\cal A}_{\tilde{M}}^*$. Puisque seule intervient la classe de $\tau$ modulo ce groupe, on peut supposer que la restriction du caractère central de $\sigma$ à $ {\cal A}_{M_{disc}}$ est triviale. Fixons comme en 1.10 une sous-algèbre de Cartan $\mathfrak{h}_{{\mathbb C}}$ de l’algèbre de Lie complexifiée de $M_{disc}({\mathbb R})$. Pour toute représentation irréductible $\sigma$ de $M_{disc}({\mathbb R})$, on choisit un élément $\mu_{\sigma}\in \mathfrak{h}_{{\mathbb C}}$ dont la classe modulo un certain groupe de Weyl paramètre le caractère infinitésimal de $\sigma$. Notons $\hat{K}$ l’ensemble des représentations irréductibles de $K$. De même, un élément $\kappa\in \hat{K}$ a un paramètre $\mu_{\kappa}\in \mathfrak{h}_{{\mathbb C}}$. Rappelons que les opérateurs qui interviennent dans l’intégrale de (3) agissent dans la représentation $Ind_{S}^G(\sigma_{\lambda})\otimes Ind_{S}^G(\sigma_{\lambda})$, où $S$ est un élément fixé de ${\cal P}(M_{disc})$. On décompose cette représentation selon les espaces isotypiques pour l’action de $K\times K$. Ces espaces sont donc indexés par $(\kappa_{1},\kappa_{2})\in \hat{K}\times \hat{K}$. On note $p_{\kappa_{1},\kappa_{2}}$ le projecteur sur l’espace isotypique indexé par $(\kappa_{1},\kappa_{2})$. Posons $$j_{\tilde{M}}^{\tilde{G}}(\pi_{\tau_{\lambda}},\varphi_{1},\varphi_{2};\kappa_{1},\kappa_{2})
=p_{\kappa_{1},\kappa_{2}}j_{\tilde{M}}^{\tilde{G}}(\pi_{\tau_{\lambda}},\varphi_{1},\varphi_{2})p_{\kappa_{1},\kappa_{2}}.$$ On a $$(2) \qquad J_{\tilde{M}}^{\tilde{G}}(\pi_{\tau_{\lambda}},f_{1},f_{2}) =\sum_{\kappa_{1},\kappa_{2}\in \hat{K}}trace(j_{\tilde{M}}^{\tilde{G}}(\pi_{\tau_{\lambda}},\varphi_{1},\varphi_{2};\kappa_{1},\kappa_{2})).$$ Remarquons que l’opérateur ${\cal X}^{\tilde{G}}_{\tilde{M}}(\pi_{\tau_{\lambda}})$ conserve les espaces isotypiques par construction, tandis que $ U_{\tau_{\lambda}}$ envoie l’espace de type $(\kappa_{1},\kappa_{2})$ sur celui de type $(\kappa'_{1},\kappa'_{2})$, où, pour $i=1,2$, $\kappa'_{i}=ad_{\gamma_{0}}(\kappa_{i}\otimes \omega)$. On a donc $$j_{\tilde{M}}^{\tilde{G}}(\pi_{\tau_{\lambda}},\varphi_{1},\varphi_{2};\kappa_{1},\kappa_{2})=
{\cal X}^{\tilde{G}}_{\tilde{M}}(\pi_{\tau_{\lambda}};\kappa_{1},\kappa_{2})\Pi_{\tau_{\lambda}}(\varphi_{1},\varphi_{2};\kappa_{1},\kappa_{2})U_{\tau_{\lambda}}(\kappa_{1},\kappa_{2}),$$ où $${\cal X}^{\tilde{G}}_{\tilde{M}}(\pi_{\tau_{\lambda}};\kappa_{1},\kappa_{2})
=p_{\kappa_{1},\kappa_{2}}{\cal X}^{\tilde{G}}_{\tilde{M}}(\pi_{\tau_{\lambda}})p_{\kappa_{1},\kappa_{2}};$$ $$\Pi_{\tau_{\lambda}}(\varphi_{1},\varphi_{2};\kappa_{1},\kappa_{2})=p_{\kappa_{1},\kappa_{2}}
\left(\overline{\Pi_{\tau_{\lambda}}(\varphi_{1})}\otimes \Pi_{\tau_{\lambda}}(\varphi_{2})\right)p_{\kappa'_{1},\kappa'_{2}};$$ $$U_{\tau_{\lambda}}(\kappa_{1},\kappa_{2})=p_{\kappa'_{1},\kappa'_{2}} U_{\tau_{\lambda}}p_{\kappa_{1},\kappa_{2}}.$$ Puisque $\tilde{\Pi}_{\boldsymbol{\tau}_{\tilde{\lambda}}}$ est unitaire, ce dernier opérateur est de norme uniformément bornée. Il est clair que les paramètres associés à $\kappa_{i}$ et $\kappa'_{i}$ sont de même norme, pour $i=1,2$. D’après \[A8\] p.174, pour tout réel $r$, il existe une fonction $c_{r}:C_{c}^{\infty}(G({\mathbb R}))\times C_{c}^{\infty}(G({\mathbb R}))\to {\mathbb R}_{\geq0}$ qui est bornée par le sup d’un ensemble fini de semi-normes et qui est telle que l’on ait la majoration $$\vert \vert \Pi_{\tau_{\lambda}}(\varphi_{1},\varphi_{2};\kappa_{1},\kappa_{2})\vert \vert \leq c_{r}(\varphi_{1},\varphi_{2})(1+\vert \vert \mu_{\sigma}\vert \vert )^{-r}(1+\vert \vert \mu_{\kappa_{1}}\vert \vert )^{-r}(1+\vert \vert \mu_{\kappa_{2}}\vert \vert )^{-r}(1+\vert \vert \lambda\vert \vert )^{-r}.$$ On montrera ci-dessous qu’il existe $C\geq0$ et un entier $D\geq0$ de sorte que l’on ait la majoration $$(3) \qquad \vert \vert {\cal X}^{\tilde{G}}_{\tilde{M}}(\pi_{\tau_{\lambda}};\kappa_{1},\kappa_{2})\vert \vert \leq C(1+\vert \vert \mu_{\sigma}\vert \vert )^{D}(1+\vert \vert \mu_{\kappa_{1}}\vert \vert )^{D}(1+\vert \vert \mu_{\kappa_{2}}\vert \vert )^{D}(1+\vert \vert \lambda\vert \vert )^{D}.$$ Donc $trace(j_{\tilde{M}}^{\tilde{G}}(\pi_{\tau_{\lambda}},\varphi_{1},\varphi_{2};\kappa_{1},\kappa_{2}))$ est bornée par le produit des deux expressions ci-dessus et des dimensions de $\kappa_{1}$ et $\kappa_{2}$. On sait que la dimension de $\kappa$ est essentiellement bornée par $(1+\vert \vert \mu_{\kappa}\vert \vert )^{D'}$ pour un entier $D'$ convenable. On en déduit pour tout réel $r$ une majoration $$\vert trace(j_{\tilde{M}}^{\tilde{G}}(\pi_{\tau_{\lambda}},\varphi_{1},\varphi_{2};\kappa_{1},\kappa_{2}))\vert \leq c'_{r}(\varphi_{1},\varphi_{2})(1+\vert \vert \mu_{\sigma}\vert \vert )^{-r}(1+\vert \vert \mu_{\kappa_{1}}\vert \vert )^{-r}(1+\vert \vert \mu_{\kappa_{2}}\vert \vert )^{-r}(1+\vert \vert \lambda\vert \vert )^{-r},$$ où $c'_{r}(\varphi_{1},\varphi_{2})$ est bornée par le sup d’un ensemble fini de semi-normes. Grâce à (2), on obtient $$\vert J_{\tilde{M}}^{\tilde{G}}(\pi_{\tau_{\lambda}},f_{1},f_{2}) \vert \leq c'_{r}(\varphi_{1},\varphi_{2}) (1+\vert \vert \mu_{\sigma}\vert \vert )^{-r}(1+\vert \vert \lambda\vert \vert )^{-r}\left(\sum_{\kappa\in \hat{K}}(1+\vert \vert \mu_{\kappa}\vert \vert )^{-r}\right)^2.$$ Si $r$ est assez grand, la dernière série est convergente. L’intégrale en $\lambda$ de l’expression ci-dessus l’est aussi et on obtient simplement $$\int_{i{\cal A}_{\tilde{M}}^*}\vert J_{\tilde{M}}^{\tilde{G}}(\pi_{\tau_{\lambda}},f_{1},f_{2}) \vert \,d\lambda\leq C'c'_{r}(\varphi_{1},\varphi_{2})(1+\vert \vert \mu_{\sigma}\vert \vert )^{-r},$$ pour une certaine constante absolue $C'$. Remarquons que chaque représentation $\sigma$ de la série discrète de $M_{disc}({\mathbb R})$ ne donne naissance qu’à un nombre uniformément borné de triplets $\tau$. La sous-somme de $J_{spec}^{\tilde{G}}(\omega,f_{1},f_{2})$ où on a fixé $\tilde{M}$ et $M_{disc}$, mais où on a remplacé la fonction que l’on intègre par sa valeur absolue, est donc essentiellement majorée par $$c'_{r}(\varphi_{1},\varphi_{2})\sum_{\sigma}(1+\vert \vert \mu_{\sigma}\vert \vert )^{-r},$$ où on somme sur les représentations irréductibles $\sigma$ de $M_{disc}({\mathbb R})$ de la série discrète et dont la restriction du caractère central à $
{\cal A}_{M_{disc}}$ est triviale. Cette dernière condition signifie que $\mu_{\sigma,M_{disc}}=0$. Comme on l’a dit en 1.10, la projection $\mu_{\sigma}^{M_{disc}}$ parcourt un réseau de $i\mathfrak{h}^{M,*}$. De plus, on sait qu’il n’y a qu’un nombre uniformément borné de séries discrètes d’un paramètre donné. Donc, si $r$ est assez grand, la série ci-dessus est convergente. Cela prouve les assertions du (i) de l’énoncé concernant $J_{spec}^{\tilde{G}}(\omega,f_{1},f_{2})$. La propriété du terme $c'_{r}(\varphi_{1},\varphi_{2})$ prouve en même temps la continuité de cette expression en $f_{1}$ et $f_{2}$.
Puique les deux membres de l’égalité du (ii) de l’énoncé sont continus en $f_{1}$ et $f_{2}$, cette égalité résulte par continuité du théorème 5.1, les fonctions lisses à support compacts et $K$-finies étant denses dans $C_{c}^{\infty}(\tilde{G}({\mathbb R}))$.
Il reste à prouver la majoration (3). Tout d’abord, en reprenant la preuve du lemme 3.25, on voit que l’on peut remplacer la $(\tilde{G},\tilde{M})$-famile $({\cal X}(\pi_{\tau_{\lambda}};\Lambda,\tilde{Q}))_{\tilde{Q}\in {\cal P}(\tilde{M})}$ par $$({\bf r}_{reg}(\pi_{\tau_{\lambda}};\Lambda,\tilde{Q}){\cal X}_{reg}(\pi_{\tau_{\lambda}};\Lambda,\tilde{Q}))_{\tilde{Q}\in {\cal P}(\tilde{M})},$$ où ${\cal X}_{reg}(\pi_{\tau_{\lambda}};\Lambda,\tilde{Q})$ ne contient que des opérateurs d’entrelacement normalisés et $${\bf r}_{reg}(\pi_{\tau_{\lambda}}:\Lambda,\tilde{Q})=r_{\bar{Q}\vert Q}(\sigma_{\lambda})r_{\bar{P}\vert P}(\sigma_{\lambda})^{-1}r_{Q\vert \bar{Q}}(\sigma_{\lambda+\Lambda})r_{P\vert \bar{P}}(\sigma_{\lambda+\Lambda})^{-1}$$ ($\tilde{P}$ est un élément fixé de ${\cal P}(\tilde{M})$, cf. 3.25). Notons que cette fonction est produit de fonctions $r_{\alpha}(\sigma_{\lambda})r_{-\alpha}(\sigma_{\lambda})^{-1}$ et $r_{\alpha}(\sigma_{\lambda+\Lambda})r_{-\alpha}(\sigma_{\lambda+\Lambda})^{-1}$ comme en 1.10. Evidemment, ${\cal X}^{\tilde{G}}_{\tilde{M}}(\pi_{\tau_{\lambda}};\kappa_{1},\kappa_{2})$ se déduit de la $(\tilde{G},\tilde{M})$-famille $({\bf r}_{reg}(\pi_{\tau_{\lambda}}:\Lambda,\tilde{Q})p_{\kappa_{1},\kappa_{2}}{\cal X}_{reg}(\pi_{\tau_{\lambda}};\Lambda,\tilde{Q})p_{\kappa_{1},\kappa_{2}})_{\tilde{Q}\in {\cal P}(\tilde{M})}$. La formule 1.7(1) montre qu’il nous suffit de majorer les valeurs en $\Lambda=0$ des dérivées en $\Lambda$ d’ordre au plus $a_{\tilde{M}}-a_{\tilde{G}}$ de tous les termes intervenant dans la définition de la $(\tilde{G},\tilde{M})$-famille ci-dessus. Les opérateurs d’entrelacement normalisés étant unitaires, cela nous ramène à majorer les opérateurs $$(4)\qquad D(p_{\kappa_{1},\kappa_{2}}R_{Q\vert P}(\sigma_{\lambda})p_{\kappa_{1},\kappa_{2}})$$ et les fonctions $$(5)\qquad D(r_{\alpha}(\sigma_{\lambda})r_{-\alpha}(\sigma_{\lambda})^{-1})$$ où $D$ est une dérivation d’ordre au plus $a_{\tilde{M}}-a_{\tilde{G}}$ s’appliquant à la variable $\lambda$. Comme on l’a dit ci-dessus, nos hypothèses entraînent $\mu_{\sigma,M_{disc}}=0$, donc $\mu_{\sigma}$ est orthogonal à $\lambda$. La majoration cherchée de (4) résulte alors du lemme 2.1 de \[A5\]. On a décrit les fonctions $r_{\alpha}(\sigma_{\lambda})r_{-\alpha}(\sigma_{\lambda})^{-1}$ en 1.10. Compte tenu de l’égalité $\mu_{\sigma,M_{disc}}=0$, elles sont produit de termes $$(6)\qquad \frac{<\mu_{\sigma},\check{\beta}>-<\lambda,\check{\beta}>} {<\mu_{\sigma},\check{\beta}>+<\lambda,\check{\beta}>}$$ et éventuellement d’un terme $$\frac{\Gamma(\frac{<\lambda,\check{\beta}>}{2})\Gamma(\frac{-<\lambda,\check{\beta}>+1}{2})}{\Gamma(\frac{-<\lambda,\check{\beta}>}{2})\Gamma(\frac{<\lambda,\check{\beta}>+1}{2})}.$$ Ce dernier terme a des dérivées à croissance modéree et ne dépend pas de $\mu_{\sigma}$. Il vérifie donc la majoration requise. Considérons une fonction (6). Il résulte de ce que l’on a rappelé en 1.10 que le produit $<\mu_{\sigma},\check{\beta}>$ appartient à un sous-groupe discret fixe de ${\mathbb R}$, tandis que $<\lambda,\check{\beta}>$ est imaginaire. Ou bien $<\mu_{\sigma},\check{\beta}>=0$ et la fonction est constante égale à $-1$. Ou bien $<\mu_{\sigma},\check{\beta}>\not=0$ et alors la valeur absolue du dénominateur est uniformément minorée par une constante strictement positive. La majoration (5) requise s’ensuit. Cela achève la preuve. $\square$
Formules de descente pour les $(\tilde{G},\tilde{M})$-familles
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On rappelle dans ce paragraphe plusieurs formules générales d’Arthur concernant les $(\tilde{G},\tilde{M})$-familles. Pour $i=1,2$, soient $\tilde{M}_{i}\in {\cal L}(\tilde{M}_{0})$ et $(x_{i}(\Lambda;\tilde{P}))_{\tilde{P}\in {\cal P}(\tilde{M}_{i})}$ une $(\tilde{G},\tilde{M}_{i})$-famille. Soit $\tilde{L}\in {\cal L}(\tilde{M}_{0})$ tel que $\tilde{M}_{i}\subset \tilde{L}$ pour $i=1,2$. De la $(\tilde{G},\tilde{M}_{i})$-famille $(x_{i}(\Lambda;\tilde{P}))_{\tilde{P}\in {\cal P}(\tilde{M}_{i})}$ se déduit une $(\tilde{G},\tilde{L})$-famille $(x_{i}(\Lambda;\tilde{Q}))_{\tilde{Q}\in {\cal P}(\tilde{L})}$. Pour $\Lambda\in i{\cal A}_{\tilde{L}}^*$, on a $x_{i}(\Lambda;\tilde{Q})=x_{i}(\Lambda;\tilde{P})$ pour $\tilde{P}\in {\cal P}(\tilde{M}_{i}) $ tel que $P\subset Q$. Notons $(y(\Lambda,\tilde{Q}))_{\tilde{Q}\in {\cal P}(\tilde{L})}$ la famille produit, c’est-à-dire $y(\Lambda,\tilde{Q})=x_{1}(\Lambda,\tilde{Q})x_{2}(\Lambda,\tilde{Q})$. De cette $(\tilde{G},\tilde{L})$-famille se déduit une fonction $y_{\tilde{L}}^{\tilde{G}}(\Lambda)$ sur $i{\cal A}_{\tilde{L}}^*$. Rappelons que, pour $i=1,2$ et pour tout $\tilde{Q}_{i}=\tilde{L}_{i}U_{Q_{i}}\in {\cal F}(\tilde{M}_{i})$, on déduit de $(x_{i}(\Lambda;\tilde{P}))_{\tilde{P}\in {\cal P}(\tilde{M}_{i})}$ une $(\tilde{L}_{i},\tilde{M}_{i})$-famille $(x_{i}^{\tilde{Q}_{i}}(\Lambda;\tilde{P}))_{\tilde{P}\in {\cal P}^{\tilde{Q}_{i}}(\tilde{M}_{i})}$, où ${\cal P}^{\tilde{Q}_{i}}(\tilde{M}_{i})$ est l’ensemble des $\tilde{P}\in {\cal P}(\tilde{M}_{i})$ tels que $\tilde{P}\subset \tilde{Q}_{i}$. D’où une fonction $x_{i,\tilde{M}_{i}}^{\tilde{Q}_{i}}(\Lambda)$ sur $i{\cal A}_{\tilde{M}_{i}}^*$. Dans l’espace ${\cal A}_{\tilde{M}_{1}}^{\tilde{G}}\oplus {\cal A}_{\tilde{M}_{2}}^{\tilde{G}}$, introduisons l’image $\Delta({\cal A}_{\tilde{L}}^{\tilde{G}})$ de ${\cal A}_{\tilde{L}}^{\tilde{G}}$ par le plongement diagonal. Notons $\Delta({\cal A}_{\tilde{L}}^{\tilde{G}})^{\bot}$ son orthogonal. Pour tout couple $(\tilde{L}_{1},\tilde{L}_{2})\in {\cal L}(\tilde{M}_{1})\times {\cal L}(\tilde{M}_{2})$, considérons les conditions suivantes
\(1) $\Delta({\cal A}_{\tilde{L}}^{\tilde{G}})\oplus({\cal A}_{\tilde{L}_{1}}^{\tilde{G}}\oplus {\cal A}_{\tilde{L}_{2}}^{\tilde{G}})={\cal A}_{\tilde{M}_{1}}^{\tilde{G}}\oplus {\cal A}_{\tilde{M}_{2}}^{\tilde{G}}$;
\(2) $\Delta({\cal A}_{\tilde{L}}^{\tilde{G}})^{\bot}\oplus({\cal A}_{\tilde{M}_{1}}^{\tilde{L}_{1}}\oplus {\cal A}_{\tilde{M}_{2}}^{\tilde{L}_{2}})={\cal A}_{\tilde{M}_{1}}^{\tilde{G}}\oplus {\cal A}_{\tilde{M}_{2}}^{\tilde{G}}$;
\(3) l’application $$\begin{array}{ccc}{\cal A}_{\tilde{M}_{1}}^{\tilde{L}_{1}}\oplus {\cal A}_{\tilde{M}_{2}}^{\tilde{L}_{2}}&\to&{\cal A}_{\tilde{L}}^{\tilde{G}}\\ (H_{1},H_{2})&\mapsto&H_{1,\tilde{L}}+H_{2,\tilde{L}}\\ \end{array}$$ est un isomorphisme.
On vérifie qu’elles sont équivalentes. Si elles le sont, on note $d^{\tilde{G}}_{\tilde{M}_{1},\tilde{M}_{2}}(\tilde{L};\tilde{L}_{1},\tilde{L}_{2})$ le jacobien de l’isomorphisme (3), chaque espace étant muni des mesures fixées en 1.2. Si les conditions ne sont pas vérifiées, on pose $d^{\tilde{G}}_{\tilde{M}_{1},\tilde{M}_{2}}(\tilde{L};\tilde{L}_{1},\tilde{L}_{2})=0$. Fixons un point $\xi\in {\cal A}_{\tilde{M}_{1}}^{\tilde{G}}\oplus {\cal A}_{\tilde{M}_{2}}^{\tilde{G}}$ en position générale. Si les conditions ci-dessus sont vérifiées, notons $(\xi_{1},\xi_{2})\in {\cal A}_{\tilde{L}_{1}}^{\tilde{G}}\oplus {\cal A}_{\tilde{L}_{2}}^{\tilde{G}}$ la projection (non orthogonale) de $\xi$ relativement à la décomposition (1). Puisque $\xi$ est en position générale, il existe pour $i=1,2$ un unique $\tilde{Q}_{i}\in {\cal P}(\tilde{L}_{i})$ tel que $\xi_{i}$ soit dans la chambre positive associée à cet espace parabolique. Cela définit une application qui, à $(\tilde{L}_{1},\tilde{L}_{2})$ tel que $d^{\tilde{G}}_{\tilde{M}_{1},\tilde{M}_{2}}(\tilde{L};\tilde{L}_{1},\tilde{L}_{2})\not=0$, associe un couple $(\tilde{Q}_{1},\tilde{Q}_{2})\in {\cal P}(\tilde{L}_{1})\times {\cal P}(\tilde{L}_{2})$.
On a l’égalité
$$(4) \qquad y_{\tilde{L}}^{\tilde{G}}(\Lambda)=\sum_{\tilde{L}_{1}\in {\cal L}(\tilde{M}_{1}),\tilde{L}_{2}\in {\cal L}(\tilde{M}_{2})}d^{\tilde{G}}_{\tilde{M}_{1},\tilde{M}_{2}}(\tilde{L};\tilde{L}_{1},\tilde{L}_{2})x_{1,\tilde{M}_{1}}^{\tilde{Q}_{1}}(\Lambda)x_{2,\tilde{M}_{2}}^{\tilde{Q}_{2}}(\Lambda)$$ pour tout $\Lambda\in i{\cal A}_{\tilde{L}}^*$.
Dans le cas non tordu, cela résulte de \[A6\] proposition 7.1, appliquée au groupe $G\times G$, à son Levi $M_{1}\times M_{2}$ et à l’espace $\mathfrak{b}=\Delta({\cal A}_{L}^G)\oplus ({\cal A}_{G}\oplus {\cal A}_{G})$. La preuve s’étend au cas tordu.
On utilisera un cas particulier de la relation (4) où les définitions se simplifient. Soient $\tilde{M},\tilde{M}_{1},\tilde{M}_{2}\in {\cal L}(\tilde{M}_{0})$ tels que $\tilde{M}\subset \tilde{M}_{i}$ pour $i=1=2$ et ${\cal A}_{\tilde{M}}^{\tilde{M}_{1}}\cap {\cal A}_{\tilde{M}}^{\tilde{M}_{2}}=0$. Il existe un unique $\tilde{L}\in {\cal L}(\tilde{M})$ tel que $${\cal A}_{\tilde{L}}={\cal A}_{\tilde{M}_{1}}\cap {\cal A}_{\tilde{M}_{2}},$$ à savoir le commutant dans $\tilde{G}$ du tore $(A_{\tilde{M}_{1}}\cap A_{\tilde{M}_{2}})^0$. On a les égalités équivalentes $${\cal A}_{\tilde{M}}^{\tilde{L}}={\cal A}_{\tilde{M}_{1}}^{\tilde{L}}\oplus {\cal A}_{\tilde{M}_{2}}^{\tilde{L}},$$ $$(5) \qquad {\cal A}_{\tilde{M}}^{\tilde{L}}={\cal A}^{\tilde{M}_{1}}_{\tilde{M}}\oplus {\cal A}^{\tilde{M}_{2}}_{\tilde{M}}.$$ On voit que les conditions équivalentes (1), (2) et (3) sont équivalentes aux deux égalités équivalentes $$(6) \qquad {\cal A}_{\tilde{M}}^{\tilde{G}}={\cal A}_{\tilde{L}_{1}}^{\tilde{G}}\oplus {\cal A}_{\tilde{L}_{2}}^{\tilde{G}},$$ $$(7)\qquad {\cal A}_{\tilde{M}}^{\tilde{G}}={\cal A}^{\tilde{L}_{1}}_{\tilde{M}}\oplus {\cal A}^{\tilde{L}_{2}}_{\tilde{M}}.$$ Supposons ces conditions vérifiées. On note $d_{\tilde{M}}^{\tilde{G}}(\tilde{L}_{1},\tilde{L}_{2})$ le jacobien de l’application somme $${\cal A}^{\tilde{L}_{1}}_{\tilde{M}}\oplus {\cal A}^{\tilde{L}_{2}}_{\tilde{M}}\to {\cal A}_{\tilde{M}}^{\tilde{G}}$$ qui est un isomorphisme d’après (7). De même, grâce (5), on définit le jacobien $d_{\tilde{M}}^{\tilde{L}}(\tilde{M}_{1},\tilde{M}_{2})$. On vérifie qu’alors $$(8) \qquad d^{\tilde{G}}_{\tilde{M}_{1},\tilde{M}_{2}}(\tilde{L};\tilde{L}_{1},\tilde{L}_{2})=d_{\tilde{M}}^{\tilde{G}}(\tilde{L}_{1},\tilde{L}_{2})d_{\tilde{M}}^{\tilde{L}}(\tilde{M}_{1},\tilde{M}_{2})^{-1}.$$ Fixons $H\in {\cal A}_{\tilde{M}}^{\tilde{G}}$ en position générale. Grâce à (6), on peut l’écrire $H=H_{1}-H_{2}$, où $H_{i}\in {\cal A}_{\tilde{L}_{i}}^{\tilde{G}}$ pour $i=1,2$. Comme précédemment, $H_{i}$ détermine un espace parabolique $\tilde{Q}_{i}\in {\cal P}(\tilde{L}_{i})$. On vérifie que $(\tilde{Q}_{1},\tilde{Q}_{2})$ coïncide avec le couple déterminé précédemment pour un choix convenable de $\xi$.
Un cas encore plus particulier est celui où $\tilde{M}_{1}=\tilde{M}_{2}=\tilde{M}$. Alors $\tilde{L}=\tilde{M}$ et la relation (4) prend la forme $$(9) \qquad y_{\tilde{M}}^{\tilde{G}}(\Lambda)=\sum_{\tilde{L}_{1},\tilde{L}_{2}\in {\cal L}(\tilde{M})}d_{\tilde{M}}^{\tilde{G}}(\tilde{L}_{1},\tilde{L}_{2})x_{1,\tilde{M}}^{\tilde{Q}_{1}}(\Lambda)x_{2,\tilde{M}}^{\tilde{Q}_{2}}(\Lambda).$$ Cf. \[A6\] corollaire 7.4.
Soient maintenant $\tilde{M},\tilde{L}\in {\cal L}(\tilde{M}_{0})$ tels que $\tilde{M}\subset \tilde{L}$ et soit $(x(\Lambda;\tilde{P}))_{\tilde{P}\in {\cal P}(\tilde{M})}$ une $(\tilde{G},\tilde{M})$-famille. On en déduit une $(\tilde{G},\tilde{L})$-famille puis une fonction $x^{\tilde{G}}_{\tilde{L}}(\Lambda)$ sur $i{\cal A}_{\tilde{L}}^*$. On a l’égalité $$(10) \qquad x^{\tilde{G}}_{\tilde{L}}(\Lambda)=\sum_{\tilde{L}'\in {\cal L}(\tilde{M})}d_{\tilde{M}}^{\tilde{G}}(\tilde{L},\tilde{L}')x_{\tilde{M}}^{\tilde{Q}'}(\Lambda),$$ où $\tilde{Q}'$ est le second terme de l’image de couple $(\tilde{L},\tilde{L}')$ par l’application décrite ci-dessus.
Application des formules de descente
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Soit $\tilde{M}\in {\cal L}(\tilde{M}_{0})$. Pour $x\in G(F)$, on définit la $(\tilde{G},\tilde{M})$-famille $(v(x;\Lambda,\tilde{P}))_{\tilde{P}\in {\cal P}(\tilde{M})}$, où $v(x;\Lambda,\tilde{P})=e^{-<\Lambda,H_{\tilde{P}}(x)>}$. On en déduit une fonction $v^{\tilde{G}}_{\tilde{M}}(x;\Lambda)$. On pose $v^{\tilde{G}}_{\tilde{M}}(x)=v^{\tilde{G}}_{\tilde{M}}(x;0)$. Pour $f\in C_{c}^{\infty}(\tilde{G}(F))$ et $\gamma\in \tilde{M}(F)\cap \tilde{G}_{reg}(F)$ tel que $\omega$ soit trivial sur $Z_{G}(\gamma,F)$, on définit l’intégrale orbitale pondérée $$(1) \qquad J_{\tilde{M}}^{\tilde{G}}(\gamma,\omega,f)=D^{\tilde{G}}(\gamma)^{1/2}\int_{Z_{G}(\gamma,F)\backslash G(F)}\omega(x)f(x^{-1}\gamma x)v^{\tilde{G}}_{\tilde{M}}(x)\,dx.$$ Si $\omega$ est non trivial sur $Z_{G}(\gamma,F)$, on pose $J_{\tilde{M}}^{\tilde{G}}(\gamma,\omega,f)=0$.
D’autre part, pour une $\omega$-représentation tempérée et de longueur finie $\tilde{\pi}$ de $\tilde{M}(F)$, on a défini en 2.7 le caractère pondéré $J_{\tilde{M}}^{\tilde{G}}(\tilde{\pi},f)$ dans le cas où $f$ est $K$-finie. Les calculs du paragraphe 5.2 montrent que cette distribution s’étend continûment à toutes les fonctions $f\in C_{c}^{\infty}(\tilde{G}(F))$.
Soient $f\in C_{c}^{\infty}(\tilde{G}(F))$ et $\tilde{P}=\tilde{M}U_{P}\in {\cal P}(\tilde{M}_{0})$. On définit une fonction $f_{\tilde{P}}$ sur $\tilde{M}(F)$ par la formule habituelle $$f_{\tilde{P}}(x)=\delta_{\tilde{P}}(x)^{1/2}\int_{U_{P}(F)}\int_{K}\omega(k)f(k^{-1}xuk)\,dk\,du.$$ Cette fonction appartient à $C_{c}^{\infty}(\tilde{M}(F))$.
0.3cm[**[Lemme]{}**]{}.
**
\(i) Soient $f_{1},f_{2}\in C_{c}^{\infty}(\tilde{G}(F))$ et $\gamma\in \tilde{M}(F)\cap \tilde{G}_{reg}(F)$. On a l’égalité $$J_{\tilde{M}}^{\tilde{G}}(\gamma,\omega,f_{1},f_{2})=\sum_{\tilde{L}_{1},\tilde{L}_{2}\in {\cal L}(\tilde{M})}d_{\tilde{M}}^{\tilde{G}}(\tilde{L}_{1},\tilde{L}_{2})\overline{J_{\tilde{M}}^{\tilde{L}_{1}}(\gamma,\omega,f_{1,\tilde{\bar{Q}}_{1}})}J_{\tilde{M}}^{\tilde{L}_{2}}(\gamma,\omega,f_{2,\tilde{Q}_{2}}).$$
\(ii) Soient $f_{1},f_{2}\in C_{c}^{\infty}(\tilde{G}(F))$, $\boldsymbol{\tau}\in {\cal E}_{disc}(\tilde{M},\omega)$ et $\tilde{\lambda}\in i\tilde{{\cal A}}_{\tilde{M},F}^*$. On a l’égalité $$J_{\tilde{M}}^{\tilde{G}}(\pi_{\tau_{\lambda}},f_{1},f_{2})=\sum_{\tilde{L}_{1},\tilde{L}_{2}\in {\cal L}(\tilde{M})}d_{\tilde{M}}^{\tilde{G}}(\tilde{L}_{1},\tilde{L}_{2})\overline{J_{\tilde{M}}^{\tilde{L}_{1}}(\tilde{\pi}_{\boldsymbol{\tau}_{\tilde{\lambda}}},f_{1,\tilde{\bar{Q}}_{1}})}J_{\tilde{M}}^{\tilde{L}_{2}}(\tilde{\pi}_{\boldsymbol{\tau}_{\tilde{\lambda}}},f_{2,\tilde{Q}_{2}}).$$
\(iii) Soient $f\in C_{c}^{\infty}(\tilde{G}(F))$, $\gamma\in \tilde{M}(F)\cap \tilde{G}_{reg}(F)$ et $\tilde{L}\in {\cal L}(\tilde{M})$. On a l’égalité $$J_{\tilde{L}}^{\tilde{G}}(\gamma,\omega,f)=\sum_{\tilde{L}'\in {\cal L}(\tilde{M})}d_{\tilde{M}}^{\tilde{G}}(\tilde{L},\tilde{L}')J_{\tilde{M}}^{\tilde{L}'}(\gamma,\omega,f_{\tilde{Q}'}).$$
\(iv) Soient $f\in C_{c}^{\infty}(\tilde{G}(F))$, $\tilde{\pi}$ une $\omega$-représentation tempérée de longueur finie de $\tilde{M}(F)$ et $\tilde{L}\in {\cal L}(\tilde{M})$. Posons $\tilde{\Pi}=Ind_{\tilde{P}}^{\tilde{L}}(\tilde{\pi})$, où $\tilde{P}$ est un élément de ${\cal P}^{\tilde{L}}(\tilde{M})$. Alors on a l’égalité $$J_{\tilde{L}}^{\tilde{G}}(\tilde{\Pi},f)=\sum_{\tilde{L}'\in {\cal L}(\tilde{M})}d_{\tilde{M}}^{\tilde{G}}(\tilde{L},\tilde{L}')J_{\tilde{M}}^{\tilde{L}'}(\tilde{\pi},f_{\tilde{Q}'}).$$
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La preuve est basée sur les formules 5.3(9) et (10) appliquées aux $(\tilde{G},\tilde{M})$-familles intervenant dans les définitions des membres de gauche. On la laisse au lecteur.
On aura besoin plus tard des propriétés suivantes. Soient $\tilde{M}\in {\cal L}(\tilde{M}_{0})$ et $\tilde{T}$ un sous-tore tordu maximal de $\tilde{M}$ tel que $\omega$ soit trivial sur $T^{\theta}(F)$. Alors
\(2) pour tout $f\in C_{c}^{\infty}(\tilde{G}(F))$, la fonction $\gamma\mapsto J_{\tilde{M}}^{\tilde{G}}(\gamma,\omega,f)$ est lisse sur $\tilde{T}(F)\cap \tilde{G}_{reg}(F)$.
En effet, au voisinage d’un point régulier, $J_{\tilde{M}}^{\tilde{G}}(\gamma,\omega,f)$ est l’intégrale sur un compact d’une fonction lisse en $\gamma$ et en la variable d’intégration.
\(3) il existe un entier $N\geq0$ et, pour tout $f\in C_{c}^{\infty}(\tilde{G}(F))$, il existe $c>0$ tel que $\vert J_{\tilde{M}}^{\tilde{G}}(\gamma,\omega,f)\vert \leq c (1+\vert log(D^{\tilde{G}}(\gamma))\vert )^N$.
Preuve. Comme on l’a déjà utilisé plusieurs fois, il résulte de la variante tordue de 1.7(1) et de 4.2(7) qu’il existe $N\geq0$ indépendant de $f$ et $c'>0$ tel que, pour $x$ contribuant à la formule (1), on ait $v^{\tilde{G}}_{\tilde{M}}(x)\leq c' (1+\vert log(D^{\tilde{G}}(\gamma))\vert )^N$. Alors $$\vert J_{\tilde{M}}^{\tilde{G}}(\gamma,\omega,f)\vert \leq c' (1+\vert log(D^{\tilde{G}}(\gamma))\vert )^ND^{\tilde{G}}(\gamma)^{1/2}\int_{Z_{G}(\gamma,F)\backslash G(F)}\vert f(x^{-1}\gamma x)\vert \,dx.$$ Il reste à appliquer 4.2(1) pour obtenir (3). $\square$
Le théorème $0$
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[0.3cm[**[Théorème]{}**]{}. [ *[Soit $f\in C_{c}^{\infty}(\tilde{G}(F))$. Supposons $I_{\tilde{G}}(\tilde{\pi},f)=0$ pour toute $\omega$-représentation irréductible tempérée $\tilde{\pi}$ de $\tilde{G}(F)$. Alors $I_{\tilde{G}}(\gamma,\omega,f)=0$ pour tout $\gamma\in \tilde{G}_{reg}(F)$.]{}*]{}0.3cm]{}
[**Remarques.**]{} (1) On n’a défini dans cet article que les intégrales orbitales relatives à des éléments $\gamma\in \tilde{G}_{reg}(F)$. Mais on sait bien qu’on peut les définir pour tout $\gamma\in \tilde{G}(F)$. Par la théorie de la descente, on montre que leur comportement local est le même que dans le cas non tordu. Cest-à-dire que les intégrales orbitales aux points singuliers s’expriment à l’aide d’intégrales orbitales aux points réguliers, soit grâce aux germes de Shalika dans le cas non-archimédien, soit par l’action d’opérateurs différentiels dans le cas archimédien. On peut donc renforcer la conclusion du théorème: $I_{\tilde{G}}(\gamma,\omega,f)=0$ pour tout $\gamma\in \tilde{G}(F)$ tel que $\omega$ soit trivial sur $Z_{G}(\gamma,F)$.
\(2) Dans le cas non-archimédien, on montre comme dans le cas non tordu qu’une fonction $f\in C_{c}^{\infty}(\tilde{G}(F))$ dont toutes les intégrales orbitales sont nulles est annulée par toute forme linéaire $l$ sur $C_{c}^{\infty}(\tilde{G}(F))$ qui vérifie la relation $l(f^x)=\omega(x)l(f)$ pour tout $f\in C_{c}^{\infty}(\tilde{G}(F))$ et tout $x\in G(F)$, où $f^x$ est la fonction $f^x(\gamma)=f(x\gamma x^{-1})$.
\(3) Dans le cas non-archimédien, le théorème a été prouvé dans \[HL\].
Preuve. On raisonne par récurence sur la dimension de $\tilde{G}$. Soit $\tilde{M} $ un espace de Levi semi-standard propre. Fixons $\tilde{P}\in {\cal P}(\tilde{M})$. Pour une $\omega$-représentation admisslble $\tilde{\pi}$ de $\tilde{M}(F)$, on a l’égalité $I_{\tilde{M}}(\tilde{\pi},f_{\tilde{P}})=I_{\tilde{G}}(\tilde{\Pi},f)$, où $\Pi=Ind_{\tilde{P}}^{\tilde{G}}(\tilde{\pi})$. La fonction $f_{\tilde{P}}$ vérifie donc la même hypothèse que $f$. Donc, par l’hypothèse de récurrence, $I_{\tilde{M}}(\gamma,\omega,f_{\tilde{P}})=0$ pour tout $\gamma\in \tilde{M}_{reg}(F)$. Mais, pour $\gamma\in \tilde{M}(F)\cap \tilde{G}_{reg}(F)$, on a l’égalité $I_{\tilde{G}}(\gamma,\omega,f)=I_{\tilde{M}}(\gamma,\omega,f_{\tilde{P}})$. Donc $I_{\tilde{G}}(\gamma,\omega,f)=0$. Cela prouve le résultat pour tout $\gamma\in \tilde{G}_{reg}(F)$ qui n’est pas elliptique. Pour traiter le cas des éléments elliptiques, considérons une fonction $f_{2}\in C_{c}^{\infty}(\tilde{G}(F))$ à support dans l’ensemble $\tilde{G}_{ell}(F)$ des éléments elliptiques réguliers de $\tilde{G}(F)$. Soient $\tilde{M}$ un espace de Levi de $\tilde{G}$, $\tau\in E_{disc}(\tilde{M},\omega)$ et $\lambda\in i{\cal A}_{\tilde{M},F}^*$. On relève $\tau$ en $\boldsymbol{\tau}\in {\cal E}_{disc}(\tilde{M},\omega)$ et $\lambda$ en $\tilde{\lambda}\in i\tilde{{\cal A}}_{\tilde{M},F}^*$. Le terme $J_{\tilde{M}}^{\tilde{G}}(\pi_{\tau_{\lambda}},f,f_{2})$ est calculé par le lemme 5.4(ii). Si $\tilde{Q}_{2}$ est un espace parabolique propre de $\tilde{G}$, l’hypothèse sur le support de $f_{2}$ entraîne que $f_{2,\tilde{Q}_{2}}=0$. La formule se simplifie donc en $$J_{\tilde{M}}^{\tilde{G}}(\pi_{\tau_{\lambda}},f,f_{2})=\overline{J_{\tilde{M}}^{\tilde{M}}(\tilde{\pi}_{\boldsymbol{\tau}_{\tilde{\lambda}}},f_{\tilde{\bar{Q}}})}J_{\tilde{M}}^{\tilde{G}}(\tilde{\pi}_{\boldsymbol{\tau}_{\tilde{\lambda}}},f_{2}),$$ où $\tilde{Q}$ est un certain élément de ${\cal P}(\tilde{M})$. On a $$J_{\tilde{M}}^{\tilde{M}}(\tilde{\pi}_{\boldsymbol{\tau}_{\tilde{\lambda}}},f_{\tilde{\bar{Q}}})=I_{\tilde{M}}^{\tilde{M}}(\tilde{\pi}_{\boldsymbol{\tau}_{\tilde{\lambda}}},f_{\tilde{\bar{Q}}})=I_{\tilde{G}}(\tilde{\Pi},f),$$ où $\tilde{\Pi}=Ind_{\tilde{\bar{Q}}}^{\tilde{G}}(\tilde{\pi}_{\boldsymbol{\tau}_{\tilde{\lambda}}})$. La représentation $\tilde{\Pi}$ est tempérée, donc $I_{\tilde{G}}(\tilde{\Pi},f)=0$. Cela prouve que $J^{\tilde{G}}_{spec}(\omega,f,f_{2})=0$. Appliquons la proposition 5.2: elle entraîne $J^{\tilde{G}}_{g\acute{e}om}(\omega,f,f_{2})=0$. Pour un espace de Levi propre $\tilde{M}$ et un élément $\tilde{S}\in T_{ell}(\tilde{M},\omega) $, on a $J_{\tilde{M},\tilde{S}}(\omega,f,f_{2})=0$: l’hypothèse sur le support de $f_{2}$ entraîne que $f_{2}(y^{-1}\gamma y)=0$ pour tout $y\in G(F)$ et tout $\gamma\in \tilde{S}(F)$. On a donc $$J_{g\acute{e}om}^{\tilde{G}}(\omega,f,f_{2})=\sum_{\tilde{S}\in T_{ell}(\tilde{G},\omega)}\vert W^G(\tilde{S})\vert ^{-1}mes(A_{\tilde{G}}(F)\backslash S^{\theta}(F))$$ $$\int_{\tilde{S}(F)/(1-\theta)(S(F))}\overline{I_{\tilde{G}}(\gamma,\omega,f)}I_{\tilde{G}}(\gamma,\omega,f_{2})\,d\gamma.$$ Soit maintenant $\gamma\in \tilde{G}_{ell}(F)$ tel que $\omega$ soit trivial sur $Z_{G}(\gamma,F)$. On veut montrer que $I_{\tilde{G}}(\gamma,\omega,f)=0$. On peut conjuguer $\gamma$ et supposer que $\gamma\in \tilde{S}(F)$ pour un $\tilde{S}$ intervenant dans la formule ci-dessus. On fait maintenant parcourir à $f_{2}$ une suite de fonctions $(f_{2,n})_{n\in {\mathbb N}}$ à valeurs positives ou nulles, non nulles en $\gamma$, et à supports dans des voisinages de plus en plus petits de $\gamma$. On voit que l’expression ci-dessus est de la forme $$\int_{S^{\theta,0}(F)}\overline{I_{\tilde{G}}(x\gamma,\omega,f)}A_{n}(x)\,dx$$ où $A_{n}$ parcourt une suite de fonctions lisses sur $S^{\theta,0}(F))$ à valeurs positives ou nulles, non nulles en $x=1$, et à supports dans des voisinages de plus en plus petits de $1$. Puisque $x\mapsto I_{\tilde{G}}(x\gamma,\omega,f)$ est lisse, la nullité de l’expression ci-dessus pour tout $n$ entraîne $I_{\tilde{G}}(\gamma,\omega,f)=0$. $\square$
La formule invariante
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Le théorème de Paley-Wiener
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Notons ${\cal K}$ le groupe de Grothendieck de l’ensemble des classes d’isomorphisme de $\omega$-représentations tempérées de longueur finie de $\tilde{G}(F)$. Notons ${\cal F}$ l’espace des fonctions linéaires de ${\cal K}$ dans ${\mathbb C}$ qui sont nulles sur toute $\omega$-représentation tempérée irréductible et non $G$-irréductible. Remarquons que ${\cal F}$ s’identifie à l’espace des fonctions à valeurs complexes sur l’ensemble des classes d’isomorphisme de $\omega$-représentations tempérées $G$-irréductibles de $\tilde{G}(F)$. Notons ${\cal PW}(\tilde{G},\omega)$ le sous-ensemble des éléments $\varphi\in {\cal F}$ vérifiant les conditions suivantes:
\(1) soient $\tilde{Q}=\tilde{L}U_{Q}$ un espace parabolique tel que $\omega$ soit trivial sur $Z_{L}(F)^{\theta}$ et $\tilde{\pi}$ une $\omega$-représentation tempérée et $L$-irréductible de $\tilde{L}(F)$; alors la fonction $\tilde{\lambda}\mapsto \varphi(Ind_{\tilde{Q}}^{\tilde{G}}(\tilde{\pi}_{\tilde{\lambda}}))$ sur $i\tilde{{\cal A}}_{\tilde{L},F}^*$ est de Paley-Wiener (cf. 2.6);
\(2) si $F$ est non-archimédien, il existe un sous-groupe ouvert compact $H$ de $\tilde{G}(F)$ tel que $\varphi(\tilde{\pi})=0$ pour toute $\omega$-représentation tempérée $\tilde{\pi}$ telle que le sous-espace des invariants par $H$ dans $\pi$ soit nul;
\(3) si $F$ est archimédien, il existe un ensemble fini $\kappa_{1},...,\kappa_{n}$ de représentations irréductibles de $K$ tel que $\varphi(\tilde{\pi})=0$ pour toute $\omega$-représentation tempérée $\tilde{\pi}$ telle que, pour tout $i$, l’espace isotypique de type $\kappa_{i}$ dans $\pi$ soit nul.
Notons $\underline{pw}_{\tilde{G}}:C_{c}^{\infty}(\tilde{G}(F),K)\to {\cal F}$ l’application linéaire qui, à $f\in C_{c}^{\infty}(\tilde{G}(F),K)$, associe la fonction $\tilde{\pi}\mapsto I_{\tilde{G}}(\tilde{\pi},f)$. Notons $I(\tilde{G}(F),K,\omega)$ le quotient de $C_{c}^{\infty}(\tilde{G}(F),K)$ par le sous-espace des $f\in C_{c}^{\infty}(\tilde{G}(F),K)$ telles que $I_{\tilde{G}}(\gamma,\omega,f)=0$ pour tout $\gamma\in \tilde{G}_{reg}(F)$.
[0.3cm[**[Théorème]{}**]{}. [ *[L’application linéaire $\underline{pw}_{\tilde{G}}$ a pour image ${\cal PW}(\tilde{G},\omega)$ et se quotiente en un isomorphisme de $I(\tilde{G}(F),K,\omega)$ sur cet espace.]{}*]{}0.3cm]{}
Quand $F$ est non-archimédien, le théorème a été démontré dans \[R\] pour $\omega=1$ et en général dans \[HL\]. Si $F={\mathbb R}$, la première assertion du théorème est démontrée dans \[DM\] pour $\omega=1$. Nous montrerons en 6.4 que le cas $\omega\not=1$ s’en déduit. Si $F={\mathbb C}$, l’assertion est équivalente à celle pour l’espace tordu sur ${\mathbb R}$ déduit de $\tilde{G}$ par restriction des scalaires. La deuxième assertion est facile: le théorème 5.5 montre que tout élément du noyau a des intégrales orbitales nulles; la réciproque provient de la locale intégrabilité des caractères des $\omega$-représentations $G$-irréductibles, cf. 2.5(2).
[**Complément au théorème.**]{} Supposons $F$ non-archimédien. Soit $H$ un sous-groupe ouvert compact de $G(F)$. Notons ${\cal PW}(\tilde{G},\omega)^H$ le sous-espace des fonctions $\varphi\in {\cal PW}(\tilde{G},\omega)$ tels que $\varphi(\tilde{\pi})=0$ pour toute $\omega$-représentation tempérée $\tilde{\pi}$ telle que le sous-espace des invariants par $H$ dans $\pi$ soit nul. Notons $C_{c}^{\infty}(H\backslash \tilde{G}(F)/H)$ le sous-espace des éléments de $C_{c}^{\infty}(\tilde{G}(F))$ qui sont biinvariants par $H$. Alors, $H$ étant donné, il existe $H'$ tel que ${\cal PW}(\tilde{G},\omega)^H$ soit contenu dans $\underline{pw}_{\tilde{G}}(C_{c}^{\infty}(H'\backslash\tilde{G}(F)/H'))$. Cf. \[HL\].
Supposons $F$ archimédien. Soit $\underline{\kappa}$ un ensemble fini de représentations irréductibles de $K$. Notons ${\cal PW}(\tilde{G},\omega)^{\underline{\kappa}}$ le sous-espace des fonctions $\varphi\in PW(\tilde{G},\omega)$ tels que $\varphi(\tilde{\pi})=0$ pour toute $\omega$-représentation tempérée $\tilde{\pi}$ telle que, pour tout $\kappa\in \underline{\kappa}$, l’espace isotypique de type $\kappa$ dans $\pi$ soit nul. Notons $C_{c}^{\infty}(\tilde{G}(F),\underline{\kappa})$ le sous-espace des $f\in C_{c}^{\infty}(\tilde{G}(F))$ tels que la représentation de $K\times K$ dans l’espace engendré par les translatés de $f$ à droite et à gauche par des éléments de $K$ n’ait pour composantes irréductibles que des éléments de $\underline{\kappa}\times \underline{\kappa}$. Alors, $\underline{\kappa}$ étant donné, il existe $\underline{\kappa}'$ tel que ${\cal PW}(\tilde{G}(F))^{\underline{\kappa}}$ soit contenu dans $\underline{pw}_{\tilde{G}}(C_{c}^{\infty}(\tilde{G}(F),\underline{\kappa}'))$. Ceci n’est pas énoncé dans \[DM\], mais résulte clairement de la preuve.
Deuxième forme du théorème de Paley-Wiener
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On rappelle que l’on a défini l’ensemble ${\cal E}(\tilde{G},\omega)$ en 2.9. Il est muni d’une action de $i\tilde{{\cal A}}_{\tilde{G},F}^*$ qui, à $\boldsymbol{\tau}\in {\cal E}(\tilde{G},\omega)$ et $\tilde{\lambda}\in i\tilde{{\cal A}}_{\tilde{G},F}^*$, associe $\boldsymbol{\tau}_{\tilde{\lambda}}$. Il est aussi muni d’une action de ${\mathbb U}$ qui, à $\boldsymbol{\tau}=(M,\sigma,\boldsymbol{\tilde{r}})$ et $z\in {\mathbb U}$, associe $z\boldsymbol{\tau}=(M,\sigma,z\boldsymbol{\tilde{r}})$. Rappelons que l’action de $z\in i{\mathbb R}/2\pi i{\mathbb Z}\in i\tilde{{\cal A}}_{\tilde{G},F}^*$ coïncide avec celle de $e^z\in {\mathbb U}$. Notons ${\cal E}_{ell}(\tilde{G},\omega)$ l’ensemble des triplets $(M,\sigma,\boldsymbol{\tilde{r}})\in {\cal E}(\tilde{G},\omega)$ tels que $(M,\sigma,\tilde{r})\in E_{ell}(\tilde{G},\omega)$, où $\tilde{r}$ est l’image de $\boldsymbol{\tilde{r}}$ dans $R^{\tilde{G}}(\sigma)$. Notons ${\cal E}_{ell}(\tilde{G},\omega)/conj$ l’ensemble des classes de conjugaison par $G(F)$ dans ${\cal E}_{ell}(\tilde{G},\omega)$. On a une surjection $({\cal E}_{ell}(\tilde{G},\omega)/conj)\to (E_{ell}(\tilde{G},\omega)/conj)$ dont les fibres sont toutes isomorphes à ${\mathbb U}$. Notons $PW_{ell}(\tilde{G},\omega)$ l’espace des fonctions $\varphi:({\cal E}_{ell}(\tilde{G},\omega)/conj)\to {\mathbb C}$ qui vérifient les conditions suivantes:
\(1) soient $\boldsymbol{\tau}\in {\cal E}_{ell}(\tilde{G},\omega)$ et $z\in {\mathbb U}$; alors $\varphi(z\boldsymbol{\tau})=z\varphi(\boldsymbol{\tau})$ (via la projection ${\cal E}_{ell}(\tilde{G},\omega)\to ({\cal E}_{ell}(\tilde{G},\omega)/conj)$, on a identifié $\varphi$ à une fonction sur ${\cal E}_{ell}(\tilde{G},\omega)$);
\(2) soit $\boldsymbol{\tau}=(M,\sigma,\boldsymbol{\tilde{r}})\in {\cal E}_{ell}(\tilde{G},\omega)$; alors la fonction $\tilde{\lambda}\mapsto \varphi(\boldsymbol{\tau}_{\tilde{\lambda}})$ est de Paley-Wiener sur $i\tilde{{\cal A}}_{\tilde{G},F}^*$;
\(3) le support de $\varphi$ est contenu dans un nombre fini d’orbites pour l’action de $i\tilde{{\cal A}}_{\tilde{G},F}^*$ dans ${\cal E}_{ell}(\tilde{G},\omega)/conj$.
Rappelons (cf. 2.12) que l’on note ${\cal L}(\tilde{M}_{0},\omega)$ l’ensemble des $\tilde{L}\in {\cal L}(\tilde{M}_{0})$ tels que $\omega$ soit trivial sur $Z_{L}(F)^{\theta}$. Soient $\tilde{L}$, $\tilde{L}'$ deux éléments de ${\cal L}(\tilde{M}_{0},\omega)$ et soit $x\in G(F)$ tel que $x\tilde{L}x^{-1}=\tilde{L}'$. De la conjugaison par $x$ se déduit un isomorphisme $T_{ell}(\tilde{L},\omega)\simeq T_{ell}(\tilde{L}',\omega)$ (cf. 2.12), puis un isomorphisme $PW_{ell}(\tilde{L},\omega)\simeq PW_{ell}(\tilde{L}',\omega)$. Notons que, si $\tilde{L}=\tilde{L}'$ et $x\in L(F)$, cet isomorphisme est l’identité. En particulier, si on note $W^G(\tilde{L})$ le quotient par $L(F)$ du normalisateur de $\tilde{L}$ dans $G(F)$, ce groupe $W^G(\tilde{L})$ agit sur $PW_{ell}(\tilde{L})$. On pose $$PW(\tilde{G},\omega)= (\oplus_{\tilde{L}\in {\cal L}(\tilde{M}_{0},\omega)}PW_{ell}(\tilde{L},\omega))^{\tilde{W}^G}$$ $$= \oplus_{\tilde{L}\in {\cal L}(\tilde{M}_{0},\omega)/\tilde{W}^G}PW_{ell}(\tilde{L},\omega)^{W^G(\tilde{L})},$$ les exposants signifiant selon l’usage que l’on prend les invariants.
[**Remarque.**]{} Soient $\tilde{L}\in {\cal L}(\tilde{M}_{0},\omega)$, $\varphi\in PW_{ell}(\tilde{L},\omega)^{W^G(\tilde{L})}$ et $\boldsymbol{\tau}\in {\cal E}_{ell}(\tilde{L},\omega)$. Le triplet $\boldsymbol{\tau}$ est par définition essentiel dans $\tilde{L}$ mais on a déjà dit qu’il ne l’était pas forcément dans $\tilde{G}$. S’il ne l’est pas, il existe $w\in W^G(\tilde{L})$ et $z\in {\mathbb U}$ tels que $w(\boldsymbol{\tau})=z\tau$, et $z\not=1$ (cf. preuve de la proposition 2.12). Cela entraîne $\varphi(\boldsymbol{\tau})=0$.
Soit $\psi\in {\cal PW}(\tilde{G},\omega)$. Pour un espace de Levi $\tilde{L}\in {\cal L}(\tilde{M}_{0},\omega)$ et un élément $\boldsymbol{\tau}\in {\cal E}_{ell}(\tilde{L},\omega)$, rappelons que l’on a défini en 2.9 une $\omega$-représentation $\tilde{\pi}_{\boldsymbol{\tau}}$ de $\tilde{L}(F)$. Fixons un espace parabolique $\tilde{Q}\in {\cal P}(\tilde{L})$, posons $\tilde{\Pi}_{\boldsymbol{\tau}}=Ind_{\tilde{Q}}^{\tilde{G}}(\tilde{\pi}_{\boldsymbol{\tau}})$. Cette représentation est tempérée et sa classe d’isomorphisme ne dépend pas de $\tilde{Q}$. Posons $\varphi(\boldsymbol{\tau})=\psi(\tilde{\Pi}_{\boldsymbol{\tau}})$. On a ainsi une fonction $\varphi$ sur $\oplus_{\tilde{L}\in {\cal L}(\tilde{M}_{0},\omega)}T_{ell}(\tilde{L},\omega)$. Cette fonction appartient à $PW(\tilde{G},\omega)$. En effet, les conditions (1) et (2) résultent de 6.1(1) tandis que la condition (3) résulte de 6.1(2) et (3). La condition d’invariance par $\tilde{W}^G$ résulte de l’égalité $\tilde{\Pi}_{w(\boldsymbol{\tau})}=\tilde{\Pi}_{\boldsymbol{\tau}}$ pour tout $w\in\tilde{W}^G$. Notons $$res:{\cal PW}(\tilde{G},\omega)\to PW(\tilde{G},\omega)$$ l’application $\psi\mapsto \varphi$ et $$pw_{\tilde{G}}:C_{c}^{\infty}(\tilde{G}(F),K)\to PW(\tilde{G},\omega)$$ la composée $pw_{\tilde{G}}=res\circ \underline{pw}_{\tilde{G}}$.
[0.3cm[**[Théorème]{}**]{}. [ *[L’application $pw_{\tilde{G}}$ se quotiente en un isomorphisme de $I(\tilde{G}(F),K,\omega)$ sur $PW(\tilde{G},\omega)$.]{}*]{}0.3cm]{}
Preuve. L’énoncé résulte du théorème 6.1 pouvu que $res$ soit bijectif. Cette bijectivité résulte aisément de la proposition 2.12. $\square$
Le théorème admet un complément similaire à celui du théorème 6.1.
Extension du théorème de Delorme et Mezo au cas $\omega\not=1$
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Dans ce paragraphe, on suppose $F={\mathbb R}$. On veut prouver
\(1) l’application $\underline{pw}_{\tilde{G}}$ a pour image ${\cal PW}(\tilde{G},\omega)$.
Delorme et Mezo ont prouvé l’assertion pour $\omega={\bf 1}$, où ${\bf 1}$ désigne le caractère trivial de $G({\mathbb R})$. Supposons d’abord qu’il existe un caractère unitaire $\mu$ de $G({\mathbb R})$ tel que $\omega=\mu\circ(1-\theta)$, cf. 2.4. Fixons un élément $\gamma_{0}\in \tilde{G}({\mathbb R})$. Comme dans la preuve de 2.5(2), on associe à toute $\omega$-représentation $\tilde{\pi}$ de $\tilde{G}({\mathbb R})$ une ${\bf 1}$-représentation $\tilde{\pi}_{1}$ définie par $\tilde{\pi}_{1}(g\gamma_{0})=\mu(g)\tilde{\pi}(\gamma_{0})$ pour tout $g\in G({\mathbb R})$. L’application $\tilde{\pi}\mapsto \tilde{\pi}_{1}$ est bijective. Pour plus de précision, on ajoute des indices $\omega$ à certains objets définis en 6.1, par exemple ${\cal F}_{\omega}$ et $\underline{pw}_{\tilde{G},\omega }$. On définit une application $$\begin{array}{ccc}{\cal F}_{\omega}&\to&{\cal F}_{{\bf 1}}\\ \varphi&\mapsto &\varphi_{1}\\ \end{array}$$ par $ \varphi_{1}(\tilde{\pi}_{1})=\varphi(\tilde{\pi})$. On définit une application $$\begin{array}{ccc}C_{c}^{\infty}(\tilde{G}({\mathbb R}),K)&\to&C_{c}^{\infty}(\tilde{G}({\mathbb R}),K)\\ f&\mapsto& f_{1} \\ \end{array}$$ par $f_{1}(g\gamma_{0})=\mu(g)^{-1}f(g\gamma_{0})$ pour tout $g\in G({\mathbb R})$. Il est clair que $I_{\tilde{G}}(\tilde{\pi}_{1},f_{1})=I_{\tilde{G}}(\tilde{\pi},f)$ pour tous $f$, $\tilde{\pi}$. Donc le diagramme suivant est commutatif: $$\begin{array}{ccc}C_{c}^{\infty}(\tilde{G}({\mathbb R}),K)&\stackrel{f\mapsto f_{1}}{\to}&C_{c}^{\infty}(\tilde{G}({\mathbb R}),K)\\ \downarrow \underline{pw}_{\tilde{G},\omega}&&\downarrow \underline{pw}_{\tilde{G},{\bf 1}}\\ {\cal F}_{\omega}&\stackrel {\varphi\mapsto \varphi_{1}}{\to}&{\cal F}_{{\bf 1}}.\\ \end{array}$$ L’application horizontale du haut est bijective. On vérifie immédiatement que celle du bas se restreint en un isomorphisme de ${\cal PW}(\tilde{G},\omega)$ sur ${\cal PW}(\tilde{G},{\bf 1})$. Alors l’assertion (1) se déduit de la même assertion pour le caractère ${\bf 1}$.
Dans le cas général, on introduit des objets $G'$, $\tilde{G}'$, $C$, $p$, $\tilde{p}$ vérifiant la proposition 2.4. On pose $\omega'=\omega\circ p$. Ces termes vérifient l’hypothèse précédente: il existe un caractère $\mu'$ de $G'({\mathbb R})$ tel que $\omega'=\mu'\circ(1-\theta')$. On note $K'$ l’unique sous-groupe compact maximal de $G'({\mathbb R})$ contenu dans $p^{-1}(K)$. Pour une $\omega$-représentation $\tilde{\pi}$ de $\tilde{G}({\mathbb R})$, on note $\tilde{\pi}'$ la $\omega'$-représentation $\tilde{\pi}\circ\tilde{p}$ de $\tilde{G}'({\mathbb R})$. Dualement, on en déduit une application $$(2) \qquad \begin{array}{ccc}{\cal F}_{\tilde{G}',\omega'}&\to&{\cal F}_{\tilde{G},\omega}\\ \varphi'&\mapsto&\varphi\\ \end{array}$$ par $\varphi(\tilde{\pi})=\varphi'(\tilde{\pi}')$ (on a ajouté des indices $\tilde{G}$ et $\tilde{G}'$ aux notations précédentes). On définit aussi une application $$\begin{array}{ccc}C_{c}^{\infty}(\tilde{G}'({\mathbb R}),K')&\to&C_{c}^{\infty}(\tilde{G}({\mathbb R}),K)\\ f'&\mapsto&f\\ \end{array}$$ par $$f(\gamma)=\int_{C({\mathbb R})}f'(c\gamma')\,dc,$$ pour tout $\gamma\in \tilde{G}({\mathbb R})$, où $\gamma'$ est un relèvement quelconque de $\gamma$ dans $\tilde{G}'({\mathbb R})$. La mesure sur $C(F)$ doit être compatible aux mesures choisies sur $G({\mathbb R})$ et $G'({\mathbb R})$. On vérifie que le diagramme suivant est commutatif $$\begin{array}{ccc}C_{c}^{\infty}(\tilde{G}'({\mathbb R}),K')&\stackrel{f'\mapsto f}{\to}&C_{c}^{\infty}(\tilde{G}({\mathbb R}),K)\\ \downarrow \underline{pw}_{\tilde{G}',\omega'}&&\downarrow \underline{pw}_{\tilde{G},\omega}\\ {\cal F}_{\tilde{G}',\omega'}&\stackrel {\varphi'\mapsto \varphi}{\to}&{\cal F}_{\tilde{G},\omega}.\\ \end{array}$$ L’application horizontale du haut est surjective. Puisque l’application $\underline{pw}_{\tilde{G}',\omega'}$ vérifie (1) d’après le cas déjà traité, il nous suffit pour conclure de montrer que l’application (2) se restreint en une surjection de ${\cal PW}(\tilde{G}',\omega')$ sur ${\cal PW}(\tilde{G},\omega)$. Que l’image de ${\cal PW}(\tilde{G}',\omega')$ par l’application (2) soit contenue dans ${\cal PW}(\tilde{G},\omega)$ résulte aisément du fait suivant. Soit $\tilde{Q}=\tilde{L}U_{Q}$ un espace parabolique de $\tilde{G}$, notons $\tilde{Q}'=\tilde{L}'U_{Q'}$ son image réciproque dans $\tilde{G}'$. L’espace $i\tilde{{\cal A}}_{\tilde{L}}^*$ s’injecte naturellement dans $i\tilde{{\cal A}}_{\tilde{L}'}^*$ et la restriction à ce sous-espace d’une fonction de Paley-Wiener sur $i\tilde{{\cal A}}_{\tilde{L}'}^*$ est encore de Paley-Wiener. Pour démontrer la surjectivité, on va construire une section $s$ de l’application (2) et montrer que $s$ envoie ${\cal PW}(\tilde{G},\omega)$ dans ${\cal PW}(\tilde{G}',\omega')$. On a une suite exacte naturelle $$0\to i\tilde{{\cal A}}_{\tilde{G}}^*\to i\tilde{{\cal A}}_{\tilde{G}'}^*\to i{\cal A}_{C}^*\to 0.$$ En choisissant une section de la dernière application, on identifie $i{\cal A}_{C}^*$ à un supplémentaire de $i\tilde{{\cal A}}_{\tilde{G}}^*$ dans $i\tilde{{\cal A}}_{\tilde{G}'}^*$. On fixe une fonction $\varphi_{C}$ de Paley-Wiener sur $i{\cal A}_{C}^*$ telle que $\varphi_{C}(0)=1$. Soit $\varphi\in {\cal F}_{\tilde{G},\omega}$. Pour une $\omega'$-représentation $\tilde{\pi}'$ de $\tilde{G}'({\mathbb R})$, tempérée et $G'$-irréductible, ou bien il n’existe aucun $\tilde{\xi}\in i{\cal A}_{C}^*$ tel que $\tilde{\pi}'_{\tilde{\xi}}$ se factorise par $\tilde{p}$. On pose alors $(s(\varphi))(\tilde{\pi}')=0$. Ou bien il existe un unique $\tilde{\xi}\in i{\cal A}_{C}^*$ tel que $\tilde{\pi}'_{\tilde{\xi}}$ se factorise par $\tilde{p}$. Pour ce $\tilde{\xi}$, on note $\tilde{\pi}$ la $\omega$-représentation de $\tilde{G}({\mathbb R})$ telle que $\tilde{\pi}'_{\tilde{\xi}}=\tilde{\pi}\circ\tilde{p}$ et on pose alors $(s(\varphi))(\tilde{\pi}')=\varphi_{C}(\tilde{\xi})\varphi(\tilde{\pi})$. La fonction $s(\varphi)$ ainsi définie appartient à ${\cal F}_{\tilde{G}',\omega'}$. Cela définit une application $s:{\cal F}_{\tilde{G},\omega}\to {\cal F}_{\tilde{G}',\omega'}$ qui est clairement une section de l’application (2). On doit montrer que, si $\varphi\in {\cal PW}(\tilde{G},\omega)$, alors $s(\varphi)\in {\cal PW}(\tilde{G}',\omega')$. La condition 6.1(3) est immédiate. Soient $\tilde{Q}'=\tilde{L}'U_{Q'}$ un espace parabolique et $\tilde{\sigma}'$ une $\omega'$-représentation tempérée et $L'$-irréductible de $\tilde{L}'(F)$. Posons $\tilde{Q}=\tilde{p}(\tilde{Q}')$, $\tilde{L}=\tilde{p}(\tilde{L}')$. Supposons d’abord que, pour tout $\tilde{\lambda}'\in i\tilde{{\cal A}}_{\tilde{L}'}^*$ et pour toute composante $G'$-irréductible $\tilde{\pi}'$ de $Ind_{\tilde{Q}'}^{\tilde{G}'}(\tilde{\sigma}'_{\tilde{\lambda}'})$, il n’existe pas de $\tilde{\xi}\in i{\cal A}_{C}^*$ tel que $\tilde{\pi}'_{\tilde{\xi}}$ se factorise par $\tilde{p}$. Alors par définition $(s(\varphi))(Ind_{\tilde{Q}'}^{\tilde{G}'}(\tilde{\sigma}'_{\tilde{\lambda}'}))=0$ pour tout $\tilde{\lambda}'$ et cette fonction de $\tilde{\lambda}'$ est bien de Paley-Wiener. Supposons au contraire qu’il existe $\tilde{\lambda}'$, $\tilde{\pi}'$ et $\tilde{\xi}$ comme ci-dessus tel que $\tilde{\pi}'_{\tilde{\xi}}$ se factorise par $\tilde{p}$. Fixons de tels objets. Quitte à remplacer $\tilde{\sigma}'$ par $\tilde{\sigma}'_{\tilde{\lambda}'+\tilde{\xi}}$, on peut supposer que $\tilde{\pi}'$ est une composante de $Ind_{\tilde{Q}'}^{\tilde{G}'}(\tilde{\sigma}')$ et qu’elle se factorise par $\tilde{p}$. Cette condition équivaut à ce que le caractère central de la représentation sous-jacente $\pi'$ se restreigne en le caractère trivial de $C({\mathbb R})$. Mais cette restriction est la même que celle du caractère central de $\sigma'$. Donc $\tilde{\sigma}'$ se factorise en une $\omega$-représentation $\tilde{\sigma}$ de $\tilde{L}({\mathbb R})$. Cela oblige $\omega$ à être trivial sur $Z_{L}({\mathbb R})^{\theta}$. On a encore l’égalité $$i\tilde{{\cal A}}_{\tilde{L}'}^*=i\tilde{{\cal A}}_{\tilde{L}}^*\oplus i{\cal A}_{C}^*.$$ Les constructions entraînent que, pour tout $\tilde{\lambda}'\in i\tilde{{\cal A}}_{\tilde{L}'}^*$, on a l’égalité $$(s(\varphi))(Ind_{\tilde{Q}'}^{\tilde{G}'}(\tilde{\sigma}'_{\tilde{\lambda}'}))=\varphi_{C}(\tilde{\xi})\varphi(Ind_{\tilde{Q}}^{\tilde{G}}(\tilde{\sigma}_{\tilde{\lambda}})),$$ où $\tilde{\lambda}$ et $\tilde{\xi}$ sont les composantes de $\tilde{\lambda}'$ selon la décomposition ci-dessus. La fonction de $\tilde{\lambda}'$ ci-dessus est de Paley-Wiener. Cela achève la preuve. $\square$
L’application $\phi_{\tilde{M}}$
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On note ${\cal H}_{ac}(\tilde{G}(F))$ l’espace des fonctions $f:\tilde{G}(F)\to {\mathbb C}$ qui vérifient les conditions suivantes:
\(1) si $F$ est non-archimédien, il existe un sous-groupe ouvert compact $H$ de $G(F)$ tel que $f$ soit biinvariante par $H$;
\(2) si $F$ est archimédien, il existe un ensemble fini $\underline{\kappa}$ de représentations irréductibles de $K$ tel que la représentation de $K\times K$ dans l’espace engendré par les translatés de $f$ à droite et à gauche par des éléments de $K$ n’ait pour composantes irréductibles que des éléments de $\underline{\kappa}\times \underline{\kappa}$;
\(3) pour toute fonction $b\in C_{c}^{\infty}( \tilde{{\cal A}}_{\tilde{G},F} )$, la fonction produit $f(b\circ \tilde{H}_{\tilde{G}})$ appartient à $C_{c}^{\infty}(\tilde{G}(F))$.
Remarquons que toute forme linéaire sur $C_{c}^{\infty}(\tilde{G}(F),K)$ dont le support a une projection compacte dans $ \tilde{{\cal A}}_{\tilde{G},F}$ s’étend à ${\cal H}_{ac}(\tilde{G}(F))$: pour une telle forme linéaire $l$ et pour $f\in {\cal H}_{ac}(\tilde{G}(F))$, on pose $l(f)=l(f(b\circ \tilde{H}_{\tilde{G}}))$, où $b$ est un élément de $C_{c}^{\infty}(\tilde{{\cal A}}_{\tilde{G},F})$ qui vaut $1$ sur un voisinage de cette projection. En particulier, les intégrales orbitales ou les intégrales orbitales pondérées sont définies sur ${\cal H}_{ac}(\tilde{G}(F))$. On note $I_{ac}(\tilde{G}(F),\omega)$ le quotient de ${\cal H}_{ac}(\tilde{G}(F))$ par le sous-espace des $f\in {\cal H}_{ac}(\tilde{G}(F))$ telles que $I_{\tilde{G}}(\gamma,\omega,f)=0$ pour tout $\gamma\in \tilde{G}_{reg}(F)$.
Les caractères de $\omega$-représentations admissibles ne s’étendent pas à l’espace ${\cal H}_{ac}(\tilde{G}(F))$ mais nous allons voir que leurs coefficients de Fourier s’y étendent. Précisément, soit $\tilde{\pi}$ une $\omega$-représentation admissible de $\tilde{G}(F)$, soit $f\in C_{c}^{\infty}(\tilde{G}(F),K)$ et soit $X\in \tilde{{\cal A}}_{\tilde{G},F}$. Pour $\tilde{\lambda}\in i\tilde{{\cal A}}_{\tilde{G},F}^*$, on définit $\tilde{\pi}_{\tilde{\lambda}}$ comme en 2.6. La fonction $\tilde{\lambda}\mapsto I_{\tilde{G}}(\tilde{\pi}_{\tilde{\lambda}},f)e^{-<\tilde{\lambda},X>}$ se descend en une fonction sur $i{\cal A}_{\tilde{G},F}^*$. Cette fonction est $C^{\infty}$, à décroissance rapide si $F$ est archimédien. Posons $$I_{\tilde{G}}(\tilde{\pi},X,f)=mes(i{\cal A}_{\tilde{G},F}^*)^{-1}\int_{i{\cal A}_{\tilde{G},F}^*}I_{\tilde{G}}(\tilde{\pi}_{\tilde{\lambda}},f)e^{-<\tilde{\lambda},X>}\,d\lambda.$$ On a
\(4) la fonction $X\mapsto I_{\tilde{G}}(\tilde{\pi},X,f)$ est à support compact et est $C^{\infty}$ dans le cas où $F$ est archimédien; son support est contenu dans la projection dans $\tilde{A}_{\tilde{G},F}$ de celui de $f$; plus précisément, pour une fonction lisse $b$ sur $\tilde{{\cal A}}_{\tilde{G},F}$, on a l’égalité $ I_{\tilde{G}}(\tilde{\pi},X,f(b\circ\tilde{H}_{\tilde{G}}))=b(X) I_{\tilde{G}}(\tilde{\pi},X,f)$.
Preuve. Par $K$-finitude, $I_{\tilde{G}}(\tilde{\pi}_{\tilde{\lambda}},f)$ est de la forme $$\sum_{i=1,...,n}<\check{v}_{i},\tilde{\pi}_{\tilde{\lambda}}(f)v_{i}>,$$ pour des éléments $v_{i}\in V_{\pi}$ et $\check{v}_{i}\in V_{\pi^{\vee}}$. Autrement dit $$I_{\tilde{G}}(\tilde{\pi}_{\tilde{\lambda}},f)=\int_{\tilde{G}(F)} B(\tilde{\lambda},\gamma)f(\gamma)\,d\gamma,$$ où $$B(\tilde{\lambda},\gamma)=\sum_{i=1,...,n}<\check{v}_{i},\tilde{\pi}_{\tilde{\lambda}}(\gamma)v_{i}>.$$ L’espace topologique $\tilde{G}(F)$ est un fibré au-dessus de $\tilde{{\cal A}}_{\tilde{G},F}$, de fibres isomorphes à $G(F)^1$ (le noyau de $H_{\tilde{G}}$). Pour tout $Y\in \tilde{{\cal A}}_{\tilde{G},F}$, notons $\tilde{G}(F;Y)$ la fibre au-dessus de $Y$. Les résultats habituels de décomposition d’intégrales sur des fibrés nous disent que l’on peut définir une intégrale $$I'_{\tilde{G}}(\tilde{\pi}_{\tilde{\lambda}},Y,f)=\int_{\tilde{G}(F;Y)} B(\tilde{\lambda},\gamma)f(\gamma)\,d\gamma$$ qui est à support compact en $Y$ et $C^{\infty}$ si $F$ est archimédien, de sorte que $$I_{\tilde{G}}(\tilde{\pi}_{\tilde{\lambda}},f)=\int_{\tilde{{\cal A}}_{\tilde{G},F}}I'_{\tilde{G}}(\tilde{\pi}_{\tilde{\lambda}},Y,f)\,dY.$$ Mais $$I'_{\tilde{G}}(\tilde{\pi}_{\tilde{\lambda}},Y,f)=e^{<\tilde{\lambda},Y>}I'_{\tilde{G}}(\tilde{\pi},Y,f).$$ Par inversion de Fourier, on en déduit l’égalité $I_{\tilde{G}}(\tilde{\pi},X,f)= I'_{\tilde{G}}(\tilde{\pi},X,f)$ et l’assertion (4) s’ensuit. $\square$
Il résulte de (4) que l’on peut définir $I_{\tilde{G}}(\tilde{\pi},X,f)$ pour $f\in {\cal H}_{ac}(\tilde{G}(F))$. Le théorème 5.5 s’étend:
\(5) un élément $f\in {\cal H}_{ac}(\tilde{G}(F))$ a une image nulle dans $I_{ac}(\tilde{G}(F),\omega)$ si et seulement si $I_{\tilde{G}}(\tilde{\pi},X,f)=0$ pour toute $\omega$-représentation irréductible et tempérée $\tilde{\pi}$ de $\tilde{G}(F)$ et tout $X\in \tilde{{\cal A}}_{\tilde{G},F}$.
Plus généralement, soit $\tilde{M}\in {\cal L}(\tilde{M}_{0})$, soit $\tilde{\pi}$ une $\omega$-représentation tempérée de longueur finie de $\tilde{M}(F)$ et soit $X\in \tilde{{\cal A}}_{\tilde{M},F}$. Pour $f\in C_{c}^{\infty}(\tilde{G}(F),K)$ et $\tilde{\lambda}\in i\tilde{{\cal A}}_{\tilde{M},F}^*$, on définit $J_{\tilde{M}}^{\tilde{G}}(\tilde{\pi}_{\tilde{\lambda}},f)$. La fonction $\tilde{\lambda}\mapsto J_{\tilde{M}}^{\tilde{G}}(\tilde{\pi}_{\tilde{\lambda}},f)e^{-<\tilde{\lambda},X>}$ se descend en une fonction sur $i{\cal A}_{\tilde{M},F}^*$. Cette fonction est $C^{\infty}$, à décroissance rapide si $F$ est archimédien. Posons $$J_{\tilde{M}}^{\tilde{G}}(\tilde{\pi},X,f)=mes(i{\cal A}_{\tilde{M},F}^*)^{-1}\int_{i{\cal A}_{\tilde{M},F}^*} J_{\tilde{M}}^{\tilde{G}}(\tilde{\pi}_{\tilde{\lambda}},f)e^{-<\tilde{\lambda},X>}\,d\lambda.$$ Comme fonction de $X$, ce terme n’est pas à support compact dans $\tilde{{\cal A}}_{\tilde{M},F}$. Toutefois
\(6) la fonction $X\mapsto J_{\tilde{M}}^{\tilde{G}}(\tilde{\pi},X,f)$ est de Schwartz; la projection dans $\tilde{{\cal A}}_{\tilde{G},F}$ de son support est contenue dans celle du support de $f$; plus précisément, pour une fonction lisse $b$ sur $\tilde{{\cal A}}_{\tilde{G},F}$, on a l’égalité $ J^{\tilde{G}}_{\tilde{M}}(\tilde{\pi},X,f(b\circ\tilde{H}_{\tilde{G}}))=b(X_{\tilde{G}}) J^{\tilde{G}}_{\tilde{M}}(\tilde{\pi},X,f)$, où $X_{\tilde{G}}$ est l’image naturelle de $X$ dans $\tilde{{\cal A}}_{\tilde{G},F}$.
Preuve. La fonction est de Schwartz car c’est la transformée de Fourier d’une fonction de Schwartz. La preuve de la deuxième assertion est la même que celle de (4), la fonction $B(\tilde{\lambda},\gamma)$ ayant maintenant la forme $$B(\tilde{\lambda},\gamma)=\sum_{i=1,...,n}<\check{v}_{i},{\cal M}_{\tilde{M}}^{\tilde{G}}(\pi_{\lambda})\tilde{\Pi}_{\tilde{\lambda}}(\gamma)v_{i}>,$$ où $\tilde{\Pi}_{\tilde{\lambda}}=Ind_{\tilde{P}}^{\tilde{G}}(\tilde{\pi}_{\tilde{\lambda}})$, cf. 2.7 pour les notations. $\square$
Il résulte de (6) que l’on peut définir $J_{\tilde{M}}^{\tilde{G}}(\tilde{\pi},X,f)$ pour $f\in {\cal H}_{ac}(\tilde{G}(F))$.
[0.3cm[**[Proposition]{}**]{}. [ *[Soit $\tilde{M}\in {\cal L}(\tilde{M}_{0},\omega)$. Pour tout $f\in {\cal H}_{ac}(\tilde{G}(F))$, il existe $\phi_{\tilde{M}}(f)\in {\cal H}_{ac}(\tilde{M}(F))$ telle que, pour toute $\omega$-représentation tempérée et de longueur finie $\tilde{\pi}$ de $\tilde{M}(F)$ et pour tout $X\in \tilde{{\cal A}}_{\tilde{M},F}$, on ait l’égalité $$I_{\tilde{M}}(\tilde{\pi},X,\phi_{\tilde{M}}(f))=J_{\tilde{M}}^{\tilde{G}}(\tilde{\pi},X,f).$$ L’image de $\phi_{\tilde{M}}(f)$ dans $I_{ac}(\tilde{M}(F),\omega)$ est uniquement déterminée.]{}*]{}0.3cm]{}
Preuve. Soient $f\in C_{c}^{\infty}(\tilde{G}(F),K)$ et $b\in C_{c}^{\infty}(\tilde{{\cal A}}_{\tilde{M},F})$. Pour une $\omega$-représentation tempérée et de longueur finie $\tilde{\pi}$ de $\tilde{M}(F)$, posons $$\varphi_{f,b}(\tilde{\pi})=\int_{\tilde{{\cal A}}_{\tilde{M},F}}J_{\tilde{M}}^{\tilde{G}}(\tilde{\pi},X,f)\overline{b(X)}\,dX.$$ Ceci est convergent d’après (6). Par transformation de Fourier, on a l’égalité $$\varphi_{f,b}(\tilde{\pi})=mes(i{\cal A}_{\tilde{M},F}^*)^{-1}\int_{i{\cal A}_{\tilde{M},F}^*}J_{\tilde{M}}^{\tilde{G}}(\tilde{\pi}_{\tilde{\lambda}},f)\overline{\hat{b}(\tilde{\lambda})}\,d\lambda,$$ où $$\hat{b}(\tilde{\lambda})=\int_{\tilde{{\cal A}}_{\tilde{M},F}}b(X)e^{<\tilde{\lambda},X>}\,dX.$$ On va montrer
\(7) la fonction $\varphi_{f,b}$ appartient à l’espace ${\cal PW}(\tilde{M},\omega)$.
D’après 2.7(1), la fonction s’identifie bien à une fonction sur le groupe de Grothendieck ${\cal K}^{\tilde{M}}$ de 6.1. D’après 2.7(2), elle annule les représentations irréductibles et non $M$-irréductibles. Les conditions (2) et (3) de 6.1 sont évidentes puisque $f$ est $K$-finie. Il faut vérifier 6.1(1). D’après la bijectivité de l’application $res$ de 6.2, il suffit de vérifier l’assertion suivante:
\(8) soit $\tilde{Q}=\tilde{L}U_{Q}\in {\cal F}^{\tilde{M}}(\tilde{M}_{0})$ et soit $\boldsymbol{\tau}\in {\cal E}_{ell}(\tilde{L},\omega)$; alors la fonction $\tilde{\mu}\mapsto \varphi_{f,b}(Ind_{\tilde{Q}}^{\tilde{M}}(\tilde{\pi}_{\boldsymbol{\tau}_{\tilde{\mu}}}))$ sur $i\tilde{{\cal A}}_{\tilde{L},F}^*$ est de Paley-Wiener.
Posons simplement $\tilde{\pi}=\tilde{\pi}_{\boldsymbol{\tau}}$. Remarquons que, pour $\tilde{\lambda}\in i\tilde{{\cal A}}_{\tilde{M},F}^*$, on a l’égalité $(Ind_{\tilde{Q}}^{\tilde{M}}(\tilde{\pi}_{\boldsymbol{\tau}_{\tilde{\mu}}}))_{\tilde{\lambda}}=Ind_{\tilde{Q}}^{\tilde{M}}(\tilde{\pi}_{\tilde{\mu}+\tilde{\lambda}})$. On utilise la formule de descente du lemme 5.4(iv) (où les rôles de $\tilde{M}$ et $\tilde{L}$ sont échangés). C’est-à-dire $$J_{\tilde{M}}^{\tilde{G}}((Ind_{\tilde{Q}}^{\tilde{M}}(\tilde{\pi}_{\tilde{\mu}+\tilde{\lambda}}),f)=\sum_{\tilde{M}'\in {\cal L}(\tilde{L})}d_{\tilde{L}}^{\tilde{G}}(\tilde{M},\tilde{M}')J_{\tilde{L}}^{\tilde{M}'}(\tilde{\pi}_{\tilde{\mu}+\tilde{\lambda}},f_{\tilde{P}'}).$$ On peut fixer $\tilde{M}'\in {\cal L}(\tilde{L})$ tel que $d_{\tilde{L}}^{\tilde{G}}(\tilde{M},\tilde{M}')\not=0$ et prouver que la fonction $\psi$ sur $i\tilde{{\cal A}}_{\tilde{L},F}^*$ définie par $$\psi(\tilde{\mu})=mes(i{\cal A}_{\tilde{M},F}^*)^{-1}\int_{i{\cal A}_{\tilde{M},F}^*}J_{\tilde{L}}^{\tilde{M}'}(\tilde{\pi}_{\tilde{\mu}+\tilde{\lambda}},f_{\tilde{P}'})\overline{\hat{b}(\tilde{\lambda})}\,d\lambda$$ est de Paley-Wiener. Grâce à (6), on peut exprimer $J_{\tilde{L}}^{\tilde{M}'}(\tilde{\pi}_{\tilde{\mu}+\tilde{\lambda}},f_{\tilde{P}'})$ par inversion de Fourier: $$J_{\tilde{L}}^{\tilde{M}'}(\tilde{\pi}_{\tilde{\mu}+\tilde{\lambda}},f_{\tilde{P}'})=\int_{\tilde{{\cal A}}_{\tilde{L},F}}J_{\tilde{L}}^{\tilde{M}'}(\tilde{\pi},Y,f_{\tilde{P}'})e^{<\tilde{\mu}+\tilde{\lambda},Y>}\,dY.$$ D’où $$\psi(\tilde{\mu})=mes(i{\cal A}_{\tilde{M},F}^*)^{-1}\int_{i{\cal A}_{\tilde{M},F}^*}\int_{\tilde{{\cal A}}_{\tilde{L},F}}J_{\tilde{L}}^{\tilde{M}'}(\tilde{\pi},Y,f_{\tilde{P}'})e^{<\tilde{\mu}+\tilde{\lambda},Y>}\overline{\hat{b}(\tilde{\lambda})}\,dY\,d\lambda.$$ Cette expression est absolument convergente. En intégrant d’abord en $\lambda$, on obtient par inversion de Fourier $$\psi(\tilde{\mu})= \int_{\tilde{{\cal A}}_{\tilde{L},F}} \Psi(Y)e^{<\tilde{\mu},Y>}\,dY,$$ où, en notant $Y_{\tilde{M}}$ la projection naturelle de $Y$ dans $\tilde{{\cal A}}_{\tilde{M},F}$, $$\Psi(Y)=J_{\tilde{L}}^{\tilde{M}'}(\tilde{\pi},Y,f^{\tilde{P}'}) \bar{b}(Y_{\tilde{M}}).$$ Ainsi, $\psi$ apparaît comme la transformée de Fourier de la fonction $\Psi$ sur $\tilde{{\cal A}}_{\tilde{L},F}$. Il s’agit de prouver que $\Psi$ est lisse et à support compact. Elle est lisse car c’est le produit de deux fonctions lisses. L’hypothèse sur $\tilde{M}'$ implique que $${\cal A}_{\tilde{L}}^{\tilde{M}}\cap {\cal A}_{\tilde{L}}^{\tilde{M}'}=0,$$ donc que la somme directe des projections $${\cal A}_{\tilde{L},F}\to {\cal A}_{\tilde{M},F}\oplus {\cal A}_{\tilde{M}',F}$$ est injective. La réplique pour les espaces affines est que le produit des projections $$\tilde{{\cal A}}_{\tilde{L},F}\to \tilde{{\cal A}}_{\tilde{M},F}\times \tilde{{\cal A}}_{\tilde{M}',F}$$ est injectif. Il est clair que c’est une immersion fermée. La projection dans $\tilde{{\cal A}}_{\tilde{M}',F}$ du support de la fonction $Y\mapsto J_{\tilde{L}}^{\tilde{M}'}(\tilde{\pi},Y,f_{\tilde{P}'})$ est compacte d’après (6). La projection dans $\tilde{{\cal A}}_{\tilde{M},F}$ du support de la fonction $Y\mapsto \bar{b}(Y_{\tilde{M}})$ est compacte puisque $b$ est à support compact. Donc $\Psi$ est à support compact. Cela démontre (8) et achève la preuve de (7).
D’après le théorème 6.1, on peut choisir une fonction $\phi_{\tilde{M}}(f,b)\in C_{c}^{\infty}(\tilde{M}(F),K^M)$ (où $K^M=K\cap M(F)$) de sorte que $I_{\tilde{M}}(\tilde{\pi},\phi_{\tilde{M}}(f,b))=\varphi_{f,b}(\tilde{\pi})$ pour toute $\omega$-représentation tempérée et de longueur finie $\tilde{\pi}$ de $\tilde{M}(F)$. On peut préciser le comportement de $\phi_{\tilde{M}}(f,b)$ par translations à droite ou à gauche par $K$. Traitons le cas non-archimédien (le cas archimédien n’en diffère que par les notations). Fixons un sous-groupe ouvert compact $H$ de $G(F)$ tel que $f$ soit biinvariante par $H$. Il existe un tel sous-groupe $H'$ de $M(F)$, ne dépendant que de $H$, tel que $J_{\tilde{M}}^{\tilde{G}}(\tilde{\pi},f)=0$ si $\pi$ n’a pas d’invariant non nul par $H'$. A fortiori, dans ce cas, $\varphi_{f,b}(\tilde{\pi})=0$ pour tout $b$. D’après le complément au théorème 6.1, on peut fixer un sous-groupe ouvert compact $H^M$ de $M(F)$, ne dépendant que de $H'$, donc ne dépendant que de $H$, et supposer que $\phi_{\tilde{M}}(f,b)$ est biinvariante par $H^M$.
Remarquons que la fonction $\tilde{\lambda}\mapsto \varphi(\tilde{\pi}_{\tilde{\lambda}})$ sur $i\tilde{{\cal A}}_{\tilde{M},F}^*$ est par construction la transformée de Fourier de la fonction $X\mapsto J_{\tilde{M}}(\tilde{\pi},X,f)\bar{b}(X)$ sur $\tilde{{\cal A}}_{\tilde{M},F}$. Par transformation de Fourier, l’égalité $I_{\tilde{M}}(\tilde{\pi},\phi_{\tilde{M}}(f,b))=\varphi_{f,b}(\tilde{\pi})$ est donc équivalente à $$(9) \qquad I_{\tilde{M}}(\tilde{\pi},X,\phi_{\tilde{M}}(f,b))=J_{\tilde{M}}^{\tilde{G}}(\tilde{\pi},X,f)\bar{b}(X)$$ pour tout $\tilde{\pi}$ et tout $X\in \tilde{{\cal A}}_{\tilde{M},F}$.
Fixons une suite $(U_{n})_{n\in {\mathbb N}}$ de sous-ensembles ouverts relativement compacts de $\tilde{{\cal A}}_{\tilde{M},F}$ de sorte que $\bar{U}_{n}\subset U_{n+1}$ (où $\bar{U}_{n}$ est la clôture de $U_{n}$) et que $\tilde{{\cal A}}_{\tilde{M},F}=\cup_{n\in {\mathbb N}}U_{n}$. En posant $U'_{n}=U_{n}-\bar{U}_{n-2}$ pour $n\geq2$ et $U'_{1}=U_{1}$, on a aussi $\tilde{{\cal A}}_{\tilde{M},F}=\cup_{n\geq1}U'_{n}$. On peut choisir une partition de l’unité $(b_{n})_{n\geq1}$ relative à ce dernier recouvrement, formée de fonctions lisses (et forcément à supports compacts). On choisit une suite $(c_{n})_{n\geq1}$ de fonctions lisses à supports compacts sur $\tilde{{\cal A}}_{\tilde{M},F}$ de sorte que $c_{n}$ vaille $1$ sur $U'_{n}$ et, pour $n\geq3$, le support de $c_{n}$ soit contenu dans $U_{n+1}-\bar{U}_{n-3}$. On choisit enfin une suite $(d_{n})_{n\geq1}$ de fonctions lisses à supports compacts sur $\tilde{{\cal A}}_{\tilde{G},F}$ de sorte que $d_{n}$ vaille $1$ sur un voisinage de la projection de $U'_{n}$ dans $\tilde{{\cal A}}_{\tilde{G},F}$. Soit $f\in {\cal H}_{ac}(\tilde{G}(F))$. Pour tout $n\geq1$, la fonction $f(d_{n}\circ\tilde{H}_{\tilde{G}})$ est à support compact, on dispose donc de $\phi_{\tilde{M}}(f(d_{n}\circ\tilde{H}_{\tilde{G}}),b_{n})$. Posons $$\phi_{\tilde{M}}(f)=\sum_{n\geq1}(c_{n}\circ\tilde{H}_{\tilde{M}})\phi_{\tilde{M}}(f(d_{n}\circ\tilde{H}_{\tilde{G}}),b_{n}).$$ Cette série est convergente: elle est localement finie d’après les propriétés de la suite $(c_{n})_{n\geq1}$. Pour la même raison, si $b\in C_{c}^{\infty}(\tilde{{\cal A}}_{\tilde{M},F})$, la fonction $\phi_{\tilde{M}}(f)(b\circ\tilde{H}_{\tilde{M}})$ est à support compact. Enfin, $\phi_{\tilde{M}}(f)$ vérifie les propriétés (1) ou (2). Par exemple, dans le cas non-archimédien, fixons un sous-groupe ouvert compact $H$ de $G(F)$ tel que $f$ soit biinvariante par $H$. Chaque fonction $f(d_{n}\circ\tilde{H}_{\tilde{G}})$ vérifie la même propriété. Comme on l’a dit ci-dessus, on peut supposer $\phi_{\tilde{M}}(f(d_{n}\circ\tilde{H}_{\tilde{G}}),b_{n})$ biinvariante par $H^M$, où $H^M$ est indépendant de $n$. Alors $\phi_{\tilde{M}}(f)$ est aussi biinvariante par $H^M$. Cela prouve que $\phi_{\tilde{M}}(f)$ appartient à ${\cal H}_{ac}(\tilde{M})$. Soit $\tilde{\pi}$ une $\omega$-représentation tempérée et de longueur finie de $\tilde{M}(F)$ et soit $X\in \tilde{{\cal A}}_{\tilde{M},F}$. En utilisant (4), on a $$I_{\tilde{M}}(\tilde{\pi},X,\phi_{\tilde{M}}(f))=\sum_{n\geq1}c_{n}(X)I_{\tilde{M}}(\tilde{\pi},X,\phi_{\tilde{M}}(f(d_{n}\circ\tilde{H}_{\tilde{G}}),b_{n})).$$ D’où, d’après (9), $$I_{\tilde{M}}(\tilde{\pi},X,\phi_{\tilde{M}}(f))=\sum_{n\geq1}c_{n}(X)b_{n}(X)J_{\tilde{M}}^{\tilde{G}}(\tilde{\pi},X,f(d_{n}\circ\tilde{H}_{\tilde{G}})).$$ D’après les choix de nos fonctions, on a les égalités $$c_{n}(X)b_{n}(X)=b_{n}(X),$$ $$b_{n}(X)J_{\tilde{M}}^{\tilde{G}}(\tilde{\pi},X,f(d_{n}\circ\tilde{H}_{\tilde{G}}))=b_{n}(X)J_{\tilde{M}}^{\tilde{G}}(\tilde{\pi},X,f).$$ Donc $$I_{\tilde{M}}(\tilde{\pi},X,\phi_{\tilde{M}}(f))=\sum_{n\geq1}b_{n}(X)J_{\tilde{M}}^{\tilde{G}}(\tilde{\pi},X,f)=J_{\tilde{M}}^{\tilde{G}}(\tilde{\pi},X,f),$$ puisque $(b_{n})_{n\geq1}$ est une partition de l’unité. C’est la propriété requise, ce qui démontre la première partie de la proposition. Que l’image de $\phi_{\tilde{M}}(f)$ dans $I_{ac}(\tilde{M}(F),\omega)$ soit uniquement déterminée résulte de (5). $\square$
Remarquons que, pour $f\in {\cal H}_{ac}(\tilde{G}(F))$ et pour une fonction lisse $b$ sur $\tilde{{\cal A}}_{\tilde{G},F}$,
\(10) les images dans $I_{ac}(\tilde{M}(F),\omega)$ de $\phi_{\tilde{M}}(f(b\circ\tilde{H}_{\tilde{G}}))$ et $(b\circ\tilde{H}_{\tilde{G}})\phi_{\tilde{M}}(f)$ coïncident.
Cela résulte de (4) et (6).
Pour $f\in {\cal H}_{ac}(\tilde{G}(F))$, on note encore $\phi_{\tilde{M}}(f)$ l’image dans $I_{ac}(\tilde{M}(F),\omega)$ de la fonction ainsi notée dans l’énoncé. Alors $\phi_{\tilde{M}}$ devient une application linéaire bien définie de ${\cal H}_{ac}(\tilde{G}(F))$ dans $I_{ac}(\tilde{M}(F),\omega)$.
Soit $\tilde{P}=\tilde{M}U_{P}\in {\cal F}(\tilde{M}_{0})$. On dispose de l’application $f\mapsto f_{\tilde{P}}$ de $C_{c}^{\infty}(\tilde{G}(F),K)$ dans $C_{c}^{\infty}(\tilde{M}(F),K^M)$. On sait que l’image de $f_{\tilde{P}}$ dans $I(\tilde{M}(F),K,\omega)$ ne dépend que de $\tilde{M}$ et pas de $\tilde{P}$. On note cette image $f_{\tilde{M}}$. Ceci s’étend à l’espace ${\cal H}_{ac}(\tilde{G}(F))$: pour $f\in {\cal H}_{ac}(\tilde{G}(F))$, on définit $f_{\tilde{M}}\in I_{ac}(\tilde{M}(F),\omega)$. Soient maintenant $\tilde{M}\in {\cal L}(\tilde{M}_{0},\omega)$ et $\tilde{L}\in {\cal L}(\tilde{M})$. Pour $f\in {\cal H}_{ac}(\tilde{G}(F))$, on a l’égalité dans $I_{ac}(\tilde{M}(F),\omega)$:
$$(11)\qquad (\phi_{\tilde{L}}^{\tilde{G}}(f))_{\tilde{M}}=\sum_{\tilde{L}'\in {\cal L}(\tilde{M})}d_{\tilde{M}}^{\tilde{G}}(\tilde{L},\tilde{L}')\phi_{\tilde{M}}^{\tilde{L}'}(f_{\tilde{Q}'}).$$
La preuve est formelle à partir de la formule de descente du lemme 5.4(iv).
Intégrales orbitales pondérées équivariantes
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Appelons forme linéaire $\omega$-équivariante sur $C_{c}^{\infty}(\tilde{G}(F),K)$, resp. ${\cal H}_{ac}(\tilde{G}(F))$, une forme linéaire qui se factorise en une forme linéaire sur $I(\tilde{G}(F),K,\omega)$, resp. $I_{ac}(\tilde{G}(F),\omega)$. Dans le cas où $F$ est non-archimédien, une forme linéaire $l$ disons sur $C_{c}^{\infty}(\tilde{G}(F))$ est $\omega$-équivariante si et seulement si elle vérifie la relation $l(^gf)=\omega(g)^{-1}l(f)$ pour toute $f\in C_{c}^{\infty}(\tilde{G}(F))$ et tout $g\in G(F)$ (cf. 5.5 remarque (2)). On identifiera souvent une forme linéaire $\omega$-équivariante à une forme linéaire sur $I(\tilde{G}(F),K,\omega)$, resp. $I_{ac}(\tilde{G}(F),\omega)$.
[0.3cm[**[Proposition]{}**]{}. [ *[Soient $\tilde{M}\in {\cal L}(\tilde{M}_{0})$ et $\gamma\in \tilde{M}(F)\cap \tilde{G}_{reg}(F)$. Il existe une unique forme linéaire $\omega$-équivariante $f\mapsto I_{\tilde{M}}^{\tilde{G}}(\gamma,\omega,f)$ sur ${\cal H}_{ac}(\tilde{G}(F))$ qui vérifie l’égalité $$I_{\tilde{M}}^{\tilde{G}}(\gamma,\omega,f)=J_{\tilde{M}}^{\tilde{G}}(\gamma,\omega,f)-\sum_{\tilde{L}\in {\cal L}(\tilde{M}), \tilde{L}\not=\tilde{G}}I_{\tilde{M}}^{\tilde{L}}(\gamma,\omega,\phi_{\tilde{L}}(f))$$ pour tout $f\in {\cal H}_{ac}(\tilde{G}(F))$.]{}*]{}0.3cm]{}
Pour donner un sens à cet énoncé, on doit raisonner par récurrence sur le rang semi-simple $rg_{ss}(G)$ de $G$. Si ce rang est nul, ou plus généralement si $\tilde{M}=\tilde{G}$, l’énoncé est tautologique: on a simplement $$I_{\tilde{G}}^{\tilde{G}}(\gamma,\omega,f)=J_{\tilde{G}}^{\tilde{G}}(\gamma,\omega,f).$$ Si $rg_{ss}(G)>0$, on suppose par récurrence que les formes linéaires $I_{\tilde{M}}^{\tilde{L}}(\gamma,\omega,.)$ sont définies pour $\tilde{L}\not=\tilde{G}$ (auquel cas $ rg_{ss}(L)<rg_{ss}(G)$) et qu’elles sont $\omega$-équivariantes. D’après la dernière assertion de la proposition précédente, le terme $I_{\tilde{M}}^{\tilde{L}}(\gamma,\omega,\phi_{\tilde{L}}(f))$ est bien défini. Il en est donc de même du membre de droite de l’égalité de l’énoncé. Cette égalité définit la forme linéaire $f\mapsto I_{\tilde{M}}^{\tilde{G}}(\gamma,\omega,f)$. L’assertion de la proposition est que celle-ci est $\omega$-équivariante. Dans le cas où $F$ est non-archimédien, on montre que l’égalité $I_{\tilde{M}}^{\tilde{G}}(\gamma,\omega,^gf)=\omega(g)^{-1}I_{\tilde{M}}^{\tilde{G}}(\gamma,\omega,f)$ est vérifiée pour tout $g\in G(F)$: la démonstration, essentiellement formelle, est la même que dans le cas non tordu. On se contente de renvoyer à \[A2\] proposition 4.1 pour ce cas. Cela suffit pour conclure. Si $F$ est archimédien, cette relation n’a plus de sens si on se limite aux fonctions $K$-finies. On peut l’adapter à de telles fonctions, mais elle ne suffit de toute façon pas à conclure. Pour l’instant, nous laissons la preuve inachevée (on la complètera en 7.1).
[**On conserve pour ce paragraphe et jusqu’à la fin de 7.1 l’hypothèse de récurrence ci-dessus, à savoir que la proposition est vérifiée si l’on remplace $\tilde{G}$ par $\tilde{G}'$ avec $rg_{ss}(G')<rg_{ss}(G)$**]{}.
Comme on l’a dit, cela suffit à définir la forme linéaire $f\mapsto I_{\tilde{M}}^{\tilde{G}}(\gamma,\omega,f)$ sur ${\cal H}_{ac}(\tilde{G}(F))$.
Il est clair que $I_{\tilde{M}}^{\tilde{G}}(\gamma,\omega,f)=0$ si $\omega$ n’est pas trivial sur $Z_{G}(\gamma,F)$.
On aura besoin des propriétés suivantes. On ne démontrera pas les deux premières, leurs preuves étant essentiellement formelles. Soient $\tilde{M},\tilde{M}'\in {\cal L}(\tilde{M}_{0})$ et $g\in G(F)$. Supposons $\tilde{M}'=g\tilde{M}g^{-1}$. Alors
\(1) pour tout $\gamma\in \tilde{M}(F)\cap \tilde{G}_{reg}(F)$ et tout $f\in {\cal H}_{ac}(\tilde{G}(F))$, on a l’égalité $$I_{\tilde{M}'}^{\tilde{G}}(g\gamma g^{-1},f)=\omega(g)I_{\tilde{M}}^{\tilde{G}}(\gamma,\omega,f).$$
Soient $\tilde{M}\in {\cal L}(\tilde{M}_{0})$ et $\tilde{L}\in {\cal L}(\tilde{M})$. Soit $\gamma\in \tilde{M}(F)\cap \tilde{G}_{reg}(F)$. Pour $f\in {\cal H}_{ac}(\tilde{G}(F))$, on a l’égalité $$(2) \qquad I_{\tilde{L}}^{\tilde{G}}(\gamma,\omega,f)=\sum_{\tilde{L}'\in {\cal L}(\tilde{M})}d_{\tilde{M}}^{\tilde{G}}(\tilde{L},\tilde{L}')I_{\tilde{M}}^{\tilde{L}'}(\gamma,\omega,f_{\tilde{L}'}).$$
Soient $\tilde{M}\in {\cal L}(\tilde{M}_{0})$, $\gamma\in \tilde{M}(F)\cap \tilde{G}_{reg}(F)$ et $b$ une fonction lisse sur $\tilde{{\cal A}}_{\tilde{G},F}$. On a l’égalité $$(3) \qquad I_{\tilde{M}}^{\tilde{G}}(\gamma,\omega,f(b\circ\tilde{H}_{\tilde{G}}))=b(\tilde{H}_{\tilde{G}}(\gamma))I_{\tilde{M}}^{\tilde{G}}(\gamma,\omega,f).$$
Preuve. La propriété analogue pour les intégrales pondérées $J_{\tilde{M}}^{\tilde{G}}(\gamma,\omega,.)$ résulte des définitions. D’autre part, pour $\tilde{L}\in {\cal L}(\tilde{M})$, on peut supposer $\phi_{\tilde{L}}( f(b\circ\tilde{H}_{\tilde{G}}))=(b\circ\tilde{H}_{\tilde{G}})\phi_{\tilde{L}}(f)$, cf. 6.4(10). Pour $\tilde{L}\not=\tilde{G}$, on peut supposer par récurrence que (3) est vrai quand on remplace $(\tilde{G},\tilde{M})$ par $(\tilde{L},\tilde{M})$. Alors, quand on remplace $f$ par $f(b\circ\tilde{H}_{\tilde{G}})$, le membre de droite de l’égalité de l’énoncé est multiplié par $b(\tilde{H}_{\tilde{G}}(\gamma))$. Donc le membre de gauche aussi. $\square$
Soient $\tilde{M}\in {\cal L}(\tilde{M}_{0})$ et $\tilde{T}$ un tore tordu maximal de $\tilde{M}$. Alors
\(4) il existe un entier $N\geq0$ et, pour $f\in {\cal H}_{ac}(\tilde{G}(F))$ et $\gamma_{0}\in \tilde{T}(F)$, il existe $c>0$ et un voisinage $\Omega$ de $\gamma_{0}$ dans $\tilde{T}(F)$ tel que l’on ait la majoration $$\vert I_{\tilde{M}}^{\tilde{G}}(\gamma,\omega,f)\vert\leq c(1+\vert log(D^{\tilde{G}}(\gamma))\vert )^N$$ pour tout $\gamma\in \Omega\cap \tilde{G}_{reg}(F)$;
\(5) pour tout $f\in {\cal H}_{ac}(\tilde{G}(F))$, la fonction $\gamma\mapsto I_{\tilde{M}}^{\tilde{G}}(\gamma,\omega,f)$ est lisse sur $\tilde{T}(F)\cap \tilde{G}_{reg}(F)$.
Preuve. En raisonnant par récurrence, il suffit de démontrer les mêmes propriétés pour $J_{\tilde{M}}^{\tilde{G}}(\gamma,\omega,f)$. Au voisinage d’un point $\gamma_{0}$, cette fonction coïncide avec $J_{\tilde{M}}^{\tilde{G}}(\gamma,\omega,f(b\circ\tilde{H}_{\tilde{G}}))$, où $b\in C_{c}^{\infty}(\tilde{{\cal A}}_{\tilde{G},F})$ vaut $1$ sur un voisinage de $\tilde{H}_{\tilde{G}}(\gamma_{0})$. Cela nous ramène au cas où $f\in C_{c}^{\infty}(\tilde{G}(F),K)$, lequel cas est traité par 5.4(2) et (3). $\square$
Le théorème
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Soient $f_{1},f_{2}\in C_{c}^{\infty}(\tilde{G}(F),K)$. Pour $\tilde{M}\in {\cal L}(\tilde{M}_{0})$, $\tilde{S}\in T_{ell}(\tilde{M},\omega)$ et pour $\gamma\in \tilde{S}(F)\cap \tilde{G}_{reg}(F)$, on pose $$I_{\tilde{M}}^{\tilde{G}}(\gamma,\omega,f_{1},f_{2})=\sum_{\tilde{L}_{1},\tilde{L}_{2}\in {\cal L}(\tilde{M})}d_{\tilde{M}}^{\tilde{G}}(\tilde{L}_{1},\tilde{L}_{2})\overline{I_{\tilde{M}}^{\tilde{L}_{1}}( \gamma,\omega,f_{1,\tilde{L}_{1}})}I_{\tilde{M}}^{\tilde{L}_{2}}(\gamma,\omega,f_{2,\tilde{L}_{2}}).$$
[**Remarque.**]{} Pour $i=1,2$, l’élément $f_{i,\tilde{L}_{i}}$ appartient à $I(\tilde{L}_{i}(F),K^L,\omega)$. Si $\tilde{L}_{i}\subsetneq \tilde{G}$, le terme $I_{\tilde{M}}^{\tilde{L}_{i}}(\gamma,\omega,f_{i,\tilde{L}_{i}})$ est bien défini d’après la proposition 6.5. Si $\tilde{L}_{i}=\tilde{G}$, on ne sait pas encore que $f\mapsto I_{\tilde{M}}^{\tilde{G}}(\gamma,\omega,f)$ se factorise par $I(\tilde{G}(F),K,\omega)$. Par convention, on suppose dans ce cas que $f_{\tilde{G}}=f$ et $I_{\tilde{M}}^{\tilde{G}}(\gamma,\omega,f_{\tilde{G}})=I_{\tilde{M}}^{\tilde{G}}(\gamma,\omega,f)$.
On pose $$I_{\tilde{M},\tilde{S}}^{\tilde{G}}(\omega,f_{1},f_{2})=\vert W^M(\tilde{S})\vert ^{-1} mes(A_{\tilde{M}}(F)\backslash S^{\theta}(F))\int_{\tilde{S}(F)/(1-\theta)(S(F))}I_{\tilde{M}}^{\tilde{G}}(\gamma,\omega,f_{1},f_{2})\,d\gamma.$$ Montrons que
\(1) cette intégrale est absolument convergente.
Preuve. D’après 6.5(4) et (5) et 4.2(2), la fonction à intégrer est localement intégrable. Il suffit de prouver qu’elle est à support compact. Puisque $\tilde{S}$ est elliptique dans $\tilde{M}$, il suffit de prouver que la projection de ce support sur $\tilde{{\cal A}}_{\tilde{M}}$ l’est. On peut encore fixer $\tilde{L}_{1},\tilde{L}_{2}$ tels que $d_{\tilde{M}}^{\tilde{G}}(\tilde{L}_{1},\tilde{L}_{2})\not=0$ et remplacer la fonction par $$\overline{I_{\tilde{M}}^{\tilde{L}_{1}}( \gamma,\omega,f_{1,\tilde{L}_{1}})}I_{\tilde{M}}^{\tilde{L}_{2}}(\gamma,\omega,f_{2,\tilde{L}_{2}}).$$ Puisque $f_{1,\tilde{L}_{1}}$ est à support compact, la relation 6.5(3) entraîne que le support de la première fonction ci-dessus a une projection compacte dans $\tilde{{\cal A}}_{\tilde{L}_{1}}$. De même, le support de la seconde fonction a une projection compacte dans $\tilde{{\cal A}}_{\tilde{L}_{2}}$. La non-nullité de $d_{\tilde{M}}^{\tilde{G}}(\tilde{L}_{1},\tilde{L}_{2})$ entraîne que le produit des projections $$\tilde{{\cal A}}_{\tilde{M}}\to\tilde{{\cal A}}_{\tilde{L}_{1}}\times\tilde{{\cal A}}_{\tilde{L}_{2}}$$ est injective. Donc le support du produit des deux fonctions a bien les propriétés requises. $\square$
Posons $$I_{\tilde{M},g\acute{e}om}^{\tilde{G}}(\omega,f_{1},f_{2})=\sum_{\tilde{S}\in T_{ell}(\tilde{M},\omega)}I_{\tilde{M},\tilde{S}}^{\tilde{G}}(\omega,f_{1},f_{2}),$$ puis $$I_{g\acute{e}om}^{\tilde{G}}(\omega,f_{1},f_{2})=\sum_{\tilde{M}\in {\cal L}(\tilde{M}_{0})}\vert \tilde{W}^M\vert \vert \tilde{W}^G\vert ^{-1}(-1)^{a_{\tilde{M}}-a_{\tilde{G}}}I_{\tilde{M},g\acute{e}om}^{\tilde{G}}(\omega,f_{1},f_{2}).$$
D’autre part, posons $$I_{disc}^{\tilde{G}}(\omega,f_{1},f_{2})=J_{\tilde{G},spec}^{\tilde{G}}(\omega,f_{1},f_{2}),$$ cf. 3.25.
[0.3cm[**[Théorème]{}**]{}. [ *[Pour tous $f_{1},f_{2}\in C_{c}^{\infty}(\tilde{G}(F),K)$, on a l’égalité $$I_{g\acute{e}om}^{\tilde{G}}(\omega,f_{1},f_{2})=I_{disc}^{\tilde{G}}(\omega,f_{1},f_{2}).$$]{}*]{}0.3cm]{}
Preuve. Cette preuve est formelle. Compte tenu de l’importance du théorème, nous la traitons en détail. On a besoin de quelques constructions, formelles comme on vient de le dire. Pour deux espaces vectoriels complexes $V$ et $V'$, notons $V\boxtimes V'$ leur produit tensoriel “sesquilinéaire”, précisément le produit tensoriel $\bar{V}\otimes V'$, où $\bar{V}$ est le conjugué complexe de $V$. Posons simplement $$\underline{C}(\tilde{G}(F))=C_{c}^{\infty}(\tilde{G}(F),K)\boxtimes C_{c}^{\infty}(\tilde{G}(F),K),$$ $$\underline{H}_{ac}(\tilde{G}(F))={\cal H}_{ac}(\tilde{G}(F))\boxtimes {\cal H}_{ac}(\tilde{G}(F)),$$ $$\underline{I}_{ac}(\tilde{G}(F),\omega)=I_{ac}(\tilde{G}(F),\omega)\boxtimes I_{ac}(\tilde{G}(F),\omega).$$
Pour $\tilde{L}\in {\cal L}(\tilde{M}_{0})$, $\tilde{L}\not=\tilde{G}$, on définit une application $$\underline{\phi}_{\tilde{L}}: \underline{H}_{ac}(\tilde{G}(F))\to \underline{I}_{ac}(\tilde{L}(F),\omega)$$ par la formule $$(2) \qquad \underline{\phi}_{\tilde{L}}(f_{1}\boxtimes f_{2})=\sum_{\tilde{L}_{1},\tilde{L}_{1}\in {\cal L}(\tilde{L})}d_{\tilde{L}}^{\tilde{G}}(\tilde{L}_{1},\tilde{L}_{2})\phi_{\tilde{L}}^{\tilde{L}_{1}}(f_{1,\tilde{\bar{Q}}_{1}})\boxtimes \phi_{\tilde{L}}^{\tilde{L}_{2}}(f_{2,\tilde{Q}_{2}}).$$ Le membre de droite dépend d’un choix de paramètre auxiliaire définissant l’application $(\tilde{L}_{1},\tilde{L}_{2})\mapsto (\tilde{Q}_{1},\tilde{Q}_{2})$. Pour que la définition soit correcte, on doit montrer qu’elle est indépendante de ce choix. Fixons des $\omega$-représentations $L$-irréductibles et tempérées $ \tilde{\pi}_{1}$ et $\tilde{\pi}_{2}$ de $\tilde{L}(F)$ et des éléments $X_{1},X_{2}\in \tilde{{\cal A}}_{\tilde{L},F}$. Considérons la forme linéaire sur $\underline{I}_{ac}(\tilde{L}(F),\omega)$ définie par $$(\varphi_{1},\varphi_{2})\mapsto \overline{I_{\tilde{L}}(\tilde{\pi}_{1},X_{1},\varphi_{1})}I_{\tilde{L}}(\tilde{\pi}_{2},X_{2},\varphi_{2}).$$ D’après 6.4(5), il suffit de prouver que cette forme linéaire prend sur le membre de droite de (2) une valeur qui ne dépend pas du choix du paramètre auxiliaire. D’après les définitions des applications $\phi_{\tilde{L}}^{\tilde{L}_{i}}$ pour $i=1,2$, cette valeur est $$\sum_{\tilde{L}_{1},\tilde{L}_{1}\in {\cal L}(\tilde{L})}d_{\tilde{L}}^{\tilde{G}}(\tilde{L}_{1},\tilde{L}_{2})\overline{J_{\tilde{L}}^{\tilde{L}_{1}}(\tilde{\pi}_{1},X_{1},f_{1,\tilde{\bar{Q}}_{1}}) }J_{\tilde{L}}^{\tilde{L}_{2}}(\tilde{\pi}_{2},X_{2},f_{2,\tilde{Q}_{2}}).$$ On peut remplacer $f_{1}$ et $f_{2}$ par leurs produits avec $b\circ\tilde{H}_{\tilde{G}}$, où $b\in C_{c}^{\infty}(\tilde{{\cal A}}_{\tilde{G},F})$ vaut $1$ sur les images de $X_{1}$ et $X_{2}$ dans $\tilde{{\cal A}}_{\tilde{G},F}$. On peut donc supposer $f_{1}$ et $f_{2}$ à support compacts. L’expression ci-dessus est alors déduite par transformation de Fourier de la fonction $$(\tilde{\lambda}_{1},\tilde{\lambda}_{2})\mapsto\sum_{\tilde{L}_{1},\tilde{L}_{1}\in {\cal L}(\tilde{L})}d_{\tilde{L}}^{\tilde{G}}(\tilde{L}_{1},\tilde{L}_{2})\overline{J_{\tilde{L}}^{\tilde{L}_{1}}(\tilde{\pi}_{1,\tilde{\lambda}_{1}},f_{1,\tilde{\bar{Q}}_{1}}) }J_{\tilde{L}}^{\tilde{L}_{2}}(\tilde{\pi}_{2,\tilde{\lambda}_{2}},f_{2,\tilde{Q}_{2}})$$ sur $i\tilde{{\cal A}}_{\tilde{L},F}^*\times i\tilde{{\cal A}}_{\tilde{L},F}^*$. Il suffit de prouver que cette fonction ne dépend pas du choix du paramètre auxiliaire. Quitte à tordre $\tilde{\pi}_{1}$ et $\tilde{\pi}_{2}$, on est ramené à montrer que l’expression $$(3) \qquad \sum_{\tilde{L}_{1},\tilde{L}_{1}\in {\cal L}(\tilde{L})}d_{\tilde{L}}^{\tilde{G}}(\tilde{L}_{1},\tilde{L}_{2})\overline{J_{\tilde{L}}^{\tilde{L}_{1}}(\tilde{\pi}_{1},f_{1,\tilde{\bar{Q}}_{1}}) }J_{\tilde{L}}^{\tilde{L}_{2}}(\tilde{\pi}_{2},f_{2,\tilde{Q}_{2}})$$ est indépendante de ce choix. Fixons $\tilde{P}\in {\cal P}(\tilde{L})$. Pour $i=1,2$, introduisons la $(\tilde{G},\tilde{L})$-famille à valeurs opérateurs $({\cal M}(\pi_{i};\Lambda,\tilde{Q}))_{\tilde{Q}\in {\cal P}(\tilde{L})}$. Posons $${\cal M}(\pi_{1}\boxtimes \pi_{2};\Lambda,\tilde{Q})={\cal M}(\pi_{1};\Lambda,\tilde{\bar{Q}})\boxtimes {\cal M}(\pi_{2};\Lambda,\tilde{Q}).$$ De la $(\tilde{G},\tilde{L})$-famille $({\cal M}(\pi_{1}\boxtimes \pi_{2};\Lambda,\tilde{Q}))_{\tilde{Q}\in {\cal P}(\tilde{L})}$ se déduit un opérateur ${\cal M}^{\tilde{G}}_{\tilde{L}}(\pi_{1}\boxtimes \pi_{2})$ comme en 2.7. On pose $$J_{\tilde{L}}^{\tilde{G}}(\tilde{\pi}_{1}\boxtimes \tilde{\pi}_{2},f_{1}\boxtimes f_{2})=trace({\cal M}^{\tilde{G}}_{\tilde{L}}(\pi_{1}\boxtimes \pi_{2})(Ind_{\tilde{P}}^{\tilde{G}}(f_{1})\boxtimes Ind_{\tilde{P}}^{\tilde{G}}(f_{2}))).$$ Le lemme 5.4(ii) se généralise à cette situation: $J_{\tilde{L}}^{\tilde{G}}(\tilde{\pi}_{1}\boxtimes \tilde{\pi}_{2},f_{1}\boxtimes f_{2})$ est égal à (3). Puisque $J_{\tilde{L}}^{\tilde{G}}(\tilde{\pi}_{1}\boxtimes \tilde{\pi}_{2},f_{1}\boxtimes f_{2})$ ne dépend d’aucun choix, il en est de même de (3), ce que l’on voulait démontrer.
Remarquons que pour $f\in {\cal H}_{ac}(\tilde{G}(F))$, l’image par $\tilde{H}_{\tilde{G}}$ du support $Supp(f)$ de $f$ est fermée: localement, c’est l’image de $Supp(f(b\circ\tilde{H}_{\tilde{G}}))$ pour une fonction $b$ à support compact convenable et ce dernier support est compact. On note $ \underline{H}_{ac}(\tilde{G}(F))^1$, resp. $\underline{H}_{ac}(\tilde{G}(F))_{0}$, le sous-espace de $\underline{H}_{ac}(\tilde{G}(F))$ engendré par les fonctions $f_{1}\otimes f_{2}$ telles que $$\tilde{H}_{\tilde{G}}(Supp(f_{1}))\cap \tilde{H}_{\tilde{G}}(Supp(f_{2}))$$ soit compact, resp. vide. On a l’égalité $$(4) \qquad \underline{H}_{ac}(\tilde{G}(F))^1=\underline{C}(\tilde{G}(F))+\underline{H}_{ac}(\tilde{G}(F))_{0}.$$ En effet, soient $f_{1},f_{2}\in {\cal H}_{ac}(\tilde{G}(F))$ tels que $\tilde{H}_{\tilde{G}}(Supp(f_{1}))\cap \tilde{H}_{\tilde{G}}(Supp(f_{2}))$ soit compact. Choisissons une fonction $b'\in C_{c}^{\infty}(\tilde{{\cal A}}_{\tilde{G}})$ valant $1$ sur un voisinage de ce compact. Posons $b''=1-b'$ et, pour $i=1,2$, $f'_{i}=f_{i}(b'\circ\tilde{H}_{\tilde{G}})$, $f''_{i}=f_{i}(b''\circ\tilde{H}_{\tilde{G}})$. On a l’égalité $$f_{1}\boxtimes f_{2}=(f'_{1}\boxtimes f'_{2})+(f'_{1}\boxtimes f''_{2})+(f''_{1}\boxtimes f'_{2})+(f''_{1}\boxtimes f''_{2}).$$ Le premier terme appartient à $\underline{C}(\tilde{G}(F))$, les trois autres à $\underline{H}_{ac}(\tilde{G}(F))_{0}$. D’où (4).
Notons $\underline{I}_{ac}(\tilde{G}(F),\omega)^1$ l’image de $\underline{H}_{ac}(\tilde{G}(F))^1$ dans $\underline{I}_{ac}(\tilde{G}(F),\omega)$. Montrons que
\(5) l’application $\underline{\phi}_{\tilde{L}}$ envoie $ \underline{H}_{ac}(\tilde{G}(F))^1$ dans $ \underline{I}_{ac}(\tilde{L}(F),\omega)^1$.
Preuve. D’après (4), il suffit de montrer que $\underline{\phi}_{\tilde{L}}(f_{1}\boxtimes f_{2})\in \underline{I}_{ac}(\tilde{L}(F),\omega)^1$ dans les deux cas suivants
\(6) $f_{i}\in C_{c}^{\infty}(\tilde{G}(F),K)$ pour $i=1,2$;
\(7) $f_{i}\in {\cal H}_{ac}(\tilde{G}(F))$ pour $i=1,2$ et $\tilde{H}_{\tilde{G}}(Supp(f_{1}))\cap \tilde{H}_{\tilde{G}}(Supp(f_{2}))=\emptyset$.
Soit $(\tilde{L}_{1},\tilde{L}_{2})$ intervenant dans la formule (2). On pose $\varphi_{1}=\phi_{\tilde{L}}^{\tilde{L}_{1}}(f_{1,\tilde{\bar{Q}}_{1}})$ et $\varphi_{2}=\phi_{\tilde{L}}^{\tilde{L}_{2}}(f_{2,\tilde{Q}_{2}})$. Dans le cas (6), les fonctions $f_{1,\tilde{\bar{Q}}_{1}} $ et $f_{2,\tilde{Q}_{2}}$ sont à supports compacts. D’après 6.4(10), on peut supposer que $\tilde{H}_{\tilde{L}_{i}}(Supp(\varphi_{i}))$ est compact pour $i=1,2$. Il en résulte que $ \tilde{H}_{\tilde{L}}(Supp(\varphi_{1}))\cap \tilde{H}_{\tilde{L}}(Supp(\varphi_{2}))$ a une projection compacte dans $\tilde{{\cal A}}_{\tilde{L}_{i}}$ pour $i=1,2$. Comme dans la preuve de (1), la non-nullité de $d_{\tilde{L}}^{\tilde{G}}(\tilde{L}_{1},\tilde{L}_{2})$ entraîne que l’ensemble $ \tilde{H}_{\tilde{L}}(Supp(\varphi_{1}))\cap \tilde{H}_{\tilde{L}}(Supp(\varphi_{2}))$ lui-même est compact. Donc $\varphi_{1}\boxtimes \varphi_{2}\in \underline{H}_{ac}(\tilde{L}(F))^1$. Dans le cas (7), puisque $\tilde{H}_{\tilde{G}}(Supp(f_{i}))$ est fermé pour $i=1,2$, on peut fixer des fonctions $b_{i}$ sur $\tilde{{\cal A}}_{\tilde{G},F}$, lisses, à supports disjoints, telles que $b_{i}$ vaille $1$ sur $\tilde{H}_{\tilde{G}}(Supp(f_{i}))$. D’après 6.4(10), on peut remplacer $\varphi_{i}$ par $\varphi_{i}(b_{i}\circ\tilde{H}_{\tilde{G}})$. Mais alors $ \tilde{H}_{\tilde{G}}(Supp(\varphi_{1}))\cap \tilde{H}_{\tilde{G}}(Supp(\varphi_{2}))$ est vide, a fortiori $ \tilde{H}_{\tilde{L}}(Supp(\varphi_{1}))\cap \tilde{H}_{\tilde{L}}(Supp(\varphi_{2}))$ l’est et $\varphi_{1}\boxtimes \varphi_{2}$ appartient à $ \underline{H}_{ac}(\tilde{L}(F))^1$. Cela prouve (5).
On a défini $\underline{\phi}_{\tilde{L}}$ pour $\tilde{L}\not=\tilde{G}$. On note simplement $\underline{\phi}_{\tilde{G}}$ l’identité de $\underline{H}_{ac}(\tilde{G}(F))$.
Les distributions $(f_{1},f_{2})\mapsto J_{g\acute{e}om}^{\tilde{G}}(\omega,f_{1},f_{2})$, $(f_{1},f_{2})\mapsto I_{g\acute{e}om}^{\tilde{G}}(\omega,f_{1},f_{2})$ et celles qui les constituent peuvent être considérées comme des formes linéaires sur $ \underline{C}(\tilde{G}(F))$. En fait elles se prolongent à l’espace $ \underline{H}_{ac}^1$: pour une telle distribution $D$ et pour $\underline{f}\in \underline{H}_{ac}^1$, on écrit $\underline{f}=\underline{f}_{c}+\underline{f}_{0}$ avec $\underline{f}_{c}\in \underline{C}(\tilde{G}(F))$ et $\underline{f}_{0}\in \underline{H}_{ac}(\tilde{G}(F))_{0}$ et on pose $D(\underline{f})=D(\underline{f}_{c})$. Pour que cette définition soit loisible, il faut évidemment montrer que $D(\underline{f})=0$ si $\underline{f}\in \underline{C}(\tilde{G}(F))\cap \underline{H}_{ac}(\tilde{G}(F))_{0}$. Les distributions en question s’expriment à l’aide des distributions basiques $\underline{f}\mapsto J_{\tilde{M}}^{\tilde{G}}(\gamma,\omega,\underline{f})$ ou $\underline{f}\mapsto I_{\tilde{M}}^{\tilde{G}}(\gamma,\omega,\underline{f})$, il suffit donc de traiter celles-ci. Chacune d’elles s’étend naturellement à $\underline{H}_{ac}(\tilde{G}(F))$ tout entier, il suffit donc de montrer que ces distributions étendues annulent $\underline{H}_{ac}(\tilde{G}(F))_{0}$. Soient donc $f_{1}$ et $f_{2} $ vérifiant (7). On choisit des fonctions $b_{i}$ comme dans la preuve de (5). Il résulte des définitions et de 6.5(3) que, pour chacune de nos deux distributions $D$ ci-dessus, on a les égalités $$D(f_{1}\boxtimes f_{2})=D(f_{1}(b_{1}\circ\tilde{H}_{\tilde{G}})\boxtimes f_{2}(b_{2}\circ\tilde{H}_{\tilde{G}}))=b_{1}(\tilde{H}_{\tilde{G}}(\gamma))b_{2}(\tilde{H}_{\tilde{G}}(\gamma))D(f_{1}\boxtimes f_{2})=0,$$ ce qu’on voulait démontrer.
Si on admet la proposition 6.5, le prolongement à $\underline{H}_{ac}(\tilde{G}(F))^1$ de la distribution $(f_{1},f_{2})\mapsto I_{g\acute{e}om}^{\tilde{G}}(\omega,f_{1},f_{2})$ se quotiente en une forme linéaire sur $\underline{I}_{ac}(\tilde{G}(F),\omega)^1$ parce les distributions $\underline{f}\mapsto I_{\tilde{M}}^{\tilde{G}}(\gamma,\omega,\underline{f})$ qui le constituent se quotientent ainsi.
La distribution $(f_{1},f_{2})\mapsto I_{disc}^{\tilde{G}}(\omega,f_{1},f_{2})$ peut elle-aussi être vue comme une forme linéaire sur $\underline{C}(\tilde{G}(F))$. Montrons qu’elle se prolonge à $\underline{H}_{ac}(\tilde{G}(F))^1$ par le même procédé que ci-dessus et que ce prolongement se quotiente en une forme linéaire sur l’espace $\underline{I}_{ac}(\tilde{G}(F),\omega)^1$. La distribution en question est combinaison linéaire de distributions $$(f_{1},f_{2})\mapsto \int_{i{\cal A}_{\tilde{G},F}^*}\overline{I_{\tilde{G}}(\tilde{\pi}_{\tilde{\lambda}},f_{1})}I_{\tilde{G}}(\tilde{\pi}_{\tilde{\lambda}},f_{2})\,d\lambda,$$ où $\tilde{\pi}$ est une $\omega$-représentation tempérée de $\tilde{G}(F)$. On transforme celles-ci par inversion de Fourier en $$(f_{1},f_{2})\mapsto mes(i{\cal A}_{\tilde{G},F}^*)\int_{\tilde{{\cal A}}_{\tilde{L},F}}\overline{I_{\tilde{G}}(\tilde{\pi},X,f_{1})}I_{\tilde{G}}(\tilde{\pi},X,f_{2})\,dX.$$ On montre grâce à 6.4(4) qu’une distribution $$(8) \qquad (f_{1},f_{2})\mapsto \overline{I_{\tilde{G}}(\tilde{\pi},X,f_{1})}I_{\tilde{G}}(\tilde{\pi},X,f_{2})$$ annule $\underline{C}(\tilde{G}(F))\cap \underline{H}_{ac}(\tilde{G}(F))_{0}$. Comme pour les distributions “géométriques”, cela permet de prolonger la distribution $(f_{1},f_{2})\mapsto I_{disc}^{\tilde{G}}(\omega,f_{1},f_{2})$ à $\underline{H}_{ac}(\tilde{G}(F))^1$. Ce prolongement se quotiente en une forme linéaire sur $\underline{I}_{ac}(\tilde{G}(F),\omega)^1$ parce les distributions (8) qui le constituent se quotientent ainsi.
Remarquons que, si l’énoncé du théorème est vrai, il s’étend en l’égalité $$I_{g\acute{e}om}^{\tilde{G}}(\omega,\underline{f})=I_{disc}^{\tilde{G}}(\omega,\underline{f})$$ pour tout $\underline{f}\in \underline{H}_{ac}(\tilde{G}(F))^1$ (ou $\underline{f}\in \underline{I}_{ac}(\tilde{G}(F),\omega)^1$ si on admet la proposition 6.5) puisque chaque terme est par définition le même terme évalué sur $\underline{f}_{c}$ où, comme plus haut $\underline{f}_{c}$ est un élément de $\underline{C}(\tilde{G}(F))$ tel que $\underline{f}\in \underline{f}_{c}+\underline{H}_{ac}(\tilde{G}(F))_{0}$.
Venons-en à la preuve du théorème. Elle se fait par récurrence sur $rg_{ss}(G)$. On suppose vérifiés le théorème et la proposition 6.5 pour les espaces $\tilde{G}'$ tels que $rg_{ss}(G')<rg_{ss}(G)$. Soient $f_{i}\in C_{c}^{\infty}(\tilde{G}(F),K)$ pour $i=1,2$, posons $\underline{f}=f_{1}\boxtimes f_{2}$. Soit $\tilde{L}\in {\cal L}(\tilde{M}_{0})$. Montrons que $$(9)\qquad J_{\tilde{L},spec}^{\tilde{G}}(\omega,f_{1},f_{2})=I^{\tilde{L}}_{disc}(\omega,\underline{\phi}_{\tilde{L}}(\underline{f})).$$ C’est tautologique si $\tilde{L}=\tilde{G}$. Supposons $\tilde{L}\not=\tilde{G}$. Les deux côtés sont des combinaisons linéaires indexées par $\tau\in(E_{disc}(\tilde{L},\omega)/conj)/i{\cal A}_{\tilde{L},F}^*$ de produits des mêmes coefficients et de certaines distributions. La distribution qui intervient dans le membre de gauche est $$(10)\qquad \int_{i{\cal A}_{\tilde{L},F}^*}J_{\tilde{L}}^{\tilde{G}}(\pi_{\tau_{\lambda}},f_{1},f_{2})\,d\lambda.$$ Relevons $\tau$ en un élément $\boldsymbol{\tau}\in {\cal E}(\tilde{L},\omega)$. Pour $X\in \tilde{{\cal A}}_{\tilde{L},F}$, notons $\underline{\varphi}\mapsto I_{\tilde{L}}(\pi_{\tau},X,\underline{\varphi})$ le prolongement à $\underline{H}_{ac}(\tilde{G})^1$ de la distribution $$(\varphi_{1},\varphi_{2})\mapsto \overline{I_{\tilde{L}}(\tilde{\pi}_{\boldsymbol{\tau}},X,\varphi_{1})}I_{\tilde{L}}(\tilde{\pi}_{\boldsymbol{\tau}},X,\varphi_{2}).$$ Alors la distribution intervenant dans le membre de droite de (9) est $$(11)\qquad mes(i{\cal A}_{\tilde{L},F}^*)\int_{\tilde{{\cal A}}_{\tilde{L},F}}I_{\tilde{L}}(\pi_{\tau},X,\underline{\phi}_{\tilde{L}}(\underline{f}))\,dX.$$ Il faut montrer que les expressions (10) et (11) sont égales. Par transformation de Fourier, le lemme 5.4(ii) entraîne que (10) est égal à $$mes(i{\cal A}_{\tilde{L},F}^*)\sum_{\tilde{L}_{1},\tilde{L}_{2}\in {\cal L}(\tilde{L})}d_{\tilde{L}}^{\tilde{G}}(\tilde{L}_{1},\tilde{L}_{2})\int_{\tilde{{\cal A}}_{\tilde{L},F}}\overline{J_{\tilde{L}}^{\tilde{L}_{1}}(\tilde{\pi}_{\boldsymbol{\tau}},X,f_{1,\tilde{\bar{Q}}_{1}})}J_{\tilde{L}}^{\tilde{L}_{2}}(\tilde{\pi}_{\boldsymbol{\tau}},X,f_{2,\tilde{Q}_{2}})\,dX.$$ D’après les définitions des applications $\phi_{\tilde{L}}^{\tilde{L}_{i}}$ pour $i=1,2$, c’est aussi $$mes(i{\cal A}_{\tilde{L},F}^*)\sum_{\tilde{L}_{1},\tilde{L}_{2}\in {\cal L}(\tilde{L})}d_{\tilde{L}}^{\tilde{G}}(\tilde{L}_{1},\tilde{L}_{2})\int_{\tilde{{\cal A}}_{\tilde{L},F}}\overline{I_{\tilde{L}}(\tilde{\pi}_{\boldsymbol{\tau}},X,\phi_{\tilde{L}}^{\tilde{L}_{1}}(f_{1,\tilde{\bar{Q}}_{1}}))}I_{\tilde{L}}(\tilde{\pi}_{\boldsymbol{\tau}},X,\phi_{\tilde{L}}^{\tilde{L}_{2}}(f_{2,\tilde{Q}_{2}}))\,dX.$$ L’égalité de cette expression avec (11) résulte alors de la définition de $\underline{\phi}_{\tilde{L}}$. Cela prouve (9).
Pour $\tilde{L}\not=\tilde{G}$, on peut par l’hypothèse de récurrence utiliser le théorème prolongé comme indiqué ci-dessus: on a $$I^{\tilde{L}}_{disc}(\omega,\underline{\phi}_{\tilde{L}}(\underline{f}))=I^{\tilde{L}}_{g\acute{e}om}(\omega,\underline{\phi}_{\tilde{L}}(\underline{f})).$$ Alors (9) implique $$J_{spec}^{\tilde{G}}(\omega,f_{1},f_{2})=I^{\tilde{G}}_{disc}(\omega,f_{1},f_{2})-I_{g\acute{e}om}^{\tilde{G}}(\omega,f_{1},f_{2})+X,$$ où $$X=\sum_{\tilde{L}\in {\cal L}(\tilde{M}_{0})}\vert \tilde{W}^{L}\vert \vert \tilde{W}^G\vert ^{-1}(-1)^{a_{\tilde{L}}-a_{\tilde{G}}}I^{\tilde{L}}_{g\acute{e}om}(\underline{\phi}_{\tilde{L}}(\underline{f}))$$ (rappelons que, par convention, $\underline{\phi}_{\tilde{G}}$ l’identité de $\underline{H}_{ac}(\tilde{G}(F))$). En utilisant le théorème 5.1, la relation cherchée $$I_{g\acute{e}om}^{\tilde{G}}(\omega,f_{1},f_{2})=I_{disc}^{\tilde{G}}(\omega,f_{1},f_{2})$$ équivaut à l’égalité $$(12) \qquad J_{g\acute{e}om}^{\tilde{G}}(\omega,f_{1},f_{2})=X.$$ Par définition $$X=\sum_{\tilde{L}\in {\cal L}(\tilde{M}_{0})}\vert \tilde{W}^{L}\vert \vert \tilde{W}^G\vert ^{-1}(-1)^{a_{\tilde{L}}-a_{\tilde{G}}}\sum_{\tilde{M}\in {\cal L}^{\tilde{L}}(\tilde{M}_{0})}\vert \tilde{W}^M\vert \vert \tilde{W}^L\vert ^{-1}(-1)^{a_{\tilde{M}}-a_{\tilde{L}}}I_{\tilde{M},g\acute{e}om}^{\tilde{L}}(\omega,\underline{\phi}_{\tilde{L}}(\underline{f}))$$ $$=\sum_{\tilde{M}\in {\cal L}(\tilde{M}_{0})}\vert \tilde{W}^M\vert \vert \tilde{W}^G\vert ^{-1}(-1)^{a_{\tilde{M}}-a_{\tilde{G}}}X_{\tilde{M}},$$ où $$X_{\tilde{M}}=\sum_{\tilde{L}\in {\cal L}(\tilde{M})}I_{\tilde{M},g\acute{e}om}^{\tilde{L}}(\omega,\underline{\phi}_{\tilde{L}}(\underline{f})).$$ En se rappelant la définition de $J_{g\acute{e}om}^{\tilde{G}}(\omega,f_{1},f_{2})$, il suffit pour démontrer (12) de fixer $\tilde{M}\in {\cal L}(\tilde{M}_{0})$ et de prouver l’égalité $$(13)\qquad J_{\tilde{M},g\acute{e}om}^{\tilde{G}}(\omega,f_{1},f_{2})=X_{\tilde{M}}.$$ Les deux membres de cette égalité sont des sommes sur $\tilde{S}\in T_{ell}(\tilde{M},\omega)$ de coefficients (qui sont les mêmes pour les deux membres) et d’intégrales sur $\tilde{S}(F)/(1-\theta)(S(F))$ de certaines fonctions. Il suffit de fixer $\tilde{S}$ et de prouver que les fonctions sont les mêmes. Fixons donc $\tilde{S}$ et un point $\gamma\in \tilde{S}(F)\cap \tilde{G}_{reg}(F)$. La valeur en $\gamma$ de la fonction relative au membre de gauche de (13) est $$\sum_{\tilde{M}_{1},\tilde{M}_{2}\in{\cal L}(\tilde{M})}d_{\tilde{M}}^{\tilde{G}}(\tilde{M}_{1},\tilde{M}_{2})\overline{J_{\tilde{M}}^{\tilde{M}_{1}}(\gamma,\omega,f_{1,\tilde{\bar{P}}_{1}})}J_{\tilde{M}}^{\tilde{M}_{2}}(\gamma,\omega,f_{2,\tilde{\bar{P}}_{2}}),$$ cela d’après le lemme 5.4(i). On peut exprimer les intégrales orbitales pondérées à l’aide d’intégrales équivariantes grâce à la proposition 6.5. On obtient $$\sum_{\tilde{M}_{1},\tilde{M}_{2}\in{\cal L}(\tilde{M})}d_{\tilde{M}}^{\tilde{G}}(\tilde{M}_{1},\tilde{M}_{2})\sum_{\tilde{L}_{1}\in {\cal L}^{\tilde{M}_{1}}(\tilde{M})}\sum_{\tilde{L}_{2}\in {\cal L}^{\tilde{M}_{2}}(\tilde{M})}\overline{I_{\tilde{M}}^{\tilde{L}_{1}}(\gamma,\omega,\phi_{\tilde{L}_{1}}^{\tilde{M}_{1}}(f_{1,\tilde{\bar{P}}_{1}})}I_{\tilde{M}}^{\tilde{L}_{2}}(\gamma,\omega,\phi_{\tilde{L}_{2}}^{\tilde{M}_{2}}(f_{2,\tilde{\bar{P}}_{2}})),$$ en posant la convention que $\phi_{\tilde{G}}^{\tilde{G}}$ est l’identité de $C_{c}^{\infty}(\tilde{G}(F),K)$. Si $\tilde{L}_{1}$ et $\tilde{L}_{2}$ sont tous deux différents de $\tilde{G}$, la distribution $$(\varphi_{1},\varphi_{2})\to \overline{I_{\tilde{M}}^{\tilde{L}_{1}}(\gamma,\omega, \varphi_{1})}I_{\tilde{M}}^{\tilde{L}_{2}}(\gamma, \omega,\varphi_{2})$$ peut être considérée comme une forme linéaire sur $I_{ac}(\tilde{L}_{1}(F),\omega)\boxtimes I_{ac}(\tilde{L}_{2}(F),\omega)$. Notons-la $\underline{\varphi}\mapsto
I_{\tilde{M}}^{\tilde{L}_{1},\tilde{L}_{2}}(\gamma,\omega,\underline{\varphi})$. Dans le cas où l’un des $\tilde{L}_{i}$ est égal à $\tilde{G}$ (ou les deux), on utilise la même notation en remplaçant la composante $I_{ac}(\tilde{L}_{i}(F),\omega)$ de l’espace de départ par ${\cal H}_{ac}(\tilde{G}(F))$. Alors l’expression précédente s’écrit $$(14) \qquad \sum_{\tilde{L}_{1},\tilde{L}_{2}\in {\cal L}(\tilde{M})}I_{\tilde{M}}^{\tilde{L}_{1},\tilde{L}_{2}}(\gamma,\omega,\underline{\varphi}(\tilde{L}_{1},\tilde{L}_{2})),$$ où $$\underline{\varphi}(\tilde{L}_{1},\tilde{L}_{2})=\sum_{\tilde{M}_{1}\in {\cal L}(\tilde{L}_{1}),\tilde{M}_{2}\in {\cal L}(\tilde{L}_{2})}d_{\tilde{M}}^{\tilde{G}}(\tilde{M}_{1},\tilde{M}_{2})\phi_{\tilde{L}_{1}}^{\tilde{M}_{1}}(f_{1,\tilde{\bar{P}}_{1}})\boxtimes \phi_{\tilde{L}_{2}}^{\tilde{M}_{2}}(f_{2,\tilde{P}_{2}}).$$
La valeur de la fonction intervenant dans le membre de droite de (13) est $$\sum_{\tilde{L}\in {\cal L}(\tilde{M})}I_{\tilde{M}}^{\tilde{L}}(\gamma,\omega,\underline{\phi}_{\tilde{L}}(\underline{f})).$$ D’après les définitions, cela s’écrit encore $$\sum_{\tilde{L}\in {\cal L}(\tilde{M})}\sum_{\tilde{L}_{1},\tilde{L}_{2}\in {\cal L}^{\tilde{L}}(\tilde{M})}d_{\tilde{M}}^{\tilde{L}}(\tilde{L}_{1},\tilde{L}_{2})I_{\tilde{M}}^{\tilde{L}_{1},\tilde{L}_{2}}(\gamma,\omega,(\underline{\phi}_{\tilde{L}}(\underline{f}))_{\tilde{L}_{1},\tilde{L}_{2}}),$$ où on a noté $\underline{\varphi}\mapsto \underline{\varphi}_{\tilde{L}_{1},\tilde{L}_{2}}$ l’application linéaire de $\underline{H}_{ac}(\tilde{L}(F))$ dans $I_{ac}(\tilde{L}_{1}(F),\omega)\boxtimes I_{ac}(\tilde{L}_{2}(F),\omega)$ qui envoie $\varphi_{1}\boxtimes \varphi_{2}$ sur $ \varphi_{1,\tilde{L}_{1}}\boxtimes \varphi_{2,\tilde{L}_{2}}$. Ici encore, on doit modifier la définition si l’un des $\tilde{L}_{i}$ est égal à $\tilde{G}$: on remplace $I_{ac}(\tilde{L}_{i}(F),\omega)$ par ${\cal H}_{ac}(\tilde{G}(F))$ et $\varphi_{i,\tilde{L}_{i}}$ par $\varphi_{i}$. L’expression ci-dessus s’écrit encore $$(15)\qquad \sum_{\tilde{L}_{1},\tilde{L}_{2}\in {\cal L}(\tilde{M})}I_{\tilde{M}}^{\tilde{L}_{1},\tilde{L}_{2}}(\gamma,\omega,\underline{\varphi}'(\tilde{L}_{1},\tilde{L}_{2})),$$ où $$\underline{\varphi}'(\tilde{L}_{1},\tilde{L}_{2})=\sum_{\tilde{L}\in {\cal L}(\tilde{M}); \tilde{L}_{1},\tilde{L}_{2}\subset \tilde{L}}d_{\tilde{M}}^{\tilde{L}}(\tilde{L}_{1},\tilde{L}_{2})(\underline{\phi}_{\tilde{L}}(\underline{f}))_{\tilde{L}_{1},\tilde{L}_{2}}.$$ On doit prouver que les expressions (14) et (15) sont égales. Il suffit de fixer $\tilde{L}_{1}$ et $\tilde{L}_{2}$ et de prouver l’égalité $$(16) \qquad \underline{\varphi}(\tilde{L}_{1},\tilde{L}_{2})= \underline{\varphi}'(\tilde{L}_{1},\tilde{L}_{2}).$$ Fixons donc $\tilde{L}_{1}$ et $\tilde{L}_{2}$. On voit d’abord que les deux membres sont nuls si la condition $$(17) \qquad {\cal A}_{\tilde{M}}^{\tilde{L}_{1}}\cap {\cal A}_{\tilde{M}}^{\tilde{L}_{2}}=\{0\}$$ n’est pas vérifiée. En effet, dans ce cas, il n’y a pas de couples $(\tilde{M}_{1},\tilde{M}_{2})$ intervenant dans la définition de $\underline{\varphi}(\tilde{L}_{1},\tilde{L}_{2})$ pour lesquels on ait $d_{\tilde{M}}^{\tilde{G}}(\tilde{M}_{1},\tilde{M}_{2})\not=0$ et il n’y a pas de $\tilde{L}$ intervenant dans la définition de $\underline{\varphi}'(\tilde{L}_{1},\tilde{L}_{2})$ pour lequel $d_{\tilde{M}}^{\tilde{L}}(\tilde{L}_{1},\tilde{L}_{2})\not=0$. On suppose donc que (17) est vérifiée. Il y a alors un unique $\tilde{L}$ intervenant dans la définition de $\underline{\varphi}'(\tilde{L}_{1},\tilde{L}_{2})$ pour lequel $d_{\tilde{M}}^{\tilde{L}}(\tilde{L}_{1},\tilde{L}_{2})\not=0$: celui pour lequel $${\cal A}_{\tilde{L}}={\cal A}_{\tilde{L}_{1}}\cap {\cal A}_{\tilde{L}_{2}},$$ cf. 5.3. Dans la suite $\tilde{L}$ désigne cet ensemble de Levi.
Supposons $\tilde{L}_{1}=\tilde{L}_{2}=\tilde{G}$. L’égalité (17) entraîne $\tilde{M}=\tilde{G}$ et on vérifie que les deux membres de (16) sont simplement égaux à $f_{1}\boxtimes f_{2}$. Supposons par exemple $\tilde{L}_{1}=\tilde{G}$ et $\tilde{L}_{2}\not=\tilde{G}$. L’égalité (17) entraîne $\tilde{L}_{2}=\tilde{M}$. On a aussi $\tilde{L}=\tilde{G}$. D’où $\underline{\varphi}'(\tilde{G},\tilde{M})=f_{1}\boxtimes f_{2,\tilde{M}}$. Seul le couple $(\tilde{M}_{1},\tilde{M}_{2})=(\tilde{G},\tilde{M})$ contribue à la définition de $\underline{\varphi}(\tilde{G},\tilde{M})$, d’où $\underline{\varphi}(\tilde{G},\tilde{M})=f_{1}\boxtimes \phi_{\tilde{M}}^{\tilde{M}}(f_{2,\tilde{P}_{2}})=f_{1}\boxtimes f_{2,\tilde{M}}$, d’où (16) dans ce cas.
On suppose maintenant $\tilde{L}_{1}$ et $\tilde{L}_{2}$ tous deux différents de $\tilde{G}$. Les deux termes de (16) appartiennent à $I_{ac}(\tilde{L}_{1}(F),\omega)\boxtimes I_{ac}(\tilde{L}_{2}(F),\omega)$. Pour prouver (16), on peut fixer pour $i=1,2$ une $\omega$-représentation tempérée $\tilde{\pi}_{i}$ de $\tilde{L}_{i}(F)$ et un élément $X_{i}\in \tilde{{\cal A}}_{\tilde{L}_{i}}$ et prouver que la forme linéaire $$(18)\qquad \varphi_{1}\boxtimes \varphi_{2}\mapsto\overline{ I_{\tilde{L}_{1}}(\tilde{\pi}_{1},X_{1},\varphi_{1})}I_{\tilde{L}_{2}}(\tilde{\pi}_{2},X_{2},\varphi_{2})$$ prend la même valeur sur les deux membres. Sa valeur sur $\underline{\varphi}(\tilde{L}_{1},\tilde{L}_{2})$ est $$\sum_{\tilde{M}_{1}\in {\cal L}(\tilde{L}_{1}),\tilde{M}_{2}\in {\cal L}(\tilde{L}_{2})}d_{\tilde{M}}^{\tilde{G}}(\tilde{M}_{1},\tilde{M}_{2})\overline{ I_{\tilde{L}_{1}}(\tilde{\pi}_{1},X_{1},\phi_{\tilde{L}_{1}}^{\tilde{M}_{i}}(f_{1,\tilde{\bar{P}}_{1}}))}I_{\tilde{L}_{2}} (\tilde{\pi}_{2},X_{2},\phi_{\tilde{L}_{2}}^{\tilde{M}_{i}}(f_{2,\tilde{\bar{P}}_{2}})).$$ En utilisant les définitions des applications $\phi_{\tilde{L}_{i}}^{\tilde{M}_{i}}$, on obtient $$\sum_{\tilde{M}_{1}\in {\cal L}(\tilde{L}_{1}),\tilde{M}_{2}\in {\cal L}(\tilde{L}_{2})}d_{\tilde{M}}^{\tilde{G}}(\tilde{M}_{1},\tilde{M}_{2})\overline{ J_{\tilde{L}_{1}}^{\tilde{M}_{i}}(\tilde{\pi}_{1},X_{1}, f_{1,\tilde{\bar{P}}_{1}})}J_{\tilde{L}_{2}}^{\tilde{M}_{i}}(\tilde{\pi}_{2},X_{2}, f_{2,\tilde{\bar{P}}_{2}}).$$ C’est la valeur en $(-X_{1},X_{2})$ (le signe $-$ provenant de la conjugaison complexe figurant dans la formule ci-dessus) de la transformée de Fourier de la fonction sur $i\tilde{{\cal A}}_{\tilde{L}_{1},F}^*\times i\tilde{{\cal A}}_{\tilde{L}_{2},F}^*$ qui, à $(\tilde{\lambda}_{1},\tilde{\lambda}_{2})$, associe $$\sum_{\tilde{M}_{1}\in {\cal L}(\tilde{L}_{1}),\tilde{M}_{2}\in {\cal L}(\tilde{L}_{2})}d_{\tilde{M}}^{\tilde{G}}(\tilde{M}_{1},\tilde{M}_{2})\overline{ J_{\tilde{L}_{1}}^{\tilde{M}_{i}}(\tilde{\pi}_{1,\tilde{\lambda}_{1}}, f_{1,\tilde{\bar{P}}_{1}})}J_{\tilde{L}_{2}}^{\tilde{M}_{i}}(\tilde{\pi}_{2, \tilde{\lambda}_{2}},f_{2,\tilde{\bar{P}}_{2}}).$$ Fixons des espaces paraboliques $\tilde{Q}_{i}\in {\cal P}(\tilde{L}_{i})$ pour $i=1,2$ contenus dans des espaces paraboliques d’espaces de Levi $\tilde{L}$ et introduisons les représentations induites $\tilde{\Pi}_{i,\tilde{\lambda}_{i}}=Ind_{\tilde{Q}_{i}}^{\tilde{G}}(\tilde{\pi}_{i,\tilde{\lambda}_{i}})$ dans l’espace $V_{i}=V_{\pi_{i},\tilde{Q}_{i}}$. On dispose des $(\tilde{G},\tilde{L}_{i})$-familles à valeurs opérateurs $({\cal M}(\pi_{i,\lambda_{i}};\Lambda,\tilde{Q}))_{\tilde{Q}\in {\cal P}(\tilde{L}_{i})}$. On vérifie que l’expression précédente n’est autre que $$(19)\qquad trace({\cal N}(\lambda_{1},\lambda_{2})(\tilde{\Pi}_{1,\tilde{\lambda}_{1}}(f_{1})\boxtimes \tilde{\Pi}_{2,\tilde{\lambda}_{2}}(f_{2})),$$ où ${\cal N}(\lambda_{1},\lambda_{2})$ est l’opérateur de $V_{1}\boxtimes V_{2}$ défini par $${\cal N}(\lambda_{1},\lambda_{2})= \sum_{\tilde{M}_{1}\in {\cal L}(\tilde{L}_{1}),\tilde{M}_{2}\in {\cal L}(\tilde{L}_{2})}d_{\tilde{M}}^{\tilde{G}}(\tilde{M}_{1},\tilde{M}_{2}){\cal M}_{\tilde{L}_{1}}^{\tilde{\bar{P}}_{1}}(\pi_{1,\lambda_{1}})\boxtimes {\cal M}_{\tilde{L}_{2}}^{\tilde{P}_{2}}(\pi_{2,\lambda_{2}}).$$ Pour calculer la valeur sur $\underline{\varphi}'(\tilde{L}_{1},\tilde{L}_{2})$ de la forme linéaire (18), on doit d’abord calculer $I_{\tilde{L}_{i}}(\tilde{\pi}_{i},X_{i},\varphi_{i,\tilde{L}_{i}})$ pour $\varphi_{i}\in {\cal H}_{ac}(\tilde{L}(F))$. On introduit une fonction $b\in C_{c}^{\infty}(\tilde{{\cal A}}_{\tilde{L},F})$ telle que $b(\tilde{H}_{\tilde{L}}(X_{i}))=1$. On a alors $$I_{\tilde{L}_{i}}(\tilde{\pi}_{i},X_{i},\varphi_{i,\tilde{L}_{i}})=I_{\tilde{L}_{i}}(\tilde{\pi}_{i},X_{i},\varphi_{i,\tilde{L}_{i}}(b\circ\tilde{H}_{\tilde{L}}))=I_{\tilde{L}_{i}}(\tilde{\pi}_{i},X_{i},(\varphi_{i} (b\circ\tilde{H}_{\tilde{L}}))_{\tilde{L}_{i}}).$$ Maintenant, $\varphi_{i} (b\circ\tilde{H}_{\tilde{L}})$ est à support compact, d’où $$I_{\tilde{L}_{i}}(\tilde{\pi}_{i},X_{i},(\varphi_{i} (b\circ\tilde{H}_{\tilde{L}}))_{\tilde{L}_{i}})=mes( i{\cal A}_{\underline{\tilde{M}}_{i},F}^*)^{-1}\int_{i{\cal A}_{\tilde{L}_{i},F}^*}
I_{\tilde{L}_{i}}(\tilde{\pi}_{i,\tilde{\lambda}_{i}},(\varphi_{i} (b\circ\tilde{H}_{\tilde{L}}))_{\tilde{L}_{i}})e^{-<\tilde{\lambda}_{i},X_{i}>}\,d\lambda_{i}.$$ En introduisant la représentation $\tilde{\pi}_{i,\tilde{\lambda}_{i}}^{\tilde{L}}=Ind_{\tilde{Q}_{i}\cap \tilde{L}}^{\tilde{L}}(\tilde{\pi}_{i,\tilde{\lambda}_{i}})$, l’expression ci-dessus devient $$mes( i{\cal A}_{\underline{\tilde{M}}_{i},F}^*)^{-1}\int_{i{\cal A}_{\tilde{L}_{i},F}^*}
I_{\tilde{L}}(\tilde{\pi}^{\tilde{L}}_{i,\tilde{\lambda}_{i}},\varphi_{i} (b\circ\tilde{H}_{\tilde{L}}))e^{-<\tilde{\lambda}_{i},X_{i}>}\,d\lambda_{i}.$$ On peut aussi l’écrire $$mes( i{\cal A}_{\underline{\tilde{M}}_{i},F}^*)^{-1}\int_{i{\cal A}_{\tilde{L}_{i},F}^*}\int_{\tilde{{\cal A}}_{\tilde{L},F}}I_{\tilde{L}}(\tilde{\pi}^{\tilde{L}}_{i,\tilde{\lambda}_{i}},Y_{i},\varphi_{i} (b\circ\tilde{H}_{\tilde{L}}))\,dY_{i}\,e^{-<\tilde{\lambda}_{i},X_{i}>}\,d\lambda_{i},$$ puis, par inversion de Fourier, $$mes( i{\cal A}_{\underline{\tilde{M}}_{i},F}^*)^{-1}mes(i{\cal A}_{\tilde{L},F}^*)\int_{i{\cal A}_{\tilde{L}_{i},F}^*/i{\cal A}_{\tilde{L},F}^*}I_{\tilde{L}}(\tilde{\pi}^{\tilde{L}}_{i,\tilde{\lambda}_{i}},X_{i,\tilde{L}},\varphi_{i} (b\circ\tilde{H}_{\tilde{L}}))e^{-<\tilde{\lambda}_{i},X_{i}>}\,d\lambda_{i},$$ où $X_{i,\tilde{L}}$ est la projection de $X_{i}$ dans $\tilde{{\cal A}}_{\tilde{L},F}$. Maintenant, on peut faire disparaître la fonction $(b\circ\tilde{H}_{\tilde{L}})$ et on obtient simplement $$mes( i{\cal A}_{\underline{\tilde{M}}_{i},F}^*)^{-1}mes(i{\cal A}_{\tilde{L},F}^*)\int_{i{\cal A}_{\tilde{L}_{i},F}^*/i{\cal A}_{\tilde{L},F}^*}I_{\tilde{L}}(\tilde{\pi}^{\tilde{L}}_{i,\tilde{\lambda}_{i}},X_{i,\tilde{L}},\varphi_{i} )e^{-<\tilde{\lambda}_{i},X_{i}>}\,d\lambda_{i}.$$ En appliquant ce calcul et les définitions, on obtient que la valeur sur $\underline{\varphi}'(\tilde{L}_{1},\tilde{L}_{2})$ de la forme linéaire (18) est $$(20) \qquad mes( i{\cal A}_{\underline{\tilde{M}}_{1},F}^*)^{-1} mes( i{\cal A}_{\underline{\tilde{M}}_{2},F}^*)^{-1}mes(i{\cal A}_{\tilde{L},F}^*)^2\int_{i{\cal A}_{\tilde{L}_{1},F}^*/i{\cal A}_{\tilde{L},F}^*}\int_{i{\cal A}_{\tilde{L}_{2},F}^*/i{\cal A}_{\tilde{L},F}^*}$$ $$I_{\tilde{L}}(\tilde{\pi}^{\tilde{L}}_{1,\tilde{\lambda}_{1}}\boxtimes \tilde{\pi}^{\tilde{L}}_{2,\tilde{\lambda}_{2}},X_{1,\tilde{L}},X_{2,\tilde{L}},\underline{\phi}_{\tilde{L}}(\underline{f}))e^{<\tilde{\lambda}_{1},X_{1}>-<\tilde{\lambda}_{2},X_{2}>}\,d\lambda_{1}\,d\lambda_{2},$$ où on a noté $\underline{\varphi}\mapsto I_{\tilde{L}}(\tilde{\pi}^{\tilde{L}}_{1,\tilde{\lambda}_{1}}\boxtimes \tilde{\pi}^{\tilde{L}}_{2,\tilde{\lambda}_{2}},X_{1,\tilde{L}},X_{2,\tilde{L}},\underline{\varphi})$ la forme linéaire $$\varphi_{1}\boxtimes \varphi_{2}\mapsto \overline{I_{\tilde{L}}(\tilde{\pi}^{\tilde{L}}_{1,\tilde{\lambda}_{1}},X_{1,\tilde{L}},\varphi_{1} )}I_{\tilde{L}}(\tilde{\pi}^{\tilde{L}}_{2,\tilde{\lambda}_{2}},X_{2,\tilde{L}},\varphi_{2} )$$ sur $\underline{H}_{ac}(\tilde{L}(F))$. Mais on a calculé la composée de $\underline{\phi}_{\tilde{L}}$ avec cette forme linéaire dans la preuve suivant la relation (2). Le résultat est le suivant. On déduit des $(\tilde{G},\tilde{L}_{i})$-familles $({\cal M}(\pi_{i,\lambda'_{i}};\Lambda,\tilde{Q}))_{\tilde{Q}\in {\cal P}(\tilde{L}_{i})}$ (où $\lambda'_{i}\in i{\cal A}_{\tilde{L}_{i},F}^*$) des $(\tilde{G},\tilde{L})$-familles et on introduit leur produit sesquilinéaire $${\cal M}(\pi_{1,\lambda'_{1}}\boxtimes \pi_{2,\lambda'_{2}};\Lambda,\tilde{Q})={\cal M}(\pi_{1,\lambda'_{1}};\Lambda,\tilde{Q})\boxtimes {\cal M}(\pi_{2,\lambda'_{2}};\Lambda,\tilde{Q})$$ pour $\tilde{Q}\in {\cal P}(\tilde{L})$. On pose $${\cal N}'(\lambda'_{1},\lambda'_{2})={\cal M}^{\tilde{G}}_{\tilde{L}}(\pi_{1,\lambda'_{1}}\boxtimes\pi_{2,\lambda'_{2}};0).$$ Alors $I_{\tilde{L}}(\tilde{\pi}^{\tilde{L}}_{1,\tilde{\lambda}_{1}}\boxtimes \tilde{\pi}^{\tilde{L}}_{2,\tilde{\lambda}_{2}},X_{1,\tilde{L}},X_{2,\tilde{L}},\underline{\phi}_{\tilde{L}}(\underline{f}))$ est la valeur en $(-X_{1},X_{2})$ de la transformée de Fourier de la fonction $$(\tilde{\mu}_{1},\tilde{\mu}_{2})\mapsto trace({\cal N}'(\lambda_{1}+\mu_{1},\lambda_{2}+\mu_{2})(\tilde{\Pi}_{1,\tilde{\lambda}_{1}+\tilde{\mu}_{1}}(f_{1})\boxtimes \tilde{\Pi}_{2,\tilde{\lambda}_{2}+\tilde{\mu}_{2}}(f_{2})))$$ sur $i\tilde{{\cal A}}_{\tilde{L},F}^*\times i\tilde{{\cal A}}_{\tilde{L},F}^*$. En insérant cette valeur dans l’expression (20), on voit que la valeur de la forme linéaire (18) sur $\underline{\varphi}'(\tilde{L}_{1},\tilde{L}_{2})$ est la valeur en $(-X_{1},X_{2})$ de la transformée de Fourier de la fonction sur sur $i\tilde{{\cal A}}_{\tilde{L}_{1},F}^*\times i\tilde{{\cal A}}_{\tilde{L}_{2},F}^*$ qui, à $(\tilde{\lambda}_{1},\tilde{\lambda}_{2})$, associe $$(21)\qquad trace({\cal N}'(\lambda_{1},\lambda_{2})(\tilde{\Pi}_{1,\tilde{\lambda}_{1}}(f_{1})\boxtimes \tilde{\Pi}_{2,\tilde{\lambda}_{2}}(f_{2}))$$ sur $i\tilde{{\cal A}}_{\tilde{L}_{1},F}^*\times i\tilde{{\cal A}}_{\tilde{L}_{2},F}^*$.
On a montré que les valeurs de la forme linéaire (18) sur les deux membres de (16) étaient les valeurs en $(-X_{1},X_{2})$ des transformées de Fourier des fonctions en $(\tilde{\lambda}_{1},\tilde{\lambda}_{2})$ définies par (19) et (21). Pour démontrer leur égalité, il suffit de prouver l’égalité de ces dernières expressions. Il suffit encore de prouver l’égalité $${\cal N}(\lambda_{1},\lambda_{2})={\cal N}'(\lambda_{1},\lambda_{2}).$$ Mais c’est ce qu’affirme la relation 5.3(1) simplifiée dans notre situation comme expliqué dans ce paragraphe (et étendue aux familles à valeurs opérateurs, ce qui ne pose pas de difficulté). Cela achève la preuve. $\square$
Variante avec caractère central
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Pour tout sous-groupe $H$ de $G$ contenant $A_{\tilde{G}}$, on pose $\underline{H}=A_{\tilde{G}}\backslash H$. Pour tout sous-variété $\tilde{H}$ de $\tilde{G}$ invariante par translations par $A_{\tilde{G}}$, on pose $\underline{\tilde{H}}=A_{\tilde{G}}\backslash \tilde{H}$. On a simplement $\underline{H}(F)=A_{\tilde{G}}(F)\backslash H(F)$, $\underline{\tilde{H}}(F)=A_{\tilde{G}}(F)\backslash \tilde{H}(F)$ puisque $A_{\tilde{G}}$ est déployé donc cohomologiquement trivial.
Pour une fonction $f$ sur $\tilde{G}(F)$ et pour $z\in A_{\tilde{G}}(F)$, définissons la fonction $f^{[z]}$ par $f^{[z]}(\gamma)=f(z\gamma)=f(\gamma z)$. Soit $\mu$ un caractère unitaire de $A_{\tilde{G}}(F)$. On note $C_{\mu}^{\infty}(\tilde{G}(F),K)$ l’espace des fonctions $f:\tilde{G}(F)\to {\mathbb C}$ qui sont lisses, $K$-finies à droite et à gauche, qui vérifient la relation $f^{[z]}=\mu(z)^{-1}f$ et dont le support est compact modulo $A_{\tilde{G}}(F)$. Soit $\pi$ une représentation irréductible et unitaire de $G(F)$. Pour $z\in A_{\tilde{G}}(F)$, $\pi(z)$ est une homothétie dont on note $\mu_{\pi}(z)$ le rapport. L’application $z\mapsto \mu_{\pi}(z)$ est un caractère unitaire. Appelons-le le $A$-caractère central de $\pi$. Soit maintenant $\tilde{\pi}$ une $\omega$-représentation unitaire de $\tilde{G}(F)$, de longueur finie. Supposons que toutes les composantes irréductibles de la représentation sous-jacente $\pi$ aient le même $A$-caractère central. On dit alors que c’est le $A$-caractère central de $\tilde{\pi}$. Supposons qu’il en soit ainsi et que ce caractère soit $\mu$. On peut alors définir l’opérateur $\tilde{\pi}(f)$ pour $f\in C_{\mu}^{\infty}(\tilde{G}(F),K)$ par $$\tilde{\pi}(f)=\int_{ A_{\tilde{G}}(F)\backslash \tilde{G}(F)}f(g)\tilde{\pi}(g)\,dg.$$ On définit aussi sa trace $I_{\tilde{\underline{G}}}(\tilde{\pi},f)=trace(\tilde{\pi}(f))$. Pour $\boldsymbol{\tau}\in {\cal E}(\tilde{G},\omega)$, la représentation $\tilde{\pi}_{\boldsymbol{\tau}}$ n’est pas $G$-irréductible, mais admet un $A$-caractère central, qui ne dépend bien sûr que de l’image $\tau$ de $\boldsymbol{\tau}$ dans $E(\tilde{G},\omega)$. Notons-le $\mu_{\tau}$. On note $E_{disc,\mu}(\tilde{G},\omega)$ l’ensemble des $\tau\in E_{disc}(\tilde{G},\omega)$ tels que $\mu_{\tau}=\mu$. On note $E_{disc,\mu}(\tilde{G},\omega)/conj$ l’ensemble des classes de conjugaison par $G(F)$ dans $E_{disc,\mu}(\tilde{G},\omega)$. Pour $\tau=(M,\sigma,\tilde{r})\in E_{disc,\mu}(\tilde{G},\omega)$ (ou $E_{disc,\mu}(\tilde{G},\omega)/conj$), notons $Stab(W^G,\tau)$ le stabilisateur de $\tau$ dans $W^G$, puis, comme en 2.9, $${\bf Stab}(W^G,\tau)=(Stab(W^G,\tau)/W^M)/W_{0}^{G}(\sigma).$$ Pour $f_{1},f_{2}\in C_{\mu}^{\infty}(\tilde{G},K)$, posons $$I_{disc}^{\tilde{\underline{G}}}(\omega,f_{1},f_{2})=\sum_{\tau\in E_{disc,\mu}(\tilde{G},\omega)/conj}\vert {\bf Stab}(W^G,\tau)\vert ^{-1}\iota(\tau)\overline{I_{\tilde{\underline{G}}}(\tilde{\pi}_{\boldsymbol{\tau}},f_{1})}I_{\tilde{\underline{G}}}(\tilde{\pi}_{\boldsymbol{\tau}},f_{1}).$$
Soient $\tilde{M}\in {\cal L}(\tilde{M}_{0})$ et $\tilde{S}\in T_{ell}(\tilde{M})$. On dispose de mesures sur $S^{\theta,0}(F)$ et sur $A_{\tilde{G}}(F)$, donc aussi sur $A_{\tilde{G}}(F)\backslash S^{\theta,0}(F)=\underline{S}^{\theta,0}(F)$. Comme en 4.1, on munit $\tilde{S}(F)/A_{\tilde{G}}(F)(1-\theta)(S(F))=\underline{\tilde{S}}(F)/(1-\theta)(\underline{S}(F))$ de la mesure telle que, pour tout $\gamma\in \tilde{S}(F)$, l’application $$\begin{array}{ccc} \underline{ S}^{\theta,0}(F)&\to&\underline{\tilde{S}}(F)/(1-\theta)(\underline{S}(F))\\ t&\mapsto &t\gamma\\ \end{array}$$ préserve localement les mesures au voisinage de l’origine. Supposons $\omega$ trivial sur $S^{\theta}(F)$. Remarquons que $C_{\mu}^{\infty}(\tilde{G},K)\subset {\cal H}_{ac}(\tilde{G})$. Les intégrales pondérées équivariantes de 6.5 sont donc définies pour $f_{1}$ et $f_{2}$ et $\gamma\in \tilde{S}(F)\cap \tilde{G}_{reg}(F)$, ainsi que les termes $I_{\tilde{M}}^{\tilde{G}}(\gamma,\omega,f_{1},f_{2})$ de 6.6. On vérifie que $$I_{\tilde{M}}^{\tilde{G}}(z\gamma,\omega,f_{1},f_{2})=I_{\tilde{M}}^{\tilde{G}}(\gamma,\omega,f_{1},f_{2})$$ pour tout $z\in A_{\tilde{G}}(F)$. Remarquons que l’inclusion $A_{\tilde{G}}(F)\backslash S^{\theta}(F)\subset \underline{S}^{\theta}(F)$ n’est pas forcément une égalité. On pose $$I_{\tilde{\underline{M}}}^{\tilde{\underline{G}}}(\gamma,\omega,f_{i})=[\underline{S}^{\theta}(F):(A_{\tilde{G}}(F)\backslash S^{\theta}(F))]^{-1}I_{\tilde{M}}^{\tilde{G}}(\gamma,\omega,f_{i})$$ pour $i=1,2$ et $$I_{\tilde{\underline{M}}}^{\tilde{\underline{G}}}(\gamma,\omega,f_{1},f_{2})=[\underline{S}^{\theta}(F):(A_{\tilde{G}}(F)\backslash S^{\theta}(F))]^{-2}I_{\tilde{M}}^{\tilde{G}}(\gamma,\omega,f_{1},f_{2}).$$
[**Remarque.**]{} On a défini les intégrales orbitales comme des intégrales sur $S^{\theta}(F)\backslash G(F)$. On doit ici les convertir en intégrales sur $\underline{S}^{\theta}(F)\backslash\underline{G}(F)$, ce qui justifie les facteurs introduits ci-dessus.
On pose, au moins formellement, $$(1) \qquad I_{\tilde{\underline{M}},\tilde{\underline{S}}}^{\tilde{\underline{G}}}(\omega,f_{1},f_{2})=\vert W^M(\tilde{S})\vert ^{-1} mes(A_{\underline{\tilde{M}}}(F)\backslash \underline{ S}^{\theta}(F))$$ $$\int_{\tilde{S}(F)/A_{\tilde{G}}(F)(1-\theta)(S(F))}I_{\tilde{\underline{M}}}^{\tilde{\underline{G}}}(\gamma,\omega,f_{1},f_{2})\,d\gamma.$$
Comme en 6.6, on définit ensuite $$I_{\tilde{\underline{M}},g\acute{e}om}^{\tilde{\underline{G}}}(\omega,f_{1},f_{2})=\sum_{\tilde{S}\in T_{ell}(\tilde{M},\omega)}I_{\tilde{\underline{M}},\tilde{\underline{S}}}^{\tilde{\underline{G}}}(\omega,f_{1},f_{2})$$ et $$I_{g\acute{e}om}^{\tilde{\underline{G}}}(\omega,f_{1},f_{2})=\sum_{\tilde{M}\in {\cal L}(\tilde{M}_{0})}\vert \tilde{W}^M\vert \vert \tilde{W}^G\vert ^{-1}(-1)^{a_{\tilde{M}}-a_{\tilde{G}}}I_{\tilde{\underline{M}},g\acute{e}om}^{\tilde{\underline{G}}}(\omega,f_{1},f_{2}).$$
[0.3cm[**[Théorème]{}**]{}. [ *[Pour $f_{1},f_{2}\in C_{\mu}^{\infty}(\tilde{G}(F),K)$, les expressions (1) sont absolument convergentes. On a l’égalité $$I_{g\acute{e}om}^{\tilde{\underline{G}}}(\omega,f_{1},f_{2})=I_{spec}^{\tilde{\underline{G}}}(\omega,f_{1},f_{2}).$$]{}*]{}0.3cm]{}
Preuve. On ne détaillera que l’aspect combinatoire de la preuve. On note $$p_{\mu}:C_{c}^{\infty}(\tilde{G}(F),K)\to C_{\mu}^{\infty}(\tilde{G}(F),K)$$ l’application qui, à $\varphi\in C_{c}^{\infty}(\tilde{G}(F),K)$, associe la fonction $f$ définie par $$f(\gamma)=\int_{A_{\tilde{G}}(F)}f(z\gamma)\mu(z)\,dz.$$ Elle est surjective. Ainsi on peut introduire pour $i=1,2$ une fonction $\varphi_{i}\in C_{c}^{\infty}(\tilde{G}(F),K)$ telle que $p_{\mu}(\varphi_{i})=f_{i}$. Le théorème 6.6 nous fournit l’égalité $$I_{g\acute{e}om}^{\tilde{G}}(\omega,\varphi_{1},\varphi_{2}^{[z]})=I_{spec}^{\tilde{G}}(\omega,\varphi_{1},\varphi_{2}^{[z]})$$ pour tout $z\in A_{\tilde{G}}(F)$. On vérifie que le membre de gauche (donc aussi celui de droite) est à support compact en $z$ et borné. On a donc l’égalité $$(2) \qquad \int_{A_{\tilde{G}}(F)}I_{g\acute{e}om}^{\tilde{G}}(\omega,\varphi_{1},\varphi_{2}^{[z]})\mu(z)\,dz\,=\int_{A_{\tilde{G}}(F)}I_{spec}^{\tilde{G}}(\omega,\varphi_{1},\varphi_{2}^{[z]})\mu(z)\,dz.$$ On va montrer que le membre de gauche est égal à $I_{g\acute{e}om}^{\tilde{\underline{G}}}(\omega,f_{1},f_{2})$, tandis que celui de droite est égal à $I_{spec}^{\tilde{\underline{G}}}(\omega,f_{1},f_{2})$.
Commençons par les termes géométriques. On peut fixer $\tilde{M}\in {\cal L}(\tilde{M}_{0})$ et $\tilde{S}\in T_{ell}(\tilde{M},\omega)$ et prouver l’égalité $$(3) \qquad mes(A_{\tilde{M}}(F)\backslash S^{\theta}(F))\int_{A_{\tilde{G}}(F)}\int_{\tilde{S}(F)/(1-\theta)(S(F))}I_{\tilde{M}}^{\tilde{G}}(\gamma,\omega,\varphi_{1},\varphi_{2}^{[z]})\,d\gamma\,dz=$$ $$mes(A_{\underline{\tilde{M}}}(F)\backslash \underline{S}^{\theta}(F))\int_{\underline{\tilde{S}}(F)/(1-\theta)(\underline{S}(F))}I_{\tilde{\underline{M}}}^{\tilde{\underline{G}}}(\gamma,\omega,f_{1},f_{2})\,d\gamma.$$ Il existe une constante $C>0$ telle que l’on ait la formule d’intégration $$(4) \qquad \int_{\tilde{S}(F)/(1-\theta)(S(F))}\phi(\gamma)\,d\gamma\,=C\int_{\underline{\tilde{S}}(F)/(1-\theta)(\underline{S}(F))}\int_{A_{\tilde{G}}(F)}\phi(z'\gamma)\,dz'\,d\gamma$$ pour toute fonction intégrable $\phi$ sur $\tilde{S}(F)/(1-\theta)(S(F))$. Le membre de gauche de (3) devient $$C mes(A_{\tilde{M}}(F)\backslash S^{\theta}(F))\int_{\underline{\tilde{S}}(F)/(1-\theta)(\underline{S}(F))}\int_{A_{\tilde{G}}(F)}\int_{A_{\tilde{G}}(F)}I_{\tilde{M}}^{\tilde{G}}(z'\gamma,\omega,\varphi_{1},\varphi_{2}^{[z]})\,dz'\,dz\,d\gamma.$$ On vérifie que $$I_{\tilde{M}}^{\tilde{G}}(z'\gamma,\omega,\varphi_{1},\varphi_{2}^{[z]})=I_{\tilde{M}}^{\tilde{G}}(\gamma,\omega,\varphi_{1}^{[z']},\varphi_{2}^{[z'z]}),$$ puis que $$\int_{A_{\tilde{G}}(F)}\int_{A_{\tilde{G}}(F)}I_{\tilde{M}}^{\tilde{G}}(z'\gamma,\omega,\varphi_{1},\varphi_{2}^{[z]})\,dz'\,dz= [\underline{S}^{\theta}(F):(A_{\tilde{G}}(F)\backslash S^{\theta}(F))]^2 I_{\tilde{\underline{M}}}^{\tilde{\underline{G}}}(\gamma,\omega,f_{1},f_{2}).$$ Alors le membre de gauche de (3) devient celui de droite, à ceci près que la constante $mes(A_{\underline{\tilde{M}}}(F)\backslash \underline{S}^{\theta}(F))$ y est remplacée par $Cmes(A_{\tilde{M}}(F)\backslash S^{\theta}(F))[\underline{S}^{\theta}(F):(A_{\tilde{G}}(F)\backslash S^{\theta}(F))]^2$. Il reste à montrer que ces deux termes sont égaux, autrement dit, on doit calculer $C$. Fixons un point base $\gamma_{0}\in \tilde{S}(F)$, ce qui permet d’identifier $S(F)$ à $\tilde{S}(F)$ par $x\mapsto x\gamma_{0}$ et de même $\underline{S}(F)$ à $\underline{\tilde{S}}(F)$. Considérons le diagramme $$\begin{array}{ccccc}&&S^{\theta,0}(F)&&\\ &\swarrow&&\searrow&\\\underline{S}^{\theta,0}(F)&&&&S(F)/(1-\theta)(S(F))\\ &\searrow&&\swarrow&\\&&\underline{S}(F)/(1-\theta)(\underline{S}(F))&&\\ \end{array}$$ Pour chaque flèche $$\begin{array}{ccc}&&X\\&\swarrow&\\Y&&\\ \end{array}$$ de ce diagramme, on définit une application $$\begin{array}{ccc}C_{c}^{\infty}(X)&\to&C_{c}^{\infty}(Y)\\ \psi_{X}&\mapsto&\psi_{Y}\\ \end{array}$$ par $$\psi_{Y}(y)=\int_{A_{\tilde{G}}(F)}\psi_{X}(zy_{X})\,dz,$$ où $y_{X}$ est une image réciproque de $y$ dans $X$. Pour chaque flèche $$\begin{array}{ccc}X&&\\&\searrow&\\&&Y\\ \end{array}$$ de ce diagramme, on définit une application $$\begin{array}{ccc}C_{c}^{\infty}(X)&\to&C_{c}^{\infty}(Y)\\ \psi_{X}&\mapsto&\psi_{Y}\\ \end{array}$$ par $\psi_{Y}(y)=\sum_{x}\psi_{X}(x)$, où $x$ parcourt l’image réciproque de $y$ dans $X$. Soit $\psi\in C_{c}^{\infty}(S^{\theta,0}(F))$. En appliquant ces constructions au côté gauche, du diagramme, on déduit de $\psi$ une fonction $\psi_{g}$ sur $\underline{S}(F)/(1-\theta)(\underline{S}(F))$. En appliquant ces constructions au côté droit du diagramme, on obtient d’abord une fonction $\phi$ sur $S(F)/(1-\theta)(S(F))$, puis une fonction $\psi_{d}$ sur $\underline{S}(F)/(1-\theta)(\underline{S}(F))$. Appliquons (4) à $\phi$. Par définition de la mesure sur $S(F)/(1-\theta)(S(F)) $, on a $$\int_{S(F)/(1-\theta)(S(F))}\phi(\gamma)\,d\gamma\,=\int_{S^{\theta,0}(F)}\psi(\gamma)\,d\gamma.$$ Par définition de la mesure sur $\underline{S}(F)/(1-\theta)(\underline{S}(F))$, c’est aussi $$\int_{\underline{ S}(F)/(1-\theta)(\underline{S}(F))}\psi_{g}(\gamma)\,d\gamma.$$ L’intégrale intérieure du membre de droite de (4) n’est autre que $\psi_{d}(\gamma)$ et ce membre de droite est égal à $$\int_{\underline{ S}(F)/(1-\theta)(\underline{S}(F))}\psi_{d}(\gamma)\,d\gamma.$$ Les fonctions $\psi_{d}$ et $\psi_{g}$ sont proportionnelles et les calculs ci-dessus montrent que la constante $C$ est celle pour laquelle $\psi_{g}=C\psi_{d}$. L’action de $A_{\tilde{G}}(F)$ sur $S^{\theta,0}(F)$ est libre, tandis que celle sur $S(F)/(1-\theta)(S(F))$ se quotiente en une action libre de $A_{\tilde{G}}(F)/(A_{\tilde{G}}(F)\cap (1-\theta)(S(F)))$. Il résulte alors des définitions que $C=\vert A_{\tilde{G}}(F)\cap (1-\theta)(S(F))\vert^{-1} $. Pour $x\in \underline{S}^{\theta}(F)$, soit $y\in S(F)$ relevant $x$, posons $z=(1-\theta)y$. L’application $x\mapsto z$ envoie $\underline{S}^{\theta}(F)$ dans $ A_{\tilde{G}}(F)\cap (1-\theta)(S(F))$. Parce que l’application $S^{\theta,0}(F)\to \underline{S}^{\theta,0}(F)$ est surjective, l’application précédente se quotiente en une application de $\underline{S}^{\theta}(F)/\underline{S}^{\theta,0}(F)$ dans $ A_{\tilde{G}}(F)\cap (1-\theta)(S(F))$. On obtient une suite d’applications $$1\to S^{\theta}(F)/S^{\theta,0}(F)\to \underline{S}^{\theta}(F)/\underline{S}^{\theta,0}(F)\to A_{\tilde{G}}(F)\cap (1-\theta)(S(F))\to 1.$$ On vérifie aisément que cette suite est exacte. Donc $$\vert A_{\tilde{G}}(F)\cap (1-\theta)(S(F))\vert=[\underline{S}^{\theta}(F):(A_{\tilde{G}}(F)\backslash S^{\theta}(F))]=[\underline{S}^{\theta}(F):\underline{S}^{\theta,0}(F)][ S^{\theta}(F):S^{\theta,0}(F)]^{-1} .$$ D’autre part, on a les égalités $$mes(A_{\underline{\tilde{M}}}(F)\backslash\underline{S}^{\theta}(F))=[\underline{S}^{\theta}(F):\underline{S}^{\theta,0}(F)]mes(A_{\underline{\tilde{M}}}(F)\backslash \underline{S}^{\theta,0}(F))$$ $$=
[\underline{S}^{\theta}(F):\underline{S}^{\theta,0}(F)]mes(A_{\tilde{M}}(F)\backslash S^{\theta,0}(F))$$ $$=[\underline{S}^{\theta}(F):\underline{S}^{\theta,0}(F)][ S^{\theta}(F):S^{\theta,0}(F)]^{-1} mes(A_{\tilde{M}}(F)\backslash S^{\theta}(F)).$$ De ces calculs résulte l’égalité $$C=mes(A_{\underline{\tilde{M}}}(F)\backslash\underline{S}^{\theta}(F))mes(A_{\tilde{M}}(F)\backslash S^{\theta}(F))^{-1}[\underline{S}^{\theta}(F):(A_{\tilde{G}}(F)\backslash S^{\theta}(F))]^{-2}$$ que l’on voulait prouver. Cela achève la preuve de (3) et l’identification du membre de gauche de (2) avec $I_{g\acute{e}om}^{\tilde{\underline{G}}}(\omega,f_{1},f_{2})$.
Montrons maintenant que le membre de droite de (2) est égal à $I_{spec}^{\tilde{\underline{G}}}(\omega,f_{1},f_{2})$. Ce membre de droite est égal à $$\sum_{\tau\in (E_{disc}(\tilde{G},\omega)/conj)/i{\cal A}_{\tilde{G},F}^*}\vert {\bf Stab}(W^G\times i{\cal A}_{\tilde{G},F}^*,\tau)\vert ^{-1}\iota(\tau)$$ $$\int_{A_{\tilde{G}}(F)}\int_{i{\cal A}_{\tilde{G},F}^*}\overline{I_{\tilde{G}}(\tilde{\pi}_{\boldsymbol{\tau}_{\tilde{\lambda}}},\varphi_{1})}I_{\tilde{G}}(\tilde{\pi}_{\boldsymbol{\tau}_{\tilde{\lambda}}},\varphi_{2}^{[z]})\,d\lambda\,\mu(z)\,dz.$$ On identifie $(E_{disc}(\tilde{G},\omega)/conj)/i{\cal A}_{\tilde{G},F}^*$ à un ensemble de représentants $\underline{E}$ dans $E_{disc}(\tilde{G},\omega)$. On a $$I_{\tilde{G}}(\tilde{\pi}_{\boldsymbol{\tau}_{\tilde{\lambda}}},\varphi_{2}^{[z]})=I_{\tilde{G}}(\tilde{\pi}_{\boldsymbol{\tau}_{\tilde{\lambda}}},\varphi_{2})\mu_{\tau}(z)^{-1}e^{-<\lambda,H_{\tilde{G}}(z)>}.$$ L’intégrale sur le plus grand sous-groupe compact de $A_{\tilde{G}}(F)$ sélectionne les $\tau\in \underline{E}$ tels que $\mu_{\tau}$ coïncide avec $\mu$ sur ce sous-groupe. Notons $\underline{E}'$ ce sous-ensemble. L’expression ci-dessus devient $$mes(A_{\tilde{G}}(F)_{c})\sum_{\tau\in \underline{E}'}\vert {\bf Stab}(W^G\times i{\cal A}_{\tilde{G},F}^*,\tau)\vert ^{-1}\iota(\tau)$$ $$\int_{{\cal A}_{A_{\tilde{G}},F}}\int_{i{\cal A}_{\tilde{G},F}^*}\overline{I_{\tilde{G}}(\tilde{\pi}_{\boldsymbol{\tau}_{\tilde{\lambda}}},\varphi_{1})}I_{\tilde{G}}(\tilde{\pi}_{\boldsymbol{\tau}_{\tilde{\lambda}}},\varphi_{2})\,d\lambda\,(\mu_{\tau}^{-1}\mu)(H)e^{-<\lambda,H>}\,dH,$$ où on a quotienté $\mu_{\tau}^{-1}\mu$ en un caractère de ${\cal A}_{A_{\tilde{G}},F}$. Par inversion de Fourier, on obtient le produit de $$(5)\qquad mes(A_{\tilde{G}}(F)_{c})mes(i{\cal A}_{\tilde{G},F}^*)[{\cal A}_{G,F}:{\cal A}_{A_{G},F}]^{-1}$$ et de $$\sum_{\tau\in \underline{E}'}\vert {\bf Stab}(W^G\times i{\cal A}_{\tilde{G},F}^*,\tau)\vert ^{-1}\iota(\tau)\sum_{\lambda\in \Lambda_{\mu}(\tau)}\overline{I_{\tilde{G}}(\tilde{\pi}_{\boldsymbol{\tau}_{\tilde{\lambda}}},\varphi_{1})}I_{\tilde{G}}(\tilde{\pi}_{\boldsymbol{\tau}_{\tilde{\lambda}}},\varphi_{2}),$$ où $\Lambda_{\mu}(\tau)$ est l’ensemble des $\lambda\in i{\cal A}_{\tilde{G},F}^*$ tels que $\mu_{\tau_{\lambda}}=\mu$, ou encore tels que $\tau_{\lambda}\in E_{disc,\mu}(\tilde{G},\omega)$. L’expression (5) vaut $1$ d’après les normalisations de 1.2. L’ensemble $\Lambda_{\mu}(\tau)$ est un espace principal homogène sous le groupe $i{\cal A}_{A_{\tilde{G}},F}^{\vee}/i{\cal A}_{\tilde{G},F}^{\vee}$. On vérifie que, pour $\lambda$ dans cet ensemble, on a l’égalité $$I_{\tilde{G}}(\tilde{\pi}_{\boldsymbol{\tau}_{\tilde{\lambda}}},\varphi_{i})=I_{\tilde{\underline{G}}}(\tilde{\pi}_{\boldsymbol{\tau}_{\tilde{\lambda}}},f_{i})$$ pour $i=1,2$. Notons $\underline{X}$ l’ensemble des couples $(\tau,\lambda)$, où $\tau\in \underline{E}'$ et $\lambda\in \Lambda_{\mu}(\tau)$. Considérons l’application de $ \underline{X}$ dans $ E_{disc,\mu}(\tilde{G},\omega)/conj$ qui, à $(\tau,\lambda)\in \underline{X}$, associe la classe de conjugaison de $\tau_{\lambda}$. On vérifie qu’elle est surjective. Notons $\underline{X}_{\tau}$ sa fibre au-dessus d’un élément $\tau\in E_{disc,\mu}(\tilde{G},\omega)/conj$. A ce point, on a transformé le membre de droite de (2) en $$(6) \qquad \sum_{\tau\in E_{disc,\mu}(\tilde{G},\omega)/conj}C(\tau)\overline{I_{\tilde{\underline{G}}}(\tilde{\pi}_{\boldsymbol{\tau}},f_{1})}I_{\tilde{\underline{G}}}(\tilde{\pi}_{\boldsymbol{\tau}},f_{1}),$$ où $$C(\tau)=\sum_{(\underline{\tau},\lambda)\in \underline{X}_{\tau}}\vert {\bf Stab}(W^G\times i{\cal A}_{\tilde{G},F}^*,\underline{\tau})\vert ^{-1}\iota(\underline{\tau}).$$ Fixons $\tau\in E_{disc,\mu}(\tilde{G},\omega)/conj$ que l’on relève en un élément de $E_{disc}(\tilde{G},\omega)$. L’ensemble $\underline{X}_{\tau}$ est celui des $(\underline{\tau},\lambda)$ tels que $\underline{\tau}\in \underline{E}'$, $\lambda\in \Lambda_{\mu}(\underline{\tau})$ et $\underline{\tau}_{\lambda}$ est conjugué à $\tau$. Cette relation entraîne que les images de $\tau$ et $\underline{\tau}$ dans $(E_{disc}(\tilde{G},\omega)/conj)/i{\cal A}_{\tilde{G},F}^*$ sont égales. Donc $\underline{\tau}$ est bien déterminé: c’est l’élément de $\underline{E}'$ qui représente l’image de $\tau$ dans $(E_{disc}(\tilde{G},\omega)/conj)/i{\cal A}_{\tilde{G},F}^*$. Cela simplifie $C(\tau)$ en $$C(\tau)=\vert \underline{X}_{\tau}\vert \vert {\bf Stab}(W^G\times i{\cal A}_{\tilde{G},F}^*,\tau)\vert ^{-1}\iota(\tau).$$ On peut fixer $\underline{\lambda}\in i{\cal A}_{\tilde{G},F}^*$ tel que $\underline{\tau}_{\underline{\lambda}}$ soit conjugué à $\tau$. Notons $\Lambda'(\tau)$ l’ensemble des $\lambda\in i{\cal A}_{\tilde{G},F}^*$ tels que $\tau_{\lambda}$ soit conjugué à $\tau$. Alors $\underline{X}_{\tau}$ est l’ensemble des $(\underline{\tau},\underline{\lambda}+\lambda)$ pour $\lambda\in \Lambda'(\tau)$. D’où $\vert \underline{X}_{\tau}\vert=\vert \Lambda'(\tau)\vert $. Remarquons que $\tau$ et $\tau_{\lambda}$ sont conjugués par $G(F)$ si et seulement s’ils le sont par $W^G$. On a défini le groupe $Stab(W^G\times i{\cal A}_{\tilde{G},F}^*,\tau)$ en 2.9, formé des $(w,\lambda)\in Stab(W^G\times i{\cal A}_{\tilde{G},F}^*,\tau)$ tels que $w\tau=\tau_{\lambda}$. L’application qui, à $(w,\lambda)$, associe $\lambda$ fournit une suite exacte $$1\to Stab(W^G,\tau)\to Stab(W^G\times i{\cal A}_{\tilde{G},F}^*,\tau)\to \Lambda'(\tau)\to 1.$$ On en déduit aisément $$\vert \Lambda'(\tau)\vert =\vert {\bf Stab}(W^G\times i{\cal A}_{\tilde{G},F}^*,\tau)\vert \vert {\bf Stab}(W^G,\tau)\vert ^{-1}.$$ D’où $$C(\tau)=\vert {\bf Stab}(W^G,\tau)\vert ^{-1}\iota(\tau).$$ Alors (6) devient $I_{disc}^{\tilde{\underline{G}}}(\omega,f_{1},f_{2})$. $\square$
Conséquences
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Fonctions cuspidales
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Rappelons que l’on note $\tilde{G}(F)_{ell}$ l’ensemble des éléments semi-simples réguliers et elliptiques de $\tilde{G}(F)$. Une fonction $f\in C_{c}^{\infty}(\tilde{G}(F),K)$ est dite cuspidale si et seulement si $I_{\tilde{G}}(\gamma,\omega,f)=0$ pour tout $\gamma\in \tilde{G}_{reg}(F)$ tel que $\gamma\not\in \tilde{G}(F)_{ell}$. Cela équivaut à ce que, pour tout espace parabolique $\tilde{P}=\tilde{M}U_{P}\in {\cal F}(\tilde{M}_{0})$, avec $\tilde{P}\not=\tilde{G}$, l’image de $f_{\tilde{P}}$ dans $I(\tilde{M}(F),K^M,\omega)$ soit nulle (en convenant que cet espace lui-même est nul si $\omega$ n’est pas trivial sur $Z_{M}(F)^{\theta}$). Grâce au théorème 5.5, cela équivaut aussi à $I_{\tilde{G}}(Ind_{\tilde{P}}^{\tilde{G}}(\tilde{\pi}),f)=0$ pour tout $\tilde{P}$ comme ci-dessus et toute $\omega$-représentation $\tilde{\pi}$ de $\tilde{M}(F)$, tempérée et de longueur finie. On note $I_{cusp}(\tilde{G}(F),K,\omega)$ l’image dans $I(\tilde{G}(F),K,\omega)$ de l’espace des fonctions cuspidales. D’après le théorème 6.2, de l’application $pw_{\tilde{G}}$ se déduit un isomorphisme $I_{cusp}(\tilde{G}(F),K,\omega)\simeq PW_{ell}(\tilde{G},\omega)$.
Dans le cas où $f_{2}$ est cuspidale, le théorème 6.6 se simplifie. Pour $\tilde{M}\in {\cal L}(\tilde{M}_{0})$, $\tilde{S}\in T_{ell}(\tilde{M},\omega)$ et $\gamma\in \tilde{S}(F)\cap \tilde{G}_{reg}(F)$, on a simplement $$I_{\tilde{M}}^{\tilde{G}}(\gamma,\omega,f_{1},f_{2})=\overline{I_{\tilde{G}}(\gamma,\omega,f_{1})}I_{\tilde{M}}^{\tilde{G}}(\gamma,\omega,f_{2}).$$ En effet, dans la somme définissant le membre de gauche (cf. 6.6), les termes pour $\tilde{L}_{2}\not=\tilde{G}$ sont nuls parce que $f_{2,\tilde{L}_{2}}=0$. Si $\tilde{L}_{2}=\tilde{G}$, on a $\tilde{L}_{1}=\tilde{M}$ et $I_{\tilde{M}}^{\tilde{M}}(\gamma,\omega,f_{1,\tilde{M}})=I_{\tilde{G}}(\gamma,\omega,f_{1})$. D’autre part, on a $$I_{disc}^{\tilde{G}}(\omega,f_{1},f_{2})=\sum_{\tau\in (E_{ell}(\tilde{G},\omega)/conj)/i{\cal A}_{\tilde{G},F}^*}\vert {\bf Stab}(W^G\times i{\cal A}_{\tilde{G},F}^*,\tau)\vert ^{-1}\iota(\tau)$$ $$\int_{i{\cal A}_{\tilde{G},F}^*}\overline{I_{\tilde{G}}(\tilde{\pi}_{\boldsymbol{\tau}_{\tilde{\lambda}}},f_{1})}I_{\tilde{G}}(\tilde{\pi}_{\boldsymbol{\tau}_{\tilde{\lambda}}},f_{2})\,d\lambda.$$ En effet, si $\tau\in E_{disc}(\tilde{G}(F))-E_{ell}(\tilde{G}(F))$, une représentation $\tilde{\pi}_{\boldsymbol{\tau}_{\tilde{\lambda}}}$ est induite à partir d’un espace parabolique propre (cf. lemmes 2.10 et 2.11), donc son caractère annule $f_{2}$. Remarquons que, pour un triplet $\tau=(M_{disc},\sigma,\tilde{r})\in E_{ell}(\tilde{G}(F))$, on a simplement $$\iota(\tau)=\vert det((1-\tilde{r})_{\vert {\cal A}_{M}^{\tilde{G}}})\vert ^{-1},$$ puisque $W_{0}^G(\sigma)=\{1\}$.
Rappelons que les caractères de $\omega$-représentations de longueur finie de $\tilde{G}(F)$ sont des distributions localement intégrables. Pour une telle $\omega$-représentation $\tilde{\pi}$, notons $\gamma\mapsto \Theta(\tilde{\pi},\gamma)$ la fonction sur $\tilde{G}(F)$ telle que $$I_{\tilde{G}}(\tilde{\pi},f)=\int_{\tilde{G}(F)}\Theta(\tilde{\pi},\gamma)f(\gamma)\,d\gamma.$$ On suppose, ainsi qu’il est loisible, que cette fonction est lisse sur $\tilde{G}_{reg}(F)$.
0.3cm[**[Théorème]{}**]{}.
**
Soient $f\in C_{c}^{\infty}(\tilde{G}(F),K)$, $\tilde{M}\in {\cal L}(\tilde{M}_{0})$ et $\gamma\in \tilde{M}(F)\cap \tilde{G}_{reg}(F)$. On suppose $f$ cuspidale. Alors
\(i) si $\gamma\not\in \tilde{M}(F)_{ell}$, $I_{\tilde{M}}^{\tilde{G}}(\gamma,\omega,f)=0$;
\(ii) si $\gamma\in \tilde{M}(F)_{ell}$, $$mes(A_{\tilde{M}}(F)\backslash Z_{G}(\gamma,F))I_{\tilde{M}}^{\tilde{G}}(\gamma,\omega,f)=(-1)^{a_{\tilde{M}}-a_{\tilde{G}}}D^{\tilde{G}}(\gamma)^{1/2}$$ $$\sum_{\tau\in (E_{ell}(\tilde{G},\omega)/conj)/i{\cal A}_{\tilde{G},F}^*}\vert {\bf Stab}(W^G\times i{\cal A}_{\tilde{G},F}^*,\tau)\vert ^{-1}\iota(\tau)
\int_{i{\cal A}_{\tilde{G},F}^*}\overline{ \Theta(\tilde{\pi}_{\boldsymbol{\tau}_{\tilde{\lambda}}},\gamma)}I_{\tilde{G}}(\tilde{\pi}_{\boldsymbol{\tau}_{\tilde{\lambda}}},f)\,d\lambda.$$
0.3cm
Preuve. Si $\gamma\not\in \tilde{M}(F)_{ell}$, on peut conjuguer $\gamma$ de sorte que $\gamma\in \tilde{L}(F)$, où $\tilde{L}\in {\cal L}(\tilde{M}_{0})$ et $\tilde{L}\subsetneq \tilde{M}$. La formule de descente 6.5(2) (où on échange les rôles de $\tilde{M}$ et $\tilde{L}$) et la cuspidalité de $f$ entraînent la conclusion de (i).
Définissons une fonction $\varphi$ sur $\tilde{G}_{reg}(F)$ de la façon suivante. Soit $\gamma\in \tilde{G}_{reg}(F)$. Si $\omega$ est non trivial sur $Z_{G}(\gamma,F)$, on pose $\varphi(\gamma)=0$. Sinon, choisissons $g\in G(F)$ et $\tilde{M}\in {\cal L}(\tilde{M}_{0})$ de sorte que $g\gamma g^{-1}\in \tilde{M}(F)_{ell}$. Posons $$\varphi(\gamma)=(-1)^{a_{\tilde{M}}-a_{\tilde{G}}}mes(A_{\tilde{M}}(F)\backslash Z_{G}(g\gamma g^{-1},F))D^{\tilde{G}}(\gamma)^{1/2}\omega(g)^{-1}I_{\tilde{M}}^{\tilde{G}}(g\gamma g^{-1},\omega,f).$$ Cette définition est loisible d’après 6.5(1). Soit $f'\in C_{c}^{\infty}(\tilde{G}(F),K)$. Comme on l’a dit ci-dessus, le terme $I^{\tilde{G}}_{g\acute{e}om}(\omega,f',f)$ se simplifie puisque $f$ est cuspidale. Grâce à la formule de Weyl (cf. 4.1), on a simplement $$I_{g\acute{e}om}^{\tilde{G}}(\omega,f',f)=\int_{\tilde{G}(F)}\overline{f'(\gamma)}\varphi(\gamma)\,d\gamma.$$ D’autre part $$I_{spec}^{\tilde{G}}(\omega,f',f)=\sum_{\tau\in (E_{ell}(\tilde{G},\omega)/conj)/i{\cal A}_{\tilde{G},F}^*}\vert {\bf Stab}(W^G\times i{\cal A}_{\tilde{G},F}^*,\tau)\vert ^{-1}\iota(\tau)$$ $$\int_{i{\cal A}_{\tilde{G},F}^*}I_{\tilde{G}}(\tilde{\pi}_{\boldsymbol{\tau}_{\tilde{\lambda}}},f)\int_{\tilde{G}(F)}\overline{\Theta(\tilde{\pi}_{\boldsymbol{\tau}_{\tilde{\lambda}}},\gamma)f'(\gamma)}\,d\gamma \,d\lambda.$$ Cette formule est absolument convergente, d’où $$I_{spec}^{\tilde{G}}(\omega,f',f)=\int_{\tilde{G}(F)}\overline{f'(\gamma)}\varphi'(\gamma)\,d\gamma,$$ où $$\varphi'(\gamma)=\sum_{\tau\in (E_{ell}(\tilde{G},\omega)/conj)/i{\cal A}_{\tilde{G},F}^*}\vert {\bf Stab}(W^G\times i{\cal A}_{\tilde{G},F}^*,\tau)\vert ^{-1}\iota(\tau)\int_{i{\cal A}_{\tilde{G},F}^*}\overline{\Theta(\tilde{\pi}_{\boldsymbol{\tau}_{\tilde{\lambda}}},\gamma)}I_{\tilde{G}}(\tilde{\pi}_{\boldsymbol{\tau}_{\tilde{\lambda}}},f)\,d\lambda.$$ Le théorème 6.6 entraîne l’égalité $$\int_{\tilde{G}(F)}\overline{f'(\gamma)}\varphi(\gamma)\,d\gamma=\int_{\tilde{G}(F)}\overline{f'(\gamma)}\varphi'(\gamma)\,d\gamma.$$ Si $F$ est non-archimédien, $f'$ est n’importe quel élément de $C_{c}^{\infty}(\tilde{G}(F))$ et cette égalité entraîne $\varphi(\gamma)=\varphi'(\gamma)$ pour tout $\gamma$, ce qui est l’assertion (2) du théorème. Si $F$ est archimédien, il y a un petit problème car $f'$ est supposée $K$-finie. Quelques calculs similaires à ceux du paragraphe 5.2 montrent que l’égalité ci-dessus se prolonge continûment à $f'\in C_{c}^{\infty}(\tilde{G}(F))$ tout entier. La conclusion est alors la même que dans le cas non-archimédien. $\square$
[**Preuve de la proposition 6.5**]{} Soient $\tilde{M}\in {\cal L}(\tilde{M}_{0})$, $\gamma\in \tilde{M}(F)\cap \tilde{G}_{reg}(F)$ et $f\in {\cal H}_{ac}(\tilde{G}(F))$ dont l’image dans $I_{ac}(\tilde{G}(F),\omega)$ soit nulle. On veut montrer que $I_{\tilde{M}}^{\tilde{G}}(\gamma,\omega,f)=0$. La relation 6.5(3) nous ramène au cas où $f\in C_{c}^{\infty}(\tilde{G}(F),K)$. Son image dans $I(\tilde{G}(F),K,\omega)$ est nulle, a fortiori $f$ est cuspidale. Si $\gamma$ n’est pas elliptique dans $\tilde{M}(F)$, l’assertion cherchée résulte du (i) du théorème ci-dessus. Si $\gamma$ est elliptique dans $\tilde{M}(F)$, elle résulte du (ii) puisque $I_{\tilde{G}}(\tilde{\pi}_{\boldsymbol{\tau}_{\tilde{\lambda}}},f)=0$ pour tous $\tau$ et $\tilde{\lambda}$. $\square$
Fonctions cuspidales, variante avec caractère central
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On introduit le groupe $\underline{G}$ de 6.7 et les notations afférentes. Soit $\mu$ un caractère unitaire de $A_{\tilde{G}}(F)$. On définit comme au paragraphe précédent la notion de cuspidalité pour une fonction $f\in C_{\mu}^{\infty}(\tilde{G}(F),K)$. On note $C_{\mu,cusp}^{\infty}(\tilde{G}(F),K)$ le sous-espace des fonctions cuspidales. On note $I_{\mu,cusp}(\tilde{G}(F),K,\omega)$ le quotient de $C_{\mu,cusp}^{\infty}(\tilde{G}(F),K)$ par le sous-espace des fonctions dont toutes les intégrales orbitales régulières sont nulles. Une variante avec caractère central du théorème 6.2 conduit à l’assertion suivante. On définit de façon évidente l’ensemble $E_{ell,\mu}(\tilde{G},\omega)$. Pour tout $\tau\in E_{ell,\mu}(\tilde{G},\omega)/conj$, fixons un élément $\boldsymbol{\tau}\in {\cal E}(\tilde{G},\omega)$ qui relève $\tau$. Considérons l’application $$\begin{array}{ccc}C_{\mu,cusp}^{\infty}(\tilde{G}(F),K)&\to&\sum_{\tau\in E_{ell,\mu}(\tilde{G},\omega)/conj}{\mathbb C}\\f&\mapsto& \oplus_{\tau\in E_{ell,\mu}(\tilde{G},\omega)/conj}I_{\tilde{\underline{G}}}(\tilde{\pi}_{\boldsymbol{\tau}},f).\\ \end{array}$$ Alors
\(1) cette application se quotiente en un isomorphisme $$I_{\mu,cusp}(\tilde{G},K,\omega)\simeq \oplus_{\tau\in E_{ell,\mu}(\tilde{G},\omega)/conj}{\mathbb C}.$$
Autrement dit, pour tout $\tau\in E_{ell,\mu}(\tilde{G},\omega)/conj$, il existe un unique pseudo-coefficient $f_{\tau}\in I_{\mu,cusp}(\tilde{G}(F),K,\omega)$ tel que, pour $\tau'\in E_{ell,\mu}(\tilde{G},\omega)/conj$, on ait $$I_{\tilde{\underline{G}}}(\tilde{\pi}_{\boldsymbol{\tau}'},f_{\tau})=\left\lbrace\begin{array}{cc}1,& \text{ si }\tau'=\tau,\\0,&\text{ si }\tau'\not=\tau.\\ \end{array}\right.$$ Et la famille $(f_{\tau})_{\tau\in E_{ell,\mu}(\tilde{G},\omega)/conj}$ est une base de $I_{\mu,cusp}(\tilde{G}(F),K)$.
[0.3cm[**[Théorème]{}**]{}. [ *[Soient $\tau\in E_{ell,\mu}(\tilde{G},\omega)/conj$, $\tilde{M}\in {\cal L}(\tilde{M}_{0})$ et $\gamma\in \tilde{M}(F)_{ell}\cap \tilde{G}_{reg}(F)$. On a l’égalité $$mes(A_{\underline{\tilde{M}}}(F)\backslash Z_{\underline{G}}(\gamma,F))I_{\tilde{\underline{M}}}^{\tilde{\underline{G}}}(\gamma,\omega,f_{\tau})=(-1)^{a_{\tilde{M}}-a_{\tilde{G}}}D^{\tilde{G}}(\gamma)^{1/2} \vert {\bf Stab}(W^G,\tau)\vert ^{-1}\iota(\tau)\overline{ \Theta(\tilde{\pi}_{\boldsymbol{\tau}},\gamma)} .$$]{}*]{}0.3cm]{}
C’est la version avec caractère central du (ii) du théorème précédent, appliqué au pseudo-coefficient $f_{\tau}$. Remarquons que, pour $\tau=(M_{disc},\sigma,\tilde{r})\in E_{ell,\mu}(\tilde{G},\omega)$, ${\bf Stab}(W^G,\tau)$ n’est autre que le commutant de $\tilde{r}$ dans $R^G(\sigma)$.
Produit scalaire elliptique
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Notons $\underline{\tilde{G}}(F)_{ell}/conj$ l’ensemble des classes de conjugaison par $G(F)$ dans $\underline{\tilde{G}}(F)_{ell} =\tilde{G}(F)_{ell}/A_{\tilde{G}}(F)$. On munit cet ensemble d’une structure de variété analytique sur $F$ de sorte que, pour tout $\gamma\in \underline{\tilde{G}}(F)_{ell}$, l’application qui, à $x\in \underline{G}_{\gamma}(F)$, associe la classe de conjugaison de $x\gamma$, se quotiente en un isomorphisme local de $\underline{G}_{\gamma}(F)$ sur $\underline{\tilde{G}}(F)_{ell}/conj$ au voisinage de $x=1$. De même, on munit $\underline{\tilde{G}}(F)_{ell}/conj$ de la mesure pour laquelle ces applications préservent la mesure au voisinage de $x=1$. On définit une fonction $m$ sur $\underline{\tilde{G}}(F)_{ell}/conj$ de la façon suivante. Pour $\gamma\in \underline{\tilde{G}}(F)$, soit $\underline{\tilde{S}}$ l’unique tore tordu maximal de $\underline{\tilde{G}}$ contenant $\gamma$. Alors $$m(\gamma)=mes(\underline{S}^{\theta}(F)).$$ Pour une fonction $\varphi\in C_{c}^{\infty}(\underline{G}(F)_{ell}/conj)$, l’intégrale $$\int_{\underline{G}(F)_{ell}/conj}m(\gamma)^{-1}\varphi(\gamma)\,d\gamma$$ ne dépend d’aucune mesure. Elle est égale à $$\sum_{S\in T_{ell}(\tilde{G})}\vert W^G(\tilde{S})\vert ^{-1}mes(\underline{S}^{\theta}(F))^{-1}\int_{\underline{\tilde{S}}(F)/(1-\theta)(\underline{S}(F))}\varphi(\gamma)\,d\gamma.$$
Soient $\mu$ un caractère unitaire de $A_{\tilde{G}}(F)$. Soient $\tilde{\pi}_{1}$ et $\tilde{\pi}_{2}$ deux $\omega$-représentations de $\tilde{G}(F)$, de longueur finie et de $A$-caractère central $\mu$. La fonction $\gamma\mapsto \overline{\Theta(\tilde{\pi}_{1},\gamma)}\Theta(\tilde{\pi}_{2},\gamma)$ sur $\tilde{G}(F)_{ell}$ se quotiente en une fonction sur $\underline{\tilde{G}}(F)_{ell}/conj$. On pose $$(\tilde{\pi}_{1},\tilde{\pi}_{2})_{ell}=\int_{\underline{\tilde{G}}(F)_{ell}/conj} m(\gamma)^{-1}D^{\tilde{G}}(\gamma)\overline{\Theta(\tilde{\pi}_{1},\gamma)}\Theta(\tilde{\pi}_{2},\gamma)\,d\gamma.$$ Cette expression ne dépend d’aucune mesure.
Rappelons que, pour tout $\tau\in E_{ell,\mu}(\tilde{G},\omega)$, on a choisi un relèvement $\boldsymbol{\tau}$ de $\tau$ dans ${\cal E}_{ell}(\tilde{G},\omega)$.
0.3cm[**[Théorème]{}**]{}.
**
\(i) Soient $\tau_{1},\tau_{2}\in {\cal E}_{ell,\mu}(\tilde{G},\omega)$. Supposons qu’ils ne sont pas conjugués par $G(F)$. Alors on a l’égalité $(\tilde{\pi}_{\boldsymbol{\tau}_{1}},\tilde{\pi}_{\boldsymbol{\tau}_{2}})_{ell}=0$.
\(ii) Soit $\tau\in {\cal E}_{ell,\mu}(\tilde{G},\omega)$. On a l’égalité $$(\tilde{\pi}_{\boldsymbol{\tau}},\tilde{\pi}_{\boldsymbol{\tau}})_{ell}=\vert {\bf Stab}(W^G,\tau)\vert \iota(\tau)^{-1}.$$
0.3cm
Preuve. En appliquant le théorème 7.2, on voit que $$(\tilde{\pi}_{\boldsymbol{\tau}_{1}},\tilde{\pi}_{\boldsymbol{\tau}_{2}})_{ell}=\vert {\bf Stab}(W^G,\tau_{1})\vert \vert {\bf Stab}(W^G,\tau_{2})\vert \iota(\tau_{1})^{-1}\iota(\tau_{2})^{-1} X,$$ où $$X=\int_{\underline{\tilde{G}}(F)_{ell}/conj} m(\gamma)\overline{I_{\tilde{\underline{G}}}(\gamma,\omega,f_{\tau_{1}})}I_{\tilde{\underline{G}}}(\gamma,\omega,f_{\tau_{2}})\,d\gamma.$$ Il résulte des définitions que $X=I_{\tilde{\underline{G}},g\acute{e}om}^{\tilde{\underline{G}}}(\omega,f_{\tau_{1}},f_{\tau_{2}})$. Par le même argument qu’en 7.1 et parce que les fonctions $f_{\tau_{1}}$ et $f_{\tau_{2}}$ sont toutes deux cuspidales, les termes $I_{\tilde{\underline{M}},g\acute{e}om}^{\tilde{\underline{G}}}(\omega,f_{\tau_{1}},f_{\tau_{2}})$ sont nuls pour $\tilde{M}\not=\tilde{G}$. Donc $X=I^{\tilde{\underline{G}}}_{g\acute{e}om}(\omega,f_{\tau_{1}},f_{\tau_{2}})$. Appliquons le théorème 6.7: $X=I^{\tilde{\underline{G}}}_{disc}(\omega,f_{\tau_{1}},f_{\tau_{2}})$. Mais cette expression se calcule grâce à la définition des pseudo-coefficients. On obtient $I^{\tilde{\underline{G}}}_{disc}(\omega,f_{\tau_{1}},f_{\tau_{2}})=0$ si $\tau_{1}$ et $\tau_{2}$ ne sont pas conjugués. Si $\tau_{1}=\tau_{2}=\tau$, on obtient $$I^{\tilde{\underline{G}}}_{disc}(\omega,f_{\tau},f_{\tau})=\vert {\bf Stab}(W^G,\tau)\vert^{-1} \iota(\tau).$$ Le théorème en résulte. $\square$
Produit elliptique pour les $\omega$-représentations irréductibles
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Soit $\mu$ un caractère unitaire de $A_{\tilde{G}}(F)$. Pour $i=1,2$, soient $M_{i}\in {\cal P}(M_{0})$ et $\sigma_{i}$ une représentation irréductible et de la série discrète de $M_{i}(F)$. On suppose que la restriction à $A_{\tilde{G}}(F)$ du caractère central de $\sigma_{i}$ est égale à $\mu$. On suppose ${\cal N}^{\tilde{G}}(\sigma_{i})$ non vide. Soit $\tilde{\rho}_{i}$ une représentation ${\cal R}^G(\sigma_{i})$-irréductible de ${\cal R}^{\tilde{G}}(\sigma_{i})$. Définissons un terme $(\tilde{\rho}_{1},\tilde{\rho}_{2})_{ell}$ de la façon suivante. Supposons d’abord que les couples $(M_{1},\sigma_{1})$ et $(M_{2},\sigma_{2})$ soient égaux. On les note simplement $(M,\sigma)$. Supposons de plus $W^G_{0}(\sigma)=\{1\}$. La fonction $\boldsymbol{\tilde{r}}\mapsto \overline{trace(\tilde{\rho}_{1}(\boldsymbol{\tilde{r}}))}trace(\tilde{\rho}_{2}(\boldsymbol{\tilde{r}}))$ sur ${\cal R}^{\tilde{G}}(\sigma)$ se descend en une fonction sur $R^{\tilde{G}}(\sigma)$. On pose $$(\tilde{\rho}_{1},\tilde{\rho}_{2})_{ell}=\vert R^G(\sigma)\vert ^{-1}\sum_{\tilde{r}\in R^{\tilde{G}}(\sigma)\cap W^{\tilde{G}}_{reg}(\sigma)}\vert det((1-\tilde{r})_{\vert {\cal A}_{M}^{\tilde{G}}})\vert trace(\tilde{\rho}_{1}(\tilde{r}))\overline{trace(\tilde{\rho}_{2}(\tilde{r}))}.$$ Supposons maintenant que $W^G_{0}(\sigma_{i})=\{1\}$ pour $i=1,2$ et que les couples $(M_{1},\sigma_{1})$ et $(M_{2},\sigma_{2})$ sont conjugués par un élément de $G(F)$. En effectuant une telle conjugaison, on remplace $(M_{2},\sigma_{2},\tilde{\rho}_{2})$ par $(M_{1},\sigma_{1},\tilde{\rho}'_{2})$. On pose $$(\tilde{\rho}_{1},\tilde{\rho}_{2})_{ell}=(\tilde{\rho}_{1},\tilde{\rho}'_{2})_{ell}.$$ Cela ne dépend pas de la conjugaison choisie. Dans les cas restants, c’est-à-dire si $W^G_{0}(\sigma_{1})$ ou $W^G_{0}(\sigma_{2})$ est non trivial, ou si $(M_{1},\sigma_{1})$ et $(M_{2},\sigma_{2})$ ne sont pas conjugués, on pose $(\tilde{\rho}_{1},\tilde{\rho}_{2})_{ell}=0$.
Pour $i=1,2$, on a associé en 2.8 à $(M_{i},\sigma_{i},\tilde{\rho}_{i})$ une représentation $G$-irréductible $\tilde{\pi}_{\tilde{\rho}_{i}}$ de $\tilde{G}(F)$.
[0.3cm[**[Corollaire]{}**]{}. [ *[On a l’égalité $(\tilde{\pi}_{\tilde{\rho}_{1}},\tilde{\pi}_{\tilde{\rho}_{2}})_{ell}=(\tilde{\rho}_{1},\tilde{\rho}_{2})_{ell}$.]{}*]{}0.3cm]{}
Preuve. Pour $i=1,2$ et pour $\tilde{r}\in R^{\tilde{G}}(\sigma_{i})$, posons $\tau_{i}(\tilde{r})=(M_{i},\sigma_{i},\tilde{r})$. Notons $R_{ess}^{\tilde{G}}(\sigma_{i})$ l’ensemble des $\tilde{r}\in R^{\tilde{G}}(\sigma_{i})$ tels que $\tau_{i}(\tilde{r})$ soit essentiel. Pour $\tilde{r}\in R_{ess}^{\tilde{G}}(\sigma_{i})$, on fixe un relèvement $\boldsymbol{\tau}_{i}(\tilde{r})=(M_{i},\sigma_{i},\boldsymbol{\tilde{r}})$ de $\tau_{i}$ dans ${\cal E}(\tilde{G},\omega)$. On a vu dans la preuve de la proposition 2.9 une formule d’inversion exprimant $\Theta(\tilde{\pi}_{\tilde{\rho}_{i}})$ en fonction des $\Theta(\tilde{\pi}_{\boldsymbol{\tau}_{i}(\tilde{r})})$ pour $\tilde{r}\in R^{\tilde{G}}_{ess}(\sigma_{i})$. Dans cette formule, on sommait sur les classes de conjugaison dans cet ensemble. On peut la récrire comme une somme sur tous les éléments de cet ensemble sous la forme $$\Theta(\tilde{\pi}_{\tilde{\rho}_{i}})=\vert R^G(\sigma_{i})\vert ^{-1}\sum_{\tilde{r}\in R^{\tilde{G}}_{ess}(\sigma_{i})} \overline{trace(\tilde{\rho}_{i}(\boldsymbol{\tilde{r}}))}\Theta(\tilde{\pi}_{\boldsymbol{\tau}_{i}(\tilde{r})}).$$ Si $W_{0}^G(\sigma_{i})\not=\{1\}$, il résulte des lemmes 2.10 et 2.11 que toutes les représentations $\tilde{\pi}_{\boldsymbol{\tau}_{i}(\tilde{r})}$ sont des induites à partir d’espaces paraboliques propres. Leur caractère est donc nul sur $\tilde{G}(F)_{ell}$. Il en résulte que, si $W^G_{0}(\sigma_{1})$ ou $W^G_{0}(\sigma_{2})$ est non trivial, $(\tilde{\pi}_{\tilde{\rho}_{1}},\tilde{\pi}_{\tilde{\rho}_{2}})_{ell}=0$. D’où l’égalité de l’énoncé dans ce cas. On suppose maintenant $W^G_{0}(\sigma_{1})=\{1\}$ et $W^G_{0}(\sigma_{2})=\{1\}$. On a alors $R^{\tilde{G}}(\sigma_{i})=W^{\tilde{G}}(\sigma_{i})$ pour $i=1,2$. Toujours d’après les lemmes 2.10 et 2.11, les représentations $\tilde{\pi}_{\boldsymbol{\tau}_{i}(\tilde{r})}$ pour $\tilde{r}\not\in W^{\tilde{G}}_{reg}(\sigma_{i})$ ont un caractère nul sur $\tilde{G}(F)_{ell}$. On obtient l’égalité $$(\tilde{\pi}_{\tilde{\rho}_{1}},\tilde{\pi}_{\tilde{\rho}_{2}})_{ell}=\vert R^G(\sigma_{1})\vert ^{-1}\vert R^G(\sigma_{2})\vert ^{-1}\sum_{\tilde{r}_{1}\in R^{\tilde{G}}_{ess}(\sigma_{1})\cap W^{\tilde{G}}_{reg}(\sigma_{1})}\sum_{\tilde{r}_{2}\in R^{\tilde{G}}_{ess}(\sigma_{2})\cap W^{\tilde{G}}_{reg}(\sigma_{2})}$$ $$trace(\tilde{\rho}_{1}(\boldsymbol{\tilde{r}_{1}}))\overline{trace(\tilde{\rho}_{2}(\boldsymbol{\tilde{r}_{2}}))}(\tilde{\pi}_{\boldsymbol{\tau}_{1}(\tilde{r}_{1})},\tilde{\pi}_{\boldsymbol{\tau}_{2}(\tilde{r}_{2})})_{ell}.$$ Si $(M_{1},\sigma_{1})$ et $(M_{2},\sigma_{2})$ ne sont pas conjugués, le (i) du théorème 7.3 dit que le membre de droite ci-dessus est nul, d’où encore l’égalité de l’énoncé. Si les deux couples sont conjugués, on peut les supposer égaux, on les note simplement $(M,\sigma)$. Le (i) du théorème 7.3 dit que seuls les couples $(\tilde{r}_{1},\tilde{r}_{2})$ formés d’éléments conjugués contribuent à la formule ci-dessus. Pour un couple d’éléments conjugués, les valeurs des différents termes intervenant sont les mêmes pour $\tilde{r}_{1}$ et $\tilde{r}_{2}$. On a déjà dit que, pour un triplet elliptique $\tau=(M,\sigma,\tilde{r})$, on a l’égalité ${\bf Stab}(W^G,\tau)=Stab(R^G(\sigma),\tilde{r})$, ce dernier groupe étant le commutant de $\tilde{r}$ dans $R^G(\sigma)$. On a aussi $\iota(\tau)^{-1}=\vert det((1-\tilde{r})_{\vert {\cal A}_{M}^{\tilde{G}}}\vert $. Grâce au (ii) du théorème 7.3, la contribution d’un couple $(\tilde{r}_{1},\tilde{r}_{2})$ d’éléments conjugués à la formule ci-dessus est donc $$\vert Stab(R^G(\sigma),\tilde{r}_{1})\vert \vert det((1-\tilde{r}_{1})_{\vert {\cal A}_{M}^{\tilde{G}}}\vert trace(\tilde{\rho}_{1}(\boldsymbol{\tilde{r}_{1}}))\overline{trace(\tilde{\rho}_{2}(\boldsymbol{\tilde{r}_{1}}))}.$$ Pour $\tilde{r}_{1}$ fixé, le nombre de $\tilde{r}_{2}$ conjugués à $\tilde{r}_{1}$ est $\vert R^G(\sigma)\vert \vert Stab(R^G(\sigma),\tilde{r}_{1})\vert^{-1}$. Notre formule devient $$(\tilde{\pi}_{\tilde{\rho}_{1}},\tilde{\pi}_{\tilde{\rho}_{2}})_{ell}=\vert R^G(\sigma)\vert ^{-1}\sum_{\tilde{r}\in R^{\tilde{G}}_{ess}(\sigma)\cap W^{\tilde{G}}_{reg}(\sigma)}\vert det((1-\tilde{r})_{\vert {\cal A}_{M}^{\tilde{G}}}\vert trace(\tilde{\rho}_{1}(\boldsymbol{\tilde{r}}))\overline{trace(\tilde{\rho}_{2}(\boldsymbol{\tilde{r}}))}.$$ C’est presque la définition de $(\tilde{\rho}_{1},\tilde{\rho}_{2})_{ell}$, à ceci près qu’ici, la somme est limitée à $\tilde{r}\in R^{\tilde{G}}_{ess}(\sigma)\cap W^{\tilde{G}}_{reg}(\sigma)$. Mais, si $\tilde{r}\not\in R^{\tilde{G}}_{ess}(\sigma)$, les termes $ trace(\tilde{\rho}_{1}(\boldsymbol{\tilde{r}}))$ et $ trace(\tilde{\rho}_{2}(\boldsymbol{\tilde{r}}))$ sont nuls. On peut donc remplacer la somme sur $R^{\tilde{G}}_{ess}(\sigma)\cap W^{\tilde{G}}_{reg}(\sigma)$ par celle sur tout $W^{\tilde{G}}_{reg}(\sigma)$. Cela achève la preuve. $\square$
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\[BT\] F. Bruhat, J. Tits: [*Groupes réductifs sur un corps local I. Données radicielles valuées*]{}, Publ. Math. IHES 41 (1972), p.5-251
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\[C\] L. Clozel: [*Characters of non-connected, reductive $p$-adic groups*]{} p.149-
\[DM\] P. Delorme, P. Mezo: [*A twisted invariant Paley-Wiener theorem for real reductive groups*]{}, Duke Math. J. 144 (2008), p.341-380
\[HC\] Harish-Chandra: [*Spherical functions on a semi-simple Lie group I*]{}, Amer. J. of Math. 80 (1958), p.241-316
\[HL\] G. Henniart, B. Lemaire: [*La transformée de Fourier tordue pour les groupes réductifs $p$-adiques*]{}, en préparation
\[LW\] J.-P. Labesse, J.-L. Waldspurger: [*La formule des traces tordue d’après le Friday morning seminar*]{}, prépublication 2012, Arxiv RT 12042888
\[L\] B. Lemaire: [*Caractères tordus des représentations admissibles*]{}, prépublication 2010.
\[R\] J. Rogawski: [*The trace Paley-Wiener theorem in the twisted case*]{}, Trans. Amer. Math. Soc. 309 (1988), p.215-229
\[W\] J.-L. Waldspurger: [*La formule de Plancherel pour les groupes $p$-adiques d’après Harish-Chandra*]{}, Journal de l’Inst. de Math. Jussieu 2 (2003), p.235-333
Institut de mathématiques de Jussieu- CNRS
2 place Jussieu, 75005 Paris
e-mail: [email protected]
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[^1] .1in .1in [a) *Variable Energy Cyclotron Centre, 1/AF Bidhan Nagar, Calcutta, India*]{}\
[b) *Saha Institute of Nuclear Physics, 1/AF Bidhan Nagar, Calcutta, India*]{}\
0.2cm
I. Introduction {#i.-introduction .unnumbered}
===============
Photons and dileptons have long been considered as excellent probes of quark-gluon plasma (QGP) expected to be formed in relativistic heavy ion collisions. However, while evaluating these type of signals, the immediate interests were to study quasi-particle scattering in dense matter. In contrast, little has been done to account for the radiative or bremsstrahlung contributions to photons and dileptons from these quasi-particles. As will be shown these contributions are dominant as long as the energy remains below 1 GeV. It has also been argued that the system formed in relativistic heavy ion collisions undergoes rapid transverse expansion during the latter stages of the collision. Since low energy photons/dileptons are produced mostly from the late stage, they will be affected by the collective transverse flow. Therefore, one can get idea about the flow by studying low energy photons and dileptons. We apply soft photon approximation (SPA) to evaluate soft photons and dileptons from quark matter as well as hadronic matter. The estimations of soft electromagnetic radiations that exist in the literature have certain discrepancies (see Ref. [@lichard] for details) which we have tried to correct in our calculations.
II. Formulation {#ii.-formulation .unnumbered}
===============
The emission of soft photons (real or virtual) occurs from external lines in any process and the probability for such emission is given by the classical result [@jackson]. The emission of photon from the interior of the scattering vertex is neglected because in the limit of very low energy of the emitted photon, its contribution is very low. The cross-section for the emission of soft real photon produced from a process $a\,b\,\rightarrow\,c\,d\,\gamma$ is given by [@drell] is given by $$q_0\,\frac{d\sigma^{\gamma}}{d^3q} =
\frac{\alpha}{4\,\pi^2}\,\frac{{\hat \sigma(s)}}{q_0^2}$$ and for the emission of a soft dilepton we have $$\frac{d\sigma^{e^+\,e^-}}{dM^2\,d^2M_T\,dy} =
\frac{\alpha}{12\,\pi^3\,M^2}\,\frac{{\hat \sigma(s)}}{q_0^2}$$ Using kinetic theory we can write down the rate of production of soft photons from a system at temperature $T$ as $$\begin{aligned}
E_{\gamma}\,\frac{dN^{\gamma}}{d^4x\,d^3q}&=&
\frac{T^6\,g_{ab}}{16\pi^4}\,\int_{z_{\mathrm {min}}}^{\infty}\,
dz\,\frac{\lambda(z^2T^2,m_a^2,m_b^2)}{T^4}\nonumber\\
&&\,\times\,\Phi(s,s_2,m_a^2,m_b^2)\,K_1(z)\,\left(q_0\,
\frac{d\sigma_{ab}^{\gamma}}{d^3q}\right),\end{aligned}$$ where $z_{\mathrm {min}} = (m_a+m_b)/T, z = \sqrt{s}/T$, and $g_{ab} = N_a\,N_b\,(2s_a+1)\,(2s_b+1)$ is the colour and spin degeneracy appropriate for the collisions. Similarly we can obtain the rate for soft dilepton emission. Here we note that ${\hat \sigma(s)}$ consists of two parts- electromagnetic and strong or elastic. To evaluate the elastic cross-section in QGP sector we follow the prescription of Ref. [@danei]. In the hadronic sector we assume a model Lagrangian [@haglin] for the calculation of strong cross-section.
Once the rates are obtained one can apply it for an evolving system to get $$\frac{dN}{d^2q_T\,dy}=\int\,\tau\,d\tau\,r\,dr\,d\phi\,d\eta\,
\left[f_Q\, q_0\frac{dN^q}{d^4x\,d^3q}+(1-f_Q)\,
q_0\frac{dN^{\pi}}{d^4x\,d^3q}\right]$$ where $f_Q(\tau)$ gives the fraction of the quark-matter[@kkmm] in the system.
III. Results {#iii.-results .unnumbered}
============
We calculate the transverse momentum distribution of soft photons and dileptons for a system undergoing transverse expansion. For this we will assume that initially a thermalised and chemically equilibrated QGP is formed in Pb-Pb collisions at proper time $\tau_i$ = 1 fm/c [@bjorken]. Cooling due to expansion leads to first order phase transition at T = 160 MeV. When the conversion to hadronic matter is complete a pure hadron phase is realised which then freezes-out at T = 140 MeV. The complete dominance of soft photon multiplicity in the region of 0.1$<p_T$(GeV)$<1.0$ is seen from fig. (1).
The result for transverse mass distribution for the low mass dileptons at RHIC energies is shown in fig. (2). We see that the pion driven processes dominate the yeild at lower $M_T$. However at larger $M_T$, the contributions of quark and pion driven processes are almost same. This is a reflection of larger temperature in the quark phase, and a larger effect of transverse flow during the hadronic phase. However, the invariant mass spectrum does not show up this type of feature. Thus we see complete dominance of bremsstrahlung dileptons in the low mass region. Similar charecteristics have been observed at SPS and LHC energies [@dpal].
-- --
-- --
[ [Figure 2(a–d): The transverse mass distribution of low mass dielectrons at RHIC energies including bremsstrahlung process and annihilation process in the quark matter and the hadronic matter. We give the results for invariant mass M equal to 0.1 GeV (a), 0.2 GeV (b), and 0.3 GeV (c) respectively. The invariant mass distribution of low mass dielectrons are also shown (d).]{}]{}
IV. Conclusion {#iv.-conclusion .unnumbered}
==============
We have calculated the transverse momentum distributions of photons and dileptons within a soft photon approximation at SPS, RHIC and LHC energies. We find that the formation of such a system may be characterised by an “intense glow” of soft electromagnetic radiations whose feature sensitively depends on the last stage of evolution once we remove the background of decay photons and dileptons and thus holds out the promise that soft electromagnetic radiations may be utilised as chroniclers of final moments of relativistic heavy ion collisions.
We have kept our discussions limited to photon energies of more than 100 MeV in the hope that Landau-Pomeranchuk-Migdal suppression there may not be substantial. However, a qualitative argument [@pradip1] shows that LPM suppression is only marginal in the hadronic sector. Also in the quark sector this suppression may not be quite severe, though may not be as justified as pion driven processes in the energy region considered here. .3cm [**This talk is an abridged version of Ref. [@dpal].**]{}
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[^1]: Based on the talk presented by Pradip Kumar Roy in ICPA’QGP-1997, Jauipur, India.
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---
abstract: 'The Galactic globular cluster $\omega$ Centauri is a prime candidate for hosting an intermediate mass black hole. Recent measurements lead to contradictory conclusions on this issue. We use VLT-FLAMES to obtain new integrated spectra for the central region of $\omega$ Centauri. We combine these data with existing measurements of the radial velocity dispersion profile taking into account a new derived center from kinematics and two different centers from the literature. The data support previous measurements performed for a smaller field of view and show a discrepancy with the results from a large proper motion data set. We see a rise in the radial velocity dispersion in the central region to 22.8$\pm$1.2 [$\rm {km}~\rm s^{-1}$]{}, which provides a strong sign for a central black hole. Isotropic dynamical models for $\omega$ Centauri imply black hole masses ranging from $3.0$ to $5.2\times10^4 M_\odot$ depending on the center. The best-fitted mass is $(4.7\pm1.0)\times10^4 M_\odot$.'
author:
- 'Eva Noyola,Karl Gebhardt, Markus Kissler-Patig, Nora Lützgendorf, Behrang Jalali, P. Tim de Zeeuw, Holger Baumgardt'
title: 'Very Large Telescope Kinematics for omega Centauri: Further Support for a Central Black Hole.[^1] '
---
Introduction {#intro}
============
Intermediate-mass black holes (IMBHs) may bridge the gap between stellar mass black holes and super-massive black holes found in the center of most galaxies. Their existence is appealing in various ways: they could extend the $M_\bullet-\sigma$ relation for galaxies [@geb00a; @fer00] down to dwarf galaxies and globular clusters, and present a potential connection to nuclear star clusters [@set10]. They could also be the seeds for super-massive black holes and alleviate problems with difficulties to account for the rapid growth necessary to explain massive QSOs at high redshift [@tan09].
The existence of an IMBH at the center of $\omega$ Centauri (NGC 5139) has been controversial. @noy08 (hereafter NGB08) obtain line-of-sight velocity dispersion (LOSVD) measurements using the Gemini-GMOS integral field unit (IFU). They find a velocity dispersion rise toward the center implying the presence of a $(4\pm1)\times10^4
M_\odot$ black hole when compared to spherical isotropic dynamical models. In contrast, @vdm10 (hereafter vdMA10), using proper motions from [*HST*]{}-ACS imaging, find a lower black hole mass of $(1.8\pm0.3)\times10^4 M_\odot$ for an isotropic model and their profile with a central cusp. Their anisotropic model sets an upper limit of $7.4\times10^3 M_\odot$. The comparison is complicated by the fact that the cluster centers between NGB08 and @and10 (hereafter AvdM10) are separated by $\sim12\arcsec$.
The nature of $\omega$ Centauri has been under discussion for a while. This object has been regarded as the largest globular cluster in the Galactic system, but the clear metallicity spread [@nor96; @sol05], as well as a double main sequence [@bed04; @pio05] has led to the suggestion that it might be the stripped core of a dwarf galaxy [@fre03; @mez05; @bek06]. $\omega$ Cen has a large central velocity dispersion of $22\pm4$ [$\rm {km}~\rm s^{-1}$]{} [@mey95], as well as a fast global rotation of 8 [$\rm {km}~\rm s^{-1}$]{} [@mer97], at 11 pc from the center. It is the most flattened Galactic globular cluster [@whi87], and has a retrograde orbit around the galaxy [@din01]. Using both radial velocities and proper motions @ven06 calculate a total mass of $2.5\times10^6 M_\odot$, making $\omega$ Cen the most massive Galactic globular cluster.
The extrapolation of the $M_\bullet-\sigma$ relation for galaxies [@tre02] predicts a $1.3\times10^4 M_{\odot}$ black hole for $\omega$ Cen. At a distance of $4.8\pm0.3$ kpc [@ven06], the sphere of influence of such a black hole is $\sim$ 5. In this Letter, we present new VLT-ARGUS data that we compare to previous measurements.
Observations and Data Reduction
===============================
We obtain central kinematics data of $\omega$ Cen using the ARGUS IFU with FLAMES on the Very Large Telescope (VLT). With a central $\sigma$ around 20 [$\rm {km}~\rm s^{-1}$]{}, a spectral resolving power of $R\sim$10,000 is sufficient to measure the dispersion from integrated stellar light. The Ca-triplet region (8450–8700Å) is well suited for kinematic analysis. The LR8 setup of the GIRAFFE spectrograph [@pas02], covering the range 820–940 nm at $R\sim$10,400 in ARGUS mode, is ideally suited for our study.
The ARGUS IFU was used in the 1:1 magnification mode providing a field of view of 11.5$\times$7.3, sampled by 0.52$\times$0.52pixels. The FLAMES observations were taken during two nights (2009 June 15 and 16). Eight different pointings were obtained at and around the two contended center determinations (see Figure \[print\]). While the pointings aimed at including both centers, position inaccuracies in the guide star catalogs made us miss the second from AvdM10. The final set of observations consists of three exposures for the first ARGUS pointings (around the NGB08 center) and two exposures for the seven other pointings, with exposure times of 1500s for the first two, 1020s for the next two (90$^\circ$ tilted, see Fig. \[print\]) and 900s for the four peripheral pointings ($\pm$45$^{\circ}$ tilted).
The first reduction steps are done with the GIRAFFE pipeline (based on the Base Line Data Reduction Software developed by the Observatoire de Genève). The pipeline recipes [*gimasterbias, gimasterdark, gimasterflat*]{} and [*giscience*]{} produce bias corrected, dark subtracted, fiber-to-fiber transmission and pixel-to-pixel variations corrected spectra. Sky subtraction and wavelength calibration are performed with our own tools, which test the wavelength solution with arc exposures and skylines.
We reconstruct the ARGUS data cubes to images in order to determine the exact location of the pointings with respect to reference *Hubble Space Telescope (HST)* images. We use a large Advanced Camera for Surveys (ACS) mosaic of $\omega$ Cen (GO-9442, PI: A. Cool), which we convolve to ground-based observed spatial resolution. The reconstructed ARGUS images are matched to the convolved ACS image and used to assign the correct location and position angle from both centers to each pixel and to identify pixels which are dominated by single stars (i.e., not suited to derive a velocity dispersion).
Kinematic Measurements {#select}
======================
Measuring kinematics of globular clusters from integral field spectroscopy is challenging. For details, we refer to NGB08. A key aspect to consider is the fact that bright stars might dominate the integrated light and increase the shot noise of the velocity dispersion . In order to minimize the shot noise from bright stars, we can choose which pixels to combine for the integrated light measure of the velocity dispersion.
There are hot stars with strong Paschen-series lines present (see Fig 2). We exclude regions dominated by these stars and those dominated by bright stars. We identify these regions by including one of these stars in the velocity template library and then exclude those regions which have a significant contribution, $\sim$5% of the pixels are excluded in this way. We also exclude those regions dominated by bright stars. After these two cuts, about 85% of the pixels remain to derive kinematics. To further minimize the effect from bright stars, we divide each spectrum by its mean value, thereby giving all pixels equal weight when combining..
We consider the shot noise from having a small number of stars contribute to a spatial bin. We calculate the shot noise using the HST $R$-band photometry from AvdM10. We use Monte Carlo simulations to generate a mock velocity data set in a given spatial bin, using magnitudes of present stars. We then estimate a velocity dispersion weighted by the fluxes of the stars. After 1000 realizations, we get sample velocity dispersion estimates from which we obtain the scatter, and hence the shot noise.
We rely on both centers by NGB08 and AvdM10, which differ by 12. AvdM10 claim that the center of NGB08 is biased toward bright stars and that these stars do not trace the center well. On the other hand, using corrected star counts biases one away from bright stars. Thus, there may be reasons to expect increased noise for the center position in both techniques. Given that we have two-dimensional (2D) kinematics, we can provide another center based on kinematics by running a kernel of 5 across the field and estimating the velocity dispersion within that kernel. From the 2D dispersion map there is a clear peak at the location highlighted in Fig. 2. It lies about 10from NGB08 and 3.5 from AvdM10. We make dynamical models using the three centers.
We use five annuli centered on each center for the dynamical analysis. We combine the pixels within each annulus using a biweight estimator [@bee90]. The average radius of the annuli are given in Table 1, they are chosen to provide a signal-to-noise ratio of at least 40 in each bin. The central annulus has about 60 pixels and the outer has 500. The shot noise in any of the outer annuli is below 3% of the velocity dispersion. In the central bins, the shot noise is 3% for the kinematic center, 6% for AvdM10 and 9% for NGB08. The uncertainties in Table 1 include the shot noise added in quadrature with the measured uncertainties.
In order to extract the kinematics from the spectra, we use the technique described in @geb00b and @pin03, also employed in NGB08. This technique provides a non-parametric estimate of the LOSVD. Starting from velocity bins of 8 [$\rm {km}~\rm s^{-1}$]{}, we adjust the height of each LOSVD bin to define a sample LOSVD. This LOSVD is convolved with a template. The parameters, bin heights and template mix are changed to minimize the $\chi^2$ fitted with the data spectrum. For the template, we use two individual stars within the IFU; these are a normal late-type giant star, and a hot star (shown in Fig. \[spec\]). The program then determines the relative weight of these two stars.
[llllllllll]{} K1 & 2.0 & 1.1 & 0.5 & 22.8 & 1.2 & 0.05 & 0.03 & -0.05 & 0.01\
K2 & 4.5 & -1.1 & 0.4 & 21.3 & 0.8 & 0.03 & 0.03 & -0.05 & 0.01\
K3 & 8.0 & 3.0 & 0.4 & 19.8 & 0.9 & 0.01 & 0.03 & -0.04 & 0.01\
K4 & 12.7 & 2.3 & 0.4 & 18.8 & 0.7 & -0.01 & 0.02 & -0.04 & 0.01\
K5 & 18.3 & 1.9 & 0.4 & 18.9 & 0.7 & -0.00 & 0.03 & -0.05 & 0.01\
N1 & 1.9 & -0.6 & 0.4 & 20.1 & 2.1 & 0.00 & 0.02 & -0.05 & 0.01\
N2 & 4.5 & 1.4 & 0.5 & 22.7 & 1.5 & -0.01 & 0.04 & -0.06 & 0.01\
N3 & 8.2 & 1.3 & 0.4 & 19.5 & 0.8 & -0.01 & 0.04 & -0.05 & 0.01\
N4 & 14.0 & 1.0 & 0.4 & 19.8 & 0.9 & 0.01 & 0.03 & -0.04 & 0.01\
N5 & 25.5 & 3.9 & 0.4 & 18.4 & 0.5 & 0.01 & 0.03 & -0.05 & 0.01\
A1 & 3.1 & 8.7 & 0.3 & 17.9 & 1.7 & 0.01 & 0.03 & -0.04 & 0.01\
A2 & 6.0 & -2.2 & 0.4 & 21.5 & 1.0 & 0.05 & 0.03 & -0.05 & 0.01\
A3 & 8.8 & -0.5 & 0.5 & 22.8 & 1.2 & 0.00 & 0.03 & -0.08 & 0.01\
A4 & 12.6 & 4.2 & 0.4 & 16.9 & 0.4 & 0.03 & 0.02 & -0.06 & 0.01\
A5 & 16.9 & 0.6 & 0.4 & 18.5 & 1.0 & -0.01 & 0.04 & -0.04 & 0.01
The non-parametric LOSVD estimate requires a smoothing parameter (see @geb00b for a discussion) in order to produce a realistic profile, otherwise, adjacent velocity bins can show large variations. We use the smallest smoothing value just before the noise in the LOSVD bins becomes large (similar to a cross-validation technique). In addition to a non-parametric estimate, we fit a Gaussian–Hermite profile including the first four moments. The second moment of both the Gauss–Hermite profile and the non-parametric LOSVD is similar, which implies that we have a good estimate for the smoothing value. We first fit all individual 4700 pixels in all dithered positions of the IFU. This step allows us to identify those pixels where hot stars provide a significant contribution. We then exclude those pixels from the combined spectra. The top spectrum in Fig. \[spec\] shows the spectral fit to the central radial bin.
The uncertainties for the LOSVD come from Monte Carlo simulations. For each spectrum, we generate a set of realizations from the best-fitted spectrum (template convolved with the LOSVD), and add noise according to the rms of the fit. We then fit a new LOSVD, varying the template mix. From the run of realizations we take the 68% confidence band to determine the LOSVD uncertainties.
Given that $\omega$ Cen contains stars with different spectral types, we also allow the equivalent widths to be an additional parameter. This parameter allows for mismatch between the stars chosen as templates and different regions of the cluster. We have tried a variety of different template stars and find no significant changes. Table 1 presents the first four moments ($v$, $\sigma$, $h_3$, and $h_4$) of a Gauss-Hermite expansion fitted to the non-parametric LOSVD. We note that the LOSVDs have statistically-significant non-zero $h_4$ components, which are important for the dynamical modeling in terms of constraining the stellar orbital properties.
Fig. \[models\] shows velocity dispersions from the LOSVDs for spectra combined around all centers. Every case shows an increase in the dispersion compared to data outside 50, while isotropic models without a black hole expect a drop in the central velocity dispersion. The dispersion profile obtained for the kinematic center shows a smooth rise, the one for the NGB08 center is still relatively smooth, while the profile for the AvdM10 center shows larger variation. While the larger scatter is not evidence that the two previous centers are not proper, it is suggestive.
Isotropic Models and Discussion
===============================
A detailed comparison with N-body simulations and orbit based models is in preparation. These models will also consider possible velocity anisotropy and contribution of dark remnants, as well as include a comparison with the large proper motion dataset in AvdM10. For the scope of this Letter, we limit ourselves to a comparison of the present data with isotropic models. These models have represented the projected quantities for globular clusters extremely well, starting with @kin66 all the way to a recent analysis by @mcl06, where the conclusion is that clusters are isotropic within their core. Thus, while isotropy needs to be explored in detail, it provides a very good basis for comparison.
For the details of the isotropic analysis, we refer to NGB08, essentially following the non-parametric method described in @geb95b. The surface brightness profile is the one obtained in NGB08, which is smoothed and deprojected assuming spherical symmetry in order to obtain a luminosity density profile [@geb96]. By assuming an $M/L$ ratio, we calculate a mass density profile, from which the potential and the velocity dispersion can be derived. We repeat the calculation adding various central point masses ranging from 0 to $7.5\times10^4M_\odot$ while keeping the global $M/L$ value fixed. Since vdMA10 obtain a density profile from star counts, we use their Nuker fit to the star-count profile to create a similar set of models. We note that the $M/L$ value needed to fit the kinematics outside the core radius is 2.7 for both profiles. For comparison, @ven06 found an $M/L$ value of 2.5.
Binaries could potentially bias a velocity dispersion measured from radial velocities, which is not an issue for proper motions. @car05 estimate an 18% binary fraction for $\omega$ Cen (with large uncertainties); this implies that at any given time, the observed fraction is about a few percent due to chance inclination and phase [@hut92]. Also, @fer06 find no mass segregation for this cluster tracing the blue straggler population with radius. Both facts imply that the expected binary contamination is low (a few percent), which at most would cause a few percent increase in the measured velocity dispersion (i.e., within our errors).
Figure \[models\] shows the comparison between the different models and the measured dispersion profiles. As in our previous study, the most relevant part of the comparison is the rise inside the core radius. As can be seen, an isotropic model with no black hole predicts a slight decline in the velocity dispersion toward the center which is not observed for any of the assumed centers. The calculated $\chi^2$ values for each model are plotted in Fig. \[chibh\], as well as a line showing $\Delta \chi^2=1$. The $\chi^2$ curve implies a best-fitted black hole mass of several $10^4 M_\odot$ in every case, but with lower $\chi^2$ for the NGB08 center. Specifically, a black hole of mass of $(5.2\pm0.5)\times10^4M_\odot$ is found for the kinematic center $(4.75\pm0.75)\times10^4M_\odot$ for the NGB08 center, and of $(3.0\pm0.4)\times10^4M_\odot$ for the AvdM10 center.
The velocity dispersion at 100 is well measured at around 17 [$\rm {km}~\rm s^{-1}$]{}. The radial velocities inward show a continual rise in the dispersion with smaller radii to the central value around 22.8 [$\rm {km}~\rm s^{-1}$]{}, which is statistically significant. This rise is now seen in multiple radial velocity datasets. It is this gradual rise that provides the significance for a central black hole. The proper motion data of AvdM10 show a slight rise in the velocity dispersion, but not all the way into their center. It is unclear why the two dispersion measurements differ.
This research was supported by the DFG cluster of excellence Origin and Structure of the Universe (www.universe-cluster.de). K.G. acknowledges support from NSF-0908639. We thank the anonymous referee for constructive comments that improved the manuscript.
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[^1]: Based on observations collected at the European Organization for Astronomical Research in the Southern Hemisphere, Chile (085.D-0928)
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---
abstract: 'Recently, video captioning has been attracting an increasing amount of interest, due to its potential for improving accessibility and information retrieval. While existing methods rely on different kinds of visual features and model structures, they do not fully exploit relevant semantic information. We present an extensible approach to jointly leverage several sorts of visual features and semantic attributes. Our novel architecture builds on LSTMs for sentence generation, with several attention layers and two multimodal layers. The attention mechanism learns to automatically select the most salient visual features or semantic attributes, and the multimodal layer yields overall representations for the input and outputs of the sentence generation component. Experimental results on the challenging MSVD and MSR-VTT datasets show that our framework outperforms the state-of-the-art approaches, while ground truth based semantic attributes are able to further elevate the output quality to a near-human level.'
author:
- |
Xiang Long\
IIIS, Tsinghua University\
[[email protected]]{}\
Chuang Gan\
IIIS, Tsinghua University\
[[email protected]]{}\
Gerard de Melo\
Rutgers University\
[[email protected]]{}\
title: 'Video Captioning with Multi-Faceted Attention'
---
Introduction
============
The task of automatically generating captions for videos has been receiving an increasing amount of attention.
On YouTube, for example, every single minute, hundreds of hours of video content are uploaded. There is no way a person could sit and watch these overwhelming amounts of videos, so new techniques to search and quickly understand them are highly sought. Generating captions, i.e., short natural language descriptions, for videos is an important technique to address this challenge, while also greatly improving their accessibility for blind and visually impaired users.
Video captioning has been studied for a long time and remains challenging, given the difficulties of video interpretation, natural language generation, and the interplay between them. Understanding a video hinges on our ability to make sense of video frames and of the relationships between consecutive frames.
The output needs to be grammatically correct sequence of words. Different parts of the output caption may pertain to different parts of the video. In previous work, 3D ConvNets [@Du2014C3D] have been proposed to capture motion information in short videos, while LSTMs [@Hochreiter1997Long] can be used to generate natural language, and a variety of different visual attention models [@Yao2015Describing; @Pan2015Hierarchical; @Yu2015Video] have been deployed, attempting to capture the relationship between caption words and the video content.
These methods, however, only make use of visual information from the video, often with unsatisfactory results. In many real-world settings, we can easily obtain additional information related to the video. Apart from sound, there may also be a title, user-supplied tags, categories, and other metadata. Both visual video features as well as attributes such as tags can be imperfect and incomplete. However, by jointly considering all available signals, we may obtain complementary information that aids in generating better captions. Humans, too, often benefit from additional context information when trying to understand what a video is portraying.
Incorporating these additional signals is not just a matter of adding additional features. While generating the sequence of words in the caption, we need to be able to flexibly attend to the relevant frames over time, the relevant parts within a given frame, and relevant additional signals to the extent that they pertain to a particular output word.
Based on these considerations, we propose a novel multi-faceted attention architecture that jointly considers multiple heterogeneous forms of inputs. This model is flexibly attends to temporal information, motion features, and semantic attributes for every channel. An example of this is given in Figure \[fig:feature\]. Each part of the attention model is an independent branch and it is straightforward to incorporate additional branches for further kinds of features, making our model highly extensible. We present a series of experiments that highlight the contribution of attributes to yield state-of-the-art results on standard datasets.
![Example video with extracted visual features, semantic attribute, and the generated caption as output. []{data-label="fig:feature"}](pics/example.pdf "fig:"){width="48.00000%"}\
Related Work
============
**Machine Translation**. Some of the first widely noted successes of deep sequence-to-sequence learning models were for the task of machine translation [@Cho2014Learning; @Cho2014On; @Sutskever2014Sequence; @Kalchbrenner2013Recurrent; @Li2015A; @Lin2015Hierarchical]. In several respects, this is actually a similar task to video caption generation, just with a rather different input modality. What they share in common is that both require bridging different representations, and that often an encoder-decoder paradigm is used with a Recurrent Neural Network (RNN) decoder to generate sentences in the target language. Many techniques for video captioning are inspired by neural machine translation ones, including soft attention mechanisms to focus on different parts of the input when generating the target sentence word by word [@Bahdanau2015Neural].
[**Image Captioning**]{}. Image captioning can be regarded as a greatly simplified case of video captioning, with videos consisting of just a single frame. Recurrent architectures are often used here as well [@Karpathy2014Large; @Kiros2014Unifying; @Chen2015Learning; @Mao2015Deep; @Vinyals2014Show]. Spatial attention mechanisms allow for focusing on different areas of an image [@Xu2015Show]. Recently, image captioning incorporating semantic concepts have achieved inspiring results. A semantic attention approach has been proposed [@You2016Image] to selectively attend to semantic concept proposals and fuse them into hidden states and outputs of RNNs, but their model is difficult to extend for multiple channels. Overall, none of these methods for image captioning need to account for temporal and motion aspects.
[**Video captioning**]{}. For video captioning, many works utilize a recurrent neural architecture to generate video descriptions, conditioned on either an average-pooling [@Venugopalan2015Translating] or recurrent encoding [@Xu2015A; @Donahue2015Long; @Venugopalan2015Sequence; @Venugopalan2016Improving] of frame-level features, or on a dynamic linear combination of context vectors obtained via temporal attention [@Yao2015Describing]. Recently, hierarchical recurrent neural encoders (HRNE) with attention mechanism have been proposed to encode video [@Pan2015Hierarchical]. A recent paper [@Yu2015Video] additionally exploits several kinds of visual attention and relies on a multimodal layer to combine them. In our work, we present a novel attention model with more effective multimodal layers that jointly models multiple heterogeneous signals, including semantic attributes, and experimentally show the benefits of this approach over previous work.
The Proposed Approach
=====================
In this section, we describe our approach for combining multiple forms of attention for video captioning. Figure \[fig:model\] illustrates the architecture of our model. The core of our model is a sentence generator based on generator is a simple Long Short Term Memory (LSTM) units [@Hochreiter1997Long]. Instead of a traditional sentence generator, which directly receives a previous word and selects the next word, our model relies on several attention layers to selectively focus on important parts of temporal, motion, and semantic features. The output words are generated via a softmax reading from a multimodal layer [@Mao2015Deep], which integrates information from the different attention layers. An additional multimodal layer integrates information before the input reaches the sentence generator to enable better hidden representations in the LSTM. We first briefly review the basic LSTM, and then describe our model in detail, including our novel multi-faceted attention mechanism to consider temporal, motion, and semantic attribute perspectives.
Long Short Term Memory Networks
-------------------------------
A Recurrent Neural Network (RNN) [@Elman1990Finding] is a neural network adding extra feedback connections to feed-forward networks, so as to be able to work with sequences. The network is updated not only based on the input but also based on the previous hidden state. RNNs can compute the hidden states $(h_1,h_2,\dots,h_m)$ given an input sequence $(x_1,x_2,\dots,x_m)$ based on recurrence of the following form: $$h_t=\phi(W_h x_t+U_h h_{t-1}+b_h),$$ where weight matrices $W$, $U$ and bias $b$ are parameters to be learned and $\phi(\cdot )$ is an element-wise activation function.
RNNs trained via unfolding have proven inferior at capturing long-term temporal information. LSTM units were introduced to avoid these challenges. LSTMs not only compute the hidden states but also maintains a cell state to account for relevant signals that have been observed. They have the ability to remove or add information to the cell state, modulated by gates.
Given an input sequence $(x_1,x_2,...,x_m)$, an LSTM unit computes the hidden state $(h_1,h_2,...,h_m)$ and cell states $(c_1,c_2,...,c_m)$ via repeated application of the following equations: $$\begin{aligned}
i_t &=\sigma(W_i x_t+U_i h_{t-1}+b_i)\\
f_t &=\sigma(W_f x_t+U_f h_{t-1}+b_f)\\
o_t &=\sigma(W_o x_t+U_o h_{t-1}+b_o)\\
g_t &=\phi(W_g x_t+U_g h_{t-1}+b_g)\\
c_t &= f_t \odot c_{t-1} + i_t \odot g_t\\
h_t &= o_t \odot c_t,
\end{aligned}$$ where $\sigma(\cdot)$ is the sigmoid function and $\odot$ denotes the element-wise multiplication of two vectors. For convenience, we denote the computations of the LSTM at each time step $t$ as $h_t$, $c_t = \operatorname{LSTM}(x_t, h_{t-1}$, $c_{t-1})$.
{width="100.00000%"}\
Input Representations
---------------------
When training a video captioning model, as a first step, we need to extract feature vectors that serve as inputs to the LSTM. For visual features, we can extract one feature vector per frame, leading to a series of what we call *temporal features*. We can also extract another form of feature vector from several consecutive frames, which we call *motion features*. Additionally, we could also extract other forms of visual features, such as features from an area of a frame, the same area of consecutive frames, etc. In this paper, we only consider temporal features, denoted by $\{ v_i \}$, and motion features, denoted by $\{ f_i \}$, which are commonly used in video captioning.
For semantic features, we need to extract a set of related attributes denoted by $\{ a_i \}$. These can be based on title, tags, etc., if available. Alternatively, we can also rely on techniques to extract or predict attributes that are not directly given. In particular, because we have captions for the videos in the training set, we can train different models to predict caption-related semantic features for videos in the validation and test sets. As the choice of semantic features is not the core contribution, we describe our specific experimental setups in Section \[sec:experiments\].
After determining a set of attributes for each video, each attribute $a_i$ in any video in the entire dataset corresponds to an entry in the vocabulary and each word $w_i$ in any caption of the training set also corresponds to an entry in the vocabulary.
An embedding matrix $E$ is used to represent both words and semantic attributes and we denote by $E[w]$ an embedding vector of a given $w$. Thus, we obtain attribute embedding vectors $\{ s_i \}$ and input word embedding vectors as: $$\begin{aligned}
s_i &= E[a_i]\\
x_t &= E[w_t]
\end{aligned}$$
Multi-Faceted Attention
-----------------------
We do not directly feed $x_t$ to the LSTM. Instead, we first apply our multi-faceted attention model to $x_t$.
Assuming that we have a series of multimodal feature vectors for a given video, we generate a caption word by word. At each step, we need to select relevant information from these feature vectors, which we from now on refer to as *context vectors* $\{ c_1,c_2,...,c_n\}$. Due to the variability of the length of videos, it is challenging to directly input all these vectors to the model at every time step. A simple strategy is to compute the average of the context vectors and input this average vector to each time step of the model. $$y_t = \frac{1}{m} \sum_{i=1}^m c_i$$ However, this strategy collapses all available information into a single vector, neglecting the inherent structure, which captures the temporal progression, among other things. Thus, this sort of folding leads to a significant loss of information. Instead, we wish to focus on the most salient parts of the features at every time step. Instead of a naive averaging of the context vectors $\{ c_1,c_2,\dots,c_n\}$, a soft attention model calculates weights $\alpha_i^t$ for each $c_i$, conditioning on the input vector $x_t$ at each time step $t$. For this, we first compute basic attention scores $e_i^t$ and then feed these through a sequential softmax layer to obtain a set of attention weights $\{ \alpha_1^t,\alpha_2^t,\dots,\alpha_n^t\}$ that quantify the relevance of $\{ c_1,c_2,\dots,c_n\}$ for $x_t$. $$\begin{aligned}
e_i^t &= x_t^T U c_i\\
\alpha_i^t &= \frac{\exp(e_i^t)}{\sum_{j=1}^n \exp(e_j^t)}\\
y_t &= \sum_{i=1}^m \alpha_i^t c_i
\end{aligned}$$ We obtain the corresponding output vectors $y_t$ as weighted averages.
This soft attention model, strictly speaking, converts an entire input sequence $(x_1,x_2,\dots,x_m)$ to an entire output sequence $(y_1,y_2,\dots,y_m)$ based on all context vectors $\{ c_1,c_2,\dots,c_n\}$. For convenience, we denote the attention model outputs at a given time step $t$ as $y_t = \operatorname{Attention}(x_t, \{ c_i\})$.
In particular, the attention model is applied to the temporal features $\{ v_i \}$, motion features $\{ f_i \}$ and semantic features $\{ s_i \}$: $$\begin{aligned}
s_t^{x} &= \operatorname{Attention}(x_t,\{ s_i \})\\
v_t^{x} &= \operatorname{Attention}(x_t,\{ v_i \})\\
f_t^{x} &= \operatorname{Attention}(x_t,\{ f_i \})
\end{aligned}$$
We then obtain the input to the LSTM $m_t^x$ via a multimodal layer $$m_t^x =\phi \left. (W^{x} \left. [x_t,s_t^{x},w_v^x \odot v_t^{x},w_f^x \odot f_t^{x}] \right. +b^{m,x}) \right.$$
Here, $w_v^x$ and $w_f^x$ facilitate capturing the relative importance of each dimension of the temporal and motion feature space [@You2016Image]. We apply dropout [@Srivastava2014Dropout] to this multimodal layer to reduce overfitting.
Subsequently, we can obtain $h_t$ via the LSTM. At the first time step, the mean values of the features are used to initialize the LSTM states to yield a general overview of the video: $$m_0^x=W^{i}[\operatorname{Mean}(\{ s_i \}),\operatorname{Mean}(\{ v_i \}),\operatorname{Mean}(\{ f_i \})]$$ $$h_0,c_0= \operatorname{LSTM}(m_0^x,0,0)$$ $$h_t,c_t= \operatorname{LSTM}(m_t^x,h_{t-1},c_{t-1})$$ where $\operatorname{Mean}(\cdot )$ denotes mean pooling of the given feature set.
We also apply the attention model to hidden states $h_t$, and use a multimodal layer to concatenate outputs of the attention model and map it into a feature space that has exactly the same dimensionality as the word embeddings. This multimodal layer is followed by a softmax layer with a dimensionality equal to the size of the vocabulary. The projection matrix from the multimodal layer to the softmax layer is set to be the transpose of the word embedding matrix: $$\begin{aligned}
s_t^{h} &= \operatorname{Attention}(h_t,\{ s_i \})\\
v_t^{h} &= \operatorname{Attention}(h_t,\{ v_i \})\\
f_t^{h} &= \operatorname{Attention}(h_t,\{ f_i \})
\end{aligned}$$ $$m_t^h=\phi \left. (W^{h} \left. [h_t,s_t^{h},w_v^h \odot v_t^{h},w_f^h \odot f_t^{h}] \right. +b^{m,h}) \right.$$ $$p_t = \operatorname{Softmax}(E^T m_t^h)$$ where $\operatorname{Softmax}(\cdot)$ denotes a sequential softmax.
By using two multimodel layers, we combine six attention layers with the core LSTM. This model is highly extensible since we can easily add extra branches for additional features.
Training and Generation
-----------------------
We can interpret the output of the softmax layer $p_t$ as a probability distribution over words: $$\operatorname{P}(w_{t+1}|w_{1:t},V,S,\Theta)$$ where $V$ denotes the corresponding video, $S$ denotes semantic attributes and $\Theta$ denotes model parameters. The overall loss function is defined as the negative logarithm of the likelihood and our goal is to learn all parameters $\Theta$ in our modal by minimizing the loss function over the entire training set: $$\min_{\Theta} \quad -\frac{1}{N} \sum_{i=1}^N \sum_{t=1}^{T_i} \operatorname{P}(w_{t+1}^i|w_{1:t}^i,V^i,S^i,\Theta)$$ where $N$ is the total number of captions in the training set, and $T_i$ is the number of words in caption $i$. During the training phase, we add a begin-of-sentence tag $\langle$BOS$\rangle$ to the start of the sentence and an end-of-sentence tag $\langle$EOS$\rangle$ to the end of sentence. We use Stochastic Gradient Descent to find the optimum with the gradient computed via Backpropagation Through Time (BPTT) [@Werbos1990Backpropagation]. Training continues until the METEOR evaluation score on the validation set stops increasing, and we optimize the hyperparameters using random search to maximize METEOR on the validation set, following previous studies that found that METEOR is more consistent with human judgments than BLEU or ROUGE [@Vedantam2015CIDEr].
After the parameters are learned, during the testing phase, we also have temporal and motion features extracted from the video as well as semantic attributes, which were either already given or are predicted using a model trained on the training set. Given a previous word, we can calculate the probability distribution of the next word $p_t$ using the model described above. Thus, we can generate captions starting from the special symbol $\langle$BOS$\rangle$ with Beam Search [@Yu2015Video].
{width="88.00000%"}\
{width="90.00000%"}\
Experimental Results {#sec:experiments}
====================
Datasets
--------
**MSVD**: We evaluate our video captioning models on the Microsoft Research Video Description Corpus [@Chen2011Collecting]. MSVD consists of 1,970 video clips typically depicting a single activity, downloaded from YouTube. Each video clip is annotated with multiple human generated descriptions in several languages. We only use the English descriptions, about 41 descriptions per video. In total, the dataset consists of 80,839 video/description pairs. Each description on average contains about 8 words. We use 1,200 videos for training, 100 videos for validation and 670 videos for testing, as provided by previous work [@Guadarrama2013YouTube2Text].
[**MSR-VTT**]{}: We also evaluate on the MSR Video-to-Text (MSR-VTT) dataset [@Xu2016MSR], a new large-scale video benchmark for video captioning. MSR-VTT provides 10,000 web video clips. Each video is annotated with about 20 natural sentences. Thus, we have 200,000 video-caption pairs in total. Our video captioning models are trained and hyper-parameters are selected using the official training and validation set, which consists of 6,513 and 497 video clips respectively. And models are evaluated using the test set of 2,990 video clips.
Preprocessing
-------------
**Visual Features**: We extract two kinds of visual features, temporal features and motion features. We use a pretrained ResNet-152 model [@He2015Deep] to extract temporal features, obtaining one fixed-length feature vector for every frame. We use a pretrained C3D [@Du2014C3D] to extract motion features. The C3D net reads in a video and emits a fixed-length feature vector every 16 frames.
**Semantic Attributes**: While MSVD and MSR-VTT are standard video caption datasets, they do not come with tags, titles, or other semantic information about the videos. Nevertheless, we can reproduce a setting with semantic attributes by extracting attributes from captions. First, we invoke the Stanford Parser [@Klein2003Accurate] to parse captions and choose the *nsubj* edges to find the subject-verb pairs for each caption. We then select the most frequent subject and verb across captions of each video as the high-quality semantic attributes. These attributes can be used to evaluate our models under a high-quality attribute condition. Next, we can use the high-quality semantic attributes of the training set to train a model to predict semantic attributes for the test set. Such attributes are used to evaluate our model under low-quality semantic attribute conditions.
For our experiments, we consider three models to predict semantic attributes. The first one (NN) is to perform a nearest-neighbor search on every frame of the training set to retrieve similar ones for every frame of each test video based on ResNet-152 features and select the most frequent attributes. The second one (SVM) is to train SVMs for the top 100 frequent attributes in the training set and predict semantic attributes for test videos based on a mean pooling of ResNet-152 features. The third one (HRNE) is to train two hierarchical recurrent neural encoders [@Pan2015Hierarchical] to predict the subject and verb separately based on temporal ResNet-152 features.
Evaluation Metrics
------------------
We rely on four standard metrics, BLEU [@Papineni2002.BLEU], METEOR [@Banerjee2005METEOR], CIDEr [@Vedantam2015CIDEr] and ROUGE-L [@Lin2004ROUGE] to evaluate our methods. These are commonly used in image and video captioning tasks, and allow us to compare our results against previous work. We use the Microsoft COCO evaluation server [@Chen2015Microsoft], which is widely used in previous work, to compute the metric scores. Across all three metrics, higher scores indicate that the generated captions are assessed as being closer to captions created by humans.
Experimental Settings
---------------------
The number of hidden units in the input multimodal layer and in the LSTM are both $512$. The activation function of the LSTM is $\tanh$ and the activation functions of both multimodal layers are $\operatorname{linear}$. The dropout rates of both the input and output multimodal layers are set to $0.5$. We use pretrained 300-dimensional GloVe [@Pennington2014Glove] vectors as our word embedding matrix. We rely on the RMSPROP algorithm [@Tieleman2012rmsprop] to update parameters for better convergence, with the learning rate $10^{-4}$. The beam size during sentence generation is set to 5. Our system is implemented using the Theano [@Bastien2012Theano; @J2010Theano] framework.
Results
-------
**Visual only**: First, for comparison, we show the result of only using visual attention at first. Specifically, we only use the temporal features and motion features (TM), by removing the semantic branch with other components of our model unchanged. To evaluate the effectiveness of different sorts of visual cues, we also report the results of using only temporal features (T) and using only motion features (M). We compare our methods with six state-of-the-art methods: LSTM-YT [@Venugopalan2015Translating], S2VT [@Venugopalan2015Sequence], TA [@Yao2015Describing], LSTM-E [@Pan2016Jointly], HRNE-A [@Pan2015Hierarchical], and h-RNN [@Yu2015Video]. Table \[tab:results1\] provides a comparison of these systems on the MSVD dataset. Since some of the previous work uses different features, we also run experiments for some of them whose source code are provided by the authors, or we re-implement the models described in their papers, and then evaluate them using our features. The corresponding extra results are marked by ‘$\ast$’.
We observe that even just with temporal features alone, we obtain fairly good results, which implies that the attention model in our approach is useful. Combining temporal and motion features, we see that our method can outperform previous work, confirming that our attention model with multimodel layers can extract useful information from temporal and motion features effectively. In fact, the TA, LSTM-E studies also employ both temporal and motion features, but do not have a separate motion attention mechanism. And the h-RNN study only considers attention after the sentence generator. Instead our attention mechanism operates both before and after the sentence generator, enabling it to attend to different aspects during the analysis and synthesis processes for a single sentence. The results on the MSR-VTT dataset are shown in Table \[tab:results3\]. They are consistent in that they also show that the combined attention for temporal and motion features obtains better results.
[**Multi-Faceted Attention**]{}: To show the influence of our multi-faceted attention with additional semantic cues, we first consider the low-quality semantic attributes. Tables \[tab:results2\] and \[tab:results3\] provide results using low-quality attributes obtained via our NN (TM-P-NN), SVM (TM-P-SVM), and HRNE (TM-P-HRNE) methods described above. We find that the results for NN and SVM are sometimes slightly worse than only using visual attention, which means that too low-quality attributes do not help in improving the quality. It appears that these methods are rather unreliable and introduce significant noise. HRNE fares slightly better than using only visual attention, as it combines top-down and bottom-up approaches to obtain more stable and reliable results.
Then, we consider the high-quality semantic attributes, subject and verb (TM-HQ-SV), derived from the ground truth captions. We also report the performance of only using the subject (TM-HQ-S) or the verb (TM-HQ-V) individually. These results, too, are included in Tables \[tab:results2\] and \[tab:results3\]. We find that our method is able to exploit high-quality subject and verb attributes to outperform other methods by very large margins. Even using just a single semantic attribute yields very strong results. Here, verb information proves slightly more informative than the subject, indicating that identifying actions in videos remains more challenging than identifying important objects in a video.
Overall, we observe that our method, with just two high-quality features, approaches human-level scores in terms of the METEOR and CIDEr metrics. For this, we randomly selected one caption for each video in the test set and evaluate this caption by removing them from the ground truth. Although not perfect, such results (Human) can be viewed as an estimation of the human-level performance. The BLEU scores of our method are in fact even greater than the human-level ones, since humans often prefer generating longer captions, which tend to obtain lower BLEU scores. Several studies, including on caption generation, have concluded that BLEU is not a sufficiently reliable metric in terms of replicating human judgment scores [@Kulkarni2013BabyTalk; @Vedantam2015CIDEr]. Figure \[fig:captions\] shows several example captions generated by our approach for MSVD videos.
To further investigate the influence of noise, we randomly select genuine high-quality subject and verb attributes and replace them with random incorrect ones. Figure \[fig:result\] provides the results on MSVD. These result show that even when adding $50\%$ noise, the results are better than just using regular visual attention. With extremely strong noise levels, the results are worse than only using visual attention, but are still maintained at a certain level. This shows that we are likely to benefit from further semantic attributes such as tags, titles, comments, and so on, which are often available for online videos, even if they are noisy.
Conclusion
==========
We have proposed a novel method for video captioning based on an extensible multi-faceted attention mechanism, outperforming previous work by large margins.
Even without semantic attributes, our method outperforms state-of-the-art approaches using visual features. With just two high-quality semantic attributes, the results become competitive with human results across a range of metrics. This opens up important new avenues for future work on exploring the large space of potential additional forms of semantic cues and attributes.
Acknowledgments
===============
This work was supported in part by the National Basic Research Program of China Grant 2011CBA00300, 2011CBA00301 and the National Natural Science Foundation of China Grant 61033001, 61361136003.
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|
{
"pile_set_name": "ArXiv"
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---
abstract: 'The extreme, time-variable Faraday rotation observed in the repeating fast radio burst (FRB) 121102 and its associated persistent synchrotron source demonstrates that some FRBs originate in dense, dynamic and possibly relativistic magneto-ionic environments. Here we show that besides rotation of the linear-polarisation vector (Faraday rotation), such media can generally convert linear to circular polarisation (Faraday conversion). We use non-detection of Faraday conversion, and the temporal variation in Faraday rotation and dispersion in bursts from FRB121102 to constrain models where the progenitor inflates a relativistic nebula (persistent source) confined by a cold dense medium (e.g. supernova ejecta). We find that the persistent synchrotron source, if composed of an electron-proton plasma, must be an admixture of relativistic and non-relativistic (Lorentz factor $\gamma<5$) electrons. Furthermore we independently constrain the magnetic field in the cold confining medium, which provides the Faraday rotation, to be between 10 and 30mG. This value is close to the equipartition magnetic field of the confined persistent source implying a self-consistent and over-constrained model that can explain the observations.'
author:
- |
H. K. Vedantham,$^{1}$[^1] V. Ravi$^{2}$\
$^{1}$ASTRON, Netherlands Institute for Radio Astronomy, Oude Hoogeveensedijk 4, 7991PD, Dwingeloo, The Netherlands\
$^{2}$Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA
date: 'Accepted XXX. Received YYY; in original form ZZZ'
title: 'Faraday conversion and magneto-ionic variations in Fast Radio Bursts'
---
\[firstpage\]
radio continuum: transients – polarization – radiative transfer
Introduction
============
By virtue of their large Faraday rotation measures, at least two Fast Radio Burst (FRB) sources [FRB121102, FRB110523; @spitler2014; @masui2015] are observed to reside in dense magneto-ionic environments. In addition to Faraday rotation, in the presence of mildly relativistic plasma ($\gamma\gtrsim 3$ typically), propagation through a magneto-ionic medium leads to Faraday conversion wherein linearly polarised light is converted to circularly polarised light and vice-versa [@sazonov1969; @pachol1970; @huang2011]. Faraday conversion is insignificant in the presence of typical interstellar magnetic fields , but is thought to result in significant circular-polarisation fractions observed in Active Galactic Nuclei jets [@hofman2001] and in SgrA$^\ast$ [@bower2002], where the radiation propagates through a relativistic media with much larger magnetic fields than the Milky Way interstellar medium.
In this paper, by taking FRB121102 as a test case, we argue that Faraday conversion is an observable effect in some FRBs and leads to upper limits on the circum-burst magnetic field and density of low Lorentz factor electrons ($3\lesssim \gamma\lesssim 100$) that are otherwise inaccessible. We show that additional constraints from temporal variations in the dispersion and Faraday rotation of FRB121102 critically constrain proposed models for the environment of FRB121102.
In §\[sec:obs\_cons\], we summarise existing observations and model-independent constraints on the magneto-ionic environment of FRB121102 and its associated persistent radio source. In §\[sec:fc\_limits\], we describe the additional constraints implied by the non-detection of circular polarisation in FRB121102, given predictions from Faraday conversion. We discuss the implications of these results for models where FRB121102 is located within the persistent radio source in §\[sec:discussion\], and conclude in §\[sec:conclusions\].
General constraints on the environment of FRB 121102 {#sec:obs_cons}
====================================================
Observations
------------
We first summarise the known properties of the magneto-ionic environment surrounding the repeating FRB121102 [@spitler2016]. It has been localised to an HII region in a galaxy at a redshift $z=0.19$ [luminosity and angular-diameter distances of $d_{\rm L} \approx 970\,$Mpc and $d_{\rm A}\approx 680\,$Mpc respectively; @chatterjee2017]. Additionally, the FRB121102 bursts are (i) co-located to within $40\,$pc (95% confidence) with a persistent flat-spectrum ($1-10$GHz) radio source with flux-density $S_\nu \approx 200\,\mu$Jy at $3\,$GHz [@marcote2017], and (ii) show very high levels of Faraday rotation, quantified by the rotation measure, ${\rm RM}=1.46 \times 10^5\,{\rm rad}\,{\rm m}^{-2}$ in the source frame. The RM reduced by $\sim10$% in seven months [@michilli2018], whereas its dispersion measure increased by just $1-3\,{\rm pc}\,{\rm cm}^{-3}$ in four years [@hessels2018].
Dispersion and Faraday rotation {#subsec:dm_rm}
-------------------------------
The dispersion measure in the entire host galaxy is constrained to be be less than $250\,{\rm pc}\,{\rm cm}^{-3}$ [@tendulkar2017]. If an amount, ${\rm DM}_{\rm RM}$, of that is in the Faraday rotating nebula, @hessels2018 obtain the following bound by requiring the magneto-ionic medium to be transparent to free-free absorption at the lowest frequency at which bursts have been observed (1GHz): $$T_4^{2.3}{\rm DM}_{\rm RM}>150\,\left(\frac{\beta}{\eta_B^2}\right)$$ where $T_4$ is the gas temperature in units of $10^4\,$K, $\beta$ is the ratio of thermal to magnetic pressure, and $\eta_B\leq 1$ is a geometric factor equal to the mean value of $\cos\theta$ along the ray path in the Faraday rotating medium where $\theta$ is the angle the ray makes with the ambient magnetic field. For an ordered field, we have $\eta_B=\cos\theta$. Significantly smaller values must be expected for a highly tangled field due to partial cancellation of positive and negative Faraday rotation. Further, for any given choice of ${\rm DM}_{\rm RM}$, the corresponding magnetic field that can generate the observed Faraday rotation is $$\label{eqn:b_dm}
B = \frac{0.18}{{\rm DM}_{\rm RM}\eta_B}\,\,{\rm G}$$ We now address the RM variations using two generic models for the magneto-ionic medium. We will address specific models in §\[sec:discussion\].
[*Expanding nebula*]{}: If the observed RM decrease is due to the expansion of a nebula of radius $R$ [e.g., @waxman2017; @margalit18], then ${\rm DM}_{\rm RM}\propto R^{-2}$ and $B\propto R^{-1.5}$, maintaining the same plasma $\beta$ and geometric factor $\eta_B$. We therefore have $\Delta{\rm RM}/{\rm RM} = \Delta B/B + \Delta {\rm DM}_{\rm RM}/{\rm DM}_{\rm RM}$, where (to first order) $\Delta B/B =-1.5\Delta R/R$, and $\Delta {\rm DM}_{\rm RM}/{\rm DM}_{\rm RM} = -2\Delta R/R$. The observed 10% variation in RM over seven months then implies $\Delta R/R\approx 0.05\,$yr$^{-1}$ and the nebula is therefore expanding on a timescale of about $\tau\approx 20$years. The implied $-5.8$% [*decrease*]{} in ${\rm DM}_{\rm RM}$ due to the expansion must be insignificant compared to the observed [*increase*]{} of $1-3\,{\rm pc}\,{\rm cm}^{-3}$ in the total DM over a longer timescale than the reported RM variations; the observed DM variations presumably occur in plasma unrelated to the Faraday-rotating plasma. Hence we can safely place the constraint ${\rm DM}_{\rm RM}< 17.5\,{\rm pc}\,{\rm cm}^{-3}$. These constraints do not differ significantly for other expansion scenarios. For instance, adiabatic expansion of a tangled field ($B\propto R^{-2}$), gives $\tau\approx 23.33\,$yr and ${\rm DM}_{\rm RM}<20\,{\rm pc}\,{\rm cm}^{-3}$.
[*Transverse motion*]{}: Alternatively, the RM variations could be due to transverse motion of the FRB source across a nebula over a characteristic timescale of $\tau \approx 5.83\,$yr. Let the line-of-sight extent of the Faraday-rotating nebula be equal to its transverse scale length. Then we have $\Delta{\rm RM}/{\rm RM} \approx \Delta {\rm DM}_{\rm RM}/{\rm DM}_{\rm RM}\approx 0.1$. The lack of significant DM variations accompanying the RM variations now gives a more stringent constraint of ${\rm DM}_{\rm RM}< 10\,{\rm pc}\,{\rm cm}^{-3}$.
Using the formalism from §4.4 of @hessels2018, we can translate the above limits on ${\rm DM}_{\rm RM}$ to the underlying plasma parameters (${\rm DM}_{10} = {\rm DM}_{\rm RM}/(10\,{\rm pc}\,{\rm cm}^{-3})$ hereafter): $$\begin{aligned}
\label{eqn:dm_rm_limits}
{\rm DM}_{10} &<&(1.75,\, 1.0) \nonumber \\
T_4&>&2.72\,\left(\frac{\beta}{\eta^2_B}\right)^{1/2.3}\,{\rm DM}_{10}^{1/2.3} \nonumber\\
B&=&18/({\rm DM}_{10}\eta_B) \,\,{\rm mG} \nonumber \\
n_e&=&9.3\times 10^6\,\left(\frac{\eta_B^2T_4}{\beta}\right)^{-1}\,{\rm DM}_{10}^{-2} \,\,{\rm cm}^{-3} \nonumber \\
l&=&1.1\times 10^{-6}\,\left(\frac{\eta_B^2 T_4}{\beta}\right)\,{\rm DM}_{10}^3 \,\,{\rm pc}\nonumber \\
\tau & = & (20,\,5.83)\,\,{\rm yr}\nonumber \\
v &=& (0.054,\,0.185)\,\left(\frac{\eta_B^2 T_4}{\beta}\right)\,{\rm DM}_{10}^3\,\,{\rm km}\,{\rm s}^{-1}\end{aligned}$$ Here the two limits within parentheses are for the expanding-nebula and transverse-motion scenarios respectively, and $l$ and $n_e$ are the thickness and electron density of the Faraday-rotating medium respectively. The velocity, $v$, in the two cases must be interpreted as the radial expansion speed of the nebula and transverse speed of the source with respect to the Faraday rotating medium, respectively. We will return to these constrains with regards to specific models in §4.
The persistent radio source {#subsec:persistent_source}
---------------------------
The properties of the persistent radio source associated with FRB121102 may be constrained independently of the Faraday-rotating medium. We assume equipartition between the relativistic gas and magnetic field as is common in synchrotron sources[^2] [@readhead1994]. The source becomes self-absorbed at $1.5\,$GHz for radius $R_{\rm per}< 0.05\,$pc; this is thus the lower bound on the source size. European VLBI Network observations of the source at 5GHz set an upper bound on the source radius of $R_{\rm per}\lesssim0.35$pc [@marcote2017]. This is consistent with the $\approx 30\%$ amplitude modulations observed in the source at 3GHz [@chatterjee2017] being caused by refractive interstellar scintillation in the Milky Way ISM [@walker1998]. For any radius within the allowed range ($0.05<R_{\rm per}/{\rm pc}<0.35$), we can determine the equipartition magnetic field, $B_{\rm eq}$, and the column of relativistic electrons, $N_{\rm rel}$, using the standard expressions for synchrotron emissivity and absorption coefficients [@rybicki_lightman their eqns. 6.36 & 6.53]. We assume a power-law energy distribution of radiating electrons with somewhat shallow index of $b=-1.5$ that can account for the relatively flat spectrum of the source [@chatterjee2017]. The peak Lorentz factor of the distribution, $\gamma_{\rm max}$ is chosen to correspond to the observed spectral break frequency of $\nu_{\rm max}=10\,$GHz. If the lower Lorentz factor cut-off corresponds to emission at $\nu_{\rm min}=1\,$GHz[^3] then the equipartition magnetic field and electron column thus determined for minimum and maximum source sizes are: $B_{\rm eq}\approx 140\,$mG, $\gamma_{
\rm min}\approx 50$, $\gamma_{\rm max}\approx 160$, $N_{\rm rel} \approx 0.95\,{\rm pc}\,{\rm cm}^{-3}$ for $R_{\rm per}=0.05\,$pc, and $B_{\rm eq}\approx 27\,$mG, $\gamma_{
\rm min}\approx 120$, $\gamma_{\rm max}\approx 370$, $N_{\rm rel} \approx 0.1\,{\rm pc}\,{\rm cm}^{-3}$ for $R_{\rm per}=0.35\,$pc. The reader can scale the equipartition field to other source sizes using $B_{\rm eq}(R)\propto R^{-6/7}$. The total energy contained in the relativistic electrons and the magnetic field (‘equipartition energy’), is $\sim 10^{49.1}$ and $\sim 10^{50.2}$erg respectively. If the relativistic electrons were injected in a one-off event, the synchrotron cooling rates at $\gamma_{\rm max}$ yield source ages of $14\,$yr for $R=0.05$pc and 60yr for $R=0.35\,$pc. The corresponding expansion velocities are $0.011\,c$ and $0.02\,c$ respectively.
Faraday conversion {#sec:faraday_conversion}
==================
![The effect of elliptical birefringence at $\nu=4\,$GHz for thermal (left panels) and power law electron populations (right panels; energy index of $-1.5$), for three different magnetic field values (different line colours). Top panels show the peak circular fraction and bottom panels show the phase angle associated with generalised Faraday rotation. An electron column of $N_e=1\,{\rm pc\,cm}^{-3}$ (which is 1 DM units) has been assumed in all plots. Dashed black line in the top panels is placed at 2% which corresponds to the observed linear fraction of $\gtrsim 98$% in FRB121102 [@michilli2018]. The deep notches in the plots are due to zero-crossings. \[fig:fc\_rel\]](rho_qv.pdf){width="\linewidth"}
\[sec:fc\_limits\]
If the FRB progenitor resides within the persistent source, then relativistically corrections to the effects of birefringence must be considered to derive the levels of Faraday rotation and conversion (collectively called generalised Faraday rotation) in the persistent source. The effect is readily visualised on the Poincaré sphere as the rotation of the polarisation vector about an axis defined by the natural modes in the medium [see @kennett1998 for details]. Two conditions must be satisfied to attain appreciable conversion of linear to circular polarisation due to propagation effects: (a) the natural modes in the medium must be sufficiently elliptical as characterised by their axial ratios, and (b) there must be a sufficient magneto-ionic column for this elliptical birefringence to have a measurable effect. In Fig. \[fig:fc\_rel\], we plot the level of generalised Faraday conversion for two commonly encountered electron distributions: a relativistic Maxwellian and a power law with an assumed index $b=-1.5$. We use the approximate expressions of @huang2011 [their eqns. 51, 58 & 59] to do so. The upper panels show the peak circular fraction allowed by the ellipticity of the natural modes (condition (a) above) and the bottom panels show the generalised Faraday rotation angle (condition (b) above). The plots assume $\nu=4\,$GHz, $\theta=\pi/4$ (giving $\eta_B\approx 0.707$), and are normalised to a total electron column of $1\,{\rm pc}\,{\rm cm}^{-3}$. It is worth noting that in the power-law case, the mode ellipticity goes from the cold-plasma limit (circular modes) to its ultra-relativistic limit (linear modes) in a rather narrow range of $2\lesssim \gamma_{\rm min}\lesssim 20$— a range that is practically inaccessible to photometric observations.
For ‘one-zone’ models where the synchrotron emission and Faraday rotation come from the same nebula, condition (b) above is satisfied by definition and condition (a) must be reconciled with the non-detection of circular polarised emission. If the electron energies are power law distributed, then Fig. \[fig:fc\_rel\] (top-right panel) shows that for $B=30,\,100\,\&250\,$G, the circular fraction is in tension with observations for $\gamma_{\rm min}>3.6,\,2.3\,\&1.7$ respectively. The one-zone nebula must therefore be an admixture of synchrotron electrons ($50\lesssim \gamma \lesssim 370$) and ‘cold’ plasma ($\gamma\lesssim 3$).
For ‘two-zone’ models, bulk of the observed Faraday rotation occurs outside the synchrotron source in presumably cold plasma that does not yield significant conversion to circular polarisation. If the radio bursts originate from within the synchrotron source, then reconciling with observations requires one to ensure that both condition (a) and (b) are not simultaneously satisfied in the synchrotron source itself. Taking equipartition solutions for the persistent source from §\[subsec:persistent\_source\] with $\nu_{\rm min}=1\,$GHz, we find that the model is in tension with the observations for $R<0.31\,$pc. If we allow $\gamma_{\rm min}$ to correspond to $\nu_{\rm min}=100\,$MHz, then the equipartition solutions are in tension with polarimetric data over the entire feasible parameter range of $0.05<R/{\rm pc}<0.35$. However, by extending the energy distribution to $\gamma_{\rm min}\lesssim 3$ (and admixture of ‘cold’ and relativistic electrons), the modes can be constructed to be sufficiently circular so as to produce $<2$% circular polarisation as in the ‘one-zone’ case.
In summary, in all models where the radio bursts pass through the persistent source powered by a power-law electron energy distribution, the distribution must extend to $\gamma_{\rm min}\lesssim 3$, failing which (i) the Faraday screen cannot be co-located with the synchrotron emitting electrons, and (ii) the synchrotron source must have a radius in a narrow range of $0.31<R/{\rm pc}<0.35$. If the electrons are all injected into the nebula are highly relativistic, then the $\gamma_{\rm min}<3$ can be attained by radiative cooling over a timescale of $275$ and $7400\,$yr for $R=0.05$ and $R=0.35\,$pc respectively. If the electrons instead cool by adiabatic expansion from an injection Lorentz factor of $\gamma\gtrsim \gamma_{\rm max}$, then they must have been injected at when the nebula was $<(9.4\times 10^{-4},\,2.8\times 10^{-3})\,$pc if the present size of the nebula is $(0.05,0.35)\,$pc. These results can be directly applied to the one-zone magnetar model of @margalit18. Consider their benchmark model with $B=0.25\,$G, injection energy of $\gamma_{\rm inj} = 200$, energy distribution of $N_\gamma\propto\gamma^{-1.3}$ for $\gamma\leq \gamma_{\rm inj}$ and nebular age of $\tau = 12.4\,$yr. Lack of observed circular polarisation at 4GHz then requires $\gamma_{\rm min}<1.45$. If this is accomplished via adiabatic expansion then the electron injection must have started when the central source was $12.4\times 1.45/200= 0.09\,$yr. Faraday conversion constraints therefore require significant magnetic and baryonic flux to be ejected from the magnetar within a month of its birth.
Discussion {#sec:discussion}
==========
Expanding relativistic nebula {#subsec:waxman_model}
-----------------------------
![Constraints on the expanding nebula model of @waxman2017 determined from eqn.\[eqn:hot\_nebula\_params\], for plasma $\beta=1$ and persistent source radius of $R_{\rm per}=0.1$. The cross-hatched parameter space (orange) is excluded primarily due to energetic considerations (See §\[sec:discussion\]). The straight-hatched region (cyan) is the allowed parameter range due to constraints from Faraday conversion and magneto-ionic variations (see §\[sec:fc\_limits\]) \[fig:hot\_nebula\_constraints\]](expand_nebula.pdf){width="\linewidth"}
We consider the generic ‘two-zone’ model of @waxman2017, where the source of FRB121102 is a compact object centrally located in a synchrotron nebula. The synchrotron nebula is confined against its tendency to relativistically expand by a much denser cold nebula. The Faraday rotation is provided by the shocked (and heated) part of the confining dense nebula. Notwithstanding the FRB generating mechanism and the nature of the compact object, this model links the velocity of the shock driven by the expanding synchrotron nebula into the surrounding colder medium and the density of the latter medium via: $v_{\rm sh}\approx \sqrt{P_{\rm sh}/(n m_p)}$, where $m_p$ is the proton mass, and $P_{\rm sh} = B^2(1+\beta)/(8\pi)$ is the pressure in the shocked part of the nebula which is similar to that in the synchrotron source. In convenient units, we have $$\label{eqn:vshock}
v_{\rm sh} \approx 9.1\left(\frac{T_4(1+\beta)}{\beta}\right)^{1/2}\,\,{\rm km\,s}^{-1}.$$ We now equate $v_{\rm sh}$ with the expansion velocity in eqn. \[eqn:dm\_rm\_limits\] to obtain the following family of models for the Faraday screen: $$\begin{aligned}
\label{eqn:hot_nebula_params}
T_4 & = & 2.94\times 10^4\,\beta(1+\beta)\,\eta_B^{-4}\,{\rm DM}_{10}^{-6}\nonumber \\
l & = & 0.0323\,{\rm DM}_{10}^{-3}\,\eta_B^{-2}\,(1+\beta)\,\,{\rm pc} \nonumber \\
n_e & = & 316.3\,{\rm DM}_{10}^4\,\eta_B^2\,(1+\beta)^{-1}\nonumber \\
E_{\rm sh} & = & 10^{49.2}\,\left(R_{\rm per}/{\rm pc}\right)^2\,{\rm DM}_{10}^{-5}\,\eta_B^{-4}\,(1+\beta)^2\,\,{\rm ergs},\end{aligned}$$ where $E_{\rm sh}$ is the combined thermal and magnetic energy in the shock-heated Faraday screen. A feasible model from the above family must additionally satisfy the following constraints. (i) To avoid violating Faraday conversion constraints, we need $T_4<10^{5}\,$K (see Fig. \[fig:fc\_rel\] bottom-left panel). (ii) The Faraday-rotating plasma is presumably shock heated by the expanding relativistic gas, which requires the total energy in the latter to be larger than that in the former. (iii) As in @waxman2017, we assume that the thickness of the Faraday screen must be smaller than the radius of the persistent source. The constraints on the family of models are graphically shown in Fig. \[fig:hot\_nebula\_constraints\] for a benchmark value of $R_{\rm per}=0.1\,$pc and $\beta=1$. We find feasible parameters ranges of ${\rm DM}_{10}\gtrsim 1.0$ and $\eta_B\gtrsim 0.4$ with a very weak dependence on $R_{\rm per}$. Taken together with the constraint of ${\rm DM}_{10}<1.75$ from §\[subsec:dm\_rm\], the allowed parameter ranges for the Faraday screen are tightly constrained: ${\rm DM}_{10}\in[1.0,\,1.75]$, $\eta_B\gtrsim 0.4$, $B\eta_B\in[10,\,18]\,{\rm mG}$, and ${\rm log}_{10}(T/{\rm K})\in[7.5,\,9]$. It is noteworthy and non-trivial that these self-consistent solutions should exist after a relationship between $v$, $T_4$ and $\beta$ due to eqn. \[eqn:vshock\] was imposed on observational constraints from eqn. \[eqn:dm\_rm\_limits\].
Dense filaments à-la Crab
-------------------------
Following the suggestion by @cordes2017 that FRB121102 is lensed by dense plasma structures similar to the cold filaments in the Crab nebula [@backer2000], we consider a model where the variable RM and DM are obtained by transverse passage of dense filaments across the line of sight to the FRB source. @hessels2018 [their §4.4 & eqns. 6 to 9] have summarised the resulting constraints in terms of the peak frequency at which lensing is apparent ($\nu_{\rm l}=8\,$GHz), the source$-$lens distance $D_{\rm sl}$ (units of pc), and the observer$-$lens distance $D_{\rm ol}\approx D_{\rm A}=622\,$Mpc. Requiring the filament to have the same transverse and line-of-sight extents, and to be transparent to free-free absorption at 1GHz, we obtain the following: $$\begin{aligned}
\label{eqn:lensing}
T_4&>&5.5\,{\rm DM}_{10}^{1.15}\,D_{\rm sl}^{-0.385}\nonumber \\
n_e&>&2.7\times 10^6\,\left(\frac{{\rm DM}_{10}}{D_{\rm sl}}\right)^{1/2}\,\,{\rm cm}^{-3}\nonumber \\
\left(\frac{\eta_BT_4}{\beta}\right)&>&1.7\,D_{\rm sl}^{1/2}\,{\rm DM}_{10}^{-2.5} \nonumber \\
l&<&3.7\times 10^{-6}\,\left(D_{\rm sl}{\rm DM}_{10}\right)^{1/2}\,\,{\rm pc}\end{aligned}$$
Anticipating large electron densities for filaments located within the persistent nebula ($d_{\rm sl}<1$), we impose an additional constraint: the Bremsstrahlung-cooling timescale should exceed $10\,$yr. Using @rybicki_lightman [their eqn. 5.15b] the cooling-timescale is $$\label{eqn:tau_ff}
\tau_{ff} = 0.26\, T_4^{1/2}\,\left(\frac{n_e}{10^6\,{\rm cm}^{-3}}\right)^{-1}\,\,{\rm yr} > 10$$ The simplest model we consider here is one where the Faraday rotation and lensing happens in the same filamentary structure/complex. To achieve this, we need plasma parameters that satisfy eqns. \[eqn:dm\_rm\_limits\], \[eqn:lensing\] and \[eqn:tau\_ff\] simultaneously. A scan through the parameter space shows that self-consistent solutions are only obtained for $\eta_B<0.2$. If we further require the putative filament to lie within the synchrotron nebula ($d_{\rm ls}<0.35\,$pc, as defined by the persistent source), then $\eta_B\lesssim 10^{-4}$ which will lead to unrealistically large magnetic fields. We therefor conclude that the same filamentary complex cannot provide the postulated plasma-lensing, and the observed Faraday rotation and variations thereof.
One could decouple Faraday-rotating and lensing plasma and readily find self-consistent solutions for two different plasma structures using eqns. \[eqn:dm\_rm\_limits\] and \[eqn:lensing\] separately. For instance, lensing can be caused by structures with plasma parameters of $T_4\sim 10^5$, ${\rm DM}_{10}\sim 1$, $n_e\sim 10^6\,{\rm cm}^{-3}$. The RM variations can be provided by transverse velocity of $\sim 30\,{\rm km/s}$ across a plasma structure with parameters of $T_4\sim 10^2$, $l\sim 2\times 10^{-4}\,{\rm pc}$ and $n_e\sim 5\times 10^4\,{\rm cm}^{-3}$.
Conclusions {#sec:conclusions}
===========
We have shown that Faraday conversion of linear to circular polarised radiation is a relevant effect for Fast Radio Bursts that propagate through dense and relativistic magneto-ionic media. For our test case of FRB121102, we conclude the following.
1. If the radio bursts pass through the synchrotron nebula then the latter must be an ad-mixture of highly relativistic and cold electrons. Specifically, if the electron energies are power law distributed then they must extend to $\gamma_{\rm min}\lesssim 3$ in order to not violate Faraday conversion limits.
2. The ‘one-zone’ family of magnetar models posited by @margalit18 are only consistent with Faraday conversion constraints of the magnetic flux diffusion from the magnetar initiates almost immediately after its birth. For their benchmark model of $B=0.25\,$G, $N_\gamma\propto \gamma^{-1.3}$, $\gamma_{\rm inj} = 200$, age of 12.4yr, the onset must be within a month of magnetar’s birth.
3. In models where the persistent source associated with FRB121102 is confined by a colder Faraday-rotating plasma shell [for e.g. @waxman2017], the latter is required to be shock-heated to $\sim 10^{7.5}-10^9\,$K, have an electron column of $10-17.5\,{\rm pc}\,{\rm cm}^{-3}$, a geometric parameter of $0.4\lesssim \eta_B\leq 1$, and magnetic field of $10\,{\rm mG}<\eta_B B < 18\,{\rm mG}$. The existence of such a self-consistent and over-constrained solution is not trivial and lends credence to the model.
4. In models involving dense filaments as in the Crab nebula, the magneto-ionic variations (DM and RM) cannot come from the same plasma structures that also act as a plasma lens which is postulated to generate certain time-frequency structures seen in FRB121102 [@cordes2017].
We emphasise that observations targeting the detection of Faraday-converted circular polarisation at $\sim 1$GHz in bursts from FRB121102 are likely of great interest. We further advocate for ‘de-rotation’ circular polarised signals in FRBs with linear polarisation fractions below unity, in the event that Faraday-converted circular polarisation has been averaged out. Finally, we anticipate that the arguments presented here will be of significance to other FRBs (particularly of a repeating nature) that are associated with radio-synchrotron sources and/or dense magneto-ionic media.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors thank (i) Andrei Grizinov and Eli Waxman for discussions on the implications of Faraday conversion, (ii) Eli Waxman for commenting on the manuscript, (iii) Jason Hessels for discussions regarding observational aspects of FRB121102, (iv) Dongzi Li for pointing out an error in our application of Farday conversion in an earlier version of the manuscript, and (v) the organisers of the 2018 Schwartz/Reisman Institute for Theoretical Physics workshop on Fast Radio Bursts for their hospitality. Numerical computations used `scipy`, `numpy`, `Python2.7`, `Python3.0` and `matplotlib` was used to render figures.
Backer, D. C., Wong, T., & Valanju, J. 2000, , 543, 740 Bower, G. C., Falcke, H., Sault, R. J., & Backer, D. C. 2002, , 571, 843 Chatterjee, S., Law, C. J., Wharton, R. S., et al. 2017, , 541, 58 Cordes, J. M., Wasserman, I., Hessels, J. W. T., et al. 2017, , 842, 35 Gajjar, V., Siemion, A. P. V., Price, D. C., et al. 2018, , 863, 2 Hessels, J. W. T., Spitler, L. G., Seymour, A. D., et al. 2018, arXiv:1811.10748 Homan, D. C., Attridge, J. M., & Wardle, J. F. C. 2001, , 556, 113 Huang, L., & Shcherbakov, R. V. 2011, , 416, 2574 Kennett, M., & Melrose, D. 1998, , 15, 211 Lorimer, D. R., Bailes, M., McLaughlin, M. A., Narkevic, D. J., & Crawford, F. 2007, Science, 318, 777 Marcote, B., Paragi, Z., Hessels, J. W. T., et al. 2017, , 834, L8 Margalit, B., & Metzger, B. D. 2018, , 868, L4 Masui, K., Lin, H.-H., Sievers, J., et al. 2015, , 528, 523 Michilli, D., Seymour, A., Hessels, J. W. T., et al. 2018, , 553, 182 Pacholczyk, A. G., & Swihart, T. L. 1970, , 161, 415 Readhead, A. C. S. 1994, , 426, 51 Rybicki, G. B., & Lightman, A. P. 1979, New York, Wiley-Interscience, 1979. 393 p., Sazonov, V. N. 1969, , 13, 396 Spitler, L. G., Cordes, J. M., Hessels, J. W. T., et al. 2014, , 790, 101 Spitler, L. G., Scholz, P., Hessels, J. W. T., et al. 2016, , 531, 202 Tendulkar, S. P., Bassa, C. G., Cordes, J. M., et al. 2017, , 834, L7 Thornton, D., Stappers, B., Bailes, M., et al. 2013, Science, 341, 53 Walker, M. A. 1998, , 294, 307 Waxman, E. 2017, , 842, 34
\[lastpage\]
[^1]: E-mail: [email protected]
[^2]: In the model of @waxman2017, which we discuss in §\[subsec:waxman\_model\], equipartition is required by dynamical and source-size (from scintillation) constraints.
[^3]: This assumption will be relaxed in §\[sec:faraday\_conversion\]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The Disk Detective citizen science project aims to find new stars with 22 $\mu$m excess emission from circumstellar dust using data from NASA’s WISE mission. Initial cuts on the AllWISE catalog provide an input catalog of 277,686 sources. Volunteers then view images of each source online in 10 different bands to identify false-positives (galaxies, background stars, interstellar matter, image artifacts, etc.). Sources that survive this online vetting are followed up with spectroscopy on the FLWO Tillinghast telescope. This approach should allow us to unleash the full potential of WISE for finding new debris disks and protoplanetary disks. We announce a first list of 37 new disk candidates discovered by the project, and we describe our vetting and follow-up process. One of these systems appears to contain the first debris disk discovered around a star with a white dwarf companion: HD 74389. We also report four newly discovered classical Be stars (HD 6612, HD 7406, HD 164137, and HD 218546) and a new detection of 22 $\mu$m excess around a previously known debris disk host star HD 22128.'
author:
- 'Marc J. Kuchner'
- 'Steven M. Silverberg'
- 'Alissa S. Bans'
- Shambo Bhattacharjee
- 'Scott J. Kenyon'
- 'John H. Debes'
- Thayne Currie
- Luciano Garcia
- Dawoon Jung
- Chris Lintott
- Michael McElwain
- 'Deborah L. Padgett'
- 'Luisa M. Rebull'
- 'John P. Wisniewski'
- Erika Nesvold
- Kevin Schawinski
- 'Michelle L. Thaller'
- 'Carol A. Grady'
- Joseph Biggs
- Milton Bosch
- Tadeas Cernohous
- 'Hugo A. Durantini Luca'
- Michiharu Hyogo
- Lily Lau Wan Wah
- Art Piipuu
- Fernanda Pineiro
title: 'Disk Detective: Discovery of New Circumstellar Disk Candidates through Citizen Science'
---
Introduction {#sec:introduction}
============
All-sky mid-infrared surveys have revolutionized the science of planet formation by discovering populations of young stars and main sequence stars with excess infrared radiation indicating the presence of dusty circumstellar disks. These disks, which include gas-rich protoplanetary disks around Young Stellar Objects (YSOs) and dusty debris disks around main sequence stars, serve as the signposts of planet formation [e.g., @2002ApJ...577L..35K]. They inform us about the timescales and the environment of planet formation [e.g., @2005ApJ...620.1010R; @2007ApJ...671.1784H; @2015ApJ...808..167J], and the present day locations and dynamics of planets [e.g., @2010ApJ...718L..87T; @2012ApJ...748L..22M; @2013ApJ...766L...1Q; @2015ApJ...798...83N; @2015ApJ...807L...7C].
The IRAS all sky survey discovered the first extrasolar debris disks [@1984ApJ...278L..23A] and provided a large sample of debris disks [e.g., @2007ApJ...660.1556R]. After IRAS, AKARI surveyed the whole sky at 9 and 18 $\mu$m with $\sim 7$ times better sensitivity than IRAS, finding many more new disks . Some disk discoveries have come from pointed studies, like the Spitzer Formation and Evolution of Planetary Systems (FEPS) survey [@2009ApJS..181..197C]. But many of the best-studied, most informative disks (like TW Hydra, Fomalhaut, etc.) are relatively isolated on the sky, requiring an all-sky survey to find them.
NASA’s Wide-field Infrared Survey Explorer (WISE) is the most recent and sensitive all-sky mid-infrared survey [@2010AJ....140.1868W], with a further factor of $\sim 80$ gain in sensitivity over AKARI in the mid-IR. Using a 16-inch mirror in a Sun-synchronous polar orbit, WISE scanned the sky at 3.4 $\mu$m, 4.6 $\mu$m, 12 $\mu$m, and 22 $\mu$m (bands W1, W2, W3, and W4 respectively). The WISE cryogenic mission, launched in 2009, lasted a little over 10 months and was followed by the first post-cyrogenic mission, NEOWISE. The AllWISE catalog[^1] combines data from both phases, making it the most comprehensive mid-infrared multi-epoch view of the sky available today.
Previous infrared surveys for debris disks have provided target lists for exoplanet searches via direct imaging [@2008ApJ...672.1196A; @2013ApJ...773...73J; @2013ApJ...773..179W; @2015ApJ...800....5M]. Debris disks found with WISE should provide crucial targets for upcoming generations of exoplanet searches. WISE could detect debris disks around main sequence A stars to a distance of 300 pc and protoplanetary disks around T Tauri stars to 1 kpc.
Indeed, many teams have used the WISE data to find new debris disks, searching a vast catalog of $>747$ million WISE sources. @2012MNRAS.427..343M cross-correlated the WISE source list with the Hipparcos catalog, finding over 86,000 stars with suspected infrared excesses. @2013MNRAS.433.2334K, @2013ApJS..208...29W and @2014ApJS..212...10P performed more careful searches for debris disks in the WISE source list using the Hipparcos catalog and found 6, 70 and 108 new debris disk candidates, respectively. Other specific surveys for debris disks have focused on stars with ages determined from chromospheric activity @2014ApJ...780..154V, white dwarfs [@2011ApJ...729....4D; @2012ApJ...759...37D], M dwarfs , G-K dwarfs [@2014MNRAS.437..391C], Kepler candidate exoplanet systems and other exoplanet catalogs [@2012ApJ...757....7M].
Likewise, the WISE data on young clusters and star-forming regions have attracted much attention. @2011ApJS..196....4R [@2014ApJ...784..126E; @2014AJ....147..133L] scoured the Taurus-Auriga Region. @2012ApJ...744..130K searched 11 outer Galaxy massive star-forming regions and three open clusters. Other studies have examined smaller regions, like the Western Circinus molecular cloud [@2011ApJ...733L...2L], the young open cluster IC 1805 the region S155 [@2014RAA....14.1269H], the Sco-Cen and $\eta$ Cha associations [@2012MNRAS.421L..97R; @2012ApJ...758...31L], nearby moving groups of young stars [@2012ApJ...751..114S] and $\lambda$ and $\sigma$ Orionis [@2015AJ....150..100K]. Still others have attempted to take in the whole sky, using color cuts or cross correlating with IRAS [@2014ApJ...784..111L]. Many of these searches were based on preliminary data releases with less sensitivity than the AllWISE release, but they have already uncovered thousands of candidate Class I, II and III YSOs and transitional disks, helping fill in our picture of the timing and progression of star formation.
Unfortunately, because of its limited spatial resolution (12 arcsec at 22 $\mu$m) contamination and confusion limit every search for disks with WISE [e.g. @2012MNRAS.426...91K]. Contamination sources include unresolved companion stars and other stars nearby on the sky, background galaxies, Galactic cirrus, and even asteroids and airplanes. For this reason, most recent searches include visual inspection of the WISE images [see e.g., @2011ApJ...729....4D; @2013ApJS..208...29W; @2014MNRAS.437..391C; @2014ApJS..212...10P]. Computer cuts alone can provide a first stage of vetting, but they generate catalogs riddled with false positives [@2012MNRAS.426...91K]; . Color cuts and source quality flags can help , but the color loci of disk candidates overlaps with the color loci of blended background galaxies [@2012ApJ...744..130K] and peaks in the Galactic dust emission. @2012MNRAS.426...91K used the IRAS 100 $\mu$m level to discard many false positive disk candidates contaminated with Galactic dust emission, but using this method prohibits searching many interesting star-forming regions.
Because of these challenges, many disks remain to be found with WISE data, even after all the efforts described above. The largest published study of debris disks [@2013ApJS..208...29W] and the still larger WISE science team disk study (Padgett et al. in prep.) are based on the Hipparcos and Tycho catalogs. These catalogs are magnitude limited in V band, so they omit a vast population of redder, late type stars[^2].
Moreover, a vast solid angle in young clusters and star-forming regions remains to be properly searched with WISE—each candidate examined by eye and followed up with spectroscopy and higher resolution imaging. When ran the all-sky data through a novel color filter to search for YSO candidates (without vetting the candidates by eye), he found a total of $\sim 10,000$ objects of interest; the WISE studies of young clusters and star-forming regions described above (which mostly included visual vetting) yielded a total of $\sim 4000$ disk candidates. The difference between these numbers provides a minimal measure of what remains for us to study with WISE: $\gtrsim 6000$ objects with colors consistent with YSOs that have not yet been visually inspected.
Here we describe a new project to scour the WISE data for new debris disks and YSOs. The Disk Detective citizen science/crowdsourcing project classifies WISE sources via a website, diskdetective.org, where volunteers examine images from WISE, the Two Micron All Sky Survey (2MASS), the Digitized Sky Survey (DSS) and when available, the Sloan Digital Sky Survey (SDSS), to check them for false positives. This approach should allow us to unleash the full potential of WISE for finding new disks, probing the cooler stars and isolated objects missed by previous debris disk searches, a catalog 8 times the size of the large @2013ApJS..208...29W survey. We describe the online vetting process in Section \[sec:approach\], our small-telescope follow-up program in Section \[sec:followup\] and we present our first list of 37 disk candidates in Section \[sec:candidates\].
Citizen Science Approach {#sec:approach}
========================
Disk Detective is a new addition to the successful Zooniverse network of Citizen Science Alliance projects [@2008MNRAS.389.1179L]. Visitors to the site (“users”) view “flipbooks” showing several images of the same source at different wavelengths. Figure \[Fig\_screenshot\] shows a sample screenshot from DiskDetective.org illustrating one frame in a flipbook.
After they view the flipbooks, users answer a question, “What best describes the object you see?”, by clicking on one or more of six buttons. The site then records the user’s choice(s) for interpretation by the Disk Detective science team and offers the user another source to classify. This approach is borrowed from another Zooniverse Project, Snapshot Serengeti (http://www.snapshotserengeti.org). Snapshot Serengeti shows users flipbooks of wildlife photographs, asking users to identify animal and bird species that appear in the images, taking advantage of the human eye’s ability to spot moving objects camouflaged by noise [@2015Swanson]. At Disk Detective, instead of identifying animals in the grasses of the Serengeti, users identify clean point sources in a forest of astrophysical and instrumental contaminants: galaxies, ISM, artifacts, etc.
The flipbooks in Disk Detective generally consist of ten images of each source: images in four WISE bands, three bands from the 2-Micron All Sky Survey [@2006AJ....131.1163S 2MASS] and three bands from the Digital Sky Survey [@1998wfsc.conf...89D DSS]. When possible, we use images from the SDSS seventh data release instead of DSS; these SDSS data cover about 1/4 of the sky [see @2006AJ....131.2332G; @2009ApJS..182..543A]. The images are independently scaled using IDL color table 1 (BLUE/WHITE), matching the color scale to the full range of the data. Hence, the flipbook approach works better than attempting to show all these bands in a single multi-color image, which would tend to be dominated by one or two bands. Overlaid on every flipbook is a circle with a radius of 10.5 arcseconds, roughly the area we must ensure is free of contamination before we can trust the photometry. (The circle’s radius was chosen conservatively; the nominal resolution of WISE at 22 $\mu$m is 12 arcseconds FWHM.) Also, overlaid on each image is a cross marking the center of the image at the WISE 3.4 $\mu$m band (W1) and indicating the expected uncertainty in position for an uncontaminated point source. The disk candidates are unresolved by WISE, so our search image is a tightly concentrated red object, possibly with diffraction spikes, that does not shift position from band to band.
Users view the flipbook by clicking on an arrow-shaped “play” button on the lower left of the screen, shown in Figure \[Fig\_screenshot\]. They also have the option to scroll through the flipbook frame by frame using the scrub bar beneath it. The frames are each labeled with the wavelength and the name of the survey that yielded the image, e.g., “2MASS K (2.16 $\mu$m)”. After a user has viewed the whole flipbook, he or she can then choose from among six classification buttons to click on, labeled “Multiple objects in the Red Circle”, “Object Moves off the Crosshairs”, “Extended beyond circle in WISE Images” “Empty Circle in WISE images”, “Not Round in DSS2 or 2MASS images” and “None of the Above/Good Candidate”. With the exception of the “None of the Above” option, the user can choose more than one description per flipbook. After at least one of these classification buttons is chosen, a button labeled “Finish” becomes active; clicking this button records the user’s choices and causes the next flipbook to appear.
Since its launch, Disk Detective has attracted a vast user community. Roughly 1.5 million classifications have been performed so far, by roughly 28,000 volunteers. Roughly half of the classifications come from an enthusiastic group of “superusers”. Fifteen superusers have already classified $>10,000$ sources each; seven have classified $> 30,000$. The superusers started their own email discussion group via Google groups to work together on Disk Detective issues. They now help train other users and form a crucial extension of the Disk Detective science team (see below).
Communication with Disk Detective users is aided by the new Zooniverse translation crowdsourcing tool. Using this online tool, volunteers have translated the site into Spanish, French, Russian, German, Hungarian, Polish, Bahasa, Romanian, Portuguese, Japanese, and Chinese (traditional and simplified characters); the translated sites are accessible via a link in the upper right corner of site.
The DiskDetective.org site is tied into the “TALK” social network common to Zooniverse sites. TALK has a page for each subject on Disk Detective that provides a simple Spectral Energy Distribution (SED) for the object composed of the 2MASS and AllWISE photometry and also a link to the SIMBAD page on the source if one exists. On TALK, users can create and maintain collections of their favorite subjects by clicking on a button labeled “collect”.
Pre-Selection of WISE Sources {#sec:approach2}
-----------------------------
To choose sources to upload to the website, we performed some initial computer-based cuts on the WISE data, informed by published debris disk searches (see above) but not limited by the Hipparcos or Tycho catalogs, for example. We utilized signal-to-noise cuts (`w4snr, w1sigmpro w4rchi2 and w4sigmpro`) and some of the AllWISE catalog flags (`cc_flags, xscprox, na, nb, n_2mass, and ext_flg`) to remove sources that were noisy or close to known extended sources. Though many searches have used sophisticated color cuts to focus on particular kinds of disks [e.g., @2012ApJ...744..130K], we kept our color cuts minimal to cast as broad a net as possible. To preselect sources with infrared excesses, we merely removed all sources with \[W4\] $>$ \[W1\]-0.25. (\[W1\] is magnitude in the WISE 3.4 $\mu$m band, and \[W4\] is magnitude in the WISE 22 $\mu$m band.) This criterion corresponds to a 26% excess over a Rayleigh-Jeans slope between those two bands. We also required that a WISE 4 excess significant at the 5-$\sigma$ level. Table \[tab:cuts\] summarizes all these initial cuts.
[l l]{} Criterion & Meaning\
w1mpro $>$ 3.5 & WISE 1 magnitude $> 3.5$\
w4mpro $<$ (w1mpro - 0.25) & WISE 4 excess of 0.25 magnitudes over W1\
w1mpro $>$ 5\*sqrt(w1sigmpro\*w1sigmpro & The WISE 4 excess is significant at the 5-$\sigma$ level.\
+ w4sigmpro\*w4sigmpro) + w4mpro &\
w4sigmpro is not null and w4rchi2 $<$ 1.3 & WISE 4 profile-fitting yielded a fit with $\chi^2 < 1.3$\
w4snr $>=$ 10 & WISE 1 profile-fit signal-to-noise ratio $>$ 10\
w4nm $>$ 5 & Source detected at WISE 4 in at least 5 individual\
& 8.8 second exposures with SNR$>3$\
na = 0 and nb = 1 & The profile-fitting did not require active deblending.\
n\_2mass = 1 & One and only one 2MASS PSC entries found\
& within a 3" radius of the WISE 1 source position.\
cc\_flags\[1\] not matches ’\[DHOP\]’ & No diffraction spike, persistence, halo or optical ghost\
cc\_flags\[4\] not matches ’\[DHOP\]’ & issues at WISE 1 or WISE 4.\
xscprox is null or xscprox $>$ 30 & No 2MASS XSC source $< 30''$ from the WISE source.\
ext\_flg = 0 & Photometry not contaminated by known\
& 2MASS extended sources.\
We launched the site on January 28, 2014 with a first batch of subjects covering only the Galactic latitudes +30 to +40, +50 to +90, and -40 to -90, with no additional magnitude limit. At first, about 20% of the initial upload was made available on the site. We soon realized that most of the volunteer effort was being spent classifying faint, extragalactic sources, so we decided to impose a magnitude limit on the search. We chose a criterion of $J < 14.5$ because the subjects brighter than this magnitude were clearly concentrated in the Galactic plane, while fainter subjects appeared to be isotropically distributed on the sky. We uploaded a second batch of 272,022 subjects on May 30, 2014. This second batch covered the rest of the sky, but was limited to $J < 14.5$. We also deactivated all the previously uploaded and active $J > 14.5$ subjects on this date, so presently the $J > 14.5$ subjects currently make up a small subset of the data for which we have classifications.
Classification Data
-------------------
We examine the classification data weekly to chart our progress. As of August 25, 2015, the selection of classification buttons had the distribution shown in Table \[tab:buttons\]. Clearly, the dominant false positive rejected by the classification process is “Multiple objects in the Red Circle”.
Note that the “Empty Circle in WISE images” button exists mainly to allow users a reasonable response option if, for example, there were a network glitch. None of the data actually had empty circles in all the WISE images, though some subjects did have empty red circles in a single band, and some users chose this “Empty Circle in WISE images” classification for such subjects. The rarity of this situation is reflected in the very low (0.6%) classification rate for this button.
[l c c]{} &\
Classification Button & J < 14.5 & J > 14.5\
Multiple objects in the Red Circle & 40.5 & 6.5\
Object Moves Off the Crosshairs & 6.5 & 6.9\
Extended beyond circle in WISE Images & 23.4 & 15.9\
Empty circle in WISE images & 0.6 & 0.4\
Not round in DSS2 or 2MASS images & 9.0 & 37.6\
None of the Above/Good Candidate & 20.0 & 33.0\
So far, we have investigated two basic algorithms for sorting the raw data into classifications: a plurality algorithm and majority algorithm. By the plurality algorithm, the classification with the most votes becomes the official classification. By the majority algorithm, the official classification is one with $\ge 50$% of the votes, if one exists.
To help test these algorithms, we prepared a “gold standard” data set of 500 subjects classified by the members of the science team. All of the subjects in this “gold standard” data set were classified by two or more science team members, and all tie votes were discarded. This process showed that science team members agree with one another at roughly the 82% level as to whether an object is “good” or not. A challenge in creating this “gold standard” set was that sometimes science team members ruled out candidates based on their expertise/outside information rather than strictly doing what the website asked for. There was also a range in terms of how conservative the science team members were in terms of crowded fields and ISM background. For example, the YSO experts (used to looking in the Galactic plane) on the science team tolerated background objects and “extended” objects more readily than the debris disk experts on the team.
The majority algorithm performed well at selecting objects deemed to be classified as “None of the Above/Good candidate”. On a randomly selected group of about 500 subjects, 94% of those classified as “None of the Above/Good candidate” by the majority algorithm agreed with the gold standard set rankings. For the other five classification buttons, the agreement between classifications selected via the plurality algorithm and the gold standard set varied: 52% for “Extended”, 54% for “Not round”, 24% for “Object Moves Off the Crosshairs”, and 7% for “Empty circle”. This lack of agreement suggests that users may be sometimes neglect to click [*all*]{} of the relevant buttons when a subject is “bad” for more than one reason. However, since our desire is simply to rule out false positives, this lack of agreement on the precise nature of the false positive does not hamper our study.
Deciding on classifications via the plurality algorithm generally leads to larger disagreements with the gold standard set and the classification data. The exception to this rule seems to be the case of “multiple objects in the red circle”; subjects classified as such by the plurality vote agree with the gold standard set 96% of the time.
Table \[tab:buttons\] shows the total selection distribution between the categories on the main site from the first $\sim 1$ million classifications. Users selected the “good” button nearly 20% of the time. Using the majority algorithm to determine a final classification, we get an 16% yield of “good” objects based on those classifications.
Some of the brighter Disk Detective subjects show diffraction spikes and noise from detector saturation, especially in DSS, Sloan, and WISE 1 images. When we launched Disk Detective we did not explicitly explain these phenomena anywhere on the site, though we readily answered questions about them in the chat forum TALK. Nonetheless, many of our first users interpreted bright stars with spikes as “oval” or in some cases “extended”. On March 31, 2014, we edited the spotters guide and the tutorial at DiskDetective.org to include examples with diffraction spikes, which greatly reduced the problem, but some of these mistakes persist in our data. We expect this confusion over diffraction spikes to get better over time, as most of our classifications now come from participants who are well aware of the problem.
Disk Detective Objects of Interest (DDOIs) {#secondvetting}
------------------------------------------
With nearly 300,000 total objects from AllWISE to classify and 18-25% yield described above, we estimate that the online classification scheme at DiskDetective.org will produce up to 75,000 good objects, still a daunting number to investigate. However, the automated online classification stage is just the beginning of our vetting process. The next stage of the process aims to harvest Disk Detective Objects of Interest (DDOIs), subjects that we consider deserving of additional follow-up observations.
For this next stage of vetting, we created a collection of Google spreadsheets which both the science team and the superusers could edit. We first populated the spreadsheets with subjects chosen as “None of the Above/Good Candidate” using the latest classification data. We chose subjects in the right portion of the sky for any upcoming follow-up observations and also populated the lists in order of agreement fractions and brightness (in J band), ensuring that the bright subjects, and those with high agreement percentages were looked at first. Then we invited the superusers to add their own favorite objects, which they collect using tools on the TALK social network. We also invited the entire Disk Detective user community to submit subjects automatically via a Google form (but not directly edit the spreadsheet).
We coached the superusers on how to research each source in SIMBAD and VizieR (and sometimes NED) to fill in information about spectral type, proper motion, variability, parallax, prior observations, and make comments on the SED, etc. Then we checked the superusers’s comments and selected the follow-up targets (DDOIs) from the list based on the following criteria.
1. SIMBAD object descriptions excluding post-AGB stars, carbon stars, novae, Cepheids, cataclysmic variables, high-mass x-ray binaries, eclipsing binaries, galaxies, Active Galactic Nuclei, planetary nebulae, reflection nebulae, rotational variables, symbiotic stars, or Wolf-Rayet stars. Note that we did keep sources with SIMBAD object descriptions Shell Star, Orion Variable, and White Dwarf.
2. No Long Period Variables (LPVs), SR+L, Slow Irregular Variables, Miras, Semi-regular Variables, Semiregular Pulsating Variables, or Carbon stars based on literature searches.
3. Including only spectral types B through M according to SIMBAD, when a type is available.
Only about half of the subjects have entries in SIMBAD. So we often relied on VizieR to help us search the relevant literature. Many of the subjects had unknown spectral types, and many were severely reddened. So at this stage of the vetting process, we often simply labeled sources as “late-type based on color.”
4. No sign that the WISE 1 photometry drops out due to saturation, based on visual inspection of the SED.
5. No known companions within $16''$, except for spectroscopic binaries.
6. For M stars and other subjects with $V-J > 1$, we require $[W1] - [W4] > 0.9$. The peak of the thermal emission from cool star photospheres may lie at long enough wavelengths that the Rayleigh-Jeans limit no longer accurately describes the photosphere’s $[W1] - [W4] $ color even in the absence of circumstellar dust. Hence, we impose this more stringent requirement of 0.9 magnitudes of 22 $\mu$m excess for these cool subjects. For some red subjects, however, we find that the SED clearly curves upwards at W4; we do not exclude these subjects.
In the process, we naturally rediscovered many known disks. So we added the following additional criterion:
7. No sources that have already been imaged by a pointed space mission (i.e. [*Spitzer*]{}, Herschel, HST) or 8-m class telescope (i.e. Keck, Gemini, VLT, Magellan) (based on the published literature), except those without good quality spectral types (SIMBAD quality C or higher).
Our search includes many sources in and near the Galactic plane and many sources with no parallax measurements from Hipparcos. But since these sources require extra care, we have added two additional criteria for the purpose of this paper:
8. No sources within $5^\circ$ of the Galactic plane.
<!-- -->
9. Only sources with parallax measurements from Hipparcos.
Most of the subjects on the vetting spreadsheets do not meet these additional eight criteria. But sources that do meet all these criteria (plus the criteria in Table \[tab:cuts\], of course) we label as DDOIs and place in our our queue for follow-up observations. So far we have collected 770 DDOIs in total that have survived the above vetting by the science team and/or by multiple superusers. Of these, 517 have classification histories via the main Disk Detective online classification tool. The remaining DDOIs have minimal classification data, since they were submitted directly by volunteers, and some volunteers choose to flip through images on TALK rather than the main site.
The yield at this stage varies greatly depending on how we rank the spreadsheets. Higher galactic latitudes provide higher yields, as do sources that are brighter in WISE 4. Also, as our users have become more educated, they have become better at selecting subjects to place on the spreadsheets, which raises the yield. But as of now, the typical yield at this stage of vetting (DDOIs per source on the vetting spreadsheet) is about 12%.
Multiplying the size of the input catalog by the automated vetting yield (24.4%) and then by the DDOI vetting yield (about 12%) gives a final yield of about 3% and an estimate of the total number of DDOIs we ultimately expect to discover of $\sim 8000$. This estimate suggests that our search is presently about 10% complete. However, it is worth noting that many of these criteria, while catching a lot of the aforementioned AGB stars that pollute our “good” candidate list, also eliminate a population of YSO disks and M stars (for example, YSOs can be irregular variables).
Using DDOIs As Quality Indicators {#ddoisquality}
---------------------------------
The DDOIs have all been carefully vetted by hand and researched in the literature by multiple scientists and/or well-trained superusers. So this list of sources serves as a reference set of subjects that we can use to make decisions about how to run the vetting.
One major consideration of any citizen science project is when to remove, or “retire,” subjects from classifications on the site. To make that decision, we need to know roughly how long it takes the user population to converge on an answer. Figure \[Fig\_agreement\] shows how the standard deviation of the agreement for majority-algorithm ruled “good” candidates in our DDOI set varies with number of total classifications. For these subjects, “agreement” is defined simply as the number of good classifications divided by the total number of classifications the subject has received. Between 10-15 classifications, the standard deviation of the agreement for this subset of “good” DDOI candidates levels off, suggesting that there would be minimal marginal benefit from requiring additional classifications beyond this point. Thus the current iteration of the site uses a conservative 15 total classifications as our benchmark for when to retire subjects.
Follow-Up Observations {#sec:followup}
======================
We obtained and analyzed spectra for our DDOIs using the FAST spectrograph [@1998PASP..110...79F] on the Tillinghast 1.5m telescope at Fred Lawrence Whipple Observatory during May–October 2014, employing the 300 mm${}^{-1}$ grating and the 3” slit. These spectra cover 3800–7500 Å at a resolution of $\sim$ 6 Å. We flux- and wavelength-calibrated the spectra using the Image Reduction and Analysis Facility (IRAF) software system. After trimming the CCD frames at each end of the slit, we corrected for the bias level, flat-fielded each frame, applied an illumination correction, and derived a full wavelength solution from calibration lamps acquired immediately after each exposure. The wavelength solution for each frame has a probable error of $\pm$0.5–1.0 Å. We extracted sources and sky spectra using the optimal extraction algorithm within APEXTRACT. The absolute flux-calibration for each night relies on observations of 2-5 standard stars [@1975ApJ...197..593H; @bar82; @1988ApJ...328..315M] and has an uncertainty of $\pm$5%–10%.
This follow-up spectroscopy has proven vital to our vetting of DDOIs. Roughly half of all DDOIs initially have no reliable spectral type. Additionally, for these objects the luminosity class is generally completely unconstrained, save for clues from parallax and proper motion measurements. This situation is more dire for red (or reddened) sources and late-type stars, which will be the subject of future work. M giants often produce their own dust, so disks around these stars are not of particular interest from the perspective of planet formation. FAST spectroscopy also allows us to screen for certain false positives, such as blended AGN.
Though our initial candidate list in this paper is small, we plan to follow-up on the entire list of DDOIs and thus will have a large number of observed spectra. Manual spectral classification of every object is highly inefficient. To speed up the process, we used the semiautomatic quantitative spectral-typing code SPTCLASS[^3], an IRAF/IDL code based on the methodology outlined in @2004AJ....127.1682H. We decided that in this first paper we would only publish DDOIs that are in Hipparcos catalogs, and consequently do have spectral types inferred from color and parallax, as a test of our pipeline.
SPTCLASS calculates spectral types of stars using spectral indices, comparing line fluxes of spectral features that are sensitive to effective temperature [e.g., @1943assw.book.....M; @1999RMxAA..35..143S; @1999yCat.6071....0C; @2001AJ....121.2148G; @1977ApJS...34..101P; @1995AJ....110.1838R and others]. SPTCLASS uses three independent spectral typing modules: indices characterizing early (OBA, 44 indices), intermediate (FG, 11 indices), and late (KM, 16 indices) spectral types. Each index is based on the equivalent width for the spectral feature, which is calculated by measuring the decrease in flux from the expected continuum due to line absorption. Indices measured by this procedure are generally insensitive to reddening, so long as each band’s wavelength coverage is relatively small. The indices have been calibrated as a function of spectral type using O8-M6 main sequence standards; this large extent for all indices ensures that degeneracy of determination is eliminated.
SPTCLASS first calculates three spectral types for each star, one for each module, by taking a weighted average of the indices used in that module, using weights estimated from the computed error for each index. It then calculates a weighted average of spectral types for all indices, using weights estimated by the computed error for each index from the measured index value and fit of the index to spectral type. The code then discards spectral indices if the spectral type they indicate deviates from the mean by $>3 \sigma$, or if their computed error is more than 6 subtypes. This minimizes possible contamination by artifacts and emission lines. We report here the final SPTCLASS spectral type, which is the new average after discarding these deviant indices. The typical 1-$\sigma$ formal uncertainty for these classifications is 2 subtypes.
First Batch of Disk Candidates {#sec:candidates}
------------------------------
Table \[tab:ddois\] lists our first batch of 50 DDOIs. Although some of the sources in Table \[tab:ddois\] are Be stars or disk candidates that have been previously reported, 37 of these sources are new disk candidates from the Disk Detective project.
We have decided to present only stars in with Hipparcos parallaxes as a test of our spectral classification pipeline. Though @2013ApJS..208...29W performed a large search for stars in the Hipparcos catalog that have excess emission in W4, they used a $[K_s] - [W4]$ cutoff, and the AllSky catalog. Our selection criteria $[W1] - [W4]$ and input catalogs (AllWISE) are different, so we find objects they missed, even in the Hipparcos catalog.
Table \[tab:ddois\] lists the photometry of these disk candidates from 2MASS and SIMBAD (V band). It also lists the spectral types we derived from our FAST spectra, informed by photometry and Hipparcos distances, and WISE photometry. Eleven of the sources in Table \[tab:ddois\] have previously been reported as disk candidates by @2012ApJ...752...58Z, @2012MNRAS.426...91K, @2013MNRAS.433.2334K, @2013ApJS..208...29W, @2013ApJ...773..179W and @2014ApJS..211...25C. We include these objects here as a check on our consistency with other searches. However, 37 of these sources have not been previously reported as disk candidates (and are not Be stars).
The WISE photometry in Table \[tab:ddois\] is taken from the All-Sky data release and corrected for saturation according to the formulae in @2014ApJS..212...10P. This corrected All-Sky WISE photometry is more accurate for sources brighter than about 8th magnitude in W1. No color correction is applied to the WISE photometry, since the sources have nearly Rayleigh-Jeans spectra. If we assumed instead that the emission were entirely from a 200 K blackbody disk, the correction to WISE 4 would be -0.016 magnitudes [@2010AJ....140.1868W], negligible compared to our 0.25 magnitude WISE 1-WISE 4 excess criterion for inclusion in Disk Detective.
The vast majority of our DDOIs did not have previously-identified luminosity classes. We assigned them using the approach of @2014ApJS..212...10P, who separated dwarfs from giants in their catalog via a simple cut on an HR diagram, retaining only stars with $M_V > 6.0(B-V)-1.5$. Fig. \[Fig\_HR\] shows an HR diagram for our candidates. The stars on Fig. \[Fig\_HR\] are color coded according to spectral type: blue=B, green=A, red=F. Most of our disk candidates are probably dwarfs since they fall below the @2014ApJS..212...10P cut, which is shown by the dashed line.
[lllcccccl]{}
AWI00055sz & 7406 & J011636.23+740136.6 & B1III & $549 \pm 103$ & 7.05 & $6.764 \pm 0.056$ & $0.619 \pm 0.052$ & a, EW$= -17.7064$\
AWI0005w41 & 218546 & J230817.21+511146.3 & B8III & $794 \pm 498$ & 8.25 & $7.918 \pm 0.020$ & $0.784 \pm 0.072$ &\
AWI0005bow & 164137 & J175911.27+135417.8 & B8III & $820 \pm 544$ & 8.02 & $7.768 \pm 0.020$ & $1.571 \pm 0.050$ &\
AWI0005ae4 & 224098 & J235449.26+742436.2 & B8V & $240 \pm 20 $ & 6.6 & $6.388 \pm 0.019$ & $0.433 \pm 0.050$ &\
AWI0000m2p & 4670 & J004848.00+181850.6 & B9IV & $386 \pm 98 $ & 7.94 & $7.841 \pm 0.020$ & $0.898 \pm 0.080$ &\
AWI00055sx & 6370 & J010652.55+743754.5 & B9IV & $286 \pm 51 $ & 8.37 & $8.168 \pm 0.029$ & $0.919 \pm 0.084$ &\
AWI0005yiz & 6612 & J010722.60+380143.9 & B9V & $316 \pm 51 $ & 7.14 & $7.156 \pm 0.020$ & $0.453 \pm 0.067$ & b, EW$= 4.09265$\
AWI00062a8 & 9985 & J013756.15+211539.9 & B9V & $202 \pm 23 $ & 7.95 & $7.824 \pm 0.018$ & $1.993 \pm 0.050$ &\
AWI0005ym7 & 23873 & J034921.76+242251.0 & B9V & $122 \pm 9 $ & 6.61 & $6.602 \pm 0.017$ & $1.138 \pm 0.057$ &\
AWI0000uj2 & 138422 & J153046.05+342756.4 & B9V & $108 \pm 4 $ & 6.81 & $6.719 \pm 0.021$ & $0.542 \pm 0.044$ & c\
AWI00004o8 & 152308 & J165204.85+145827.2 & B9V & $124 \pm 7 $ & 6.5 & $6.504 \pm 0.034$ & $0.494 \pm 0.048$ &\
AWI00062fh & 207888 & J215221.25-031028.9 & B9V & $184 \pm 17 $ & 6.6 & $6.688 \pm 0.026$ & $1.235 \pm 0.045$ &\
AWI0000bs0 & 2830 & J003140.76-014737.3 & A0V & $107 \pm 6 $ & 7.07 & $6.917 \pm 0.019$ & $1.015 \pm 0.054$ & c\
AWI0004ne5 & 9590 & J013525.89+560237.3 & A0V & $162 \pm 12 $ & 7.02 & $6.791 \pm 0.018$ & $0.655 \pm 0.059$ &\
AWI0005yjp & 14893 & J022515.75+370707.9 & A0V & $192 \pm 23 $ & 7.34 & $7.341 \pm 0.018$ & $1.037 \pm 0.067$ &\
AWI0005ylw & 22614 & J033906.73+244209.8 & A0V & $120 \pm 11 $ & 7.09 & $6.979 \pm 0.024$ & $0.614 \pm 0.079$ &\
AWI00000wz & 74389 & J084546.93+485243.4 & A0V & $111 \pm 7 $ & 7.48 & $7.296 \pm 0.023$ & $0.678 \pm 0.067$ &\
AWI0000u8s & 85672 & J095359.12+274143.5 & A0V & $107 \pm 7 $ & 7.58 & $7.215 \pm 0.024$ & $1.011 \pm 0.056$ & c, d\
AWI0000w9x & 129584 & J144313.04+014928.7 & A0V & $156 \pm 20 $ & 7.33 & $7.260 \pm 0.024$ & $0.785 \pm 0.052$ &\
AWI0000tz1 & 140101 & J154030.20+370101.1 & A0V & $165 \pm 13 $ & 7.17 & $7.109 \pm 0.023$ & $0.688 \pm 0.051$ & c\
AWI0000gjb & 214982 & J224206.62-032824.4 & A0V & $124 \pm 9 $ & 7.16 & $7.101 \pm 0.021$ & $0.655 \pm 0.065$ &\
AWI0000kg4 & 218155 & J230533.05+145732.5 & A0V & $106 \pm 5 $ & 6.77 & $6.712 \pm 0.020$ & $0.579 \pm 0.059$ &\
AWI0000fye & 224155 & J235537.71+081323.7 & A0V & $128 \pm 6 $ & 6.82 & $6.770 \pm 0.026$ & $0.462 \pm 0.066$ &\
AWI00062h1 & 224429 & J235746.21+112827.6 & A0V & $95 \pm 4 $ & 6.65 & $6.581 \pm 0.023$ & $0.426 \pm 0.058$ & c\
AWI0002mhd & 173056 & J184305.97+071626.4 & A0.5V & $203 \pm 32 $ & 8.24 & $7.913 \pm 0.027$ & $1.504 \pm 0.061$ &\
AWI0005yjn & 14685 & J022317.32+381509.7 & A1IV & $254 \pm 35 $ & 7.14 & $7.037 \pm 0.018$ & $0.706 \pm 0.062$ & e, EW$= 4.01698$\
AWI0005abf & 213290 & J222753.27+704806.0 & A1IV & $326 \pm 44 $ & 7.7 & $7.318 \pm 0.023$ & $0.627 \pm 0.050$ &\
AWI0005mry & 3051 & J003412.66+540359.0 & A1V & $229 \pm 32 $ & 7.6 & $7.350 \pm 0.019$ & $0.354 \pm 0.073$ &\
AWI0000phh & 18271 & J025614.05+040254.2 & A1V & $116 \pm 14 $ & 7.71 & $7.664 \pm 0.032$ & $0.974 \pm 0.073$ &\
AWI0005zz5 & 25466 & J040238.47-004803.7 & A1V & $113 \pm 8 $ & 6.93 & $6.835 \pm 0.020$ & $0.606 \pm 0.053$ &\
AWI0000uji & 134854 & J151147.67+101259.8 & A1V & $113 \pm 8 $ & 6.87 & $6.799 \pm 0.018$ & $0.405 \pm 0.065$ &\
AWI0005zx4 & 11085 & J014928.21+244048.7 & A2V & $151 \pm 22 $ & 8.31 & $8.013 \pm 0.019$ & $0.641 \pm 0.088$ &\
AWI0004nfu & 21375 & J032853.67+490412.8 & A2V & $160 \pm 21 $ & 7.47 & $7.186 \pm 0.026$ & $0.468 \pm 0.068$ & e\
AWI00062aj & 12445 & J020221.16+192323.6 & A3V & $227 \pm 43 $ & 8.4 & $8.090 \pm 0.024$ & $0.944 \pm 0.084$ &\
AWI0000tgc & 84870 & J094902.82+340506.9 & A3V & $88 \pm 5 $ & 7.19 & $6.752 \pm 0.032$ & $0.398 \pm 0.063$ & c, f\
AWI0005bps & 165507 & J180533.55+182643.9 & A3V & $193 \pm 33 $ & 8.17 & $7.952 \pm 0.019$ & $0.672 \pm 0.099$ &\
AWI0005vyx & 204829 & J212959.78+413037.3 & A3V & $175 \pm 13 $ & 7.34 & $6.744 \pm 0.021$ & $0.338 \pm 0.067$ &\
AWI0005w29 & 212556 & J222412.79+484918.7 & A3V & $150 \pm 11 $ & 7.61 & $7.415 \pm 0.029$ & $0.469 \pm 0.060$ &\
AWI0005a9r & 208410 & J215305.45+682955.0 & A3V & $189 \pm 15 $ & 7.48 & $7.297 \pm 0.027$ & $0.539 \pm 0.053$ &\
AWI0005ykd & 21062 & J032448.99+283908.6 & A4V & $102 \pm 6 $ & 7.12 & $6.838 \pm 0.018$ & $0.612 \pm 0.058$ &\
AWI0000v1z & 138214 & J152954.11+234901.6 & A5V & $138 \pm 13 $ & 7.58 & $7.124 \pm 0.024$ & $0.322 \pm 0.061$ &\
AWI0006222 & 201377 & J210916.04-001405.6 & A7V & $101 \pm 5 $ & 6.66 & $6.355 \pm 0.032$ & $0.574 \pm 0.064$ &\
AWI0005w7o & 20994 & J032429.84+341709.9 & A8V & $245 \pm 60 $ & 8.67 & $7.931 \pm 0.027$ & $1.191 \pm 0.074$ &\
AWI0005c01 & 199392 & J205143.50+730449.3 & A8V & $169 \pm 18 $ & 8.28 & $8.004 \pm 0.023$ & $0.402 \pm 0.071$ &\
AWI00002ms & 71988 & J083100.44+185806.0 & F0V & $83 \pm 4 $ & 7.42 & $6.977 \pm 0.032$ & $0.358 \pm 0.080$ &\
AWI00062bl & 22128 & J033337.91-072453.8 & F0IV & $166 \pm 20 $ & 7.56 & $6.919 \pm 0.024$ & $0.853 \pm 0.053$ &\
AWI0005yk3 & 19257 & J030651.95+303136.8 & F0V & $79 \pm 4 $ & 7.06 & $6.470 \pm 0.020$ & $2.093 \pm 0.037$ & g\
AWI000048c & 87827 & J100719.80-152718.9 & F2V & $107 \pm 8 $ & 8.12 & $7.425 \pm 0.020$ & $0.968 \pm 0.060$ & c\
AWI0000hjr & 221853 & J233536.20+082256.9 & F2V & $68 \pm 3 $ & 7.34 & $6.559 \pm 0.019$ & $1.335 \pm 0.042$ & c,f,h\
AWI00002yt & 157165 & J172007.53+354103.6 & F6V & $100 \pm 7 $ & 8.27 & $7.496 \pm 0.020$ & $0.427 \pm 0.064$ &\
\[tab:ddois\]
If we conservatively assume that the quality of our spectra corresponds to the “C” quality flag in SIMBAD (“A” is the highest quality), then of the 50 targets with new spectral types that we present here, 48 show clear improvements in quality over the published literature, while the remaining nine targets have spectral types of the same quality as the literature. The root mean square change in spectral types versus SIMBAD for all 50 targets was 2.64, with 27 objects shifted at least one subtype toward lower temperature and 12 objects shifted at least one subtype toward higher temperature.
Figures \[Fig\_radec\] and \[Fig\_dist\] summarize some more properties of the stars in Table \[tab:ddois\]: their distribution on the sky and their distance distribution. Their distance distribution peaks at roughly 110 pc; at this distance, a large telescope with adaptive optics like the Gemini Planet Imager could image an analog to the HR 8799 planetary system and comfortably detect at least two of the planets. For comparison, @2014ApJS..212...10P limited their study to sources with distances < 120 pc. But the distance distribution of the DDOIs is grossly similar to that of the disk candidates in @2013ApJS..208...29W, which peaks at about 90 pc. The distribution of our DDOIs also includes a long tail of objects beyond 200 pc; roughly 1/4 of our DDOIs are in this long tail, which consists mainly of B and early A dwarfs. Fig. \[Fig\_dist\] also shows the distances to some well-known disks for references. The distribution of the DDOIs on the sky is shaped mainly by the range of declinations accessible to FAST and our decision to add sources to the website grouped by Galactic latitude.
Figure \[Fig\_Wu\] compares our disk candidates with the @2013ApJS..208...29W color selection criteria. We did not impose such a criterion, but all of our disk candidates fall to the right of the dotted line in this figure, showing that they meet the @2013ApJS..208...29W criterion anyway. On average, the disk candidates in this paper are somewhat redder than those in Wu et al. (2013).
We fit simple models to the SEDs consisting of a stellar photosphere plus a blackbody dust component using a Levenberg-Marquardt algorithm. These fits yield constraints on the dust temperature and the fractional infrared luminosity, $f$, the total bolometric power emitted by our single-temperature blackbody disk model divided by the total bolometric power emitted by our blackbody stellar model for each source. For stars with excess at only WISE 4, these fits yield only an upper limit on the dust temperature and a lower limit on $f$.
Figure \[Fig\_SEDs\] shows spectral energy distributions (SEDs) for six of the new disk candidates, together with a simple model for the flux. Our SEDs employ WISE 1 and WISE 2 fluxes from the All-Sky survey catalog corrected for saturation using the formulae in @2014ApJS..212...10P. For these models, we supplemented the WISE and 2MASS photometry with UBVRI photometry from SIMBAD when it was available. Dashed lines show blackbody models fit to the star and to the disk component; the solid line shows the total model flux.
Table \[tab:diskcandidates\] summarizes our SED modeling of all the 39 new disk candidates reported in Table \[tab:ddois\]. The temperature uncertainties are $1-\sigma$ uncertainties from the shape of the two-parameter $\chi^2$ surface near the minimum. The fractional infrared excesses, $f$, listed in this table should all be considered lower limits. When the temperature listed is an upper limit, the listed fractional infrared excess corresponds to a blackbody disk with the listed upper limit temperature.
[lccc]{} 3051 & $ < 248 $ & <$4.7\times 10^{-5}$\
4670 & $ 156$ +21/-19 & $ 5.0\times 10^{-5}$\
6370 & $< 168 $ & <$6.9\times 10^{-5}$\
6612 & $< 189 $ & <$ 2.1\times 10^{-5}$\
9590 & $ < 200 $ & <$5.8\times 10^{-5}$\
9985 & $< 136 $ & <$2.8\times 10^{-4}$\
11085 & $< 200 $ & <$7.2\times 10^{-5}$\
12445 & $< 157 $ & <$8.9\times 10^{-5}$\
14893 & $ < 117 $ & <$3.4\times 10^{-5}$\
18271 & $ < 104 $ & <$8.7\times 10^{-5}$\
20994 & $ 187 $ +25/-19 & $2.5\times 10^{-4}$\
21062 & $ < 207 $ & <$8.0\times 10^{-4}$\
22128 & $< 74 $ & <$6.8\times 10^{-4}$\
22614 & $ < 187 $ & <$6.8\times 10^{-5}$\
23873 & $< 131 $ & <$7.3\times 10^{-5}$\
25466 & $ < 107 $ & <$3.3\times 10^{-5}$\
71988 & $ < 221 $ & <$1.1\times 10^{-4}$\
74389 & $ < 136 $ & <$5.7\times 10^{-5}$\
129584 & $ < 170 $ & <$4.2\times 10^{-5}$\
134854 & $ < 251 $ & <$4.6\times 10^{-5}$\
138214 & $ < 216 $ & <$9.6\times 10^{-5}$\
152308 & $< 228 $ & <$2.6\times 10^{-5}$\
157165 & $ < 245 $ & <$1.4\times 10^{-4}$\
158419 & $< 244 $ & <$4.1\times 10^{-5}$\
165507 & $ < 205 $ & <$7.2\times 10^{-5}$\
173056 & $ 177 $ +20/-18 & $2.0\times 10^{-4}$\
199392 & $ < 252 $ & <$8.4\times 10^{-5}$\
201377 & $ < 131 $ & <$1.7\times 10^{-5}$\
204829 & $ < 263 $ & <$5.3\times 10^{-5}$\
207888 & $ <136 $ & <$6.8\times 10^{-5}$\
208410 & $ < 212 $ & <$7.4\times 10^{-5}$\
212556 & $ < 225 $ & <$6.2\times 10^{-5}$\
213290 & $ < 198 $ & <$7.3\times 10^{-5}$\
214982 & $ < 193 $ & <$6.3\times 10^{-5}$\
218155 & $ < 193 $ & <$6.3\times 10^{-5}$\
224098 & $ < 272 $ & <$3.3\times 10^{-5}$\
224155 & $ < 242 $ & <$4.4\times 10^{-5}$\
\[tab:diskcandidates\]
W3 Excesses
-----------
All of the diskdetective.org subjects were pre-selected to have \[W1\]-\[W4\] excesses significant at the 5-sigma level or better, based on the AllWISE catalog. We also checked the All-Sky catalog of WISE photometry for these sources, correcting this photometry via the correction factors in @2014ApJS..212...10P. All of the sources in Table \[tab:diskcandidates\] still have \[W1\]-\[W4\] excesses significant at the 5-sigma level or better using the corrected All-Sky photometry.
Additionally, eight of our DDOIs also turned out to have significant ($>3\sigma$) W3 excess based on corrected All-Sky WISE photometry, as you can read in Table \[tab:w3\]: HD 4670, HD 7406, HD 14685, HD 19257, HD 20994, HD 164137, HD 173056, and HD 218546.
[lllccc]{}
AWI00055sz & 7406 & J011636.23+740136.6 & B1III & $549 \pm 103$ & $0.292 \pm 0.042$\
AWI0005w41 & 218546 & J230817.21+511146.3 & B8III & $794 \pm 498$ & $0.102 \pm 0.029$\
AWI0005bow & 164137 & J175911.27+135417.8 & B8III & $820 \pm 544$ & $0.646 \pm 0.029$\
AWI0000m2p & 4670 & J004848.00+181850.6 & B9IV & $386 \pm 98$ & $0.115 \pm 0.030$\
AWI0002mhd & 173056 & J184305.97+071626.4 & A0.5V & $203 \pm 32$ & $0.114 \pm 0.030$\
AWI0005yjn & 14685 & J022317.32+381509.7 & A1IV & $254 \pm 35$ & $0.205 \pm 0.035$\
AWI0005w7o & 20994 & J032429.84+341709.9 & A8V & $245 \pm 60$ & $0.174 \pm 0.030$\
AWI0005yk3 & 19257 & J030651.95+303136.8 & F0V & $79 \pm 4$ & $0.718 \pm 0.031$\
\[tab:w3\]
Binary Companions and Variability
---------------------------------
Five of the stars with newly reported excesses have known companions or secondaries. None of these is apparent in the WISE images or likely to be cool enough to create the observed WISE infrared excesses.
HD 7406
: The separation of this binary is $\sim 1\arcmin$ , wide enough that the companion probably does not contaminate the WISE photometry.
HD 74389
: HD 74389 has several nearby companion stars which could conceivably contaminate the SED. This star has a white dwarf companion discovered by @1990PASP..102..440S, type DA1.3, V=14.62 located $20.11\arcsec$ (2230 AU projected separation) to the west [@2013MNRAS.435.2077H], which is readily visible in the DSS Blue, Red and IR images. @2013MNRAS.435.2077H also report a nearby M dwarf companion to the west and a nearby sdB companion to the north. Indeed, the same DSS Blue, Red and IR images bands also show a second background object $\sim 13\arcsec$ to the north of the star, probably the sdB.
However, none of these objects is visible in the WISE or even the 2MASS images, which look like unsaturated clean point sources, probing a resolution at K band of $<4\arcsec$. Moreover, we only see a significant excess at W4, but not at W3. The W3-W4 color of an M dwarf or sdB photosphere is $\sim$0 since the SEDs of these objects are in the Rayleigh Jeans limit beyond 10$\micron$, so the lack of a resolved companion at K band rules out a significant contribution to the W4 excess from the companion’s photospheric emission. An M dwarf or sdB unresolved by 2MASS might still be cause for concern if it were itself dusty. But since the A star is the most luminous object in the system, it seems most likely that we are observing reprocessed light from dust around the A star.
See below for further discussion of this interesting object.
HD 23873
: is a member of the Pleiades, and listed as a binary in the Hipparcos Input Catalog [@1993BICDS..43....5T]. The separation of the binary is 11000 AU according to , corresponding to roughly $1.5\arcmin$.
HD 207888
: was identified as a visual binary by . The separation is wide enough ($\sim 1\arcmin$) that the companion probably does not comtaminate the WISE photometry.
HD 22128
: is a double-lined spectroscopic binary system with semimajor axis roughly 9 Solar radii. We found a spectral type of F0IV for the combined light, but in fact, the two components are A stars; see @2013MNRAS.433.3336F for a detailed spectral analaysis. This star was first identified as a debris disk host star by @2000PhDT........17S based on ISO photometry (60–100$\mu$m). We are the first to report excess emission from this source at WISE 4, excess at the 8-$\sigma$ level. @2007ApJ...663..365W considered the object to be an “anomalous system” based on the high luminosity of its inferred disk given the star’s age. We note that the system appears slightly extended in 2MASS K, but not in other bands. No 18 $\mu$m flux measurement for this object appears in the Akari catalogue .
HD 224098
: This B8V star is listed as having faint optical secondary, type F0V, in . No data on the separation of the secondary is provided.
Two of our new disk candidates have known stellar variability of $<=0.04$ magnitudes in V band. This degree of variability is too small to affect our selection process.
HD 152308
: is classified as an $\alpha2$ CVn variable star by @2011MNRAS.414.2602D and as type A0 Cr Eu according to . The variability is 0.04 magnitudes in the Hipparcos system, with a period of 0.94 days. Our classification yielded B9V.
HD 218155
: This star varies over a range of $\sim0.03$ magnitudes in V band [@2006SASS...25...47W].
Be Stars
--------
Aside from normal main sequence stars, our program reveals four new Be stars: HD 6612, HD 7406, HD 164137, and HD 218546. None of these stars appears in the Be Star Spectra (BeSS) database [@2011AJ....142..149N] or in the catalog of H$\alpha$ emission line stars, which cataloged stars with galactic latitude $|b|<10^{\circ}$. All show prominent H$\alpha$ emission and the strong upper level Balmer absorption lines characteristic of classical Be stars . Two of these stars, HD 7406 and HD 218546, show significant W3 excess, in addition to their W4 excess. The B1e star HD 7406 has additional strong He I absorption lines; we detect weak He I absorption in the other stars. Our high S/N spectra reveal no trace of other permitted or forbidden emission lines, confirming our classification of them as classical Be stars rather than pre-main sequence Herbig Ae/Be stars .
These stars illustrate the promise of Disk Detective for identifying new classical Be stars. Although Be stars have large infrared excesses due to free-free emission , most have been identified from optical spectroscopic surveys optical photometric surveys [e.g., @2006ApJ...652..458W] and IR spectroscopic surveys [e.g., @2015AJ....149....7C]. Uniform selection based on infrared excess from WISE data provides a flux-limited survey fairly independent of interstellar reddening.
With only four Be stars, it is premature to analyze spectroscopic properties and statistics for the WISE sample of classical Be stars. However, it is worth noting that the frequency of late-type Be stars (three B8–B9 stars and one B1 star) is somewhat larger than expected from current samples where B4 and earlier stars are much more common . We plan a detailed analysis of a larger sample in a future paper (Bans et al., in prep).
HD 74389: A Star With a Candidate Debris Disk and White Dwarf Companion
-----------------------------------------------------------------------
One of our new candidate debris disks appears to orbit HD 74389 A, an A0V star with a white dwarf companion discovered by @1990PASP..102..440S. It also has a possible M dwarf companion and sdB companion reported by @2013MNRAS.435.2077H. There are three known planetary systems with white dwarfs as distant companions: Gl 86 , HD 27442 , and HD 147513 . However, the HD 74389 system appears to contain the first debris disk around a star with a white dwarf companion.
The white dwarf companion, HD 74389 B, has V mag 14.62, and is located 20.11 arcsec to the west [@2013MNRAS.435.2077H]. It is readily visible in the DSS Blue, Red and IR images. These bands also show a background object $\sim 13 \arcsec$ to the north of the star, possibly the M dwarf companion reported by @2013MNRAS.435.2077H.
It is interesting to ponder the origin of the disk around HD 74389 A and how the post-main sequence evolution of HD 74389 B may have affected it. Though the white dwarf may presently have a projected separation of 2230 AU, it could have been 2-4$\times$ closer in when it was on the Main Sequence, thanks to stellar mass loss. The disk may have merely survived the evolution of the higher-mass star, or it may represent a signature of dynamical changes to the system, like planet exchange [e.g., @2012ApJ...753...91K]. It could even have been built from mass loss by the companion, via the process described by @2013ApJ...764..169P.
DISCUSSION {#sec:conclusions}
==========
We have outlined and demonstrated a novel process for identifying new candidate circumstellar disks in the WISE survey data. This paper reports only results from the first 10% of the search, so it might be premature to try to derive any statistically meaningful inferences about the population of debris disk from this limited sample. But our list of 37 new, well-vetted disk candidates demonstrates the utility of crowdsourcing analysis of WISE images. One of our disk candidate systems appears to contain the first debris disk discovered around a star with a white dwarf companion: HD 74389. We also report four newly discovered classical Be stars (HD 6612, HD 7406, HD 164137, and HD 218546) and a new detection of 22 $\mu$m excess around previously known debris disk host star HD 22128.
We decided to only publish in this paper candidates that are in the Hipparcos catalog. Since the @2013ApJS..208...29W cross-correlated the WISE archive with the Hipparcos catalog, they could conceivably have identified all of the candidates that we are announcing. However, @2013ApJS..208...29W used a \[2MASS\] - \[W4\] color criterion, as opposed to our \[W1\] - \[W4\] color criterion, and used the WISE AllSky data rather than the WISE ALLWISE data. Yet all of the candidates presented in this paper would have also been selected by the @2013ApJS..208...29W color criterion. More importantly, while we examined candidates by eye to discard objects potentially contaminated by nearby stars and galaxies, @2013ApJS..208...29W used a statistical likelihood-ratio (LR) technique to accomplish this goal. Perhaps their statistical technique was more conservative than our more labor intensive approach, leaving these candidates unidentified.
We have several further improvements to Disk Detective project underway, which we will describe in upcoming papers, including:
- New ways to retire the sources after fewer classifications.
- Spectroscopy of Southern hemisphere DDOIs via the CASLEO telescope in Argentina
- Imaging follow-up of DDOIs with the Robo-AO [@2014ApJ...790L...8B] instrument at the Palomar Observatory 60-inch telescope to check for background contaminants located closer to the star than DSS can probe.
With its high sensitivity and angular resolution in the mid-infrared, we expect that the James Webb Space Telescope (JWST) will be an important tool for following up disks discovered via Disk Detective. So we aim to have the project mostly completed by the time JWST launches in the fall of 2018.
We acknowledge support from grant 14-ADAP14-0161 from the NASA Astrophysics Data Analysis Program. Marc Kuchner acknowledges funding from the NASA Astrobiology Program via the Goddard Center for Astrobiology.
Development of the Disk Detectives site was supported by a grant from the Alfred P. Sloan foundation, and the Zooniverse platform is supported by a Google Global Impact award.
WISE is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory (JPL)/California Institute of Technology (Caltech), funded by NASA. 2MASS is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center (IPAC) at Caltech, funded by NASA and the NSF. This paper uses data products produced by the OIR Telescope Data Center, supported by the Smithsonian Astrophysical Observatory.
The Digitized Sky Survey was produced at the Space Telescope Science Institute under U.S. Government grant NAG W-2166. The images of these surveys are based on photographic data obtained using the Oschin Schmidt Telescope on Palomar Mountain and the UK Schmidt Telescope. The plates were processed into the present compressed digital form with the permission of these institutions. This work has made use of the BeSS database, operated at LESIA, Observatoire de Meudon, France: http://basebe.obspm.fr
Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http://www.sdss.org/.
The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington.
The data presented in this paper result from the efforts of the Disk Detective volunteers, without whom this work would not have been possible. The following volunteers helped classify the stars listed in this paper: *13lueAngeL, A.Brodersen, abienvenu, AlarmingAlarm, AlbeeLou, alinastart, anowi, anoxie, Arlon, artman40, arvintan, atmosferah, bc2callhome, bellapagano, berlherm, biggsjrex, billweiler, bmw.996, BraehlerM, brainell, breckwilhite, Bromista, cazze74, Chinabob, chloecollins , christania10110, clairwallis, clayzer-ev, Cosmic Jerk, cstickmaker, daisy1000, Deke scott T, dianaz, diplomacy42, djperkins$@$live.co.uk, DNiergarth, dooces, Ercydive, feder, ferrertiago, fireandice, firewatermoonearth, Fletrik, ftpol, GBav8r, geckzilla, Gez Quiruga, giazira, golfmadman, Grubenm, Haian, hardan, hearzadrsos, HelmutU, i-mac, iponenubs, IrvSet, Ivan3, [email protected], jacedjohnson, jacobus.valk$@$polk.de, janiceashdown, JasonJason, jellybelly123, JessicaElizabeth, jgreyes, jmeyers314, JoJeFree, jorge96, josa310, justinsbuzz, kafter, kecsap, keel, KI4FYP, kiarash, kijkuit, Killoch, kmk, lambrosliamis, lehensuge, ljhinton21, Mack777, manemag, marca, marchl, marcin.s, Maria Mar, mattlou117, MDLW, michiharu, miertje, miltonbosch, mlavall2$@$uwo.ca, Mric116, MylesAtkins, N5bz, namorris, nipper10, nirajsanghvi, norbertf, Noreal, nunolanca, ohiopugmom, OMHans, onetimegolfer, orionsam1, Pamela Foster, PaulRo, peterw143, Petrusperes, Pini2013, pixelfixx, planetari7, plutoexpress, ptrip2010, QGR, Raymond Hall, RefugeZero, Ridence, RobhJ, Rocketman93, RoLeCa, rt26556$@$wdmtigers.or, russ jones, ryangeho, ryanstone87, sandrisvi, scottwferg, sheilaandpeter, Shigeru, shocko61, Shroomzz, silviug, Siver, SoloSlayer9, Star hunter 1, starbase3, steve bourne, SUMO2011, SunJinx, symaski62, TED91, Tigrincs, Timothy Fitzgerald, tom.luthe, Troomander, Trumanator, Tsgt, turtle0920, Victory1, Vinokurov, Vonkohon, voyager1682002, Vulpi, weric1, WizardHowl, Woomaster, WXdestroyer, xantipa, Yiska, Zealex*
We also thank Christoph Baranec and Katharina Doll for providing useful comments on this paper.
APPENDIX: Additional Notes on Individual Disk Candidates {#sec:appendix}
========================================================
HD 9985
: All four WISE images appear slightly ($\approx 2 \arcsec$ ) offset from the DSS and 2MASS images to the SW, a common feature of slightly saturated images in AllWISE.
HD 14893
: @2014yCat....1.2023S lists this star as a B9.5 V. We find type A0V.
HD 22614
: Possible member of the Pleiades cluster [@2001ApJS..132..253W].
HD 173056
: found a spectral type of A1V; we find A0.5V.
HD 213290
: DSS Blue, Red and IR show an unresolved background object roughly 14 arcsec north of the star.
HD 25466
: Our spectral classification (A1V) matches that of for this star.
HD 134854
: The SDSS images of this star are highly distorted.
HD 201377
: Using higher resolution spectra (R=42000), find that this star is type A3 with logg=3.93. (We find A7V.)
HD 20994
: @2015AJ....150...95A identified this star in the Perseus OB 2 association as an AGB star based on its K-WISE 4 color alone. However, our spectral typing (A8V) shows that this object is more likely to be a debris disk.
HD 19257
: The WISE 3 and WISE 4 excesses of this star have been reported by @2013MNRAS.433.2334K. We are the first to report a luminosity class for it; we find a spectral type of F0V, which disagrees with SIMBAD but is consistent with @2013MNRAS.433.2334K.
HD 87827
: All four WISE images appear slightly ($\approx 2 \arcsec$ ) offset from the DSS and 2MASS images to the SW, a common feature of slightly saturated images in AllWISE.
HD 221853
: We find a spectral type F2V for this well-known debris disk host star, consistent with the age estimate of 100 Myr in @2015ApJ...798...87M.
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[^1]: Available at http://irsa.ipac.caltech.edu/Missions/wise.html
[^2]: The initial candidates listed in this paper are also all in Hipparcos, but our full dataset does not cross-correlate with any stellar catalogues.
[^3]: http://dept.astro.lsa.umich.edu/ hernandj/SPTCLASS/sptclass.html
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We study thermal transport in one dimensional spin systems both in the presence and absence of impurities. In the absence of disorder, all these spin systems display a temperature dependent Drude peak in the thermal conductivity. In gapless systems, the low temperature Drude weight is proportional to temperature and to the central charge which characterizes the conformal field theory that describes the system at low energies. On the other hand, the low temperature Drude weight of spin gap systems shows an activated behavior modulated by a power law. For temperatures higher than the spin gap, one recovers the linear $T$ behavior akin to gapless systems. For temperatures larger than the exchange coupling, the Drude weight decays as $T^{-2}$. We argue that this behavior is a generic feature of quasi one dimensional spin gap systems with a relativistic-like low energy dispersion. We also consider the effect of a magnetic field on the Drude weight with emphasis on the commensurate-incommensurate transition induced by it. We then study the effect of nonmagnetic impurities on the thermal conductivity of the dimerized XY chain and the spin-$\frac12$ two leg ladder. Impurities destroy the Drude peak and the thermal conductivity exhibits a purely activated behavior at low temperature, with an activation gap renormalized by disorder. The relevance of these results for experiments is briefly discussed.'
author:
- 'E. Orignac'
- 'R. Chitra'
- 'R. Citro'
title: 'Thermal transport in one-dimensional spin gap systems'
---
Introduction
============
The past many years have seen a resurgence of interest, both theoretical and experimental in quasi-one dimensional spin gap systems. Well known examples of systems with a gap are spin chains with dimerization, frustration and anisotropy[@haldane_dimerized; @nijs_equivalence]. Another interesting example is the two leg spin $S=\frac12$ ladder which was proposed as a toy model for the pseudogap phase in high temperature superconductors.[@rice_srcuo] Renewed interest in these systems was triggered by the availability of anisotropic materials[@takano_spingap; @chaboussant_cuhpcl; @uchara_SrCaCuO] in which the magnetic properties of the insulating phase could be ascribed to one or quasi-one dimensional spin systems. The dynamical properties of these quasi-1d spin phases have been extensively studied using standard techniques like neutron scattering and NMR measurements.[@dagotto_2ch_review; @dagotto_supra_ladder_review] More recently, heat transport is being used as a complementary probe to study low dimensional spin systems. Measurements of thermal conductivity have been carried out in systems such as the spin chain materials $\mathrm{SrCuO_2}$ and $\mathrm{Sr_2CuO_3}$[@sologubenko_therm_sr2cuo3; @sologubenko_spinchains_th], the spin Peierls system $\mathrm{CuGeO_3}$[@ando_sP_thermal] and the spin ladder materials $\mathrm{(Sr,Ca,La)_{14}Cu_{24}O_{41}}$ [@sologubenko_thermal_ladder; @hess_thermal_ladder; @kudo_thermal_ladder]. The huge anisotropy seen in the thermal conductivity in the directions parallel and perpendicular to the chains or the ladders, indicates that magnetic excitations of these quasi-1D systems do play an important role in heat transport. This is further confirmed by measurements in the presence of a magnetic field[@ando_sP_thermal].
Various attempts have been made to extract from these measurements, the purely magnetic contribution to the thermal conductivity. This is often done by subtracting a phonon background calculated within a Debye model. However, in order to account for the entire magnetic contribution, one needs to understand the interactions of the spin excitations of the low dimensional spin systems with themselves and, with defects and phonons. This is a non-trivial problem since the spin excitations are not necessarily weakly interacting, and the form of interaction of these spin excitations with phonons or defects is usually rather complicated. Consequently, experimental results have been fitted using various phenomenological kinetic theory expressions for non-interacting spinons or magnons. This effort to obtain the purely magnetic contribution to the thermal conductivity has stimulated theoretical studies of thermal transport in spin chain and spin ladder systems.[@alvarez_ladder; @saito_thermal; @heidrich_frustrated] In the absence of extrinsic scattering such as phonons or defects, some studies[@alvarez_ladder] showed that the frequency dependent thermal conductivity $\kappa(\omega,T)=\pi\tilde{\kappa}(T)\delta(\omega)$, where $\tilde{\kappa}(T)$ is the thermal Drude weight. However, the Drude weight extracted from finite size zig-zag ladders [@heidrich_frustrated] seems to be at odds with the idea of an infinite thermal conductivity in spin systems without disorder.
In this paper, we use analytical methods to revisit the problem of the thermal conductivity for various quasi 1d spin systems with special emphasis on the two leg spin-$\frac12$ ladder. In the absence of impurities, we present results which should be valid for spin gap systems possessing low energy triplet excitations and gapless systems irrespective of the details of the nature of the interaction. A schematic representation of the thermal Drude weight for a spin gap system is shown in Fig.\[fig:weight\]. We also study the effect of one/many impurities on the thermal conductivity of the spin ladder and show that in the presence of impurities, the thermal conductivity is not simply given by the Drude weight times a temperature independent scattering time.
The paper is organized as follows: in Sec. \[sec:pure-case\], we discuss the temperature dependence of the Drude weight for different spin systems ranging from gapless integrable spin chains to spin gap systems like the spin ladder and the spin$-1$ chain. In Sec. \[sec:disorder-case\], we discuss the effect of impurities on the thermal conduction. In particular, we use the Landauer approach [@landauer_formula] to evaluate the effect of a single non-magnetic impurity on the thermal conductivity of the ladder and the XY-chain. We then study the effect of a finite concentration of impurities on the ladder. Finally, we present a comparison of our results to experiments and other theoretical work on the subject.
Translationally invariant systems {#sec:pure-case}
=================================
In this section, we briefly outline the general definitions of the thermal current and the thermal conductivity calculated within linear response theory. We then use this formalism to calculate the dc thermal conductivity of various spin systems with and without a gap to low energy excitations. The examples considered are: gapless integrable spin chains described by a conformal fixed point, the spin-$\frac12$ ladder, the dimerized XY chain and lastly the case of massive bosons. We also consider the effect of a magnetic field on the thermal Drude weight.
Definition of thermal current and thermal conductivity
------------------------------------------------------
We consider a system defined by a Hamiltonian density ${\cal H}(x)$ so that the total Hamiltonian is $H=\int dx {\cal H}(x)$. Conservation of energy leads to the continuity equation $$\begin{aligned}
\label{eq:continuity}
\partial_t {\cal H}(x,t)+\partial_x j_e(x,t)=0,\end{aligned}$$ where $j_e$ is the energy(thermal) current of the system. In the absence of charged excitations, the energy and thermal current are equivalent. Eq.(\[eq:continuity\]) permits a definition of the thermal current in terms of the Hamiltonian density. Within linear response theory[@luttinger_thermal], the energy current response function at temperature $T>0$ reads: $$\label{eq:response-def}
\chi(\omega,T)= \int dx \int_0^\infty dt e^{i\omega t} \langle
[j_e(x,t), j_e(0,0)] \rangle,$$ where $\langle \ldots \rangle$ indicates both quantum and thermal averaging. It is often easier to use the imaginary time formalism to calculate $\chi(i\omega_n)$ where $\omega_n$ are the Matsubara frequencies and then analytically continue to real frequencies $i\omega_n\to \omega +i0$ to obtain $\chi(\omega,T)$.[@mahan_book] The frequency dependent thermal conductivity is then given by [@luttinger_thermal]: $$\label{eq:conduc-def}
\kappa(\omega,T)=\frac 1 {i \omega T} \left[ \chi(0,T) -
\chi(\omega,T)\right],$$ In general, in the absence of phonons or impurities, the total thermal current $J_e(t)=\int dx j_e(x,t)$ is conserved. This conservation permits an alternative but equivalent formulation of the thermal conductivity (\[eq:conduc-def\]) $$\kappa(\omega,T)= \frac 1 {2 L T^2} \int_0^\infty \langle \{
J_e(t),J_e(0)\} \rangle e^{i\omega t} dt,$$ where $L$ is the system size. Since $J_e$ is conserved, the total current is time independent and $$\label{eq:drude-thermal}
\kappa(\omega,T)= \frac{\pi}{L T^2} \langle J_e^2 \rangle \delta(\omega) =
\tilde{\kappa}(T) \delta(\omega).$$ This implies an infinite dc thermal conductivity, with a temperature dependent Drude weight $\tilde{\kappa}(T)$ which vanishes at zero temperature. This thermal Drude weight has been studied numerically for some spin gap systems in Refs.. In the following sections, we present an analytical discussion of the behavior of the Drude weight in various gapless and gapped quasi-one dimensional spin systems.
Gapless Integrable Spin Chains {#sec:pure-conformal}
-------------------------------
In this section, we present results for the thermal Drude weight of integrable spin chains which are characterized by a vanishing singlet-triplet gap. One example of such a system, is the integrable spin-$\frac12$ Heisenberg model, which is known to have a Drude weight that vanishes linearly as temperature goes to zero ${\tilde \kappa}= \pi^2vT/3$ [@kluemper_xxz], where $v$ is the velocity of spin excitations. Other interesting systems, are the various integrable generalizations of the spin-$\frac 12$ Heisenberg spin chain, like the spin$-S$ chain models[@takhtajan_spin_s; @babujian_spin_s] and $SU(N)$ spin chain models[@sutherland_su3_ba; @uimin_su3_ba; @lai_su3_ba]. This description allows one to easily obtain the low temperature thermal conductivity of these chains. The long wavelength behavior of these integrable systems are described by a conformal invariant fixed point [@witten_wz; @affleck_wz; @affleck_strongcoupl]. This description allows one to easily obtain the low temperature thermal Drude weight of these chains. The effective Hamiltonian of these systems has the generic form $$\begin{aligned}
\label{eq:wz-models}
H=\int dx [{\cal H}_R(x)+{\cal H}_L(x)]\end{aligned}$$ where ${\cal H}_R$ and ${\cal H}_L$ describe right and left moving chiral modes. In addition, chirality imposes the constraints ${\cal H}_R(x,t)={\cal H}_R(x-vt)$ and ${\cal H}_L(x,t)={\cal H}_L(x+vt)$. This leads to the following relation $$\begin{aligned}
\label{eq:contin-wz}
\partial_t ({\cal H}_R(x,t)+{\cal H}_L(x,t))=-v\partial_x({\cal
H}_R(x,t)-{\cal H}_L(x,t))\end{aligned}$$ which results in a thermal current density: $$\begin{aligned}
\label{eq:energy-curr-wz}
J_e=v \int dx [:{\cal H}_R(x):-:{\cal H}_L(x):]\end{aligned}$$ As before, since $[H,J_e]=0$, the Drude weight is given by $$\begin{aligned}
\label{eq:drude-wz}
\tilde{\kappa}(T)=\frac{\pi}{L T^2} \langle J_e^2 \rangle\end{aligned}$$ Moreover, since there is no interaction between the right and the left moving modes, $\langle J_e^2 \rangle =v^2 \langle H^2 \rangle$. One thus immediately obtains the result $\tilde{\kappa}_{WZ}(T)=\pi C_v(T)
v^2$, where $C_v$ is the specific heat of these modes. For conformally invariant modes with a central charge $c$, the specific heat is given by $C_v(T)=\frac{\pi T}{3v}c$, leading to a thermal Drude weight: $$\begin{aligned}
\label{eq:weight-conformal}
\tilde{\kappa}(T)=\frac{\pi^2 Tv}{3}c\end{aligned}$$ For the integrable spin-1 chain at the Takhtajan-Babujian point [@takhtajan_spin_s; @babujian_spin_s], this weight can also be recovered from explicit calculations using the Majorana formalism to be discussed in the forthcoming sections. For a theory described by a free massless boson like the spin-$\frac12$ Heisenberg chain, which has a central charge $c=1$, this weight is $\pi^2 Tv/3$, which can also be checked by direct calculations of the thermal susceptibility.[@heidrich_frustrated] For systems with a Luttinger liquid like description[@schulz_houches_revue] with a Luttinger exponent $K$, the present derivation illustrates clearly that the weight $\tilde{\kappa}$ is independent of the Luttinger exponent or equivalently, the compactification radius of the free bosonic Luttinger field. Considering the case of $XXZ$ chains, this result implies that the thermal Drude weight is independent of the anisotropy $J_z/J_{xy}$ which is in agreement with Bethe Ansatz calculations on the XXZ spin chain in the Luttinger liquid regime.[@kluemper_xxz]
Spin-$\frac 12$ ladder {#sec:pure-ladder}
----------------------
Here and in the following sections, we focus exclusively on spin gap systems. We first apply the formalism of Sec. \[sec:pure-case\] to the clean two leg spin ladder. The Hamiltonian of the two leg spin ladder is $$\begin{aligned}
\label{eq:lattice-hamiltonian}
H=J_\parallel \sum_{i\atop {p=1,2}} {\bf S}_{i,p} \cdot {\bf
S}_{i+1,p} + J_\perp \sum_i {\bf S}_{i,1}\cdot {\bf S}_{i,2},\end{aligned}$$ where the ${\bf S}_{i,p}$ are spin-$\frac 12$ operators, and the exchange constants $J_\parallel,J_\perp>0$. For weak interchain coupling $J_\perp\ll
J_\parallel$, the spin ladder can be described by a continuum theory of spinless Majorana fermions [@shelton_spin_ladders]. The continuum Hamiltonian reads: $$\begin{aligned}
\label{eq:continuum-hamiltonian}
H&=&\sum_{a=0}^3 \int dx {\cal H}^a(x),\\
{\cal H}^a(x)&=&
\frac{-iv}{4}\lbrack \xi^a_R(x) \partial_x \xi^a_R(x)-(\partial_x
\xi^a_R)(x) \xi^a_R(x)\nonumber \\ & & -\xi^a_L(x)
\partial_x \xi^a_L(x) + (\partial_x \xi^a_L)(x)\xi^a_L(x)\rbrack +i m^a \xi^a_R(x)\xi^a_L(x)\end{aligned}$$
where the velocity of the Majorana fermions $v=\frac \pi 2
J_\parallel a$ ($a$ is the lattice spacing). Physically, the Majorana modes $\xi^a_{R,L}$ $(a=1,2,3)$ with masses $m_{1,2,3}=J_\perp/(2\pi)\equiv \Delta$ describe triplet excitations with a gap $\Delta$ and $\xi^0_{R,L}$ with mass $m_0=-3J_\perp/(2\pi)=-3\Delta$ describe singlet excitations. We remark that the bosonized version of the low energy Hamiltonian (\[eq:continuum-hamiltonian\]) describes more general spin ladder models than the one considered in (\[eq:lattice-hamiltonian\]).[@nersesyan_biquadratic; @kim] Using (\[eq:continuity\]), the energy current for the ladder takes the form $$\begin{aligned}
\label{eq:majorana-therm-curr}
j_e(x)&=&\sum_{a=0}^3 j_e^a(x), \\ j_e^a(x)&=& \frac{-iv^2}{4} \left[
\xi^a_R \partial_x \xi^a_R - (\partial_x \xi^a_R ) \xi^a_R + \xi^a_L
\partial_x \xi^a_L- (\partial_x \xi^a_L ) \xi^a_L \right].\nonumber\end{aligned}$$
From (\[eq:majorana-therm-curr\]) and (\[eq:drude-thermal\]), the total Drude weight for the spin ladder is found to be $$\tilde{\kappa}(T)=\sum_a
\tilde{\kappa}^a(T)=\tilde{\kappa}^0(T)+3\tilde{\kappa}^1(T).$$ Since the Majorana fermions are essentially free, the correspondence between Majorana and Dirac fermions can be used to evaluate the Drude weight $\tilde{\kappa}^a(T)$ $$\label{eq:kappa-t}
\tilde{\kappa}^a(T)= \frac 1 {8T^2} \int_{-\Lambda }^{\Lambda } dk \frac
{v^4 k^2}{\cosh^2 \left( \frac{\epsilon_a(k)}{2T}\right) },$$ where the energy dispersion $\epsilon_a(k)=\sqrt{(vk)^2+m_a^2}$ and $\Lambda \sim
\frac {2\pi}{a}$ is the lattice induced ultra-violet cutoff. The details of the calculation are presented in Appendix \[app:weight\]. The thermal Drude weight of the spin ladder is now given by $$\begin{aligned}
\label{eq:ladder-kappa}
\tilde{\kappa}(T)= \frac 3 {8T^2} \int_{-\Lambda }^{\Lambda } dk \frac
{v^4 k^2}{\cosh^2 \left( \frac{\epsilon_1(k)}{2T}\right) } +
\frac 1 {8T^2} \int_{-\Lambda }^{\Lambda } dk \frac {v^4 k^2}{\cosh^2
\left( \frac{\epsilon_0(k)}{2T}\right) }\end{aligned}$$ At low enough temperatures $T\ll J_\perp/(2\pi)$, the triplet excitations are the dominant carriers of heat and $$\label{eq:tcond-low}
\tilde{\kappa}(T) = 3 \sqrt{\frac{\pi \Delta^2} {2T}} v
e^{-\Delta/T}$$ For temperatures $\Delta\ll T\ll J_{\parallel}$, the coupling between the two spin half chains becomes irrelevant and we recover $\tilde{\kappa} \propto T $ i.e., it is the sum of the Drude weights of two independent spin$-\frac12$ chains cf. Sec. \[sec:pure-conformal\]. For temperatures $T\gg J_{\parallel}$, the temperature dependence in the integrand of (\[eq:ladder-kappa\]) becomes negligible and since the $k$ integral is bounded on a lattice, the thermal Drude weight decays as ${\tilde \kappa} \propto T^{-2}$. The prefactor depends on the cutoff $\Lambda$ and it is reasonable to assume that the continuum theory over-estimates this prefactor. To summarize, the Drude weight of the spin ladder has three regimes :i) at very low temperatures, $T\ll \Delta$, we obtain the exponential behavior (\[eq:tcond-low\]) ii) for intermediate temperatures, $\Delta \ll T \ll
J_\parallel$, $\tilde{\kappa}\propto T$ and iii) for $T\gg J_{\parallel}$,$\tilde{\kappa} \sim
1/T^2$. Since $\tilde{\kappa}(T \to 0)=0$, this implies the presence of at least one maximum in $\tilde{\kappa}$ at a finite temperature for a lattice model. We expect $\tilde{\kappa}$ to have a peak in the vicinity of $T\sim J_\parallel$ (cf. Fig.\[fig:weight\]). We note that the numerical results for $\tilde{\kappa}(T)$ for the ladder presented in Ref. confirm our picture.
To study the effect of an applied magnetic field $h$ on the thermal conductivity, we first note that the effect of the magnetic field is to alter the dispersion of the triplet. The degenerate triplet dispersion $\epsilon_1(k)$ now splits into three branches $\epsilon_1(k)
+h$,$\epsilon_1(k)$ and $\epsilon_1(k)-h$ and the singlet dispersion $\epsilon_0(k)$ remains unaltered. The Drude weight in the presence of the field is now given by:
$$\label{eq:weight-field}
\tilde{\kappa} = \frac 1 {8 T^2} \int_{- \Lambda }^{\Lambda } dk
\left(\frac{\partial \epsilon_1(k)}{\partial k}\right)^2
\left\{ \frac {(\epsilon_1(k)-h)^2} {\cosh^2\left(\frac{\epsilon_1(k) - h}{2T}\right)}
+\frac {(\epsilon_1(k) + h)^2} {\cosh^2\left(\frac{\epsilon_1(k) + h}{2T}\right)} + \frac {\epsilon_1(k)^2}
{\cosh^2\left(\frac{\epsilon_1(k)}{2T}\right)} \right\} + \frac 1 {8
T^2} \int_{- \Lambda }^{\Lambda } dk \frac {v^4 k^2}
{\cosh^2\left(\frac{\epsilon_0(k)}{2T}\right)}$$
There are now two regimes of interest: $h \ll \Delta$ and $h > \Delta$. In the former case, the effective gap $\Delta-h$ dominates the thermal conductivity and $\tilde{\kappa}\propto
e^{-(\Delta-h)/T}/\sqrt{T}$. The magnetic field leads to a sufficient enhancement of the low temperature thermal conductivity. The physical reason is that the increase of the number of triplet excitations with $S^z=+1$ strongly dominates the diminution of the number of excitations having $S^z=-1$. For $h\sim \Delta$, the dispersion of the Majorana fermions describing the $S_z=+1$ sector is no longer relativistic-like but quadratic, $\epsilon(k) \propto k^2$, resulting in $\tilde{\kappa} \sim T^{3/2}$. Finally, for $h>\Delta$, the gap in spin ladder is closed[@chitra_spinchains_field; @sachdev_qaf_magfield], and the fermionic excitations have an effective linear dispersion, leading to $\tilde{\kappa}(T)=\frac{\pi^2 T}{3} {\tilde v}(h)$, where the effective Fermi velocity ${\tilde v}(h)=v \sqrt{1-(\Delta/h)^2}$.
Let us note that the above results (\[eq:ladder-kappa\]), (\[eq:weight-field\]) are also relevant for spin-1 chains. Indeed, spin-1 chains are also described at low energy by massive Majorana fermions, with a Hamiltonian similar to (\[eq:continuum-hamiltonian\]), except that the singlet mode $\xi^0$ is absent[@tsvelik_field]. This mapping to massive Majorana fermions originally derived for a spin-1 chain with bi-quadratic interactions in the vicinity of the Takhtajan-Babujian point[@takhtajan_spin_s; @babujian_spin_s], is also expected to provide a qualitatively description of the low energy properties of the Heisenberg spin-1 chain. Therefore, the thermal conductivity of the spin-1 chain is easily obtained by taking the limit $\epsilon_0 \to
\infty$ in Eqs. (\[eq:kappa-t\]) and (\[eq:weight-field\]). For the spin-1 chain, the low temperature behavior of the Drude weight in the thermal conductivity is still given by (\[eq:tcond-low\]). The main difference between the ladder and the spin-1 chain stems from the fact that while in the former the gap to triplet excitations is small, in the latter the gap $\Delta$ is of the order of the Heisenberg exchange $J$ ($\Delta=0.41 J$). Consequently, the intermediate regime of linear temperature dependence of $\tilde{\kappa}$ can hardly be observed in the spin-1 chain. However, reasonably strong bi-quadratic interactions can reduce the gap appreciably rendering an observation of an intermediate linear regime possible. This predicted linear behavior might in fact be observable in the compound $\mathrm{LiV_2GeO_6}$ which is expected to have sizeable biquadratic interactions[@millet_biquad].
Dimerized XY Chain {#sec:pure-XY}
------------------
We consider a spin-$\frac 12 $ XY chain with alternating exchange in an external magnetic field $h$, described by the Hamiltonian: $$\begin{aligned}
\label{eq:XYchain-ham}
H&=&\sum_n J_1 (S_{2n}^xS_{2n+1}^x+S_{2n}^yS_{2n+1}^y)\nonumber \\
&& + \sum_n J_2 (S_{2n}^xS_{2n-1}^x+S_{2n}^yS_{2n-1}^y) -h S_n^z\end{aligned}$$ Using the Jordan-Wigner transformation[@jordan_transformation], $$\begin{aligned}
\label{eq:jordan-wigner}
S^+_{n}&=&a^\dagger_n \cos\left(\sum_{m<n} a^\dagger_m a_m\right),\nonumber \\
S^z_n&=&a^\dagger_n a_n-\frac 1 2,\end{aligned}$$ where the $a,a^\dagger$ are fermion annihilation and creation operators, the Hamiltonian (\[eq:XYchain-ham\]) can be rewritten as: $$\begin{aligned}
\label{eq:XY-fermionized}
H=J_1\sum_n (a^\dagger_{2n+1}a_{2n}+ \text{H.c.}) + J_2 \sum_n (a^\dagger_{2n-1}a_{2n}+ \text{H.c.})\end{aligned}$$ Diagonalizing the above Hamiltonian, we obtain $$\begin{aligned}
\label{eq:XY-diagonalized}
H=\sum_k [E(k)-h] a^\dagger_{k,+} a_{k,+} - [E(k)+h]a^\dagger_{k,-}
a_{k,-},\end{aligned}$$ where $E(k)=\sqrt{(J_1-J_2)^2+4 J_1 J_2 \cos^2 k}$. Clearly, the dimerization induces a gap in the dispersion. Using the results of the previous sections and Appendix \[app:mag-clean\], the thermal Drude weight of this dimerized chain is $$\begin{aligned}
\label{eq:XY-weight}
\tilde{\kappa}_{XY}(T,h)&=&\frac 1 {8T^2} \int_{-\frac \pi a}^{\frac{\pi}{a}} dk \left[ \frac
{(E(k)-h)^2}{\cosh^2\left(\frac{E(k)-h}{2T}\right)}\right. \nonumber \\ && \left. + \frac
{(E(k)+h)^2}{\cosh^2\left(\frac{E(k)+h}{2T}\right)}\right]
\left(\frac{\partial E(k)}{\partial k}\right)^2\end{aligned}$$ For $T\ll |J_1-J_2|$, and $|J_1-J_2| \ll \sqrt{J_1 J_2}$ the physics is similar to that of the continuum model of the weakly coupled ladder discussed in Sec. \[sec:pure-ladder\]. The fact that the model is defined on a lattice allows us to verify that for $h=0$ and at very high temperatures the thermal conductivity indeed decays as $T^{-2}$. For $T\gg \sqrt{J_1 J_2}$, and $h=0$, since the energy spectrum of the XY chain is bounded, one has: $$\begin{aligned}
\label{eq:XY-highT}
\tilde{\kappa}_{XY}(T)=\frac 1 {4T^2} \int_{-\frac \pi a}^{\frac{\pi}{a}} dk E(k)^2
\left(\frac{\partial E(k)}{\partial k}\right)^2\end{aligned}$$ This result is in fact more general. Since, at high temperatures, $\langle
J_e^2 \rangle $ is finite for a lattice model, we have the asymptotic behavior ${\tilde \kappa} \sim\langle
J_e^2 \rangle_{T=\infty} /T^2 $. Another interesting limit is when $\sqrt{J_1 J_2}\ll
|J_1-J_2|$, i.e. when the spin gap is much larger than the bandwidth of magnetic excitations. In this case, for $\sqrt{J_1J_2}\ll
T \ll |J_1-J_2|$, replacing $E(k)$ in (\[eq:XY-weight\]) by $|J_1-J_2|\equiv \Delta$ we obtain $$\begin{aligned}
\label{eq:schottky}
\tilde{\kappa}(T)\simeq \frac{\pi J_1^2 J_2^2} {a T^2 \cosh^2\left(\frac \Delta
{2T}\right)}\end{aligned}$$ Note that the Drude weight can be recast in the form $\tilde{\kappa}(T)=\pi C_v(T)
v_{\text{eff.}}$, where $C_v(T)$ is the specific heat of a fermion that can occupy two levels separated by $|J_1-J_2|$ and with an effective velocity $v_{eff.}\sim J_1 J_2/|J_1-J_2|$. However, this analogy cannot be extended systematically to other spin gapped systems.
Turning to the effect of the magnetic field, the Drude weight can again be rewritten as (see Eq. (\[eq:kinetic-interpretation\])) : $$\begin{aligned}
\tilde{\kappa}(T,h)= \int_{-\frac \pi a}^{\frac{\pi}{a}} dk [C_v(\epsilon(k)-h)+C_v(\epsilon(k)+h)]v^2(k),\end{aligned}$$ where $C_v(\epsilon)$ is the specific heat of a single fermion of energy $\epsilon$ and the velocity $v(k)= \epsilon \partial\epsilon/ \partial k$. This form helps us derive a kind of sum rule for the thermal conductivity. For a free fermion, one has: $$\begin{aligned}
\int_0^\infty \frac{C_v(T)}{T} dT=S(T=\infty)-S(T=0)=k_B \ln 2,\end{aligned}$$ And thus: $$\begin{aligned}
\label{eq:sumrule}
\int_0^\infty \frac{\tilde{\kappa}(T,h)}{T} dT=k_B \ln 2 \int dk v^2(k).\end{aligned}$$ We note that the integral is independent of the magnetic field, so we have a kind of “sum rule”. Since in the presence of the magnetic field, it is easily seen that the low temperature thermal weight is enhanced by a factor $e^{h/T}$, this necessarily implies that for higher temperatures, the thermal weight must decrease when a magnetic field is applied This scenario is confirmed by Fig. \[fig:kappa-h\].
We also note that for high magnetic fields, a double peak structure appears in the thermal weight as seen in Fig. \[fig:double-peak\].
The double peak results from the low temperature shift of the maximum of the contribution of the up spins, in a region in which the contribution of the down spins in negligible, and the high temperature shift of the maximum of the contribution of down spins. A similar double peak is also visible in the heat capacity. It would be interesting to investigate whether such a double peak is also present in other spin gap systems. It is known that a double peak is present in the specific heat of zig-zag spin ladder in a magnetic field.[@maeshima_zigzag] To summarize, the sum rule (\[eq:sumrule\]) for the thermal Drude weight, holds for all spin systems which can be described by an effective theory of non-interacting fermions.
The Massive Boson Model {#sec:pure-boson}
-----------------------
We now consider the massive triplet boson model which was proposed as a phenomenological model for the spin-1 Heisenberg chain. This model can be obtained from the nonlinear sigma model[@haldane_gap] that describes integer spin$-S$ chains in the limit $S\to \infty$ by softening the constraint on the $O(3)$ fields. More precisely, this model is characterized by a Hamiltonian density [@affleck_field] $$\begin{aligned}
\label{eq:massive-bosons}
{\cal H}(x)= \frac u 2 \sum_{\alpha=1}^3 \left[ \Pi_\alpha^2 +(\partial_x
\phi_\alpha)^2\right] + u V({\bf {{{\mbox{\boldmath $\phi$}}}}}),\end{aligned}$$ where $V({{{\mbox{\boldmath $\phi$}}}})=\frac{\Delta^2} {2u^2} {\bf {{{\mbox{\boldmath $\phi$}}}}^2} +
\frac{\lambda}{4}({\bf {{{\mbox{\boldmath $\phi$}}}}^2})^2$, and $[\phi_\alpha(x),\Pi_\beta(x')]=i\delta(x-x')\delta_{\alpha,\beta}$. The energy current takes the simple form [@li_thermal] $$\begin{aligned}
\label{eq:boson-thermal-current}
j_e(x)=-u^2 \sum_{\alpha=1}^3 \Pi_\alpha \partial_x\phi_\alpha,\end{aligned}$$ Note that this current is independent of the potential $V({{{\mbox{\boldmath $\phi$}}}})$ and up to a prefactor, it is just the momentum density of the boson field[@messiah_field_chapter]. Consequently, translation invariance implies that the total thermal current $J_e$ is conserved. This allows us to use the Eq. (\[eq:drude-thermal\]) to obtain the thermal Drude weight. Since the bosons are weakly interacting, we can consider the case $\lambda=0$, to obtain the Drude weight $$\begin{aligned}
\label{eq:boson-weight}
\tilde{\kappa}(T)=\frac 3 {8T^2} \int_{-\Lambda}^{\Lambda} dk \frac {u^4
k^2}{\sinh^2\left(\frac{\sqrt{(uk)^2+\Delta^2}}{2T}\right)}\end{aligned}$$ As before, the limit $T\ll \Delta$ again leads to the result (\[eq:tcond-low\]) for $\tilde{\kappa}(T)$ and for $T\gg
\Delta$, we recover a linear weight $\tilde{\kappa}(T)= \pi^2 u T$. We note that in the case of the Takhtajan-Babujian spin-$S$ chains (which are described by $SU(2)_{2S}$ WZNW models at low energy[@affleck_strongcoupl]), the weight is given by: $$\tilde{\kappa}(T)=\frac{\pi^2}{3} \frac{3S}{S+1} u T,$$ for $S\to \infty$, this weight is the same as the one of the triplet of bosons. This is consistent with the fact that the non-linear sigma model describes spin$-S$ chains in the limit $S\to \infty$.
Discussion
----------
In the preceding sections, we have seen that all the spin gap systems studied in this paper exhibit the same generic behavior for the thermal conductivity. The reason is that for gapped 1D systems that can be bosonized, the low energy theory is Lorenz invariant, and excitations are described by massive particles having relativistic-like dispersions $\epsilon_\alpha(p)=\sqrt{(vp)^2+m_\alpha^2}$, with the gap $\Delta$ being the mass of the lightest particle. When these excitations are spin triplets, the lowest excited state contains exactly one of these particles, and the total energy current is $\epsilon(p) \frac {\partial \epsilon}{\partial p}$ which then yields a weight: $$\begin{aligned}
\label{eq:relativistic-kappa}
\tilde{\kappa}(T)\sim \int dp \epsilon(p)^2 \left(\frac {\partial
\epsilon}{\partial p}\right)^2 e^{-\epsilon(p)/T}\end{aligned}$$ Since Lorentz invariance dictates that $\epsilon(p) \frac {\partial
\epsilon}{\partial p}=p$, one obtains the same thermal weight as in (\[eq:tcond-low\]) in the low-temperature regime. Examples of systems possessing this triplet branch are the alternating spin-$\frac12$ chain[@haldane_dimerized; @barnes_dimerized], the two-leg spin ladder[@barnes_ladder; @knetter_ladder] and the Heisenberg spin-1 chain[@takahashi_spin1]. We therefore, expect that the above mentioned systems will exhibit a finite thermal Drude weight. However, this result could differ in the case of the zig-zag ladder or the frustrated spin $\frac12$ chain. This stems chiefly from the fact that though the zig-zag ladder has a gapful spectrum, the low energy excitations having a relativistic dispersion, are spinons[@haldane_dimerized; @allen_spinons] carrying a spin $\frac12$. Another example with spinonic excitations is the XXZ chain in the Ising phase[@nijs_equivalence]. Since the total spin of the system can only vary by an integer, the spinons occur in pairs. Consequently, the interaction between these spinons has a strong influence on the thermal weight. In the case of the XXZ chain, since the spinons are non-interacting, the current of a given excited state is conserved, and one expects to recover a finite Drude weight. However, in the case of the zig-zag ladder or the frustrated spin $\frac12$ chain, the interaction between the spinons can lead to a non-conservation of the current of the two spinon state, resulting in the suppression of the thermal Drude weight[@heidrich_frustrated].
It would be worthwhile to compare our predictions for the Drude weight for various systems with numerical simulations[@heidrich_frustrated] or with other analytical techniques on the lines of Ref. in the case of integrable models. However, in the former case, the extraction of the power law prefactor in the activated thermal Drude weight from numerical data might prove very difficult.
Effect of impurities {#sec:disorder-case}
====================
We have seen in Sec. \[sec:pure-case\] that the thermal conductivity in clean systems has a Drude peak as a result of the translational invariance of the system. In a real system, we expect this Drude peak to be replaced by a finite thermal conductivity, due to the finite lifetime of eigenstates of the Hamiltonian induced by phonon or impurity scattering. In the present section, we study the effect of impurity scattering on the gapped systems we discussed in sections \[sec:pure-ladder\] and \[sec:pure-XY\]. We will begin with a calculation of the conductance of the system with a single impurity, and then we will turn to a system with a nonzero concentration of impurities.
Single-impurity problem
-----------------------
The thermal conductivity of a system with a single-impurity can be calculated using the simple Landauer approach[@landauer_formula] provided, the elementary excitations are non interacting. The basic idea[@fazio_thermal_1d] is to consider two reservoirs at temperature $T_1$ and $T_2$ (with $T_1>T_2$) in presence of a barrier (the impurity potential). Reservoir 1 emits a particle with momentum $k>0$, energy $\epsilon (k)$ and velocity $\partial \epsilon (k)/\partial k$. The probability to traverse the barrier is given by the square of the transmission coefficient $|t(k)|^2$. The current flowing from reservoir 1 to reservoir 2 is:
$$\label{curr1}
J_{1\to 2}=\int_0^\infty \frac{dk}{2 \pi} n_1(k,T_1) |t(k)|^2
\epsilon(k) \frac{\partial \epsilon (k)}{\partial k},$$
and similarly the current flowing from to reservoir 2 to 1 is:
$$\label{curr2}
J_{2\to 1}=\int_0^\infty \frac{dk}{2 \pi} n_2(k,T_2) |t(k)|^2
\epsilon(k) \frac{\partial \epsilon (k)}{\partial k},$$
where $n_{1,2}(k,T_{1,2})$ are the fermion distribution functions at temperature $T_{1,2}$. In the limit $T_1\simeq T_2$, the net current flowing through the barrier is:
$$\begin{aligned}
J&=&J_{1\to 2}-J_{2\to 1}=\int_0^\infty \frac{dk}{2 \pi}
\frac{|t(k)|^2 \epsilon^2(k)}{4 \cosh (\frac{\epsilon(k)}{2 T_2})}\frac{T_1-T_2}{T^2_2} \frac{\partial \epsilon
(k)}{\partial k}\nonumber \\&=&{\cal K}(T_1)(T_1-T_2),\end{aligned}$$
Hence a knowledge of the transmission probability $\vert
t^2\vert$ permits us to obtain the thermal conductance
$$\label{tsi}
{\cal K}(T)=\int_0^\infty \frac{dk}{2 \pi} |t(k)|^2 \frac{
\epsilon^2(k)}{4 T^2\cosh (\frac{ \epsilon(k)}{2 T})}
\frac{\partial \epsilon (k)}{\partial k},$$
We now apply this general formula (\[tsi\]) to two spin gap systems in which the elementary excitations are non-interacting.
### Ladder with a defect {#sec:1imp-ladder}
We consider a two leg spin 1/2 ladder with a defect on a rung, described by the Hamiltonian: $$\label{eq:ham-ladder-1imp}
H=J_\parallel \sum_{n\atop p=1,2} {\bf S}_{n,p}\cdot {\bf S}_{n+1,p}
+ J_\perp \sum_{n \neq 0} {\bf S}_{n,1}\cdot {\bf S}_{n,2} +
J_\perp' {\bf S}_{0,1}\cdot {\bf S}_{0,2}$$ This Hamiltonian can be fermionized following Ref. . The perturbation to the ladder becomes: $$\begin{aligned}
\label{eq:perturbation}
(J_\perp'-J_\perp)a^2 ({\bf J}_1 + {\bf n}_1)(0)\cdot ({\bf J}_2 +
{\bf n}_2)(0),\end{aligned}$$ where ${\bf J}_{1,2}$ and ${\bf n}_{1,2}$ are the uniform and staggered spin densities, respectively, and the most relevant contribution is $(J_\perp'-J_\perp) a^2 {\bf
n}_1(0)\cdot {\bf n}_2(0)$. This contribution can be fermionized, so that the resulting low energy Hamiltonian of the ladder with a rung defect reads: $$\begin{aligned}
\label{eq:ladder-1imp-majorana}
H&=&-\frac{iv}{2} \sum_{a=0}^4 \int dx (\xi_R^a \partial_x \xi_R^a -\xi_L^a
\partial_x \xi_L^a)\nonumber \\ && + i \int dx m(x) \left(\sum_{a=0}^3
\xi_R^a\xi_L^a -3 \xi_R^a\xi_L^a\right),\end{aligned}$$ where $m(x)=m + g \delta(x)$, with $m=J_\perp/(2\pi)$ and $g=(J_\perp'-J_\perp)a/(2\pi)$. Clearly, each Majorana mode is scattered independently from the barrier, so that their contributions to the thermal conductivity is additive. As discussed in in the preceding sections and in App. \[app:weight\], we use the correspondence between the Majorana and Dirac fermions to calculate the thermal conductivity with the barrier. The First Quantized Hamiltonian for the Dirac fermions reads: $$\begin{aligned}
\label{eq:ladder-1imp-dirac}
H=-iv \sigma_3 \partial_x + m(x) \sigma_2,\end{aligned}$$ where $\sigma_i$ are Pauli matrices. Solving the Schr[ö]{}dinger equation with appropriate boundary conditions for the wavefunction at the barrier, we obtain the transmission probability $$\label{eq:transmission-dirac}
|t(k)|^2=\cos^2 \psi \frac{k^2}{k^2+K^2},$$
where $K=m/v \sin \psi$ and $\tan(\frac{\psi}{2})=\frac{g}{2v}=(J_\perp'-J_\perp)/(2\pi^2 J_\parallel)$, Eq.(\[tsi\]) becomes: $$\begin{aligned}
{\cal K}(T)&=&\int_0^\infty \frac{dk}{2 \pi} \cos^2 \psi
\frac{k^2}{k^2+K^2} \frac{(vk)^2+m^2}{4 T^2\cosh (\frac{
\sqrt{(vk)^2+m^2}}{2T})} \nonumber \\ && \times\frac{(vk)}{\sqrt{(vk)^2+m^2}},\end{aligned}$$ where we have used $\epsilon(k)=\sqrt{(vk)^2+m^2}$. In the limit $T\to 0$, the transmission probability is dominated by momentum $k\ll K$ for which $|t(k)|^2\sim k^2/K^2$ i. e. the barrier is a strong scatterer, and $$\label{eq:conductance-ladder-low}
{\cal K}(T) = \frac {3m} {2\pi} e^{-m/T} \cot^2 \psi ,$$ where we have taken into account the triplet of Majorana modes . One can obtain an estimate of the temperature $T^*$ below which this result is valid by noting that for $T\ll m$, one has $\langle (vk)^2\rangle =
mT$, so that the criterion for low temperature is $mT\ll m^2 \sin^2
\psi$, i. e. $T\ll T^*=J_\perp (J_\perp'-J_\perp)^2/J_\parallel^2$. This temperature is clearly much smaller than the gap $m$. For higher temperatures, $T^*\ll T \ll m$, the thermal conductance is obtained by making the approximation $|t(k)|^2\sim \cos^2 \psi$, leading to: $$\begin{aligned}
{\cal K}(T)&=& \frac {3\cos^2 \psi}{4\pi} T \int_{m/T}^\infty dx
\left( \frac {x/2}{\cosh (x/2)}\right)^2 \nonumber \\
&&+\frac {\cos^2 \psi}{4\pi}
T \int_{3 m/T}^\infty dx
\left( \frac {x/2}{\cosh (x/2)}\right)^2\end{aligned}$$ For $T^*\ll T \ll m$, one finds ${\cal K}(T) \sim \frac{3m^2}{4\pi T}
e^{-m/T}$, and for $T\gg m$, ${\cal K}(T)=\frac{2\pi T}{3} \cos^2
\psi$. Contrary to the result (\[eq:tcond-low\]) for the pure ladder, the thermal conductance (\[eq:conductance-ladder-low\]) for $T\ll T^*$ is purely activated without any $T$ dependent prefactor. Therefore, the Drude weight in the thermal conductivity for the pure system is not an accurate indication on the behavior of the thermal conductivity in a system with impurities. The reason for that is clear from (\[eq:transmission-dirac\]), namely low energy modes experience much stronger impurity scattering than the high-energy ones. It is only in the high temperature limit $T\gg m$ that the replacement $\delta(\omega) \to \tau$ is justified. We will see in the following section that this result is not restricted to the spin ladder.
### XY chain with a defect {#sec:1imp-XY}
We consider again the XY-chain with alternating exchange of Sec. \[sec:pure-XY\]. We now suppose that the bond strength $J_1$ between the sites $0$ and $1$ is replaced by $J'_1$. This bond acts as a barrier and using the results of Appendix \[app:eigen-XY\], the transmission probability across this barrier is given by $$\label{eq:transmission-XY}
|t(k)|^2=\frac{4J_1^2 (J'_1)^2 \sin^2 \phi_k}{(J_1^2-(J'_1)^2)^2+ 4J_1^2
(J'_1)^2 \sin^2 \phi_k}$$
In particular, we can show that when $k\simeq\pi/2$ we have: $$|t(k)|^2=\frac{16 J_1^2 (J'_1)^2
J_2^2 (k-\pi/2)^2}{(J_1^2-(J'_1)^2)^2(J_1-J_2)^2+ 16 J_1^2 (J'_1)^2 J_2^2 (k-\pi/2)^2},$$ which indicates that for low temperatures $T\ll \sqrt{J_1 J_2}$, the behavior of the thermal conductivity in the XY chain with a bond defect is identical to the behavior of the thermal conductivity in the ladder discussed in Sec. \[sec:1imp-ladder\]. For high temperatures, $T\gg
\sqrt{J_1 J_2}$, we can neglect the variation of the transmission coefficient with the energy, and assume that all states have the same probability of occupation. Then, the thermal conductance reads: $${\cal K}(T)=T \int_{(J_1-J_2)/T}^{(J_1+J_2)/T} d\epsilon \epsilon^2
\langle |t(\epsilon)|^2\rangle \sim 1/T^2$$
Many impurities case
--------------------
In this section, we consider the effect of a finite concentration of impurities on the thermal conductivity of the ladder. As before, the disorder we consider is a random rung coupling. The Hamiltonian of the disordered ladder reads: $$\begin{aligned}
H=J_\parallel \sum_{i\atop {p=1,2}} {\bf S}_{i,p} \cdot {\bf
S}_{i+1,p} + \sum_i J^i_\perp {\bf S}_{i,1}\cdot {\bf S}_{i,2},\end{aligned}$$ where $J_\perp^i= J_\perp+\eta_i$. We have $\vert\eta_i\vert<J_\perp$, so that all rung interactions remain antiferromagnetic. This Hamiltonian can be analyzed by mapping onto a random mass Majorana fermions model[@gogolin_disordered_ladder; @steiner_random_mass].
$$\begin{aligned}
H&=&-\sum_{a=1}^4 \int dx \{ \frac{iv}{2}\lbrack \xi^a_R(x) \partial_x
\xi^a_R(x)-\xi^a_L(x) \partial_x \xi^a_L(x) \nonumber \\ &&+im^a(x)
\xi^a_R(x)\xi^a_L(x) \rbrack \},\end{aligned}$$
with $m^{1,2,3}(x)=m(x)$ for the triplet magnetic excitation, $m^0=-3|m|(x)$ for the singlet excitation and $m(x)=m+\eta(x)$ where $\overline{\eta(x)\eta(x')}=D\delta(x-x')$. We note that disorder does not mix the different flavors of Majorana fermions. Consequently, the contribution of the Majorana modes to the thermal conductivity remains additive. As before, to calculate the disorder induced self-energy, it is useful to recast the above problem in terms of Dirac Fermions. $$\begin{aligned}
\label{eq:random_dirac}
H&=&-i v \int dx (\psi^\dagger_R \partial_x \psi_R - \psi^\dagger_L
\partial_x\psi_L) \nonumber \\ &&+ m(x) \int dx (\psi^\dagger_R\psi_L +
\psi^\dagger_L\psi_R),
$$ We note that the Hamiltonian (\[eq:random\_dirac\]) can also be derived from a dimerized XY chain with bond defects.[@mckenzie_random_xy] We define the $2\times2$ matrix disordered averaged Green’s function $\hat{G}$ by its components, $$G_{\alpha \beta}(x,\tau)= - \langle T_\tau \psi_\alpha(x,\tau)
\psi^\dagger_\beta(0,0) \rangle,$$ where $\alpha,\beta \in \{R,L\}$. The Hamiltonian (\[eq:random\_dirac\]) can be rewritten in matrix form as: $$H=\int dx \Psi^\dagger(x)[-i v \tau_3 \partial_x + m(x) \tau_1]\Psi(x) ,$$ where $\tau_{1,3}$ are Pauli matrices. The impurity self-energy matrix $\hat{\Sigma}$ can be calculated within the Self Consistent Born Approximation (SCBA)[@mori_scba] and satisfies the Dyson equation for the disorder averaged Green’s function: $$\label{eq:self}
(i \omega_n - vk \tau_3 - m\tau_1 - {\Sigma}){G} = 1,$$ In this approximation, the self-energy is independent of momentum and is determined self-consistently by $${\Sigma(i\omega_n)} = D \int \frac{dk}{2\pi} \tau_1 [i\omega_n - v
k \tau_3 - m\tau_1 -{\Sigma(i\omega_n)}]^{-1}\tau_1$$ Clearly, the self energy possesses the following structure, ${\Sigma}(i\omega_n) = i \sigma(i\omega_n) + V(i\omega_n)\tau_1$ leading to the following self-consistent equations for $\sigma$ and $V$[@mori_scba]: $$\begin{aligned}
\label{eq:scba}
\sigma(i\omega_n)= \frac{D}{2v}
\frac{\sigma(i\omega_n)-\omega_n}{\sqrt{(\sigma(i\omega_n)-\omega_n)^2
+ (m + V(i\omega_n))^2}} \nonumber \\
V(i\omega_n)=\frac{D}{2v}
\frac{V(i\omega_n) -m}{\sqrt{(\sigma(i\omega_n)-\omega_n)^2 +
(m + V(i\omega_n))^2}}\end{aligned}$$ Introducing the dimensionless variables: $s=i\sigma/m$, $t = V/m$, $x= i\omega_n/m$ and $\lambda= D/(2vm)$ the above self-consistent equations simplify to: $$\begin{aligned}
&& t=\frac{s}{x-2s}, \\ && s^4 -s^3 x - s^2 (\frac{1-x^2} 4 -
\lambda^2) - \lambda^2 sx + \frac{\lambda^2} 4 x^2 =0.\end{aligned}$$
This quartic equation can be solved numerically. Some sample curves are shown in Fig.\[fig:self-energy\]. We find that the disorder renormalizes the gap in the spectrum and a sufficiently strong disorder ($\lambda=0.5$) closes the gap indicating a disorder induced phase transition within the SCBA. A plot of the renormalized gap $\omega_c$ as a function of disorder strength is shown on Fig. \[fig:graph3\]. In the ensuing calculation, we only consider disorder strengths for which the renormalized gap is non-zero.
Using the results of the previous sections, the thermal conductivity for the disordered ladder can be rewritten as [@mahan_book]: $$\label{eq:mahan_formula}
\kappa(T)=\frac 1 {T} \int \frac{d\omega }{2\pi}
\left(-\frac{\partial n_F}{\partial \omega }\right) [
P(\omega-i0,\omega+i0) -\mathrm{Re} P(\omega+i0,\omega+i0)],$$ where: $$\label{eq:pww}
P(\omega,\omega')=\int \frac{dk}{2\pi} (v^2k)^2 \mathrm{Tr}\left[
G(k,\omega) G(k,\omega')\right],$$ Vertex corrections to (\[eq:mahan\_formula\]) are negligible in the weak disorder limit. Contrary to the suggestion in Ref. that $\mathrm{Re} P(\epsilon+i0,\epsilon+i0)$ in Eq. (\[eq:mahan\_formula\]) can be neglected, we find that this term is indeed crucial to take into account the presence of a gap in the energy spectrum. To proceed with the calculation of $\kappa$, we first note that for weak disorder i.e., $D\ll v m$, since the off-diagonal self-energy $V$ always occurs in the combination $m+V$ (\[eq:self\]), it is reasonable to neglect $V$ in the Green’s function $G$ which can then be approximated as $$G(k,\omega )=\frac{\omega -\sigma(\omega) +vk \sigma_3 +m \sigma_1}{[\omega -\sigma(\omega)]^2
-\epsilon(k)^2},$$ where $\epsilon(k)=\sqrt{v^2k^2 + m^2}$. This yields $$\begin{aligned}
G(k,\omega+i0)&=&\left(\frac 1 2 + \frac{vk \sigma_3 +
m\sigma_1}{2\epsilon(k)}\right) \frac 1
{\omega-\sigma(\omega)-\epsilon(k)}\nonumber \\ && + \left(\frac 1 2 - \frac{vk
\sigma_3 + m\sigma_1}{2\epsilon(k)}\right) \frac 1
{\omega-\sigma(\omega)+\epsilon(k)},\nonumber \\
G(k,\omega-i0)&=&\left(\frac 1 2 + \frac{vk \sigma_3 +
m\sigma_1}{2\epsilon(k)}\right) \frac 1
{\omega-\sigma^*(\omega)-\epsilon(k)}\nonumber \\ && + \left(\frac 1 2 - \frac{vk
\sigma_3 + m\sigma_1}{2\epsilon(k)}\right) \frac 1
{\omega-\sigma^*(\omega)+\epsilon(k)}.\nonumber\end{aligned}$$ Substituting the above in (\[eq:pww\]), we obtain $$P(\omega+i0,\omega-i0)-\mathrm{Re}P(\omega+i0,\omega+i0) = \int
\frac{dk}{2\pi} (v^2k)^2 K(\omega,k)$$ where $$\begin{aligned}
\label{eq:definition_K}
K(\omega,k)&=&\frac{(\mathrm{Im} \sigma(\omega))^2}{\left[
(\omega - \epsilon(k) -\mathrm{Re}\sigma(\omega))^2+
(\mathrm{Im}\sigma(\omega))^2\right]^2} \nonumber \\ &+& \frac{(\mathrm{Im}
\sigma(\omega))^2}{\left[ (\omega + \epsilon(k)
-\mathrm{Re}\sigma(\omega))^2+
(\mathrm{Im}\sigma(\omega))^2\right]^2},\end{aligned}$$ This expression can now be used in (\[eq:mahan\_formula\]) to obtain the thermal conductivity. At low temperatures, the derivative of the Fermi function in (\[eq:mahan\_formula\]) decays exponentially as $\exp-(\omega/T)$ indicating that frequencies much larger than $T$ can be neglected in the integral for the thermal conductivity $\kappa(T)$. Consequently, the low temperature behavior of $\kappa (T)$ is completely dictated by the the low frequency behavior of $K(\omega ,k)$. We now analyze the behavior of $K$. Firstly, since the diagonal self energy $\mathrm{Im}\sigma(\omega+i0)=0$, for $\omega< \omega_c$ i.e., for frequencies smaller than the disorder renormalized gap, $K(\omega ,k)$ is identically zero for all $\omega< \omega_c$ . A typical plot of $K$ as a function of $\omega $ for two different values of $k$ is shown on figure \[fig:graph2\].
Clearly, the dominant contribution to $\kappa$ for temperatures $T<\omega_c$ comes from the behavior of $K(\omega ,k)$ in the vicinity of $\omega_c$. This behavior has been analyzed numerically, and we find that for all $k$, $K$ can be developed as a series in $\omega-\omega_c$: $$\begin{aligned}
\label{eq:fit-K}
K(\omega,k)&=&\alpha(k)(\omega-\omega_c)\nonumber \\
&& +\beta(k)(\omega-\omega_c)^2+\ldots\end{aligned}$$ We find that $\alpha,\beta$ are fast decreasing functions of $|k|$, such that $\int dk k^2 \alpha(k) <\infty$ and $\int dk k^2 \beta(k)
<\infty$. Substituting (\[eq:fit-K\]) in (\[eq:mahan\_formula\]), we obtain $$\begin{aligned}
\kappa(T)&=&\frac 1 {4T^{2}} \int dk \int d\omega \; k^2
e^{-\omega/T}\Theta(\omega-\omega_c)
\left[\alpha(k)(\omega-\omega_c)\right. \nonumber \\ &&\left. +\beta(k)
(\omega-\omega_c)^2+\ldots \right]\end{aligned}$$ leading to, $$\label{eq:kappa}
\kappa(T) \sim \tilde{\alpha}
e^{-\omega_c/T}+ \tilde{\beta} T e^{-\omega_c/T} + o(T
e^{-\omega_c/T})$$ $\tilde{\alpha}, \tilde{\beta}$ are temperature independent constants. We find that the results are similar to those for a single impurity (\[eq:conductance-ladder-low\]) with the difference that a finite concentration of impurities renormalizes the gap. At very high temperatures one recovers the usual $T^{-2}$ decrease of the thermal conductivity. It would be interesting to obtain the cross-over behavior from the the low to high temperature regimes. However, the rather complicated form of the self energies makes analytical calculations very difficult for these intermediate temperatures and this is left for future work. For small magnetic fields $h\ll \omega_c$ the same result holds with the substitution $\omega_c \to \omega_c
- h$.
Discussion
==========
We now highlight the connection between our approach and that of the Boltzmann equation. A Boltzmann like equation for the thermal conductivity can be recovered from the SCBA in the limit of small self energies. For $\sigma(\omega )\ll m$, the function $K$ defined by (\[eq:definition\_K\]), takes the simpler form $$\label{eq:boltzmann}
K(\omega ,k)=\pi [\delta(\omega-\epsilon(k))
+\delta(\omega-\epsilon(k))] (\rm{Im} \sigma(\omega))^{-1}$$ Inserting the above result in (\[eq:mahan\_formula\]), we obtain: $$\label{eq:kappa-boltzmann}
\kappa(T)=\frac 1 {T} \int \frac{dk}{2\pi} \left(\epsilon(k)
\frac{\partial\epsilon(k)}{\partial k}\right)^2\left(-\frac{\partial
n_F}{\partial \epsilon}\right)(\epsilon(k)) (\rm{Im} \sigma(\epsilon(k)))^{-1}$$ We see that in the limit of very small self energies, we recover the Boltzmann equation result for the thermal conductivity[@ziman_solid_book]. Comparing (\[eq:kappa-boltzmann\]) with (\[eq:drude-thermal\]) and (\[eq:ladder-kappa\]), we see that if $Im \sigma(\epsilon(k))= \tau^{-1}$ where $\tau$ is a constant independent of $\epsilon$, the thermal conductivity can be written as a product of the Drude weight in the absence of impurities and the mean relaxation time $\tau$ as was proposed in Ref.. The underlying assumption there, was that all the eigenstates of the system have the same lifetime $\tau$ independent of the energy of the eigenstate. As shown above, the explicit energy dependence of the self energy found in Sec. \[sec:disorder-case\], i.e., the fact that the low energy spin excitations are much more scattered by impurities than high energy excitations, shows that any assumption of energy independent lifetime is invalid even for the simplest models. As a result, the thermal Drude weight can atmost yield a heuristic behavior of the thermal conductance in a real system in which spin excitations are interacting with impurities and/or phonons due to the different extrinsic lifetimes of current carrying states.
We present a brief comparison of our results with experiments. Since disorder is ubiquitous in real systems, it is reasonable to compare our results for the two leg ladder with impurities with experimental measurements. One of the systems studied extensively is the spin gap compound $\mathrm{Sr_{14-x}(La,Ca)_xCu_{24}O_{41}}$ [@sologubenko_thermal_ladder; @hess_thermal_ladder; @kudo_thermal_ladder]. In the insulating phase, these systems can be well described by an array of two-leg spin ladders[@uchara_SrCaCuO]. In this system, the phonon subtracted thermal conductivity was shown to have an exclusive spin contribution. At low temperatures, a fit for spin thermal conductivity yielded a $\kappa(T) \sim e^{-\Delta/T}$. This low temperature fit is in good accord with our prediction of $\kappa(T) \sim e^{-\omega_c/T}$. However, a full quantitative comparison requires an understanding of the effect of the disorder in the material on the spins, the effect of spin-phonon interactions and other exchange interactions in the ladder.
Conclusion
==========
In the present paper, we have calculated the thermal conductivity of gapless spin chains and spin gap systems including the two-leg spin-$\frac12$ ladder and the dimerized XY spin-$\frac12$ chain. In the absence of disorder, the thermal Drude weight of gapless spin chains vanishes linearly with temperature. On the other hand, for the ladder and the XY chain, which can both be represented as free fermions, we find that the thermal Drude weight $\tilde{\kappa} \sim T^{-1/2} e^{-m/T}$, where $m$ is the gap to the lowest triplet excitation. For intermediate temperatures, ${\tilde \kappa}
\propto T$ and decays as $T^{-2}$ at very high temperatures. We argue that this behaviour is generic to all quasi-one dimensional spin gapped systems having low energy triplet excitations with a relativistic-like dispersion. We have also considered the effect of a magnetic field which results in a substantial enhancement of the low temperature thermal conductivity. Furthermore, in the case of dimerized XY chains, a double peak can be obtained in the thermal conductivity for large enough fields. We have also studied the effect of impurities on spin gap systems like the ladder and the XY chain. Impurities destroy the Drude peak resulting in a finite thermal conductivity at zero frequency. This thermal conductivity has a generic form $e^{-\tilde{m}/T}$ at low temperatures where, $\tilde{m}$ is the effective gap of the system. It would be interesting to include the effects of magnetic impurities and also scattering from phonons. These and other questions are left for future work.
Acknowledgments
===============
We thank the authors of Ref. for their comments on a previous version of this manuscript. R.C. acknowledges [É]{}cole Normale Sup[é]{}rieure for kind hospitality during the completion of the present work.
Calculation of the thermal conductivity for Majorana fermions {#app:weight}
=============================================================
Thermal conductivity of Dirac fermions
--------------------------------------
We consider massive Dirac fermions with the following Hamiltonian density: $$\begin{aligned}
\label{eq:hamiltonian-dirac}
{\cal H}(x)&=&-i \frac v 2 (\psi^\dagger_R \partial_x \psi_R -
(\partial_x \psi_R^\dagger) \psi_R - \psi^\dagger_L
\partial_x\psi_L +(\partial_x\psi_L^\dagger)\psi_L ) \nonumber \\ &&
+ m (e^{i\varphi} \psi^\dagger_R\psi_L +e^{-i\varphi} \psi^\dagger_L\psi_R),\end{aligned}$$ In this case, the energy current reads: $$\label{eq:current-dirac}
j_e(x)=-i \frac{v^2} 2 (\psi^\dagger_R \partial_x \psi_R -
(\partial_x \psi^\dagger_R) \psi_R + \psi^\dagger_L \partial_x
\psi_L - (\partial_x \psi^\dagger_L) \psi_L)(x)$$ We note that using the transformation $\psi_L \to \varphi \psi_L$, we can reduce the Hamiltonian (\[eq:hamiltonian-dirac\]) to the case $\phi=0$, while leaving the current (\[eq:current-dirac\]) invariant. Therefore, for the purpose of calculating the thermal transport we can without loss of generality restrict to the case $\phi=0$ in (\[eq:hamiltonian-dirac\]).
Using the Fourier decomposition $$\begin{aligned}
\label{eq:fourier-dirac}
\psi_{\nu}(x)=\frac 1 {\sqrt{L}} \sum_k c_{k,\nu} e^{ik x}\end{aligned}$$ the Dirac Hamiltonian $H=\int dx {\cal H}(x)$ can be diagonalized to obtain $$\begin{aligned}
\label{eq:diagonal-dirac}
H=\sum_k \epsilon(k) (c^\dagger_{k,+} c_{k,+} -c^\dagger_{k,-}
c_{k,-})\end{aligned}$$ where the fermionic operators $c_{k,\pm}$ are linear combinations of the $c_{k,R/L}$ such that $c^\dagger_{k,+}
c_{k,+}+c^\dagger_{k,-} c_{k,-}=c^\dagger_{k,R} c_{k,R}
+c^\dagger_{k,L} c_{k,L}$ and $\epsilon(k)=\sqrt{(vk)^2+m^2}$. This allows us to rewrite the total energy current $J_e= \int dx j_e(x) =\sum_k v^2 k (c^\dagger_{k,R} c_{k,R} +
c^\dagger_{k,L} c_{k,L} -1)$ as: $$\begin{aligned}
J_e=\sum_k v^2 k (c^\dagger_{k,+} c_{k,+} + c^\dagger_{k,-} c_{k,-}
-1).\end{aligned}$$ Using (\[eq:diagonal-dirac\]), one easily obtains $$\begin{aligned}
\langle J_e^2 \rangle = 2 \sum_k (v^2 k)^2 \langle n_+(k) \rangle
(1- \langle n_+(k) \rangle)\end{aligned}$$ where the Fermi distribution function $ \langle n_+(k) \rangle = (e^{\beta\epsilon(k)}+1)^{-1}$. From (\[eq:drude-thermal\]), the thermal conductivity $\kappa(\omega,T)=\tilde{\kappa}\delta(\omega)$ with a Drude weight $$\begin{aligned}
\label{eq:conductivity-dirac}
\tilde{\kappa}(T)= \frac 1 {4T^2} \int_{-\infty}^{\infty} dk \frac
{v^4 k^2}{\cosh^2 \left( \frac{\epsilon(k)}{2T}\right) }\end{aligned}$$ The above result has a simple interpretation in terms of kinetic theory. Consider the expression for the specific heat: $$C_v(T)=\frac 1 {T^2} \int \frac{dk}{2\pi} \frac {\epsilon(k)^2}
{\cosh^2 \left( \frac{\epsilon(k)}{2T}\right) } \equiv \int \frac{dk}{2\pi}
c_v(k,T),$$ i.e., a mode of momentum $k$ contributes $c_v(k)$ to the specific heat. Such a mode has a velocity $v(k)=v^2
k/\epsilon(k)$. This now permits us to rewrite (\[eq:kappa-t\]) as: $$\label{eq:kinetic-interpretation}
\tilde{\kappa} = \int \frac{dk}{2\pi} c_v(k,T) v^2(k),$$ i.e., the contribution of each mode $k$ to the the Drude weight is just the product of its specific heat and square of the velocity. This is similar to the kinetic theory result that the thermal Drude weight is given by the product of the specific heat and the square of the velocity of the free modes.
Majorana fermions
------------------
It is well known that the Dirac Hamiltonian (\[eq:hamiltonian-dirac\]) can be re-expressed in terms of two Majorana fermions fields defined by $\psi_\nu=(\zeta_\nu^1+i\zeta_\nu^2)/\sqrt{2}$. The Hamiltonian can be written as a sum of two Majorana Hamiltonians $$\begin{aligned}
H_{\text{Dirac}}=H_M[\zeta^1]+H_M[\zeta^2]\end{aligned}$$ Similarly, the energy current (\[eq:current-dirac\]) can be written as the sum of two energy currents, each associated with one Majorana field: $j^D_e(x)=j^1_e(x)+j^2_e(x)$. The thermal conductivity of the Dirac Hamiltonian can then be written as the sum of the conductivities associated with the two Majorana field i.e., $\kappa_{\text{Dirac}}(\omega,T)=\kappa^1(\omega,T)+\kappa^2(\omega,T)$. The expression of the currents and the Hamiltonian being identical for the two species of Majorana fermions, it is clear that $\kappa^1(\omega,T)=\kappa^2(\omega,T)$. Thus, one obtains the generic result that $$\begin{aligned}
\label{eq:majorana-conduction}
\kappa_{\text{Majorana}}(\omega,T)=\kappa_{\text{Dirac}}(\omega,T)/2.\end{aligned}$$ This result shows that it suffices to calculate the thermal conductivity of the corresponding Dirac Hamiltonian to obtain the Majorana thermal conductivity. This correspondence holds provided there are no interactions between the various species of Majorana fermions.
Thermal current in the presence of an applied magnetic field {#app:mag-clean}
============================================================
In the presence of an applied magnetic field, the Hamiltonian density is ${\cal H}(x)={\cal
H}(x)^{{\bf h}=0}-{\bf h}\cdot {\bf m}(x)$, where ${\bf m}(x)$ is the magnetization density and ${\bf
h}$ is the external magnetic field. Using the continuity equation for the Hamiltonian density and the equation of conservation of the moment $\partial_t {\bf m}+\partial_x {\bf j}_s=0$, the thermal current is now given by $$\begin{aligned}
\label{eq:mag-current}
j_{\text{th.}}(x)=j_e(x)-{\bf h}\cdot {\bf j}_s(x),\end{aligned}$$ where $j_e$ is the energy current for ${\bf h}=0$ and ${\bf j}_s(x)$ is the magnetization current.
For the specific case of the ladder with a magnetic field along the $z$ direction, the Pauli coupling is $$\begin{aligned}
\label{eq:ladder-field}
H_{\text{mag.}}=-ih \int dx (\xi_R^1 \xi_R^2+\xi_L^1 \xi_L^2)\end{aligned}$$ Note that the contribution to the thermal conductivity arising from the $\xi^0_{R,L},\xi^3_{R,L}$ is not changed by the application of the magnetic field. To obtain the thermal conductivity coming from the modes $\xi^{1,2}$ it is convenient to turn to the Dirac Fermions[@shelton_spin_ladders; @orignac_2spinchains] $\psi_{\nu,s}=(\xi_\nu^1+i\xi_\nu^2)/\sqrt{2}$. Then, one can rewrite $H_{\text{mag.}}$ as: $$\begin{aligned}
\label{eq:ladder-field-dirac}
H_{\text{mag.}}=-h \int dx
(\psi^\dagger_{R,s}\psi_{R,s}+\psi^\dagger_{L,s}\psi_{L,s})\end{aligned}$$ The expression of the total thermal current then reads: $$\begin{aligned}
\label{eq:thermal-current-field}
J_e&=& \sum_k \left[ (\epsilon(k)-h) \frac{\partial \epsilon}{\partial
k} (c^\dagger_{k,+}c_{k,+}-\langle
c^\dagger_{k,+}c_{k,+}\rangle)\right.\nonumber \\ && \left. -(\epsilon(k)+h)\frac{\partial \epsilon}{\partial
k} (c^\dagger_{k,-}c_{k,-}-\langle
c^\dagger_{k,-}c_{k,-}\rangle)\right]\end{aligned}$$ The contribution of the $\xi^{1,2}$ modes to the Drude weight in the thermal conductivity is then calculated to be $$\begin{aligned}
\label{eq:drude-field}
\tilde{\kappa}^1(T,h)+\tilde{\kappa}^2(T,h)&=&\frac 1 {8T^2}\int dk
\left[
\frac{(\epsilon(k)-h)^2}{\cosh\left(\frac{\epsilon(k)-h}{T}\right)}\right. \nonumber\\ && \left. + \frac{(\epsilon(k)+h)^2}{\cosh\left(\frac{\epsilon(k)+h}{T}\right)}\right]\end{aligned}$$
Eigenvalues and Eigenstates of the fermionized XY chain {#app:eigen-XY}
=======================================================
Translational Invariant Case
----------------------------
The eigenstates of the Hamiltonian (\[eq:XY-fermionized\]) are obtained by solving the equations: $$\begin{aligned}
J_1 A_{2n}+J_2A_{2n+2}=E A_{2n+1} \label{eq:eig-odd} \\
J_2 A_{2n-1}+J_1 A_{2n+1}=E A_{2n}\label{eq:eig-even}\end{aligned}$$
One finds positive energy solutions: $$\label{eq:positive-eig}
\left(\begin{array}{c} A_{2n} \\ A_{2n+1}\end{array} \right)=
e^{2ikn} \left( \begin{array}{c} e^{-i\phi_k/2} \\
e^{i\phi_k/2}\end{array}\right)$$ with $E(k)=\sqrt{(J_1-J_2)^2+4 J_1 J_2 \cos^2 k}$ and $J_1+J_2e^{2ik}=E(k) e^{i\phi_k}$, and negative energy solutions: $$\label{eq:negative-eig}
\left(\begin{array}{c} A_{2n} \\ A_{2n+1}\end{array} \right)=
e^{2ikn} \left( \begin{array}{c} - e^{-i\phi_k/2} \\
e^{i\phi_k/2}\end{array}\right)$$ with $E(k)=-\sqrt{(J_1-J_2)^2+4 J_1 J_2 \cos^2 k}$ and $J_1+J_2e^{2ik}=|E(k)| e^{i\phi_k}$,
Single impurity case
--------------------
Clearly, the solutions with momentum $k$ and $-k$ are degenerate in energy, thus we search the solution as a linear combination of these solutions. The system of equations to solve reads: $$\begin{aligned}
\label{eq:defect}
J_1 A_{2n} + J_2 A_{2n+2}=E A_{2n+1} (n\neq 0) \label{eq:odd-gen}\\
J_2 A_{2n-1}+J_1 A_{2n+1}=E A_{2n} (n\neq 0) \label{eq:even-gen}\\
J'_1 A_0+J_2 A_2=E A_1 (n=0)\label{eq:odd-zero}\\
J_2 A_{-1}+J'_1A_1=E A_0 (n=0) \label{eq:even-zero}\end{aligned}$$
We search for solutions of the form: $$\label{eq:left-sol}
\left(\begin{array}{c} A_{2n} \\ A_{2n+1}\end{array} \right)=
e^{2ikn} \left( \begin{array}{c} e^{-i\phi_k/2} \\
e^{i\phi_k/2}\end{array}\right) + r(k) e^{-2ikn} \left( \begin{array}{c} e^{i\phi_k/2} \\
e^{-i\phi_k/2}\end{array}\right)$$ for $n\leq -1$ and: $$\label{eq:right-sol}
\left(\begin{array}{c} A_{2n} \\ A_{2n+1}\end{array} \right)= t(k) e^{2ikn} \left( \begin{array}{c} e^{-i\phi_k/2} \\
e^{i\phi_k/2}\end{array}\right)$$ for $n\geq 1$. Applying equations (\[eq:odd-gen\]) for $n=-1$ and (\[eq:even-gen\]) for $n=1$ we obtain respectively: $$\begin{aligned}
\label{eq:particular}
&& A_0=e^{-i\phi_k/2}+r(k) e^{i\phi_k/2} \nonumber \\
&& A_1=t(k) e^{i\phi_k/2}\end{aligned}$$ The equations that determine $t,r$ are obtained from (\[eq:odd-zero\]) and (\[eq:even-zero\]). They read: $$\begin{aligned}
\label{eq:rtsystem}
&& J_2(e^{-2ik} e^{i\phi_k/2} + r(k) e^{2ik} e^{-i \phi_k/2} + J'_1
t(k) e^{i\phi_k/2} =E(e^{-i\phi_k/2}+r(k) e^{i\phi_k/2}) \\
&& J_1'(e^{-i\phi_k/2} +r(k) e^{i\phi_k/2}) +J_2 t(k) e^{2ik}
e^{-i\phi_k/2} =E t(k) e^{i\phi_k/2}\end{aligned}$$ Using the relation $J_1+J_2e^{i2k}=E(k)e^{i\phi_k}$ these equations are simplified as follows: $$\begin{aligned}
-J_1 e^{-i\phi_k/2} r(k)+J'_1 e^{i\phi_k/2}t(k)=J_1 e^{i\phi_k/2} \\
J'_1 e^{i\phi_k/2} r(k) -J_1 e^{-i\phi_k/2} t(k) =-J'_1 e^{-i\phi_k/2}\end{aligned}$$
We obtain the transmission amplitude $t(k)$ and the reflection amplitude $r(k)$ as: $$\begin{aligned}
\label{eq:rt-coeff}
r(k)=\frac{(J'_1)^2-J_1^2}{J_1^2 e^{-i\phi_k}-(J'_1)^2e^{i\phi_k}} \\
t(k)=\frac{-2i J_1 J'_1 \sin \phi_k}{J_1^2 e^{-i\phi_k}-(J'_1)^2e^{i\phi_k}}\end{aligned}$$
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In this paper, we show an approximation in law of the complex Brownian motion by processes constructed from a stochastic process with independent increments. We give sufficient conditions for the characteristic function of the process with independent increments that ensure the existence of the approximation. We apply these results to Lévy processes. Finally we extend this results to the $m$-dimensional complex Brownian motion.'
author:
- 'Xavier Bardina[^1] and Carles Rovira[^2]'
title: Approximations of a complex Brownian motion with processes constructed from a process with independent increments
---
$^*$[Departament de Matemàtiques, Facultat de Ciències, Edifici C, Universitat Autònoma de Barcelona, 08193 Bellaterra]{}. [*Email:* ]{}[[email protected]]{} [*Tel:(34) 935868563*]{} $\mbox{ }$ $^\dagger$[Facultat de Matemàtiques, Universitat de Barcelona, Gran Via 585, 08007 Barcelona]{}. [*Email:* ]{}[[email protected]]{} [*Tel:(34) 934034740*]{}
[^3]
Introduction and main result.
=============================
The purpose of this paper is to investigate a weak approximation of a complex Brownian motion. The most typical processes taken as approximations to Gaussian processes are usually based on Donsker approximations ( the functional central limit theorem) or on Kac-Stroock type approximations. In this paper, we will deal with this family of approximations.
In 1974, Kac [@K] described the solution of the telegrapher’s equation in terms of a Poisson process. Eight years later, Stroock [@S] showed the weak convergence of this solution to a Brownian motion. More precisely, given $\{N_t, t \ge 0\}$ a standard Poisson process, the laws of the processes $x_{\varepsilon}$ $$\{x_{\varepsilon}(t)=\varepsilon\int_0^{\frac{t}{\varepsilon^{2}}}(-1)^{N_s}ds,\quad t \in [0,T]\}$$ converge weakly towards the law of a standard Brownian motion in the space of continuos functions on $[0,T]$.
This result have been extended in order to obtain approximations of other processes as, among others: $m$-dimensional Brownian process [@BR1], SPDE driven by Gaussian white noise [@BJQ], fractional SDE [@BNRT], multiple Wiener integrals [@BJT] or complex Brownian motion [@Ba].
Although all this cases are built begining from a Poisson process, a detailed study of the proofs shows that the authors use only some properties of the Poisson process that can be found in a bigger class of processes as Lévy processes. In this paper we will show how to do this extension.
More precisely, we will deal with approximations of the complex Brownian motion built from an unique stochastic process with independent increments. Let us recall that $\{B_{t},\, t\in[0,T]\}$ is a complex Brownian motion if it takes values on $\mathbb C$ and its real part and its imaginary part are two independent standard Brownian motions.
We consider the processes $$\{x_{\varepsilon}^{\theta}(t)=c(\theta)\varepsilon\int_{0}^{\frac{2t}{\varepsilon^2}}e^{i\theta
X_s}ds,\quad t\in[0,T]\} ,\label{eq1}$$ where $\{X_t, t \ge 0\}$ is a stochastic processes with independent increments and $c(\theta)$ is a constant, depending on $\theta$, that we will determine latter. Let us recall that our approximations can be written as $$x_{\varepsilon}^{\theta}(t)=\varepsilon c(\theta)\int_{0}^{\frac{2t}{\varepsilon^2}}\cos(\theta
X_s)ds + i\varepsilon
c(\theta)\int_{0}^{\frac{2t}{\varepsilon^2}}\sin(\theta X_s)ds.$$
Notice that when $X$ is a Poisson process it has been proved in [@Ba] that for $\theta \not= 0$ and $\theta \not= \pi$ the limit is a complex Brownian motion. When $\theta=\pi$ we obtain an alternative version of Stroock’s results since in this case $$e^{i\theta X_s}=(-1)^{X_s}.$$
The aim of this paper is to study the weak limits of the processes (\[eq1\]) when $\varepsilon$ tends to zero depending on the value of $\theta$, showing that Lévy processes can be used to approximate a complex Brownian motion.
In section 2, we recall some basic facts on Lévy processes and we present the classical methodology to get weak approximations of Gaussian processes.
Section 3 is devoted to give the main results of the paper. First we give some conditions on the characteristic functions of the process $X$ that ensures the weak convergence of (\[eq1\]) to a complex Brownan motion. Then, we show when the characteristic functions of Lévy processes satisfy such conditions.
In section 4 we study the $m$-dimesional case, showing how we can obtain a $m$-dimensional complex Brownian motion from an unique Lévy process.
Along the paper $K$ denote positive constants, not depending on $\varepsilon$, which may change from one expression to another one. The real part and the imaginary part of a complex number will be denoted by $Re[\cdot]$ and $Im[\cdot]$.
Preliminaries
=============
Lévy processes
--------------
Set $\{X_s,\, s\geq 0\}$ a Lévy process, that is, X has stationary and independent increments, is continuous in probability, is càdlàg and $X_0 = 0$, and it is defined on a complete probability space $(\Omega, \cal{F},P)$. There are many important examples of Lévy processes: Brownian motion, Poisson Process, jump-diffusion processes, stable processes, subordinators, etc.
Consider $\phi_{X_t}(u)$ its characteristic function. Remember that it can be written as $$\phi_{X_t}(u)=E\left(e^{iuX_t}\right)=e^{-t \psi_X(u)},$$ where $\psi_X(u)$ is called the Lévy exponent of $X$.
It is well known that the Lévy exponent can be expressed, by the Lévy-Khinchine formula, as $$\label{khin}\psi_{X}(u)=-aiu+\frac12\sigma^2 u^2-\int_{{\mathbb{R}}\setminus\{0\}}(e^{iux}-1-iuxI_{|x|<1})\eta(dx),$$ where $a\in{\mathbb{R}}$, $\sigma\geq0$ and $\eta$ is a Lévy measure, that is, $\int_{{\mathbb{R}}\setminus\{0\}}\min\{x^2,1\}\eta(dx)<\infty$.
For notation and simplicity along the paper we will write $$a(u):=Re[\psi_X(u)]=\frac12\sigma^2u^2-\int_{{\mathbb{R}}\setminus\{0\}}(\cos(ux)-1)\eta(dx),\label{mm1}$$ and $$b(u)=Im[\psi_X(u)]=-au-\int_{{\mathbb{R}}\setminus\{0\}}(\sin(ux)-uxI_{|x|<1})\eta(dx).\label{mm2}$$ Notice that $a(-u)=a(u)$ and $b(-u)=-b(u)$.
We refer the reader to [@Sato] for more information about Lévy processes.
Weak approximations of the complex Brownian motion {#sweak}
--------------------------------------------------
For any $\varepsilon >0$, set $
\{x_{\varepsilon}(t), t\in[0,T]\}$ a complex stochastic process with $x_\varepsilon(0)=0$. Consider $P_{\varepsilon}$ the image law of $x_{\varepsilon}$ in the Banach space $\mathcal C([0,T],\mathbb
C)$ of continuous functions on $[0,T]$.
In order to prove that $P_\varepsilon$ converges weakly as $\varepsilon$ tends to zero towards the law on $\mathcal C([0,T],\mathbb C)$ of a complex Brownian motion we have to check that the family $P_{\varepsilon}$ is tight and that the law of all possible weak limits of $P_\varepsilon$ is the law of two independent standard Brownian motions.
In order to prove that the family $P_\varepsilon$ is tight, we need to prove that the laws corresponding to the real part and the imaginary part of the processes $x_{\varepsilon}$ are tight. Using the Billingsley criterium (see Theorem 12.3 of [@B]) and that our processes are null on the origin, it suffices to prove that there exists a constant $K$ such that for any $s<t$ $$\sup_{\varepsilon}\big(E((Re[x_\varepsilon(t)-x_\varepsilon(t)])^{4}) + E((Im[x_\varepsilon(t)-x_\varepsilon(t)])^{4})\big)\leq K(t-s)^{2}.\label{ademo1}$$
The second part of the proof consists in the identification of the limit law. Let $\{P_{\varepsilon_n}\}_{n}$ be a subsequence of $\{P_{\varepsilon}\}_{\varepsilon}$ (that we will also denote by $\{P_{\varepsilon}\}$) weakly convergent to some probability $P$. We want to see that the canonical process $Y=\{Y_{t}(x)=:y(t)\}$ is a complex Brownian motion under the probability $P$, that is, the real part and the imaginary part of this process are two independent Brownian motions. Using Paul Lévy’s theorem it suffices to prove that under $P$, the real part and the imaginary part of the canonical process are both martingales with respect to the natural filtration, $\{\mathcal
F_{t}\}$, with quadratic variations $<Re[Y],Re[Y]>_{t}=t$, $<Im[Y],Im[Y]>_{t}=t$ and covariation $<Re[Y],Im[Y]>_{t}=0$.
To see that under $P$ the real part and the imaginary part of the canonical process $X$ are martingales with respect to its natural filtration $\{\mathcal
F_{t}\}$, we have to prove that for any $s_{1}\leq s_{2}\leq\cdots\leq s_{n}
\leq s$ and for any bounded continuous function $\varphi:\mathbb C^{n}
\longrightarrow\mathbb R$, $$E_{P}\big[\varphi(X_{s_{1}},...,X_{s_{n}})(Re[Y_{t}]-Re[Y_{s}])\big]=0,$$ $$E_{P}\big[\varphi(X_{s_{1}},...,X_{s_{n}})(Im[Y_{t}]-Im[Y_{s}])\big]=0.$$ Since $P_{\varepsilon}\stackrel{w}{\Rightarrow}P$, we have that, $$\begin{aligned}
&& \lim_{\varepsilon\to0}E_{P_{\varepsilon}}
\big[\varphi(y(s_{1}),...,y(s_{n}))(Re[y(t)]-Re[y(s)])\big] \cr
&&\cr &=& E_{P}
\big[\varphi(y(s_{1}),...,y(s_{n}))(Re[y(t)]-Re[y(s)])\big],\cr\end{aligned}$$ and we get the same with the imaginary part. So, it suffices to see that $$\begin{aligned}
& &\lim_{\varepsilon \to 0}E\big(\varphi(x_\varepsilon(s_{1}),...,x_\varepsilon(s_{n})) \big(Re[x_\varepsilon(t)] - Re[x_\varepsilon(s)]\big)\big)=0,\label{ademo2}\\
& &\lim_{\varepsilon \to 0}E\big(\varphi(x_\varepsilon(s_{1}),...,x_\varepsilon(s_{n}))\big( Im[x_\varepsilon(t)] - Im[x_\varepsilon(s)]\big)\big)=0.\label{ademo3}\end{aligned}$$ To chek the quadratic variation, it is enough to prove that for any $s_{1}\leq s_{2}\leq\cdots\leq s_{n} \leq s$ and for any bounded continuous function $\varphi:\mathbb C^{n}
\longrightarrow\mathbb R$, $$\begin{aligned}
& &\lim_{\varepsilon \to 0}E\big[\varphi(x_\varepsilon(s_{1})...,x_\varepsilon
(s_{n}))\big((Re[x_\varepsilon(t)]-Re[x_\varepsilon(s)])^2-(t-s)\big)\big]=0,\label{ademo4}\\
& &\lim_{\varepsilon \to 0}E\big[\varphi(x_\varepsilon(s_{1})...,x_\varepsilon(s_{n}))\big((Im[x_\varepsilon(t)]-Im[x_\varepsilon(s)])^2-(t-s)\big)\big]=0.\label{ademo5}\end{aligned}$$ Finally to prove that $<Re[Y],Im[Y]>_{t}=0$, it suffices to check that for any $s_{1}\leq s_{2}\leq\cdots\leq s_{n} \leq s$ and for any bounded continuous function $\varphi:\mathbb C^{n}
\longrightarrow\mathbb R$, $$\lim_{\varepsilon \to 0}E\big[\varphi(x_\varepsilon(s_{1})...,x_\varepsilon(s_{n}))
(Re[x_\varepsilon(t)]-Re[x_\varepsilon(s)])
(Im[x_\varepsilon(t)]-Im[x_\varepsilon(s)])\big]=0\label{ademo6}.$$
Approximations to a complex Brownian motion
===========================================
As we have explained, we built our approximations from a stochastic process X with independent increments. We will deal with $X$ using the study of its characteristic function $\phi_X$. Let us introduce a set of usefull hypothesis ($H^{\theta}$) for the characteristic function $\phi_X$ of a process $X$:
- there exists a constant $K(\theta)$ such that $$\varepsilon^2\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}
\int_{\frac{2s}{\varepsilon^2}}^{y}\|\phi_{X_{y}-X_{x}}(\theta)\|dxdy
\le K(\theta) (t-s),$$
- there exists a constant $c(\theta)$ such that $$\lim_{\varepsilon \to 0} \varepsilon^2c(\theta)^2\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}\int_{\frac{2s}{\varepsilon^2}}^{x}
[\phi_{X_x-X_y}(\theta)+\phi_{X_x-X_y}(-\theta)]dydx=2(t-s),$$
- $$\lim_{\varepsilon \to 0} \varepsilon^2
\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}\int_{\frac{2s}{\varepsilon^2}}^{y}
\|\phi_{X_y-X_x}(\theta)\|\|\phi_{X_x-X_{\frac{2s}{\varepsilon^2}}}(2\theta)\|dxdy = 0.$$
In the next Theorem we give some sufficient conditions on the characteristic function of the process $\{X_s, s \ge 0\}$ to get the convergence to a complex Brownian motion.
\[teogen\] Let $\{X_s,\, s\geq 0\}$ be a stochastic process with independent increments and characteristic function $\phi_X$. Set $C_X=\{\theta,
{ \mbox{ such that }} \phi_X {\mbox{ satisfies }}
(H^{\theta}) \}$.
Define for any $\varepsilon>0$ and $\theta\in C_X$ $$\{x_{\varepsilon}^{\theta}(t)=\varepsilon c(\theta) \int_{0}^{\frac{2t}{\varepsilon^2}}e^{i\theta
X_s}ds,\quad t\in[0,T]\}$$ where $c(\theta)$ is the constant given by hypothesis $(H^{\theta}2)$.
Consider $P_{\varepsilon}^{\theta}$ the image law of $x_{\varepsilon}^{\theta}$ in the Banach space $\mathcal C([0,T],\mathbb
C)$ of continuous functions on $[0,T]$. Then, $P_\varepsilon^{\theta}$ converges weakly as $\varepsilon$ tends to zero, towards the law on $\mathcal C([0,T],\mathbb C)$ of a complex Brownian motion.
We will follow the method explained in Subsection \[sweak\].
[*Step1: Tightness.*]{} We have to check (\[ademo1\]), that is, that there exists a constant $K(\theta)$ such that for any $s<t$ $$\sup_{\varepsilon}\big(E(\varepsilon c(\theta)\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}\cos(\theta
N_x)dx)^{4} + E(\varepsilon
c(\theta)\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}\sin(\theta
N_x)dx)^{4}\big)\leq K(\theta)(t-s)^{2}.$$ From the properties of the complex numbers we have that $$\begin{aligned}
&&\!\!E(\varepsilon c(\theta)\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}\cos(\theta
X_x)dx)^{4} +
E(\varepsilon c(\theta)\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}\sin(\theta
X_x)dx)^{4} \nonumber\cr &&\nonumber\cr
&\leq&2E\|x_\varepsilon^{\theta}(t)-x_\varepsilon^{\theta}(s)\|^4\nonumber\\
&=&2c(\theta)^4\varepsilon^4E\left(\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}
e^{i\theta X_v}dv
\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}
e^{-i\theta X_u}du\right)^2\nonumber\\
&=&2c(\theta)^4\varepsilon^4\int_{[\frac{2s}{\varepsilon^2},\frac{2t}{\varepsilon^2}]^4}
E\left(e^{i\theta[(X_{v_1}-X_{u_1})+(X_{v_2}-X_{u_2})]}\right)dv_1dv_2du_1du_2.\label{eqtight1}\end{aligned}$$ Using that for $x_1<x_2<x_3<x_4$ and $\rho_i \in \{0,1\}$ for $i=1,2,3,4$ with $\sum_{i=1}^4 \rho_i=2$ we can write $$\begin{aligned}
&& \!\! (-1)^{\rho_4} X_{x_4} + (-1)^{\rho_3} X_{x_3} + (-1)^{\rho_2} X_{x_2} + (-1)^{\rho_1} X_{x_1} \\
& = & (-1)^{\rho_4} (X_{x_4} -X_{x_3}) + \big((-1)^{\rho_4} + (-1)^{\rho_3}\big) ( X_{x_3} - X_{x_2} ) \\
& & +\big((-1)^{\rho_4} + (-1)^{\rho_3} + (-1)^{\rho_2}\big) ( X_{x_2} - X_{x_1}),\end{aligned}$$ and the last expression (\[eqtight1\]) can be written as the sum of 24 integrals of the type $$\begin{aligned}
2c(\theta)^4\varepsilon^4\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}
\int_{\frac{2s}{\varepsilon^2}}^{x_4}
\int_{\frac{2s}{\varepsilon^2}}^{x_3}
\int_{\frac{2s}{\varepsilon^2}}^{x_2}
&E\left(e^{i\theta[c_1(X_{x_4}-X_{x_3})+c_2(X_{x_3}-X_{x_2})+c_3(X_{x_2}-X_{x_1})]}\right) \nonumber\\
& \times dx_1dx_2dx_3dx_4.\label{eqtight2} \end{aligned}$$ where $c_1\in\{1,-1\}$, $c_2\in\{-2,0,2\}$ and $c_3\in\{1,-1\}$. Notice that since the process $X$ has independent increments, we have that $$\begin{aligned}
&&E\left(e^{i\theta[c_1(X_{x_4}-X_{x_3})+c_2(X_{x_3}-X_{x_2})+c_3(X_{x_2}-X_{x_1})]}\right)\\
&=&E\left(e^{i\theta c_1(X_{x_4}-X_{x_3})}\right)E\left(e^{i\theta
c_2(X_{x_3}-X_{x_2})}\right)E\left(e^{i\theta
c_3(X_{x_2}-X_{x_1})]}\right),\end{aligned}$$ and we obtain, $$\begin{aligned}
&&\|E\left(e^{i\theta[c_1(X_{x_4}-X_{x_3})+c_2(X_{x_3}-X_{x_2})+c_3(X_{x_2}-X_{x_1})]}\right)
\|\\&\leq&\|\phi_{X_{x_4}-X_{x_3}}(c_1\theta)\|\|\phi_{X_{x_2}-X_{x_1}}(c_3\theta)\|\\
&\leq&\|\phi_{X_{x_4}-X_{x_3}}(\theta)\|\|\phi_{X_{x_2}-X_{x_1}}(\theta)\|,\end{aligned}$$ where we have used that for any random variable $Z$, $\|\phi_Z(-u)\|=\|\phi_Z(u)\|$.
So, each one of the 24 integrals of the type (\[eqtight2\]) is bounded by $$c(\theta)^4\varepsilon^2\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}
\int_{\frac{2s}{\varepsilon^2}}^{x_4}\|\phi_{X_{x_4}-X_{x_3}}(\theta)\|dx_3dx_4
\varepsilon^2\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}
\int_{\frac{2s}{\varepsilon^2}}^{x_2}\|\phi_{X_{x_2}-X_{x_1}}(\theta)\|dx_1dx_2.$$ Clearly, hypothesis ($H^\theta$1) completes the proof of this step.
[*Step 2: Martingale property.*]{} It is enough to check (\[ademo2\]) and (\[ademo3\]). So, it suffices to see that $$E\big(\varphi(x_\varepsilon^{\theta}(s_{1}),...,x_\varepsilon^{\theta}(s_{n}))\varepsilon c(\theta)\int_{\frac{2s}{\varepsilon^{2}}}
^{\frac{2t}{\varepsilon^2}}\cos(\theta X_x)dx\big)$$ and, $$E\big(\varphi(x_\varepsilon^{\theta}(s_{1}),...,x_\varepsilon^{\theta}(s_{n}))\varepsilon c(\theta) \int_{\frac{2s}{\varepsilon^{2}}}
^{\frac{2t}{\varepsilon^2}}\sin(\theta X_x)dx\big)$$ converge to zero when $\varepsilon$ tends to zero.
Thus, it is enough to prove that $$\|E\big(\varphi(x_\varepsilon^{\theta}(s_1),...,x_\varepsilon^{\theta}(s_n))
\varepsilon c(\theta)\int_{\frac{2s}{\varepsilon^{2}}}
^{\frac{2t}{\varepsilon^2}}e^{i\theta X_x}dx\big)\|$$ converges to zero when $\varepsilon$ tends to zero.
But applying the Schwartz inequality and using that the function $\varphi$ is bounded it is enough to prove the convergence to zero of $$\begin{aligned}
&&\|E\big(\varepsilon
c(\theta)\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}e^{i\theta
X_x}dx\big)^2\|\\
&=&\|E\big(\varepsilon^2
c(\theta)^2\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}\int_{\frac{2s}{\varepsilon^2}}^{y}
e^{i\theta(X_x+X_y)}dxdy\big)\|\\
&=&\|E\big(\varepsilon^2
c(\theta)^2\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}\int_{\frac{2s}{\varepsilon^2}}^{y}
e^{i\theta(X_y-X_x)+2i\theta(X_x-X_{\frac{2s}{\varepsilon^2}})+2i\theta X_{\frac{2s}{\varepsilon^2}}}dxdy\big)\|\\
&=&\|E\big[e^{2i\theta X_{\frac{2s}{\varepsilon^2}}}\big]
\varepsilon^2
c(\theta)^2\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}\int_{\frac{2s}{\varepsilon^2}}^{y}
\phi_{X_y-X_x}(\theta)\cdot\phi_{X_x-X_{\frac{2s}{\varepsilon^2}}}(2\theta)dxdy\|.\end{aligned}$$ Notice that this last expression can be bounded by $$\varepsilon^2
c(\theta)^2\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}\int_{\frac{2s}{\varepsilon^2}}^{y}
\|\phi_{X_y-X_x}(\theta)\|\|\phi_{X_x-X_{\frac{2s}{\varepsilon^2}}}(2\theta)\|dxdy$$ that from ($H^\theta$3) converges to zero when $\varepsilon$ goes to zero.
[*Step 3: Quadratic variations.*]{} It is enough to check (\[ademo4\]) and (\[ademo5\]), that is to prove that for any $s_{1}\leq s_{2}\leq\cdots\leq s_{n} \leq s$ and for any bounded continuous function $\varphi:\mathbb C^{n}
\longrightarrow\mathbb R$, $$a_\varepsilon:=E\big[\varphi(x_\varepsilon^{\theta}(s_{1})...,x_\varepsilon^{\theta}
(s_{n}))\big((Re[x_\varepsilon^{\theta}(t)]-Re[x_\varepsilon^{\theta}(s)])^2-(t-s)\big)\big]$$ and $$b_\varepsilon:=E\big[\varphi(x_\varepsilon^{\theta}(s_{1})...,x_\varepsilon^{\theta}(s_{n}))\big((Im[x_\varepsilon^{\theta}(t)]-Im[x_\varepsilon^{\theta}(s)])^2-(t-s)\big)\big]$$ converge to zero when $\varepsilon$ tends to zero.
In order to prove that $a_\varepsilon$ and $b_\varepsilon$ converge to zero, when $\varepsilon$ goes to zero, it is enough to show that $a_\varepsilon+b_\varepsilon$ and $a_\varepsilon-b_\varepsilon$ converge to zero. But, $$\begin{aligned}
& &a_\varepsilon+b_\varepsilon\\
&=&E\big[\varphi(x_\varepsilon^{\theta}(s_{1})...,x_\varepsilon^{\theta}
(s_{n}))\big(\|x_\varepsilon^{\theta}(t)-x_\varepsilon^{\theta}(s)\|^2-2(t-s)\big)\big]\\
&=&E\big[\varphi(x_\varepsilon^{\theta}(s_{1})...,x_\varepsilon^{\theta}
(s_{n}))\big(\varepsilon^2c(\theta)^2\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}
e^{i\theta(X_v-X_u)}dvdu-2(t-s)\big)\big]\\
&=&E\big[\varphi(x_\varepsilon^{\theta}(s_{1})...,x_\varepsilon^{\theta}
(s_{n})\big])E\big(\varepsilon^2c(\theta)^2\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}
e^{i\theta(X_v-X_u)}dvdu
-2(t-s)\big)\\
&=&E\big[\varphi(x_\varepsilon^{\theta}(s_{1})...,x_\varepsilon^{\theta}
(s_{n})\big])\big[E\big(\varepsilon^2c(\theta)^2\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}\int_{\frac{2s}{\varepsilon^2}}^{v}
e^{i\theta(X_v-X_u)}dudv\big)\\&&
+E\big(\varepsilon^2c(\theta)^2\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}\int_{\frac{2s}{\varepsilon^2}}^{u}
e^{-i\theta(X_u-X_v)}dvdu\big)
-2(t-s)\big)\big]\\
&=&E\big[\varphi(x_\varepsilon^{\theta}(s_{1})...,x_\varepsilon^{\theta}
(s_{n})\big])\\&&\times\big[\varepsilon^2c(\theta)^2\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}\int_{\frac{2s}{\varepsilon^2}}^{x}
[\phi_{X_x-X_y}(\theta)+\phi_{X_x-X_y}(-\theta)]dydx -2(t-s)\big].\end{aligned}$$ Clearly, ($H^\theta$2) yields that $\lim_{\varepsilon \to 0}
(a_\varepsilon+b_\varepsilon)=0.$
It remains to see that $a_\varepsilon-b_\varepsilon$ converges to zero. But $$\begin{aligned}
a_\varepsilon-b_\varepsilon&=&E\big[\varphi(x_\varepsilon^{\theta}(s_{1})...,x_\varepsilon^{\theta}
(s_{n}))\big[\big(\varepsilon
c(\theta)\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}\cos(\theta
X_x)dx\big)^2\nonumber\\\nonumber&&-\big(\varepsilon
c(\theta)\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}\sin(\theta
X_x)dx\big)^2\big]\\\nonumber
&=&\frac12E\big[\varphi(x_\varepsilon^{\theta}(s_{1})...,x_\varepsilon^{\theta}
(s_{n}))\big[\big(\varepsilon
c(\theta)\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}e^{i\theta
X_x}dx\big)^2\\&&+\big(\varepsilon
c(\theta)\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}e^{-i\theta
X_x}dx\big)^2\big],\label{dostermes}\end{aligned}$$ where in the last step we have used that $2(\alpha^2-\beta^2)=(\alpha+\beta i)^2+(\alpha-\beta i)^2$. We will show that this two last terms go to zero. For the first one we have that, $$\begin{aligned}
&&\frac12E\big[\varphi(x_\varepsilon^{\theta}(s_{1})...,x_\varepsilon^{\theta}
(s_{n}))\big(\varepsilon
c(\theta)\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}e^{i\theta
X_x}dx\big)^2\big]\\
&=&E\big[\varphi(x_\varepsilon^{\theta}(s_{1})...,x_\varepsilon^{\theta}
(s_{n}))\varepsilon^2
c(\theta)^2\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}\int_{\frac{2s}{\varepsilon^2}}^{y}
e^{i\theta(X_x+X_y)}dxdy\big]\\
&=&E\big[\varphi(x_\varepsilon^{\theta}(s_{1})...,x_\varepsilon^{\theta}
(s_{n})) \\ & & \qquad\qquad \times \varepsilon^2
c(\theta)^2\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}\int_{\frac{2s}{\varepsilon^2}}^{y}
e^{i\theta(X_y-X_x)+2i\theta(X_x-X_{\frac{2s}{\varepsilon^2}})+2i\theta X_{\frac{2s}{\varepsilon^2}}}dxdy\big]\\
&=&E\big[\varphi(x_\varepsilon^{\theta}(s_{1})...,x_\varepsilon^{\theta}
(s_{n}))e^{2i\theta X_{\frac{2s}{\varepsilon^2}}}\big] \\ & & \qquad\qquad \times \varepsilon^2
c(\theta)^2\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}\int_{\frac{2s}{\varepsilon^2}}^{y}
\phi_{X_y-X_x}(\theta)\cdot\phi_{X_x-X_{\frac{2s}{\varepsilon^2}}}(2\theta)dxdy.\end{aligned}$$ Notice that this last expression can be bounded by $$K\varepsilon^2
c(\theta)^2\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}\int_{\frac{2s}{\varepsilon^2}}^{y}
\|\phi_{X_y-X_x}(\theta)\|\|\phi_{X_x-X_{\frac{2s}{\varepsilon^2}}}(2\theta)\|dxdy$$ that from ($H^\theta$3) converges to zero when $\varepsilon$ goes to zero. Following the same computations, and using that, in general, for any random variable $Z$, $\|\phi_Z(-u)\|=\|\phi_Z(u)\|$ we obtain the same bound and the convergence to zero, for the second term of expression (\[dostermes\]).
[*Step 4: Quadratic covariation.*]{} It is enough to check (\[ademo6\]). Using that $$\alpha\beta=\frac14i[(\alpha-\beta i)^2-(\alpha+\beta i)^2],$$ the term in the right side of (\[ademo6\]) is equal to $$\begin{aligned}
&&E\big(\varphi(x_\varepsilon^{\theta}(s_1),...,x_\varepsilon^{\theta}(s_n))(\varepsilon
c(\theta)\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}
\cos(\theta X_x)dx)(\varepsilon
c(\theta\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}
\sin(\theta X_x)dx) \big)\cr &=& \frac14i
E\big[\varphi(x_\varepsilon^{\theta}(s_{1})...,x_\varepsilon^{\theta}
(s_{n}))\big[\big(\varepsilon
c(\theta)\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}e^{-i\theta
X_x}dx\big)^2\\&&-\big(\varepsilon
c(\theta)\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}e^{i\theta
X_x}dx\big)^2\big].\end{aligned}$$ We have already shown in the study of (\[dostermes\]) that this term goes to zero.
$\Box$
Let us state now the main result of the paper. We prove that the approximations built from a Lévy process converge to a complex Brownian motion.
\[teopar\] Define for any $\varepsilon>0$ $$\{x_{\varepsilon}^{\theta}(t)=\varepsilon c(\theta) \int_{0}^{\frac{2t}{\varepsilon^2}}e^{i\theta
X_s}ds,\quad t\in[0,T]\}$$ where $\{X_s,\, s\geq 0\}$ is a Lévy process with Lévy exponent $\psi_X$ and $$c(u)=\sqrt{\frac{\| \psi_X(u) \|^2 }{2 Re[\psi_X(u)]}}.$$
Consider $P_{\varepsilon}^{\theta}$ the image law of $x_{\varepsilon}^{\theta}$ in the Banach space $\mathcal C([0,T],\mathbb
C)$ of continuous functions on $[0,T]$. Then, for $\theta$ such that $Re[\psi_X(\theta)]Re[\psi_X(2\theta)] \not= 0$, $P_\varepsilon^{\theta}$ converges weakly as $\varepsilon$ tends to zero, towards the law on $\mathcal C([0,T],\mathbb C)$ of a complex Brownian motion.
The results follows as a particular case of Theorem \[teogen\]. It suffices to check that the characteristic function $\phi_X$ of the Lévy process $X$ satisfies ($H^{\theta}$) for any $\theta$ such that $a(\theta)a(2\theta) \not= 0$ (recall definitions (\[mm1\]) and (\[mm2\])).
[*Proof of* ]{}($H^\theta$1): We can write $$\begin{aligned}
\varepsilon^2\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}
\int_{\frac{2s}{\varepsilon^2}}^{y}\|\phi_{X_{y}-X_{x}}(\theta))\|dxdy&=&
\varepsilon^2\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}
\int_{\frac{2s}{\varepsilon^2}}^{y}e^{-(y-x)a(\theta)}dxdy\\
\leq \frac{2}{a(\theta)}(t-s). \end{aligned}$$ Using that $a(\theta)>0$ we complete the proof of ($H^\theta$1).
[*Proof of* ]{}($H^\theta$2): Note first that $$\begin{aligned}
&&\varepsilon^2c(\theta)^2\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}\int_{\frac{2s}{\varepsilon^2}}^{x}
\phi_{X_x-X_y}(\theta)dydx\\
&=&\varepsilon^2c(\theta)^2\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}\int_{\frac{2s}{\varepsilon^2}}^{x}
e^{-(x-y)(a(\theta)+b(\theta)i)}dydx\\
&=&\varepsilon^2\frac{c(\theta)^2}{a(\theta)+b(\theta)i}\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}
\big(1-
e^{-(x-\frac{2s}{\varepsilon^2})(a(\theta)+b(\theta)i)}\big)dx\\
&=&o(\varepsilon)+2(t-s)\frac{c(\theta)^2}{a(\theta)+b(\theta)i}.\end{aligned}$$ Following the same computations and taking into account that $a(-\theta)=a(\theta)$, and that $b(-\theta)=-b(\theta)$ we obtain that $$\begin{aligned}
&&\varepsilon^2c(\theta)^2\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}\int_{\frac{2s}{\varepsilon^2}}^{x}
\phi_{X_x-X_y}(-\theta)dydx\\
&=&o(\varepsilon)+2(t-s)\frac{c(\theta)^2}{a(\theta)-b(\theta)i}.\end{aligned}$$ So $$\begin{aligned}
&&\varepsilon^2c(\theta)^2\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}\int_{\frac{2s}{\varepsilon^2}}^{x}
[\phi_{X_x-X_y}(\theta)+\phi_{X_x-X_y}(-\theta)]dydx\\&=&o(\varepsilon)+2(t-s)\big(
\frac{c(\theta)^2}{a(\theta)+b(\theta)i}+\frac{c(\theta)^2}{a(\theta)-b(\theta)i}
\big)\\&=&o(\varepsilon)+2(t-s),\end{aligned}$$ and ($H^\theta$2) is clearly true.
[*Proof of* ]{}($H^\theta$3): Notice that $$\begin{aligned}
&&K\varepsilon^2
\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}\int_{\frac{2s}{\varepsilon^2}}^{y}
\|\phi_{X_y-X_x}(\theta)\|\|\phi_{X_x-X_{\frac{2s}{\varepsilon^2}}}(2\theta)\|dxdy\\
&=&K\varepsilon^2
\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}\int_{\frac{2s}{\varepsilon^2}}^{y}
e^{-(y-x)a(\theta)}e^{-(x-\frac{2s}{\varepsilon^2})a(2\theta)}dxdy\\
&\leq&K\varepsilon^2\frac{
1}{a(\theta)}\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}
e^{-(x-\frac{2s}{\varepsilon^2})a(2\theta)}dx\\
&\leq& K\varepsilon^2\frac{ 1}{a(\theta)a(2\theta)},\end{aligned}$$ that converges to zero when $\varepsilon$ goes to zero.
$\Box$
Given a Lévy process with characteristic function given by the Lévy-Khinchine formula (\[khin\]), the condition $Re[\psi_X(\theta)]=0$ is equivalent to $$\frac12\sigma^2\theta^2-\int_{{\mathbb{R}}\setminus\{0\}}(\cos(\theta x)-1)\eta(dx)=0,$$ that is, $\sigma=0$ and $$\int_{{\mathbb{R}}\setminus\{0\}}(\cos(\theta x)-1)\eta(dx)=0.$$
So, the condition $Re[\psi_X(\theta)]Re[\psi_X(2\theta)] \not= 0$ can be written as $\sigma \not=0$ or $$(\int_{{\mathbb{R}}\setminus\{0\}}(\cos(\theta x)-1)\eta(dx))(\int_{{\mathbb{R}}\setminus\{0\}}(\cos(2\theta x)-1)\eta(dx)) \not=0.$$
When we consider $\{X_t, t \ge 0\}$ a standard Poisson process it is well-known that it is a Lévy process with Lévy exponent $$\psi_X(u)=-(\cos(u)-1) -i \sin(u)$$ that corresponds to the Lévy-Khinchine formula (\[khin\]) with $a=0, \sigma=0$ and $\eta=\delta_{\{1\}}$. Then the condition $Re[\psi_X(\theta)]Re[\psi_X(2\theta)] \not= 0$ yields that $\theta\not= k \pi$ for any $k \ge 1$.
When $\theta= (2k+1) \pi$ , we have that $$x_{\varepsilon}^{\theta}(t)=c((2k+1) \pi)\varepsilon \int_{0}^{\frac{2t}{\varepsilon^2}}cos((2k+1) \pi
X_s)ds=\varepsilon \int_{0}^{\frac{2t}{\varepsilon^2}}(-1)^{X_s}ds,\label{part1}$$ that is a real process that can not converge to a complex Brownian motion. Nevertheless part of the same proof done in Theorem \[teogen\] (steps 1 and 2 and study of $a_\varepsilon$, note that $b_\varepsilon=0$) works to prove tant the processes defined by (\[part1\]) converge weakly to a standard Brownian motion.
On the other hand, when $\theta= 2k \pi$ , we have that $$x_{\varepsilon}^{\theta}(t)=c(2k \pi)\varepsilon \int_{0}^{\frac{2t}{\varepsilon^2}}cos( 2k \pi
X_s)ds=0.$$
The $m$-dimensional case
========================
The aim of this section is to extend this result to a $m$-dimensional case for any $m\ge 1$. We will give the extensions of Theorem \[teogen\] and \[teopar\].
We define for any $\varepsilon>0$ and for any $1\leq j\leq m$ $$\left\{x_{\varepsilon}^{\theta_j}(t)=\varepsilon\int_{0}^{\frac{2t}{\varepsilon^2}}e^{i\theta_jX_s}ds,\,t\in[0,T]\right\},$$ where $\{X_s,\,s\geq0\}$ is a stochastic process with independent increments and we consider $$\left\{x_{\varepsilon}^{\theta}(t)=\left(x_{\varepsilon}^{\theta_1},\dots,x_{\varepsilon}^{\theta_m}\right)(t),\,t\in[0,T]\right\}.$$
In order to simplify calculus and notation we will denote by $\theta$ the $m$ values $\theta_1,
\theta_2,\dots,\theta_m$ . Since we have to control more quadratic covariations we will need to introduce new hypothesis on $\theta$, ($\bar H^{\theta_j,\theta_h}$) for a characteristic function $\phi_X$:
$(\bar H^{\theta_j,\theta_h})$ For any $c_1 \in \{-1,1\}$ $$\lim_{\varepsilon \to 0} \varepsilon^2
\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}\int_{\frac{2s}{\varepsilon^2}}^{y}
\|\phi_{X_y-X_x}
(\theta_j)\|\|\phi_{X_x-X_{\frac{2s}{\varepsilon^2}}}(\theta_j +c_1
\theta_h )\|dxdy = 0.$$
Then, the extension of Theorem \[teogen\], reads as follows:
\[teo2\] Let $\{X_s,\, s\geq 0\}$ be a stochastic process with independent increments and characteristic function $\phi_X$. Set $C_X^m=\{\theta \in {\mathbb{R}}^m,$ such that $\phi_X$ satisfies ($H^{\theta_j}$) for any $ j=1,\ldots,m$ and satisfies $ (\bar H^{\theta_j,\theta_h})$ for any $h \not= j \}$.
Define for any $\varepsilon>0$ and for any $1\leq j \leq m$ $$\{x_{\varepsilon}^{\theta_j}(t)=\varepsilon c(\theta_j) \int_{0}^{\frac{2t}{\varepsilon^2}}e^{i\theta_j
X_s}ds,\quad t\in[0,T]\},$$ where $c(\theta_j)$ is the constant given by hypothesis $(H^{\theta}2)$.
Consider $P_{\varepsilon}^{\theta}$ the image law of $x_{\varepsilon}^{\theta}=\left(x_{\varepsilon}^{\theta_1},\dots,x_{\varepsilon}^{\theta_m}\right)$ in the Banach space $\mathcal C([0,T],\mathbb
C^m)$ of continuous functions on $[0,T]$. Then, if $\theta\in C_X^m$, $P_\varepsilon^{\theta}$ converges weakly as $\varepsilon$ tends to zero towards the law on $\mathcal C([0,T],\mathbb C^{m})$ of a $m$-dimensional complex Brownian motion.
The proof follows applying the computations done for the one-dimensional case combined to the method used in [@BR1]. We will only give some hints of the proof.
Notice that the proof of the tightnes, the martingale property of each component ans the quadratic variations can be done following exactly the proof of the one-dimensional case. So, it remains only to study all the covariations. As it can be seen in Section 3.1 in [@BR1], it suffices to prove that for $j\neq h$ and for any $s_{1}\leq
s_{2}\leq\cdots\leq s_{k} \leq s<t$ and for any bounded continuous function $\varphi:\mathbb C^{mk} \longrightarrow\mathbb R$,
$$\begin{aligned}
& &E\left(\varphi\left(x_{\varepsilon}^{\theta}(s_1),\dots,x_{\varepsilon}^{\theta}(s_k)\right)
\left(\varepsilon\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}c(\theta_j)\cos(\theta_j
X_x)dx\right) \right.\\ & & \qquad \qquad \qquad \qquad \qquad \qquad \times \left.
\left(\varepsilon\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}c(\theta_h)\cos(\theta_h
X_y)dy\right)\right),
\\
& &E\left(\varphi\left(x_{\varepsilon}^{\theta}(s_1),\dots,x_{\varepsilon}^{\theta}(s_k)\right)
\left(\varepsilon\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}c(\theta_j)\sin(\theta_j
X_x)dx\right) \right.\\ & & \qquad \qquad \qquad \qquad \qquad \qquad \times \left.
\left(\varepsilon\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}c(\theta_h)\sin(\theta_h
X_y)dy\right)\right)\end{aligned}$$
and $$\begin{aligned}
& &
E\left(\varphi\left(x_{\varepsilon}^{\theta}(s_1),\dots,x_{\varepsilon}^{\theta}(s_k)\right)
\left(\varepsilon\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}c(\theta_j)\cos(\theta_j
X_x)dx\right) \right.\\ & & \qquad \qquad \qquad \qquad \qquad \qquad \times \left.
\left(\varepsilon\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}c(\theta_h)\sin(\theta_h
X_y)dy\right)\right)\end{aligned}$$ converge to zero when $\varepsilon$ tends to zero. But, using that $\cos(\theta)=\frac{e^{i\theta}+e^{-i\theta}}{2}$ and $\sin(\theta)=\frac{e^{i\theta}-e^{-i\theta}}{2i}$ , and the symmetry between $x$ and $y$ (interchanging the roles of $j$ and $h$), it is enough to show that $$\label{otra}
\lim_{\varepsilon \to 0} \varepsilon^2 \| E\left(\varphi\left(x_{\varepsilon}^{\theta}(s_1),\dots,x_{\varepsilon}^{\theta}(s_k)\right)
\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}
\int_{\frac{2s}{\varepsilon^2}}^{y}e^{i(c_1 \theta_jX_x+c_2 \theta_hX_y)}dxdy\right)\| = 0,$$ for any $c_1, c_2 \in \{-1,1\}.$ But, $$\begin{aligned}
&& \| E\left(\varphi\left(x_{\varepsilon}^{\theta}(s_1),\dots,x_{\varepsilon}^{\theta}(s_k)\right)
\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}
\int_{\frac{2s}{\varepsilon^2}}^{y}e^{i(c_1\theta_jX_x+c_2\theta_h X_y)}dxdy\right)\| \\
&=&\| E\left(\varphi\left(x_{\varepsilon}^{\theta}(s_1),\dots,x_{\varepsilon}^{\theta}(s_k)\right)
\right. \\ && \left. \quad \times
\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}
\int_{\frac{2s}{\varepsilon^2}}^{y}e^{ ic_2\theta_h(X_y-X_x)}
e^{i(c_1\theta_j+c_2\theta_h)(X_x-X_{\frac{2s}{\varepsilon^2}})}e^{i(c_1\theta_j+c_2\theta_h)X_{\frac{2s}{\varepsilon^2}}}
dxdy\right)\|\\
&\leq&K \int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}
\int_{\frac{2s}{\varepsilon^2}}^{y} \|
\phi_{(X_y-X_x)}(c_2\theta_h)\|
\|\phi_{(X_x-X_{\frac{2s}{\varepsilon^2}})}(c_1\theta_j+c_2\theta_h)
\| dxdy, \\&\leq&K
\int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}
\int_{\frac{2s}{\varepsilon^2}}^{y} \| \phi_{(X_y-X_x)}(\theta_h)\|
\|\phi_{(X_x-X_{\frac{2s}{\varepsilon^2}})}(\theta_j+c_3\theta_h) \|
dxdy,\end{aligned}$$ for $c_3\in\{-1,1\}$ and (\[otra\]) follows from ($\bar
H^{\theta_j,\theta_h}$). $\Box$
Finally, we state the extension of Theorem \[teopar\].
\[te4r\] Assume now that $\{X_s,\, s\geq 0\}$ is a Lévy process with Lévy exponent $\psi_X$ and set $$c(u)=\sqrt{\frac{\| \psi_X(u) \|^2 }{2 Re[\psi_X(u)]}}.$$
Consider $P_{\varepsilon}^{\theta}$ the image law of $x_{\varepsilon}^{\theta}$ in the Banach space $\mathcal
C([0,T],\mathbb C^m)$ of continuous functions on $[0,T]$. Then, for $\theta$ such that $Re[\psi_X(\theta_j)]Re[\psi_X(2\theta_j)] \not=
0$ for all $j \in \{1,\ldots,m\}$ and $Re[\psi_X(\theta_j + c_1
\theta_h)]\not=0$ for all $j,h \in \{1,\ldots,m\}$ and $c_1 \in
\{-1,1\}$, $P_\varepsilon^{\theta}$ converges weakly as $\varepsilon$ tends to zero, towards the law on $\mathcal
C([0,T],\mathbb C^m)$ of a $m$-dimensional complex Brownian motion.
As in the one-dimensional case it suffices to check that the characteristic function $\phi_X$ of the Lévy process satisfies ($H^{\theta_j}$) for any $ j=1,\ldots,m$ and satisfies $(\bar
H^{\theta_j,\theta_h})$ for any $ j \not= h$. It remains only to see the second part and can be easily checked that, for $c_1 \in \{-1,1\}$ $$\begin{aligned}
&& \int_{\frac{2s}{\varepsilon^2}}^{\frac{2t}{\varepsilon^2}}
\int_{\frac{2s}{\varepsilon^2}}^{y} \| \phi_{(X_y-X_x)}(\theta_j) \|
\| \phi_{(X_x-X_{\frac{2s}{\varepsilon^2}})}(\theta_j+c_1 \theta_h)
\|
dxdy\\
&\leq&K\varepsilon^2\frac{1}{a(\theta_j)a(\theta_j+c_1 \theta_h)}.\end{aligned}$$
$\Box$
[99]{} Bardina, X. The complex Brownian motion as a weak limit of processes constructed from a Poisson process. In: Stochastic Analysis and Related Topics VII. Proceedings of the 7th Silivri Workshop, Kusadasi 1998, pp. 149-158. [*Progress in Probability, Birkhäuser.*]{} (2001) Bardina, X., Jolis, M., Quer-Sardanyons, Ll. Weak convergence for the stochastic heat equation driven by Gaussian white noise. [*Electron. J. Probab.*]{} [**15**]{} (2010), no. 39, 1267-1295. Bardina, X., Jolis, M., Tudor, C. On the convergence to the multiple Wiener-Ito integral. [*Bull. Sci. Math.*]{} [**133**]{} (2009), no. 3, 257-271. Bardina, X., Nourdin, I., Rovira, C., Tindel, S. Weak approximation of a fractional SDE. [*Stochastic Process. Appl.*]{} [**120**]{} (2010), no. 1, 39-65. Bardina,X., Rovira, C. A d-dimensional Brownian motion as a weak limit from a one-dimesional Poisson process. [*Lithuanian Mathematical Journal*]{} [**53**]{} (2013), 17–26. Billingsley, P. Convergence of Probability Measures. [*John Wiley and Sons.*]{} (1968). Kac, M. A stochastic model related to the telegraphers equation [*Rocky Moutain J. Math.*]{} [**4** ]{}(1974), 497-509. Sato, K. Lévy processes and infinitely divisible distributions. Cambridge Studies in Advanced Mathematics, 68. [*Cambridge University Press, Cambridge*]{} (1999). Stroock, D. Topics in Stochastic Differential Equations (Tata Institute of Fundamental Research, Bombay.) [*Springer Verlag.*]{} (1982).
[^1]: X. Bardina is supported by the grant MTM2012-33937 from SEIDI, Ministerio de Economia y Competividad.
[^2]: C. Rovira is supported by the grant MTM2012-31192 from SEIDI, Ministerio de Economia y Competividad.
[^3]: MSC2010: 60F17, 60G15.
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|
---
abstract: 'Predictivity of the Kohn-Sham approach to dynamical problems, when regarded as an initial value problem in a time-dependent density functional framework, is analysed for a class of models for which the argument devised in the work of Maitra *et al.* [@neepacausality] for the standard electronic many-body problem does not apply. The original argument is here extended and revised. As a result, predictivity for this class of problems seems possible only at the price of introducing extra unknown functionals in the corresponding Kohn-Sham equation. Furthermore, the same argument, when applied to original electronic problem, suggests that the Hartree-exchange-correlation potential is not unambiguously identified by the contemporary and past densities and initial states, but also requires knowledge of the divergence of the contemporary Kohn-Sham current.'
author:
- Walter Tarantino
bibliography:
- 'bibliography.bib'
date: May 2015
title: 'On the Kohn-Sham Approach to Time-Dependent Problems in a Density-Functional Framework'
---
Introduction {#introduction .unnumbered}
============
Time-dependent density functional theory (TDDFT) in the Kohn-Sham approach allows, in principle, to calculate the charge density of the electronic many-body problem by solving the equation of motion of a system of non-interacting electrons [@rungegross; @vL1999]. The same approach has been adopted for other many-body Hamiltonians and expectation value of local operators, usually refer to as ‘densities’. Predictivity of the original approach was questioned in [@sch] and immediately clarified in [@neepacausality].
This work is intended as an extension of the argument of [@neepacausality] and moves a step forth towards a complete description of the mathematical foundations of the Kohn-Sham approach within TDDFT as an initial-value problem.
In the following I will improve the argument of [@neepacausality] for the standard TDDFT case and explain how it does not hold for some other class of many-body problems, including some models of physical interest. I will discuss the consequences of the failure of such an argument, showing how the issue of ‘predictivity’ can be discussed in terms of existence of a unique solution of the equations of motion of the Kohn-Sham system. I shall then characterize such a class of problems and, finally, explain how in some cases the approach can still be made ‘predictive’, as long as new (unknown) density-functionals are introduced in the relevant equations.
In Section \[Sprocedure\] the Kohn-Sham approach within the framework of Density-Functional Theory for time-dependent problems is recalled; in Section \[Shubbard\] a simple model (Hubbard dimer) for which such a procedure is problematic is presented; in Section \[Sprocedure2\] a tool for understanding whether a model suffers of the same problem is devised and a simple modification of the original recipe that fixes the problem for the Hubbard case is generalized.
The Density-Functional Time-Dependent Kohn-Sham Approach (1 of 2) {#Sprocedure}
=================================================================
TDDFT in its KS declination represents an alternative to the time-dependent Schrödinger equation for describing the evolution of the charge density of a system of interacting electrons [@ullrich]. In fact such a reformulation of the electronic problem can be generalized and used for other time-dependent problems as well. The common way to perform such a generalization and apply it to other models is the object of this section.
We start with a generic time-evolution quantum problem, formulated in second quantization and characterized by the equation of motion \[MBeom\] i\_t =( ++(t)), which includes a kinetic term $\hat{T}$, an interaction term $\hat{W}$ and the action of an external classical field via $\hat{V}(t)$, and by a given state $\ket{\Psi_0}$ such that \[MBiv\] =. I assume that the first order differential equation (\[MBeom\]) has one and only one solution $\ket{\Psi(t)}$ satisfying (\[MBiv\]). I shall refer to this as the ‘time-dependent many-body problem’, or, simply, ‘the many-body (MB) problem’.
If we are not interested in finding the state vector that solves the problem, but only in the expectation values of a finite number of local operators (from now on ‘densities’), we can attempt to approach the problem via the Kohn-Sham method, which consists in solving an auxiliary differential equation that is only linear in the field and characterized by a set of effective potentials such that the densities evaluated in such a system match their value in the original problem. Let us write $\hat{V}(t)=\sum {v_{\rm ext}}(t) \hat{n}$, where the sum is a shorthand notation for a integration/sum over all residual degrees of freedom (space/lattice points, vectorial/tensorial/spin components...) and $\hat{n}$ is a local operator bilinear in the field. For sake of argument, I consider the case in which we are interested only in the expectation value of $\hat{n}$, namely $\{\braket{\Psi (t)|\hat{n}|\Psi(t)}\;|\;t\in [0,+\infty)\}$. The Konh-Sham system for this problem is then identified by the equation of motion \[KSeom\] i\_t =( +\_s(t)) with $\hat{V}_s(t) \equiv \sum v_s(t) \hat{n}$, and a certain initial state \[KSiv\] = such that $\braket{\Psi (t)|\hat{n}|\Psi(t)}=\braket{\Phi (t)|\hat{n}|\Phi(t)}$ for all $t\in [0,+\infty)$. The existence of such a Kohn-Sham system is in general not known and it becomes crucial to prove it even under restrictive assumptions. To proceed with my argument, I assume that a theorem similar to the van Leeuwen-Runge-Gross theorem [@rungegross; @vL1999] can in fact be proven for the model under exam. Such a theorem ensures existence and uniqueness of the system, under specific conditions, and allows to rewrite the problem in a density-functional framework. It can be stated in the following terms.
Given $f(t)$ a generic function of time, we distinguish the set $\{f(t)|\;t\in[0,+\infty)\}$ denoted by simply $f$, from the value of the function *at the specific time $t$* denoted by $f(t)$. A *set of functions* is then denoted by $\mathfrak{F}\equiv\{f,g,...\}$. The expectation value of an observable ${\hat{\mathcal{O}}}$ is denoted by $\mathcal{O}_{MB}(t)$, if sandwiched by $\ket{\Psi(t)}$, or by $\mathcal{O}_{KS}(t)$, if sandwiched by $\ket{\Phi(t)}$. Moreover, $n(t)\equiv n_{MB}(t)=n_{KS}(t)$. Then, given a certain set of external potentials $\{{v_{\rm ext}},{v_{\rm ext}}',{v_{\rm ext}}'',...\}\equiv \mathfrak{V_e}$, we denote by $\mathfrak{N}\equiv\{n,n',n'',...\}$ the set of densities obtained by solving the problem (\[MBeom\]-\[MBiv\]) with any element of $\mathfrak{V_e}$ (giving rise to the set of ‘v-representable densities’). The first part of the theorem is then stated as following:
There is a one-to-one correspondence between elements of $\mathfrak{V_e}$ and elements of $\mathfrak{N}$, up to gauge transformations.
Moreover, given the set $\mathfrak{M}$ of densities obtained by solving (\[KSeom\]-\[KSiv\]) by means of some effective potentials, the second part of the theorem reads
There exists a set $\mathfrak{V_s}\equiv \{v_s,v_s',v_s'',...\}$ such that $\mathfrak{M}=\mathfrak{N}$ and, moreover, elements of $\mathfrak{V_s}$ are in one-to-one correspondence with $\mathfrak{N}$, up to gauge transformations.
From now on I shall refer to these two statements as to ‘vLRGl’ (van Leeuwen-Runge-Gross-like theorem). For the electronic many-body problem the theorem has been proved under some restrictive assumptions [@rungegross; @vL1999; @fixed], which for the moment are not of our concern.
Once the existence of the Kohn-Sham system is ensured by vLRGl, one has to find a viable way to calculate the effective potential $v_s$. Looking back at the inputs of the original problem, $v_s(t)$ can be regarded as a functional of $\ket{\Psi_0}$, $\ket{\Phi_0}$ an ${v_{\rm ext}}$. This will be denoted by $v_s(t)=F_t[{v_{\rm ext}}]$, in which the explicit dependence on initial states is omitted but always understood. If we were given this functional, then the problem (\[KSeom\]-\[KSiv\]) would have been well defined and admit, like the problem (\[MBeom\]-\[MBiv\]), one and only one solution.
A *density*-functional formulation of the problem, which we are allowed to do thanks to vLRGl, implies the definition of quantities that depend on the density and not on the external potential. We cannot use directly $v_s(t)=\tilde{F}_t[n]$, even though vLRGl ensures its existence and uniqueness, since the problem (\[KSeom\]-\[KSiv\]) would have no information at all about the external potential that drives the evolution of (\[MBeom\]).
To feed in some information about the external potential, one usually [@rungegross] writes \[vhxcdef\] v\_s(t)=[v\_[ext]{}]{}(t)+[v\_[Hxc]{}]{}(t) which in fact defines a new quantity, ${v_{\rm Hxc}}(t)$. As long as vLRGl holds, we can write ${v_{\rm Hxc}}(t)=G_t[n]$, which expresses the functional dependence of ${v_{\rm Hxc}}(t)$ on $n$.
It should be emphasized that vLRGl states that, given *the entire set* $\{n(t)|\;t\in[0,+\infty)\}$, *the entire set* $\{{v_{\rm Hxc}}(t)|\;t\in[0,+\infty)\}$ is unambiguously identified; however, it does not specify how single elements of these sets are linked. More specifically, it does not tell whether information on future densities is necessary to construct ${v_{\rm Hxc}}(t)$ or not. As discussed in [@sch; @neepacausality], it appears that this might threaten the predictivity of the approach. For the standard electronic many-body problem of [@rungegross], an argument to prove that these worries are unjustified was presented in [@neepacausality], in which it was explained how ${v_{\rm Hxc}}(t)$ requires no information on the future to be completely determined. However, this argument is specific for the Hamiltonian and the choice of densities of [@rungegross]. Moreover, part of it is in conflict with the initial assumptions of [@rungegross], as I shall soon explain. In the following section I shall present a simpler model, which will allows me to enlighten some limits of the original argument of [@neepacausality] and suggest possible solutions.
The case of the Hubbard Dimer {#Shubbard}
=============================
Definition of the model and corresponding Kohn-Sham system
----------------------------------------------------------
I now consider the Hubbard dimer in presence of an external field. Such a model has been analyzed in a TDDFT framework in several works [@fuks2013; @fuks2014a; @fuks2014b] but its relevance to the realistic electronic many-body problem is not of our concern, as I use it only as prototype of a class of problems for which the argument of [@neepacausality] does not apply. The model is defined by \[MBH\] -\_[=,]{}( \^\_[L]{}\_[R]{} + \^\_[R]{}\_[L]{} )+ U(\_[L]{}\_[L]{}+ \_[R]{}\_[R]{})+ v(t) where $\hat{n}_i\equiv\sum_\sigma \hat{c}^\dagger_{i\sigma}\hat{c}_{i\sigma}$, and $\{\hat{c}_{i\sigma},\hat{c}^\dagger_{j\rho}\}= \delta_{ij} \delta_{\sigma\rho}$, with $i,j=R,L$ and $\sigma,\rho=\uparrow,\downarrow$, all other anticommutators being zero, and by a given initial state $\ket{\Psi(t=0)}=\ket{\Psi_0}$.
If one is interested only in the value of $\braket{\hat{n}_L}$ and $\braket{\hat{n}_R}$, one can attempt to solve the problem by means of the approach above outlined. In fact, the value of $\braket{\hat{n}_L+\hat{n}_R}$ is only determined by the initial state, since $[\hat{n}_L+\hat{n}_R,H(t)]=0$. We can then concentrate on the observable $n(t)$, where . Following [@fuks2014b], I chose the corresponding Kohn-Sham to be characterized by the following Hamiltonian: \[KSH\] \_s-\_[=,]{}( \^\_[L]{}\_[R]{} + \^\_[R]{}\_[L]{} ) + v\_s(t). For simplicity, I consider $\ket{\Phi(t=0)}=\ket{\Psi_0}$. Then, following the procedure outlined in the previous section, I define \[vhxcdef1\] [v\_[Hxc]{}]{}(t)v\_s(t)- v(t). Such a function is determined by the condition \[n=n\] n\_[MB]{}(t)= n\_[KS]{}(t). This condition can be recast as {
[c]{}
n\_[KS]{}(0)= n\_[MB]{}(0)\
\_[KS]{}(0)=\_[MB]{}(0)\
\_[KS]{}(t)=\_[MB]{}(t),
. The first two identities are satisfied no matter what effective potential is chosen, as long as the two initial states are taken to be the same. So we can say that ${v_{\rm Hxc}}(t)$ is in fact defined by the condition \[ddot\] \_[KS]{}(t)=\_[MB]{}(t). The quantity $\ddot {n}(t)$ can be connected with the contemporary external potential by means of the equations of motion (\[KSeom\]) and (\[MBeom\]): \[MBlfe\] [v\_[ext]{}]{}(t)\_[MB]{}=(t)+4 \^2 n(t)+\_[MB]{}. Similarly, for the KS system we have: \[KSlfe\] v\_s(t)\_[KS]{}=(t)+4 \^2 n(t). In fact, using (\[ddot\],\[MBlfe\],\[KSlfe\],\[vhxcdef1\]) it is possible to derive that \[vhxcdef2\] [v\_[Hxc]{}]{}(t)= v(t)(-1) -. Vice versa, as long as (\[vhxcdef2\]) holds, equation (\[ddot\]) and hence (\[n=n\]) follow.
Taylor-expandability and Causality
----------------------------------
Adapting the constructive algorithm of [@vL1999], it is possible to prove that a set of ${v_{\rm Hxc}}$ satisfying (\[vhxcdef2\]) does exist, at least if we restrict our attention to the set of external potentials that are Taylor-expandable in the time variable (‘Te’ potentials, from now on). In fact, under this restriction, adapting the entire argument of [@vL1999] allows to prove the vLRGl for this case. This also allows us to density-functionalize the theory. We can then correctly regard ${v_{\rm Hxc}}(t)$ as a functional of the density only. I now want to investigate the property of ‘predictivity’ of the approach, in the sense of uniqueness of the solution of the KS system (\[KSeom\],\[KSiv\]), with ${v_{\rm Hxc}}(t)=G_t[n]$.
Following [@neepacausality], one might start by wondering about the dependence of ${v_{\rm Hxc}}(t)$ on preceding ($0\leq t'<t$), contemporary ($t'=t$), and successive ($t'>t$) densities $n(t')$. In [@neepacausality] it was argued that, for any $T>0$, ${v_{\rm Hxc}}(t)$ with $0<t<T$ cannot depend on densities $n(t)$ with $t>T$, using the Runge-Gross theorem and the fact that two external potentials that are the same in $[0,T]$ but differ for $t>T$ must give different densities at times $t>T$. However, for Te external potentials this very first step is problematic, for two Te functions equal in a finite interval $0<\epsilon_1<t<\epsilon_2<T$ are in fact equal everywhere within the radius of convergence of the Taylor series. This means that, when Taylor expandable external potentials are considered, as we did in order to prove the vLRGl, two external potentials that were the same in a finite interval in the past will be necessarily the same also in the future (up to the radius of convergence, i.e. until vLRGl holds). Equivalently, within the radius of convergence of the Taylor expansion of the density, any dependence on the density on a finite interval in the past can be recast as a dependence on the density on a finite interval in the future, and vice versa. In this case, the intuitive notion of ${v_{\rm Hxc}}(t)$ being ‘causal’, in the sense that does not depend on future densities, becomes meaningless.
Nonetheless, a meaningful concept of ‘causality’ is not necessary to discuss the predictivity of the approach, that remains a non-trivial problem. In order to proceed, we look back at equation (\[vhxcdef2\]) and consider separately the objects in the form of $\braket{{\hat{\mathcal{O}}}}$ from $v(t)$.
Density Functionalization (1of2)
--------------------------------
The contemporary expectation values $\braket{{\hat{\mathcal{O}}}}$ in (\[vhxcdef2\]) are unambiguously identified by the state vectors $\ket{\Psi(t)}$ and $\ket{\Phi(t)}$. In order to argue that these objects, when regarded as functionals of the density only, pose no threat to the time propagation of the Kohn-Sham system, we extend the second part of the argument of [@neepacausality], based on a propagation on a finite time-grid.
More specifically, we consider a discrete time variable and a simple rule for translating a differentiation on a continuous variable, namely \[rule\] . To compare results on the time-grid with the continuous limit, we consider only terms of leading order in $\Delta$, since the limit $\Delta\rightarrow 0$ makes (\[rule\]) an identity.
If we denote the many-body and Kohn-Sham state at the time step $i$ with $\psi_i$ and $\phi_i$, respectively, we can write \_[i+1]{}&=&F(v\_i,\_i)\
\_[i+1]{}&=&G(v\^s\_i,\_i)\[phi\]\
v\_i&=&V(n\_[i+2]{},n\_[i+1]{},\_i,\_i)\
v\_i\^s&=&V\^s(v\_i,\_i,\_i)\[vs\]\
n\_[i+1]{}&=&N(n\_i,J\_i) with $F(.)$ obtained from (\[MBeom\],\[MBH\],\[rule\]), $G(.)$ from (\[KSeom\],\[KSH\],\[rule\]), $V(.)$ from (\[MBlfe\],\[rule\]),[^1] $V^s(.)$ from (\[vhxcdef1\],\[vhxcdef2\]), and $N(.)$ from the continuity equation \[cont\] (t)=- , with $\hat{J}\equiv i[\hat{n},\hat{T}]$. Then, for the two terms appearing in (\[vhxcdef2\]) we can write \[OO\] L(\_i,\_i) for which
L(\_i,\_i)&=&L(F(v\_[i-1]{},\_[i-1]{}),G(v\^s\_[i-1]{},\_[i-1]{}))=\
&=&L(F(V(n\_[i+1]{},n\_[i]{},\_[i-1]{},\_[i-1]{}),\_[i-1]{}), G(V\^s(V(n\_[i+1]{},n\_[i]{},\_[i-1]{},\_[i-1]{}),\_[i-1]{},\_[i-1]{}),\_[i-1]{}))=\
&&M(n\_[i+1]{},n\_[i]{},\_[i-1]{},\_[i-1]{})=\
&=&M(N(n\_i,J\_i),n\_[i]{},\_[i-1]{},\_[i-1]{})=\
&&L\_[(1)]{}(J\_i,n\_[i]{},\_[i-1]{},\_[i-1]{})=\
&=&L\_[(1)]{}(J\_i,n\_[i]{},F(v\_[i-2]{},\_[i-2]{}),G(v\^s\_[i-2]{},\_[i-2]{}))=\
&=&L\_[(1)]{}(J\_i,n\_[i]{},F(V(n\_[i]{},n\_[i-1]{},\_[i-2]{},\_[i-2]{}),\_[i-2]{}), G(V\^s(V(n\_[i]{},n\_[i-1]{},\_[i-2]{},\_[i-2]{}),\_[i-2]{},\_[i-2]{}),\_[i-2]{}))=\
&&L\_[(2)]{}(J\_i,n\_[i]{},n\_[i-1]{},\_[i-2]{},\_[i-2]{})=\
&&\
&=&L\_[(i)]{}(J\_i,n\_[i]{},n\_[i-1]{},...,n\_1,\_[0]{},\_[0]{}).\[L\]
This means that on a time-grid these terms are unambiguously determined by: the Kohn-Sham state vector at same time step (which enters through $J_i$), the set of contemporary and previous densities, and the initial states. Since at a given time step, these ingredients are in principle all at our disposal, they pose no threat to the time-propagation of such a discretized KS system. While rigorous for the discretized problem, this argument is only a first step towards a complete proof for the continuous time problem.
Nonetheless, when applied to the electronic problem of [@rungegross], the argument supports the claim of [@neepacausality] that terms in the form of (\[OO\]) pose no threat to the time-propagation of the corresponding KS system.
In fact, following farther the argument, we are also led to conclude that the ${v_{\rm Hxc}}({\bf r},t)$ of [@rungegross] cannot be determined by sole knowledge of initial states, contemporary and past densities, but also requires knowledge of the divergence of the contemporary Kohn-Sham current ${\bf \nabla}\cdot {\bf j}(t)$, which appears in the equivalent of (\[cont\]). The importance of such a term was recognized in [@neepacausality], but only at $t=0$, when its value can be obtained from the initial KS state. Pushing farther their own argument, as done above, suggests that the dependence of ${v_{\rm Hxc}}$ on ${\bf \nabla}\cdot {\bf j}$ for finite times, rather than remaining a dependence on the initial ${\bf \nabla}\cdot {\bf j}(t=0)$, it becomes a dependence on the *contemporary* ${\bf \nabla}\cdot {\bf j}$.
Such a dependence is not incompatible with vLRGl, for which we expect that complete knowledge of the density suffices to identify the system. $J_i$ has indeed been introduced to encode the information contained in $n_{i+1}$, which, in the language of the continuous time case, translates into $\dot{n}(t)$. Choosing of expressing such an information in terms of $J(t)$, which one can express in terms of $\ket{\Phi(t)}$, rather than $\dot{n}(t)$, avoids the complications of having a time derivative of the state on the right-hand side of the equation of motion (\[KSeom\]).
Density Functionalization (2of2) {#Sdft2}
--------------------------------
Once the $n$ dependence of the terms $\braket{{\hat{\mathcal{O}}}}$ has been clarified, to some extend at least, we can go back to (\[vhxcdef2\]) and discuss the presence of $v(t)$. This needed to not to be discussed in [@neepacausality], because of no explicit $v(t)$ dependence in the corresponding equation. More precisely, in the standard TDDFT case, the equivalent of (\[MBlfe\]) and (\[KSlfe\]) reads +\_[MB]{} and +\_[MB]{} with ${\hat{\mathcal{O}}}_2$ and ${\hat{\mathcal{O}}}_1$ two appropriate operators, $n({\bf r},t)$ the electronic charge-density, $v({\bf r},t)$ the external field and $v_s({\bf r},t)$ the effective field of the corresponding KS system [@vL1999]. Since both potentials are multiplied by the electronic charge-density, a quantity that is the same for the many-body and the KS system, the explicit $v(t)$-dependence in the corresponding ${v_{\rm Hxc}}({\bf r},t)$ cancels out.
The fact that in general $\braket{\hat{T}}_{MB}/\braket{\hat{T}}_{KS}$ does not simplify to $1$, leading in our case to the missed cancellation, can be proved by considering the specific case of $\tau=U=1$, $v(t)=\sin (t)$, and $\ket{\Psi_0}$ being the half-filled ground-state of the Hamiltonian at $t=0$, for which
[l]{} \_[MB]{}=-2+t\^2-t\^4+t\^6+(t\^8)\
\_[KS]{}=-2+t\^4+t\^6+(t\^8),
as one can prove by using the constructive algorithm of [@vL1999].
The expression on the right-hand of (\[vhxcdef2\]) side is therefore not ‘universal’, in the sense that it explicitly depends on the system via $v(t)$. However, this dependence can be removed if we use again equation (\[MBlfe\]) to extract the information about the density $n$ contained in $v(t)$. This leads to \[vddot\] v(t)=F\_t\[n\]+G\_t\[n\] . with $F$ and $G$ some suitable functionals. While for these two functionals the argument of the previous section applies, the presence of $\ddot{n}(t)$ needs a dedicated analysis.
Even though one could discuss the consequences of the presence of $\ddot {n}(t)$ on the time-grid, in fact we do not need to, as they can be cleared enough by directly looking at the original problem in continuous time. When (\[vddot\]) is plugged back into the equation of motion (\[KSeom\]), an explicit dependence on the second-time derivative of $\ket{\Phi(t)}$ is introduced. Even assuming that the conclusions of the previous sections can be safely generalised to the continuous time case, the presence of a double time-derivative of $\ket{\Phi(t)}$ changes the status of the equation (\[KSeom\]) from first to second-order. This implies that the initial value (\[KSiv\]) is no longer sufficient to ensure one single solution to the problem. In other words, the density-functionalization of the Kohn-Sham system (\[KSH\]) leads to a problem that, despite being an actual exact reformulation of the many-body problem, does not admit only one solution, and hence it fails to unambiguously characterize the solution of the initial many-body problem, even if all functionals were known. Using the terminology of [@sch; @neepacausality] one could say that this Kohn-Sham system *is not predictive*.
Restoring Predictivity
----------------------
Applying the recipe of Section \[Sprocedure\] to the Hubbard dimer above defined led to a reformulation of the original problem that, despite being exact, cannot be used to calculate the wanted densities.
This problem can in fact be avoided if we use a slightly modified recipe, which however has the cost of introducing a new (unknown) functional of the density. More specifically, the way information about ${v_{\rm ext}}(t)$ is extracted from $v_s(t)$ has to be modified as follows: \[ffdef\] {
[l]{} v\_s(t)= v\_1(t) [v\_[ext]{}]{}(t)+ v\_2(t)\
v\_1(t)\
v\_2(t)-.
. When $v_1$ and $v_2$ are regarded as functionals of $n$, the argument used above for $\braket{{\hat{\mathcal{O}}}}$ applies again and the problems cause by the presence of $\ddot{n}(t)$ are avoided.
The Density-Functional Time-Dependent Kohn-Sham Approach (2 of 2) {#Sprocedure2}
=================================================================
A New Recipe
------------
The case of the Hubbard dimer here considered is not an isolated exception, as we shall see in the next section. In fact we can use what learned from this example to identify a *class* of problems that, if approached in the standard way, present the same complications.
Given an Hamiltonian and a certain choice of densities for which the vLRGl holds, we can connect the external potential(s) to the effective one(s) by differentiating the densities and using the equations of motion (\[MBeom\]-\[KSeom\]). This would lead to an expression that in the simplified notation of Section \[Sprocedure\] would read \[testeq\] \_[KS]{} +\_[KS]{}= \_[MB]{} +\_[MB]{} with ${\hat{\mathcal{O}}}_i[f(t)]$ acting as an operator on the Hilbert space of state vectors and possibly a differential operator on the argument $f(t)$. If \_[KS]{} = \_[MB]{}, then we can say that the problem (Hamiltonian$+$choice of densities) is *RG-like* and the KS system is as predictive as the one of standard TDDFT [@rungegross]; otherwise, we can say that it is *RG-unlike* and considering the Hartree-exchange-correlation potential defined by ${v_{\rm Hxc}}(t) \equiv v_s (t) - {v_{\rm ext}}(t)$ as a functional of the density only leads to a reformulation of the many-body problem that does not admit one unique solution, making the density-functionalized Kohn-Sham system not predictive.
In case the operator ${\hat{\mathcal{O}}}_i[f(t)]$ is purely multiplicative on its argument: ${\hat{\mathcal{O}}}_i[f(t)]\rightarrow {\hat{\mathcal{O}}}_i f(t)$, like in our example of the Hubbard dimer, it is possible to fix this problem by simply considering {
[l]{} v\_s(t)= v\_1(t) [v\_[ext]{}]{}(t)+ v\_2(t)\
v\_1(t)\
v\_2(t)
. and regarding $v_1(t)$ and $v_2(t)$ as functionals of the density separately.
When the density(/ies) is(/are) chosen to be the expectation value of the local operator(s) conjugate to the external potential(s) in the many-body Hamiltonian, one has &&=\
&&=\
&&=\
&&=. by virtue of $[\hat{V},\hat{n}]=0$.
Relativistic Electrons
----------------------
Another example of the class of *RG-unlike* problems is provided by the Kohn-Sham approach to QED defined in [@rmb], to which I refer the reader for the notation of this section. In that work the four-potential $J^\mu$ is chosen as ‘density’. As one can easily verify from equation (31) of [@rmb], our equation (\[testeq\]) becomes \_[KS]{}a\^[s]{}\_+\_[KS]{} =\_[QED]{}a\^[ext]{}\_+\_[QED]{} with \^(\^\^0 \^-\^\^0 \^) where $\hat{\psi}$ is the Dirac field operator, $\psib\equiv \psi^\dagger\gamma^0$ and $\gamma^\mu$ the gamma matrices acting on the spin degrees of freedom of $\psi$, and in which explicit spacetime dependence has been omitted. Since $\braket{\hat{{\hat{\mathcal{O}}}}^{\mu\nu}}_{KS}\neq \braket{\hat{{\hat{\mathcal{O}}}}^{\mu\nu}}_{MB}$ the approach has the problem of predictivity here discussed, if $a^\mu_{\rm Hxc}(x)$ is regarded as a functional of the four-current only.[^2]
The case of QED is particularly instructive because it shows that the predictivity issue here discussed does not seem to be necessarily inherent the many-body Hamiltonian considered, for the choice of densities and the Kohn-Sham system can also be crucial. In [@rugge2] a different set of densities was chosen for the same QED problem, namely the four quantities $\psi^\dagger \gamma^\mu \psi\equiv P^\mu$. In this case, the equivalent of (\[testeq\]) calculated in the Coulomb gauge (using equations (63) and (65) of [@rugge2]) leads to $ \braket{\hat{{\hat{\mathcal{O}}}}_1[v_s(t)]}_{KS} \rightarrow P^0 \vec{a}_s $ and $ \braket{\hat{{\hat{\mathcal{O}}}}_3[{v_{\rm ext}}(t)]}_{MB} \rightarrow P^0 \vec{a}_{\rm ext} $. Being $P^0$ one of the chosen densities, the identity $ \braket{\hat{{\hat{\mathcal{O}}}}_1[{v_{\rm ext}}(t)]}_{KS} = \braket{\hat{{\hat{\mathcal{O}}}}_3[{v_{\rm ext}}(t)]}_{MB} $ is in fact fulfilled making the system of [@rugge2] *RG-like*.
Conclusions {#Sconclusions .unnumbered}
===========
In this work the arguments used in [@neepacausality] to support the predictive character of the Kohn-Sham system as an initial-value problem within a TDDFT framework were reviewed and extended.
It was pointed out that the original argument cannot be applied if external potentials that are Taylor expandable in time are considered. Then, in order to study the ‘predictivity’ of a density-functionalized Kohn-Sham system, identified with the property of such a system of admitting one and only one solution when regarded as an initial-value problem, an argument on a time-grid of infinitesimal pace has been developed. When applied to the standard TDDFT case, such an argument, on one hand, confirms the claims of [@neepacausality] about predictivity of standard TDDFT; on the other hand, it suggests that, in order to preserve the differential structure of the Kohn-Sham equation and have a predictive Kohn-Sham system, ${v_{\rm Hxc}}({\bf r},t)$ must be regarded as functional of contemporary and previous densities, the many-body and Kohn-Sham initial state and *the divergence of the contemporary Kohn-Sham current.* The argument is however rigorous only on a time-grid and the problem demands for further investigations.
The specificity of the original argument of [@neepacausality] to the TDDFT problem has also been enlightened by showing another many-body problem, a time-dependent Hubbard dimer, whose Kohn-Sham system is in fact not predictive, as long as the corresponding ${v_{\rm Hxc}}(t)\equiv v_s(t)-{v_{\rm ext}}(t)$ is regarded as a functional of the density only. It was proved that the Kohn-Sham equation resulting from a density-functionalization of ${v_{\rm Hxc}}$ was requiring more boundary-conditions then provided, failing to being characterized by a unique solution, and hence failing to make a ‘predictive’ system. The origin of such a problem was identified, allowing to derive a criteria to fulfill for a Kohn-Sham system for not being affected by the same predictivity issue.
In some simple cases, like for the Hubbard dimer here considered, predictivity can be restored by modifying the definition of the quantities to be considered functionals of the density. More specifically, it was argued that, if the effective potential is decomposed as $v_s(t)=v_1(t) {v_{\rm ext}}(t)+v_2(t)$, with $v_1(t)$ and $v_2(t)$ unambiguously defined as expectation values of some specific operators, and the two potentials $v_1(t)$ and $v_2(t)$ are regarded as functionals of the density, the Kohn-Sham initial-value problem has in fact only one solution.
Acknowledgements {#acknowledgements .unnumbered}
================
The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC grant agreement no. 320971.
[^1]: In fact, an entry in $V(.)$ for $\phi_i$ is redundant, since $v_i$ is only determined by $n_{i+2}$, $n_{i+1}$ and $\psi_i$. However, this is specific for the model here considered, while in general also $\phi_i$ may be required.
[^2]: In applying the argument of Section \[Sdft2\] one might think that using again equation (31) of [@rmb] to connect $a^\mu_{\rm ext}(x)$ to $\dot{j}^\mu$, the resulting Dirac-Kohn-Sham equation would not become of second order, but remaining of first. If so, the argument on the number of solutions would no longer apply. However, only two components of $a^\mu_{\rm ext}(x)$ are determined by (31), since the kernel of the matrix $\braket{{\hat{\mathcal{O}}}^{\mu\nu}}$ is of dimension 2. Even if one component is fixed by a gauge condition, another component remains to fix and for this one should necessarily look at higher derivatives.
|
{
"pile_set_name": "ArXiv"
}
|
[<span style="font-variant:small-caps;">André Joyal</span> and <span style="font-variant:small-caps;">Joachim Kock</span>]{}
[Université du Québec à Montréal]{}
------------------------------------------------------------------------
**Abstract.** We define weak units in a semi-monoidal $2$-category ${\mathscr{C}}$ as cancellable pseudo-idempotents: they are pairs $(I,\alpha)$ where $I$ is an object such that tensoring with $I$ from either side constitutes a biequivalence of ${\mathscr{C}}$, and $\alpha: I {\otimes}I \to I$ is an equivalence in ${\mathscr{C}}$. We show that this notion of weak unit has coherence built in: Theorem \[thmA\]: $\alpha$ has a canonical associator $2$-cell, which automatically satisfies the pentagon equation. Theorem \[thmB\]: every morphism of weak units is automatically compatible with those associators. Theorem \[thmC\]: the $2$-category of weak units is contractible if non-empty. Finally we show (Theorem \[thmE\]) that the notion of weak unit is equivalent to the notion obtained from the definition of tricategory: $\alpha$ alone induces the whole family of left and right maps (indexed by the objects), as well as the whole family of Kelly $2$-cells (one for each pair of objects), satisfying the relevant coherence axioms.
------------------------------------------------------------------------
Introduction {#introduction .unnumbered}
============
The notion of tricategory, introduced by Gordon, Power, and Street [@Gordon-Power-Street] in 1995, seems still to represent the highest-dimensional explicit weak categorical structure that can be manipulated by hand (i.e. without methods of homotopy theory), and is therefore an important test bed for higher-categorical ideas. In this work we investigate the nature of weak units at this level. While coherence for weak associativity is rather well understood, thanks to the geometrical insight provided by the Stasheff associahedra [@Stasheff:1963], coherence for unit structures is more mysterious, and so far there seems to be no clear geometric pattern for the coherence laws for units in higher dimensions. Specific interest in weak units stems from Simpson’s conjecture [@Simpson:9810], according to which strict $n$-groupoids with weak units should model all homotopy $n$-types.
In the present paper, working in the setting of a strict $2$-category ${\mathscr{C}}$ with a strict tensor product, we define a notion of weak unit by simple axioms that involve only the notion of equivalence, and hence in principle make sense in all dimensions. Briefly, a weak unit is a cancellable pseudo-idempotent. We work out the basic theory of such units, and compare with the notion extracted from the definition of tricategory. In the companion paper [*Weak units and homotopy $3$-types*]{} [@Joyal-Kock:traintracks] we employ this notion of unit to prove a version of Simpson’s conjecture for $1$-connected homotopy $3$-types, which is the first nontrivial case. The strictness assumptions of the present paper should be justified by that result.
By cancellable pseudo-idempotent we mean a pair $(I,\alpha)$ where $I$ is an object in ${\mathscr{C}}$ such that tensoring with $I$ from either side is an equivalence of $2$-categories, and $\alpha: I {\otimes}I {\stackrel{\raisebox{0.1ex}[0ex][0ex]{\(\sim\)}} {\raisebox{-0.15ex}[0.28ex]{\(\rightarrow\)}}}I$ is an equi-arrow (i.e. an arrow admitting a pseudo-inverse). The remarkable fact about this definition is that $\alpha$, viewed as a multiplication map, comes with canonical higher order data built in: it possesses a canonical associator ${\ensuremath{\mathsf{A}}}$ which automatically satisfies the pentagon equation. This is our Theorem \[thmA\]. The point is that the arrow $\alpha$ alone, thanks to the cancellability of $I$, induces all the usual structure of left and right constraints with all the $2$-cell data that goes into them and the axioms they must satisfy.
As a warm-up to the various constructions and ideas, we start out in Section \[sec:dim1\] by briefly running through the corresponding theory for cancellable-idempotent units in monoidal $1$-categories. This theory has been treated in detail in [@Kock:0507349].
The rest of the paper is dedicated to the case of monoidal $2$-categories. In Section \[sec:dim2-main\] we give the definitions and state the main results: Theorem \[thmA\] says that there is a canonical associator $2$-cell for $\alpha$, and that this $2$-cell automatically satisfies the pentagon equation. Theorem \[thmB\] states that unit morphisms automatically are compatible with the associators of Theorem \[thmA\]. Theorem \[thmC\] states that the $2$-category of units is contractible if non-empty. Hence, ‘being unital’ is, up to homotopy, a property rather than a structure.
Next follow three sections dedicated to proofs of each of these three theorems. In Section \[sec:left-right\] we show how the map $\alpha:II{\stackrel{\raisebox{0.1ex}[0ex][0ex]{\(\sim\)}} {\raisebox{-0.15ex}[0.28ex]{\(\rightarrow\)}}}I$ alone induces left and right constraints, which in turn are used to construct the associator and establish the pentagon equation. The left and right constraints are not canonical, but surprisingly the associator does not depend on the choice of them. In Section \[sec:catArr\] we prove Theorem \[thmB\] by interpreting it as a statement about units in the $2$-category of arrows, where it is possible to derive it from Theorem \[thmA\]. In Section \[sec:contractibility\] we prove Theorem \[thmC\]. The key ingredient is to use the left and right constraints to link up all the units, and to show that the unit morphisms are precisely those compatible with the left and right constraints; this makes them ‘essentially unique’ in the required sense.
In Section \[sec:classical\] we go through the basic theory of classical units (i.e. as extracted from the definition of tricategory [@Gordon-Power-Street]). Finally, in Section \[sec:comparison\] we show that the two notions of unit are equivalent. This is our Theorem \[thmE\]. A curiosity implied by the arguments in this section is that the left and right axioms for the $2$-cell data in the Gordon-Power-Street definition (denoted TA2 and TA3 in [@Gordon-Power-Street]) imply each other. (We have no Theorem D.)
This notion of weak units as cancellable idempotents is precisely what can be extracted from the more abstract, Tamsamani-style, theory of fair $n$-categories [@Kock:0507116] by making an arbitrary choice of a fixed weak unit. In the theory of fair categories, the key object is a contractible space of all weak units, rather than any particular point in that space, and handling this space as a whole bypasses coherence issues. However, for the sake of understanding what the theory entails, and for the sake of concrete computations, it is interesting to make a choice and study the ensuing coherence issues, as we do in this paper. The resulting approach is very much in the spirit of the classical theory of monoidal categories, bicategories, and tricategories, and provides some new insight to these theories. To stress this fact we have chosen to formulate everything from scratch in such classical terms, without reference to the theory of fair categories.
In the case of monoidal $1$-categories, the cancellable-idempotent viewpoint on units goes back to Saavedra [@Saavedra]. The importance of this viewpoint in higher categories was first suggested by Simpson [@Simpson:9810], in connection with his weak-unit conjecture. He gave an ad hoc definition in this style, as a mere indication of what needed to be done, and raised the question of whether higher homotopical data would have to be specified. The surprising answer is, at least here in dimension $3$, that specifying $\alpha$ is enough, then the higher homotopical data is automatically built in.
This paper was essentially written in 2004, in parallel with [@Joyal-Kock:traintracks]. We are ourselves to blame for the delay of getting it out of the door. The present form of the paper represents only half of what was originally planned to go into the paper. The second half should contain an analysis of strong monoidal functors (along the lines of what was meanwhile treated just in the $1$-dimensional case [@Kock:0507349]), and also a construction of the ‘universal unit’, hinted at in [@Kock:0507116]. We regret that these ambitions should hold back the present material for so long, and have finally decided to make this first part available [*as is*]{}, in the belief that it is already of some interest and can well stand alone.
[Acknowledgements.]{} We thank Georges Maltsiniotis for pointing out to us that the cancellable-idempotent notion of unit in dimension $1$ goes back to Saavedra [@Saavedra], and we thank Josep Elgueta for catching an error in an earlier version of our comparison with tricategories. The first-named author was supported by the NSERC. The second-named author was very happy to be a CIRGET postdoc at the UQAM in 2004, and currently holds support from grants MTM2006-11391 and MTM2007-63277 of Spain.
Units in monoidal categories {#sec:dim1}
============================
It is helpful first briefly to recall the relevant results for monoidal categories, referring the reader to [@Kock:0507349] for further details of this case.
[Semi-monoidal categories.]{} A [*semi-monoidal category*]{} is a category ${\mathscr{C}}$ equipped with a tensor product (which we denote by plain juxtaposition), i.e. an associative functor $$\begin{aligned}
{\mathscr{C}}\times{\mathscr{C}}& \longrightarrow & {\mathscr{C}}\\
(X,Y) & \longmapsto & XY .
\end{aligned}$$ For simplicity we assume strict associativity, $X(YZ)=(XY)Z$.
[Monoidal categories.]{} (Mac Lane [@MacLane:naturalAssociativity].) A semi-monoidal category ${\mathscr{C}}$ is a [*monoidal category*]{} when it is furthermore equipped with a distinguished object $I$ and natural isomorphisms
I X & \^[\_X]{} & X & \^[\_X]{} & X I
obeying the following rules (cf. [@MacLane:naturalAssociativity]): $$\begin{aligned}
\lambda_I &= \rho_I \\
\lambda_{XY} &= \lambda_X Y \\
\rho_{XY} &= X\rho_Y \\
X\lambda_Y &= \rho_X Y \label{Kelly} \end{aligned}$$
Naturality of $\lambda$ and $\rho$ implies $$\label{naturality1}
\lambda_{IX} = I\lambda_X , \qquad \rho_{XI} = \rho_X I ,$$ independently of Axioms (1)–(4).
Tensoring with $I$ from either side is an equivalence of categories.
[(Kelly [@Kelly:MacLanesCoherence].)]{} Axiom (4) implies axioms (1), (2), and (3).
\(4) implies (2): Since tensoring with $I$ on the left is an equivalence, it is enough to prove $I \lambda_{XY} = I\lambda_X Y$. But this follows from Axiom (4) applied twice (swap $\lambda$ out for a $\rho$ and swap back again only on the nearest factor): $$I\lambda_{XY} = \rho_I XY = I \lambda_X Y .$$ Similarly for $\rho$, establishing (3).
\(4) and (2) implies (1): Since tensoring with $I$ on the right is an equivalence, it is enough to prove $\lambda_I I = \rho_I I$. But this follows from (2), (5), and (4): $$\lambda_I I = \lambda_{II} = I \lambda_I = \rho_I I .$$
The following alternative notion of unit object goes back to Saavedra [@Saavedra]. A thorough treatment of the notion was given in [@Kock:0507349].
[Units as cancellable pseudo-idempotents.]{}\[unit1\] An object $I$ in a semi-monoidal category ${\mathscr{C}}$ is called [*cancellable*]{} if the two functors ${\mathscr{C}}\to{\mathscr{C}}$ $$\begin{aligned}
X & \longmapsto & IX\\
X & \longmapsto & XI
\end{aligned}$$ are fully faithful. By definition, a [*pseudo-idempotent*]{} is an object $I$ equipped with an isomorphism $\alpha : II {\stackrel{\raisebox{0.1ex}[0ex][0ex]{\(\sim\)}} {\raisebox{-0.15ex}[0.28ex]{\(\rightarrow\)}}}I$. Finally we define a [*unit object*]{} in ${\mathscr{C}}$ to be a cancellable pseudo-idempotent.
\[Kelly-lemma\] [[@Kock:0507349]]{} Given a unit object $(I,\alpha)$ in a semi-monoidal category ${\mathscr{C}}$, for each object $X$ there are unique arrows
I X & \^[\_X]{} & X & \^[\_X]{} & X I
such that $$\begin{aligned}
({\ensuremath{\mathsf{L}}}) & & I\lambda_X = \alpha X \\
({\ensuremath{\mathsf{R}}}) & & \rho_X I = X \alpha .
\end{aligned}$$ The $\lambda_X$ and $\rho_X$ are isomorphisms and natural in $X$.
Let $\mathbb{L}: {\mathscr{C}}\to {\mathscr{C}}$ denote the functor defined by tensoring with $I$ on the left. Since $\mathbb{L}$ is fully faithful, we have a bijection $${\operatorname{Hom}}(IX,X) \to {\operatorname{Hom}}(IIX,IX).$$ Now take $\lambda_X$ to be the inverse image of $\alpha X$; it is an isomorphism since $\alpha X$ is. Naturality follows by considering more generally the bijection $${\operatorname{Nat}}(\mathbb{L},{\operatorname{id}}_{\mathscr{C}}) \to {\operatorname{Nat}}(\mathbb{L}\circ \mathbb{L}, \mathbb{L})
;$$ let $\lambda$ be the inverse image of the natural transformation whose components are $\alpha X$. Similarly on the right.
\[Kelly-dim1\] [[@Kock:0507349]]{} For $\lambda$ and $\rho$ as above, the Kelly axiom holds: $$X \lambda_Y = \rho_X Y .$$ Therefore, by Lemma \[Kelly-lemma\] a semi-monoidal category with a unit object is a monoidal category in the classical sense.
In the commutative square
XIIY & \^[XI\_Y]{} & XIY\
\
XIY & \_[X\_Y]{} & XY
the top arrow is equal to $X \alpha Y$, by $X$ tensor ([$\mathsf{L}$]{}), and the left-hand arrow is also equal to $X \alpha Y$, by ([$\mathsf{R}$]{}) tensor $Y$. Since $X\alpha Y$ is an isomorphism, it follows that $X \lambda_Y = \rho_X Y$.
\[ass+eq\] For a unit object $(I,\alpha)$ we have: (i) The map $\alpha:II\to I$ is associative. (ii) The two functors $X\mapsto IX$ and $X\mapsto XI$ are equivalences.
Since $\alpha$ is invertible, associativity amounts to the equation $I \alpha
= \alpha I$, which follows from the previous proof by setting $X=Y=I$ and applying ${\ensuremath{\mathsf{L}}}$ and ${\ensuremath{\mathsf{R}}}$ once again. To see that $\mathbb{L}$ is an equivalence, just note that it is isomorphic to the identity via $\lambda$.
[Uniqueness of units.]{} Just as in a semi-monoid a unit element is unique if it exists, one can show [@Kock:0507349] that in a semi-monoidal category, any two units are uniquely isomorphic. This statement does not involve $\lambda$ and $\rho$, but the proof does: the canonical isomorphism $I {\stackrel{\raisebox{0.1ex}[0ex][0ex]{\(\sim\)}} {\raisebox{-0.15ex}[0.28ex]{\(\rightarrow\)}}}J$ is the composite $I \stackrel{\rho_I{^{\scriptscriptstyle -1}}}{\rTo} IJ \stackrel{\lambda_J}{\rTo} J$.
Units in monoidal $2$-categories: definition and main results {#sec:dim2-main}
=============================================================
In this section we set up the necessary terminology and notation, give the main definition, and state the main results.
[$2$-categories.]{} We work in a strict $2$-category ${\mathscr{C}}$. We use the symbol $\#$ to denote composition of arrows and horizontal composition of $2$-cells in ${\mathscr{C}}$, always written from the left to the right, and occasionally decorating the symbol $\#$ by the name of the object where the two arrows or $2$-cells are composed. By an [*equi-arrow*]{} in ${\mathscr{C}}$ we understand an arrow $f$ admitting an (unspecified) pseudo-inverse, i.e. an arrow $g$ in the opposite direction such that $f \# g$ and $g \# f$ are isomorphic to the respective identity arrows, and such that the comparison $2$-cells satisfy the usual triangle equations for adjunctions.
We shall make extensive use of arguments with pasting diagrams [@Kelly-Street:2cat]. Our drawings of $2$-cells should be read from bottom to top, so that for example
X && \^h && Z\
&\_f & & \_g &\
& & Y & &
denotes ${\ensuremath{\mathsf{U}}} : f \,\underset Y \#\, g \Rightarrow h$. The symbol ${\copyright}$ will denote identity $2$-cells.
The few $2$-functors we need all happen to be strict. By [*natural transformation*]{} we always mean pseudo-natural transformation. Hence a natural transformation $u: F \Rightarrow G$ between two $2$-functors from ${\mathscr{D}}$ to ${\mathscr{C}}$ is given by an arrow $u_X: FX\to GX$ for each object $X\in {\mathscr{D}}$, and an invertible $2$-cell
FX & \^[u\_X]{} & GX\
\
FX’ & \_[u\_[X’]{}]{} & GX’
for each arrow $x: X\to X'$ in ${\mathscr{D}}$, subject to the usual compatibility conditions [@Kelly-Street:2cat]. The modifications we shall need will happen to be invertible.
[Semi-monoidal $2$-categories.]{}\[semimonoidal\] By [*semi-monoidal $2$-category*]{} we mean a $2$-category ${\mathscr{C}}$ equipped with a tensor product, i.e. an associative $2$-functor $$\begin{aligned}
{\otimes}: {\mathscr{C}}\times{\mathscr{C}}& \longrightarrow & {\mathscr{C}}\\
(X,Y) & \longmapsto & XY ,\end{aligned}$$ denoted by plain juxtaposition. We already assumed ${\mathscr{C}}$ to be a strict $2$-category, and we also require ${\otimes}$ to be a strict $2$-functor and to be strictly associative: $(XY)Z=X(YZ)$. This is mainly for convenience, to keep the focus on unit issues.
Note that the tensor product of two equi-arrows is again an equi-arrow, since its pseudo-inverse can be taken to be the tensor product of the pseudo-inverses.
[Semi-monoids.]{} A [*semi-monoid*]{} in ${\mathscr{C}}$ is a triple $(X,\alpha,{\text{\rm \textsf{\AA}}})$ consisting of an object $X$, a multiplication map $\alpha: X X \to X$, and an invertible $2$-cell ${\text{\rm \textsf{\AA}}}$ called the [*associator*]{}, $$\begin{diagram}[w=6ex,h=4.5ex,tight]
XXX & \rTo^{\alpha X} & X X \\
\dTo<{X\alpha} & {\scriptstyle {\text{\rm \textsf{\AA}}}} & \dTo>\alpha \\
X X & \rTo_{\alpha} & X
\end{diagram}$$ required to satisfy the ‘pentagon equation’: $$\begin{diagram}[w=40pt,h=30pt,tight,scriptlabels,objectstyle=\scriptstyle,hug]
XXXX&\rTo^{\alpha XX}&XXX&&&&\\
\dTo<{XX\alpha}&\rdTo>{X\alpha X}&\psr{{\text{\rm \textsf{\AA}}}X}&\rdTo>{\alpha X}&&&\\
XXX&\psr{X{\text{\rm \textsf{\AA}}}}&XXX&\rTo^{\alpha X}&XX\\
&\rdTo_{X\alpha}&\dTo<{X\alpha}&{\text{\rm \textsf{\AA}}}&\dTo>{\alpha}\\
&&XX&\rTo_{\alpha}&X
\end{diagram}
\qquad = \qquad
\begin{diagram}[w=40pt,h=30pt,tight,scriptlabels,objectstyle=\scriptstyle,hug]
XXXX&\rTo^{\alpha XX}&XXX&&&&\\
\dTo<{XX\alpha}&{\copyright}&\dTo>{X\alpha}&\rdTo>{\alpha X}&&&\\
XXX&\rTo_{\alpha X}&XX&\psr{{\text{\rm \textsf{\AA}}}}&XX\\
&\rdTo_{X\alpha}&\psr{{\text{\rm \textsf{\AA}}}}&\rdTo<{\alpha}&\dTo>{\alpha}\\
&&XX&\rTo_{\alpha}&X
\end{diagram}$$ In the applications, $\alpha$ will be an equi-arrow, and hence we will have $${\text{\rm \textsf{\AA}}}= {\ensuremath{\mathsf{A}}} \underset{XX}{\#} \alpha$$ for a some unique invertible $${\ensuremath{\mathsf{A}}}: X\alpha \Rightarrow \alpha X,$$ which it will more convenient to work with. In this case, the pentagon equation is equivalent to the more compact equation $$\label{shortApentagon}
\begin{diagram}[w=6ex,h=6ex,tight]
XXXX&\rTo^{\alpha XX}&XXX\\
{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(-9,-25)(-23,0)(-9,25)
\put(-20,0){\makebox(0,0)[r]{$\scriptstyle {XX\alpha}$}}
\put(-8.7,-25.6){\vector(1,-2){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${X{\ensuremath{\mathsf{A}}}}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(9,-25)(23,0)(9,25)
\put(20,0){\makebox(0,0)[l]{$\scriptstyle {X\alpha X}$}}
\put(8.7,-25.6){\vector(-1,-2){0}}
\end{picture}
}&{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put({20},0){\makebox(0,0)[b]{${{\text{\rm \textsf{\AA}}}X}$}}
\end{picture}
}&{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(9,-25)(23,0)(9,25)
\put(20,0){\makebox(0,0)[l]{$\scriptstyle {\alpha X}$}}
\put(8.7,-25.6){\vector(-1,-2){0}}
\end{picture}
}\\
XXX&\rTo_{\alpha X}&XX
\end{diagram}
\qquad\qquad = \qquad\qquad
\begin{diagram}[w=6ex,h=6ex,tight]
XXXX&\rTo^{\alpha XX}&XXX\\
{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(-9,-25)(-23,0)(-9,25)
\put(-20,0){\makebox(0,0)[r]{$\scriptstyle {XX\alpha}$}}
\put(-8.7,-25.6){\vector(1,-2){0}}
\end{picture}
}&{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put({-20},0){\makebox(0,0)[b]{${{\copyright}}$}}
\end{picture}
}&{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(-9,-25)(-23,0)(-9,25)
\put(-20,0){\makebox(0,0)[r]{$\scriptstyle {X\alpha}$}}
\put(-8.7,-25.6){\vector(1,-2){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{A}}}}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(9,-25)(23,0)(9,25)
\put(20,0){\makebox(0,0)[l]{$\scriptstyle {\alpha X}$}}
\put(8.7,-25.6){\vector(-1,-2){0}}
\end{picture}
}\\
XXX&\rTo_{\alpha X}&XX
\end{diagram}$$ which we shall also make use of.
[Semi-monoid maps.]{}\[semimonoid-map\] A [*semi-monoid map*]{} $f: (X,\alpha,{\text{\rm \textsf{\AA}}}) \to (Y,\beta,{\ensuremath{\mathsf{B}}})$ is the data of an arrow $f:X\to Y$ in ${\mathscr{C}}$ together with an invertible $2$-cell
X X & \^[f f]{} & Y Y\
\
X & \_f & Y
such that this cube commutes: $$\def\sdf#1{\rotstart{-36.87 rotate}\hbox to0pt {\vsize 0pt \hss\(\scriptstyle #1\)\hss}\rotfinish}
\begin{diagram}[w=40pt,h=30pt,tight,scriptlabels,objectstyle=\scriptstyle,hug]
&&YYY&\rTo^{\beta Y}& YY \\
& \ruTo^{fff} &\sdf{{\ensuremath{\mathsf{F}}}f}& \ruTo^{ff} & \dTo>\beta \\
XXX&\rTo^{\alpha X}&XX&\sdf{{\ensuremath{\mathsf{F}}}}&Y\\
\dTo<{X\alpha}&{\text{\rm \textsf{\AA}}}&\dTo>{\alpha}&\ruTo_f&\\
XX&\rTo_{\alpha}&X&&
\end{diagram}
\qquad = \qquad
\begin{diagram}[w=40pt,h=30pt,tight,scriptlabels,objectstyle=\scriptstyle,hug]
&&YYY&\rTo^{\beta Y}&YY\\
& \ruTo^{fff}&\dTo<{Y \beta}&{\ensuremath{\mathsf{B}}}&\dTo>{\beta}\\
XXX&\sdf{f{\ensuremath{\mathsf{F}}}}&YY&\rTo_{\beta}&Y \\
\dTo<{X\alpha}&\ruTo_{ff}&\sdf{{\ensuremath{\mathsf{F}}}}&\ruTo_f&\\
XX&\rTo_{\alpha}&X&&
\end{diagram}$$ When $\beta$ is an equi-arrow, the cube equation is equivalent to the simpler equation: $$\label{shortSemimonoidMap}
\begin{diagram}[w=6ex,h=6ex,tight]
XXX&\rTo^{fff}&YYY\\
{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(-9,-25)(-23,0)(-9,25)
\put(-20,0){\makebox(0,0)[r]{$\scriptstyle {X\alpha}$}}
\put(-8.7,-25.6){\vector(1,-2){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{A}}}}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(9,-25)(23,0)(9,25)
\put(20,0){\makebox(0,0)[l]{$\scriptstyle {\alpha X}$}}
\put(8.7,-25.6){\vector(-1,-2){0}}
\end{picture}
}&{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put({20},0){\makebox(0,0)[b]{${{\ensuremath{\mathsf{F}}}f}$}}
\end{picture}
}&{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(9,-25)(23,0)(9,25)
\put(20,0){\makebox(0,0)[l]{$\scriptstyle {\beta Y}$}}
\put(8.7,-25.6){\vector(-1,-2){0}}
\end{picture}
}\\
XX&\rTo_{ff}&YY
\end{diagram}
\qquad\qquad = \qquad\qquad
\begin{diagram}[w=6ex,h=6ex,tight]
XXX&\rTo^{fff}&YYY\\
{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(-9,-25)(-23,0)(-9,25)
\put(-20,0){\makebox(0,0)[r]{$\scriptstyle {X\alpha}$}}
\put(-8.7,-25.6){\vector(1,-2){0}}
\end{picture}
}&{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put({-20},0){\makebox(0,0)[b]{${f{\ensuremath{\mathsf{F}}}}$}}
\end{picture}
}&{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(-9,-25)(-23,0)(-9,25)
\put(-20,0){\makebox(0,0)[r]{$\scriptstyle {Y\beta}$}}
\put(-8.7,-25.6){\vector(1,-2){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{B}}}}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(9,-25)(23,0)(9,25)
\put(20,0){\makebox(0,0)[l]{$\scriptstyle {\beta Y}$}}
\put(8.7,-25.6){\vector(-1,-2){0}}
\end{picture}
}\\
XX&\rTo_{ff}&YY
\end{diagram}$$ which will be useful.
[Semi-monoid transformations.]{}\[semimonoid-transf\] A [*semi-monoid transformation*]{} between two parallel semi-monoid maps $(f,{\ensuremath{\mathsf{F}}})$ and $(g,{\ensuremath{\mathsf{G}}})$ is a $2$-cell ${\ensuremath{\mathsf{T}}}: f \Rightarrow g$ in ${\mathscr{C}}$ such that this cylinder commutes: $$\begin{diagram}[w=6ex,h=6ex,tight]
XX&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,25)(28,11)
\put(0,21){\makebox(0,0)[b]{$\scriptstyle {gg}$}}
\put(28.6,10.7){\vector(2,-1){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{TT}}}}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-18)(28,-4)
\put(0,-14){\makebox(0,0)[t]{$\scriptstyle {ff}$}}
\put(28.6,-3.7){\vector(2,1){0}}
\end{picture}
}&YY\\
\dTo<{\alpha}&{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{-12}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{F}}}}$}}
\end{picture}
}&\dTo>{\beta}\\
X&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-18)(28,-4)
\put(0,-14){\makebox(0,0)[t]{$\scriptstyle {f}$}}
\put(28.6,-3.7){\vector(2,1){0}}
\end{picture}
}&Y
\end{diagram}
\qquad = \qquad
\begin{diagram}[w=6ex,h=6ex,tight]
XX&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,25)(28,11)
\put(0,21){\makebox(0,0)[b]{$\scriptstyle {gg}$}}
\put(28.6,10.7){\vector(2,-1){0}}
\end{picture}
}&YY\\
\dTo<{\alpha}&{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{12}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{G}}}}$}}
\end{picture}
}&\dTo>{\beta}\\
X&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,25)(28,11)
\put(0,21){\makebox(0,0)[b]{$\scriptstyle {g}$}}
\put(28.6,10.7){\vector(2,-1){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{T}}}}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-18)(28,-4)
\put(0,-14){\makebox(0,0)[t]{$\scriptstyle {f}$}}
\put(28.6,-3.7){\vector(2,1){0}}
\end{picture}
}&Y
\end{diagram}
\vspace{24pt}$$
\[inv\] Let $f:X \to Y$ be a semi-monoid map. If $f$ is an equi-arrow (as an arrow in ${\mathscr{C}}$) with quasi-inverse $g:Y\to X$, then there is a canonical $2$-cell ${\ensuremath{\mathsf{G}}}$ such that $(g,{\ensuremath{\mathsf{G}}})$ is a semi-monoid map.
The $2$-cell ${\ensuremath{\mathsf{G}}}$ is defined as the mate [@Kelly-Street:2cat] of the $2$-cell ${\ensuremath{\mathsf{F}}}{^{\scriptscriptstyle -1}}$. It is routine to check the cube equation in \[semimonoid-map\].
[Pseudo-idempotents.]{}\[idempotent\] A [*pseudo-idempotent*]{} is a pair $(I,\alpha)$ where $\alpha:II\to I$ is an equi-arrow. A [*morphism of pseudo-idempotents*]{} from $(I,\alpha)$ to $(J,\beta)$ is a pair $(u,{\ensuremath{\mathsf{U}}})$ consisting of an arrow $u:I\to J$ in ${\mathscr{C}}$ and an invertible $2$-cell
II & \^[uu]{} & JJ\
\
I & \_u & J .
If $(u,{\ensuremath{\mathsf{U}}})$ and $(v,{\ensuremath{\mathsf{V}}})$ are morphisms of pseudo-idempotents from $(I,\alpha)$ to $(J,\beta)$, a [*$2$-morphism of pseudo-idempotents*]{} from $(u,{\ensuremath{\mathsf{U}}})$ to $(v,{\ensuremath{\mathsf{V}}})$ is a $2$-cell ${\ensuremath{\mathsf{T}}}: u \Rightarrow v$ satisfying the cylinder equation of \[semimonoid-transf\].
[Cancellable objects.]{}\[cancellable\] An object $I$ in ${\mathscr{C}}$ is called [*cancellable*]{} if the two $2$-functors ${\mathscr{C}}\to{\mathscr{C}}$ $$\begin{aligned}
X & \longmapsto & IX\\
X & \longmapsto & XI
\end{aligned}$$ are fully faithful. (Fully faithful means that the induced functors on hom categories are equivalences.) A [*cancellable morphism*]{} between cancellable objects $I$ and $J$ is an equi-arrow $u:I \to J$. (Equivalently it is an arrow such that the functors on hom cats defined by tensoring with $u$ on either side are equivalences, cf. \[unitmap\].) A [*cancellable $2$-morphism*]{} between cancellable arrows is any invertible $2$-cell.
We are now ready for the main definition and the main results.
[Units.]{}\[main-def\] A [*unit object*]{} is by definition a cancellable pseudo-idempotent. Hence it is a pair $(I,\alpha)$ consisting of an object $I$ and an equi-arrow $\alpha : II \to I$, with the property that tensoring with $I$ from either side define fully faithful $2$-functors ${\mathscr{C}}\to{\mathscr{C}}$.
A [*morphism*]{} of units is a cancellable morphism of pseudo-idempotents. In other words, a unit morphism from $(I,\alpha)$ to $(J,\beta)$ is a pair $(u,{\ensuremath{\mathsf{U}}})$ where $u: I \to J$ is an equi-arrow and ${\ensuremath{\mathsf{U}}}$ is an invertible $2$-cell
II & \^[uu]{} & JJ\
\
I & \_u & J .
A [*$2$-morphism*]{} of units is a cancellable $2$-morphism of pseudo-idempotents. Hence a $2$-morphism from $(u,{\ensuremath{\mathsf{U}}})$ to $(v,{\ensuremath{\mathsf{V}}})$ is a $2$-cell ${\ensuremath{\mathsf{T}}}: u \Rightarrow v$ such that $$\begin{diagram}[w=6ex,h=6ex,tight]
II&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,25)(28,11)
\put(0,21){\makebox(0,0)[b]{$\scriptstyle {vv}$}}
\put(28.6,10.7){\vector(2,-1){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{TT}}}}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-18)(28,-4)
\put(0,-14){\makebox(0,0)[t]{$\scriptstyle {uu}$}}
\put(28.6,-3.7){\vector(2,1){0}}
\end{picture}
}&JJ\\
\dTo<{\alpha}&{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{-12}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{U}}}}$}}
\end{picture}
}&\dTo>{\beta}\\
I&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-18)(28,-4)
\put(0,-14){\makebox(0,0)[t]{$\scriptstyle {u}$}}
\put(28.6,-3.7){\vector(2,1){0}}
\end{picture}
}&J
\end{diagram}
\qquad = \qquad
\begin{diagram}[w=6ex,h=6ex,tight]
II&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,25)(28,11)
\put(0,21){\makebox(0,0)[b]{$\scriptstyle {vv}$}}
\put(28.6,10.7){\vector(2,-1){0}}
\end{picture}
}&JJ\\
\dTo<{\alpha}&{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{12}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{V}}}}$}}
\end{picture}
}&\dTo>{\beta}\\
I&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,25)(28,11)
\put(0,21){\makebox(0,0)[b]{$\scriptstyle {v}$}}
\put(28.6,10.7){\vector(2,-1){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{T}}}}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-18)(28,-4)
\put(0,-14){\makebox(0,0)[t]{$\scriptstyle {u}$}}
\put(28.6,-3.7){\vector(2,1){0}}
\end{picture}
}&J
\end{diagram}
\vspace{24pt}$$
This defines the [*$2$-category of units*]{}.
In the next section we’ll see how the notion of unit object induces left and right constraints familiar from standard notions of monoidal $2$-category. It will then turn out (Lemmas \[unitmap\] and \[unit2map\]) that unit morphisms and $2$-morphisms can be characterised as those morphisms and $2$-morphisms compatible with the left and right constraints.
[Theorem \[thmA\] (Associativity).]{}
*Given a unit object $(I,\alpha)$, there is a canonical invertible $2$-cell*
III & \^[I]{} & II\
\
II & \_& I
which satisfies the pentagon equation
$$\label{pentaA}
\begin{diagram}[w=40pt,h=30pt,tight,scriptlabels,objectstyle=\scriptstyle,hug]
IIII&\rTo^{\alpha II}&III&&&&\\
\dTo<{II\alpha}&\rdTo>{I\alpha I}&\psr{{\text{\rm \textsf{\AA}}}I}&\rdTo>{\alpha I}&&&\\
III&\psr{I{\text{\rm \textsf{\AA}}}}&III&\rTo^{\alpha I}&II\\
&\rdTo_{I\alpha}&\dTo<{I\alpha}&{\text{\rm \textsf{\AA}}}&\dTo>{\alpha}\\
&&II&\rTo_{\alpha}&I
\end{diagram}
\qquad = \qquad
\begin{diagram}[w=40pt,h=30pt,tight,scriptlabels,objectstyle=\scriptstyle,hug]
IIII&\rTo^{\alpha II}&III&&&&\\
\dTo<{II\alpha}&{\copyright}&\dTo>{I\alpha}&\rdTo>{\alpha I}&&&\\
III&\rTo_{\alpha I}&II&\psr{{\text{\rm \textsf{\AA}}}}&II\\
&\rdTo_{I\alpha}&\psr{{\text{\rm \textsf{\AA}}}}&\rdTo<{\alpha}&\dTo>{\alpha}\\
&&II&\rTo_{\alpha}&I
\end{diagram}$$
In other words, a unit object is automatically a semi-monoid. The $2$-cell ${\ensuremath{\mathsf{A}}}$ is characterised uniquely in \[uniqueA\].
[Theorem \[thmB\].]{} [ *A unit morphism $(u,{\ensuremath{\mathsf{U}}}): (I,\alpha) \to (J,\beta)$ is automatically a semi-monoid map, when $I$ and $J$ are considered semi-monoids in virtue of Theorem A.*]{}
[Theorem \[thmC\] (Contractibility).]{} [ *The $2$-category of units in ${\mathscr{C}}$ is contractible, if non-empty.* ]{}
In other words, between any two units there exists a unit morphism, and between any two parallel unit morphisms there is a unique unit $2$-morphism. Theorem C shows that units objects are unique up to homotopy, so in this sense ‘being unital’ is a property not a structure.
The proofs of these three theorems rely on the auxiliary structure of left and right constraints which we develop in the next section, and which also displays the relation with the classical notion of monoidal $2$-category: in Section \[sec:comparison\] we show that the cancellable-idempotent notion of unit is equivalent to the notion extracted from the definition of tricategory of Gordon, Power, and Street [@Gordon-Power-Street]. This is our Theorem \[thmE\].
Left and right actions, and associativity of the unit (Theorem \[thmA\]) {#sec:left-right}
========================================================================
Throughout this section we fix a unit object $(I,\alpha)$.
\[LR\] For each object $X$ there exists pairs $(\lambda_X, {\ensuremath{\mathsf{L}}}_X)$ and $(\rho_X, {\ensuremath{\mathsf{R}}}_X)$, $$\begin{aligned}
\lambda_X : IX \to X, & &
{\ensuremath{\mathsf{L}}}_X : I\lambda_X \Rightarrow \alpha X \\
\rho_X : XI \to X, & &
{\ensuremath{\mathsf{R}}}_X : X \alpha \Rightarrow \rho_X I
\end{aligned}$$ where $\lambda_X$ and $\rho_X$ are equi-arrows, and ${\ensuremath{\mathsf{L}}}_X$ are ${\ensuremath{\mathsf{R}}}_X$ are invertible $2$-cells.
For every such family, there is a unique way to assemble the $\lambda_X$ into a natural transformation (this involves defining $2$-cells $\lambda_f$ for every arrow $f$ in ${\mathscr{C}}$) in such a way that ${\ensuremath{\mathsf{L}}}$ is a natural modification. Similarly for the $\rho_X$ and ${\ensuremath{\mathsf{R}}}_X$.
The $\lambda_X$ is an action of $I$ on each $X$, and the $2$-cell ${\ensuremath{\mathsf{L}}}_X$ expresses an associativity constraint on this action. Using these structures we will construct the associator for $\alpha$, and show it satisfies the pentagon equation. Once that is in place we will see that the actions $\lambda$ and $\rho$ are coherent too (satisfying the appropriate pentagon equations).
We shall treat the left action. The right action is of course equivalent to establish.
[Construction of the left action.]{} Since tensoring with $I$ is a fully faithful $2$-functor, the functor $${\operatorname{Hom}}(IX,X) \to {\operatorname{Hom}}(IIX,IX)$$ is an equivalence of categories. In the second category there is the canonical object $\alpha X$. Hence there is a pseudo pre-image which we denote $\lambda_X : I X \to X$, together with an invertible $2$-cell ${\ensuremath{\mathsf{L}}}_X : I \lambda_X \Rightarrow \alpha X$:
IIX&
by
(0,0)(0,0) (-28,11)(0,25)(28,11) (0,21)[(0,0)\[b\][$\scriptstyle {\alpha X}$]{}]{} (28.6,10.7)[(2,-1)[0]{}]{}
(0,0)(0,0) (0,[-2]{})[(0,0)\[b\][${{\ensuremath{\mathsf{L}}}_X}$]{}]{}
by
(0,0)(0,0) (-28,-4)(0,-18)(28,-4) (0,-14)[(0,0)\[t\][$\scriptstyle {I\lambda_X}$]{}]{} (28.6,-3.7)[(2,1)[0]{}]{}
&IX
Since $\alpha$ is an equi-arrow, also $\alpha X$ is equi, and since ${\ensuremath{\mathsf{L}}}_X$ is invertible, we conclude that also $I
\lambda_X$ is an equi-arrow. Finally since the $2$-functor ‘tensoring with $I$’ is fully faithful, it reflects equi-arrows, so already $\lambda_X$ is an equi-arrow.
[Naturality.]{}\[naturality2\] A slight variation in the formulation of the construction gives directly a natural transformation $\lambda$ and a modification ${\ensuremath{\mathsf{L}}}$: Let $\mathbb{L}:
{\mathscr{C}}\to{\mathscr{C}}$ denote the $2$-functor ‘tensoring with $I$ on the left’. Since $\mathbb{L}$ is fully faithful, there is an equivalence of categories $${\operatorname{Nat}}(\mathbb{L},{\operatorname{Id}}_{\mathscr{C}}) \to {\operatorname{Nat}}(\mathbb{L}\circ \mathbb{L}, \mathbb{L}) .$$ Now in the second category we have the canonical natural transformation whose $X$-component is $\alpha X$ (and with trivial components on arrows). Hence there is a pseudo pre-image natural transformation $\lambda : \mathbb{L} \to
{\operatorname{id}}_{\mathscr{C}}$, together with a modification ${\ensuremath{\mathsf{L}}}$ whose $X$-component is ${\ensuremath{\mathsf{L}}}_X : I \lambda_X \Rightarrow \alpha X$.
However, we wish to stress the fact that the construction is completely object-wise. This fact is of course due to the presence of the isomorphism ${\ensuremath{\mathsf{L}}}_X$: something isomorphic to a natural transformation is again natural. More precisely, to provide the $2$-cell data $\lambda_f$ needed to make $\lambda$ into a natural transformation, just pull back the $2$-cell data from the natural transformation $\alpha X$. In detail, we need invertible $2$-cells
I X & \^[\_X]{} & X\
f\
I Y & \_[\_Y]{} & Y
To say that the ${\ensuremath{\mathsf{L}}}_X$ constitute a modification (from $\lambda$ to the identity) is to have this compatibility for every arrow $f:X\to Y$: $$\begin{diagram}[w=6ex,h=6ex,tight]
IIX&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,25)(28,11)
\put(0,21){\makebox(0,0)[b]{$\scriptstyle {\alpha X}$}}
\put(28.6,10.7){\vector(2,-1){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{L}}}_X}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-18)(28,-4)
\put(0,-14){\makebox(0,0)[t]{$\scriptstyle {I\lambda_X}$}}
\put(28.6,-3.7){\vector(2,1){0}}
\end{picture}
}&IX\\
\dTo<{IIf}&{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{-12}){\makebox(0,0)[b]{${I\lambda_f}$}}
\end{picture}
}&\dTo>{If}\\
IIY&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-18)(28,-4)
\put(0,-14){\makebox(0,0)[t]{$\scriptstyle {I\lambda_Y}$}}
\put(28.6,-3.7){\vector(2,1){0}}
\end{picture}
}&IY
\end{diagram}
\qquad = \qquad
\begin{diagram}[w=6ex,h=6ex,tight]
IIX&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,25)(28,11)
\put(0,21){\makebox(0,0)[b]{$\scriptstyle {\alpha X}$}}
\put(28.6,10.7){\vector(2,-1){0}}
\end{picture}
}&IX\\
\dTo<{IIf}&{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{12}){\makebox(0,0)[b]{${{\copyright}}$}}
\end{picture}
}&\dTo>{If}\\
IIY&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,25)(28,11)
\put(0,21){\makebox(0,0)[b]{$\scriptstyle {\alpha Y}$}}
\put(28.6,10.7){\vector(2,-1){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{L}}}_Y}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-18)(28,-4)
\put(0,-14){\makebox(0,0)[t]{$\scriptstyle {I\lambda_Y}$}}
\put(28.6,-3.7){\vector(2,1){0}}
\end{picture}
}&IY
\end{diagram}
\vspace{24pt}$$ (Here the commutative cell is actually the $2$-cell part of the natural transformation $\alpha X$.) Now the point is that each $2$-cell $\lambda_f$ is uniquely defined by this compatibility: indeed, since the other three $2$-cells in the diagram are invertible, there is a unique $2$-cell that can fill the place of $I\lambda_f$, and since $I$ is cancellable this $2$-cell comes from a unique $2$-cell $\lambda_f$. The required compatibilities of $\lambda_f$ with composition, identities, and $2$-cells now follows from its construction: $\lambda_f$ is just the translation via ${\ensuremath{\mathsf{L}}}$ of the identity $2$-cell $\alpha X$.
[Uniqueness of the left constraints.]{}\[lambda-lambda\] There may be many choices for $\lambda_X$, and even for a fixed $\lambda_X$, there may be many choices for ${\ensuremath{\mathsf{L}}}_X$. However, between any two pairs $(\lambda_X,{\ensuremath{\mathsf{L}}}_X)$ and $(\lambda'_X,{\ensuremath{\mathsf{L}}}'_X)$ there is a unique invertible $2$-cell ${\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{left}}}}_X:
\lambda_X \Rightarrow \lambda'_X$ such that this compatibility holds: $$\xymatrixrowsep{50pt}
\xymatrixcolsep{42pt}
\xymatrix @!=0pt {
I\lambda_X \ar@{=>}[rr]^{I {\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{left}}}}_X} \ar@{=>}[dr]_{{\ensuremath{\mathsf{L}}}_X} && I \lambda'_X \ar@{=>}[dl]^{{\ensuremath{\mathsf{L}}}'_X} \\
& \alpha X &
}$$ Indeed, this diagram defines uniquely an invertible $2$-cell $I\lambda_X
\Rightarrow I \lambda_X'$, and since $I$ is cancellable, this $2$-cell comes from a unique $2$-cell $\lambda_X \Rightarrow
\lambda_X'$ which we then call ${\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{left}}}}_X$.
There is of course a completely analogous statement for right constraints.
[Construction of the associator.]{} We define ${\ensuremath{\mathsf{A}}} : I \alpha \Rightarrow \alpha I$ as the unique $2$-cell satisfying the equation $$\label{A}
\begin{diagram}[w=56pt,h=42pt,tight]
IIII & {\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-26,6)(0,14)(26,6)
\put(0,12){\makebox(0,0)[b]{$\scriptstyle {I\alpha I}$}}
\put(26.6,5.7){\vector(3,-1){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{R}}}_I{^{\scriptscriptstyle -1}}I}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-26,-1)(0,-9)(26,-1)
\put(0,-7){\makebox(0,0)[t]{$\scriptstyle {\rho II}$}}
\put(26.6,-0.7){\vector(3,1){0}}
\end{picture}
} & III \\
{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(-3,-25)(-16,0)(-3,25)
\put(-13,0){\makebox(0,0)[r]{$\scriptstyle {I\alpha I}$}}
\put(-2.5,-25.6){\vector(1,-2){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${I{\ensuremath{\mathsf{L}}}_I{^{\scriptscriptstyle -1}}}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(3,-25)(16,0)(3,25)
\put(13,0){\makebox(0,0)[l]{$\scriptstyle {II\lambda}$}}
\put(2.7,-25.6){\vector(-1,-2){0}}
\end{picture}
} &
{\copyright}& {\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(-3,-25)(-16,0)(-3,25)
\put(-13,0){\makebox(0,0)[r]{$\scriptstyle {I\lambda}$}}
\put(-2.5,-25.6){\vector(1,-2){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{L}}}_I}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(3,-25)(16,0)(3,25)
\put(13,0){\makebox(0,0)[l]{$\scriptstyle {\alpha I}$}}
\put(2.7,-25.6){\vector(-1,-2){0}}
\end{picture}
} \\
III & {\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-26,6)(0,14)(26,6)
\put(0,12){\makebox(0,0)[b]{$\scriptstyle {\rho I}$}}
\put(26.6,5.7){\vector(3,-1){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{R}}}_I}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-26,-1)(0,-9)(26,-1)
\put(0,-7){\makebox(0,0)[t]{$\scriptstyle {I\alpha}$}}
\put(26.6,-0.7){\vector(3,1){0}}
\end{picture}
} & II
\end{diagram}
\qquad\qquad = \qquad
\begin{diagram}[w=30pt,h=22.5pt,tight,hug]
IIII &&&& \\
&\rdTo^{I\alpha I} &&&& \\
&& III && \\
&&&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\psr{\begin{picture}(0,0)(0,0)
\qbezier(-34,6)(0,16)(30,6)
\put(0,13){\makebox(0,0)[b]{$\scriptstyle {\alpha I}$}}
\put(30.6,5.7){\vector(3,-1){0}}
\end{picture}}
}
{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\psr{\begin{picture}(0,0)(0,0)
\qbezier(-34,1)(0,-8)(30,1)
\put(0,-6){\makebox(0,0)[t]{$\scriptstyle {I\alpha}$}}
\put(30.6,1.3){\vector(3,1){0}}
\end{picture}}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\psr{\begin{picture}(0,0)(0,0)
\put(2,8){\makebox(0,0)[t]{${\ensuremath{\mathsf{A}}}$}}
\end{picture}}
} & \\
&&&& II
\end{diagram}
\vspace{24pt}$$ This definition is meaningful: since $I\alpha I$ is an equi-arrow, pre-composing with $I\alpha I$ is a $2$-equivalence, hence gives a bijection on the level of $2$-cells, so ${\ensuremath{\mathsf{A}}}$ is determined by the left-hand side of the equation. Note that ${\ensuremath{\mathsf{A}}}$ is invertible since all the $2$-cells in the construction are.
The associator ${\text{\rm \textsf{\AA}}}$ is defined as ${\ensuremath{\mathsf{A}}} \text{-followed-by-}\alpha$: $${\text{\rm \textsf{\AA}}}\ {\: {\raisebox{0.255ex}{\normalfont\scriptsize :\!\!}}=}\ {\ensuremath{\mathsf{A}}} \ \underset{II}{\#} \ \alpha ,$$ but it will be more convenient to work with ${\ensuremath{\mathsf{A}}}$.
\[independence\] The definition of ${\ensuremath{\mathsf{A}}}$ does not depend on the choices of left constraint $(\lambda,{\ensuremath{\mathsf{L}}})$ and right constraint $(\rho,{\ensuremath{\mathsf{R}}})$.
Write down the left-hand side of in terms of different left and right constraints. Express these cells in terms of the original ${\ensuremath{\mathsf{L}}}_I$ and ${\ensuremath{\mathsf{R}}}_I$, using the comparison $2$-cells ${\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{left}}}}_I$ and ${\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{right}}}}_I$ of \[lambda-lambda\]. Finally observe that these comparison cells can be moved across the commutative square to cancel each other pairwise.
[Uniqueness of ${\ensuremath{\mathsf{A}}}$.]{}\[uniqueA\] Equation may not appear familiar, but it is equivalent to the following ‘pentagon’ equation (after post-whiskering with $\alpha$): $$\label{penta-LRK}
\begin{diagram}[w=40pt,h=30pt,tight,scriptlabels,objectstyle=\scriptstyle,hug]
IIII&\rTo^{\rho II}&III&&&&\\
\dTo<{II\lambda}&\rdTo>{I\alpha I}&\psr{({\ensuremath{\mathsf{R}}} I)\# (\alpha I)}&\rdTo>{\alpha I}&&&\\
III&\psr{(I{\ensuremath{\mathsf{L}}})\#(I\alpha)}&III&\rTo^{\alpha I}&II\\
&\rdTo_{I\alpha}&\dTo<{I\alpha}&{\ensuremath{\mathsf{A}}}\#\alpha&\dTo>{\alpha}\\
&&II&\rTo_{\alpha}&I
\end{diagram}
\quad = \quad
\begin{diagram}[w=40pt,h=30pt,tight,scriptlabels,objectstyle=\scriptstyle,hug]
IIII&\rTo^{\rho II}&III&&&&\\
\dTo<{II\lambda}&{\copyright}&\dTo>{I\lambda}&\rdTo>{\alpha I}&&&\\
III&\rTo_{\rho I}&II&\psr{{\ensuremath{\mathsf{L}}}\#\alpha}&II\\
&\rdTo_{I\alpha}&\psr{{\ensuremath{\mathsf{R}}} \#\alpha}&\rdTo<{\alpha}&\dTo>{\alpha}\\
&&II&\rTo_{\alpha}&I
\end{diagram}$$ From this pentagon equation we shall derive the pentagon equation for ${\ensuremath{\mathsf{A}}}$, asserted in Theorem \[thmA\]. To this end we need comparison between $\alpha$, $\lambda_I$, and $\rho_I$, which we now establish, in analogy with Axiom (1) of monoidal category: the left and right constraints coincide on the unit object, up to a canonical $2$-cell:
\[lambda-rho\] There are unique invertible $2$-cells $$\rho_I \stackrel{{\ensuremath{\mathsf{E}}}}{\Rightarrow}
\alpha \stackrel{{\ensuremath{\mathsf{D}}}}{\Rightarrow} \lambda_I ,$$ such that $$\label{SAT}
\begin{diagram}[w=7ex,h=6ex,tight]
III&
{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,33)(28,11)
\put(0,24){\makebox(0,0)[b]{$\scriptstyle {\alpha I}$}}
\put(28.6,10.7){\vector(1,-1){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{12}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{L}}}}$}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{-12}){\makebox(0,0)[b]{${I{\ensuremath{\mathsf{D}}}}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\put(15,4){\makebox(0,0)[b]{$\scriptstyle {I\lambda}$}}
\put(-21,3){\vector(1,0){44}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-26)(28,-4)
\put(0,-17){\makebox(0,0)[t]{$\scriptstyle {I\alpha}$}}
\put(28.6,-3.7){\vector(1,1){0}}
\end{picture}
}
&II
\end{diagram}
\quad = \quad
\begin{diagram}[w=6ex,h=6ex,tight]
III&
{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,25)(28,11)
\put(0,21){\makebox(0,0)[b]{$\scriptstyle {\alpha I}$}}
\put(28.6,10.7){\vector(2,-1){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{A}}}}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-18)(28,-4)
\put(0,-14){\makebox(0,0)[t]{$\scriptstyle {I\alpha}$}}
\put(28.6,-3.7){\vector(2,1){0}}
\end{picture}
}
&II
\end{diagram}
\quad = \quad
\begin{diagram}[w=7ex,h=6ex,tight]
III&
{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,33)(28,11)
\put(0,24){\makebox(0,0)[b]{$\scriptstyle {\alpha I}$}}
\put(28.6,10.7){\vector(1,-1){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{12}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{E}}} I}$}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{-12}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{R}}}}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\put(15,4){\makebox(0,0)[b]{$\scriptstyle {\rho I}$}}
\put(-21,3){\vector(1,0){44}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-26)(28,-4)
\put(0,-17){\makebox(0,0)[t]{$\scriptstyle {I\alpha}$}}
\put(28.6,-3.7){\vector(1,1){0}}
\end{picture}
}
&II
\end{diagram}
\vspace{30pt}$$
The left-hand equation defines uniquely a $2$-cell $I \alpha \Rightarrow
I\lambda_I$, and since $I$ is cancellable, this cell comes from a unique $2$-cell $\alpha \Rightarrow \lambda_I$ which we then call ${\ensuremath{\mathsf{D}}}$. Same argument for ${\ensuremath{\mathsf{E}}}$.
\[thmA\] Given a unit object $(I,\alpha)$, there is a canonical invertible $2$-cell
III & \^[I]{} & II\
\
II & \_& I
which satisfies the pentagon Equation .
On each side of the cube equation , paste the cell ${\ensuremath{\mathsf{E}}}
II$ on the top, and the cell $II{\ensuremath{\mathsf{D}}}$ on the left. On the left-hand side of the equation we can use Equations directly, while on the right-hand side we first need to move those cells across the commutative square before applying . The result is precisely the pentagon cube for ${\text{\rm \textsf{\AA}}}= {\ensuremath{\mathsf{A}}} \# \alpha$.
[Coherence of the actions.]{} We have now established that $(I,\alpha,{\text{\rm \textsf{\AA}}})$ is a semi-monoid, and may observe that the left and right constraints are coherent actions, i.e. that their ‘associators’ ${\ensuremath{\mathsf{L}}}$ and ${\ensuremath{\mathsf{R}}}$ satisfy the appropriate pentagon equations. For the left action this equation is: $$\begin{diagram}[w=40pt,h=30pt,tight,scriptlabels,objectstyle=\scriptstyle,hug]
IIIX&\rTo^{\alpha IX}&IIX&&&&\\
\dTo<{II\lambda}&\rdTo>{I\alpha X}&\psr{{\text{\rm \textsf{\AA}}}X}&\rdTo>{\alpha X}&&&\\
IIX&\psr{(I{\ensuremath{\mathsf{L}}}) \# (I\lambda)}&IIX&\rTo^{\alpha X}&IX\\
&\rdTo_{I\lambda}&\dTo<{I \lambda}&{\ensuremath{\mathsf{L}}} \#\lambda&\dTo>{\lambda}\\
&&IX&\rTo_{\lambda}&X
\end{diagram}
\qquad = \qquad
\begin{diagram}[w=40pt,h=30pt,tight,scriptlabels,objectstyle=\scriptstyle,hug]
IIIX&\rTo^{\alpha IX}&IIX&&&&\\
\dTo<{II\lambda}&{\copyright}&\dTo>{I\lambda}&\rdTo>{\alpha X}&&&\\
IIX&\rTo_{\alpha X}&IX&\psr{{\ensuremath{\mathsf{L}}} \#\lambda}&IX\\
&\rdTo_{I\lambda}&\psr{{\ensuremath{\mathsf{L}}} \#\lambda}&\rdTo<{\lambda}&\dTo>{\lambda}\\
&&IX&\rTo_{\lambda}&X
\end{diagram}$$ Establishing this (and the analogous equation for the right action) is a routine calculation which we omit since we will not actually need the result. We also note that the two actions are compatible—i.e. constitute a two-sided action. Precisely this means that there is a canonical invertible $2$-cell
I X I & \^[\_X I]{} & X I\
\
I X & \_[\_X]{} & X
This $2$-cell satisfies two pentagon equations, one for $IIXI$ and one for $IXII$.
Units in the $2$-category of arrows in ${\mathscr{C}}$, and Theorem \[thmB\] {#sec:catArr}
============================================================================
In this section we prove Theorem \[thmB\], which asserts that a morphism of units $(u,{\ensuremath{\mathsf{U}}}) : (I,\alpha) \to (J,\beta)$ is automatically a semi-monoid map (with respect to the canonical associators ${\ensuremath{\mathsf{A}}}$ and ${\ensuremath{\mathsf{B}}}$ of the two units). We have to establish the cube equation of \[semimonoid-map\], or in fact the reduced version . The strategy to establish Equation is to interpret everything in the $2$-category of arrows of ${\mathscr{C}}$. The key point is to prove that a morphism of units is itself a unit in the $2$-category of arrows. Then we invoke Theorem \[thmA\] to get an associator for this unit, and a pentagon equation, whose short form will be the sought equation.
[The $2$-category of arrows.]{} The [*$2$-category of arrows*]{} in ${\mathscr{C}}$, denoted ${{\mathscr{C}}^{\mathbf{2}}}$, is the $2$-category described as follows. The objects of ${{\mathscr{C}}^{\mathbf{2}}}$ are the arrows of ${\mathscr{C}}$, $$X_0 \rTo^x X_1 .$$ The arrows from $(X_0,X_1,x)$ to $(Y_0,Y_1,y)$ are triples $(f_0,f_1,F)$ where $f_0:X_0 \to Y_0$ and $f_1:X_1\to Y_1$ are arrows in ${\mathscr{C}}$ and $F$ is a $2$-cell
X\_0 & \^[f\_0]{} & Y\_0\
y\
X\_1 & \_[f\_1]{} & Y\_1
If $(g_0,g_1,G)$ is another arrow from $(X_0,X_1,x)$ to $(Y_0,Y_1,y)$, a $2$-cell from $(f_0,f_1,F)$ to $(g_0,g_1,G)$ is given by a pair $(m_0,m_1)$ where $m_0: f_0\Rightarrow g_0$ and $m_1: f_1 \Rightarrow g_1$ are $2$-cells in ${\mathscr{C}}$ compatible with $F$ and $G$ in the sense that this cylinder commutes: $$\begin{diagram}[w=6ex,h=6ex,tight]
X_0&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,25)(28,11)
\put(0,21){\makebox(0,0)[b]{$\scriptstyle {g_0}$}}
\put(28.6,10.7){\vector(2,-1){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${m_0}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-18)(28,-4)
\put(0,-14){\makebox(0,0)[t]{$\scriptstyle {f_0}$}}
\put(28.6,-3.7){\vector(2,1){0}}
\end{picture}
}&Y_0\\
\dTo<{x}&{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{-12}){\makebox(0,0)[b]{${F}$}}
\end{picture}
}&\dTo>{y}\\
X_1&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-18)(28,-4)
\put(0,-14){\makebox(0,0)[t]{$\scriptstyle {f_1}$}}
\put(28.6,-3.7){\vector(2,1){0}}
\end{picture}
}&Y_1
\end{diagram}
\qquad = \qquad
\begin{diagram}[w=6ex,h=6ex,tight]
X_0&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,25)(28,11)
\put(0,21){\makebox(0,0)[b]{$\scriptstyle {g_0}$}}
\put(28.6,10.7){\vector(2,-1){0}}
\end{picture}
}&Y_0\\
\dTo<{x}&{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{12}){\makebox(0,0)[b]{${G}$}}
\end{picture}
}&\dTo>{y}\\
X_1&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,25)(28,11)
\put(0,21){\makebox(0,0)[b]{$\scriptstyle {g_1}$}}
\put(28.6,10.7){\vector(2,-1){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${m_1}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-18)(28,-4)
\put(0,-14){\makebox(0,0)[t]{$\scriptstyle {f_1}$}}
\put(28.6,-3.7){\vector(2,1){0}}
\end{picture}
}&Y_1
\end{diagram}
\vspace{24pt}$$ Composition of arrows in ${{\mathscr{C}}^{\mathbf{2}}}$ is just pasting of squares. Vertical composition of $2$-cells is just vertical composition of the components (the compatibility is guaranteed by pasting of cylinders along squares), and horizontal composition of $2$-cells is horizontal composition of the components (compatibility guaranteed by pasting along the straight sides of the cylinders). Note that ${{\mathscr{C}}^{\mathbf{2}}}$ inherits a tensor product from ${\mathscr{C}}$: this follows from functoriality of the tensor product on ${\mathscr{C}}$.
\[map-canc\] If $I_0$ and $I_1$ are cancellable objects in ${\mathscr{C}}$ and $i: I_0 \to
I_1$ is an equi-arrow, then $i$ is cancellable in ${{\mathscr{C}}^{\mathbf{2}}}$.
We have to show that for given arrows $x:X_0 \to X_1$ and $y: Y_0\to
Y_1$, the functor $${\operatorname{Hom}}_{{{\mathscr{C}}^{\mathbf{2}}}}(x,y) \to {\operatorname{Hom}}_{{{\mathscr{C}}^{\mathbf{2}}}}(ix,iy)$$ defined by tensoring with $i$ on the left is an equivalence of categories (the check for tensoring on the right is analogous).
Let us first show that this functor is essentially surjective. Let
I\_0 X\_0 & \^[s\_0]{} & I\_0 Y\_0\
\
I\_1 X\_1 & \_[s\_1]{} & I\_1 Y\_1
be an object in ${\operatorname{Hom}}_{{{\mathscr{C}}^{\mathbf{2}}}}(ix,iy)$. We need to find a square
X\_0 & \^[k\_0]{} & Y\_0\
\
X\_1 & \_[k\_1]{} &Y\_1
and an isomorphism $(m_0,m_1)$ from $(s_0,s_1,S)$ to $(I_0 k_0, I_1 k_1, i
K)$, i.e. a cylinder $$\begin{diagram}[w=6ex,h=6ex,tight]
I_0 X_0&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,25)(28,11)
\put(0,21){\makebox(0,0)[b]{$\scriptstyle {I_0 k_0}$}}
\put(28.6,10.7){\vector(2,-1){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${m_0}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-18)(28,-4)
\put(0,-14){\makebox(0,0)[t]{$\scriptstyle {s_0}$}}
\put(28.6,-3.7){\vector(2,1){0}}
\end{picture}
}&I_0 Y_0\\
\dTo<{ix}&{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{-12}){\makebox(0,0)[b]{${S}$}}
\end{picture}
}&\dTo>{iy}\\
I_1 X_1&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-18)(28,-4)
\put(0,-14){\makebox(0,0)[t]{$\scriptstyle {s_1}$}}
\put(28.6,-3.7){\vector(2,1){0}}
\end{picture}
}&I_1 Y_1
\end{diagram}
\qquad = \qquad
\begin{diagram}[w=6ex,h=6ex,tight]
I_0 X_0&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,25)(28,11)
\put(0,21){\makebox(0,0)[b]{$\scriptstyle {I_0 k_0}$}}
\put(28.6,10.7){\vector(2,-1){0}}
\end{picture}
}&I_0 Y_0\\
\dTo<{ix}&{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{12}){\makebox(0,0)[b]{${i K}$}}
\end{picture}
}&\dTo>{iy}\\
I_1 X_1&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,25)(28,11)
\put(0,21){\makebox(0,0)[b]{$\scriptstyle {I_1 k_1}$}}
\put(28.6,10.7){\vector(2,-1){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${m_1}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-18)(28,-4)
\put(0,-14){\makebox(0,0)[t]{$\scriptstyle {s_1}$}}
\put(28.6,-3.7){\vector(2,1){0}}
\end{picture}
}&I_1 Y_1
\end{diagram}
\vspace{24pt}$$ Since $I_0$ is a cancellable object, the arrow $s_0$ is isomorphic to $I_0 k_0$ for some $k_0: X_0 \to Y_0$. Let the connecting invertible $2$-cell be denoted $m_0: s_0 \Rightarrow I_0 k_0$. Similarly we find $k_1$ and $m_1: s_1
\Rightarrow I_1 k_1$. Since $m_0$ and $m_1$ are invertible, there is a unique $2$-cell
I\_0 X\_0 & \^[I\_0 k\_0]{} & I\_0 Y\_0\
\
I\_1 X\_1 & \_[I\_1 k\_1]{} & I\_1 Y\_1
that can take the place of $iK$ in the cylinder equation; it remains to see that $T$ is of the form $iK$ for some $K$. But this follows since the map $$\begin{aligned}
{\rm 2Cell}_{{\mathscr{C}}}(k_0\# y, x\# k_1) & \longrightarrow & {\rm 2Cell}_{{\mathscr{C}}}(i(k_0\# y),
i(x\# k_1)) \notag \\
K & \longmapsto & iK \label{iK}
\end{aligned}$$ is a bijection. Indeed, the map factors as ‘tensoring with $I_0$ on the left’ followed by ‘post-composing with $i Y_1$’; the first is a bijection since $I_0$ is cancellable, the second is a bijection since $i$ (and hence $i Y_1$) is an equi-arrow). Now for the fully faithfulness of ${\operatorname{Hom}}_{{{\mathscr{C}}^{\mathbf{2}}}}(x,y) \to {\operatorname{Hom}}_{{{\mathscr{C}}^{\mathbf{2}}}}(ix,iy)$. Fix two objects in the left-hand category, $P$ and $Q$: $$\begin{diagram}
X_0 & \rTo^{p_0} & Y_0 \\
\dTo<x & P & \dTo>y \\
X_1 & \rTo_{p_1} & Y_1
\end{diagram}
\qquad
\begin{diagram}
X_0 & \rTo^{q_0} & Y_0 \\
\dTo<x & Q & \dTo>y \\
X_1 & \rTo_{q_1} & Y_1
\end{diagram}$$ The arrows from $P$ to $Q$ are pairs $(m_0,m_1)$ consisting of $$m_0: p_0 \Rightarrow q_0 \qquad m_1: p_1 \Rightarrow q_1$$ cylinder-compatible with the $2$-cells $P$ and $Q$. The image of these two objects are $$\begin{diagram}
I_0 X_0 & \rTo^{I_0 p_0} & I_0 Y_0 \\
\dTo<{ix} & iP & \dTo>{iy} \\
I_1 X_1 & \rTo_{I_1 p_1} & I_1 Y_1
\end{diagram}
\qquad
\begin{diagram}
I_0 X_0 & \rTo^{I_0 q_0} & I_0 Y_0 \\
\dTo<{ix} & iQ & \dTo>{iy} \\
I_1 X_1 & \rTo_{I_1 q_1} & I_1 Y_1
\end{diagram}$$ The possible $2$-cells from $iP$ to $iQ$ are pairs $(n_0,n_1)$ consisting of $$n_0: I_0 p_0 \Rightarrow I_0 q_0 \qquad n_1: I_1 p_1 \Rightarrow I_1 q_1$$ cylinder-compatible with the $2$-cells $iP$ and $iQ$. Now since $I_0$ is cancellable, every $2$-cell $n_0$ like this is uniquely of the form $I_0 n_0$ for some $n_0$. Hence there is a bijection between the possible $m_0$ and the possible $n_0$. Similarly for $m_1$ and $n_1$. So there is a bijection between pairs $(m_0, m_1)$ and pairs $(n_0,
n_1)$. Now by functoriality of tensoring with $i$, all images of compatible $(m_0,m_1)$ are again compatible. It remains to rule out the possibility that some $(n_0,n_1)$ pair could be compatible without $(m_0, m_1)$ being so, but this follows again from the argument that ‘tensoring with $i$ on the left’ is a bijection on hom sets, just like argued for .
\[f0f1-equi\] An arrow in ${{\mathscr{C}}^{\mathbf{2}}}$,
X\_0 & \^[f\_0]{} & Y\_0\
y\
X\_1 & \_[f\_1]{} & Y\_1
is an equi-arrow in ${{\mathscr{C}}^{\mathbf{2}}}$ if the components $f_0$ and $f_1$ are equi-arrows in ${\mathscr{C}}$ and $F$ is invertible.
We can construct an explicit quasi-inverse by choosing quasi-inverses to the components.
If $(I_0,\alpha_0)$ and $(I_1, \alpha_1)$ are units in ${\mathscr{C}}$, and $(u,{\ensuremath{\mathsf{U}}}): I_0 \to I_1$ is a unit map between them, then $$u: I_0 \to I_1$$ is a unit object in ${{\mathscr{C}}^{\mathbf{2}}}$ with structure map
I\_0 I\_0 & \^[\_0]{} & I\_0\
u\
I\_1 I\_1 & \_[\_1]{} & I\_1 .
The object $u$ is cancellable by Lemma \[map-canc\], and the morphism $(\alpha_0,\alpha_1,{\ensuremath{\mathsf{U}}}{^{\scriptscriptstyle -1}})$ from $uu$ to $u$ is an equi-arrow by Lemma \[f0f1-equi\].
\[thmB\] Let $(I_0,\alpha_0)$ and $(I_1,\alpha_1)$ be units, with canonical associators ${\ensuremath{\mathsf{A}}}_0$ and ${\ensuremath{\mathsf{A}}}_1$, respectively. If $(u,{\ensuremath{\mathsf{U}}})$ is a unit map from $I_0$ to $I_1$ then it is automatically a semi-monoid map. That is, $$\begin{diagram}[w=6ex,h=6ex,tight]
I_0 I_0 I_0&\rTo^{uuu}&I_1 I_1 I_1\\
{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(-9,-25)(-23,0)(-9,25)
\put(-20,0){\makebox(0,0)[r]{$\scriptstyle {I_0\alpha_0}$}}
\put(-8.7,-25.6){\vector(1,-2){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{A}}}_0}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(9,-25)(23,0)(9,25)
\put(20,0){\makebox(0,0)[l]{$\scriptstyle {\alpha_0 I_0}$}}
\put(8.7,-25.6){\vector(-1,-2){0}}
\end{picture}
}&{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put({20},0){\makebox(0,0)[b]{${{\ensuremath{\mathsf{U}}}u}$}}
\end{picture}
}&{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(9,-25)(23,0)(9,25)
\put(20,0){\makebox(0,0)[l]{$\scriptstyle {\alpha_1 I_1}$}}
\put(8.7,-25.6){\vector(-1,-2){0}}
\end{picture}
}\\
I_0 I_0&\rTo_{uu}&I_1 I_1
\end{diagram}
\qquad\qquad = \qquad\qquad
\begin{diagram}[w=6ex,h=6ex,tight]
I_0 I_0 I_0&\rTo^{uuu}&I_1 I_1 I_1\\
{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(-9,-25)(-23,0)(-9,25)
\put(-20,0){\makebox(0,0)[r]{$\scriptstyle {I_0\alpha_0}$}}
\put(-8.7,-25.6){\vector(1,-2){0}}
\end{picture}
}&{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put({-20},0){\makebox(0,0)[b]{${u{\ensuremath{\mathsf{U}}}}$}}
\end{picture}
}&{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(-9,-25)(-23,0)(-9,25)
\put(-20,0){\makebox(0,0)[r]{$\scriptstyle {I_1\alpha_1}$}}
\put(-8.7,-25.6){\vector(1,-2){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{A}}}_1}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(9,-25)(23,0)(9,25)
\put(20,0){\makebox(0,0)[l]{$\scriptstyle {\alpha_1 I_1}$}}
\put(8.7,-25.6){\vector(-1,-2){0}}
\end{picture}
}\\
I_0 I_0&\rTo_{uu}&I_1 I_1
\end{diagram}$$
By the previous Corollary, $(u,{\ensuremath{\mathsf{U}}}{^{\scriptscriptstyle -1}})$ is a unit object in ${{\mathscr{C}}^{\mathbf{2}}}$. Hence there is a canonical associator $${\ensuremath{\mathsf{B}}} : u{\ensuremath{\mathsf{U}}}{^{\scriptscriptstyle -1}}\Leftrightarrow {\ensuremath{\mathsf{U}}}{^{\scriptscriptstyle -1}}u .$$ By definition of $2$-cells in ${{\mathscr{C}}^{\mathbf{2}}}$, this is a pair of $2$-cells in ${\mathscr{C}}$ $${\ensuremath{\mathsf{B}}}_0 : I_0 \alpha_0 \Rightarrow \alpha_0 I_0
\qquad
{\ensuremath{\mathsf{B}}}_1 : I_1 \alpha_1 \Rightarrow \alpha_1 I_1 ,$$ fitting the cylinder equation $$\begin{diagram}[w=6ex,h=6ex,tight]
I_0 I_0 I_0&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,25)(28,11)
\put(0,21){\makebox(0,0)[b]{$\scriptstyle {\alpha_0 I_0}$}}
\put(28.6,10.7){\vector(2,-1){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{B}}}_0}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-18)(28,-4)
\put(0,-14){\makebox(0,0)[t]{$\scriptstyle {I_0 \alpha_0}$}}
\put(28.6,-3.7){\vector(2,1){0}}
\end{picture}
}&I_0 I_0\\
\dTo<{uuu}&{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{-12}){\makebox(0,0)[b]{${u {\ensuremath{\mathsf{U}}}{^{\scriptscriptstyle -1}}}$}}
\end{picture}
}&\dTo>{uu}\\
I_1 I_1 I_1&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-18)(28,-4)
\put(0,-14){\makebox(0,0)[t]{$\scriptstyle {I_1 \alpha_1}$}}
\put(28.6,-3.7){\vector(2,1){0}}
\end{picture}
}&I_1 I_1
\end{diagram}
\qquad = \qquad
\begin{diagram}[w=6ex,h=6ex,tight]
I_0 I_0 I_0&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,25)(28,11)
\put(0,21){\makebox(0,0)[b]{$\scriptstyle {\alpha_0 I_0}$}}
\put(28.6,10.7){\vector(2,-1){0}}
\end{picture}
}&I_0 I_0\\
\dTo<{uuu}&{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{12}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{U}}}{^{\scriptscriptstyle -1}}u}$}}
\end{picture}
}&\dTo>{uu}\\
I_1 I_1 I_1&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,25)(28,11)
\put(0,21){\makebox(0,0)[b]{$\scriptstyle {\alpha_1 I_1}$}}
\put(28.6,10.7){\vector(2,-1){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{B}}}_1}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-18)(28,-4)
\put(0,-14){\makebox(0,0)[t]{$\scriptstyle {I_1 \alpha_1}$}}
\put(28.6,-3.7){\vector(2,1){0}}
\end{picture}
}&I_1 I_1
\end{diagram}
\vspace{24pt}$$ This is precisely the cylinder diagram we are looking for—provided we can show that ${\ensuremath{\mathsf{B}}}_0 = {\ensuremath{\mathsf{A}}}_0$ and ${\ensuremath{\mathsf{B}}}_1 = {\ensuremath{\mathsf{A}}}_1$. But this is a consequence of the characterising property of the associator of a unit: first note that as a unit object in ${{\mathscr{C}}^{\mathbf{2}}}$, $u$ induces left and right constraints: for each object $x: X_0 \to X_1$ in ${{\mathscr{C}}^{\mathbf{2}}}$ there is a left action of the unit $u$, and this left action will induce a left action of $(I_0,\alpha_0)$ on $X_0$ and a left action of $(I_1,\alpha_1)$ on $X_1$ (the ends of the cylinders). Similarly there is a right action of $u$ which induces right actions at the ends of the cylinder. Now the unique ${\ensuremath{\mathsf{B}}}$ that exists as associator for the unit object $u$ compatible with the left and right constraints induces ${\ensuremath{\mathsf{B}}}_0$ and ${\ensuremath{\mathsf{B}}}_1$ at the ends of the cylinder, and these will of course be compatible with the induced left and right constraints. Hence, by uniqueness of associators compatible with left and right constraints, these induced associators ${\ensuremath{\mathsf{B}}}_0$ and ${\ensuremath{\mathsf{B}}}_1$ must coincide with ${\ensuremath{\mathsf{A}}}_0$ and ${\ensuremath{\mathsf{A}}}_1$. Note that this does not dependent on choice of left and right constraints, cf. Proposition \[independence\].
Contractibility of the space of weak units (Theorem \[thmC\]) {#sec:contractibility}
=============================================================
The goal of this section is to prove Theorem \[thmC\], which asserts that the $2$-category of units in ${\mathscr{C}}$ is contractible if non-empty. First we describe the unit morphisms and unit $2$-morphisms in terms of compatibility with left and right constraints. This will show that there are not too many $2$-cells. Second we use the left and right constraints to connect any two units.
The following lemma shows that just as the single arrow $\alpha$ induces all the $\lambda_X$ and $\rho_X$, the single $2$-cell ${\ensuremath{\mathsf{U}}}$ of a unit map induces families ${\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{left}}}}_X$ and ${\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{right}}}}_X$ expressing compatibility with $\lambda_X$ and $\rho_X$.
\[unitmap\] Let $(I,\alpha)$ and $(J,\beta)$ be units, and let $(u,{\ensuremath{\mathsf{U}}})$ be a morphism of pseudo-idempotents from $(I,\alpha)$ to $(J,\beta)$. The following are equivalent.
$u$ is an equi-arrow (i.e. $u$ is a morphism of units).
$u$ is left cancellable, i.e. tensoring with $u$ on the left is an equivalence of categories ${\operatorname{Hom}}(X,Y) \to {\operatorname{Hom}}(IX,JY)$.
$u$ is right cancellable, i.e. tensoring with $u$ on the right is an equivalence of categories ${\operatorname{Hom}}(X,Y) \to {\operatorname{Hom}}(XI,YJ)$.
For fixed left actions $(\lambda_X,{\ensuremath{\mathsf{L}}}_X)$ for the unit $(I,\alpha)$ and $(\ell_X,{\ensuremath{\mathsf{L}}}'_X)$ for the unit $(J,\beta)$, there is a unique invertible $2$-cell ${\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{left}}}}_X$, natural in $X$:
IX & \^[uX]{} & JX\
\
X & \_X & X
such that this compatibility holds: $$\label{P}
\begin{diagram}[w=6ex,h=6ex,tight]
IIX&\rTo^{uuX}&JJX\\
{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(-9,-25)(-23,0)(-9,25)
\put(-20,0){\makebox(0,0)[r]{$\scriptstyle {I\lambda_X}$}}
\put(-8.7,-25.6){\vector(1,-2){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{L}}}_X}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(9,-25)(23,0)(9,25)
\put(20,0){\makebox(0,0)[l]{$\scriptstyle {\alpha X}$}}
\put(8.7,-25.6){\vector(-1,-2){0}}
\end{picture}
}&{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put({20},0){\makebox(0,0)[b]{${{\ensuremath{\mathsf{U}}}X}$}}
\end{picture}
}&{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(9,-25)(23,0)(9,25)
\put(20,0){\makebox(0,0)[l]{$\scriptstyle {\beta X}$}}
\put(8.7,-25.6){\vector(-1,-2){0}}
\end{picture}
}\\
IX&\rTo_{uX}&JX
\end{diagram}
\qquad\qquad = \qquad\qquad
\begin{diagram}[w=6ex,h=6ex,tight]
IIX&\rTo^{uuX}&JJX\\
{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(-9,-25)(-23,0)(-9,25)
\put(-20,0){\makebox(0,0)[r]{$\scriptstyle {I\lambda_X}$}}
\put(-8.7,-25.6){\vector(1,-2){0}}
\end{picture}
}&{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put({-20},0){\makebox(0,0)[b]{${u{\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{left}}}}_X}$}}
\end{picture}
}&{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(-9,-25)(-23,0)(-9,25)
\put(-20,0){\makebox(0,0)[r]{$\scriptstyle {J\ell_X}$}}
\put(-8.7,-25.6){\vector(1,-2){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{L}}}'_X}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(9,-25)(23,0)(9,25)
\put(20,0){\makebox(0,0)[l]{$\scriptstyle {\beta X}$}}
\put(8.7,-25.6){\vector(-1,-2){0}}
\end{picture}
}\\
IX&\rTo_{uX}&JX
\end{diagram}$$
For fixed right actions $(\rho_X,{\ensuremath{\mathsf{R}}}_X)$ for the unit $(I,\alpha)$ and $(r_X,{\ensuremath{\mathsf{R}}}'_X)$ for the unit $(J,\beta)$, there is a unique invertible $2$-cell ${\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{right}}}}_X$, natural in $X$:
XI & \^[Xu]{} & XJ\
\
X & \_X & X
such that this compatibility holds: $$\label{Q}
\begin{diagram}[w=6ex,h=6ex,tight]
XII&\rTo^{Xuu}&XJJ\\
{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(-9,-25)(-23,0)(-9,25)
\put(-20,0){\makebox(0,0)[r]{$\scriptstyle {X\alpha}$}}
\put(-8.7,-25.6){\vector(1,-2){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{R}}}_X}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(9,-25)(23,0)(9,25)
\put(20,0){\makebox(0,0)[l]{$\scriptstyle {\rho_X I}$}}
\put(8.7,-25.6){\vector(-1,-2){0}}
\end{picture}
}&{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put({20},0){\makebox(0,0)[b]{${{\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{right}}}}_X u}$}}
\end{picture}
}&{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(9,-25)(23,0)(9,25)
\put(20,0){\makebox(0,0)[l]{$\scriptstyle {r_X J}$}}
\put(8.7,-25.6){\vector(-1,-2){0}}
\end{picture}
}\\
XI&\rTo_{Xu}&XJ
\end{diagram}
\qquad\qquad = \qquad\qquad
\begin{diagram}[w=6ex,h=6ex,tight]
XII&\rTo^{Xuu}&XJJ\\
{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(-9,-25)(-23,0)(-9,25)
\put(-20,0){\makebox(0,0)[r]{$\scriptstyle {X\alpha}$}}
\put(-8.7,-25.6){\vector(1,-2){0}}
\end{picture}
}&{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put({-20},0){\makebox(0,0)[b]{${X{\ensuremath{\mathsf{U}}}}$}}
\end{picture}
}&{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(-9,-25)(-23,0)(-9,25)
\put(-20,0){\makebox(0,0)[r]{$\scriptstyle {X\beta}$}}
\put(-8.7,-25.6){\vector(1,-2){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{R}}}'_X}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(9,-25)(23,0)(9,25)
\put(20,0){\makebox(0,0)[l]{$\scriptstyle {r_X J}$}}
\put(8.7,-25.6){\vector(-1,-2){0}}
\end{picture}
}\\
XI&\rTo_{Xu}&XJ
\end{diagram}$$
\(i) implies (ii): ‘tensoring with $u$’ can be done in two steps: given an arrow $X \to Y$, first tensor with $I$ to get $IX \to IY$, and then post-compose with $uY$ to get $IX \to JY$. The first step is an equivalence because $I$ is a unit, and the second step is an equivalence because $u$ is an equi-arrow.
\(ii) implies (iii): In Equation (\[P\]), the $2$-cell labelled $u{\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{left}}}}_X$ is uniquely defined by the three other cells, and it is invertible since the three other cells are. Since tensoring with $u$ on the left is an equivalence, this cell comes from a unique invertible cell ${\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{left}}}}_X$, justifying the label $u{\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{left}}}}_X$.
\(iii) implies (i): The invertible $2$-cell ${\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{left}}}}_X$ shows that $uX$ is isomorphic to an equi-arrow, and hence is an equi-arrow itself. Now take $X$ to be a right cancellable object (like for example $I$) and conclude that already $u$ is an equi-arrow.
Finally, the equivalence (i)[$\Rightarrow$]{}(ii’)[$\Rightarrow$]{}(iii’)[$\Rightarrow$]{}(i) is completely analogous.
Note that for $(u,{\ensuremath{\mathsf{U}}})$ the identity morphism on $(I,\alpha)$, we recover Observation \[lambda-lambda\].
\[unit2map\] Let $(I,\alpha)$ and $(J,\beta)$ be units; let $(u,{\ensuremath{\mathsf{U}}})$ and $(v,{\ensuremath{\mathsf{V}}})$ be morphisms of pseudo-idempotents from $I$ to $J$; and consider a $2$-cell ${\ensuremath{\mathsf{T}}}: u\Rightarrow v$. Then the following are equivalent.
${\ensuremath{\mathsf{T}}}$ is an invertible $2$-morphism of pseudo-idempotents.
${\ensuremath{\mathsf{T}}}$ is a left cancellable $2$-morphism of pseudo-idempotents (i.e., induces a bijection on hom sets (of hom cats) by tensoring with ${\ensuremath{\mathsf{T}}}$ from the left).
${\ensuremath{\mathsf{T}}}$ is a right cancellable $2$-morphism of pseudo-idempotents (i.e., induces a bijection on hom sets (of hom cats) by tensoring with ${\ensuremath{\mathsf{T}}}$ from the right).
For fixed left actions $(\lambda_X, {\ensuremath{\mathsf{L}}}_X)$ for $(I,\alpha)$ and $(\ell_X,{\ensuremath{\mathsf{L}}}'_X)$ for $(J,\beta)$, with induced canonical $2$-cells ${\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{left}}}}_X$ and ${\ensuremath{\mathsf{V}}}{^{\text{\tiny \rm{left}}}}_X$ as in \[unitmap\], we have: $$\label{TXP}
\begin{diagram}[w=6ex,h=6ex,tight]
IX&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,25)(28,11)
\put(0,21){\makebox(0,0)[b]{$\scriptstyle {vX}$}}
\put(28.6,10.7){\vector(2,-1){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{T}}} X}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-18)(28,-4)
\put(0,-14){\makebox(0,0)[t]{$\scriptstyle {uX}$}}
\put(28.6,-3.7){\vector(2,1){0}}
\end{picture}
}&JX\\
\dTo<{\lambda_X}&{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{-12}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{left}}}}_X}$}}
\end{picture}
}&\dTo>{\ell_X}\\
X&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-18)(28,-4)
\put(0,-14){\makebox(0,0)[t]{$\scriptstyle {X}$}}
\put(28.6,-3.7){\vector(2,1){0}}
\end{picture}
}&X
\end{diagram}
\qquad = \qquad
\begin{diagram}[w=6ex,h=6ex,tight]
IX&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,25)(28,11)
\put(0,21){\makebox(0,0)[b]{$\scriptstyle {vX}$}}
\put(28.6,10.7){\vector(2,-1){0}}
\end{picture}
}&JX\\
\dTo<{\lambda_X}&{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{12}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{V}}}{^{\text{\tiny \rm{left}}}}_X}$}}
\end{picture}
}&\dTo>{\ell_X}\\
X&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,25)(28,11)
\put(0,21){\makebox(0,0)[b]{$\scriptstyle {X}$}}
\put(28.6,10.7){\vector(2,-1){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${{\copyright}}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-18)(28,-4)
\put(0,-14){\makebox(0,0)[t]{$\scriptstyle {X}$}}
\put(28.6,-3.7){\vector(2,1){0}}
\end{picture}
}&X
\end{diagram}
\vspace{24pt}$$
For fixed right actions $(\rho_X,{\ensuremath{\mathsf{R}}}_X)$ for $(I,\alpha)$ and $(r_X,{\ensuremath{\mathsf{R}}}'_X)$ for $(J,\beta)$, with induced canonical $2$-cells ${\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{right}}}}_X$ and ${\ensuremath{\mathsf{V}}}{^{\text{\tiny \rm{right}}}}_X$ as in \[unitmap\], we have: $$\label{TXQ}
\begin{diagram}[w=6ex,h=6ex,tight]
XI&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,25)(28,11)
\put(0,21){\makebox(0,0)[b]{$\scriptstyle {Xv}$}}
\put(28.6,10.7){\vector(2,-1){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${X{\ensuremath{\mathsf{T}}}}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-18)(28,-4)
\put(0,-14){\makebox(0,0)[t]{$\scriptstyle {Xu}$}}
\put(28.6,-3.7){\vector(2,1){0}}
\end{picture}
}&XJ\\
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\begin{picture}(0,0)(0,0)
\put(0,{-12}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{right}}}}_X}$}}
\end{picture}
}&\dTo>{r_X}\\
X&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-18)(28,-4)
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\put(28.6,-3.7){\vector(2,1){0}}
\end{picture}
}&X
\end{diagram}
\qquad = \qquad
\begin{diagram}[w=6ex,h=6ex,tight]
XI&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,25)(28,11)
\put(0,21){\makebox(0,0)[b]{$\scriptstyle {Xv}$}}
\put(28.6,10.7){\vector(2,-1){0}}
\end{picture}
}&XJ\\
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\begin{picture}(0,0)(0,0)
\put(0,{12}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{V}}}{^{\text{\tiny \rm{right}}}}_X}$}}
\end{picture}
}&\dTo>{r_X}\\
X&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,25)(28,11)
\put(0,21){\makebox(0,0)[b]{$\scriptstyle {X}$}}
\put(28.6,10.7){\vector(2,-1){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${{\copyright}}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-18)(28,-4)
\put(0,-14){\makebox(0,0)[t]{$\scriptstyle {X}$}}
\put(28.6,-3.7){\vector(2,1){0}}
\end{picture}
}&X
\end{diagram}
\vspace{24pt}$$
It is obvious that (i) implies (ii). Let us prove that (ii) implies (iii), so assume that tensoring with ${\ensuremath{\mathsf{T}}}$ on the left defines a bijection on the level of $2$-cells. Start with the cylinder diagram for compatibility of tensor $2$-cells (cf. \[semimonoid-transf\]). Tensor this diagram with $X$ on the right to get $$\begin{diagram}[w=6ex,h=6ex,tight]
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\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,25)(28,11)
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\put(28.6,10.7){\vector(2,-1){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
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\put(0,{0}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{TT}}}X}$}}
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}{\setlength{\unitlength}{0.003\DiagramCellWidth}
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\qbezier(-28,-4)(0,-18)(28,-4)
\put(0,-14){\makebox(0,0)[t]{$\scriptstyle {uuX}$}}
\put(28.6,-3.7){\vector(2,1){0}}
\end{picture}
}&JJX\\
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\begin{picture}(0,0)(0,0)
\put(0,{-12}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{U}}}X}$}}
\end{picture}
}&\dTo>{\beta X}\\
IX&{\setlength{\unitlength}{0.003\DiagramCellWidth}
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\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-18)(28,-4)
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\put(28.6,-3.7){\vector(2,1){0}}
\end{picture}
}&JX
\end{diagram}
\qquad = \qquad
\begin{diagram}[w=6ex,h=6ex,tight]
IIX&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,25)(28,11)
\put(0,21){\makebox(0,0)[b]{$\scriptstyle {vvX}$}}
\put(28.6,10.7){\vector(2,-1){0}}
\end{picture}
}&JJX\\
\dTo<{\alpha X}&{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{12}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{V}}}X}$}}
\end{picture}
}&\dTo>{\beta X}\\
IX&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,25)(28,11)
\put(0,21){\makebox(0,0)[b]{$\scriptstyle {vX}$}}
\put(28.6,10.7){\vector(2,-1){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{T}}}X}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-18)(28,-4)
\put(0,-14){\makebox(0,0)[t]{$\scriptstyle {uX}$}}
\put(28.6,-3.7){\vector(2,1){0}}
\end{picture}
}&JX
\end{diagram}
\vspace{24pt}$$ On each side of this equation, paste an ${\ensuremath{\mathsf{L}}}_X$ along $\alpha X$, apply Equation on each side, and cancel the ${\ensuremath{\mathsf{L}}}'_X$ that appear on the other side of the square. The resulting diagram $$\begin{diagram}[w=6ex,h=6ex,tight]
IIX&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
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\put(0,21){\makebox(0,0)[b]{$\scriptstyle {vvX}$}}
\put(28.6,10.7){\vector(2,-1){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{TT}}}X}$}}
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}{\setlength{\unitlength}{0.003\DiagramCellWidth}
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\qbezier(-28,-4)(0,-18)(28,-4)
\put(0,-14){\makebox(0,0)[t]{$\scriptstyle {uuX}$}}
\put(28.6,-3.7){\vector(2,1){0}}
\end{picture}
}&JJX\\
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\begin{picture}(0,0)(0,0)
\put(0,{-12}){\makebox(0,0)[b]{${u {\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{left}}}}_X}$}}
\end{picture}
}&\dTo>{J \ell_X}\\
IX&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-18)(28,-4)
\put(0,-14){\makebox(0,0)[t]{$\scriptstyle {uX}$}}
\put(28.6,-3.7){\vector(2,1){0}}
\end{picture}
}&JX
\end{diagram}
\qquad = \qquad
\begin{diagram}[w=6ex,h=6ex,tight]
IIX&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,25)(28,11)
\put(0,21){\makebox(0,0)[b]{$\scriptstyle {vvX}$}}
\put(28.6,10.7){\vector(2,-1){0}}
\end{picture}
}&JJX\\
\dTo<{I\lambda_X}&{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{12}){\makebox(0,0)[b]{${v{\ensuremath{\mathsf{V}}}{^{\text{\tiny \rm{left}}}}_X}$}}
\end{picture}
}&\dTo>{J \ell_X}\\
IX&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,25)(28,11)
\put(0,21){\makebox(0,0)[b]{$\scriptstyle {vX}$}}
\put(28.6,10.7){\vector(2,-1){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{T}}}X}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-18)(28,-4)
\put(0,-14){\makebox(0,0)[t]{$\scriptstyle {uX}$}}
\put(28.6,-3.7){\vector(2,1){0}}
\end{picture}
}&JX
\end{diagram}
\vspace{24pt}$$ is the tensor product of ${\ensuremath{\mathsf{T}}}$ with the promised equation . Since ${\ensuremath{\mathsf{T}}}$ is cancellable, we can cancel it away to finish.
\(iii) implies (i): the arguments in (ii)[$\Rightarrow$]{}(iii) can be reverted: start with , tensor with ${\ensuremath{\mathsf{T}}}$ on the left, and apply to arrive at the axiom for being a $2$-morphism of pseudo-idempotents. Since both ${\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{left}}}}_X$ and ${\ensuremath{\mathsf{V}}}{^{\text{\tiny \rm{left}}}}_X$ are invertible, so is ${\ensuremath{\mathsf{T}}} X$. Now take $X$ to be a right cancellable object, and cancel it away to conclude that already ${\ensuremath{\mathsf{T}}}$ is invertible.
Finally, the equivalence (i)[$\Rightarrow$]{}(ii’)[$\Rightarrow$]{}(iii’)[$\Rightarrow$]{}(i) is completely analogous.
\[unique-2-cell\] Given two parallel morphisms of units, there is a unique unit $2$-morphism between them.
The $2$-cell is determined by the previous lemma.
Next we aim at proving that there is a unit morphism between any two units. The strategy is to use the left and right constraints to produce a unit morphism $$I \rTo IJ \rTo J.$$ As a first step towards this goal we have:
\[IJunit\] Let $I$ and $J$ be units, and pick a left constraint $\lambda$ for $I$ and a right constraint $r$ for $J$. Put $$\gamma {\: {\raisebox{0.255ex}{\normalfont\scriptsize :\!\!}}=}r_I \lambda_J : IJIJ \to IJ$$ Then $(IJ, \gamma)$ is a unit.
Since $I$ and $J$ are cancellable, clearly $IJ$ is cancellable too. Since $\lambda_J$ and $r_I$ are equi-arrows, $\gamma$ is too.
\[lambdasemimonoidmap\] There is a $2$-cell
IJIJ & \^[\_J \_J]{} & JJ\
\
IJ & \_[\_J]{} & J .
Hence $(\lambda_J,{\ensuremath{\mathsf{Z}}})$ is a unit map. (And there is another $2$-cell making $r_I$ a unit map.)
The $2$-cell ${\ensuremath{\mathsf{Z}}}$ is defined like this:
IJIJ & &\
\
IJ & \_[\_J]{} & J
where the $2$-cell ${\ensuremath{\mathsf{K}}}^\lambda$ is constructed in Lemma \[lambdaXY\].
\[exist\] Given two units, there exists a unit morphism between them.
Continuing the notation from above, by Lemma \[IJunit\], $(IJ,\gamma)$ is a unit, and by Lemma \[lambdasemimonoidmap\], $\lambda: IJ \to J$ is a morphism of units. Similarly, $r: IJ \to I$ is a unit morphism, and by Lemma \[inv\] any chosen pseudo-inverse $r{^{\scriptscriptstyle -1}}: I \to IJ$ is again a unit morphism. Finally we take $$I \stackrel{r{^{\scriptscriptstyle -1}}}\rTo IJ \stackrel{\lambda}\rTo J.$$
\[thmC\] The $2$-category of units in ${\mathscr{C}}$ is contractible, if non-empty. In other words, between any two units there exists a unit morphism, and between any two parallel unit morphisms there is a unique unit $2$-morphism.
By Lemma \[exist\] there is a unit morphism between any two units (an equi-arrow by definition), and by Lemma \[unique-2-cell\] there is a unique unit $2$-morphism between any two parallel unit morphisms.
Classical units {#sec:classical}
===============
In this section we review the classical theory of units in a monoidal $2$-category, as extracted from the definition of tricategory of Gordon, Power, and Street [@Gordon-Power-Street]. In the next section we compare this notion with the cancellable-idempotent approach of this work. The equivalence is stated explicitly in Theorem \[thmE\].
[Tricategories.]{} The notion of tricategory introduced by Gordon, Power, and Street [@Gordon-Power-Street] is is roughly a weak category structure enriched over bicategories: this means that the structure maps (composition and unit) are weak $2$-functors satisfying weak versions of associativity and unit constraints. For the associativity, the pentagon equation is replaced by a specified pentagon $3$-cell (TD7), required to satisfy an equation corresponding to the $3$-dimensional associahedron. This equation (TA1) is called the nonabelian $4$-cocycle condition. For the unit structure, three families of $3$-cells are specified (TD8): one corresponding to the Kelly axiom, one left variant, and one right variant (those two being the higher-dimensional analogues of Axioms (2) and (3) of monoidal category). Two axioms are imposed on these three families of $3$-cells: one (TA2) relating the left family with the middle family, and one (TA3) relating the right family with the middle family. These are called left and right normalisation. (These two axioms are the higher-dimensional analogues of the first argument in Kelly’s lemma \[Kelly-lemma\].) It is pointed out in [@Gordon-Power-Street] that the middle family together with the axioms (TA2) and (TA3) completely determine the left and right families if they exist.
[Monoidal $2$-categories.]{} By specialising the definition of tricategory to the one-object case, and requiring everything strict except the units, we arrive at the following notion of monoidal $2$-category: a [*monoidal $2$-category*]{} is a semi-monoidal $2$-category (cf. \[semimonoidal\]) equipped with an object $I$, two natural transformations $\lambda$ and $\rho$ with equi-arrow components $$\lambda_X: IX \to X$$ $$\rho_X : XI \to X$$ and (invertible) $2$-cell data $$\begin{diagram}
I X & \rTo^{\lambda_X} & X \\
\dTo<{I f} & \lambda_f & \dTo>f \\
I Y & \rTo_{\lambda_Y} & Y
\end{diagram}
\qquad\qquad
\begin{diagram}
XI & \rTo^{\rho_X} & X \\
\dTo<{f I} & \rho_f & \dTo>f \\
YI & \rTo_{\rho_Y} & Y ,
\end{diagram}$$ together with three natural modifications ${\ensuremath{\mathsf{K}}}$, ${\ensuremath{\mathsf{K}}}^\lambda$, and ${\ensuremath{\mathsf{K}}}^\rho$, with invertible components $$\begin{aligned}
{\ensuremath{\mathsf{K}}} \; : X \lambda_Y &\Rightarrow \rho_X Y \\
{\ensuremath{\mathsf{K}}}^\lambda : \lambda_{XY} &\Rightarrow \lambda_X Y \\
{\ensuremath{\mathsf{K}}}^\rho : X \rho_Y &\Rightarrow \rho_{XY} .\end{aligned}$$ We call ${\ensuremath{\mathsf{K}}}$ the [*Kelly cell*]{}. These three families are subject to the following two equations: $$\label{TA2}
\xymatrixrowsep{50pt}
\xymatrixcolsep{42pt}
\xymatrix @!=0pt {
X \lambda_{YZ} \ar@{=>}[rr]^{X{\ensuremath{\mathsf{K}}}^\lambda_{Y,Z}} \ar@{=>}[dr]_{{\ensuremath{\mathsf{K}}}_{X,YZ}} && X\lambda_Y Z \ar@{=>}[dl]^{{\ensuremath{\mathsf{K}}}_{X,Y} Z} \\
& \rho_X YZ &
}$$ $$\label{TA3}
\xymatrixrowsep{50pt}
\xymatrixcolsep{42pt}
\xymatrix @!=0pt {
X\rho_Y Z \ar@{=>}[rr]^{{\ensuremath{\mathsf{K}}}^\rho_{X,Y} Z} \ar@{<=}[dr]_{X{\ensuremath{\mathsf{K}}}_{Y,Z}} &&
\rho_{XY} Z \ar@{<=}[dl]^{{\ensuremath{\mathsf{K}}}_{XY,Z}} \\
& XY\lambda_Z &
}$$
We have made one change compared to [@Gordon-Power-Street], namely the direction of the arrow $\rho_X$: from the viewpoint of $\alpha$ it seems more practical to work with $\rho_X:XI\to X$ rather than with the convention of $\rho_X : X \to XI$ chosen in [@Gordon-Power-Street]. Since in any case it is an equi-arrow, the difference is not essential. (Gurski in his thesis [@Gurski:PhD] has studied a version of tricategory where all the equi-arrows in the definition are equipped with specified pseudo-inverses. This has the advantage that the definition becomes completely algebraic, in a technical sense.)
\[Icanc\] The object $I$ is cancellable (independently of the existence of ${\ensuremath{\mathsf{K}}}$, ${\ensuremath{\mathsf{K}}}^\lambda$, and ${\ensuremath{\mathsf{K}}}^\rho$.)
We need to establish that ‘tensoring with $I$ on the left’, $$\mathbb{L} : {\operatorname{Hom}}(X,Y) \to {\operatorname{Hom}}(IX,IY) ,$$ is an equivalence of categories. But this follows since the diagram
(X,Y) & \^ & (IX,IY)\
\
(X,Y) & \_[\_X \# \_]{} & (IX,Y)
is commutative up to isomorphism: the component at $f:X\to Y$ of this isomorphism is just the naturality square $\lambda_f$. Since the functors $\lambda_X \,\#\, \_$ and $ \_ \,\#\, \lambda_Y$ are equivalences, it follows from this isomorphism that $\mathbb{L}$ is too.
[Coherence of the Kelly cell.]{}\[Klambda-from-K\] As remarked in [@Gordon-Power-Street], if the ${\ensuremath{\mathsf{K}}}^\lambda$ and ${\ensuremath{\mathsf{K}}}^\rho$ exist, they are determined uniquely from ${\ensuremath{\mathsf{K}}}$ and the two axioms. Indeed, the two equations $$\label{KKK}
\xymatrixrowsep{50pt}
\xymatrixcolsep{42pt}
\xymatrix @!=0pt {
I \lambda_{YZ} \ar@{=>}[rr]^{I{\ensuremath{\mathsf{K}}}^\lambda_{Y,Z}} \ar@{=>}[dr]_{{\ensuremath{\mathsf{K}}}_{I,YZ}} && I\lambda_Y Z \ar@{=>}[dl]^{{\ensuremath{\mathsf{K}}}_{I,Y} Z} \\
& \rho_I YZ &
}
\qquad\qquad
\xymatrixrowsep{50pt}
\xymatrixcolsep{42pt}
\xymatrix @!=0pt {
X\rho_Y I \ar@{=>}[rr]^{{\ensuremath{\mathsf{K}}}^\rho_{X,Y} I} \ar@{=>}[dr]_{X{\ensuremath{\mathsf{K}}}_{Y,I}} && \rho_{XY} I \ar@{=>}[dl]^{{\ensuremath{\mathsf{K}}}_{XY,I}} \\
& XY\lambda_I &
}$$ which are just special cases of and uniquely determine ${\ensuremath{\mathsf{K}}}^\lambda$ and ${\ensuremath{\mathsf{K}}}^\rho$, by cancellability of $I$. But these two special cases of the axioms do not imply the general case.
We shall take the Kelly cell ${\ensuremath{\mathsf{K}}}$ as the main structure, and say that ${\ensuremath{\mathsf{K}}}$ is [*coherent on the left*]{} (resp. [*on the right*]{}) if Axiom (resp. ) holds for the induced cell ${\ensuremath{\mathsf{K}}}^\lambda$ (resp. ${\ensuremath{\mathsf{K}}}^{\rho}$). We just say [*coherent*]{} if both hold. We shall see (\[KLR\]) that in fact coherence on the left implies coherence on the right and vice versa.
[Naturality.]{}\[natK\] The Kelly cell is a modification. For future reference we spell out the naturality condition satisfied: given arrows $f: X \to X'$ and $g: Y \to Y'$, we have $$\begin{diagram}[w=6ex,h=6ex,tight]
XIY&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,25)(28,11)
\put(0,21){\makebox(0,0)[b]{$\scriptstyle {\rho_X Y}$}}
\put(28.6,10.7){\vector(2,-1){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{K}}}_{X,Y}}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-18)(28,-4)
\put(0,-14){\makebox(0,0)[t]{$\scriptstyle {X\lambda_Y}$}}
\put(28.6,-3.7){\vector(2,1){0}}
\end{picture}
}&XY\\
\dTo<{fIg}&{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{-12}){\makebox(0,0)[b]{${f \lambda_g}$}}
\end{picture}
}&\dTo>{fg}\\
X'IY'&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-18)(28,-4)
\put(0,-14){\makebox(0,0)[t]{$\scriptstyle {X'\lambda_{Y'}}$}}
\put(28.6,-3.7){\vector(2,1){0}}
\end{picture}
}&X'Y'
\end{diagram}
\qquad = \qquad
\begin{diagram}[w=6ex,h=6ex,tight]
XIY&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,25)(28,11)
\put(0,21){\makebox(0,0)[b]{$\scriptstyle {\rho_X Y}$}}
\put(28.6,10.7){\vector(2,-1){0}}
\end{picture}
}&XY\\
\dTo<{fIg}&{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{12}){\makebox(0,0)[b]{${\rho_f g}$}}
\end{picture}
}&\dTo>{fg}\\
X'IY'&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,25)(28,11)
\put(0,21){\makebox(0,0)[b]{$\scriptstyle {\rho_{X'}Y'}$}}
\put(28.6,10.7){\vector(2,-1){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{K}}}_{X',Y'}}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-18)(28,-4)
\put(0,-14){\makebox(0,0)[t]{$\scriptstyle {X'\lambda_{Y'}}$}}
\put(28.6,-3.7){\vector(2,1){0}}
\end{picture}
}&X'Y'
\end{diagram}
\vspace{24pt}$$
Particularly useful is naturality of $\lambda$ with respect to $\lambda_X$ and naturality of $\rho$ with respect to $\rho_X$. In these cases, since $\lambda_X$ and $\rho_X$ are equi-arrows, we can cancel them and find the following invertible $2$-cells: $$\begin{aligned}
{\ensuremath{\mathsf{N}}}^\lambda : I \lambda_X &\Rightarrow \lambda_{IX} \\
{\ensuremath{\mathsf{N}}}^\rho\,: \,\rho_{XI} &\Rightarrow X\rho_I ,
\end{aligned}$$ in analogy with Observation (5) of monoidal categories.
The following lemma holds for ${\ensuremath{\mathsf{K}}}$ independently of Axioms and :
\[NK=KN\] The Kelly cell ${\ensuremath{\mathsf{K}}}$ satisfies the equation $$\begin{diagram}[w=9ex,h=7ex,tight]
XIIY&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,33)(28,11)
\put(0,24){\makebox(0,0)[b]{$\scriptstyle {\rho_X IY}$}}
\put(28.6,10.7){\vector(1,-1){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{15}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{K}}}_{X,IY}}$}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{-12}){\makebox(0,0)[b]{${X{\ensuremath{\mathsf{N}}}^\lambda}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\put(15,4){\makebox(0,0)[b]{$\scriptstyle {X\lambda_{IY}}$}}
\put(-21,3){\vector(1,0){44}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-26)(28,-4)
\put(0,-17){\makebox(0,0)[t]{$\scriptstyle {XI\lambda_Y}$}}
\put(28.6,-3.7){\vector(1,1){0}}
\end{picture}
}&XIY
\end{diagram}
\qquad = \qquad
\begin{diagram}[w=9ex,h=7ex,tight]
XIIY&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,33)(28,11)
\put(0,24){\makebox(0,0)[b]{$\scriptstyle {\rho_X IY}$}}
\put(28.6,10.7){\vector(1,-1){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{15}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{N}}}^\rho Y}$}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{-15}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{K}}}_{XI,Y}}$}}
\end{picture}
}
{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\put(15,4){\makebox(0,0)[b]{$\scriptstyle {\rho_{XI}Y}$}}
\put(-21,3){\vector(1,0){44}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-26)(28,-4)
\put(0,-17){\makebox(0,0)[t]{$\scriptstyle {XI\lambda_Y}$}}
\put(28.6,-3.7){\vector(1,1){0}}
\end{picture}
}&XIY
\end{diagram}
\vspace{30pt}$$
It is enough to establish this equation after post-whiskering with $X\lambda_Y$. The rest is a routine calculation, using on one side the definition of the cell ${\ensuremath{\mathsf{N}}}^\lambda$, then naturality of ${\ensuremath{\mathsf{K}}}$ with respect to $f=X$ and $g=\lambda_Y$. On the other side, use the definition of ${\ensuremath{\mathsf{N}}}^\rho$ and then naturality of ${\ensuremath{\mathsf{K}}}$ with respect to $f=\rho_X$ and $g=Y$. In the end, two ${\ensuremath{\mathsf{K}}}$-cells cancel.
Combining the $2$-cells described so far we get $$\rho_I I \stackrel{{\ensuremath{\mathsf{K}}}{^{\scriptscriptstyle -1}}}{\Rightarrow} I\lambda_I
\stackrel{{\ensuremath{\mathsf{N}}}^\lambda}{\Rightarrow} \lambda_{II} \stackrel{{\ensuremath{\mathsf{K}}}^\lambda}{\Rightarrow} \lambda_I I$$ and hence, by cancelling $I$ on the right, an invertible $2$-cell $${\ensuremath{\mathsf{P}}} : \rho_I \Rightarrow \lambda_I .$$ Now we could also define ${\ensuremath{\mathsf{Q}}} : \rho_I \Rightarrow \lambda_I$ in terms of $$I\rho_I \stackrel{{\ensuremath{\mathsf{K}}}^\rho} \Rightarrow
\rho_{II} \stackrel{{\ensuremath{\mathsf{N}}}^\rho} \Rightarrow
\rho_I I \stackrel{{\ensuremath{\mathsf{K}}}{^{\scriptscriptstyle -1}}}\Rightarrow
I\lambda_I .$$ Finally, in analogy with Axiom (1) for monoidal categories:
\[U=V\] We have ${\ensuremath{\mathsf{P}}} = {\ensuremath{\mathsf{Q}}}$. (This is true independently of Axioms and .)
Since $I$ is cancellable, it is enough to show $I{\ensuremath{\mathsf{P}}} I = I{\ensuremath{\mathsf{Q}}} I$. To establish this equation, use the constructions of ${\ensuremath{\mathsf{P}}}$ and ${\ensuremath{\mathsf{Q}}}$, then substitute the characterising Equations for the auxiliary cells ${\ensuremath{\mathsf{K}}}^\lambda$ and ${\ensuremath{\mathsf{K}}}^\rho$, and finally use Lemma \[NK=KN\].
[The $2$-category of GPS units.]{}\[GPS\] For short we shall say [*GPS unit*]{} for the notion of unit just introduced. In summary, a GPS unit is a quadruple $(I, \lambda, \rho, {\ensuremath{\mathsf{K}}})$ where $I$ is an object, $\lambda_X$ and $\rho_X$ are natural transformations with equi-arrow components, and ${\ensuremath{\mathsf{K}}} : X\lambda_Y
\Rightarrow \rho_X Y$ is a good Kelly cell (natural in $X$ and $Y$, of course).
A [*morphism of GPS units*]{} from $(I,\lambda,\rho,{\ensuremath{\mathsf{K}}})$ to $(J,\ell, r,
{\ensuremath{\mathsf{H}}})$ is an arrow $u: I \to J$ equipped with natural families of invertible $2$-cells $$\begin{diagram}
IX & \rTo^{uX} & JX \\
\dTo<{\lambda_X} & {\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{left}}}}_X & \dTo>{\ell_X} \\
X & \rTo_X & X
\end{diagram}
\qquad\qquad
\begin{diagram}
XI & \rTo^{Xu} & XJ \\
\dTo<{\rho_X} & {\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{right}}}}_X & \dTo>{r_X} \\
X & \rTo_X & X
\end{diagram}$$ satisfying the equation $$\label{PK}
\begin{diagram}[w=6ex,h=6ex,tight]
XIY&\rTo^{XuY}&XJY\\
{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(-9,-25)(-23,0)(-9,25)
\put(-20,0){\makebox(0,0)[r]{$\scriptstyle {X\lambda_Y}$}}
\put(-8.7,-25.6){\vector(1,-2){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{K}}}}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(9,-25)(23,0)(9,25)
\put(20,0){\makebox(0,0)[l]{$\scriptstyle {\rho_X Y}$}}
\put(8.7,-25.6){\vector(-1,-2){0}}
\end{picture}
}&{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put({20},0){\makebox(0,0)[b]{${{\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{right}}}}_X Y}$}}
\end{picture}
}&{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(9,-25)(23,0)(9,25)
\put(20,0){\makebox(0,0)[l]{$\scriptstyle {r_X Y}$}}
\put(8.7,-25.6){\vector(-1,-2){0}}
\end{picture}
}\\
XY&\rTo_{XY}&XY
\end{diagram}
\qquad\qquad = \qquad\qquad
\begin{diagram}[w=6ex,h=6ex,tight]
XIY&\rTo^{XuY}&XJY\\
{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(-9,-25)(-23,0)(-9,25)
\put(-20,0){\makebox(0,0)[r]{$\scriptstyle {X\lambda_Y}$}}
\put(-8.7,-25.6){\vector(1,-2){0}}
\end{picture}
}&{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put({-20},0){\makebox(0,0)[b]{${X{\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{left}}}}_Y}$}}
\end{picture}
}&{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(-9,-25)(-23,0)(-9,25)
\put(-20,0){\makebox(0,0)[r]{$\scriptstyle {X\ell_Y}$}}
\put(-8.7,-25.6){\vector(1,-2){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{H}}}}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(9,-25)(23,0)(9,25)
\put(20,0){\makebox(0,0)[l]{$\scriptstyle {r_X Y}$}}
\put(8.7,-25.6){\vector(-1,-2){0}}
\end{picture}
}\\
XY&\rTo_{XY}&XY
\end{diagram}$$ Finally, a [*$2$-morphism of GPS unit maps*]{} is a $2$-cell ${\ensuremath{\mathsf{T}}}: u
\Rightarrow v$ satisfying the compatibility conditions and of Lemma \[unit2map\].
[Remarks.]{} Note first that $u$ is automatically an equi-arrow. Observe also that ${\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{left}}}}$ and ${\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{right}}}}$ completely determine each other by Equation , as is easily seen by taking respectively $X$ to be a left cancellable object and $Y$ to be a right cancellable object. Finally note that there are two further equations, expressing compatibility with ${\ensuremath{\mathsf{K}}}^\lambda$ and ${\ensuremath{\mathsf{K}}}^\rho$, but they can be deduced from Equation , independently of the coherence Axioms and . Here is the one for ${\ensuremath{\mathsf{K}}}^\lambda$ for future reference: $$\label{KP=PH}
\begin{diagram}[w=6ex,h=6ex,tight]
IXY&\rTo^{uXY}&JXY\\
{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(-9,-25)(-23,0)(-9,25)
\put(-20,0){\makebox(0,0)[r]{$\scriptstyle {\lambda_{XY}}$}}
\put(-8.7,-25.6){\vector(1,-2){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{K}}}^\lambda}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(9,-25)(23,0)(9,25)
\put(20,0){\makebox(0,0)[l]{$\scriptstyle {\lambda_X Y}$}}
\put(8.7,-25.6){\vector(-1,-2){0}}
\end{picture}
}&{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put({20},0){\makebox(0,0)[b]{${{\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{left}}}}_X Y}$}}
\end{picture}
}&{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(9,-25)(23,0)(9,25)
\put(20,0){\makebox(0,0)[l]{$\scriptstyle {\ell_X Y}$}}
\put(8.7,-25.6){\vector(-1,-2){0}}
\end{picture}
}\\
XY&\rTo_{XY}&XY
\end{diagram}
\qquad\qquad = \qquad\qquad
\begin{diagram}[w=6ex,h=6ex,tight]
IXY&\rTo^{uXY}&JXY\\
{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(-9,-25)(-23,0)(-9,25)
\put(-20,0){\makebox(0,0)[r]{$\scriptstyle {\lambda_{XY}}$}}
\put(-8.7,-25.6){\vector(1,-2){0}}
\end{picture}
}&{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put({-20},0){\makebox(0,0)[b]{${{\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{left}}}}_{XY}}$}}
\end{picture}
}&{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(-9,-25)(-23,0)(-9,25)
\put(-20,0){\makebox(0,0)[r]{$\scriptstyle {\ell_{XY}}$}}
\put(-8.7,-25.6){\vector(1,-2){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{H}}}^\ell}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(9,-25)(23,0)(9,25)
\put(20,0){\makebox(0,0)[l]{$\scriptstyle {\ell_X Y}$}}
\put(8.7,-25.6){\vector(-1,-2){0}}
\end{picture}
}\\
XY&\rTo_{XY}&XY
\end{diagram}$$
Comparison with classical theory (Theorem \[thmE\]) {#sec:comparison}
===================================================
In this section we prove the equivalence between the two notions of unit.
[From cancellable-idempotent units to GPS units.]{} We fix a unit object $(I,\alpha)$. We also assume chosen a left constraint $\lambda_X : IX \to X$ with ${\ensuremath{\mathsf{L}}}_X:
I\lambda_X \Rightarrow \alpha X$, and a right constraint $\rho_X : XI \to X$ with ${\ensuremath{\mathsf{R}}}_X: X \alpha \Rightarrow \rho_X I$. First of all, in analogy with Axioms (2) and (3) of monoidal categories we have:
\[lambdaXY\] There are unique natural invertible $2$-cells $$\begin{aligned}
{\ensuremath{\mathsf{K}}}^\lambda : \lambda_{XY} &\Rightarrow \lambda_X Y \\
{\ensuremath{\mathsf{K}}}^\rho: X \rho_Y &\Rightarrow \rho_{XY}
\end{aligned}$$ satisfying $$\label{IIAB}
\begin{diagram}[w=6ex,h=6ex,tight]
IIXY&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,25)(28,11)
\put(0,21){\makebox(0,0)[b]{$\scriptstyle {I\lambda_X Y}$}}
\put(28.6,10.7){\vector(2,-1){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${I{\ensuremath{\mathsf{K}}}^\lambda}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-18)(28,-4)
\put(0,-14){\makebox(0,0)[t]{$\scriptstyle {I\lambda_{XY}}$}}
\put(28.6,-3.7){\vector(2,1){0}}
\end{picture}
}&IXY
\end{diagram}
\qquad = \qquad
\begin{diagram}[w=8ex,h=6ex,tight]
IIXY&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,33)(28,11)
\put(0,24){\makebox(0,0)[b]{$\scriptstyle {I\lambda_X Y}$}}
\put(28.6,10.7){\vector(1,-1){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{12}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{L}}}{^{\scriptscriptstyle -1}}Y}$}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{-12}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{L}}}}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\put(15,4){\makebox(0,0)[b]{$\scriptstyle {\alpha XY}$}}
\put(-21,3){\vector(1,0){44}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-26)(28,-4)
\put(0,-17){\makebox(0,0)[t]{$\scriptstyle {I\lambda_{XY}}$}}
\put(28.6,-3.7){\vector(1,1){0}}
\end{picture}
}&IXY
\end{diagram}
\vspace{30pt}$$
$$\label{XYII}
\begin{diagram}[w=6ex,h=6ex,tight]
XYII&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,25)(28,11)
\put(0,21){\makebox(0,0)[b]{$\scriptstyle {\rho_{XY}I}$}}
\put(28.6,10.7){\vector(2,-1){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{K}}}^\rho I}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-18)(28,-4)
\put(0,-14){\makebox(0,0)[t]{$\scriptstyle {X\rho_Y I}$}}
\put(28.6,-3.7){\vector(2,1){0}}
\end{picture}
}&XYI
\end{diagram}
\qquad = \qquad
\begin{diagram}[w=8ex,h=6ex,tight]
XYII&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,33)(28,11)
\put(0,24){\makebox(0,0)[b]{$\scriptstyle {\rho_{XY}I}$}}
\put(28.6,10.7){\vector(1,-1){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{12}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{R}}}}$}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{-12}){\makebox(0,0)[b]{${X{\ensuremath{\mathsf{R}}}{^{\scriptscriptstyle -1}}}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\put(15,4){\makebox(0,0)[b]{$\scriptstyle {XY\alpha}$}}
\put(-21,3){\vector(1,0){44}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-26)(28,-4)
\put(0,-17){\makebox(0,0)[t]{$\scriptstyle {X\rho_Y I}$}}
\put(28.6,-3.7){\vector(1,1){0}}
\end{picture}
}&XYI
\end{diagram}
\vspace{30pt}$$
The condition precisely defines the $2$-cell, since $I$ is cancellable.
For fixed left constraint $(\lambda,{\ensuremath{\mathsf{L}}})$ and fixed right constraint $(\rho,{\ensuremath{\mathsf{R}}})$, there is a canonical family of invertible $2$-cells (the Kelly cell) $${\ensuremath{\mathsf{K}}} : X \lambda_Y \Rightarrow \rho_X Y ,$$ natural in $X$ and $Y$.
This is analogous to the construction of the associator: ${\ensuremath{\mathsf{K}}}$ is defined as the unique $2$-cell ${\ensuremath{\mathsf{K}}}: X\lambda_Y
\Rightarrow \rho_X Y$ satisfying the equation $$\label{RYXL}
\begin{diagram}[w=52pt,h=39pt,tight]
XIIY & {\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-26,6)(0,14)(26,6)
\put(0,12){\makebox(0,0)[b]{$\scriptstyle {X\alpha
Y}$}}
\put(26.6,5.7){\vector(3,-1){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${X{\ensuremath{\mathsf{L}}}}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-26,-1)(0,-9)(26,-1)
\put(0,-7){\makebox(0,0)[t]{$\scriptstyle {XI\lambda_Y}$}}
\put(26.6,-0.7){\vector(3,1){0}}
\end{picture}
} &
XIY \\
{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(-3,-25)(-16,0)(-3,25)
\put(-13,0){\makebox(0,0)[r]{$\scriptstyle {X\alpha Y}$}}
\put(-2.5,-25.6){\vector(1,-2){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{R}}}Y}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(3,-25)(16,0)(3,25)
\put(13,0){\makebox(0,0)[l]{$\scriptstyle {\rho_X IY}$}}
\put(2.7,-25.6){\vector(-1,-2){0}}
\end{picture}
} &
{\copyright}& \dTo>{\rho_X Y} \\
XIY & \rTo_{X\lambda_Y} & XY
\end{diagram}
\qquad = \qquad
\begin{diagram}[w=30pt,h=22.5pt,tight,hug]
XIIY &&&& \\
&\rdTo^{X\alpha Y} &&&& \\
&& XIY && \\
&&&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\psr{\begin{picture}(0,0)(0,0)
\qbezier(-34,6)(0,16)(30,6)
\put(0,13){\makebox(0,0)[b]{$\scriptstyle {\rho_X Y}$}}
\put(30.6,5.7){\vector(3,-1){0}}
\end{picture}}
}
{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\psr{\begin{picture}(0,0)(0,0)
\qbezier(-34,1)(0,-8)(30,1)
\put(0,-6){\makebox(0,0)[t]{$\scriptstyle {X\lambda_Y}$}}
\put(30.6,1.3){\vector(3,1){0}}
\end{picture}}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\psr{\begin{picture}(0,0)(0,0)
\put(2,8){\makebox(0,0)[t]{${\ensuremath{\mathsf{K}}}$}}
\end{picture}}
} & \\
&&&& XY
\end{diagram}
\vspace{24pt}$$ This makes sense since $X\alpha Y$ is an equi-arrow, so we can cancel it away. Clearly ${\ensuremath{\mathsf{K}}}$ is invertible since ${\ensuremath{\mathsf{L}}}$ and ${\ensuremath{\mathsf{R}}}$ are.
We constructed ${\ensuremath{\mathsf{K}}}^\lambda$ and ${\ensuremath{\mathsf{K}}}^\rho$ directly from ${\ensuremath{\mathsf{L}}}$, and ${\ensuremath{\mathsf{R}}}$. Meanwhile we also constructed ${\ensuremath{\mathsf{K}}}$, and we know from classical theory (\[Klambda-from-K\]) that this cell determines the two others. The following proposition shows that all these constructions match up, and in particular that the constructed Kelly cell is coherent on both sides:
\[Kcoh\] The families of $2$-cells constructed, ${\ensuremath{\mathsf{K}}}$, ${\ensuremath{\mathsf{K}}}^\lambda$ and ${\ensuremath{\mathsf{K}}}^\rho$ satisfy the GPS unit axioms and : $$\xymatrixrowsep{50pt}
\xymatrixcolsep{42pt}
\xymatrix @!=0pt {
X \lambda_{YZ} \ar@{=>}[rr]^{X{\ensuremath{\mathsf{K}}}^\lambda_{Y,Z}} \ar@{=>}[dr]_{{\ensuremath{\mathsf{K}}}_{X,YZ}} && X\lambda_Y Z \ar@{=>}[dl]^{{\ensuremath{\mathsf{K}}}_{X,Y} Z} \\
& \rho_X YZ &
}
\qquad\qquad
\xymatrixrowsep{50pt}
\xymatrixcolsep{42pt}
\xymatrix @!=0pt {
X\rho_Y Z \ar@{=>}[rr]^{{\ensuremath{\mathsf{K}}}^\rho_{X,Y} Z} \ar@{<=}[dr]_{X{\ensuremath{\mathsf{K}}}_{Y,Z}} &&
\rho_{XY} Z \ar@{<=}[dl]^{{\ensuremath{\mathsf{K}}}_{XY,Z}} \\
& XY\lambda_Z &
}$$
We treat the left constraint (the right constraint being completely analogous). We need to establish $$\begin{diagram}[w=10ex,h=6ex,tight]
XIYZ&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,33)(28,11)
\put(0,24){\makebox(0,0)[b]{$\scriptstyle {\rho_X YZ}$}}
\put(28.6,10.7){\vector(1,-1){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{15}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{K}}}_{X,Y}Z}$}}
\end{picture}
}
{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{-17}){\makebox(0,0)[b]{${X{\ensuremath{\mathsf{K}}}^\lambda_{Y,Z}}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\put(15,4){\makebox(0,0)[b]{$\scriptstyle {X\lambda_Y Z}$}}
\put(-21,3){\vector(1,0){44}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-26)(28,-4)
\put(0,-17){\makebox(0,0)[t]{$\scriptstyle {X\lambda_{YZ}}$}}
\put(28.6,-3.7){\vector(1,1){0}}
\end{picture}
}&XYZ
\end{diagram}
\qquad = \qquad
\begin{diagram}[w=8ex,h=6ex,tight]
XIYZ&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,25)(28,11)
\put(0,21){\makebox(0,0)[b]{$\scriptstyle {\rho_X YZ}$}}
\put(28.6,10.7){\vector(2,-1){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{-2}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{K}}}_{X,YZ}}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-18)(28,-4)
\put(0,-14){\makebox(0,0)[t]{$\scriptstyle {X\lambda_{YZ}}$}}
\put(28.6,-3.7){\vector(2,1){0}}
\end{picture}
}&XYZ
\end{diagram}
\vspace{30pt}$$ and it is enough to establish this equation pre-whiskered with $X\alpha Y Z$. In the diagram resulting from the left-hand side: $$\begin{diagram}[w=10ex,h=6ex,tight]
XIIYZ & \rTo^{X\alpha YZ} &
XIYZ&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,33)(28,11)
\put(0,24){\makebox(0,0)[b]{$\scriptstyle {\rho_X YZ}$}}
\put(28.6,10.7){\vector(1,-1){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{15}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{K}}}_{X,Y}Z}$}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{-17}){\makebox(0,0)[b]{${X{\ensuremath{\mathsf{K}}}^\lambda_{Y,Z}}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\put(15,4){\makebox(0,0)[b]{$\scriptstyle {X\lambda_Y Z}$}}
\put(-21,3){\vector(1,0){44}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-26)(28,-4)
\put(0,-17){\makebox(0,0)[t]{$\scriptstyle {X\lambda_{YZ}}$}}
\put(28.6,-3.7){\vector(1,1){0}}
\end{picture}
}&XYZ
\end{diagram}
\vspace{30pt}$$ we can replace $(X\alpha YZ) \# ({\ensuremath{\mathsf{K}}}_{X,Y}Z)$ by the expression that defined ${\ensuremath{\mathsf{K}}}_{X,Y}Z$ (cf. \[RYXL\]), yielding altogether
& & XIYZ & &\
& [ by ]{} [ by ]{}[ by ]{} & & \^[\_X YZ]{} &\
XIIYZ & & [©]{}& & XYZ\
& [ by ]{} [ by ]{}[ by ]{} & &[ by ]{} [ by ]{}[ by ]{} &\
&& XIYZ
Here we can move the cell $X{\ensuremath{\mathsf{K}}}^\lambda_{Y,Z}$ across the square, where it becomes $XI {\ensuremath{\mathsf{K}}}^\lambda_{Y,Z}$ and combines with $X{\ensuremath{\mathsf{L}}}_Y Z$ to give altogether $X {\ensuremath{\mathsf{L}}}_{YZ}$ (cf. ). The resulting diagram
& & XIYZ & &\
& [ by ]{} [ by ]{}[ by ]{} & & \^[\_X YZ]{} &\
XIIYZ & & [©]{}& & XYZ\
& [ by ]{} [ by ]{}[ by ]{} & & [ by ]{} &\
&& XIYZ
is nothing but $$\begin{diagram}[w=8ex,h=6ex,tight]
XIIYZ & \rTo^{X\alpha YZ} &
XIYZ&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,25)(28,11)
\put(0,21){\makebox(0,0)[b]{$\scriptstyle {\rho_X YZ}$}}
\put(28.6,10.7){\vector(2,-1){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{-2}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{K}}}_{X,YZ}}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-18)(28,-4)
\put(0,-14){\makebox(0,0)[t]{$\scriptstyle {X\lambda_{YZ}}$}}
\put(28.6,-3.7){\vector(2,1){0}}
\end{picture}
}&XYZ
\end{diagram}
\vspace{30pt}$$ (by Equation again) which is what we wanted to establish.
Hereby we have concluded the construction of a GPS unit from $(I,\alpha)$. We will also need a result for morphisms:
\[Kelly-compatibility\] Let $(u,{\ensuremath{\mathsf{U}}}) :(I,\alpha) \to (J,\beta)$ be a morphism of units in the sense of \[main-def\], and consider the two canonical $2$-cells ${\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{left}}}}$ and ${\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{right}}}}$ constructed in Lemma \[unitmap\]. Then Equation holds: $$\begin{diagram}[w=6ex,h=6ex,tight]
XIY&\rTo^{XuY}&XJY\\
{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(-9,-25)(-23,0)(-9,25)
\put(-20,0){\makebox(0,0)[r]{$\scriptstyle {X\lambda_Y}$}}
\put(-8.7,-25.6){\vector(1,-2){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{K}}}}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(9,-25)(23,0)(9,25)
\put(20,0){\makebox(0,0)[l]{$\scriptstyle {\rho_X Y}$}}
\put(8.7,-25.6){\vector(-1,-2){0}}
\end{picture}
}&{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put({20},0){\makebox(0,0)[b]{${{\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{right}}}}_X Y}$}}
\end{picture}
}&{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(9,-25)(23,0)(9,25)
\put(20,0){\makebox(0,0)[l]{$\scriptstyle {r_X Y}$}}
\put(8.7,-25.6){\vector(-1,-2){0}}
\end{picture}
}\\
XY&\rTo_{XY}&XY
\end{diagram}
\qquad\qquad = \qquad\qquad
\begin{diagram}[w=6ex,h=6ex,tight]
XIY&\rTo^{XuY}&XJY\\
{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(-9,-25)(-23,0)(-9,25)
\put(-20,0){\makebox(0,0)[r]{$\scriptstyle {X\lambda_Y}$}}
\put(-8.7,-25.6){\vector(1,-2){0}}
\end{picture}
}&{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put({-20},0){\makebox(0,0)[b]{${X{\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{left}}}}_Y}$}}
\end{picture}
}&{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(-9,-25)(-23,0)(-9,25)
\put(-20,0){\makebox(0,0)[r]{$\scriptstyle {X\ell_Y}$}}
\put(-8.7,-25.6){\vector(1,-2){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{H}}}}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(9,-25)(23,0)(9,25)
\put(20,0){\makebox(0,0)[l]{$\scriptstyle {r_X Y}$}}
\put(8.7,-25.6){\vector(-1,-2){0}}
\end{picture}
}\\
XY&\rTo_{XY}&XY
\end{diagram}$$ (Hence $(u,{\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{left}}}},{\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{right}}}})$ is a morphism of GPS units.)
It is enough to prove the equation obtained by pasting the $2$-cell $X{\ensuremath{\mathsf{U}}}
Y$ on top of each side of the equation. This enables us to use the characterising equation for ${\ensuremath{\mathsf{K}}}$ and ${\ensuremath{\mathsf{H}}}$. After this rewriting, we are in position to apply Equations and , and after cancelling ${\ensuremath{\mathsf{R}}}$ and ${\ensuremath{\mathsf{L}}}$ cells, the resulting equation amounts to a cube, where it is easy to see that each side is just ${\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{right}}}}_X {\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{left}}}}_Y$.
[From GPS units to cancellable-idempotent units.]{} Given a GPS unit $(I, \lambda, \rho, {\ensuremath{\mathsf{K}}})$, just put $$\alpha {\: {\raisebox{0.255ex}{\normalfont\scriptsize :\!\!}}=}\lambda_I ,$$ then $(I,\alpha)$ is a unit object in the sense of \[main-def\]. Indeed, we already observed that $I$ is cancellable (\[Icanc\]), and from the outset $\lambda_I$ is an equi-arrow. That’s all! To construct it we didn’t even need the Kelly cell, or any of the auxiliary cells or their axioms.
[Left and right actions from the Kelly cell.]{}\[LRfromK\] Start with natural left and right constraints $\lambda$ and $\rho$ and a Kelly cell ${\ensuremath{\mathsf{K}}} : X\lambda_Y \Rightarrow \rho_X Y$ (not required to be coherent on either side). Construct ${\ensuremath{\mathsf{K}}}^\lambda$ as in \[Klambda-from-K\], put $\alpha {\: {\raisebox{0.255ex}{\normalfont\scriptsize :\!\!}}=}\lambda_I$, and define left and right actions as follows. We define ${\ensuremath{\mathsf{L}}}_X$ as $$I\lambda_X \stackrel{{\ensuremath{\mathsf{N}}}^\lambda}{\Rightarrow} \lambda_{IX}
\stackrel{{\ensuremath{\mathsf{K}}}^\lambda}{\Rightarrow} \lambda_I X = \alpha X ,$$ while we define ${\ensuremath{\mathsf{R}}}_X$ simply as $$X\alpha = X\lambda_I \stackrel{{\ensuremath{\mathsf{K}}}_{X,I}}{\Rightarrow} \rho_X I .$$
\[KLR\] For fixed $(I,\lambda,\rho,{\ensuremath{\mathsf{K}}})$, the following are equivalent:
The left and right $2$-cells ${\ensuremath{\mathsf{L}}}$ and ${\ensuremath{\mathsf{R}}}$ just constructed in \[LRfromK\] are compatible with the Kelly cell in the sense of Equation .
The Kelly cell ${\ensuremath{\mathsf{K}}}$ is coherent on the left (i.e. satisfies Axiom ).
The Kelly cell ${\ensuremath{\mathsf{K}}}$ is coherent on the right (i.e. satisfies Axiom ).
Proposition \[Kcoh\] already says that (i) implies both (ii) and (ii’). To prove (ii)[$\Rightarrow$]{}(i), we start with an auxiliary observation: by massaging the naturality equation $$\begin{diagram}[w=6ex,h=6ex,tight]
XIIY&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,25)(28,11)
\put(0,21){\makebox(0,0)[b]{$\scriptstyle {\rho_X IY}$}}
\put(28.6,10.7){\vector(2,-1){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{K}}}_{X,IY}}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-18)(28,-4)
\put(0,-14){\makebox(0,0)[t]{$\scriptstyle {X\lambda_{IY}}$}}
\put(28.6,-3.7){\vector(2,1){0}}
\end{picture}
}&XIY\\
\dTo<{XI\lambda_Y}&{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{-12}){\makebox(0,0)[b]{${X\lambda_{\lambda_Y}}$}}
\end{picture}
}&\dTo>{X\lambda_Y}\\
XIY&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-18)(28,-4)
\put(0,-14){\makebox(0,0)[t]{$\scriptstyle {X\lambda_Y}$}}
\put(28.6,-3.7){\vector(2,1){0}}
\end{picture}
}&XY
\end{diagram}
\qquad = \qquad
\begin{diagram}[w=6ex,h=6ex,tight]
XIIY&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,25)(28,11)
\put(0,21){\makebox(0,0)[b]{$\scriptstyle {\rho_X IY}$}}
\put(28.6,10.7){\vector(2,-1){0}}
\end{picture}
}&XIY\\
\dTo<{XI\lambda_Y}&{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{12}){\makebox(0,0)[b]{${{\copyright}}$}}
\end{picture}
}&\dTo>{X\lambda_Y}\\
XIY&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,25)(28,11)
\put(0,21){\makebox(0,0)[b]{$\scriptstyle {\rho_X Y}$}}
\put(28.6,10.7){\vector(2,-1){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{K}}}_{X,Y}}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-18)(28,-4)
\put(0,-14){\makebox(0,0)[t]{$\scriptstyle {X\lambda_Y}$}}
\put(28.6,-3.7){\vector(2,1){0}}
\end{picture}
}&XY
\end{diagram}
\vspace{24pt}$$ we find the equation $$\label{tailored}
\begin{diagram}[w=7ex,h=7ex,tight]
XIIY&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,25)(28,11)
\put(0,21){\makebox(0,0)[b]{$\scriptstyle {X \lambda_{IY}}$}}
\put(28.6,10.7){\vector(2,-1){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${X{\ensuremath{\mathsf{N}}}^\lambda}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-18)(28,-4)
\put(0,-14){\makebox(0,0)[t]{$\scriptstyle {XI\lambda_Y}$}}
\put(28.6,-3.7){\vector(2,1){0}}
\end{picture}
}&XIY\\
\dTo<{\rho_X IY}&{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{-12}){\makebox(0,0)[b]{${{\copyright}}$}}
\end{picture}
}&\dTo>{\rho_X Y}\\
XIY&\rTo_{X\lambda_Y}&XY
\end{diagram}
\qquad = \qquad
\begin{diagram}[w=7ex,h=7ex,tight]
XIIY&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,25)(28,11)
\put(0,21){\makebox(0,0)[b]{$\scriptstyle {X \lambda_{IY}}$}}
\put(28.6,10.7){\vector(2,-1){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{K}}}{^{\scriptscriptstyle -1}}_{X,IY}}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-18)(28,-4)
\put(0,-14){\makebox(0,0)[t]{$\scriptstyle {\rho_X IY}$}}
\put(28.6,-3.7){\vector(2,1){0}}
\end{picture}
}&XIY\\
\dTo<{\rho_X IY}&{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{-12}){\makebox(0,0)[b]{${{\copyright}}$}}
\end{picture}
}&
{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(-9,-25)(-23,0)(-9,25)
\put(-20,0){\makebox(0,0)[r]{$\scriptstyle {X\lambda_Y}$}}
\put(-8.7,-25.6){\vector(1,-2){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{K}}}_{X,Y}}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(9,-25)(23,0)(9,25)
\put(20,0){\makebox(0,0)[l]{$\scriptstyle {\rho_X Y}$}}
\put(8.7,-25.6){\vector(-1,-2){0}}
\end{picture}
}\\
XIY&\rTo_{X\lambda_Y}&XY
\end{diagram}
\vspace{24pt}$$ tailor-made to a substitution we shall perform in a moment.
Now for the main computation, assuming first that ${\ensuremath{\mathsf{K}}}$ is coherent on the left, i.e. that Axiom holds. Start with the left-hand side of Equation , and insert the definitions we made for ${\ensuremath{\mathsf{L}}}$ and ${\ensuremath{\mathsf{R}}}$ to arrive at $$\begin{diagram}[w=60pt,h=50pt,tight]
XIIY & {\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,33)(28,11)
\put(0,24){\makebox(0,0)[b]{$\scriptstyle {X\lambda_I Y}$}}
\put(28.6,10.7){\vector(1,-1){0}}
\end{picture}
}
{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{15}){\makebox(0,0)[b]{${X{\ensuremath{\mathsf{K}}}^\lambda}$}}
\end{picture}
}
{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\put(15,4){\makebox(0,0)[b]{$\scriptstyle {X\lambda_{IY}}$}}
\put(-21,3){\vector(1,0){44}}
\end{picture}
}
{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{-15}){\makebox(0,0)[b]{${X{\ensuremath{\mathsf{N}}}^\lambda}$}}
\end{picture}
}
{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-26)(28,-4)
\put(0,-17){\makebox(0,0)[t]{$\scriptstyle {XI\lambda_Y}$}}
\put(28.6,-3.7){\vector(1,1){0}}
\end{picture}
} &
XIY \\
{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(-3,-25)(-16,0)(-3,25)
\put(-13,0){\makebox(0,0)[r]{$\scriptstyle {X\lambda_I Y}$}}
\put(-2.5,-25.6){\vector(1,-2){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{K}}}Y}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(3,-25)(16,0)(3,25)
\put(13,0){\makebox(0,0)[l]{$\scriptstyle {\rho_X IY}$}}
\put(2.7,-25.6){\vector(-1,-2){0}}
\end{picture}
} &
{\copyright}& \dTo>{\rho_X Y} \\
XIY & \rTo_{X\lambda_Y} & XY
\end{diagram}
\vspace{24pt}$$ in which we can now substitute to get $$\begin{diagram}[w=72pt,h=60pt,tight]
XIIY & {\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,11)(0,33)(28,11)
\put(0,24){\makebox(0,0)[b]{$\scriptstyle {X\lambda_I Y}$}}
\put(28.6,10.7){\vector(1,-1){0}}
\end{picture}
}
{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{15}){\makebox(0,0)[b]{${X{\ensuremath{\mathsf{K}}}^\lambda}$}}
\end{picture}
}
{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\put(15,4){\makebox(0,0)[b]{$\scriptstyle {X\lambda_{IY}}$}}
\put(-21,3){\vector(1,0){44}}
\end{picture}
}
{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{-15}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{K}}}{^{\scriptscriptstyle -1}}_{X,IY}}$}}
\end{picture}
}
{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,0)
\qbezier(-28,-4)(0,-26)(28,-4)
\put(0,-17){\makebox(0,0)[t]{$\scriptstyle {\rho_X IY}$}}
\put(28.6,-3.7){\vector(1,1){0}}
\end{picture}
} &
XIY \\
{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(-3,-25)(-16,0)(-3,25)
\put(-13,0){\makebox(0,0)[r]{$\scriptstyle {X\lambda_I Y}$}}
\put(-2.5,-25.6){\vector(1,-2){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{K}}}_{X,I}Y}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(3,-25)(16,0)(3,25)
\put(13,0){\makebox(0,0)[l]{$\scriptstyle {\rho_X IY}$}}
\put(2.7,-25.6){\vector(-1,-2){0}}
\end{picture}
} &
{\copyright}& {\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(-9,-25)(-23,0)(-9,25)
\put(-20,0){\makebox(0,0)[r]{$\scriptstyle {X\lambda_Y}$}}
\put(-8.7,-25.6){\vector(1,-2){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{K}}}_{X,Y}}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(9,-25)(23,0)(9,25)
\put(20,0){\makebox(0,0)[l]{$\scriptstyle {\rho_X Y}$}}
\put(8.7,-25.6){\vector(-1,-2){0}}
\end{picture}
} \\
XIY & \rTo_{X\lambda_Y} & XY
\end{diagram}
\vspace{24pt}$$ Here finally the three $2$-cells incident to the $XIIY$ vertex cancel each other out, thanks to Axiom , and in the end, remembering $\alpha=\lambda_I$, we get $$\begin{diagram}[w=36pt,h=28pt,tight,hug]
XIIY &&&& \\
&\rdTo^{X\alpha Y} &&&& \\
&& XIY && \\
&&&{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\psr{\begin{picture}(0,0)(0,0)
\qbezier(-34,6)(0,16)(30,6)
\put(0,13){\makebox(0,0)[b]{$\scriptstyle {\rho_X Y}$}}
\put(30.6,5.7){\vector(3,-1){0}}
\end{picture}}
}
{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\psr{\begin{picture}(0,0)(0,0)
\qbezier(-34,1)(0,-8)(30,1)
\put(0,-6){\makebox(0,0)[t]{$\scriptstyle {X\lambda_Y}$}}
\put(30.6,1.3){\vector(3,1){0}}
\end{picture}}
}{\setlength{\unitlength}{0.003\DiagramCellWidth}
\multiply \unitlength by \scaleFactor
\psr{\begin{picture}(0,0)(0,0)
\put(2,8){\makebox(0,0)[t]{${\ensuremath{\mathsf{K}}}$}}
\end{picture}}
} & \\
&&&& XY
\end{diagram}
\vspace{24pt}$$ as required to establish that ${\ensuremath{\mathsf{K}}}$ satisfies Equation . Hence we have proved (ii)[$\Rightarrow$]{}(i), and therefore altogether (ii)[$\Rightarrow$]{}(ii’). The converse, (ii’)[$\Rightarrow$]{}(ii) follows now by left-right symmetry of the statements. (But note that the proof via (i) is not symmetric, since it relies on the definition $\alpha=\lambda_I$. To spell out a proof of (ii’)[$\Rightarrow$]{}(ii), use rather $\alpha=\rho_I$, observing that the intermediate result (i) would refer to different ${\ensuremath{\mathsf{L}}}$ and ${\ensuremath{\mathsf{R}}}$.)
We have now given a construction in each direction, but both constructions involved choices. With careful choices, applying one construction after the other in either way gets us back where we started. It is clear that this should constitute an equivalence of $2$-categories. However, the involved choices make it awkward to make the correspondence functorial directly. (In technical terms, the constructions are ana-$2$-functors.) We circumvent this by introducing an intermediate $2$-category dominating both. With this auxiliary $2$-category, the results we already proved readily imply the equivalence.
[A correspondence of $2$-categories of units.]{} Let ${\mathscr{U}}$ be following $2$-category. Its objects are septuples $$(I,\alpha,\lambda,\rho,{\ensuremath{\mathsf{L}}}, {\ensuremath{\mathsf{R}}},{\ensuremath{\mathsf{K}}}),$$ with equi-arrows $$\alpha:II\to I, \quad
\lambda_X: IX \to X , \quad
\rho_X : XI \to X,$$ (and accompanying naturality $2$-cell data), and natural invertible $2$-cells $${\ensuremath{\mathsf{L}}} : I \lambda_X \Rightarrow \alpha X, \quad
{\ensuremath{\mathsf{R}}}: X\alpha \Rightarrow \rho_X I, \quad
{\ensuremath{\mathsf{K}}}: X\lambda_Y\Rightarrow \rho_X Y .$$ These data are required to satisfy Equation (compatibility of ${\ensuremath{\mathsf{K}}}$ with ${\ensuremath{\mathsf{L}}}$ and ${\ensuremath{\mathsf{R}}}$). The arrows in ${\mathscr{U}}$ from $(I,\alpha,\lambda,\rho,{\ensuremath{\mathsf{L}}}, {\ensuremath{\mathsf{R}}},{\ensuremath{\mathsf{K}}})$ to $(J,\beta,\ell,r,{\ensuremath{\mathsf{L}}}', {\ensuremath{\mathsf{R}}}',{\ensuremath{\mathsf{H}}})$ are quadruples $$(u, {\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{left}}}}, {\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{right}}}}, {\ensuremath{\mathsf{U}}}) ,$$ where $u:I \to J$ is an arrow in ${\mathscr{C}}$, ${\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{left}}}}$ and ${\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{right}}}}$ are as in \[GPS\], and ${\ensuremath{\mathsf{U}}}$ is a morphism of pseudo-idempotents from $(I,\alpha)$ to $(J,\beta)$. These data are required to satisfy Equation (compatibility with Kelly cells) as well as Equations and in Lemma \[unitmap\] (compatibility with the left and right $2$-cells).
Finally a $2$-cell from $(u, {\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{left}}}}, {\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{right}}}}, {\ensuremath{\mathsf{U}}})$ to $(v, {\ensuremath{\mathsf{V}}}{^{\text{\tiny \rm{left}}}}, {\ensuremath{\mathsf{V}}}{^{\text{\tiny \rm{right}}}}, {\ensuremath{\mathsf{V}}})$ is a $2$-cell $${\ensuremath{\mathsf{T}}} : u \Rightarrow v$$ required to be a $2$-morphism of pseudo-idempotents (compatibility with ${\ensuremath{\mathsf{U}}}$ and ${\ensuremath{\mathsf{V}}}$ as in \[semimonoid-transf\]), and to satisfy Equation (compatibility with ${\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{left}}}}$ and ${\ensuremath{\mathsf{V}}}{^{\text{\tiny \rm{left}}}}$) as well as Equation (compatibility with ${\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{right}}}}$ and ${\ensuremath{\mathsf{V}}}{^{\text{\tiny \rm{right}}}}$).
Let ${\mathscr{E}}$ denote the $2$-category of cancellable-idempotent units introduced in \[main-def\], and let ${\mathscr{G}}$ denote the $2$-category of GPS units of \[GPS\]. We have evident forgetful (strict) $2$-functors
&&\
& \^ && \^\
&&&& .
\[thmE\] The $2$-functors $\Phi$ and $\Psi$ are $2$-equivalences. More precisely they are surjective on objects and strongly fully faithful (i.e. isomorphisms on hom categories).
The $2$-functor $\Phi$ is surjective on objects by Lemma \[LR\] and Proposition \[Kcoh\]. Given an arrow $(u,{\ensuremath{\mathsf{U}}})$ in ${\mathscr{E}}$ and overlying objects in ${\mathscr{U}}$, Lemma \[unitmap\] says there are unique ${\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{left}}}}$ and ${\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{right}}}}$, and Proposition \[Kelly-compatibility\] ensures the required compatibility with Kelly cells (Equation ). Hence $\Phi$ induces a bijection on objects in the hom categories. Lemma \[unit2map\] says we also have a bijection on the level of $2$-cells, hence $\Phi$ is an isomorphism on hom categories. On the other hand, $\Psi$ is surjective on objects by \[LRfromK\] and Proposition \[KLR\]. Given an arrow $(u,{\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{left}}}},{\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{right}}}})$ in ${\mathscr{G}}$, Lemma \[W\] below says that for fixed overlying objects in ${\mathscr{U}}$ there is a unique associated ${\ensuremath{\mathsf{U}}}$, hence $\Psi$ induces a bijection on objects in the hom categories. Finally, Lemma \[unit2map\] gives also a bijection of $2$-cells, hence $\Psi$ is strongly fully faithful.
\[W\] Given a morphism of GPS units
(I,,,K) & \^[(u,[$\mathsf{U}$]{}[\^]{},[$\mathsf{U}$]{}[\^]{})]{} & (J,,r,H)
fix an equi-arrow $\alpha: II {\stackrel{\raisebox{0.1ex}[0ex][0ex]{\(\sim\)}} {\raisebox{-0.15ex}[0.28ex]{\(\rightarrow\)}}}I$ with natural families ${\ensuremath{\mathsf{L}}}_X: I\lambda_X \Rightarrow \alpha X$ and ${\ensuremath{\mathsf{R}}}_X: \alpha X
\Rightarrow \rho_X I$ satisfying Equation (compatibility with ${\ensuremath{\mathsf{K}}}$), and fix an equi-arrow $\beta: JJ {\stackrel{\raisebox{0.1ex}[0ex][0ex]{\(\sim\)}} {\raisebox{-0.15ex}[0.28ex]{\(\rightarrow\)}}}J$ with natural families ${\ensuremath{\mathsf{L}}}'_X: I\ell_X \Rightarrow \beta X$ and ${\ensuremath{\mathsf{R}}}'_X: \beta X
\Rightarrow r_X I$ also satisfying Equation (compatbility with ${\ensuremath{\mathsf{H}}}$). Then there is a unique $2$-cell
II & \^[uu]{} & JJ\
\
I & \_u & J .
satisfying Equations and (compatibility with ${\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{left}}}}$ and the left $2$-cells, as well as compatibility with ${\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{right}}}}$ and the right $2$-cells).
Working first with left $2$-cells, define a family ${\ensuremath{\mathsf{W}}}_X$ by the equation $$\begin{diagram}[w=6ex,h=6ex,tight]
IIX&\rTo^{uuX}&JJX\\
{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(-9,-25)(-23,0)(-9,25)
\put(-20,0){\makebox(0,0)[r]{$\scriptstyle {I\lambda_X}$}}
\put(-8.7,-25.6){\vector(1,-2){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{L}}}_X}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(9,-25)(23,0)(9,25)
\put(20,0){\makebox(0,0)[l]{$\scriptstyle {\alpha X}$}}
\put(8.7,-25.6){\vector(-1,-2){0}}
\end{picture}
}&{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put({20},0){\makebox(0,0)[b]{${{\ensuremath{\mathsf{W}}}_X}$}}
\end{picture}
}&{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(9,-25)(23,0)(9,25)
\put(20,0){\makebox(0,0)[l]{$\scriptstyle {\beta X}$}}
\put(8.7,-25.6){\vector(-1,-2){0}}
\end{picture}
}\\
IX&\rTo_{uX}&JX
\end{diagram}
\qquad\qquad = \qquad\qquad
\begin{diagram}[w=6ex,h=6ex,tight]
IIX&\rTo^{uuX}&JJX\\
{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(-9,-25)(-23,0)(-9,25)
\put(-20,0){\makebox(0,0)[r]{$\scriptstyle {I\lambda_X}$}}
\put(-8.7,-25.6){\vector(1,-2){0}}
\end{picture}
}&{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put({-20},0){\makebox(0,0)[b]{${u {\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{left}}}}_X}$}}
\end{picture}
}&{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(-9,-25)(-23,0)(-9,25)
\put(-20,0){\makebox(0,0)[r]{$\scriptstyle {J\ell_X}$}}
\put(-8.7,-25.6){\vector(1,-2){0}}
\end{picture}
}{\setlength{\unitlength}{1pt}
\begin{picture}(0,0)(0,0)
\put(0,{0}){\makebox(0,0)[b]{${{\ensuremath{\mathsf{L}}}'_X}$}}
\end{picture}
}{\setlength{\unitlength}{0.003\DiagramCellHeight}
\multiply \unitlength by \scaleFactor
\begin{picture}(0,0)(0,-4)
\qbezier(9,-25)(23,0)(9,25)
\put(20,0){\makebox(0,0)[l]{$\scriptstyle {\beta X}$}}
\put(8.7,-25.6){\vector(-1,-2){0}}
\end{picture}
}\\
IX&\rTo_{uX}&JX
\end{diagram}$$ It follows readily from Equation that the family has the property $${\ensuremath{\mathsf{W}}}_{XY} = {\ensuremath{\mathsf{W}}}_X Y$$ for all $X, Y$, and it is a standard argument that since a unit object exists, for example $(I,\lambda_I)$, this implies that $${\ensuremath{\mathsf{W}}}_X = {\ensuremath{\mathsf{U}}} X$$ for a unique $2$-cell
II & \^[uu]{} & JJ\
\
I & \_u & J ,
and by construction this $2$-cell has the required compatibility with ${\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{left}}}}$ and the left constraints. To see that this ${\ensuremath{\mathsf{U}}}$ is also compatible with ${\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{right}}}}$ and the right constraints we reason backwards: $(u,{\ensuremath{\mathsf{U}}})$ is now a morphisms of units from $(I,\alpha)$ to $(J,\beta)$ to which we apply the right-hand version of Lemma \[unitmap\] to construct a new ${\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{right}}}}$, characterised by the compatibility condition. By Proposition \[Kelly-compatibility\] this new ${\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{right}}}}$ is compatible with ${\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{left}}}}$ and the Kelly cells ${\ensuremath{\mathsf{K}}}$ and ${\ensuremath{\mathsf{H}}}$ (Equation ), and hence it must in fact be the original ${\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{right}}}}$ (remembering from \[GPS\] that ${\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{left}}}}$ and ${\ensuremath{\mathsf{U}}}{^{\text{\tiny \rm{right}}}}$ determine each other via ). So the $2$-cell ${\ensuremath{\mathsf{U}}}$ does satisfy both the required compatibilities.
[10]{}
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. . J. Algebra [**1**]{} (1964), 397–402.
. . In [*Category Seminar (Proc. Sem., Sydney, 1972/1973)*]{}, no. 420 in Lecture Notes in Mathematics, pp. 75–103. Springer-Verlag, Berlin, 1974.
. . IMRP Internat. Math. Res. Papers [**2006**]{} (2006), 1–54. ArXiv:math.CT/0507116.
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. . Trans. Amer. Math. Soc. [**108**]{} (1963), 275–292; 293–312.
\[lastpage\]
|
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"pile_set_name": "ArXiv"
}
|
---
abstract: 'We investigate the formal deformation theory of (rank 1) branes on generalized complex (GC) manifolds. This generalizes, for example, the deformation theory of a complex submanifold in a fixed complex manifold. For each GC brane $\B$ on a GC manifold $(X,\mathbb{J})$, we construct a formal (pointed) groupoid $\textbf{Def}^{\mathcal{B}}(X,\mathbb{J})$ (defined over a certain category of real Artin algebras) that encodes the formal deformations of $\mathcal{B}$. We study the geometric content of this groupoid in a number of different situations. Using the theory of (bi)semicosimplicial differential graded Lie algebras (DGLAs), we construct for each brane $\mathcal{B}$ a DGLA $L_{\mathcal{B}}$ that governs the “locally trivializable" deformations of $\mathcal{B}$. As a concrete application of this construction, we prove an unobstructedness result.'
author:
- 'Braxton L. Collier'
title: Deformations of generalized complex branes
---
Introduction
============
Let $X$ be a complex manifold, and $Z\subset X$ a complex submanifold. A classical problem in complex geometry is to understand the possible deformations of $Z$ in $X$, i.e. the collection of complex submanifolds of $X$ that are “close to" $Z$, in an appropriate sense. In the most well-behaved situation, there exists a *universal family* of such deformation; this is described, for example, in [@K] (where Kodaira calls it a “maximal family"). Such a family consists of an auxiliary complex manifold $\mathcal{M}$ with basepoint $0\in\mathcal{M}$, and a complex submanifold $\hat{Z}\subset X\times \mathcal{M}$, such that, for each $m\in\mathcal{M}$, the fiber $\hat{Z}_m:=\hat{Z}\cap (X\times\{m\})$ is a complex submanifold of $X\times\{m\}\cong X$; the fiber over $0\in\mathcal{M}$ corresponds to $Z$ itself. For every other family $(\mathcal{M}',\hat{Z}')$, there exists a neighborhood $U\subset \mathcal{M}'$ of the basepoint, and a unique (pointed) holomorphic map $\varphi:U\to \mathcal{M}$ such that, for each $m'\in U$ we have $\hat{Z}'_{m'}=\hat{Z}_{\varphi(m')}$ (regarded as complex submanifolds of $Z$). We may regard $\mathcal{M}$ (or more precisely the universal family) as a (local) *moduli space* for the deformations of $Z$. For a given $X$ and $Z$, such a moduli space may or may not exist.
At a formal level, the possible deformations of $Z$ in $X$ may be encoded as a functor $$\textit{Def}_Z:\textrm{Art}_{\C}\to \Set,$$ where $\textrm{Art}_{\C}$ is the category of local Artin algebras over $\C$ (with residue field $\C$) [@Kol][@M]. Heuristically, given an Artin algebra $A\in\textrm{Art}_{\C}$, we may view an element of $\textit{Def}_Z(A)$ as a family of complex submanifolds of $X$ deforming $Z$, which is parameterized by $\textrm{Spec}(A)$ (with basepoint the unique geometric point of $\textrm{Spec}(A)$, i.e. the maximal ideal). In the situation that there is a well-defined moduli space $\mathcal{M}$ as above, we may identify $\textit{Def}_Z(A)$ with the set of (pointed) maps $\textrm{Spec}(A)\to \mathcal{M}$; for example, the value of $\textit{Def}_Z$ on the so-called “dual numbers" $A=\C[\epsilon]/(\epsilon^2)$ gives the (geometric) tangent space of $\mathcal{M}$ at $0$. Appealing to the functor of points philosophy, we may view $\textit{Def}_Z$ itself as a formal stand-in for the moduli space that can always be defined. Even when $\mathcal{M}$ itself does not exist, the “tangent space" $\textit{Def}_Z(\C[\epsilon]/(\epsilon^2))$ still has an interesting geometric interpretation. We summarize this as a theorem (although the result is a straightforward consequence of the definition of $\df_Z$).
\[Z tangent\] There is a one-to-one correspondence between elements of\
$\textit{Def}_Z(\C[\epsilon]/(\epsilon^2))$ and holomorphic sections of the normal bundle $NZ$. In other words, there is a natural bijection $$\textit{Def}_Z(\C[\epsilon]/(\epsilon^2))\cong H^0(Z;\mathcal{O}_{NZ}).$$
In fact, by a general argument (independent of Theorem \[Z tangent\]), $\textit{Def}_Z(\C[\epsilon]/(\epsilon^2))$ inherits the structure of a $\C$-vector space; Theorem \[Z tangent\] is then better formulated as an isomorphism of vector spaces.
Another way in which $\textit{Def}_Z$ encodes the geometry of the putative moduli space involves the *obstruction* properties of $\textit{Def}_Z$. Given an Artin algebra $A\in\textrm{Art}_{\C}$, a deformation $\hat{Z}\in\textit{Def}_Z(A)$, and a surjective map $\mu: A'\to A$ in $\textrm{Art}_{\C}$, consider the problem of finding an extension of $\hat{Z}$ to $\textit{Def}_Z(A')$, i.e. an element $\hat{Z}'\in\textit{Def}_Z(A')$ satisfying $\textit{Def}_Z(\mu)(\hat{Z}')=\hat{Z}$. In general, there will be an obstruction to the existence of such an extension. If a smooth moduli space $\mathcal{M}$ exists, however, then it is always possible to solve this extension problem and we say $\textit{Def}_Z$ is *unobstructed*. From a different perspective, we might allow $\mathcal{M}$ to be a more general (possible singular) type of space than a complex manifold; the obstruction problem for $\textit{Def}_Z$ then relates to the smoothness of $\mathcal{M}$ at $0$. This explains the importance of the following well-known result [@Kol][@M].
\[Z obstructions\] Let $Z\subset X$ be a complex submanifold satisfying the condition $$H^1(Z;\mathcal{O}_{NZ})=0.$$ Then the deformation functor $\textit{Def}_{Z}$ is unobstructed.
Compared to Theorem \[Z tangent\], which has a very clear geometric interpretation, Theorem \[Z obstructions\] is perhaps harder to understand intuitively (and is correspondingly more difficult to prove).
The goal of the present paper is to extend the above constructions and results as far as possible into the realm of *generalized* complex geometry [@H][@G]. A generalized complex structure on a manifold $X$ is an endomorphism of the vector bundle $TX\oplus T\uv X$ (the direct sum of the tangent bundle and cotangent bundle), which squares to minus the identity, preserves the natural pairing of $TX\oplus T\uv X$ with itself, and satisfies an integrability condition defined with respect to the *Dorfman bracket* $$\ll(\xi,a),(\eta,b)\rr:=([\xi,\eta],\pounds(\xi)b-\iota(\eta)da).$$ As the name suggests, an ordinary complex structure $J:TX\to TX$ may be viewed as a particular example of a GC structure; a symplectic structure on $X$ also gives an example, however, so the subject could just as well be called “generalized symplectic geometry". Indeed, a fundamental appeal of GC geometry is that it provides a framework for treating complex and symplectic geometry in a unified way. The subject also admits a wide variety of interesting examples: these range from examples that may be viewed as hybrids (or deformations) of complex and symplectic manifolds, to more exotic structures that exhibit features not present in either of the two “classical" cases [@G].
To motivate the generalization of complex submanifolds whose deformation theory we study in this paper, as well as to explain some of the issues that arise in generalizing from the case of complex submanifolds, consider the following situation. Let $(X,\omega)$ be a symplectic manifold, and consider the problem of constructing a well-behaved moduli space of *Lagrangian* submanifolds of $X$. An issue that immediately arises–not present in the complex case–is the need to incorporate the so-called *Hamiltonian symmetries* of $(X,\omega)$. Recall that, to any smooth function $f:X\to \R$ we may associate its *Hamiltonian vector field*, which is characterized (up to a sign convention) by the equation [@C] $$\iota(X_f)\omega=-df.$$ Every such vector field is an infinitesimal symmetry of the symplectic structure; in particular the flow $\{\varphi_t\}_{t\in\R}$ generated by $X_f$ (when it exists) satisfies $$\varphi_t^*\omega=\omega$$ for each $t\in \R$. The symplectomorphisms of $(X,\omega)$ that arise in this way are known as *Hamiltonian symmetries*,[^1] and they form a subgroup of the group of all symplectomorphisms of $(X,\omega)$. When forming a moduli space of Lagrangian submanifolds, it is custumary to identity two Lagrangians $L,L'\subset X$ if $L'=\varphi(L)$ for some Hamiltonian symmetry $\varphi$. This is necessary, for example, to have any hope of constructing a finite dimensional moduli space [@F1].
From the point of view of mirror symmetry (and string theory), it is actually more natural to consider a slightly different moduli space; namely, the collection of all pairs $(Z,\L)$, where $Z\subset X$ is a Lagrangian submanifold, and $\L$ is a Hermitian line bundle with a flat connection supported on $Z$ [@Kon][@F1; @F2]. Adopting terminology from physics [@KO], such an object may be called a (rank 1) *Lagrangian brane*. A nice property of this moduli space (when it can be defined), is that it carries a natural complex structure. An added subtlety is the need to keep track of the possible equivalences between line bundles with connection.
On any GC manifold $(X,\J)$, a (rank 1) *GC brane* is similarly defined[^2] as a submanifold $Z\subset X$, together with a Hermitian line bundle with unitary connection $\L$ supported on $Z$; the pair $\B=(Z,\L)$ must satisfy a certain condition defined with respect to $\J$. In the case of a complex manifold $X$, the compatibility condition implies that $Z$ must be a complex submanifold, and the curvature form $F\in\Omega^2(Z)$ of $\L$ must be of type $(1,1)$ with respect to the induced complex structure on $Z$ (i.e. the connection must induce a holomorphic structure on $\L$) [@G][@KL]. In particular, any complex submanifold–equipped with the trivial line bundle–gives an example of a GC brane. In the symplectic case, the Lagrangian branes described above give one type of example, but there are actually others as well. These so-called “coisotropic" A-branes were first introduced by Kapustin and Orlov in [@KO], motivated by considerations coming from string theory and homological mirror symmetry; the fact that these more exotic objects arise naturally in the context of GC geometry is another appealing feature of this framework [@G].
In this paper we will study the formal deformation theory of GC branes. In analogy to the symplectic case described above, it will be important to incorporate the action of the *generalized Hamiltonian* symmetries of $(X,\J)$ on GC branes. On a GC manifold $(X,\J)$, one may associate to any *complex*-valued function $f:X\to \C$ its *generalized Hamiltonian vector field*, which is a section $\xx_f\in\Cinf(TX\oplus T\uv X)$. As in the symplectic case, $\xx_f$ is an infinitesimal symmetry of $(X,\J)$, and may (at least locally) be integrated to a family of symmetries (a “flow"). Such a symmetry consists of a pair $(\varphi,u)$, where $\varphi:X{\xymatrix@C=1.5pc{ \ar[r]^-{\cong} & }}X$ is a diffeomorphism and $u\in\Omega^1(X)$ is a 1-form. Roughly speaking, the diffeomorphism component acts on a brane by pulling it back, and the 1-form component acts by changing the connection on the line bundle.
Let describe the main constructions and results of the paper. For each GC brane $\B$, we construct a functor[^3] $$\textit{Def}_{\B}:\Art\to \textrm{Set}$$ encoding the formal deformations of $\B$ (Definition \[brane deformation functor\]). This functor is defined as the truncation of a certain formal groupoid $\De\uB(X,\J)$ (Definition \[full deformation groupoid\]), which encodes all the relevant notions of equivalence for deformations (including those induced by generalized Hamiltonian symmetries). The construction of this functor is more intricate than in the case of complex submanifolds discussed above, and is done in several steps.
As a first step in studying the deformations of a GC brane $\B$, we consider the value of the functor $\textit{Def}_{\B}$ on the dual numbers $\DN$; these are the first-order deformations of $\B$. It was argued in [@KM] that such first-order deformations should correspond to elements of a certain Lie algebroid cohomology group associated to $\B$; we recover their result in a rigorous framework.
\[brane tangent\] For every GC brane $\B$, there is a natural bijection between elements of $\textit{Def}_{\B}(\R[\epsilon]/(\epsilon^2))$ and elements of the Lie algebroid cohomology group $H^1(\B)$.
The calculation needed to prove this result is somewhat involved–and different from the one in [@KM]–so the fact that we reach the same conclusion may be regarded as a check on our Definition \[brane deformation functor\]. To explain the connection to Theorem \[Z tangent\] stated above, suppose $Z\subset X$ is a complex submanifold, regarded as a GC brane $\B_Z$. There is a natural isomorphism $$H^1(\B_Z)\cong H^0(Z;\mathcal{O}_{NZ})\oplus H^0(Z;\Omega^1(Z)),$$ where $\Omega^1(Z)$ denotes the sheaf of holomorphic one-forms on $Z$. The first component $H^0(Z;\mathcal{O}_{NZ})$ corresponds to the possible deformations of the submanifold $Z$, as discussed above. The second component $H^0(Z;\Omega^1(Z))$ corresponds to deformations of the trivial holomorphic line bundle on $Z$. From a different perspective, if we denote by $\mathcal{E}_Z$ the coherent sheaf of $\mathcal{O}_X$-modules determined by $Z$ (the push-forward of its structure sheaf), then there are natural isomorphisms $$H^k(\B_Z)\cong \textrm{Ext}^k(\mathcal{E}_Z,\mathcal{E}_Z)$$ for each $k$. In particular, first-order deformations of $\B_Z$ as a GC brane correspond to first-order deformations of $\mathcal{E}_Z$ as a coherent sheaf (since these are parameterized by $\textrm{Ext}^1(\mathcal{E}_Z,\mathcal{E}_Z)$).
To go deeper into the geometric content of the deformation functor (and formal groupoid) associated to a GC brane, we next take a closer look at the relationship between symmetries of the GC manifold $(X,\J)$ and the deformations of $\B$. It is here we encounter the first fundamental difference from the complex and Lagrangian cases described above. In the case of a complex submanifold $Z$ of a complex manifold $(X,J)$, for instance, every deformation of $Z$ is locally induced by a (local) symmetry of $(X,J)$. To explain this, consider the case of 1st order deformations, which by Theorem \[Z tangent\] correspond to holomorphic sections of the normal bundle $NZ$. Given such a section $\xi$, for each $z\in Z$ it is always possible to extend $\xi$ to a holomorphic vector field $\tilde{\xi}$ on some neighborhood $U\subset X$ of $z$; heuristically, the corresponding deformation is then given near $z$ by “flowing" $Z$ along $\tilde{\xi}$ for an infinitesimal time. An analogous construction in the case of an arbitrary $GC$ brane is *not* always possible, however. We call deformations that have this property *locally trivializable*. Although an arbitrary GC brane may have deformations which are not locally trivializable, we introduce a large class of branes, which we call *leaf-wise Lagrangian* (LWL) branes, and prove that their deformations are always locally trivializable (Theorem \[essentially surjective\]). This is a non-trivial statement about the geometry of such branes, and relies on the existence of a local normal form (which we prove in Theorem \[LWL normal form\]).
The final part of the paper is devoted to proving a version of Theorem \[Z obstructions\] for GC branes. The result we prove applies only to those branes with locally trivializable deformations (in particular, by Proposition \[essentially surjective\], to holds for LWL branes). We prove the theorem using the machinery of (bi)semicosimplicial differential graded Lie algebras (DGLAs), as developed in [@FMM][@I][@BM]. Specifically, by adapting a construction given in [@I], we construct, for each brane $\B$, a DGLA $L_{\B}$ (depending on a choice). We prove that $L_{\B}$ governs the locally trivializable deformations of $\B$. A concrete calculation involving $L_{\B}$ then allows us to prove the following result.
\[brane obstructions\] Let $\B$ be a leaf-wise Lagrangian brane on a GC manifold (or more generally a brane with locally trivializable deformations). If the Lie algebroid cohomology group $H^2(\B)$ vanishes, then the functor $\textit{Def}_{\B}$ is unobstructed.
This result can clearly be sharpened in various situations: for example, in the case of a Lagrangian brane $\B$, it is not hard to see that the DGLA $L_{\B}$ is (homotopy) abelian, so that in this case $\df_{\B}$ is always unobstructed. We leave the problem of strengthening Theorem \[brane obstructions\]–as well as extending it to arbitrary branes–for future work.
Organization of the paper
-------------------------
In §\[Courant algebroid\] through §\[LWL submanifolds\], we present the definitions and results in GC geometry that will be needed for the rest of the paper. Although much of this material is well-known, there are a few definitions and results that have not–to the best of our knowledge–appeared before. This includes, for example, the notion of a leaf-wise Lagrangian submanifold (or brane), which we introduce in Definition \[definition LWL submanifold\]. Our treatment of the symmetries of the standard Courant algebroid is also somewhat non-standard. In §\[section formal\], we recall the basic definitions and results about Artin algebras and nilpotent Lie algebras we will need to formulate the definition of the functor $\df_{\B}$ associated to a GC brane (Definition \[brane deformation functor\]). In particular, we introduce a “formal" (infinitesimal) version of the group of symmetries of the Courant algebroid, and establish some of its basic properties. In §\[section deformations\], we construct the formal groupoid $\De\uB(X,\J)$ of deformations of a GC brane (Definition \[full deformation groupoid\]); the deformation functor $\df_{\B}:\Art\to \Set$ of $\B$ is formed by taking $\pi_0$ of this groupoid (Definition \[brane deformation functor\]).
After defining the deformation functor (and formal groupoid), we proceed in §\[first order deformations\] to prove Theorem \[brane tangent\] about first-order deformations. In §\[section functoriality\] we study the behavior of the deformation groupoid under equivalences between different branes, and in particular prove the invariance of the deformation functor under such equivalences. In §\[LWL branes example\] we consider the deformations of leaf-wise Lagrangian branes; the main result of this section is Theorem \[essentially surjective\], which implies that every deformation of a LWL brane is locally induced by a symmetry of the ambient GC manifold. In §\[induced deformations\]-\[cosimplicial groupoids\] we investigate the relationship between deformations and symmetries in more detail.
The remainder of the paper is devoted to applying the theory of DGLAs to the deformation theory of GC branes. In §\[DGLA theory\], we recall some material from [@FMM][@I][@BM] concerning DGLAs, the construction of the Deligne groupoid, and semicosimplicial objects. We then use this setup to construct a DGLA $L_{\B}$ governing the locally trivializable deformations of a GC brane $\B$. In the final section (§\[cohomology\]), we use this construction to prove Theorem \[brane obstructions\]. Finally, there is a short appendix in which we prove a technical result stated in the main body of the paper.
**Acknowledgements:** I would like to thank Florian Schätz, Michael Bailey, Andrei Caldararu, Uli Bunke, and Dan Freed for useful discussions related to the subject matter of this paper. I would also like to acknowledge the useful questions and feedback received in response to several talks I gave at the University of Regensburg. Special thanks are also due to Simone Diverio for pointing me towards the reference [@I]; this was in response to a question I posed on the mathoverflow website, and I would also like to thank Domenico Fiorenza and Urs Schreiber for useful answers.
The standard Courant algebroid and its symmetries {#Courant algebroid}
=================================================
Let $X$ be a smooth manifold, with tangent bundle $TX$ and cotangent bundle $T\uv X$. We denote by $\TT X$ the direct sum $TX\oplus T\uv X$. We have the natural pairing $$\begin{aligned}
\la \cdot,\cdot \ra: \TT X\oplus \TT X & \to X\times \R \\ (\xi,a), (\eta,b) & \mapsto \frac{1}{2}(\iota(\xi) b+\iota(\eta)a),\end{aligned}$$ as well as the Dorfman bracket $$\begin{aligned}
\label{Dorfman Bracket} \ll\cdot,\cdot\rr: &\Cinf(\TTX)\times\Cinf(\TTX) \to \Cinf(\TTX) \\ & (\xi,a),(\eta,b) \mapsto ([\xi,\eta],\pounds(\xi)b-\iota(\eta)da),\notag\end{aligned}$$ which is closely related to the Courant bracket [@G]. The vector bundle $\TTX$ equipped with these structures, together with the projection $\pi:\TTX\to TX$ have the structure of an *exact Courant algebroid* [@G; @G2]. In this paper we work with this “standard" Courant algebroid, leaving for future work the extension of our results to a more general setting. Let $\Diff(X)$ denote the group of diffeomorphisms of $X$, and $G(X)$ denote the opposite group $\Diff(X)^{op}$. Given a diffeomorphism $\varphi:X {\xymatrix@C=1.5pc{ \ar[r]^-{\cong} & }}X$, we denote the corresponding element of $G(X)$ by $\varphi\us$, so that multiplication in $G(X)$ is given by $\varphi\us\psi\us=(\psi\varphi)\us$. There is a left action of $G(X)$ on $\Omb(X)\oplus \Cinf(TX)$ by pullback, where by definition the pull-back of a vector field $\xi$ by a diffeomorphism $\varphi$ is the push-forward of $\xi$ by $\varphi^{-1}$: $$\varphi\us\xi:=(\varphi^{-1})_*\xi.$$ This action is compatible with the differential graded commutative algebra structure on $\OmbX$, the Lie bracket on $\Cinf(TX)$, and the operation of contracting a vector field with a differential form. By the formula (\[Dorfman Bracket\]) for the Dorman bracket, it follows the action is also compatible with the Courant algebroid structure on $\TTX$ in the following sense: given any sections $\xx,\yy\in\Cinf(\TTX)$, and any $g=\varphi\us\in G(X)$, we have $$\label{sym 1}\la g\xx,g\yy\ra=g\la\xx,\yy\ra,$$ $$\label{sym 2} \ll g\xx,g\yy\rr=g\ll\xx,\yy\rr,$$ and $$\label{sym 3} \pi(g\xx)=g\pi\xx.$$
A key feature of the Courant algebroid $(\TTX,\la\cdot,\cdot\ra,\ll\cdot,\cdot\rr,\pi)$ is that it admits a group of symmetries which is larger than $\Diff(X)$ [@G]. Namely, given a closed 2-form $B\in \Omega^2(X)$, the bundle map $e^B:\TTX\to \TTX$ given by $$(\xi,a)\mapsto (\xi,a-\iota(\xi)B)$$ is compatible with the Dorfman bracket and preserves the pairing and the projection map. In particular, any 1-form $u\in\Omega^1(X)$ determines a symmetry of the Courant algebroid by setting $B=du$. This motivates the following definition.
\[Definition cG\] Let $\cG(X)$ denote the semi-direct product $G(X)\ltimes \Om^1(X)$, i.e. as a set $\cG(X)=G(X)\times \Om^1(X)$, and the group multiplication is given by the formula $$\label{cGmult}(\varphi\us,u)(\psi\us,w)=((\psi\varphi)\us,u+\varphi\us w).$$
\[leftgroupaction\] There is a left action of $\cG(X)$ on $\Cinf(\TTX)$ given as follows: for each $g=(\varphi\us,u)\in\cG(X)$, and each $\xx\in \Cinf(\TTX)$, we define $$\label{group action}g\cdot\xx=e^{du}\varphi\us\xx.$$ Moreover, for each $g\in \cG(X)$, the identities (\[sym 1\]), (\[sym 2\]), and (\[sym 3\]) hold.
\[2 group\] Since the action of a group element $(\varphi\us,u)\in \cG(X)$ on $\Cinf(\TTX)$ depends only on the diffeomorphism $\varphi$ and the 2-form $du\in\Omega^2(X)$, one might ask why we did not define $\cG(X)$ to be the semi-direct product of $\textrm{Diff}(X)^{op}$ with the space of closed 2-forms on $X$ (as is perhaps more standard, see for example [@G2]). The reason is that this group would not act on the collection of branes on $X$. As discussed in the introduction, such a brane consists of a pair $\B=(Z,\L)$, where $Z\subset X$ is a submanifold, and $\L$ is a Hermitian line bundle with unitary connection supported on $Z$. As discussed below in Remark \[geometric brane 2\], a diffeomorphisms acts on such a brane $\B$ by pull-back, whereas an element of the form $(0,u)\in \cG(X)$ acts by altering the connection on $\L$ (by adding $-2\pi iu|_Z$).
From a more fundamental point of view, we should view $\cG(X)$ as a stand-in for a certain 2-group $\mathcal{G}$ (a group object in categories). An object of $\mathcal{G}$ is a pair $(\varphi^*,\mathcal{E})$, where $\varphi$ is a diffeomorphism of $X$, and $\mathcal{E}$ is a Hermitian line bundle with unitary connection (or equivalently, a principal $U(1)$-bundle) over $X$. The bundle component $\mathcal{E}$ acts on a brane $(Z,\L)$ by first restricting $\mathcal{E}\uv$ to $Z$ and then tensoring with $\L$. This 2-group $\G$ is the symmetry group of the trivial gerbe (with connective structure) on $X$; such symmetries naturally act on the category of branes on X.
Given an element $(\varphi^*,u)\in\cG(X)$, for example, we may view it as an object $(\varphi\us,\mathcal{E})$ of $\G$, where $\mathcal{E}$ is the trivial Hermitian line bundle on $X$ with connection $\nabla=d+2\pi i$. Viewed in this way, a *morphism* between two elements $(\varphi^*,u)$ and $(\varphi^*,u')$ of $\cG(X)$ (with the same diffeomorphism component) consists of a function $g:X\to U(1)$ satisfying $$g^{-1}dg=2\pi i(u'-u).$$ Roughly speaking, the elements of $\cG(X)$ correspond to the objects of the “identity component" of $\mathcal{G}$; for the purposes of this paper these are all the symmetries we need.
Because we treat only the standard Courant algebroid $TX\oplus T\uv X$ in this paper (corresponding to the trivial gerbe on $X$), the extra categorical structure of the 2-group will not be needed explicitly. In order to extend to the case of more general Courant algebroids, however, it will almost certainly be necessary to replace $\cG(X)$ with the relevant 2-group.
We now return to the main exposition, and prove Proposition \[leftgroupaction\].
Let $\xx=(\xi,a)$. We have $$\begin{aligned}
(\varphi\us,u)\cdot((\psi\us,w)\cdot\xx) &= (\varphi\us,u)\cdot(\psi\us\xi,\psi\us a-\iota(\psi\us\xi)dw) \\
&= (\varphi\us\psi\us\xi,\varphi\us\psi\us a-\varphi\us(\iota(\psi\us\xi)dw)-\iota(\varphi\us\psi\us\xi)du) \\
&= ((\psi\varphi)\us\xi,(\psi\varphi)\us a-\iota(\varphi\us\psi\us\xi)d(\varphi\us w)-\iota(\varphi\us\psi\us\xi)du)\\
&= ((\psi\varphi)\us\xi,(\psi\varphi)\us a-\iota((\psi\varphi)\us\xi))d(u+\varphi\us w)) \\
&= ((\psi\varphi)\us,u+\varphi\us w)\cdot \xx \\
&= ((\varphi\us,u)(\psi\us,w))\cdot\xx\end{aligned}$$
Since for each 1-form $u\in\Omega^1(X)$ its exterior derivative $du\in\Omega^2(X)$ is closed, it follows that equations (\[sym 1\]), (\[sym 2\]), and (\[sym 3\]) hold.
\[symmetry groupoid\] More generally, we may define a groupoid $\cG$ whose objects are smooth manifolds, such that a morphism from a manifold $X$ to a manifold $Y$ is a pair $(\varphi^*,u)$, where $\varphi: Y{\xymatrix@C=1.5pc{ \ar[r]^-{\cong} & }}X$ is a diffeomorphism and $u\in\Omega^1(Y)$. Composition of morphisms in this groupoid is given by the formula (\[cGmult\]). For a fixed manifold $X$, the group $\cG(X)$ is then recovered as the automorphism group of $X$. Note also that a morphism from $X\to Y$ in $\cG$ determines an isomorphism $\Cinf(\TTX)\to \Cinf(\TT Y)$ using the same formula (\[group action\]).
For each $g=(\varphi\us,u)\in\cG(X)$, the action (\[group action\]) on $\Cinf(\TTX)$ is induced by a bundle map $\TTX\to\TTX$ covering the diffeomorphism $\varphi^{-1}:X\xymatrix@C=1.5pc{ \ar[r]^-{\cong} & } X$. In particular, for every bundle endomorphism $F:\TTX\to \TTX$, the map $\Cinf(\TTX)\to \Cinf(\TTX)$ given by $$\xx\mapsto g\cdot F(g^{-1}\cdot \xx)$$ is induced by a unique bundle endomorphism $\TTX\to\TTX$, which we denote by $g\cdot F$. Clearly, the map $F\mapsto g\cdot F$ defines an action of $\cG(X)$ on $\Cinf(\End(\TTX))$.
We next introduce a Lie algebra that is the infinitesimal version of $\cG(X)$, in the same way that the infinitesimal version of $G(X)=\textrm{Diff}^{op}(X)$ is the Lie algebra of vector fields on $X$.
\[definition cg\] Let $\g(X)$ denote $\Cinf(TX)$, viewed as a real Lie algebra with respect to the Jacobi-Lie bracket. We define $\cg(X)$ to be the semi-direct product $\g(X)\ltimes \Omega^1(X)$, where the action of $\g(X)$ on $\Omega^1(X)$ is given by the Lie derivative. Explicitly, as a vector space we have $$\cg(X)=\g(X)\oplus \Omega^1(X)\cong\Cinf(\TTX),$$ and the bracket is given by the formula $$\label{ghatbracket}[(\xi,a),(\eta,b)]=([\xi,\eta],\pounds(\xi)b-\pounds(\xi)a)$$ for each $\xi,\eta\in\g(X)$ and $a,b\in\Omega^1(X)$.
Returning briefly to the discussion in Remark \[2 group\], it is actually more natural to view elements of $\cg(X)$ as objects of a certain Lie 2-algebra, corresponding to the *infinitesimal* symmetries of the trivial gerbe on $X$. Such symmetries were introduced and studied in the author’s Ph.D. thesis [@Co]. As mentioned in Remark \[2 group\], the fact that we deal only with the standard Courant algebroid $TX\oplus T\uv X$ allows us to avoid the use of the full Lie 2-algebra structure explicitly. Even in this case, however, the reader may find the categorical perspective conceptually useful.
Note that, although the underlying vector space of $\cg(X)$ is isomorphic to $\Cinf(\TTX)$, the Lie bracket (\[ghatbracket\]) is *not* the same as the Dorfman bracket (\[Dorfman Bracket\]); in particular, the latter is not even a Lie bracket [@G]. On the other hand, the two brackets are compatible in a certain sense, as explained in the following proposition.
\[Courant action prop\]
1. There is a left action of the Lie algebra $\cg(X)$ on $\Cinf(\TTX)$ given by the same formula as the Dorfman bracket (\[Dorfman Bracket\]), i.e. for each $\xi=(\xi,a)\in\cg(X)$ and each $\alpha=(\tau,c)\in \Cinf(\TTX)$ we define $$\xx\cdot\alpha = ([\xi,\tau],\pounds(\xi)c-\iota(\tau)da).$$
2. Let $F:\TTX\to \TTX$ be a bundle map. For each $\xi\in \cg(X)$, the map $\Cinf(\TTX)\to \Cinf(\TTX)$ given by $$\label{end action}\alpha\mapsto \xx\cdot(F\alpha)-F(\xx\cdot\alpha)$$ for all $\alpha\in\Cinf(\TTX)$ is induced by a unique bundle endomorphism $\TTX\to \TTX$, which we denote by $\xx\cdot F$. The map $F\mapsto \xx\cdot F$ defines a Lie algebra action of $\cg(X)$ on $\Cinf(\End(\TTX))$.
To prove the first part, we must show that for every $\xx,\yy\in \cg(X)$, and every $\alpha\in\Cinf(\TTX)$, we have $$\label{lieaction1}\xx\cdot(\yy\cdot \alpha)-\yy\cdot(\xx\cdot\alpha)=[\xx,\yy]\cdot \alpha.$$ Writing $\xx=(\xi,a),\yy=(\eta,b)\in \cg(X)$, and $\alpha=(\tau,c)\in \Cinf(\TTX)$, we calculate $$\begin{aligned}
(\xi,a)\cdot((\eta,b)\cdot(\tau,c)) & =(\xi,a)\cdot([\eta,\tau,\pounds(\eta)c-\iota(\tau)db)\\
& =([\xi,[\eta,\tau]],\pounds(\xi)\pounds(\eta)c-\pounds(\xi)\iota(\tau)db-\iota([\eta,\tau])da.\end{aligned}$$ Therefore $$\begin{aligned}
\label{expression} & \xx\cdot (\yy\cdot\alpha)-\yy\cdot(\xx\cdot\alpha) \nonumber \\
& = ([\xi,[\eta,\tau]]-[\eta,[\xi,\tau]],\pounds(\xi)\pounds(\eta)c-\pounds(\eta)\pounds(\xi)c-\pounds(\xi)\iota(\tau)db+\pounds(\eta)\iota(\tau)da-\iota([\eta,\tau])da+\iota([\xi,\tau])db. \end{aligned}$$ By the Jacobi identity for vector fields, we have $$\label{calc 1}[\xi,[\eta,\tau]]-[\eta,[\xi,\tau]]=[[\xi,\eta],\tau].$$ To calculate the one-form part, recall the following identities: given vector fields $\xi,\eta$, we have $[\pounds(\xi),\pounds(\eta)]=\pounds([\xi,\eta])$, $\iota([\xi,\eta])=[\pounds(\xi),\iota(\eta)]$, and $\pounds(\xi)=d\iota(\xi)+\iota(\xi)d$. The one-form component of (\[expression\]) is therefore equal to $$\begin{aligned}
\label{calc 2} & [\pounds(\xi),\pounds(\eta)]c-\pounds(\xi)\iota(\tau)db+\pounds(\eta)\iota(\tau)da-\pounds(\eta)\iota(\tau)da+\iota(\tau)\pounds(\eta)da+\pounds(\xi)\iota(\tau)db-\iota(\tau)\pounds(\xi)db \nonumber\\
& =\pounds([\xi,\eta])c+\iota(\tau)(\pounds(\eta)da-\pounds(\xi)db) \nonumber\\
&= \pounds([\xi,\eta])c+\iota(\tau)(d\iota(\eta)da-d\iota(\xi)db) \nonumber\\
&= \pounds([\xi,\eta])c-\iota(\tau)d(\pounds(\xi)b-\pounds(\eta)a).\end{aligned}$$ Combining (\[calc 1\]) and (\[calc 2\]) we see that (\[expression\]) is equal to $$([\xi,\eta],\tau),( \pounds([\xi,\eta])c-\iota(\tau)d(\pounds(\xi)b-\pounds(\eta)a))=[(\xi,a),(\eta,b)]\cdot (\tau,c).$$ This includes the proof of the first part of the Proposition.
As for the second part, let $\xx=(\xi,a)\in\g(X)$, $T\in\Cinf(\End(\TTX)$, and $\alpha=(\eta,b)\in \Cinf(\TTX)$. We have $$\begin{aligned}
\xx\cdot(f\alpha) & =([\xi,f\eta],\pounds(\xi)(fb)-\iota(f\eta)da) \\
& = (f[\xi,\eta]+\xi(f)\eta,f\pounds(\xi)b+\xi(f) b-f\iota(\eta)da) \\
& = f\xx\cdot\alpha+\xi(f)\alpha.\end{aligned}$$ Since $T$ is linear over $\Cinf(X)$, we see that $$\begin{aligned}
(\xx\cdot T)(f\alpha) & =f\xx\cdot T(\alpha)+\xx(f)T\alpha-fT\xx\cdot\alpha-\xx(f)T\alpha\\
& = f(\xx\cdot T)(\alpha)\end{aligned}$$ This shows that the map $\Cinf(\TTX)\to \Cinf(\TTX)$ defined by the right-hand side of equation (\[end action\]) is linear over functions, and therefore is induced by a well-defined section of $\Cinf(\End(\TTX))$. It is also easy to see that equation (\[lieaction1\]) implies that $$\xx\cdot(\yy\cdot T)-\yy\cdot(\xx\cdot T)=[\xx,\yy]\cdot T$$ holds for every $\xx,\yy\in\cg(X)$.
Recall that a vector field $\xi\in\g(X)=\Cinf(TX)$ is called *complete* if it generates a flow $$\begin{aligned}
\Phi: & X\times \R\to X \\ & (x,t) \mapsto \varphi_t(x).\end{aligned}$$ Given such a vector field $\xi\in \g(X)$, we define $$e^{t\xi}=\varphi\us_t\in G(X).$$ The (left) action of $\g(X)$ on $\Omb(X)\oplus \Cinf(TX)$ by Lie derivative is the infinitesimal version of the (left) action of $G(X)=\textrm{Diff}^{op}(X)$ on $\Omb(X)\oplus \Cinf(TX)$ in the sense that for every complete $\xi\in\g(X)$, and every $\alpha\in\Omb(X)\oplus \Cinf(TX)$, we have $$\frac{d}{ds}|_{s=t}e^{s\xi}\alpha=\pounds(\xi)e^{t\xi}\alpha,$$ where the derivative with respect to $s$ is defined point-wise.
Similarly, we say that an element $\xx=(\xi,a)\in\cg(X)$ is *complete* if the vector field $\xi$ is. In this case, define $$a^{t\xi}=\int_0^te^{s\xi}ads,$$ and $$e^{t\xx}=(e^{t\xi},a^{t\xi})\in \cG(X).$$ We then have the following result, similar to [@G2 Prop 2.3].
\[exp Dorfman\] Let $\xx\in\cg(X)$ be complete. Then
1. for every $t,t'\in\R$, we have $$e^{t\xx}e^{t'\xx}=e^{(t+t')\xx}.$$
2. For every every $\alpha\in\Cinf(\TT X)$ and every $t\in \R$, we have $$\frac{d}{ds}|_{s=t}e^{s\xx}\cdot \alpha=\xx\cdot(e^{t\xi}\cdot \alpha),$$ where the derivative with respect to $s$ is defined point-wise.
Given $\xx=(\xi,a)$, we have $$\label{multiplication}e^{t\xx}e^{t'\xx}=(\varphi_t^*\varphi_{t'}^*,a^{t\xi}+\varphi_t^*a^{t'\xi}).$$ We have $$\varphi_t^*a^{t'\xi}=\int_0^{t'}\varphi^*_{t+s}ads=\int_{t}^{t+t'}\varphi^*_{s'}ads',$$ where the second equality follows from the change of variables $s'=s+t$. Therefore $$\begin{aligned}
a^{t\xi}+\varphi_t^*a^{t'\xi} & =\int_0^t\varphi_s^*ads+\int_{t}^{t+t'}\varphi_s^*ads \\ &= \int_0^{t+t'}\varphi_s^*ads=a^{(t+t')\xi}.\end{aligned}$$ Substituting into equation (\[multiplication\]) we obtian the desired result $$\label{exponential rule}e^{t\xx}e^{t'\xx}=e^{(t+t')\xx}.$$ Given $\yy=(\eta,b)\in\Cinf(\TT X)$, consider $$\label{t derivative} \frac{d}{dt}|_{t=0}(e^{t\xx}\cdot \yy)=(\frac{d}{dt}|_{t=0}(\varphi_t^*\eta), \frac{d}{dt}|_{t=0}(\varphi_t^*b)-\frac{d}{dt}|_{t=0}(\iota(\varphi_t^*\eta)d\int_0^{t}\varphi_s^*ads).$$ We have $$\label{t derivative 1}\frac{d}{dt}|_{t=0}(\varphi_t^*\eta)=[\xi,\eta]$$ and $$\label{t derivative 2}\frac{d}{dt}|_{t=0}(\varphi_t^*b)=\pounds(\xi)b.$$ Since $a^{t\xi}$ vanishes at $t=0$, the Leibnitz rule together with the fundamental theorem of calculus implies that implies that $$\label{t derivative 3}\frac{d}{dt}|_{t=0}(\iota(\varphi_t^*\eta)d\int_0^{t}\varphi_s^*ads=\iota(\eta)da.$$ Substituting equations (\[t derivative 1\]), (\[t derivative 2\]), and (\[t derivative 3\]) into (\[t derivative\]), we see that $$\label{t derivative 4} \frac{d}{dt}|_{t=0}(e^{t\xx}\cdot \yy)=\ll \xx,\yy\rr.$$
Finally, combining (\[exponential rule\]) with (\[t derivative 4\]), easily verify the second part of the proposition.
Generalized complex structures {#GC structures}
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Basic definitions and examples
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A good reference for the material in this subsection is [@G]. As described in the introduction, a *generalized complex structure* on a manifold $X$ is an endomorphism $$\J:\TTX\to\TTX$$ that preserves the pairing $\la\cdot,\cdot\ra$, satisfies $\J^2=-id_{\TTX}$, and satisfies a certain integrability condition. To describe this condition, note that, since $\J$ squares to minus the identity, we may decompose $$\TTX\otimes\C=L\oplus\bar{L},$$ where $L$ and $\bar{L}$ are the $+i$ and $-i$ eigen-bundles of (the complex-linear extension of) $\J$, respectively. We require that $\Cinf(L)$ be involutive with respect to the Dorfman bracket, i.e. we require that $$\ll\Cinf(L),\Cinf(L)\rr\subset\Cinf(L).$$
Given a GC structure $\J$ on $X$, the restriction of the Dorfman bracket to $\Cinf(L)$ endows $L$ with the structure of a complex Lie algebroid over $X$, with anchor map given by projection $\pi:L\to TX\otimes\C$. In particular, we may associate to $\J$ its *generalized Dolbeault* complex $$\xymatrix{ \Cinf(\Lambda^0L\uv) \ar[r]^{\delta_L} & \Cinf(\Lambda^1L\uv) \ar[r]^{\delta_L} & \Cinf(\Lambda^2L\uv) \ar[r]^-{\delta_L} & \cdots}.$$ For future reference, let us explicitly describe the first two differentials in this complex. Given $f:X\to \C$ (viewed as a section of $\Lambda^0L\uv$), the section $\delta_Lf\in\Cinf(\Lambda L\uv)$ is given by $$\delta_Lf(\xx)=\pi(\xx)\cdot f$$ for every $\xx\in\Cinf(L)$, where we recall $\pi:L\to TX\otimes\C$ is the projection (anchor) map. Given $\alpha\in\Cinf(\Lambda L\uv)$, for every $\xx,\yy\in\Cinf(L)$ we have $$\delta_L\alpha(\xx,\yy)=\pi(\xx)\cdot \alpha(\yy)-\pi(\yy)\cdot\alpha(\xx)-\alpha([\xx,\yy]).$$
A useful observation is that, since both $L$ and $\bar{L}$ are maximally isotropic sub-bundles of $\TTX\otimes \C$ (with respect to the $\C$-linear extension of the pairing), the pairing determines an isomorphism $$\bar{L}\xymatrix@C=1.5pc{ \ar[r]^-{\cong} & } L^{\vee}.$$
\[complex\] Any ordinary complex structure $J$ on $X$ determines a GC structure given by $$\label{Jcomplex}\J_{J}:=\left(\begin{array}{cc} -J & 0 \\ 0 & J^{\vee} \end{array}\right).$$ In this case we have $$L=(TX)^{0,1}\oplus(T\uv X)^{1,0}$$ and $$L\uv\cong\bar{L}=(TX)^{1,0}\oplus(T\uv X)^{0,1}.$$ The generalized Dolbeault complex is isomorphic to $\Omega^{0,\bullet}(X;\Lambda^{\bullet}(TX)^{1,0})$ with differential the $\bar{\partial}$-operator corresponding to the standard holomorphic structure on $\Lambda^{\bullet}(TX)^{1,0}$ [@G].
\[symplectic\] Let $\omega$ be a symplectic structure on $X$, viewed as an isomorphism $TX\to T\uv X$. This determines a GC structure given by $$\label{Jsymplectic} \J_{\omega}:=\left(\begin{array}{cc} 0 & -\omega^{-1} \\ \omega & 0 \end{array}\right).$$ In this case, we have $$L=e^{i\omega}(TX\otimes\C)=\{(\xi,-i\omega\xi):\xi\in TX\}.$$ The bundle map $$e^{i\omega}:TX\otimes\C\to L$$ is an isomorphism of complex Lie algebroids, and in particular the generalized Dolbeualt complex is isomorphic to $(\Omega^{\bullet}(X;\C),d_{dR})$, the ordinary de-Rham complex of $X$ with complex coefficients.
Given a GC manifolds $(X,\J)$ and $(X',\J')$, the product $X\times X'$ inherits a natural GC structure $\J\times\J'$.
Given a GC manifold $(X,\J)$ with $+i$ eigen-bundle $L\subset \TTX\otimes \C$, the *type* of $\J$ at a point $x\in X$ is the complex dimension of $L\cap (T\uv X\otimes \C)$; equivalently, it is the complex codimension of the projection $\pi_{TX\otimes\C}(L)\subset TX\otimes \C$. The GC manifold is said to be *regular* at a point $x\in X$ if the type of $\J$ is constant on some neighborhood of $x$.
An equivalent definition of the type can be given as follows. Decompose $\J$ as $$\J=\left(\begin{array}{cc} \J_{11} & \J_{12} \\ \J_{21} & \J_{22}\end{array}\right),$$ where $\J_{11}:TX\to TX$, $\J_{21}:T X\to T\uv X$, $\J_{12}:T\uv X\to TX$, and $\J_{22}:T\uv X\to T\uv X$. The fact that $\J$ preserves the natural pairing and squares to minus the identity implies that $P:=\J_{12}:T\uv X\to TX$ is skew-symmetric, i.e. for every $a,b\in T\uv X$ we have $$a(P(b))=-b(P(a)).$$ In fact, the integrability of $\J$ implies that $P$ is a Poisson structure on $X$. Defining $R=\Ker(P)\subset T\uv$, we easily check that $\J_{22}:T\uv\to T\uv$ preserves $R$ and in fact restricts to a complex structure on $R$. It is then easy to check that the type of $\J$ is equal $\textrm{Dim}_{\C}(R)=\frac{1}{2}\textrm{Dim}_{\R}(R)$. In particular, $\J$ is regular at a point $x\in X$ if and only if the Poisson structure $P$ is regular at $x$, i.e. if and only if $\textrm{Dim}_{\R}(P(T\uv X))$ is constant on a neighborhood of $x\in X$.
As an example, given a symplectic manifold $(X,\omega)$ and a complex manifold $(Y,J)$ of complex dimension $k$, the product $(X\times Y,\J_{\omega}\times\J_J)$ is of type $k$ at every point; in particular, it is everywhere regular. Conversely, the *generalized Darboux theorem*, due to Gualtieri [@G], says if $x$ is a regular point of an arbitrary GC manifold $(X,\J)$, then there exists some neighborhood of $x$ on which $\J$ is equivalent to such a product. Let us give a precise statement of this result in a form convenient for our purposes. Endow $\R^{2m}\cong (\R^m)\uv$ with the standard symplectic structure $$\omega=dx^{m+1}\wedge dx^1+\cdots +dx^{2m}\wedge dx^{m},$$ and $\R^{2n}\cong \C^n$ with the standard complex structure. Let $X_0^{m,n}=(\R^{2m+2n},\J_0^{m,n}=\J_{\omega}\times \J_J)$ be the product GC manifold. We may then state (a slight variation) of the generalized Darboux theorem [@G Th. 4.35].
\[generalized Darboux\] Let $(X,\J)$ be a GC manifold of dimension $2(m+n)$, and let $x\in X$ be a regular point such that $\J$ is of type $n$ at $x$. Then there exists a neighborhood $U$ of $x$, a neighborhood $U_0$ of the origin in $X^{m,n}_0$, a diffeomorphism $\Phi:U{\xymatrix@C=1.5pc{ \ar[r]^-{\cong} & }}U_0$ taking $x$ to the origin, and a 1-form $u\in \Omega^1(U)$, such that $$\Phi^*(\J^{m,n}_0|_{U_0})=e^u\cdot(\J|_{U}).$$
Generalized Holomorphic Vector Fields
-------------------------------------
Let $\J$ be a GC structure on $X$, and consider the projection maps $$\pi_L:\TTX\otimes\C\to L$$ given by $$\xx\mapsto \xx^{(1,0)}:= \frac{1}{2}(\xx-i\J\xx).$$ and $$\pi_{\bar{L}}:\TTX\otimes\C\to \bar{L}$$ given by $$\xx\mapsto \xx^{(0,1)}:= \frac{1}{2}(\xx+i\J\xx).$$
Also define $\mu:\TTX\to L\uv$ by $$\label{mudef} \mu(\xx)= 2\la\xx,\cdot\ra|_{L}.$$ We easily check that $$\J^{\vee}\mu(\xx)=\mu(-\J\xx),$$ so that if we regard $\TTX$ as a complex vector bundle with complex structure $-\J$, $\mu$ is a map of complex vector bundles. We also note that, since $\xx=\xx^{(1,0)}+\xx^{(0,1)}$ and $L$ is isotropic, we have $$\mu(\xx)=2\la \xx^{(0,1)},\cdot\ra|_L.$$ Since $\pi_{\bar{L}}:(\TTX,-\J)\xymatrix@C=1.5pc{ \ar[r]^-{\cong} & } \bar{L}$ is an isomorphism of complex vector bundles, and $\bar{L}$ pairs non-degenerately with $L$, it follows that $\mu$ is an isomorphism.
\[generalized holomorphic vector field\] A section $\xx\in \cg(X)\cong C^{\infty}(\TTX)$ is a *generalized holomorphic vector field* if it satisfies $$\label{ghol}\delta_{L}\mu(\xx)=0.$$ We denote the space of all generalized holomorphic vector fields by $\T(X)\subset \cg(X)$.
The space of generalized holomorphic vector fields of course depends on the GC structure $\J$. When necessary, will use the more precise notation $\T(X,\J)$ to indicate which GC structure is appearing in the condition (\[ghol\]).
The next proposition shows that we may view the space of generalized holomorphic vector fields as the infinitesimal symmetries of the GC structure $\J$.
\[holomorphic symmetries\] For each $\xx\in\cg(X)$, we have $$\xx\cdot\J=0$$ if and only if $$\delta_{L}\mu(\xx)=0.$$ In particular, if $\xx$ is complete then $$e^{\xx}\J=\J.$$
It is easy to see from this Proposition that the subspace $\T(X)\subset \cg(X)$ is closed under the Lie bracket (\[ghatbracket\]).
The following lemma will be useful for the proof of the proposition.
For each $\xx\in\cg(X)$, we have $\xx\cdot\J=0$ if and only if $$\xx\cdot\Cinf(L)\subset\Cinf(L).$$
Suppose $\xx\cdot\J=0$. Then for every $v\in\Cinf(L)$ we have $$0=\xx\cdot(\J v)-\J(\xx\cdot v)=i\xx\cdot v-\J(\xx\cdot v)$$ so that $$\J(\xx\cdot v)=i\xx\cdot v,$$ so we see that $\xx\cdot v\in\Cinf(L)$. Conversely, suppose that $\xx\cdot\Cinf(L)\subset\Cinf(L)$. Since $\xx$ is real it follows that $\xx\cdot\Cinf(\bar{L})\subset\Cinf(\bar{L})$ also. For arbitrary $v\in\Cinf(\TTX\otimes\C)$, we see that $$\xx\cdot\pi_L(v)=\pi_L(\xx\cdot\pi_L(v))=\pi_L(\xx\cdot v)$$ and $$\xx\cdot\pi_{\bar{L}}(v)=\pi_{\bar{L}}(\xx\cdot\pi_{\bar{L}}(v))=\pi_{\bar{L}}(\xx\cdot v).$$ Therefore $$\begin{aligned}
(\xx\cdot\J)(v) &= \xx\cdot(\J v)-\J(\xx\cdot v) \\
& =i\xx\cdot\pi_L(v)-i\xx\cdot\pi_{\bar{L}}(v)-i\pi_L(\xx\cdot v)+i\pi_{\bar{L}}(\xx\cdot v)\\
&= 0.\end{aligned}$$
By the lemma, to prove the Proposition, we must show that $\delta_L\mu(\xx)=0$ if and only if $\xx\cdot\Cinf(L)\subset\Cinf(L)$. Since $L$ is a maximal isotropic subspace of $\TTX\otimes \C$, we have $\xx\cdot\Cinf(L)\subset\Cinf(L)$ if and only if for every pair of sections $v,w\in\Cinf(L)$ we have $$\label{preservation condition}\la\xx\cdot v,w\ra=0.$$ We claim that the left-hand side of equation (\[preservation condition\]) is equal to $\frac{1}{2}\delta_L(\mu(\xx))(w,v)$, so that it vanishes for arbitrary $v$ and $w$ if and only if $\xx\in\T(X)$. To see this, recall the following two identities satisfied by the Dorfman bracket (see e.g. [@G §3.2])
1. For every $A,B\in \Cinf(\TTX)$ $$\label{Dorfman1}\ll A,B\rr=-\ll B,A\rr +2(0,d\la A,B\ra).$$
2. For every $A,B,C\in\Cinf(\TTX)$ $$\label{Dorfman2}\pi(A)\la B,C\ra=\la\ll A,B\rr,C\ra+\la B,\ll A,C\rr\ra.$$
Using these we have $$\begin{aligned}
\la \xx\cdot v,w\ra &= \la \ll\xx,v\rr,w\ra \\
&= -\la\ll v,\xx\rr,w\ra+2\la d\la \xx,v\ra,w\ra \\
&= -(\pi(v)\la\xx,w\ra-\la\xx,[v,w]\ra)+\pi(w)\la\xx,v\ra \\
&= \pi(w)\la\xx,v\ra-\pi(v)\la\xx,w\ra-\la\xx,[w,v]\ra \\
&=\frac{1}{2}\delta_L(\mu(\xx))(w,v).\end{aligned}$$
Given a complex structure $J$ on $X$, let $\J_J$ be the induced GC structure described in Example \[complex\]. It is straightforward to check that a vector field $\xi\in \Cinf(TX)$ and a 1-form $u\in \Omega^1(X)$ determine a generalized holomorphic vector field $\xx=(\xi,u)$ on $(X,\J_J)$ if and only if $$\label{holomorphic 1} \bar{\partial}\xi^{1,0}=0$$ and $$\label{holomorphic 2} \bar{\partial}u^{0,1}=0.$$
Note that condition (\[holomorphic 1\]) is equivalent to requiring $$\pounds(\xi)J=0,$$ whereas condition (\[holomorphic 2\]) is equivalent to requiring that $du$ be of complex type $(1,1)$.
Given a symplectic structure $\omega$ on $X$, let $\J_{\omega}$ be the induced GC structure described in Example \[symplectic\]. In this case, a vector field $\xi\in \Cinf(TX)$ and 1-form $u\in \Omega^1(X)$ determine a generalized holomorphic vector field $\xx=(\xi,u)$ on $(X,\J_{\omega})$ if and only if $$\pounds(\xi)\omega=0$$ and $$du=0.$$ Thus, an infinitesimal symmetry of $\J_{\omega}$ consists of an infinitesimal symplectomorphism $\xi$ together with a closed 1-form $u$.
Generalized Hamiltonian Vector Fields
-------------------------------------
Given a symplectic manifold $(X,\omega)$ and a real-valued function $f:X\to \R$, recall that the *Hamiltonian vector field* associated to $f$ is defined (up to a sign convention) by $$\label{ordinary hamiltonian} X_f=-\omega^{-1}(df).$$ This vector field is an infinitesimal symmetry of the symplectic structure in the sense that $$\pounds_{X_f}\omega=0,$$ and the collection of all Hamiltonian vector fields form a sub-algebra of $\Cinf(TX)$. We next introduce a generalization of this construction. Namely, given a GC manifold $(X,\J)$, for every *complex*-valued function $f:X\to \C$ we will define infinitesimal symmetry $\xx_f$ of $\J$.
\[genhamdef\] Let $(X,\J)$ be a GC manifold. For every smooth function $f:X\to \C$, the *generalized Hamiltonian vector field* $\xx_f\in\Cinf(\TTX)$ associated to $f$ is given by $$\label{genham}\xx_f=-\textrm{Re}(2i(0,df)^{(0,1)})=\textrm{Re}(\J(0,df)-(0,idf)).$$
The author learned of this construction (for real-valued $f$) in [@H2 Prop 6], where it is shown that $\xx_f$ is an infinitesimal symmetry of $(X,\J)$. We give a different proof of this result (for complex-valued $f$) as part (1) of the following proposition. To the best of our knowledge, part (2) of Proposition \[hamiltonian\] has not appeared before.
\[hamiltonian\]
1. For every $f:X\to \C$, the generalized Hamiltonian vector field $\xx_f$ is an element of $\T(X)$, i.e. satisfies $$\delta_{L}\mu(\xx_f)=0.$$
2. The collection $\H$ of generalized Hamiltonian vector fields is a Lie sub-algebra of $\T(X)$.
Given $f:X\to \C$, one easily calculates using Definition \[genhamdef\] and and (\[mudef\]) that $$\mu(\xx_f)=-i\delta_{L}f,$$ where on the right-hand side $f$ is regarded as a section of $\Lambda^0 L\uv$. The first part of the proposition therefore follows from the identity $\delta_{\J}^2=0$. To prove the second part, we must show that given any $f,g:X\to \C$, there exists $h:X\to\C$ such that $[\xx_f,\xx_g]=\xx_h$. First, suppose that $f$ and $g$ are purely imaginary. Writing $f=if_I$ and $g=ig_I$, we have $$[\xx_f,\xx_g]=[(0,df_I),(0,dg_I)]=0,$$ so the result is trivial in this case. Next, suppose that $f$ is purely real and $g=ig_I$ is purely imaginary. Decomposing the GC structure as $$\label{J decom}\J=\left(\begin{array}{cc} J & P \\ \sigma & K\end{array}\right),$$ we have $$[\xx_f,\xx_g]=[(P df,Kdf_r),(0,dg_I)]=(0,\pounds(P df)dg_I).$$ Using the Cartan formula for the Lie derivative, we see that this is equal to $$(0,d\iota(P df)dg_I)=\xx_{i\iota(P df)dg_I}.$$ The last case, where both $f$ and $g$ are purely real, is more difficult, and uses the integrability of $\J$ in a crucial way. It will be useful to rewrite the integrability condition for $\J$. To do so, first recall the definition of the *Courant Bracket* on sections of $\TT X$, which is given by a formula closely related to that for the Dorfman bracket [@G]: $$[(\xi,a),(\eta,b)]_C=([\xi,\eta],\pounds(\xi)b-\pounds(\eta)a-\frac{1}{2}d(\iota(\xi)b-\iota(\eta)a)).$$ Given an almost generalized complex structure $\J$, we define the *Nijenhuis tensor* $\Nij:\TT X\otimes \TT X\to \TT X$ of $\J$ by the formula $$\Nij(A,B)=[\J A,\J B]_C-\J[\J A,B]_C-\J[A,\J B]_C-[A,B]_C$$ for every pair of sections $A,B\in\Cinf(\TT X)$. Integrability of $\J$ is equivalent to the vanishing of $\Nij$ [@AB].
Returning to the proof of Proposition \[hamiltonian\], the integrability of $\J$ implies that, for every pair of real-valued functions $f$ and $g$ we have $$\begin{aligned}
\label{nij0}0 & =\Nij((0,df),(0,dg)) \notag\\
& =[\J(0,df),\J(0,dg)]_C-\J[\J(0,df),(0,dg)]_C-\J[(0,df),\J(0,dg)]_C-[(0,df),(0,dg)]_C.\end{aligned}$$ In terms of the decomposition (\[J decom\]) of $\J$, we have $[\J(0,df),\J(0,dg)]_C$= $$\label{eqn11} ([P df, P dg],\pounds(P df)Kdg-\pounds(P dg)Kdf-\frac{1}{2}d\iota(P df)dg+\frac{1}{2}d\iota(P dg)df).$$ It is easy to check that $P:T\uv X\to TX$ must be skew-symmetric, in the sense that for every $a,b\in \Omega^1(X)$ we have $a(P(b))=-b(P(a)$. Therefore the expression (\[eqn11\]) is equal to $$\label{nij1} ([P df, P dg],\pounds(P df)Kdg-\pounds(P dg)Kdf-d\iota(P df)dg).$$ Similarly, we calculate $$\begin{aligned}
\label{nij2} \J[\J(0,df),(0,dg)]_C & = \J(0,\pounds(P df)dg-\frac{1}{2}d\iota(P df)dg) \notag\\
& = (P\pounds(P df)dg-\frac{1}{2}P d\iota(P df)dg, K\pounds(P df)dg-\frac{1}{2}K d\iota(P df)dg) \notag \\
& = (P d\iota(P df)dg-\frac{1}{2}P d\iota(P df)dg,K d\iota(P df)dg-\frac{1}{2}Kd\iota(P df)dg) \notag \\
& =(\frac{1}{2}P d\iota(P df)dg-,\frac{1}{2}K d\iota(P df)dg) ,\end{aligned}$$ and $$\begin{aligned}
\label{nij3} \J[(0,df),\J(0,dg)]_C & =(-P \pounds(P dg)df+\frac{1}{2}P d\iota(P dg)df, -K \pounds(P dg)df+\frac{1}{2}Kd\iota(P dg)df)\notag \\
& = (-P d\iota(P dg)df+\frac{1}{2}P d\iota(P dg)df,-K d\iota(P dg)df+\frac{1}{2}K d\iota(P dg)df) \notag\\
& = (-\frac{1}{2}P d\iota(P dg)df,-\frac{1}{2}K d\iota(P dg)df) .\end{aligned}$$ Also, we easily see that $[(0,df),(0,dg)]_C=0$. Adding the 1-form components of (\[nij1\]), (\[nij2\]), and (\[nij3\]) (and again using the skew symmetry of $P$), the 1-form component of equation (\[nij0\]) yields: $$0=\pounds(P df)Kdg-\pounds(P dg)Kdf-d(\iota(P df)Kdg)-Kd\iota(P dg)df,$$ or equivalently $$\label{nij4}\pounds(P df)Kdg-\pounds(P dg)Kdf=d(\iota(P df)Kdg)+Kd\iota(P dg)df.$$ On the other hand, adding the vector field components of (\[nij1\]), (\[nij2\]), and (\[nij3\]) we see that $$\label{nij5}[P d\iota(P df)dg,P d\iota(P dg)df)=P d\iota(P dg)df.$$ With equations (\[nij4\]) and (\[nij5\]) in hand, we calculate $$\begin{aligned}
[\xx_f,\xx_g] & = [(P(0,df),K(0,df),(P(0,df),K(0,df))] \notag \\
& = ([P(0,df),P(0,dg)],\pounds(P (0,df))K(0,dg)-\pounds(P (0,dg))K(0,df)) \notag \\
& = (P d\iota(P dg)df, Kd\iota(P dg)df)+(0,d\iota(P df)Kdg).\end{aligned}$$ If we define $h=\iota(P dg)df+i\iota(P df)Kdg$, we see that $$[\xx_f,\xx_g]=\xx_h.$$
In the case that $\J$ is induced by an ordinary complex structure $J$, the generalized Hamiltonian vector field associated to $f:X\to \C$ is given by $$(0,2\textrm{Re}(\bar{\partial}f)).$$
Suppose $\J$ is induced by a symplectic structure $\omega$. Given $f:X\to \C$, write $f=f_R+if_I$ for real-valued functions $f_R,f_I$. The generalized Hamiltonian vector field associated to $f$ is given by $$\xx_f=(X_{f_R},df_I),$$ where $X_{f_R}$ is the ordinary Hamiltonian vector field for the function $f_R:X\to \R$ given by formula (\[ordinary hamiltonian\]).
Generalized complex submanifolds and branes {#GC submanifolds}
===========================================
Generalized submanifolds
------------------------
The following definition is a special case of [@G Def. 7.4].
A *generalized submanifold* of a smooth manifold $X$ is a pair $(Z,F)$, where $Z\subset X$ is a submanifold, and $F\in \Omega^2(Z)$ is a closed 2-form.
\[geometric brane\] As discussed in the introduction, we will primarily be interested in a related structure, which we call (following [@KL]) a (rank 1) *brane* on $X$. Such an brane is a pair $\B=(Z,\L)$, where $Z\subset X$ is a submanifold, and $\L$ is a Hermitian line bundle with unitary connection supported on $Z$. In particular, by setting $F\in\Omega^2(Z)$ to be the curvature form of $\L$, every such $\B$ determines a generalized submanifold (which we sometimes call the *underlying* generalized submanifold of $\B$) . When we study the deformation theory of such branes later in the paper, it will actually be conventient to use a slightly different–but essentially equivalent–definition (Definition \[brane definition 1\])
Given a submanifold $Z\subset X$, let $i:Z\hookrightarrow X$ denote the inclusion, and let $(TX)|_Z$ and $(T\uv X)|_Z$ denote the restrictions to $Z$ of the tangent and cotangent bundles of $X$. We then have the push-forward $$i_*:TZ\to (TX)|_Z$$ and pull-back $$i^*:(T\uv X)|_Z\to T\uv Z.$$
\[gen tangent def\] [@G Def. 7.5] The *generalized tangent bundle* $\TT(Z,F)$ of a generalized submanifold $(Z,F)$ is the sub-bundle of $\TTX|_Z$ whose fiber at $z\in Z$ is given by $$\TT_z(Z,F)=\{(i_*\xi,a)\in T_zX\oplus T\uv_zX:\xi\in T_zZ, i^*a=\iota(\xi)F\}.$$
The generalized tangent bundle $\TT(Z,F)$ is a maximal isotropic sub-bundle of $\TTX|_Z$ with respect to the (restriction of) the pairing $\la\cdot,\cdot\ra$. It fits into an exact sequence $$\xymatrix{ 0 \ar[r] & \Ann(TZ) \ar[r] & \TT(Z,F) \ar[r] & TZ \ar[r] & 0}.$$ When $F=0$, there is a natural splitting of this sequence $TZ\to \TT(Z,F)$ given by $$\xi\mapsto (i_*\xi,0),$$ which exhibits $\TT(Z,F)$ as the direct sum $TZ\oplus\Ann(TZ)$.
\[definition K\] Let $$r:\cg(X)=\Cinf(\TTX)\to \Cinf(\TTX|_Z)$$ denote the restriction map. We define $K^{(Z,F)}\subset \cg(X)$ by $$K^{(Z,F)}=r^{-1}(\Cinf(\TT(Z,F))).$$ In other words, $K^{(Z,F)}$ consists of those section of $\TTX$ which extend section of $\TT(Z,F)$.
\[K closed under bracket\] The subspace $K^{(Z,F)}\subset \Cinf(\TTX)$ is closed under the Dorfman bracket.
Let $\xx=(\xi,v)$, $\yy=(\eta,w)$ be elements of $\KK^{(Z,F)}$. By the assumption that $\xx$ and $\yy$ are elements of $\KK^{(Z,F)}$, the vector fields $\xi$ and $\eta$ are tangent to $Z$, so there exist unique vector fields $\tau,\zeta\in C^{\infty}(TZ)$ such that the restriction of $\xi$ to $Z$ is equal to $i_*\tau$ and the restriction of $\eta$ to $Z$ is equal to $i_*\zeta$. Furthermore, we have $i^*v=\iota(\tau)F$ and $i^*w=\iota(\zeta)F$. Since $\xi$ and $\eta$ are both tangent to $Z$, their Lie bracket $[\xi,\eta]$ is as well; in fact its restriction to $Z$ is equal to $i_*[\tau,\zeta]$. We have $$\ll\xx,\yy\rr= ([\xi,\eta],\pounds(\xi)w-\iota(\eta)dv),$$ so we need to show that $$i^*(\pounds(\xi)w-\iota(\eta)dv)=\iota([\tau,\zeta])F.$$ Expanding the right hand side using the identity $\iota([\tau,\zeta])=[\pounds(\tau),\iota(\zeta)]$ and the Cartan formula we obtain $$\begin{aligned}
\iota([\tau,\zeta])F & = \pounds(\tau)\iota(\zeta)F-\iota(\zeta)\pounds(\tau)F \\
& = \pounds(\tau)\iota(\zeta)F-\iota(\zeta)d\iota(\tau)F-\iota(\zeta)\iota(\tau)df \\
& = \pounds(\tau)\iota(\zeta)F-\iota(\zeta)d\iota(\tau)F \\
& = \pounds(\tau)i^*v-\iota(\zeta)di^*w\\
& = i^*(\pounds(\xi)v-\iota(\eta)dw).\end{aligned}$$
The following definition is taken from [@KM].
The *generalized normal bundle* of the generalized submanifold $(Z,F)$ is the quotient $$\N(Z,F)=\TTX|_Z/\TT(Z,F).$$
Let $q:\TTX|_Z\to \N(Z,F)$ denote the quotient map. Since $\TT(Z,F)\subset \TTX|_Z$ is maximal isotropic, it follows that the there is a well-define pairing of $\N(Z,F)$ with $\TT(Z,F)$ given by $$\la q(\xx),\yy\ra=\la\xx,\yy\ra,$$ which identifies $\N(Z,F)$ with the dual of $\TT(Z,F)$.
Compatibility with a GC structure
---------------------------------
\[GC submanifold\] [@G Def. 7.6] Let $(X,\J)$ be a GC manifold. A generalized submanifold $(Z,F)$ of $X$ is *compatible with $\J$* if $$\label{compatibility} \J|_Z(\TT(Z,F))=\TT(Z,F).$$ In the case we say that $(Z,F)$ is a *generalized complex submanifold* of $(X,\J)$.
\[geometric brane 1\] Given a brane $\B=(Z,\L)$ on $X$, as described in Remark \[geometric brane\], we say that $\B$ is compatible with $\J$ if its underlying generalized submanifold $(Z,F)$ is. In this case, we simply call $\B$ a *generalized complex (GC) brane*. As mentioned above, the definition we use later (Definition \[GC brane definition 1\]) is actually slightly different.
\[rem1\]It will be convenient to recast the compatibility condition (\[compatibility\]) in a slightly different form. Given a GC manifold $(X,\J)$, define $$Q_{\J}:\TTX\times\TTX\to X\times\R$$ by $$\label{Q def}Q_{\J}(\xx,\yy)=\la\J\xx,\yy\ra.$$We easily verify that $Q_{\J}$ is skew-symmetric and non-degenerate. Furthermore, since $\TT(Z,F)$ is a maximal isotropic sub-bundle of $\TTX|_{Z}$, it follows that $\TT(Z,F)$ is preserved by $\J$ if and only if the restriction of $Q_{\J}$ to $\TT(Z,F)$ vanishes. If we define $$\label{I def}\II^Z(X)=\{f\in \Cinf(X):f|Z=0\},$$ then a generalized submanifold $(Z,F)$ is compatible with $\J$ if and only if $$Q_{\J}(K^{(Z,F)},K^{(Z,F)})\subset \II^Z.$$
\[symplectic sub\] In the case where $\J=\J_J$ comes from a complex structure $J$ on $X$, it is shown in [@G Ex. 7.7] that a generalized submanifold $(Z,F)$ is compatible with $\J$ if and only if $Z$ is a complex submanifold of $X$, and $F$ is of type $(1,1)$ with respect to the induced complex structure on $Z$.
\[complex sub\]Suppose $\J=\J_{\omega}$ comes from a symplectic structure on $X$. If $Z\subset X$ is a Lagrangian submanifold, then $(Z,F)$ is compatible with $\J$ if and only if $F=0$. Conversely, a generalized submanifold of the form $(Z,0)$ is compatible with $\J$ if and only if $Z$ is Lagrangian. On the other hand, there exist more exotic examples where $F$ is non-zero, and $Z$ is coisotropic but of dimension greater than $\textrm{Dim}_{\R}(X)/2$ [@G Ex. 7.8]
The action symmetries on GC submanifolds
-----------------------------------------
Let $(Z,F)$ be a generalized submanifold of a manifold $X$. Let $Y$ be another manifold, $\varphi:Y\to X$ , a diffeomorphism, and $u\in \Omega^1(Y)$ a 1-form. Let $g=e^u\varphi^*:X\to Y$ be the morphism in the groupoid $\cG$ introduced in Remark \[symmetry groupoid\]. Consider the submanifold $\varphi^{-1}(Z)\subset Y$, and let $\varphi_Z:\varphi^{-1}(Z)\to Z$ denote the diffeomorphism induced by $\varphi$. We define $g\cdot (Z,F)$ to be the generalized submanifold of $Y$ given by $(\varphi^{-1}(Z),(\varphi_Z)^*F-di_{\varphi^{-1}(Z)}^*u).$
\[action on GC submanifolds\]
1. $$g\cdot K^{(Z,F)}=K^{g\cdot(Z,F)}.$$
2. Given a GC structure $\J$ on $X$, a generalized submanifold $(Z,F)$ on $X$ is compatible with $\J$ if and only if $g\cdot (Z,F)$ is compatible with the GC structure $g\cdot \J:=g\cdot\J\cdot g^{-1}$ on $Y$.
First, let consider the case where $g$ is of the form $g=(\textrm{id}_X,u)$ for some $u\in\Omega^1(X)$; we then have $$g\cdot (Z,F)=(Z,F+di_Z^*u).$$ Set $F'=F+di_Z^*u$. Given $\xx=(\xi,a)\in K^{(Z,F)}$, let $\xx'=g\cdot\xx=(\xi',a')$; we then have $\xi'=\xi$ and $a'=a-\iota(\xi)du$. Since $\xx\in K^{(Z,F)}$, $\xi$ is tangent to $Z$. Denoting the restriction of $\xi$ to $Z$ by $\tau\in\Cinf(TZ)$, we than have $$i^*a=\iota(\tau)F.$$ This implies that $$i^*a'=\iota(\tau)F-\iota(\tau)di^*u=\iota(\tau)F',$$ and we therefore have $\xx'\in K^{(Z,F')}$.
Set $Z'=\varphi^{-1}(Z)$ and $F'=\varphi_Z^*F\in\Omega^2(Z')$, so that $g\cdot (Z,F)=(Z',F')$. Let $i_Z:Z\hookrightarrow X$ and $i_{Z'}:Z'\hookrightarrow X$ the inclusion maps, and note that by construction we have $$\label{diff com1}i_{Z}\varphi_Z=\varphi i_{Z'}$$ and $$\label{diff com}i_{Z'}\varphi^{-1}_Z=\varphi^{-1} i_Z.$$ Given $z\in Z$, and $z'=\varphi^{-1}(z)\in Z'$, let $\Psi_z:\TT_zX\to \TT_{z'}$ denote the linear isomorphism induced by the pair $(\varphi,u)$; in other words, for each section $\xx\in\Cinf(\TT X)$, we have $$(g\cdot \xx)_{z'}=\Psi(\xx_z).$$ Part (1) of the Proposition is equivalent to the statement that, for each $z\in Z$ we have $$\Psi_z(\TT_z(Z,F))=\TT_{z'}(Z',F').$$ Given $\xx=(\xi,a)\in\T_z(Z,F)$, by definition we have $\xi=(i_Z)_*\tau$ for some $\tau\in T_zZ$, $i_Z^*a=\iota(\tau)F$. Let us write $(\xi',a')=\Psi_z(\xi,a)\in \TT_{z'}X$. Defining $\tau'=(\varphi_Z^{-1})_*\tau\in T_{z'}Z'$, the equality (\[diff com\]) implies that $$\begin{aligned}
\xi' & = \varphi^{-1}_*(\iota_Z)_*\tau \\
& = (\varphi^{-1}\iota_Z)_*\tau \\
& = (i_{Z'}\varphi_Z^{-1})_*\tau \\
& = (i_{Z'})_*\tau'.\end{aligned}$$ Similarly, using (\[diff com1\]) we see that $$\begin{aligned}
i_{Z'}^*a' & = i_{Z'}^*\varphi^*a \\
& = (\varphi i_{Z'})^*a \\
& = (i_Z\varphi_Z)*a \\
& = \varphi_Z^*i_Z^*a\\
& = \varphi_Z^*(\iota(\tau)F) \\
& = \iota(\tau')F'.\end{aligned}$$ This completes the proof of part (1).
Let $\J'$ denote $g\cdot \J$. Using part (1) as well as Proposition \[leftgroupaction\] we have $$\la \J'K^{(Z',F')},K^{(Z',F')}\ra = \la g\J g^{-1}gK^{(Z,F)},gK^{(Z,F)} \ra= \varphi^* \la \J K^{(Z,F)},K^{(Z,F)}\ra.$$ On the other hand, we clearly have $\varphi^*I^Z=I^{Z'}$, so it follows that $\la \J'K^{(Z',F')},K^{(Z',F')}\ra\subset I^{Z'}$ if and only if $\la \J K^{(Z,F)},K^{(Z,F)}\ra\subset I^Z$. Recalling the discussion in Remark \[rem1\], this proves part (2) of the Proposition.
\[geometric brane 2\] Continuing with Remarks \[geometric brane\] and \[geometric brane 1\], we may define a similar action of $\cG(X)$ on the set of branes on $X$. Given a brane $\B=(Z,\L)$ and a diffeomorphism $\varphi:X{\xymatrix@C=1.5pc{ \ar[r]^-{\cong} & }}X$, define $$g\cdot\B:=(\varphi^{-1}(Z),\varphi_Z^*\L),$$ where as above $\varphi_Z:\varphi^{-1}(Z){\xymatrix@C=1.5pc{ \ar[r]^-{\cong} & }}Z$ denotes the diffeomorphism induced by $\varphi$. Given a 1-form $u\in\Omega^1(X)$, the group element $(\textrm{id}_X,u)$ acts on $\B$ changing the connection $\nabla$ on $\L$ according to $$\nabla\mapsto \nabla-2\pi i u|_Z.$$ Using Proposition \[action on GC submanifolds\], it is then easy to check that for any $g\in G$, if the brane $\B$ is compatible with a GC structure $\J$ then $g\cdot \B$ is compatible with the GC structure $g\cdot \J$. In particular, the symmetries of a fixed $\J$ act on the set of $GC$ branes on $(X,\J)$.
The Lie algebroid complex of a GC submanifold {#Lie alg cohomology}
---------------------------------------------
Let $(Z,F)$ be a GC submanifold of $(X,\J)$. Since $\J$ restricts to an endomorphism of $\TT(X,F)$ which squares to minus the identity, we have the decomposition $$\TT(Z,F)\otimes \C=l\oplus \bar{l},$$ where $l$ is the $+i$-eigenbundle of $\TT(Z;F)\otimes \C$ and $\bar{l}$ the $-i$-eigenbundle. By construction, $l$ is a sub-bundle of $L|Z$. Furthermore, as described in [@G], the Lie algebroid bracket on $L$ (i.e. the restriction of the Dorman bracket to $L$) induces a well-defined bracket on sections of $l$, giving $l$ the structure of a complex Lie algebroid over $Z$ with anchor map the projection $\pi:l\to TZ\otimes\C$. Explicitly, given sections $\xx,\yy\in\Cinf(l)$, choose sections $\tilde{\xx},\tilde{\yy}\in\Cinf(L)$ that extend $\xx$ and $\yy$, i.e. such that $r(\tilde{\xx})=\xx$ and $r(\tilde{\yy})=\yy$. Then it is straightforward to check that $r(\ll\tilde{\xx},\tilde{\yy}\rr)\in\Cinf(L|_Z)$ lies in $\Cinf(l)$ and is independent of the choice of extensions $\tilde{\xx}$ and $\tilde{\yy}$. We therefore have a well-defined bracket given by the formula $$\label{brane bracket}\ll\xx,\yy\rr_{\B}=r(\ll\tilde{\xx},\tilde{\yy}\rr).$$ Just as we did for the Lie algebroid $L$, we construct from $l$ a differential complex $$\xymatrix{ \Cinf(\Lambda^0l\uv) \ar[r]^{\delta_l} & \Cinf(\Lambda^1l\uv) \ar[r]^{\delta_l} & \Cinf(\Lambda^2l\uv) \ar[r]^{\delta_l} & \cdots}.$$
Recall that we defined a linear isomorphism $\mu:\TT X\to L^{\uv}$ on any GC manifold. Similarly, for any GC submanifold $(Z,F)$, we have a linear bijection $$\mu:\N^{(Z,F)}\to l\uv$$ characterized by the equation $$\mu(q(\xx))(v)=\la \xx,v\ra$$ for any pair of sections $\xx\in \Cinf(\TT X|_Z)$ and $v\in \Cinf(l)$, where $q:\TT X|_Z\to \N^{(Z,F)}$ is the quotient map introduced above.
A *generalized holomorphic section* of the generalized normal bundle is a section $\xx\in\Cinf(\N Z)$ satisfying $$\delta_l\mu(\xx)=0.$$
The following proposition is an easy consequence of the definitions, and we omit the proof.
\[induced holomorphic\] Let $\xx\in\Cinf(\TT X)$ be a generalized holomorphic vector field on $X$. Then $r(\xx)\in\Cinf(\N(Z,F))$ is a generalized holomorphic section of $\N(Z,F)$.
Leaf-Wise Lagrangian GC Submanifolds {#LWL submanifolds}
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\[LWL submanifolds\]
Let $\J$ be an a generalized complex structure on a manifold $X$. As discussed in $\S\ref{GC structures}$, $\J$ induces a Poisson structure $P:T\uv X\to TX$. Let $S=P(T\uv X)\subset TX$ be the (not necessarily constant rank) distribution induced by $P$. The distribution $S$ inherits symplectic structure, i.e. a non-degenerate, skew-symmetric pairing $\omega$ (which is “closed" in an appropriate sense) defined by $$\label{sym dis}\omega(\xi,\eta)=a(\xi),$$ where $a\in
\Cinf(T\uv X)$ is any section with $P(a)=\eta$. For example, when $\J$ is induced by a symplectic structure as in Example \[symplectic\], then $S=TX$ and the induced pairing $\omega$ is given by the original symplectic form.
\[definition LWL submanifold\] Let $(Z,F)$ be a GC submanifold of $(X,\J)$, such that $\J$ is regular at each point of $Z$. We say that $(Z,F)$ is *leaf-wise* Lagrangian if for each $z\in Z$ the intersection $T_zZ\cap S_z$ is a Lagrangian subspace of $S_z$ with respect to the pairing (\[sym dis\]).
\[standard brane\] Let $(X^{m,n}_0=\R^{2m+2n},\J_0^{m,n}=\J_{\omega}\times\J_J)$ be the “standard" GC manifold that appeared in Theorem \[generalized Darboux\]. Introduce coordinates $(s_0,\cdots, s_{2m},t_0,\cdots,t_{2n})$ on $X_0^{m,n}$, where $s_0,\cdots s_{2m}$ are standard coordinates on $\R^{2m}$, and $t_0,\cdots t_{2n}$ are standard coordinates on $\R^{2n}\cong \C^n$. For a natural number $k\leq n$, and let $Z^k_0\subset X^{m,n}_0$ be the product of the Lagrangian submanifold $\R^m\subset \R^{2m}$ with the complex submanifold $\C^k\subset \C^n$; explicitly, $Z^k_0$ is defined by the conditions $s_{m+1}=\cdots= s_{2m}=t_{2k+1}=\cdots = t_{2n}=0$. We easily check that $Z^k_0$, equipped with the zero 2-form, is a leaf-wise Lagrangian GC submanifold of $(X_0,\J_0)$.
The following result, giving a local normal form for LWL GC submanifolds, is a variation on the generalized Darboux theorem (Theorem \[generalized Darboux\]).
\[LWL normal form\] Let $(Z,F)$ be a LWL GC sub manifold of $(X,\J)$, where $X$ is of dimension $2(m+n)$, $Z$ is of dimension $m+2k$, and $\J$ is of type $n$ at each point of $Z$. Then for each $z\in Z$, there exists a neighborhood $U$ of $z$ in $X$, a 1-form $u\in \Omega^1(U)$, a neighborhood $U_0$ of the origin in $X^{m,n}_0$, and a diffeomorphism $\Phi:U\to U_0$ such that
1. $\Phi(Z\cap U)=Z^k_0\cap U_0$ and $\Phi(z)=0$,
2. $\Phi^*(\J^{m,n}_0|_{U_0})=e^u\cdot\J|_{U}$, and
3. $F|_{U\cap Z}=d\rho (u)$, where $\rho:\Omega^1(U)\to \Omega^1(U\cap Z)$ is the pull-back (restriction) map.
For simplicity write $(X_0,\J_0)=(X_0^{m,n},\J_0^{m,n})$ and $Z_0^k=Z_0$. By the generalized Darboux Theorem \[generalized Darboux\], we may assume without loss of generality that $X$ is an open subset of $X_0$, and $z=0$. Decompose that tangent space of $X_0=\R^{2m}\times\C^n$ at the origin as $$T_0 (\R^{2m}\times\C^n)=T_0\R^{2m}\oplus T_0\C^n.$$ Define $E=T_0Z\cap T_0\R^{2m}$, and let $K\subset T_0\C^n$ be the projection of $T_0Z$ onto $T_0\C^n$. By assumption, $E$ is a Lagrangian subspace of $T_0\R^{2m}\cong \R^{2m}$.
$K$ is a complex subspace of $T_0\C^n\cong\C^n$.
Introduce the notation $S=T_0\R^{2m}$, $W= T_0\C^{2m}$, so that we have a decomposition $\TT_0X=S\oplus W\oplus S\uv\oplus W\uv$. For every $v\in K$, there exists $u\in S$, $a\in S\uv$ and $b\in W\uv$ such that $((u,v),(a,b))\in \TT_0(Z,F)$. We have $$\J(((u,v),(a,b)))=((-\omega^{-1}(a),-J(v)),(\omega(u),J\uv(b))).$$ Since by assumption this is again an element of $\TT_0(Z,F)$, we must have $$(-\omega^{-1}(a),-J(v))\in T_0Z.$$ Therefore $$-J(v)=\pi_{T_0\C^n}(-\omega^{-1}(a),-J(v))\in K,$$ so we see that $J(K)=K$.
Recall that the set of linear symplectomorphisms of $\R^{2m}$ acts transitively on the set of Lagrangian subspaces, and that the set of complex linear automorphisms of $\C^n$ acts transitively on the set of complex subspaces of fixed dimension. Therefore, by applying a linear change of coordinates on $X_0$, we may assume without loss of generality that $E$ is the subspace spanned by $\{\frac{\partial}{\partial s^i}\}_{i=1}^m$, and $K$ is the subspace spanned by $\{\frac{\partial}{\partial t^i}\}_{i=1}^{2k}$. In particular, if we let $\pi:Z\to Z_0$ denote the projection map, then we see that $$\pi_*TZ\to TZ_0$$ is an isomorphism at the origin. We may therefore choose a neighborhood $\tilde{Z}\subset Z$ of the origin, such that the restriction of $\pi$ to $\tilde{Z}$ give a diffeomorphism from $\tilde{Z}$ to $\tilde{Z}_0:=\pi(\tilde{Z})$.
Introduce the alternate coordinates $\{x^1,\cdots x^{d:=m+2k},y^1,\cdots y^{d':=m+2n-2k}\}$ given by $x_1=s_1,\cdots x_m=s_m,x_{m+1}=t_1,\cdots, x_{m+2k}=t_{2k}$ and $y_1=s_{m+1}\cdots y_m=s_{2m}, y_{m+1}=t_{2k+1}\cdots y_{m+2n-sk}=t_{2n}$. These coordinates are compatible with the decomposition $X_0\cong Z_0\times Y$, where $(x^1,\cdots,x^d)$ are coordinates on $Z_0$ and $(y^1,\cdots ,y^{d'})$ are coordinates on $Y\cong \R^{m+2n-2k}$. In particular, the restriction of $(x^1,\cdots, x^d)$ give coordinates on $\tilde{Z}$, and we may write $$\tilde{Z}=\{(x^1,\cdots, x^d,\psi^1(x^1,\cdots x^d),\cdots \psi^{d'}(x^1,\cdots, x^d):(x^1,\cdots x^d))\in \tilde{Z}_0\}$$ for unique smooth functions $\{\psi^I\in \Cinf(\tilde{Z}_0)\}$. Introduce the notation $\tilde{X}_0=\tilde{Z}_0\times Y\subset X_0$, and define $\Psi:\tilde{X}_0 \to \tilde{X}_0$ by $$(x^1,\cdots x^d,y^1,\cdots y^{d'}) \mapsto (x^1,\cdots x^d,y^1-\psi^1(x),\cdots, y^{d'}- \psi^{d'}(x)).$$ By construction, we have $$\Psi^{-1}(\tilde{Z}_0)=\tilde{Z}.$$ Choose an $\epsilon$-ball $\tilde{Y}\subset Y$ around the origin such that $$\Psi^{-1}(\tilde{Z}_0\times \tilde{Y})\cap Z=\tilde{Z}.$$ Define $$U_0= \tilde{Z}_0\times \tilde{Y},$$ $$U=\Psi^{-1}(\tilde{Z}_0\times \tilde{Y}),$$ and let the diffeomorphism $$\Phi:U\to U_0$$ be the restriction of $\Psi$ to $U$.
Let $\tilde{F}$ be the restriction of $F$ to $\tilde{Z}$. By shrinking $\tilde{Z}$ if necessary, we can find $w\in\Omega^1(\tilde{Z})$ such that $dw=F$. Let $\pi_{\tilde{Z}}:\tilde{X}_0\to \tilde{Z}$ be the projection, and define $u'=\pi_{\tilde{Z}}^*w$; if we denote by $\rho_{\tilde{Z}}:\Omega\ub(\tilde{X}_0)\to \Omega\ub(\tilde{Z}_0)$, then clearly $\rho(u')=w$. Finally, let $u\in\Omega^1(U)$ be the restriction of $u'$ to $U$.
Consider the quadruple $(U,U_0,\Phi,u)$. By construction, we have $\Phi(Z\cap U)=Z_0\cap U_0$, $\Phi(0)=0$, and $d\rho_{Z\cap U}(u)=F|_{Z\cap U}$. Therefore, the proof of Theorem \[LWL normal form\] will be complete if we can show that $$\label{pullback of gc} e^u\Phi^*(\J_0|_{U_0})=\J_0|_U.$$ Defining $g=e^{u'}\Psi^*\in\cG(\tilde{X}_0)$ and $\tilde{\J}_0=\J_0|_{\tilde{X}_0}$, we will show that $$g\cdot \tilde{J}_0=\tilde{J}_0,$$ which clearly implies the condition (\[pullback of gc\]).
Let $R\subset \TTX$ be the sub-bundle spanned by $\{(\frac{\partial}{\partial y^I},0),(0,dx^i)\}_{i,I}$ and $S\subset \TTX$ the sub-bundle spanned by $\{(\frac{\partial}{\partial x^i},0),(0,dy^I)\}_{i,I}$. The following facts are clear by inspection:
1. $\TTX=R\oplus S$. Furthermore, both $R$ and $S$ are maximal isotropic so the pairing gives an identification $S\cong R\uv$ and $R\cong S\uv$.
2. $\J_0(R)=R$ and $\J_0(S)=S$.
3. $S|_{Z_0}=\TT (Z_0,0)$.
Let $\Y$ be the set of vector fields on $X_0$ of the form $\sum_{I}c^I\frac{\partial}{\partial y^I}$ for real constants $\{c^I\}_I$. Define also $\RR\subset \Cinf(R)$ and $\SS\subset \Cinf(S)$ to be the subspaces of elements $\xx\in \Cinf(R),\Cinf(S)$ satisfying $(\xi,0)\cdot\xx=0$ for all $\xi\in \Y$.
\[basic facts\]
(1) $\ll\RR,\SS\rr\subset \RR$,
(2) $\ll\RR,\RR\rr=0$.
(3) $\J_0(\SS)=\SS$, $\J_0(\RR)=\RR$.
(4) For every $\xx\in\RR$, we have $(\xx\cdot\J_0)(\RR)=0 , (\xx\cdot\J_0)(\SS)\subset \RR$, and $\xx\cdot(\xx\cdot\J_0)=0$.
(5) Every $\xx\in \RR$ is complete, and satisfies $$\label{simple action}e^{\xx}\J_0=\J_0+\xx\cdot\J_0.$$
Every $\xx\in \RR$ is of the form $$\label{R form}\xx=(\xi^I(x)\frac{\partial}{\partial y^I},a_i(x)dx^i),$$ whereas every element $\yy\in\SS$ is of the form $$\label{S form}\yy=(\eta^i(x)\frac{\partial}{\partial x^I},b_I(x)dy^I).$$ We calculate $$\label{brack1}[\xi^I(x)\frac{\partial}{\partial y^I},\eta^i(x)\frac{\partial}{\partial x^i}]=-\eta^i\frac{\partial\xi^I}{\partial x^i}\frac{\partial}{\partial y^I},$$ $$\begin{aligned}
\label{brack2} \pounds(\xi^I(x)\frac{\partial}{\partial y^I})b_I(x)dy^I& = \iota(\xi^I(x)\frac{\partial}{\partial y^I})(\frac{\partial b_I}{\partial x^i}dx^i\wedge dy^I)+d(\xi^Ib_I) \notag \\ & = -\xi^I\frac{\partial b_I}{\partial x^I}dx^i+\frac{\partial \xi_I}{\partial x^i}b_Idx^i+\xi^I\frac{\partial b_I}{\partial x^i}dx^i \notag \\
& = \frac{\partial \xi_I}{\partial x^i}b_Idx^i,\end{aligned}$$ and $$\label{brack3}\iota(\eta^i\frac{\partial}{\partial x^i})d(a_idx^i) = \eta^j(\frac{\partial a^i}{\partial x^j}-\frac{\partial a^j}{\partial x^i})dx^i.$$ Using (\[brack1\]), (\[brack2\]), and (\[brack3\]), we see that for $\xx$ given by (\[R form\]) and $\yy$ by (\[S form\]) the Dorfman bracket $\ll \xx,\yy\rr$ is equal to $$(-\eta^i\frac{\partial\xi^I}{\partial x^i}\frac{\partial}{\partial y^I}, \frac{\partial \xi_I}{\partial x^i}b_Idx^i+\eta^j(\frac{\partial a^i}{\partial x^j}-\frac{\partial a^j}{\partial x^i})dx^i) \in \RR.$$ This verifies part (1) of the lemma. Part (2) is verified by a similar computation, which we omit.
Part (3) follows from an easy calculation using the explicit form of $\J_0=\J_{\omega}\times \J_J$ given by (\[Jsymplectic\]) and (\[Jcomplex\]). For example, for $i=1,\cdots, m$ we have $$\begin{aligned}
& \J_0((\frac{\partial}{\partial x^i},0))=(0,-dy^i),\\ & \J_0((\frac{\partial}{\partial y^i},0))=(0,dx^i),\\ & \J_0((0,dx^i))=(-\frac{\partial}{\partial y^i},0),\\ &\J_0((0,dy^i))=(\frac{\partial}{\partial x^i}).\end{aligned}$$ Part (4) follows from combining parts (1), (2), and (3).
To prove part (5), recall that $\xx=(\xi,a)\in\Cinf(\TT X)$ is complete if and only its vector field component $\xi\in\Cinf(TX)$ is complete, i.e. if and only if it generates a well-defined flow $\{\varphi_t:X\to X\}$ for all $t\in \R$. If $\xx\in\RR$, its vector field component is of the form $\xi^i(x)\frac{\partial}{\partial y^i}$, for which the flow is explicitly given by the formula $$\varphi_t(x^1,\cdots, x^d,y^1,\cdots, y^{d'})=(x^1,\cdots, x^d,y^1+\xi^1(x),\cdots y^{d'}+\xi^{d'}(x).$$ Given $\yy\in\Cinf(\TT X)$, for each $t\in\R$ define $\yy_t=e^{t\xx}\cdot\yy$. It follows from Proposition \[exp Dorfman\] that the map $t\mapsto \yy_t$ is characterized by the conditions $\yy_0=\yy$, and $$\label{deriv} \frac{d}{ds}|_{s=t}\yy_s=\ll\xx,\yy_t\rr.$$ We claim that, if $\yy$ is an element of either $\RR$ or $\SS$, then $$\label{integrate}\yy_t=\yy+t\ll\xx,\yy\rr,$$ and in particular $$\label{int1}e^{\xx}\yy=\yy+\ll\xx,\yy\rr.$$ To see this, note that $\yy_0=\yy$, and that the left-hand side of (\[deriv\]) is equal to $\ll\xx,\yy\rr$ for all $t\in\R$. If $\yy \in \RR$, then by part (4) of the Lemma $\ll\xx,\yy\rr=0$, so that both sides of (\[deriv\]) vanish. If $\yy\in \SS$, then $\ll\xx,\yy\rr\in \RR$ and therefore $\ll\xx,\ll\xx,\yy\rr\rr=0$, so that the right-hand side of (\[deriv\]) is equal to $$\ll\xx,\yy+\ll\xx,\yy\rr\rr=\ll\xx,\yy\rr,$$ and again we see that equation (\[deriv\]) holds.
By inspection, $e^{\xx}\J_0$ is clearly determined by its action $\RR\oplus\SS\subset \Cinf(\TTX)$, so that part (5) of the lemma follows from equation (\[int1\]).
Consider the vector field on $V\times Y$ given by $\xi(x,y)=\psi^I(x)\frac{\partial}{\partial y^I}$. It is easy to check that $e^{(\xi,0)}=\varphi^*$ in $\cG(V\times Y)$. Furthermore, since $(\xi,0)$ and $(0,u)$ are both elements of $\RR$, it follows from Lemma \[basic facts\] that $$(0,\pounds(\xi)u)=\ll(\xi,0),(0,u)\rr=0,$$ so that $e^{\xi}u=u$. Therefore, defining $\xx=(\xi,u)\in \RR$, we see that $$g:=e^u\varphi^*=e^{\xx}.$$ Therefore, to complete the proof of the Proposition, it will be sufficient to show that $$\xx\cdot \J_0=0.$$
Actually, we will prove the equivalent statement $-\xx\cdot \J_0=0$. Since $-\xx\in \RR$, it follows from Lemma \[basic facts\] that $-\xx\cdot \J$ vanishes if and only if $$\la(-\xx\cdot\J_0)\SS,\SS\ra = 0;$$ since $g^{-1}\cdot\J_0=\J_0-\xi\cdot\J_0$ and $\J_0(\SS)=\SS$, this is equivalent to $$\la(g^{-1}\cdot\J)\SS,\SS\ra =0,$$ which is in turn equivalent to $$\la \J (g\cdot\SS),g\cdot\SS\ra =0.$$ It follows from Proposition \[action on GC submanifolds\] that $$g\cdot \KK^{(Z_0,0)}=\KK^{(Z,F)}.$$ Since $\SS\subset \KK^{(Z_0,0)}$ and $(Z,F)$ is compatible with $\J$, it follows that $$\la \J (g\cdot\SS),g\cdot\SS\ra\subset I^Z.$$ On the other hand, since for each $x,y\in \SS$ the quantity $\la \J (g\cdot x),g\cdot y\ra$ is invariant under the flow generated by the vector field $\xi$, it follows that if $\la \J (g\cdot x),g\cdot y\ra$ vanishes on $Z$ then it must vanish on all of $\tilde{X}_0$, so we see that $$\la \J (g\cdot\SS),g\cdot\SS\ra=0.$$
Formal symmetries of generalized complex structures {#section formal}
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The remainder of the paper will be devoted to studying the deformation theory of generalized complex branes. As discussed in the introduction, our first task will be to define, for every GC brane $\B$, a functor $$\De_{\B}:\Art\to \Set$$ encoding the formal deformations of $\B$; we do this in Definition \[brane deformation functor\]. Before we can formulate Definition \[brane deformation functor\], however, it will be necessary to introduce a framework that includes certain “formal" versions of geometric structures studied in §\[Courant algebroid\]-§\[LWL submanifolds\]. We therefore begin by recalling some basic facts about local Artin algebras, as well as nilpotent Lie algebras and their exponentiation. We then use this theory to recast some of the definitions and results about GC geometry presented in the previous sections.
Artin algebras, nilpotent Lie algebras, and exponentiation
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Recall that a ring is called *Artinian* if it satisfies the descending chain condition on ideals.
Let $\Art$ denote the category of local, unital Artinian $\R$-algebras with residue field $\R$.
For a discussion of Artin algebras and their use in deformation theory, see [@KS][@Ha]. The basic facts stated below about $\Art$ may be found in these references.
Every $A\in\Art$ may be decomposed, as a real vector space, as $$A=\R\cdot 1_A\oplus \m,$$ where $\m\subset A$ is the unique maximal ideal. Furthermore, $\m$ is finite dimensional over $\R$, and coincides with the set of nilpotent elements of $A$. Any $\R$-algebra homomorphism $\varphi:A'\to A$ for $A,A'\in\Art$ necessarily satisfies $\varphi(\m_{A'})\subset \m_{A}$.
For each natural number $N$, the $\R$-algebra $$A=\R[\epsilon]/(\epsilon^{N+1})$$ is an object of $\Art$, with maximal ideal $(\epsilon^N)$.
Given $A\in\Art$, a *small extension* of $A$ is a pair $(A',\pi)$, where $A'$ is an object of $\Art$, and $\pi:A'\to A$ is a surjective map of $\R$-algebras, such that the kernel of $\pi$ is a principal ideal $I\subset A'$ satisfying $\m_{A'}I=0$.
Continuing with the previous example, for each natural number $N$ we have a small extension $$\R[\epsilon]/(\epsilon^{N+2})\to \R[\epsilon]/(\epsilon^{N+1}).$$
The following basic result will enable us to prove a number of results by inducting on small extensions [@Ha].
\[induct on small extensions\] Any surjective map in $\Art$ can be factored through a finite sequence of small extensions. In particular, for any $A\in\Art$, the quotient map $A\to \R$ can be factored through a finite sequence of small extensions.
Let $\g$ be a real Lie algebra. Then $\g$ determines a functor from $\Art$ to the category of real Lie algebras, given by $$\label{vect functor} A\mapsto \g_A:=\m\otimes_{\R}\g.$$ We will refer to such a functor as a *Lie algebra over $\Art$*. As a slight abuse of notation, we will sometimes use the same symbol “$\g$" to refer both to the Lie algebra itself and the Lie algebra over $\Art$ it determines; the meaning will always be clear from context.
\[over art\] More generally, for any category $\CC$, we may define an object of $\CC$ over $\Art$ to be functor $\Art\to \CC$, and a morphism (over $\Art$) between such objects to be a natural transformation. For example, a real vector space $V$ determines two different vector spaces over $\Art$, given respectively by $$A\mapsto A\otimes_{\R}V$$ or $$A\mapsto \m\otimes_{\R}V.$$ A representation of a group $G$ over $\Art$ on a vector space $V$ over $\Art$ consists of a representation of $G\da$ on $V\da$ for each $A\in\Art$, the collection of which must be suitably compatible with morphisms $A\to A'$ in $\Art$.
The groups over $\Art$ we consider in this paper all have the property that their value on the trivial Artin algebra $\R\in\Art$ is the trivial group. We will refer to such a group as a *formal group over $\Art$*, or more succinctly as a *formal group*.
To each real Lie algebra $\g$, recall that we can associate a formal group over $\Art$ as follows. For each $A\in\Art$, the Lie algebra $\g\da:=\m\otimes_{\R}\g$ is nilpotent, and may therefore be “exponentiated" it to form a group $e^{\g\da}$. By definition, there is a canonical bijective map from the underlying set of $\g\da$ to the underlying set of $e^{\g\da}$, which we write as $x\mapsto e^{x}$. For each pair of elements $x,y\in\g\da$, the product of $e^x$ with $e^y$ is defined using the Baker-Campbell-Hausdorff formula $$e^x e^y=e^{x+y+\frac{1}{2}[x,y]+\cdots},$$ where the formally infinite sum $x+y+\frac{1}{2}[x,y]\cdots$ terminates after a finite number of terms due to the nilpotence. This construction $\g\mapsto e^{\g}$ is functorial: every homomorphism of Lie algebras $\varphi:\g\to\g'$ induces a homomorphism of formal groups $e^{\varphi}:e^{\g}\to e^{\g'}$, given for each Artin algebra $A\in\Art$ and each $x\in\g\da$ by the formula $e^{x}\mapsto e^{\varphi\da(x)}$.
We record here for later use some well-known facts about this construction.
\[exp prop\] Let $\g$ be a real Lie algebra with corresponding formal group $e^{\g}$.
1. Every action of $\g$ on a real vector space $V$ induces an action of $e^{\g}$ on $V$ (viewed as a vector space $A\mapsto A\otimes_{\R} V$ over $\Art$ ). This action is given by the exponential formula: for each Artin algebra $A\in\Art$, each $x\in\g\da$, and each $v\in A\otimes_{\R}V$, we set$$\label{exp action}e^{x}v=v+x\cdot v+\frac{1}{2}x\cdot(x\cdot v)\cdots.$$
2. Consider the action of $e^{\g}$ on $\g$ obtained by exponentiating the adjoint representation of $\g$ on itself, i.e. for every Artin algebra $A$ and every $x,y\in A\otimes_{\R}\g$ have $$e^{x}y=y+[x,y]+\frac{1}{2}[x,[x,y]]+\cdots.$$ Then for every $x,y\in\g\da$, we have the identity $$e^x e^y e^{-x}=e^{e^{x}y}.$$
3. Fix $A\in\Art$. Every group homomorphism $\varphi:\R\to e^{\g\da}$ is of the form $$\label{1parsubformal}t\mapsto e^{tx}$$ for a unique $x\in\g\da$. Conversely, for each $x\in\g\da$, the formula (\[1parsubformal\]) determines a homomorphism $\R\to e^{\g\da}$.
The formal symmetries of a Courant algebroid
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Fix a smooth manifold $X$. As in §\[Courant algebroid\], we denote by $\g(X)$ the real Lie algebra of vector fields on $X$, and by $\cg(X)=\g(X)\ltimes \Omega^1(X)$ the semi-direct product Lie algebra introduced in Definition \[definition cg\]. Following the convention introduced above, we may also regard $\g(X)$ and $\cg(X)$ as a Lie algebras over $\Art$, which assign to each $A\in\Art$ with maximal ideal $\m\subset A$ the nilpotent Lie algebras $\g\da(X):=\m\otimes\g(X)$ and $\cg\da(X)=\m\otimes_{\R}\cg(X)$, respectively. More generally, we have sheaves of Lie algebras over $\Art$, which assign to each open set $U\subset X$ and each $A\in\Art$ the Lie algebras $\g\da(U)=\m\otimes_{\R}\g(U)$ and $\cg(U)=\m\otimes_{\R}\g(U)$.
We also introduce the notation for differential forms $$\Omega\ub\da(X):=A\otimes_{\R}\Omega\ub(X)$$ and $$\Omega\ub\dm(X):=\m\otimes_{\R}\Omega\ub(X),$$ and similarly $$\Cinf\da(TX):=A\otimes_{\R} \Cinf(TX),$$ $$\Cinf\dm(TX):=\m\otimes_{\R} \Cinf (TX),$$ $$\Cinf\da(\TTX):=A\otimes_{\R}\Cinf(\TTX),$$ and $$\Cinf\dm(\TTX):=\m\otimes_{\R}\Cinf(\TTX).$$ A potentially confusing point about this notation: while $\Cinf(TX)$ and $\g(X)$ both denote the space of vector fields on $X$, given $A\in\Art$ the spaces $\Cinf\da(TX)$ and $\g\da(X)$ are not the same. This corresponds to the different roles played by the space of vector fields in our development.
Consider the formal group $e^{\g(X)}$, whose elements may be viewed as infinitesimal diffeomorphisms of $X$. The action of $\g(X)$ on the spaces $\Omega\ub(X)$ and $\Cinf(TX)$ via the Lie derivative may be exponentiated (as described in Proposition \[exp prop\]) to an action of $e^{\g(X)}$. Unsurprisingly, this action is compatible with the relevant structures on $\Omega\ub(X)$, and $\Cinf(TX)$, such as the wedge product of differential forms, and the contraction operation between vector fields and forms. This is summarized in the following Proposition, whose proof is deferred to the appendix.
\[formal courant action\] Fix an Artin algebra $A\in\Art$. For each $g\in e^{\g\da(X)}$, each $a,b\in A\otimes_{\R} \Omb(X)$, and each $\xi,\eta \in A\otimes_{\R} \Cinf(TX)$ we have $$g(da)=dg(a),$$ $$g(a\wedge b)=g(a)\wedge g(b),$$ $$g(\iota(\xi)a)=\iota(g(\xi))g(a),$$ and $$g([\xi,\eta])=[g(\xi),g(\eta)].$$
Turning next to the formal group $e^{\cg(X)}$, we may similarly exponentiate the action of $\cg(X)$ on $\Cinf(\TTX)$ (described in Proposition \[Courant action prop\]) to give an action of $e^{\cg(X)}$. Using Proposition \[formal courant action\]), we may then easily prove the following.
\[action\] For each $A\in\Art$, the action of $e^{\cg\da(X)}$ on $\Cinf\da(\TTX)$ is compatible with the (A-linear extensions of the) Dorfman bracket (\[Dorfman Bracket\]), the natural pairing $\la\cdot,\cdot\ra$, and the projection $\pi:\Cinf\da(\TTX)\to\Cinf\da(TX)$.
It will be convenient to have a formal version of Proposition (\[exp Dorfman\]), which gives an explicit formula for the exponential map $\cg\da(X)\mapsto e^{\cg\da(X)}$ in terms of the subgroups $e^{\g\da(X)}$ and $e^{\m\otimes_{\R}\Omega^1(X)}$; introducing the notation $\h(X)$ for $\Omega^1(X)$ (regarded as an abelian Lie algebra), the latter group may be denoted by $e^{\h\da(X)}$. Adapting the same notation used in the statement of Proposition (\[exp Dorfman\]), for each $\xi\in\g\da(X)$ and each $a\in\h\da(X)$, we define $$\label{formal a def}a^{\xi}=\int_0^1e^{t\xi}(a)dt,$$ where the precise meaning of the right-hand side is as follows: for each $\xi\in \g\da(X)$ and $a\in \h\da(X)$, the expression $$e^{t\xi}(a):=a+t\pounds(\xi)a+\frac{t^2}{2}\pounds(\xi)\pounds(\xi)a+\cdots$$ defines a polynomial in $t$, i.e. an element of $\h\da(X)[t]$. The integral with respect to $t$ in (\[formal a def\]) is then defined algebraically: for each natural number $n$ and each $b\in\h\da$ we set $$\int_0^1 bt^ndt:=\frac{b}{n+1}.$$
\[exponential\] For all $(\xi,a)\in \g\da(X)\ltimes\h\da(X)$, we have $$e^{(\xi,a)}=e^{(0,a^{\xi})}e^{(\xi,0)}.$$
Given $(\xi,a)\in\cg\da(X)$, consider the map $$\varphi:\R\to e^{\cg\da(X)}$$ given by $$t\mapsto e^{\int_0^te^{s\xi}ads}e^{t\xi}.$$ By Proposition \[exp prop\], for each $\xi\in\g\da(X)$ and each $a\in \h\da(X)$ we have $$e^{\xi}e^{a}e^{-\xi}=e^{e^{\xi}(a)};$$ the same calculation used in the proof of Proposition \[exp Dorfman\] may then be used to show that $\varphi:\R\to e^{\cg\da(X)}$ is a group homomorphism. Therefore, by part (2) of Proposition \[exp prop\], it follows that $\varphi$ is of the form $$\label{poly1}t\mapsto e^{t\xx}$$ for a unique $\xx\in\cg\da(X)$. Note that $$a^{t\xi}=\int_0^te^{s\xi}ads=ta+t\sum_{k=1}^{\infty}\frac{t^k}{(k+1)!}\xi^k\cdot a:=ta+th(t),$$ where $h(t)$ is an element of $\cg\da(X)[t]$ (depending on $\xi$ and $a$), so that by the Baker-Campbell-Hausdorff formula we have $$\label{poly2} e^{a^{t\xi}}e^{t\xi}=e^{t(\xi+a)+t^2\tilde{h}(t)},$$ where $\tilde{h}(t)$ is an element of $\cg\da(X)[t]$. Comparing equations (\[poly1\]) and (\[poly2\]), we see that $\tilde{h}(t)=0$ and $\xx=(\xi,a)$.
By construction, we have inclusions of $e^{\g\da(X)}$ and $e^{\h\da(X)}$ as sub-groups of $e^{\g\da(X)\ltimes\h\da(X)}$. Proposition \[exponential\] implies that these subgroups (which have trivial intersection) generate $e^{\g\da\ltimes\h\da(X)}$, so that $e^{\g\da(X)\ltimes\h\da(X)}$ may be identified with the semi-direct product $e^{\g\da(X)}\ltimes e^{\h\da(X)}$ (this is of course a special case of a more general result about the exponentiation functor). In particular, we have the following corollary, recorded here for later use.
\[exp formula cor\] For each $\xi\in\g\da(X)$, the linear map $\h\da(X)\to\h\da(X)$ mapping $a\mapsto a^{\xi}$ is a bijection.
Next, let $\J$ be a GC structure on $X$. Recall from §\[GC structures\] the Lie algebra $\T(X)$ of generalized holomorphic vector fields (Definition \[generalized holomorphic vector field\]), which are the infinitesimal symmetries of $\J$, as well as the sub-algebra $\H(X)\subset \T(X)$ of generalized Hamiltonian vector fields (Definition \[genhamdef\]). For each $A\in\Art$ we may then define the nilpotent Lie algebra $\T\da(X)$ as well as the subalgebra $\H\da(X)\subset \T\da(X)$. By inspection of the exponential formula (\[exp action\]), we immediately see the following
\[formal holomorphic symmetries\] For each $\xx\in\T\da(X)$ we have $$e^{\xx}\cdot\J=J.$$
The deformation functor of a generalized complex brane {#section deformations}
======================================================
Using the framework introduced in the last section, we now to turn to the problem of defining for each $GC$ brane $\B$ a functor $$\df_{\B}:\Art\to \Set$$ encoding the formal deformations of $\B$. As mentioned in the introduction, this functor (given in Definition \[brane deformation functor\]), will be constructed in terms of a formal groupoid $\De\uB(X,\J)$ over $\Art$ (Definition \[full deformation groupoid\]), which encodes the appropriate notions of equivalence between deformations of $\B$.
Deformations of a Hermitian line bundle with unitary connection
---------------------------------------------------------------
Given a brane $\B=(Z,\L)$, we may simultaneously deform both $Z$ and $\L$. As a preliminary to studying the general situation, we first consider the deformations of $\L$ only, i.e. the deformations of a Hermitian line bundle with unitary connection over a fixed space.
Let $X$ be a smooth manifold, and $\mathcal{W}=\{W_I\}$ an open cover of $X$. We denote by $W_{I_1 I_2\cdots I_k}$ the $k$-fold intersection $W_{I_1}\cap \cdots W_{I_k}$.
The set $\Herm(X)$ consists of triples $\L=(\{W\si\},\{c\sij\},\{a\si\})$, where $\{W\si\}$ is an open cover of $X$, $\{c_{IJ}\in \Cinf(W_{IJ})\}$ is a collection of real-valued functions satisfying $$c_{JK}-c_{IK}+c_{IJ}\in \Z,$$ and $\{a_{I}\in \Omega^1(W_I)\}$ is a collection of (real) 1-forms satisfying $$a_J-a_I=dc_{IJ}$$ on the intersections $W_{IJ}$.
\[gluing bundles\] Given an element $\L=(\W,\{c\sij\},\{a\si\})\in \Herm(X)$ as above, we may construct a Hermitian line bundle with unitary connection on $X$ by gluing. The line bundle is constructed by gluing trivial (Hermitian) line bundles on each $W_I$ using transition functions $g\sij=e^{2\pi i c_{IJ}}$. The unitary connection is then locally specified on each $W\si$ by $$\nabla=d+2\pi i.$$ Furthermore, for each manifold $X$, there exists an open cover $\W$ such that every Hermitian line bundle with unitary connection is isomorphic to one of this form. More formally, the gluing construction may be extended to an equivalence of categories (defined on objects as above).
For the most part, we will work with a fixed open cover $\{W_I\}$ and omit it from the notation, i.e. we will simply write $\L=(\{c_{IJ}\},\{a\si\})$ to specify an element of $\Herm(X)$.
An element $\L=(\{c\sij\},\{a\si\})\in \Herm(X,\mathcal{W})$ determines a closed 2-form $F\in \Omega^2(X)$, which is given by $F|_{W_I}=da_I$ on each open set $W\si$.
Given an open set $U\subset X$, there is a natural restriction map $\Herm(X)\to \Herm(U)$, which we denote by $\L\mapsto \L|_U$ for $\L\in\Herm(X)$.
\[groupoid over art\] Recall from Remark \[over art\], that for any category $\CC$ we defined an *object of $\CC$ over $\Art$* to be a functor $\Art\to \CC$. A similar definition can be made when $\CC$ is a 2-category, for example the 2-category $\textrm{Gpd}$ of (small) groupoids. For example a (strict) *groupoid over $\Art$* may be defined as a (strict) functor $\Art\to \textrm{Gpd}$; explicitly, this consists of a groupoid $\G\da$ for every $A\in\Art$, and for every homomorphism $A\to A'\in \Art$ a functor $\G\da\to\G_{A'}$. For our present purposes, we require the composition of these functors to be strictly compatible with composition in $\Art$. The groupoids over $\Art$ we consider will all have the property that they map the trivial Artin algebra $\R\in\Art$ to a trivial groupoid, i.e. a groupoid with a single object and a single morphism. We will refer these as *formal groupoids* (over $\Art$). The notion of a (strict) functor between formal groupoids is defined in the obvious way. We will sometimes drop the modifier “formal" when it is clear from context.
To streamline the exposition, we will usually define specific formal groupoids (and functors between them) by simply describing their value on each $A\in\Art$. The additional structure needed to make things completely precise will always be clear from the context.
\[Herm def\] Given $\L=(\{c\sij\},\{a\si\})\in\Herm(X)$, we define a formal groupoid $\De^{\L}(X)$ (over $\Art$) as follows. For each $A\in\Art$, an object of $\De^{\L}_A(X)$, which we call an *A-deformation* of $\L$, is a pair $\hL=(\{\hc\sij\},\{\ha\si\})$, where
(i) each $\ha\si\in\Omega^1_A(W\si)$, and is of the form $$\ha\si=a\si+u\si$$ for $u\si\in\Omega^1\dm(U\si)$,
(ii) each $\hc\sij\in \Cinf\da(W\sij)$ is of the form $$\hc\sij=c\sij+f\sij$$ for $f\sij\in \Cinf\dm(W\sij)$,
(iii) on each overlap $W\sij$ we have $$\ha\sj-\ha\si=d\hc\sij,$$ or equivalently $u\sj-u\si=df\sij$,
(iv) on each 3-fold intersection $W_{IJK}$ we have $$\hc_{JK}-\hc_{IK}+\hc_{IJ}\in \Z,$$ or equivalently $f_{JK}-f_{IK}+f_{IJ}=0$.
Given $A$-deformations $\hL=(\{\hc\sij\},\{\ha\si\})$ and $\hL'=(\{\hc'\sij\},\{\ha'\si\})$ of $\L$, an *isomorphism* $\hL\to\hL'$ is a collection $g=\{g\si\in \Cinf\dm(W\si)\}$ such that
1. $\ha'\si=\ha\si+dg\si$ ($\Leftrightarrow u'\si=u\si+dg\si$) and
2. $\hc'\sij=\hc\sij+g\sj-g\si$ ($\Leftrightarrow f'\sij=f\sij+g\sj-g\si$).
The composition of isomorphisms is given by addition. For any $A$-deformation of $\L$, the identity isomorphism is given by $\{g_I=0\}$.
\[bundle def sheaf\] Given an open set $U\subset X$, let us introduce the notation $$\De^{\L}(U):=\De^{\L|_U}(U).$$ For every inclusion of open subset $V\subset U$, we then have a natural restriction functor (between formal groupoids) $$\De^{\L}(U)\to\De^{\L}(V).$$ Altogether, we obtain a (strict) sheaf of formal groupoids on $X$ sending $U$ to the formal groupoid $\De^{\L}(U)$.
\[pi zero bundle\]Every $A$-deformation of $\L$ is isomorphic to one of the form $\hL=(\{c\sij\},\{\ha\si\})$, i.e. to one with undeformed transition functions $\hat{c}\sij=c\sij$.
Given $\hL=(\{c_{IJ}+f_{IJ}\},\{a_I+u_I\})$, by part (II) of Definition \[Herm def\] we see that $$f_{JK}-f_{IK}+f_{IJ}=0.$$ The functions $f_{IJ}\in \Cinf(W_{IJ})\otimes \m\da$ therefore define a Cech cocycle. Since the sheaf $\underline{C}^{\infty}$ of smooth, real-valued functions on $X$ admit partitions of unity, so does the sheaf $\underline{C}^{\infty}\otimes \m\da$. We may therefore choose $\{g_I\in\Cinf\dm(W_I)\}$ satisfying $$g_J-g_I=f_{IJ}$$ on the overlaps $W_{IJ}$. This implies that $\hL$ is isomorphic to $\hL'=(\{c_{IJ}\},\{a_I+dg_I\})$.
\[group action on category\] Let $\CC$ be a groupoid, and $G$ a group. A *strict left action of $G$ on $\CC$* is a collection of functors $\{F_g\}_{g\in G}$ satisfying the following:
1. For every $g,g'\in G$, we have $F_g\circ F_{g'}=F_{gg'}$.
2. The functor $F_1$ induced by the identity element $1\in G$ is the identity functor.
A strict *right* action is defined similarly, except that for every $g,g'\in G$ we require that $F_{g}\circ F_{g'}=F_{g'g}$.
Given a formal group $G$ and a formal groupoid $\CC$, we may define an action of $G$ on $\CC$ to consist of an action of $G\da$ on $\CC\da$ for each $A\in\Art$, subject to some obvious compatibility relations.
Given a strict left action of $G$ on $\CC$, we will use the notation $$F_g(x)=g\cdot x$$ for each $g\in G$ and each object $x\in \CC$. Similarly, given a morphism $\varphi:x\to y$ in $\CC$ we denote $F_g(\varphi)$ by $g\cdot\varphi$. Similarly, if $G$ acts on $\CC$ on the right, we will use the notation $x\cdot g$ and $\varphi\cdot g$.
We now describe an action of $e^{\cg(X)}$ (the formal symmetries of the Courant algebroid) on $\De^{\L}(X)$.
\[bundle action\] Given $A\in\Art$, $\hL=(\{\hc\sij\},\{\ha\si\})\in\De^{\L}_A(X)$, and $x=e^ue^{\tau}\in e^{\cg\da(Z)}$, let $$x\cdot\hL=(\{e^{\tau|_{W\sij}}\hc\sij\},\{e^{\tau|_{W\si}}\ha_I-u|_{W\si}\})\in \De^{\L}_A(X).$$ Given an isomorphism $\{g\si\}:\hL\to\hL'$ in $\De^{\L}\da(X)$, define $$x\cdot \{g\si\}=\{e^{\tau}g\si\}:x\cdot\hL\to x\cdot\hL'$$
We then have the following easy result, the proof of which is omitted.
\[bundle action\]Definition \[bundle action\] determines a strict left action of $e^{\cg(X)}$ on $\De^{\L}(X)$.
Deformations of branes
----------------------
We now turn to the deformations of GC branes. For each GC brane $\B$ on a GC manifold $(X,\J)$, we will define a formal groupoid $\De^{\B}(X,\J)$ encoding the infinitesimal deformations of $\B$ and their equivalences (Definition \[full deformation groupoid\]). By passing to equivalence classes of deformations, we construct from $\B$ a functor (Definition \[brane deformation functor\]) $$\df_{\B}:=\pi_0(\De^{\B}(X,\J)):\Art\to \Set.$$ We will see that in many situations it is necessary to work with the formal groupoid $\De\ub(X,\J)$ itself, and not the functor $\df_{\B}$.
The construction of $\De\uB(X,\J)$ will be given in several steps. First, we ignore the GC structure $\J$ and define a formal groupoid $\Dt\uB(X)$ that does not encode any compatibility condition with respect to $\J$. We then define a sub-groupoid $\Dt\uB(X,\J)$ whose objects are those deformations compatible with $\J$. The formal group $e^{\T(X)}$ of symmetries of $(X,\J)$ acts on $\Dt\uB(X,\J)$; in particular, there is an action of $e^{\H(X)}$, the formal generalized Hamiltonian symmetries. Incorporating this action leads to the formal groupoid $\De\ub(X,\J)$.
\[brane definition 1\] Let $X$ be a smooth manifold. A (rank 1) *brane* on $X$ is a pair $\B=(Z,\L)$, where $Z\subset X$ is a smooth submanifold and $\L$ is an element of $\Herm(Z)$. We denote the collection of all such branes on $X$ by $\Br(X)$.
Given such a brane $(Z,\L)$, applying the gluing construction described in Remark \[gluing bundles\] to $\L$ determines a GC brane in the sense of Remark \[geometric brane\].
\[definition par def\] Given a manifold $X$, and a brane $\B\in \Br(X)$, let $\Dt^{\B}(X)$ be the following formal groupoid.
1. For each $A\in\Art$, an object of $\Dt_A^{\B}(X)$ is a pair $\hB=(\hrho,\hL)$, where
1. $\hrho:\Omega\ub\da(X)\to \Omega\ub\da(Z)$ is of the form $\hrho=\rho e^{\xi}$ for some $\xi\in \g\da(X)$, where $\rho:\Omega\ub(X)\to\Omega\ub(Z)$ denotes the pull-back map for the inclusion $i:Z\hookrightarrow X$.
2. $\hL$ is an object of $\De^{\L}_A(Z)$, as in Definition \[Herm def\].
We will refer to such an object as an *$A$-deformation of $\B$*.
2. Given two $A$-deformations $\hB=(\hrho,\hL)$ and $\hB=(\hrho',\hL')$, an *equivalence* $\hB{\xymatrix@C=1.5pc{ \ar[r]^-{\cong} & }}\hB'$ is a pair $\Psi=(e^{\tau},\{g\si\})$, where
1. $e^{\tau}\in e^{\g\da(Z)}$ satisfies $$\hrho'=e^{\tau}\hrho,$$ and
2. $\{g\si\}$ is a morphism $$\psi:\hL'\to e^{\tau}\cdot\hL$$ in the groupoid $\Def^{\L}_A(Z)$.
3. Given equivalences $\Psi=(e^{\tau},\{g_I\}):\hB{\xymatrix@C=1.5pc{ \ar[r]^-{\cong} & }}\hB'$ and $\Psi'=(e^{\tau'},\{g'\si\}):\hB{\xymatrix@C=1.5pc{ \ar[r]^-{\cong} & }}\hB''$, their composition is defined by $$\Psi'\circ\Psi=(e^{\tau'}e^{\tau},\{e^{\tau'}g\si+g'\si\}):\hB{\xymatrix@C=1.5pc{ \ar[r]^-{\cong} & }}\hB''.$$ For any $A$-deformation $\hB$ of $\B$, the *identity* isomorphism $\hB\to \hB$ is given by $(1,id_{\hL})$.
Let us examine the composition in $\Dt^{\B}_A(X)$ in a little more detail (and in particular verify that it is well-defined). Suppose $\Psi=(e^{\tau},\{g\si\})$ is an equivalence from $\B=(\hrho,\hat{\L})$ to $\B'=(\hrho',\hL')$ and $\Psi'=(e^{\tau'}, \{g'\si\})$ is an equivalence from $\hB'$ to $\hB''=(\hrho'',\hL'')$. By definition, we have $\hrho'=e^{\tau}\hrho$, $\hrho''=e^{\tau'}\hrho'$, $\{g\si\}$ is an equivalence $\hL'\to e^{\tau}\hL$ and $\{g'\si\}$ is an equivalence $\hL''\to e^{\tau'}\hL'$. Defining $e^{\tau''}=e^{\tau'}e^{\tau}$, it follows that $\hrho''=e^{\tau''}\hrho$. Furthermore, $e^{\tau'}\{g_I\}$ is an equivalence from $e^{\tau'}\hL\to e^{\tau'}e^{\tau}\L=e^{\tau''}\L$, so that the composition $(e^{\tau'}\{g\si\})\circ \{g'\si\})$ is an equivalence from $\hL''\to e^{\tau''}\cdot\hL$; therefore $$\label{comp identity} (e^{\tau''},(e^{\tau'}\{g\si\})\circ\{g'\si\})=(e^{\tau'}e^{\tau},\{e^{\tau'}g_I+g'\si\})$$ does in fact define a morphism in $\Dt^{\B}_A(X)$ from $\hB$ to $\hB''$. Writing the composition as on the left-hand side of (\[comp identity\]) makes it easy to check its associativity using Proposition \[bundle action\].
\[right action par\] There is a strict right action of the formal group $e^{\cg(X)}$ on $\widetilde{\De}^{\B}(X)$, given as follows:
1. For each $A\in\Art$, each $g=e^{(0,w)}e^{(\xi,0)}\in e^{\cg\da(X)}$, and each $\hB=(\hrho,\hL)\in \widetilde{\De}^{\B}\da(X)$, define $$\hB\cdot g=(\hrho e^{\xi}, e^{-\hrho(w)}\cdot\hL),$$ where we recall that by definition $$e^{-\hrho(w)}\cdot\L=(\{\hat{c}\sij\},\{\hat{a}\si+\hrho(w)|_{W_I}\}).$$
2. For each equivalence $\Psi=(e^{\tau},\{g_I\}):\hB_1{\xymatrix@C=1.5pc{ \ar[r]^-{\cong} & }}\hB_2$, define $$\Psi\cdot g:\hB_1\cdot g\to \hB_2\cdot g$$ to be given by the same pair $(e^{\tau},\{g\si\})$, regarded as an equivalence $\hB{\xymatrix@C=1.5pc{ \ar[r]^-{\cong} & }}\hB'$.
First, let us check that if $\Psi=(e^{\tau},\{g\si\})$ is an equivalence $\hB_1{\xymatrix@C=1.5pc{ \ar[r]^-{\cong} & }}\hB_2$, then the same pair $(e^{\tau},\{g\si\})$ does in fact define an equivalence $\hB_1\cdot g{\xymatrix@C=1.5pc{ \ar[r]^-{\cong} & }}\hB_2\cdot g$. Let us write $\hB_1=(\hrho_1,\hL_1), \hB_2=(\hrho_2,\hL_2)$, $\hB'_1:=\hB_1\cdot g=(\hrho'_1,\hL'_1)$, $\hB'_2:=\hB_2\cdot g=(\hrho'_2,\hL_2')$, and similarly write $\Psi=(e^{\tau},\psi)$ with $\psi=\{g_I\}:\hL_2\to\hL_2$, and $\Psi'=\Psi\cdot g = (e^{\tau'},\psi')$ with $\tau'=\tau$ and $\psi'=\{g\si\}$. Note that since $\hrho_2=e^{\tau}\hrho_1$, we have $\hrho_2'=\hrho_2e^{\xi}=e^{\tau}\hrho_1e^{\xi}=e^{\tau}\hrho'_1=e^{\tau'}\hrho'_1$. Furthermore, since $\psi$ is an equivalence from $\hL_2\to e^{\tau}\hL_1$ in $\Def^{\L}_A(Z)$, it follows that $e^{-\hrho_2(u)}\cdot\psi$ is an equivalence from $\L_2'=e^{-\hrho_2(u)}\cdot\hL_2$ to $e^{-\hrho_2(u)}e^{\tau}\hL_1$. But we have $$e^{-\hrho_2(u)}e^{\tau}\hL_1=e^{-e^{\tau}\hrho_1(u)}e^{\tau}\hL_1=e^{\tau}e^{-\hrho_1(u)}\hL_1=e^{\tau'}\hL'_1,$$ so that $e^{-\hrho_2(u)}\psi$ is a morphism from $\hL'_2\to e^{\tau'}\hL'_1$ in $\Def^{\L}\da(Z)$. Thus, we see that $\Psi\cdot g= \Psi'$ does indeed define an equivalence from $\hB_1\cdot g$ to $\hB_2\cdot g$, as claimed.
Since $g$ acts trivially on morphisms, the functoriality property $(\Psi'\cdot g)\circ(\Psi\cdot g)=(\Psi'\circ \Psi)\cdot g$ holds trivially. Furthermore, by construction, the identity element of $e^{\cg(X)}$ acts as the identity functor. Therefore, to finish the proof of the proposition we just need to check that $(\hB\cdot g)\cdot g'=\hB\cdot (gg')$ hold for each object $\hB\in\Dt^{\B}_A(X)$ and each $g,g'\in e^{\g\da(X)}$.
Let $\hB=(\hrho,\hL)\in\Dt^{\B}\da(X)$ and $g=e^ue^{\xi}$, $g'=e^{u'}e^{\xi'}\in e^{\cg\da(X)}$. We have $$\label{ggprime}(\hB\cdot g)\cdot g'=(\hrho e^{\xi},e^{-\hrho(u)}\hL)g'=(\hrho e^{\xi}e^{\xi'},e^{-\hrho e^{\xi}u'-\hrho u}\hL).$$ Since $gg'=e^{u+e^{\xi}u'}e^{\xi}e^{\xi'}$, we see that the right-hand side of (\[ggprime\]) is equal to $\hB\cdot (gg')$.
Compatibility with a generalized complex structure
--------------------------------------------------
Every $\B=(Z,\L)\in\Br(X)$ determines a generalized submanifold $(Z,F)$ on $X$, where $F$ is the curvature form of $\L$. In particular, we have the *generalized tangent bundle* $\TT\B$ and the *generalized normal bundle* $\N\B$, which by definition are the generalized tangent bundle and generalized normal bundle of $(Z,F)$ (as defined in §\[GC submanifolds\]). Similarly, we denote by $K\uB$ the space $K^{(Z,F)}$ introduced in Definition \[definition K\], and by $H\ub(\B)$ the Lie algebroid cohomology groups associated to $(Z,F)$ (described in §\[Lie alg cohomology\]). Recall also the notation $$I^Z:=\{f\in\Cinf(X):f|_Z=0\}.$$
\[GC brane definition 1\] Let $(X,\J)$ be a GC manifold. A (rank 1) *GC brane* on $(X,\J)$ is an element $\B\in\Br(X)$ as in Definition \[brane definition 1\], such that the underlying generalized submanifold $(Z,F)$ of $\B$ is compatible with $\J$. We denote the collection of all such GC branes by $\Br(X,\J)$.
\[compatibility1\] Given $\B=(Z,\L)\in\Br(X,\J)$, and an Artin algebra $A\in\Art$, an element $g\in e^{\cg(X)}$ is said to be*compatible* with $\J$ with respect to $\B$ if $$Q_{g\cdot\J}(K_A^{\B},K_A^{\B})\subset I^Z\da,$$ where $Q_{g\cdot\J}$ is the $A$-linear extension of pairing defined by (\[Q def\]), $K\uB_A:=A\otimes_{\R}K\uB$ and $I^Z\da:=A\otimes_{\R}I^Z$.
\[well-defined\] Let $\B\in\Br(X,\J)$, and let $g=e^ue^{\xi}$ and $g'=e^{u'}e^{\xi'}$ be elements of $e^{\cg\da(X)}$ such that
1. $\rho e^{\xi}=\rho e^{\xi'}$ and
2. $\rho u=\rho u'$.
Then $g$ is compatible with $\J$ with respect to $\B$ if and only if $g'$ is.
Suppose we are given $\xi\in\Cinf\dm(TX)$ such that $\rho e^{\xi}=\rho:\Omega\ub\da(X)\to\Omega\ub\da(Z)$; this is equivalent to requiring that $0=\rho\pounds(\xi):\Omega\da\ub(X)\to\Omega\da\ub(Z)$. Suppose also that we are given $\eta\in\Cinf\da(TX)$ that is tangent to $Z$, with $\rho(\eta)=\tau\in\Cinf\da(Z)$; this means that $\rho(\pounds(\xi)v)=\pounds(\tau)\rho v$ for every $v\in \Omega\ub\da(X)$. We claim that in this case $$e^{\xi}\eta\in\Cinf\da(TX)$$ is still tangent to $Z$ with $\rho(e^{\xi}\eta)=\tau$. To see this, note that for every $v\in\Omega\ub\da(X)$ we have $$\begin{aligned}
\rho(\pounds(e^{\xi}\eta)v) & = \rho(e^{\xi}(\pounds(\eta)e^{-\xi}v)) \\
& = \rho(\pounds(\eta)e^{-\xi}v) \\
& = \pounds(\tau)\rho e^{-\xi}v \\
& = \pounds(\tau)\rho v.\end{aligned}$$
Next, suppose we are given $h=e^{(0,u)}e^{(\xi,0)}\in e^{\cg\da(X)}$ such that $\rho e^{\xi}=\rho$ and $\rho u=0$. We claim that in this case $$h\cdot K\da^{\B}=K\da^{\B}.$$ To see this, let $\xx=(\eta,a)$ be an arbitrary element of $K\da^{\B}$. By definition, this means that $\eta$ is tangent to $Z$, say with $\rho(\eta)=\tau\in\Cinf\da(TZ)$, and $\rho(a)=\iota(\tau)F$. Write $h\cdot \xx=(\tilde{\eta},\tilde{a})$, with $$\tilde{\eta}=e^{\xi}\eta$$ and $$\tilde{a}=e^{\xi}a-\iota(\tilde{\eta})du.$$ We have already seen that $\tilde{\eta}$ is tangent to $Z$ with $\rho(\tilde{\eta})=\tau$. Furthermore, we see that $$\begin{aligned}
\rho\tilde{a} & = \rho(e^{\xi}a)-\rho(\iota(\tilde{\eta})du) \\
& = \rho a-\iota(\tau)d\rho(u) \\
& = \rho a = \iota(\tau)F,\end{aligned}$$ so that $h\cdot \xx\in\KK\da^{\B}$, as claimed.
To complete the proof, let $g=e^{u}e^{\xi}$ and $g'=e^{u'}e^{\xi'}$ with $\rho e^{\xi}=\rho e^{\xi'}$ and $\rho u=\rho u'$. Defining $h=g'g^{-1}$, we see that $$h=e^{u'-e^{\xi'}e^{-\xi}u}e^{\xi'}e^{-\xi}.$$ Defining $\tilde{\xi}$ by the equation $$e^{\tilde{\xi}}=e^{\xi'}e^{-\xi}$$ and $\tilde{u}=u'-e^{\tilde{\xi}}u$, we see that $h=e^{\tilde{u}}e^{\tilde{\xi}}$, and that $$\rho e^{\tilde{\xi}}=\rho$$ and $$\rho \tilde{u}=0;$$ as shown above this implies $$h\cdot K\da^{\B}=K\da^{\B}.$$ Therefore, we have $$\begin{aligned}
\label{Q equal} Q_{g\cdot\J}(K\da^{\B},K\da^{\B}) & = \la g\J g^{-1} K\da^{\B},K\da^{\B}\ra \notag\\
& = \la h^{-1} g'\J (g')^{-1}hK\da^{\B},K\da^{\B}\ra \notag\\
& = h^{-1}\la (g'\cdot\J)hK\da^{\B},hK\da^{\B}\ra \notag\\
& = h^{-1}\la (g'\cdot\J)K\da^{\B},K\da^{\B}\ra\notag\\
& = h^{-1} Q_{g'\cdot\J}(K\da^{\B},K\da^{\B}).\end{aligned}$$ Since $\rho h=\rho$, in particular both $\tilde{\xi}$ and $-\tilde{\xi}$ are tangent to $Z$, so that it follows that $h(I\da^Z)=h^{-1}(I\da^Z)=I\da^Z$. Combining this with the equality (\[Q equal\]), we obtain the desired result.
With Proposition \[well-defined\] in hand, we are ready to define what it means for a deformation of GC brane to be compatible with the GC structure. Let $\B\in\Br(X,\J)$, and let $\hB=(\hrho,\hL)$ be an $A$-deformation of $\B$ as defined in Definition \[definition par def\]. Choose $\xi\in \cg\da(X)$ such that $\hrho=\rho e^{\xi}$. Also, for each $I$, let $\tilde{W}_I\subset X$ be an open set such that $\tilde{W}_I\cap Z=W_I,$ and choose $u_I\in \Omega^1(\tilde{W}_I)$ such that $\hat{a}_I=a_I+\rho(u_I)$. Set $\xi_I=\xi|_{\tilde{W}_I}$, and define $g_I=e^{u_I}e^{\xi_I}\in e^{\cg\da(\tilde{W}_I)}.$
\[compatibility def\] We say that $\hB\in\Dt\da\uB(X,\J)$ is *compatible* with $\J$ if each $g_I$ (chosen as above) is compatible with $\J|_{\tilde{W}_I}$ with respect to $\B|_{\tilde{W}_I}$. We denote by $\Dt^{\B}(X,\J)$ the formal subgroupoid of $\Dt^{\B}\d(X)$, whose objects for each $A\in\Art$ are the $A$-deformations compatible with $\J$.
It is clear from Proposition \[well-defined\] that the condition in Definition \[compatibility def\] is well-defined, i.e. does not depend on the particular choices of $\xi$, $\{\tilde{W}_I\}$ or $\{u_I\}$.
\[brane def sheaf\] Similarly to the situation described in Remark \[bundle def sheaf\], the formal groupoid $\Dt\uB(X,\J)$ extends in a natural way to a presheaf of formal groupoids on $X$. The fact that this is actually a sheaf (satisfies the descent condition) is Proposition \[descent property\].
\[compatibility under isomorphism\] Let $\hB,\hB'$ be isomorphic elements of $\Dt^{\B}_A(X)$. Then $\hB$ is compatible with $\J$ if and only if $\hB'$ is compatible with $\J$.
Write $\hB=(\hrho,\hL)$ and $\hB'=(\hrho',\hL')$ (with the same underlying submanifold $Z\subset X$) with $\hL=(\{\hc_{IJ}\},\{\ha_{IJ}\})$ and $\hL'=(\{\hc'_{IJ}\},\{\ha'_{IJ}\})$, and let $\Psi=(e^{\tau},\{g\si\})$ be an isomorphism $\hB\to\hB'$. As in Definition \[compatibility def\], let $\{\tilde{W}_I\}$ be open sets in $X$ with $W_I=Z\cap \tilde{W}\si$, and choose $\xi\in\g\da(X)$ and $\{u_I\in\Omega^1\dm(\tilde{W}_I)\}$ such that $\hrho=\rho e^{\xi}$ and $\ha_I=a_I+\rho(u_I)$. Define $x_I=e^{u\si}e^{\xi\si}$, where $\xi\si$ is the restriction of $\xi$ to $\tilde{W}\si$. According to Definition \[compatibility def\], $\hB$ is compatible with $\J$ if and only if, for each $I$ we have $$\label{compat1}\la (x_I\cdot\J)K^{\B}\da,K^{\B}\da\ra\subset I^{Z}\da,$$ where $K\uB\da$ and $I^Z\da$ are defined here with respect to $\tilde{W}\si$. Let $\eta\in\g\da(X)$ be an extension of $\tau$, and define $\xi'\in\g\da(X)$ by $e^{\eta}e^{\xi}=e^{\xi'}$. Then $$\hrho'=e^{\tau}\hrho=e^{\tau}\rho e^{\xi}=\rho e^{\eta}e^{\xi}=\rho e^{\xi'}.$$ Furthermore, we have $$dg_I=e^{\tau}\ha\si-\ha'\si=e^{\tau}a_I+e^{\tau}\rho(u\si)-\ha'\si$$ which implies that $$\ha'\si=a\si+(e^{\tau}a\si-a\si+e^{\tau}\rho(u\si)-dg_I).$$ Choose $\tilde{a}\si\in\Omega\dm^1(\tilde{W}\si)$ such that $\rho(\tilde{a}\si)=a\si$, and $\tilde{g}\si\in\Cinf\dm(\tilde{W}\si)$ such that $\rho(\tilde{g}\si)=g\si$. Then $$e^{\tau}a\si-a\si-e^{\tau}\rho(u\si)-dg_I)=\rho(e^{\eta}\tilde{a}\si-\tilde{a}\si+e^{\eta}u\si-d\tilde{g}\si).$$ Set $$u'\si=e^{\eta}\tilde{a}\si-\tilde{a}\si+e^{\eta}u\si-d\tilde{g}\si,$$ and $x'\si=e^{u'\si}e^{\xi'\si}$, where $\xi'\si$ is the restriction of $\xi'$ to $\tilde{W}\si$. Then $\hB'$ is compatible with $\J$ if and only if for each $I$ we have $$\label{compat2}\la (x'\si\cdot \J)K^{\B}\da,K^{\B}\da\ra\subset I^{Z}\da.$$ We calculate $$\begin{aligned}
x'\si x^{-1}\si & =e^{e^{\eta}\tilde{a}\si-\tilde{a}\si+e^{\eta}u\si-d\tilde{g}\si}e^{\eta}e^{\xi} e^{-\xi}e^{-u\si} \\
& = e^{e^{\eta}\tilde{a}\si-\tilde{a}\si-d\tilde{g}\si}e^{\eta}
\end{aligned}$$
We have $$e^{\eta}\tilde{a}_I-\tilde{a}_I=d\int_0^1e^{t\eta}\iota(\eta)\tilde{a}_Idt+\int_0^1e^{t\eta}\iota(\eta)d\tilde{a}\si dt.$$ Defining $h\si=\int_0^1e^{t\eta}\iota(\eta)\tilde{a}_Idt-\tilde{g}\si$ we have $$x\si'x^{-1}\si=e^{dh\si}e^{(\eta,\iota(\eta)d\tilde{a}_I)}.$$ Note that $\xx:=(\eta,\iota(\eta)d\tilde{a}\si)$ is an element of $\KK^{\B}\da$, so that by Lemma \[K closed under bracket\] we have $$e^{-\xx}\KK^{\B}\da=\KK^{\B}\da.$$ Using the fact that $e^{dh\si}$ acts trivially on $\Cinf\da(\TTX)$, and writing $\KK:=\KK^{\B}\da$, we see that $$\begin{aligned}
\la (x'\si\cdot \J)\KK,\KK\ra & = \la (e^{\xx}x\si\cdot \J)\KK,\KK\ra \\
& = \la e^{\xx}(x\si\cdot\J)e^{-\xx}\KK,\KK\ra \\
& = e^{\eta}\la (x\si\cdot\J)e^{-\xx}\KK,e^{-\xx}\KK\ra\\
& = e^{\eta}\la (x\si\cdot\J)\KK,\KK\ra.
\end{aligned}$$ Since $\eta$ is tangent to $Z$, it follows that $$e^{\eta}I^{Z}\da=I^{Z}\da,$$ so we see that condition (\[compat1\]) is indeed equivalent to condition (\[compat2\]).
\[action of formal symmetries\] Given $\hB\in\Dt\da^{\B}(X)$ and $g\in e^{\T\da(X)}$, the deformation $\hB\cdot g$ is compatible with $\J$ if and only if $\hB$ is. In particular, the action of $e^{\cg(X)}$ on $\Dt^{\B}(X)$ restricts to give a well-defined right action of the formal subgroup $e^{\T(X)}$ on $\Dt^{\B}(X,\J)$.
Given $\hB=(\hrho,\hL)\in\Dt\da^{\B}(X)$ compatible with $\J$, choose $\{\tilde{W}_I\}$, $\{u\si\}$, and $\xi$, as in Definition \[compatibility def\], and set $\xi\si=\xi|_{\tilde{W}_I}$. By definition, for each $I$ we have $$\label{compat1}\la (e^{u\si}e^{\xi\si}\J)K^{\B}\da,\KK^{\B}_{\da}\ra\subset I^Z\da.$$ Given $g=e^{w}e^{\eta}\in e^{\T\da(X)}$, write $\hB':=\hB\cdot g=(\hrho',\hL')$. Then we have $\hrho'=\hrho e^{\eta}=\rho e^{\xi}e^{\eta}$ and $\hL'=(\{\hc_{IJ}\},\{a\si+\rho u\si+(\rho e^{\xi}w)|_{W\si}\})=(\{\hc_{IJ}\},\{a\si+\rho u'\si\})$ with $u'\si=u\si+e^{\xi\si} w|_{\tilde{W}_I}$. Defining $\xi'_I$ by $e^{\xi'_I}=e^{\xi_I}e^{\eta|_{\tilde{W}_I}}$, we see that the compatibility of $\hB'$ with $\J$ is equivalent to the conditions $$\label{compat2}\la (e^{u'\si}e^{\xi'\si}\J)\KK^{\B}\da,\KK^{\B}_{\da}\ra\subset I^Z\da.$$ Letting $g_I=g_{\tilde{W}_I}=e^{w|_{\tilde{W}_I}}e^{\eta|_{\tilde{W}_I}}$, we see that $$e^{u'\si}e^{\xi'\si}=e^{u\si}e^{\xi\si} g_I.$$ Since by assumption $g$ is a symmetry of the GC structure $\J$, it follows that the left-hand sides of (\[compat1\]) and (\[compat2\]) are equal.
Recall from Definition \[genhamdef\] and Proposition \[hamiltonian\] the subalgebra $\H(X)\subset \T(X)$ of generalized Hamiltonian vector fields on $(X,\J)$. We correspondingly have the formal subgroup$e^{\H(X)}\subset e^{\T(X)}$ of formal generalized Hamiltonian symmetries.
\[full deformation groupoid\] Given $\B\in\Br(X,\J)$, let $\De^{\B}(X,\J)$ be the following formal groupoid (over $\Art$). For each $A\in\Art$, the groupoid $\De_A^{\B}(X,\J)$ has the same objects as $\widetilde{\De}^{\B}_A(X,\J)$. Given objects $\hB$ and $\hB'$, a morphism in $\De^{\B}_A(X,\J)$ from $\hB$ to $\hB'$ is a pair $(\psi,z)$, where $z\in e^{\H\da(X)}$ and $\psi$ is a morphism in $\widetilde{\De}^{\B}_A(X,\J)$ from $\hB$ to $\hB'\cdot z$. Given $(\psi,z):\hB\to\hB'$ and $(\psi',z'):\hB'\to\hB''$, the composition is defined by $$\label{composition}(\psi',z')\circ (\psi,z)=(\psi'\psi,z'z):\hB\to \hB''.$$ The identity morphism of any $\hB$ is the pair $(id_{\hB},1)$, where $id_{\hB}$ denotes the identity morphism of $\hB$ considered as an object in $\widetilde{\De}^{\B}_A(X,\J)$.
One easily checks using Proposition \[right action par\] that the composition given by (\[composition\]) is well-defined, unital, and associative.
\[brane deformation functor\] The *deformation functor* $$\df_{\B}:\Art\to \textrm{Set}$$ of $\B\in\Br(X,\J) $ is given by $$A\mapsto \pi_0(\De_A^{\B}(X,\J)).$$
First order deformations and Lie algebroid cohomology {#first order deformations}
=====================================================
In [@KM], an argument was given that first-order deformations of a GC brane $\B$ should correspond to elements of the Lie algebroid cohomology group $H^1(\B)$. In our framework, first-order deformations of $\B$ are encoded as elements of $\df_{\B}(\R[\epsilon](\epsilon^2))$; therefore, a natural expectation motivated by [@KM] is that elements of $\D_{\B}( \R[\epsilon](\epsilon^2))$ should correspond to classes in $H^1(\B)$. In this section, we verify this by unpacking Definition \[brane deformation functor\] in the special case $A=\DN$. On the hand hand, this result (stated as Theorem \[brane tangent\] in the introduction) can be regarded as a rigorous verification of the result obtain in [@KM]. Going in the other direction, we may view Theorem \[brane tangent\] as a check that our Definition \[brane deformation functor\] is a reasonable one from a geometric point of view.
We will actually prove Theorem \[brane tangent\] as part of a slightly better statement, given below as Proposition \[first order induced\]. Namely, we will construct an explicit map $H^1(\B)\to \df_{\B}(\DN)$, and prove that it is both well-defined and bijective.
We now turn to the construction. Let $\B=(Z,\L)\in\Br(X,\J)$ be a GC brane, with generalized tangent bundle $\TT\B$. Recall that, since $\B$ is compatible with $\J$, the GC structure $\J$ preserves $\TT\B\subset \TT X|_{Z}$, and we define $l\subset \TT\B\otimes\C$ to be the $+i$-eigenbundle, which is a complex sub-bundle of $L|_{Z}$. Recall also that the generalized normal bundle $\N\B$ is defined as the quotient of $\TT X|_Z$ by $\TT\B$ (with quotient map $q:\TTX|_Z\to \N\B$). Since $\TT\B$ is a maximal isotropic subbundle of $\TT X|_Z$, the pairing on $\TT X|_Z$ gives a well-defined non-degenerate pairing of $\TT\B$ with $\N \B$. Furthermore, since $\J$ preserves $\TT\B$, it determines a well-defined complex structure $\J_{\N\B}$ on $\N\B$; regarding $\N\B$ as a complex vector bundle with complex structure $-\J_{\N\B}$, we have an isomorphism of complex vector bundles $$\mu:\N\B\to l\uv$$ given by $$\mu(q(\xx))(v)=\la\xx,v\ra$$ for every section $\xx$ of $\TT X|_Z$. Given $\xx=(\xi,u)\in \Cinf(\TT X|_Z)$, choose a section $\tilde{\xx}=(\tilde{\xi},\tilde{u})\in \Cinf(\TT X)$ extending $\xx$, i.e. such that $r(\tilde{\xx})=\xx$. As described in Proposition \[right action par\], the generalized vector field $\tilde{\xx}$ determines an object of $\De^{\B}_{\DN}(X)$ given by $$\B\cdot e^{\epsilon\tilde{\xx}}=(\rho e^{\epsilon\tilde{\xi}},e^{-\epsilon\rho(\tilde{u})}\cdot\L)\in \De_{\DN}^{\B}(X,\J).$$
The deformation $\B\cdot e^{\epsilon\tilde{\xx}}$ depends only on $\xx$, i.e. not on the choice of the extension $\tilde{\xx}$.
Suppose $\eta\in\Cinf(TX)$ satisfies $r(\eta)=0$, i.e. $\eta$ vanishes at each point of $Z$. Then for every $w\in\Omega\ub(X)$, we have $$\rho(\iota(\eta)w)=0.$$ This implies that, for every $v\in\Omega\ub(X)$, we have $$\rho(\pounds(\eta)v)=d\rho(\iota(\eta)v)+\rho(\iota(\eta)dv)=0.$$ Therefore, given two different extensions $\tilde{\xi},\tilde{\xi}'\in\Cinf(TX)$ of $\xi$, we have $$\rho\pounds(\tilde{\xi})=\rho\pounds(\tilde{\xi}'):\Omega\ub(X)\to\Omega\ub(Z).$$ This in turn implies that $\rho e^{\epsilon\tilde{\xi}}=\rho+\epsilon\rho\pounds(\tilde{\xi})$ is equal to $\rho e^{\epsilon\tilde{\xi}'}=\rho+\epsilon\rho\pounds(\tilde{\xi}')$. It is also clear that $\rho(\tilde{u})$ depends only on $r(\tilde{u})$, so that $\rho(\tilde{u})$ is independent of the choice of extension.
Since the deformation depends only on $\xx$, we will use the notation $\hB^{\xx}=\B\cdot e^{\epsilon\tilde{\xx}}$. Note that the section $\xx$ also determines a section $q(\xx)\in\Cinf(\N\B)$ as well as a section $\mu q(\xx)\in\Cinf(l\uv)$.
\[first order induced\]
(1) The deformation $\hB^{\xx}$ is compatible with $\J$ if and only if $\delta_l(\mu q(\xx))=0$, i.e. if and only if $q(\xx)$ is a generalized holomorphic section of $\N\B$.
(2) Given $\xx,\xx'\in\Cinf(\TTX|_Z)$ such that both $q(\xx)$ and $q(\xx')$ are generalized holomorphic sections of $\N\B$, the deformations $\B^{\xx}$ and $\B^{\xx'}$ are isomorphic in $\De^{\B}_{\DN}(X,\J)$ if and only if the elements $[\mu q(\xx)],[\mu q(\xx')]\in H^1(\B)$ are equal. In other words, there is a well-defined injective map $H^1(\B)\to \df_{\B}(\DN)$ mapping $[\mu q(\xx)]$ to the equivalence class of $\B^{\xx}$.
(3) Every element of $\De^{\B}_{\DN}(X,\J)$ is isomorphic to one of the form $\B^{\xx}$ for some $\xx\in \Cinf(\TT X|_{Z})$. In other words, the injective map $H^1(\B){\xymatrix@C=1.5pc{ \ar[r]^-{\cong} & }}\df_{\B}(\DN)$ is actually a bijection.
Let us begin with part (1). By Definition \[compatibility def\], we see that $\B^{\xx}$ will be compatible with $\J$ if and only if $e^{\epsilon\tilde{\xx}}$ is compatible with $\J$ in the sense of Definition \[compatibility1\], i.e. if and only if we have $$\la (e^{\epsilon\tilde{\xx}}\cdot\J)K^{\B}\da,K^{\B}\da\ra \subset I^Z\da.$$ Since $e^{\epsilon\tilde{\xx}}\cdot\J=\J+\epsilon\tilde{\xx}\cdot\J$, this is equivalent to $$\la (\tilde{\xx}\cdot\J)K^{\B}\da,K^{\B}\da\ra\subset I^Z\da.$$ By definition, this means that for every $A,B\in K\da^{\B}$ we have $$\label{compatibility cond1}\rho(\la \ll\tilde{\xx},\J A\rr-\J\ll\tilde{\xx},A\rr,B\ra =\rho(\la\ll\tilde{\xx},\J A\rr,B\ra+\la \ll\tilde{\xx},A\rr,\J B\ra)=0.$$
Suppose condition (\[compatibility cond1\]) holds, and let $u,v\in\Cinf(l)$ be arbitrary sections. Choose sections $\tilde{u},\tilde{v}\in\Cinf(L)$ which extend $u,v$, i.e. such that $r(\tilde{u})=u$ and $r(\tilde{v})=v$. Since $l\subset \TT\B\otimes \C$, condition (\[compatibility cond1\]) implies that $$\label{compatibility condit2}0=\rho(\la\ll\tilde{\xx},\J \tilde{u}\rr,\tilde{v}\ra+\la \ll\tilde{\xx},\tilde{u}\rr,\J \tilde{v}\ra)=2i\rho(\la\ll\tilde{\xx},\tilde{u}\rr,\tilde{v}\ra).$$ Making use of the identities (\[Dorfman1\]) and (\[Dorfman2\]) satisfied by the Dorfman bracket, we calculate the the right-hand side of (\[compatibility condit2\]) is equal to $$\begin{aligned}
\label{compatibility condit3} & 2i\rho(\pi(\tilde{v})\cdot\la\tilde{\xx},\tilde{u}\ra-\pi(\tilde{u})\la\tilde{\xx},\tilde{v}\ra-\la\tilde{\xx},\ll\tilde{u},\tilde{v}\rr) \notag\\
= & 2i(\pi(v)\cdot\la\xx,u\ra-\pi(u)\cdot\la\xx,v\ra-\la\xx,\ll u,v\rr_{\B}) \notag \\
= & 2i\delta_l(\mu q(\xx)(v,u)),
\end{aligned}$$ where in the second line the expression $\la\xx,v\ra-\la\xx,\ll u,v\rr_{\B}$ denotes the Lie algebroid bracket of the sections $u,b\in\Cinf(l)$, as defined in (\[brane bracket\]). Thus, we see that if $\B^{\xx}$ is compatible with $\J$, then $q(\xx)$ is a generalized holomorphic section of $\N\B$.
Conversely, suppose we have $\delta_l(\mu q(\xx))=0$. Given $u,v\in K^{\B}\da$, define $A,B\in \Cinf(L)$ by $A=u-i\J u$ and $B=v-i\J v$, and note that $r(A),r(B)\in\Cinf(l)$. Using the above calculation (\[compatibility condit3\]), we see that $$\begin{aligned}
0 = & \rho(\la\ll\tilde{\xx}, A\rr,B\ra) \\
= & \rho(\la\ll\tilde{\xx},u-i\J u\rr,v-i\J v\ra) \\
= & \rho(\la\ll\tilde{\xx},u\rr,v\rr+\la\ll\tilde{\xx},\J u\rr,\J v\ra)+i\rho(\la \ll\tilde{\xx},\J u\rr,v\ra+\la\ll\tilde{\xx},u\rr,\J v\ra).
\end{aligned}$$ In particular, this implies $$\rho(\la \ll\tilde{\xx},\J u\rr,v\ra+\la\ll\tilde{\xx},u\rr,\J v\ra)=0$$ holds for arbitrary $u,v\in K^{\B}\da$; as previously mentioned, this condition is equivalent to the compatibility of $\B^{\xx}$ with $\J$. This completes the proof of part (1) of Proposition \[first order induced\].
We will prove part (2) of Proposition \[first order induced\] via a series of lemmas.
\[coboundary isomorphism\] Suppose we are given $\xx,\xx'\in \Cinf(\TTX|_Z)$ such that $q(\xx)=q(\xx')$. Then there exists an isomorphism $\hB^{\xx}{\xymatrix@C=1.5pc{ \ar[r]^-{\cong} & }}\hB^{\xx'}$ in $\Dt^{\B}_{\DN}(X)$.
Let us write $\B=(Z,\L)$, with $\L=(\{c\sij\},\{a\si\})$, and also $\xx=(\xi,u)$ and $\xx'=(\xi',u')$. Also, let $\tilde{\xx}=(\tilde{\xi},\tilde{u})\in\Cinf(\TTX)$ be a choice of extension of $\xx$ and $\tilde{\xx}'=(\tilde{\xi}',\tilde{u}')\in\Cinf(\TTX)$ be a choice of extension of $\xx'$. By definition, the assumption $q(\xx)=q(\xx')$ implies that $\xx'-\xx$ is a section of $\TT\B$. Thus, we have that $\xi'-\xi$ is tangent to $Z$, and $$\rho(u'-u)=\iota(\tau)F,$$ where $\tau=\rho(\xi'-\xi)\in\Cinf(TX)$ and $F$ is the curvature form of $\L$. Defining $g=\{g_I=\epsilon \iota(\tau)a_I\}$, we claim that $$\Psi=(e^{\epsilon\tau},g)$$ is an isomorphism $\hB^{\xx}{\xymatrix@C=1.5pc{ \ar[r]^-{\cong} & }}\hB^{\xx'}$ in $\De^{\B}_{\DN}(X)$. Recall that $$\hB^{\xx}=(\rho e^{\epsilon\tilde{\xi}},(\{c_{IJ}\},\{a_I+\epsilon(\rho\tilde{u})|_{W_I}\})$$ and $$\hB^{\xx}=(\rho e^{\epsilon\tilde{\xi}'},(\{c_{IJ}\},\{a_I+\epsilon(\rho\tilde{u'})|_{W_I}\}).$$ According to Definition \[definition par def\], we need to check three conditions to show that $\Psi$ does indeed define an isomorphism from $\hB^{\xx}\to\hB^{\xx'}$:
1. $\rho e^{\epsilon\tilde{\xi}'}=e^{\epsilon\tau}\rho e^{\epsilon\tilde{\xi}}$,
2. $g_J-g_I=e^{\epsilon\tau}c_{IJ}-c_{IJ}$
3. $dg_I=e^{\epsilon\tau}(a_I+\epsilon\rho(\tilde{u})|_{W_I})-(a_I+\epsilon\rho(\tilde{u}')|_{W_I}),$
Condition (1) is equivalent to $\rho(\xi'-\xi)=\tau\rho$, which holds by construction. To check condition (2), first note that, since we are working over $A=\DN$, we have $$e^{\epsilon\tau}c_{IJ}-c_{IJ}=\epsilon\pounds(\tau)c_{IJ}.$$ On the other hand, we calculate that $$g_J-g_I=\epsilon\iota(\tau)(a_J-a_I)=\epsilon \iota(\tau)dc_{IJ}=\epsilon\pounds(\tau)c\sij.$$ To check condition (3), first note that it is equivalent to $$dg_I=\epsilon(\pounds(\tau)a_I+\rho(u)|_{W_I}-\rho(u')|_{W_I}).$$ We then calculate $$\begin{aligned}
dg_I=\epsilon d(\iota(\tau)a_I) &= \epsilon(\pounds(\tau)a_I-\iota(\tau)da_I)\\
& = \epsilon(\pounds(\tau)a_I-\iota(\tau)F|_{W_I}) \\
& = \epsilon(\pounds(\tau)a_I-\rho(u'-u)|_{W_I}) \\
& = \epsilon(\pounds(\tau)a_I+\rho(u)|_{W_I}-\rho(u')|_{W_I}).\end{aligned}$$
\[restrict function\] For any smooth function $f:X\to \C$, we have the equality $$\mu qr(\xx_f)=-i\delta_l(\rho(f)),$$ where $\xx_f\in\Cinf(\TTX)$ is the generalized Hamiltonian vector field associated to $f$.
For any section $v\in\Cinf(l)$, the vector field component $\pi(v)\in\Cinf(TX|Z\otimes \C)$ is tangent to $Z$. Therefore, if $f:X\to \C$ is any smooth function, we have $$\la r(0,df),v\ra=\frac{1}{2}\pi(v)\cdot\rho(f).$$
Writing $f=f_R+if_I$, recall that the corresponding generalized Hamiltonian vector field is given by the formula $$\xx_f=\J(0,df_R)+(0,df_I).$$ Given a section $v\in\Cinf(l)$, we then calculate $$\begin{aligned}
\mu qr(\xx_f)(v) & = 2\la r(\xx_f),v\ra \\
& = 2\la \J r(0,df_R),v\ra+2\la r(0,df_I),v\ra \\
& = -2\la r(0,df_R),\J v\ra+2\la r(0,df_I),v\ra \\
& = -2i\la r(0,df_R),v)+2\la r(0,df_I),v\ra \\
& = -i\pi(v)\cdot\rho(f_R)+\pi(v)\cdot \rho(f_I) \\
& = -i\pi(v)\cdot(f_R+if_I)\\
& -i\delta_l\rho(f)(v).\end{aligned}$$
Returning to the proof of part (2) of Proposition \[first order induced\], suppose $[\mu q(\xx)]=[\mu q(\xx')]\in H^1(\B)$. Then there exists a smooth function $f:Z\to \C$ such that $$\mu q(\xx')-\mu q(\xx)=-i\delta_l f.$$ Choose a smooth function $\tilde{f}:X\to \C$ that extends $f$, i.e. such that $\rho(\tilde{f})=f$. By Lemma \[restrict function\], we have $$\mu q(\xx)=\mu q(\xx'+r(\xx_{\tilde{f}})),$$ or equivalently $$q(\xx)=q(\xx'+r(\xx_{\tilde{f}})).$$ This implies that $$\hB^{\xx'+r(\xx_{\tilde{f}})}=\hB^{\xx'}\cdot e^{\xx_{\epsilon\tilde{f}}}.$$ Lemma \[coboundary isomorphism\] therefore implies that $\B^{\xx}$ is isomorphic to $\B^{\xx'}\cdot e^{\xx_{\epsilon}\tilde{f}}$ in $\Dt^{\B}_{\DN}(X,\J)$, so that by definition $\B^{\xx}$ and $\B^{\xx'}$ are isomorphic in $\De_{\DN}^{\B}(X,\J)$. Therefore $\B^{\xx}$ and $\B^{\xx'}$ determine the same element of the set $\df_{\B}(\DN):=\pi_0(\De_{\DN}^{\B}(X,\J)).$
To finish the proof of part (2) of Proposition \[first order induced\], it remains to show that if $\hB^{\xx}$ and $\hB^{\xx'}$ are isomorphic in $\De^{\B}_{\DN}(X,\J)$, then $[\mu q(\xx)]=[\mu q (\xx')]\in H^1(\B)$. Let $(\Phi,z)$ be an isomorphism from $\hB^{\xx}$ to $\hB^{\xx'}$ in $\De^{\B}_{\DN}(X,\J)$; here $z=e^{\xx_{\epsilon f}}$ is an element of $e^{\H_{\DN}(X)}$, and $$\Phi=(e^{\epsilon \tau},\{\epsilon g_I\}):\hB^{\xx}{\xymatrix@C=1.5pc{ \ar[r]^-{\cong} & }}\hB^{\xx'}\cdot z$$ is an isomorphism in $\Dt^{\B}_{\DN}(X,\J)$. Write $\xx=(\xi,u)$, $\xx'=(\xi',u')$, and$\xx_f=(\eta_f,v_f)$, a calculation similar to that used in the proof of Lemma \[coboundary isomorphism\] implies the following three conditions: $$\label{condition11}\xi'+r(\eta)-\xi=\iota_*\tau,$$ $$\label{condition22} g_J-g_I=\pounds(\tau)c_{IJ},$$ and $$\label{condition33}dg\si=\pounds(\tau)a_I+\rho(u-u'-r(v_f))|_{W_I}.$$
Define $h_I:W_I\to \R$ by $h_I=\iota(\tau)a_I-g_I$. Using the assumption that $a_J-a_I=dc_{IJ}$, together with condition (\[condition22\]), we see that on each overlap $W_I\cap W_J$ we have $$h_J-h_I=0,$$ so that there exists a unique function $h:Z\to \R$ with $h_I=h|_{W_I}$ on each $W\si$. Using the Cartan formula $\pounds(\tau)a_I=d\iota(\tau)a_I+\iota(\tau)da_I$, we may then rewrite condition (\[condition33\]) as $$\label{hI condition}-dh|_{W_I}=(\iota(\tau)F)|_{W\si}+\rho(u-u'-r(v_f))|_{W_I}.$$ Choosing a smooth function $\tilde{h}:X\to \R$ that extends $h$, equation (\[hI condition\]) is equivalent to $$\label{hI condition1}\rho(u'+r(v_f-d\tilde{h})|_{W_I}-\rho(u)=\iota(\tau)F.$$ Defining $f'=f-i\tilde{h}$, we have $$\xx_{f'}=\xx_f-(0,d\tilde{h})=(\eta_f,v_f-d\tilde{h}).$$ Equation (\[condition11\]) together with equation (\[hI condition1\]) imply that $\xx'+r(\xx_{f'})-\xx$ is a section of $\TT\B$, so that $$q(\xx')-q(\xx)=q(r(\xx_{f'})),$$ and therefore $$\mu q(\xx')-\mu q(\xx)=-\mu qr(\xx_{f'})=\delta_l(i\rho(f')),$$ where in the last equality we used Lemma \[restrict function\]. Therefore, we see that $$[\mu q(\xx')]=[\mu q(\xx)]\in H^1(\B),$$ as claimed.
Finally, part (3) of Proposition \[first order induced\] follows easily from Proposition \[pi zero bundle\].
Functoriality and Invariance {#section functoriality}
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In this section we explain how various equivalence between GC branes $\B$ and $\B'$ induce equivalences between the corresponding formal deformation groupoids. To do so, it will be convenient to rephrase the definition of the formal groupoid $\De^{\B}(X,\J)$ from the action of $e^{\H(X,\J)}$ on $\Dt^{\B}(X,\J)$ in terms of a general construction.
\[definition action groupoid\] Let $\CC$ be a groupoid, and $G$ a group with a strict right action on $\CC$. We define a new category $\CC//G$, called the (right) *action groupoid of $G$ acting on $\CC$*, which has the same objects as $\CC$, and such that for pair of objects $x,y$ we have $$\Hom_{\CC//G}(x,y)=\{(\varphi,g):g\in G, \varphi\in \Hom_{\CC}(x,y\cdot g)\}.$$ Composition is defined by $$\label{compo def}(\varphi,g)\circ (\psi,h)=((\varphi\cdot h)\circ \psi,gh).$$ For each object $x$, the identity morphism from $x$ to itself in $\CC//G$ is defined to be $(id_x,1)$, where $id_x\in \Hom_{\CC}(x,x)$ is the identity morphism in $\CC$, and $1\in G$ is the identity group element.
It is an easy exercise using Definition \[group action on category\] to check that the composition given by equation (\[compo def\]) is well-defined, associative, and unital.
Given a formal group $G$ (over $\Art$) with a right action on a formal groupoid $\CC$, we similarly define a formal groupoid $\CC//G$, which associates to each $A\in\Art$ the groupoid $\CC\da//G\da$ defined as above. For example, translating Definition \[full deformation groupoid\] into this language, we see that $$\De\uB(X,\J)=\Dt\uB(X,\J)//e^{\H(X,\J)}.$$
The proof of the following Proposition follows easily from the definitions, and is omitted.
\[induced equivalence\] Let $\CC$ be a groupoid equipped with a strict right action of the group $G$, and $\CC'$ a groupoid equipped with a strict right action of a group $G'$. Let $\varphi:G{\xymatrix@C=1.5pc{ \ar[r]^-{\cong} & }}G'$ be a group isomorphism, and let $\Phi:\CC\to \CC'$ be a functor that intertwines the group actions in the sense that, for every $x\in \CC$ and $g\in G$ we have $\Phi(x\cdot g)=\Phi(x)\cdot \varphi(g)$, and for every $\psi:x\to y$ in $\CC$ we have $\Phi(\psi\cdot g)=\Phi(\psi)\cdot \varphi(g)$. Then there is a well-defined functor $\Phi//G$ which agrees with $\Phi$ on objects, and which maps a morphism $(\psi,g):x\to y$ in $\CC//G$ to $(\Phi(\psi),\varphi(g)):\Phi(x)\to \Phi(y)$ in $\CC'//G$. Furthermore, $\Phi//G$ is an equivalence if and only if $\Phi$ is.
This result has an immediate extension to the case that $\CC$ and $\CC'$ are formal groupoids and $\Phi,\Phi'$ are formal groups.
We now return to our main topic. Let $\B=(Z,\L)$ and $\B'=(Z,\L')$ be two elements of $\Br(X,\J)$ supported on the same submanifold $Z$. Furthermore, suppose that $\L$ and $\L'$ are defined with respect to the same open cover $\W$ of $Z$. Write $\L=(\{c_{IJ}\},\{a\si\})$ and $\L'=(\{c'\sij\},\{a'\si\})$. Suppose we are given $\gamma=\{g_I:W_I\to \R\}$ satisfying $$c'\sij-c\sij=g_J-g\si$$ and $$a'\si-a\si=dg\si.$$ We say that $\gamma$ is an equivalence $\L{\xymatrix@C=1.5pc{ \ar[r]^-{\cong} & }}\L'$.
\[invariance 1\] The equivalence $\gamma:\L{\xymatrix@C=1.5pc{ \ar[r]^-{\cong} & }}\L'$ induces an equivalence of formal groupoids $$\tilde{\Phi}^{\gamma}:\Dt^{\B}(X,\J){\xymatrix@C=1.5pc{ \ar[r]^-{\cong} & }}\Dt^{\B'}(X,\J)$$ defined as follows. Given $A\in \Art$ and $\hB=(\hrho,\hL)\in \Dt^{\B}_A(X,\J)$, write $\hL=(\{c\sij+f\sij\},\{a\si+v\si\})$, and define $$\label{onobjects}\tilde{\Phi}\da^{\gamma}(\hB):=(\hrho,(\{c'\sij+f\sij\},\{a'\si+v\si\})).$$ Given a morphism $$\psi=(e^{\tau},\{h\si\},z):\hB_1\to \hB_2$$ in $\Def^{\B}_A(X,\J)$, we define $$\label{gamma mor}\tilde{\Phi}\da^{\gamma}(\psi)=(e^{\tau},\{h\si+e^{\tau}g\si-g\si\},z):\tilde{\Phi}\da^{\gamma}(\B_1)\to\tilde{\Phi}\da^{\gamma}(\B_2).$$
First, note that it is clear by inspection that the right-hand side of (\[onobjects\]) defines an $A$-deformation of $\B'$. Next, we must check that if $\psi=(e^{\tau},\{h\si\})$ is a morphisms from $\hB_1$ to $\hB_2$, then $\tilde{\Phi}\da^{\gamma}(\psi)$ as defined by equation (\[gamma mor\]) does in fact define a morphisms from $\tilde{\Phi}\da^{\gamma}(\hB_1)$ to $\tilde{\Phi}\da^{\gamma}(\hB_2)$. Writing $\hB_1=(\hrho_1,(\{c_{IJ}+f_{IJ,1}\},\{a_I+v_{I,1}\}))$ and $\hB_2=(\hrho_2,(\{c_{IJ}+f_{IJ,2}\},\{a_I+v_{I,2}\}))$, by Definition \[definition par def\], this means that $$\label{psi1} \hrho_2=e^{\tau}\hrho_1,$$ $$\label{psi2} e^{\tau}(c_{IJ}+f_{IJ,1})-c_{IJ}-f_{IJ,2}=h_J-h_I,$$ and $$\label{psi3} e^{\tau}(a_I+v_{I,1})-a_I-v_{I,2}=dh_I.$$ We need to check that these same equations hold when $c_{IJ}$ is replaced by $c'_{IJ}$, $a_I$ is replaced by $a'_I$, and $h_I$ is replaced by $h_I':=h_I+e^{\tau}g_I-g_I$. Equation (\[psi1\]) is unchanged under these replacements, and thus continues to hold automatically. We calculate $$\begin{aligned}
h'_J-h'_I & = h_J+e^{\tau}g_J-g_J-h_I-e^{\tau}g_I+g_I \\
& = h_J-h_I+e^{\tau}(g_J-g_I)-(g_J-g_I) \notag \\
& = e^{\tau}(c_{IJ}+f_{IJ,1})-c_{IJ}-f_{IJ,2}+e^{\tau}(c'_{IJ}-c_{IJ})-(c_{IJ}'-c_{IJ}) \\
& = e^{\tau}(c_{IJ}+f_{IJ,1}+c'_{IJ}-c_{IJ})-(c_{IJ}+f_{IJ,2}+c'_{IJ}-c_{IJ}) \\
& = e^{\tau}(c'_{IJ}+f_{IJ,1})-(c_{IJ}'+f_{IJ,2}),\end{aligned}$$ so we see that the “primed" version of equation (\[psi2\]) holds. Finally, we have $$\begin{aligned}
dh'_I & = dh_I+de^{\tau}g_I-dg_I \\
& = e^{\tau}(a_I+v_{I,1})-a_I-v_{I,2}+e^{\tau}(a'_I-a_I)-(a'_I-a_I) \\
& = e^{\tau}(a'_I+v_{I,1})-a'_I-v_{I,2},\end{aligned}$$ so the “primed" version of equation (\[psi3\]) also holds. This finishes the proof that $\tilde{\Phi}\da^{\gamma}(\psi)$ does in fact define an isomorphism from $\tilde{\Phi}\da^{\gamma}(\hB_1)$ to $\tilde{\Phi}\da^{\gamma}(\hB_2)$.
We need to verify that $\tilde{\Phi}\da^{\gamma}$ is functorial, i.e. we must check that, given isomorphisms $\psi:\hB_1\to\hB_2$ and $\tilde{\psi}:\hB_2\to \hB_3$ that $\tilde{\Phi}\da^{\gamma}(\tilde{\psi})\circ \Phi\da^{\gamma}(\psi)=\Phi\da^{\gamma}(\tilde{\psi}\circ\psi)$. If we write $\psi=(e^{\tau},\{h_I\})$ and $\tilde{\psi}=(e^{\tilde{\tau}},\{\tilde{h_I}\})$, then by Definition \[definition par def\] we have $$\tilde{\psi}\circ\psi = (e^{\tilde{\tau}}e^{\tau},\{\tilde{h}_I+e^{\tilde{\tau}}h_I\}).$$ and therefore $$\Phi\da^{\gamma}(\tilde{\psi}\circ\psi)=(e^{\tilde{\tau}}e^{\tau},\{\tilde{h}_I+e^{\tilde{\tau}}h_I+e^{\tilde{\tau}}e^{\tau}g_I-g_I\}).$$ On the other hand, we calculate $$\begin{aligned}
\tilde{\Phi}\da^{\gamma}(\tilde{\psi})\circ\tilde{\Phi}\da^{\gamma}(\psi) & = (e^{\tilde{\tau}},\{\tilde{h}_I+e^{\tilde{\tau}}g_I-g_I\})\circ (e^{\tau},\{h_I+e^{\tau}g_I-g_I\}) \\
& =(e^{\tilde{\tau}}e^{\tau}, \{\tilde{h}_I+e^{\tilde{\tau}}g_I-g_I+e^{\tilde{\tau}}(h_I+e^{\tau}g_I-g_I)\}) \\
& = (e^{\tilde{\tau}}e^{\tau},\{\tilde{h}_I+e^{\tilde{\tau}}h_I+e^{\tilde{\tau}}e^{\tau}g_I-g_I\}).\end{aligned}$$
To complete the proof, consider $\gamma^{-1}:=\{-g_I\}$, which is an equivalence from $\L'\to\L$. It is clear by inspection that $$\tilde{\Phi}\da^{\gamma^{-1}}:\Dt_A^{\B'}(X,\J)\to \Dt_A^{\B}(X,\J)$$ is a (strict) inverse for $\tilde{\Phi}\da^{\gamma}$; in particular, $\tilde{\Phi}\da^{\gamma}$ is an equivalence of groupoids.
The equivalence $\gamma:\L{\xymatrix@C=1.5pc{ \ar[r]^-{\cong} & }}\L'$ induces an equivalence of formal groupoids $$\Phi^{\gamma}:\De^{\B}(X,\J){\xymatrix@C=1.5pc{ \ar[r]^-{\cong} & }}\De^{\B'}(X,\J),$$ where $$\Phi^{\gamma}=\tilde{\Phi}^{\gamma}//e^{\H(X,\J)}$$ is defined as in Proposition \[induced equivalence\]. In particular, this induces a well-defined natural isomorphism of functors $$\df_{\B}{\xymatrix@C=1.5pc{ \ar[r]^-{\cong} & }}\df_{\B'}.$$
Let $\W=\{W_I\}_{I\in \mathcal{I}}$ be an open cover of $Z$ with indexing set $\mathcal{I}$. Recall that a *refinement* of $\W$ is a pair $(\W',\sigma)$, where $\W'=\{W'_{I'}\}_{I'\in \mathcal{I}'}$ is an open cover of $Z$, and $\sigma:\mathcal{I'}\to \mathcal{I}$ is a map of indexing sets such that for each $I'\in \mathcal{I}'$ we have $$W'_{I'}\subset W_{\sigma(I')}.$$ Given $\L\in \Herm(Z)$ defined with respect to the open cover $\W$, we obtain a new element $\sigma^*\L\in \Herm(Z)$ defined with respect to the open cover $\W'$. Explicitly $\sigma^*\L=(\{c'_{I'J'}\},\{a'_{I'}\})$ with $c'_{I'J'}=(c_{\sigma(I')\sigma(J')})|_{W'_{I'J'}}$ and $a'_{I'}=(a_{\sigma(I')})|_{W_{I'}}$. In particular, if $\B=(Z,\L)$ is a GC brane on $(X,\J)$, then the refinement $(\W',\sigma)$ determines $\sigma^*\B=(Z,\sigma^*2\L)$, which is again a GC brain on $(X,\J)$. Furthermore, there is a natural functor of formal groupoids $$\sigma^*:\De^{\B}(X,\J)\to \De^{\B}(X,\J).$$
\[invariance 2\] The functor $$\sigma^*\De^{\sigma^*\B}_A(X,\J)\to \De^{\sigma^*\B}_A(X,\J)$$ is an equivalence of formal groupoids. In particular, it induces a natural isomorphism of functors $$\df_{\B}{\xymatrix@C=1.5pc{ \ar[r]^-{\cong} & }}\df_{\sigma^*\B}.$$
Fix $A\in\Art$, and let $\hB\in \Def^{\sigma^*\B}\da(X)$. Using Propositions \[pi zero bundle\] and, it follows that $\hB$ is isomorphic to an object of the form $\sigma\da^*\B\cdot e^{\xx}$ for some $\xx\in \cg\da(X)$; by definition, $e^{\xx}$ is compatible with $\J$ as in Definition \[compatibility1\]. It is also clear that $\sigma\da^*\B\cdot e^{\xx}=\sigma\da^*(\B\cdot e^{\xx})$. This shows that $\sigma^*$ is essentially surjective. On the other hand, it is also clear that $\sigma\da^*$ is fully faithful, and therefore an equivalence.
Finally, let $\B$ be a GC brane on $(X,\J)$, and let $u\in \Omega^1(X)$ be a 1-form. This determines a new GC structure $\J'=e^{u}\cdot\J$. Writing $\B=(Z,\L)$, define $$\B'=e^{u}\cdot\B=(Z,e^{\rho(u)}\cdot\L);$$ explicitly, if we write $\L=(\{c\sij\},\{a\si\})$, we have $$\L'=e^{\rho(u)}\cdot \L=(\{c\sij\},\{a\si-\rho(u)|_{W_I}\}).$$ Using Proposition \[action on GC submanifolds\], it follows that $\B'\in\Br(X,\J')$. For each $A\in\Art$, let us define $$\tilde{\Phi}\da^u:\Dt_A^{\B}(X,\J)\to \Dt_A^{\B'}(X,\J')$$ as follows: given $\hB=(\hrho,\hL)\in \De_A^{\B}(X,\J)$, we define $$\tilde{\Phi}\da^u(\hB)=(\hrho,e^{\hrho(u)}\cdot\hL).$$
\[B transform on deformations\]
(1) $\tilde{\Phi}\da^u(\hB)$ is an object of $\De^{\B'}_A(X,\J')$ for each $\hB\in \Dt_A^{\B}(X,\J)$.
(2) For each equivalence $\psi=(e^{\tau},\{g\si\}):\hB_1{\xymatrix@C=1.5pc{ \ar[r]^-{\cong} & }}\hB_2$ in $\Dt_A^{\B}(X,\J)$, $\psi$ also determines an equivalence $\tilde{\Phi}\da^u(\hB_1){\xymatrix@C=1.5pc{ \ar[r]^-{\cong} & }}\tilde{\Phi}\da^u(\hB_2)$ in $\Dt_A^{\B'}(X,\J')$, which we denote by $\Phi^u\da(\psi)$.
(3) The map taking $\hB\in\Dt_A^{\B}(X,\J)$ to $\tilde{\Phi}^u\da(\hB)$ and $\psi:\hB\to\hB'$ in $\Dt_A^{\B}(X,\J')$ to $\tilde{\Phi}^u\da(\psi)$ defines an equivalence of formal groupoids $$\Dt^{\B}(X,\J){\xymatrix@C=1.5pc{ \ar[r]^-{\cong} & }}\Dt^{\B}(X,\J').$$
The proof of part (2) is a straightforward computation (similar to that in the proof of Proposition \[invariance 1\]), and is therefore omitted.
To verify part (1), let $\{\tilde{W}_I\}$ be a collection of open sets in $X$ chosen as in Definition \[compatibility def\]. Also choose $\xi\in\Cinf\da(X)$ such that $\hrho=\rho e^{\xi}$ and $w_I\in\Omega\dm(\tilde{W}_I)$ such that $\hat{a}_I=a_I+\rho(w_I)$. According to Definition \[compatibility def\], the compatibility of $\hB$ with $\J$ means that, for each $I$ we have $$\label{comp1}\la (e^{w_I}e^{\xi}\cdot \J)K^{\B}\da,K_{A}^{\B}\ra\subset I\da^Z.$$
By construction, $\hB':=\tilde{\Phi}\da^u(\hB)$ is given by $$\begin{aligned}
& (\rho e^{\xi},(\{c_{IJ}\},\{a_I+\rho(w_I)-\rho e^{\xi}(u|_{\tilde{W_I}})\})
\\ = & (\rho e^{\xi},(\{c_{IJ}\},\{a_I-\rho(u)|_{W_I}+\rho(w_I+ (-e^{\xi}u+u)|_{\tilde{W}_I}))\}) \\
= & (\rho e^{\xi},(\{c_{IJ}\},\{a'_I+\rho(w'_I)\},)\end{aligned}$$ where we have defined $a'_I:=a_I+\rho(u)|_{W_I}$ and $w_I':=w_I+ (u-e^{\xi}u)|_{\tilde{W}_I}$. Therefore, to verify that $\hB'$ is compatible with $\J'$, we must check that for each $I$ we have $$\label{comp2}\la (e^{w'_I}e^{\xi\si}\cdot\J')K_A^{\B'},K_A^{\B'}\ra\subset I\da^Z.$$ We have $$e^{w'_I}e^{\xi\si}\cdot \J' = e^{w_I+ u-e^{\xi\si}u}e^{\xi\si}e^{u}\J=e^{w_I+u}e^{\xi\si}\J.$$ Also, by Proposition \[action on GC submanifolds\] we have $K_A^{\B'}=e^{u}\cdot K\da^\B$, so that the left hand side of (\[comp2\]) is equal to $$\begin{aligned}
\la (e^{u}\cdot e^{w_I}e^{\xi\si}\cdot\J)e^uK\da^{\B},e^{u}K\da^{\B}\ra & = \la e^u (e^{w_I}e^{\xi\si}\J)e^{-u}e^uK\da^{\B},e^uK\da^{\B}\ra \\ & = \la (e^{w_I}e^{\xi\si}\J)K\da^{\B},K\da^{\B}\ra \subset I\da^Z.\end{aligned}$$ Therefore, we see that condition (\[comp1\]) is equivalent to condition (\[comp1\]), so the assumption that the former is satisfied implies that the latter is as well. This completes the proof of part (1).
To see why part (3) of the Proposition is true, first note that the functoriality of $\Phi_u$ follows trivially from its definition. Also, by inspection $\Phi_u$ is a bijection on both objects and morphisms, and is in particular an equivalence of categories. Indeed, the functor $\Phi^{-u}\da:\Dt^{\B'}_A(X,\J')\to \Dt^{\B}\da(X,\J)$ is easily verified to be the inverse of $\Phi\da^u$.
Given $u\in\Omega^1(X)$ and $\xx=(\xi,a)\in \g\da(X)$, we have $$[(0,u),\xx]=(0,-\pounds(\xi)u)$$ and $$[(0,u),[(0,u),\xx]]=0.$$ The adjoint action of $e^u$ on $\cg\da(X)$ is therefore given by $$\label{adjoint}\Ad_{e^u}(\xi,a)=(\xi,a-\pounds(\xi)u).$$ If $\J$ is a GC structure on $X$, we clearly have $e^ue^{\T\da(X,\J)}e^{-u}=e^{\T\da(X,e^u\cdot\J)}$.
$e^u e^{\H\da(X,\J)}e^{-u}=e^{\H\da(X,e^u\cdot\J)}.$
Recall from Proposition \[exp prop\] the identity $e^{u}e^{\xx}e^{-u}=e^{\textrm{Ad}_u(\xx)}.$ Therefore, we need to show that for every $f:X\to\C$ there exists $f':X\to \C$ such that $\textrm{Ad}_u\xx^{\J}_f=\xx^{\J'}_{f'}$. Given $f=f_R+if_I:X\to \C$, recall from Definition \[genhamdef\] that $$\label{Ham1}\xx_f^{\J}=\J(0,df_R)+(0,df_I).$$ Therefore, setting $\J'=e^u\cdot\J$ we have $$\label{Ham2}\xx_f^{\J'}=e^u\J e^{-u}\cdot (0,df_R)+(0,df_I).$$ If $f=if_I$ is purely imaginary, we see from (\[adjoint\]) that $$\Ad_{e^u}\xx^{\J}_f=\Ad_{e^u}(0,df_I)=(0,df_I)=\xx^{\J'}_f.$$ If $f$ is real, write $(\xi,a)=\xx_f^{\J}=\J(0,df)$, then we see that $$\xx^{\J'}_f=e^u\cdot\J e^{-u}(0,df)=e^u\cdot\J(0,df)=(\xi,a-\iota(\xi)du).$$ On the other hand, we have $$\begin{aligned}
\Ad_{e^u}\xx^{\J}_f & =(\xx,a-\pounds(\xi)u)\\
& = (\xx,a-\iota(\xi)du)-(0,d\iota(\xi)u)=\xx^{\J'}_{f-i\iota(\xi)u}.\end{aligned}$$
Let $\Psi:e^{\H(X,\J)}\to e^{\H(X,e^u\J)}$ be the isomorphism of formal groups given for each $A\in\Art$ and $z\in e^{\H\da(X,\J)}$ by $$z\mapsto e^uze^{-u}.$$ For fixed $z\in e^{\H\da(X,\J)}$, let $R_z:\Dt^{\B}_A(X,\J)\to\Dt^{\B}_A(X,\J)$ be the functor $\hB\mapsto\hB\cdot z$ determined by the right action of $e^{\H\da(X,\J)}$ on $\Dt^{\B}_A(X,\J)$, and similarly let $R_{\Psi(z)}:\Dt^{e^u\cdot\B}_A(X,e^u\cdot\J)\to \Dt^{e^u\cdot\B}_A(X,e^u\cdot\J)$ be the functor $\hB\mapsto \hB\cdot \Psi(z)$.
$\tilde{\Phi}\da^u\circ R_z=R_{\Psi(z)}\tilde{\Phi}\da^u$.
The result follows from an explicit calculation very similar to that used in the proof of Proposition \[right action par\].
Therefore, using the construction described in Proposition \[induced equivalence\], we may extend the functor $\tilde{\Phi}^u:\Dt^{\B}(X,\J)\to \Dt^{e^u\cdot\B}(X,e^u\cdot\J)$ to a functor $\Phi^u:\De^{\B}(X,\J)\to \De^{e^u\cdot\B}(X,e^u\cdot\J)$. Furthermore, since $\tilde{\Phi}^u$ is an equivalence, Proposition \[induced equivalence\] implies the following result.
The functor $\Phi^u:\De^{\B}(X,\J)\to \De^{e^u\cdot\B}(X,e^u\cdot\J)$ is an equivalence of formal groupoids.
Example: Leaf-wise Lagrangian branes {#LWL branes example}
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Let $(X,J)$ be a complex manifold, and $\varphi:X{\xymatrix@C=1.5pc{ \ar[r]^-{\cong} & }}X$ a diffeomorphism satisfying $\varphi^*J=J$, i.e. a symmetry of $(X,J)$. Given any complex submanifold $Z\subset X$, the submanifold $\varphi^{-1}(Z)\subset X$ will again be compatible with $J$: symmetries of $(X,J)$ act on its collection of complex submanifolds. Similarly, for every Artin algebra $A$ and every holomorphic vector field $\xi\in \m\otimes_{\R}\Cinf(TX)$, the action of the formal symmetry $e^{\xi}$ on $Z$ induces a deformation of $Z$, i.e. an element $\hat{Z}$ of $\df_Z(A)$. Since applying the inverse $e^{-\xi}$ to $\hat{Z}$ gives the trivial deformation (Z itself), we say that $\hat{Z}$ is trivializable. Of course, not every deformation is trivializable in this way; however, for each point $z\in Z$, we can find an open set $U\subset X$ containing $z$, so that every deformation of $Z\cap U$ in $(U,J|_U)$ is induced by a formal symmetry of $(U,J|_U)$.
Similarly, given a GC brane $\B\in (X,\J)$ and a formal symmetry $g=e^{\xx}\in e^{\cg\da(X)}$ for some $A\in \Art$, we saw in Proposition \[action of formal symmetries\] that $e^{\xx}$ induces an $A$-deformation $\B\cdot e^{\xx}$ of $\B$. Motivated by the complex case described above, a natural question is whether every $A$-deformation is locally isomorphic to one of this form.
\[definition locally extendible\] Let $\B=(Z,\L)\in \Br(X,\J)$ be a GC brane. We say that $\B$ has *trivializable* deformations if for each $A\in \Art$, and each $A$-deformation $\hB$ of $\B$, there exists a formal symmetry $e^{\xx}\in e^{\T\da(X)}$ such that $\hB$ is isomorphic to $\B\cdot e^{\xx}$ in $\Dt\da\uB(X,\J)$. Similarly, we say that $\B$ has *locally trivializable deformations* if for each $z\in Z$ there exists an open set $U\subset X$ containing $z$, such that $\B|_U\in\Br(U,J|_U)$ has trivializable deformations.
Examination of examples shows that–unlike in the complex case discussed above–not every GC brane has locally trivializable deformations. However, in this section we will prove the following result.
\[essentially surjective\] Let $\B\in\Br(X,\J)$ be a leaf-wise Lagrangian brane (that is, a brane whose underlying GC submanifold is leaf-wise Lagrangian as in Definition \[definition LWL submanifold\]). Then $\B$ has locally trivializable deformations.
Let $\B=(Z,\L)$ be a LWL brane, and $z\in Z$. Given a neighborhood $U\subset X$ of $z$ and a 1-form $u\in\Omega^1(U)$, let $\J'=e^{-u}\J|_U$ and $\B'=\B|_U\cdot e^u$. By Proposition \[action on GC submanifolds\], we have $\B'\in\Br(U,\J'|_U)$. Suppose that for every $A\in\Art$ and $\hB'\in\Dt^{\B'}_A(U,\J')$ we can find $x'\in e^{\T\da(U,\J')}$ such that $\hB$ is isomorphic to $\B'\cdot x$. Then by applying Proposition \[B transform on deformations\], together with the fact that $$e^{u}e^{\T\da(U,\J')}e^{-u}=e^{\T\da(U,\J|_U)},$$ it follows that same is true when $\B'$ is replaced by $\B$ and $\J'$ is replaced by $\J$. Using Theorem \[LWL normal form\], it is therefore sufficient to consider the case where $X$ is a neighborhood of the origin in $X^{m,n}_0=\R^{2m}\times\C^{n}$, that $Z=Z^k_0\cap X$ for natural numbers $m,n,k$ (with $k\leq n$), and where the curvature form $F$ of $\L$ is zero; here $Z^k_0\subset X_0$ is submanifold described in Example \[standard brane\]. Furthermore, by applying Propositions \[invariance 1\] and \[invariance 2\], it is enough to consider the case that $\L$ is the trivial element of $\Herm(Z)$.
The rest of the proof will be similar to the proof of Theorem \[LWL normal form\], and we begin by recalling some of the notation introduced there. We introduce coordinates $(s_0,\cdots, s_{2m},t_0,\cdots,t_{2n})$ on $X_0=\R^{2m}\times \C^n$, where $(s_0,\cdots,s_{2m})$ are coordinates on $\R^{2m}$ and $(t_0,\cdots, t_{2n})$ are (real) coordinates on $\C^n\cong\R^{2n}$. We then relable these to obtain new coordinates $(x^1,\cdots x^{d},y^1,\cdots y^{d'})$, with $d=m+k$ and $d'=m+2n-2k$, defined by $$x_1=s_1,\cdots x_m=s_m,x_{m+1}=t_1,\cdots, x_{m+2k}=t_{2k},$$ and $$y_1=s_{m+1}\cdots y_m=s_{2m}, y_{m+1}=t_{2k+1}\cdots y_{m+2n-sk}=t_{2n}.$$ In terms of these coordinates, $Z$ is defined by the equations $y^I=0$ for $I=1,\cdots, d'$.
Let $\hB=(\hrho,\hL)$ be an object of $\De^{\B}_A(U,\J)$. Let $\pi:X\to Z$ be the projection map $(x,y)\mapsto x$, and $\pi^*:\Cinf_{Z}\to \Cinf_{X}$ the pull-back homomorphism.
There exists a unique $\tau\in \g\da(Z)$ such that $$\hrho\circ\pi^*=e^{\tau}.$$
Suppose that $\mu:A\to A'$ is a small extension in $\Art$ with kernel $I$, and let $\sigma:A'\to A$ be an $\R$-linear splitting of $\mu$. Inductively, assume that the result holds for the Artin algebra $A'$. Write $\Phi=\hrho\circ\pi^*:\Cinf\da(Z)\to \Cinf\da(Z)$, and let $\Phi'=\mu(\Phi):\Cinf_{A'}(Z)\to \Cinf_{A'}(Z)$ , then exists a unique $\tau'\in \g_{A'}(Z)$ such that $e^{\tau'}=\Phi'$. Let $\tilde{\tau}=\sigma(\tau')$, i.e. for each $f\in \Cinf\da(Z)$ we have $$\tilde{\tau}(f)=\sigma(\tau'\mu(f)).$$ Define $\eta\in I\otimes_{\R}\End_{\R}(\Cinf(Z))$ such that for every $f\in\Cinf\da(Z)$ we have $$e^{\tilde{\tau}}(f)=\Phi(f)-\eta(f).$$ Define $\tau=\tilde{\tau}+\eta$. Noting that $I\otimes\End_{\R}(\Cinf(Z))$ lies in the center of $A\otimes_{\R}\End_R(\Cinf(Z))$, we see that $$\begin{aligned}
e^{\tau}f & = e^{\eta}e^{\tilde{\tau}}(f) \\
& = e^{\eta}(\Phi(f)-\eta(f)) \\
& = \Phi(f)-\eta(f)+\eta(\Phi(f)-\eta(f)) \\
& =\Phi(f)-\eta(f)+\eta(f)\\
& = \Phi(f).\end{aligned}$$ Furthermore, we claim that $\eta$–and therefore $\tau$–is a derivation. To see this, note that by Proposition \[action\] $\tilde{\Phi}=e^{\tilde{\tau}}$ is a homomorphism (actually automorphism) of $R$-algebras. Therefore we have $$\begin{aligned}
\Phi(f)\Phi(g)& =(\tilde{\Phi}(f)-\eta(f))(\tilde{\Phi}(g)-\eta(g)) \\
& = \tilde{\Phi}(f)\tilde{\Phi}(g)-f\eta(g)-g\eta(f) \\
& = \tilde{\Phi}(fg)-f\eta(g)-g\eta(f).\end{aligned}$$ By comparing this with the equation $$\Phi(fg)=\tilde{\Phi}(fg)-\eta(fg),$$ we see that indeed $\eta(fg)=f\eta(g)+\eta(f)g$ for every $f,g\in\Cinf\da(X)$.
Therefore $e^{-\tau}\hrho\circ \pi^*$ is the identity. Since $\hB$ is isomorphic in $\Dt^{\B}\da(U,\J)$ to $(e^{-\tau}\hrho,e^{-\tau}\hat{\L})$, we may assume without loss of generality that $$\label{projection} \hrho\circ\pi^*=id_{\Cinf\da(Z)}.$$
In terms of the coordinates $(x,y)$, this means that $$\hrho(x^i)=x^i$$ and $$\hrho(y^I)\in \Cinf\dm(Z).$$ Let us write $\phi^I=\hrho(y^I)\in \Cinf\dm(Z)$, and define $$\xi=(\pi^*\phi^I)\frac{\partial}{\partial y^I}\in \g\da(X).$$
$\hrho=\rho e^{\xi}$.
By definition, there exists some $\eta\in \g\da(X)$ such that $\hrho=\rho e^{\eta}.$ Defining $\zeta\in\cg\da(X)$ by $e^{\zeta}=e^{\eta}e^{-\xi}$, we see that the desired result is equivalent to the equation $$\rho e^{\zeta}=\rho.$$ Introducing the notation $\{w^{\alpha}\}$ for the collective coordinates $\{x^i,y^I\}$ on $U\subset X_0$, by assumption we have $\rho e^{\zeta}w^{\alpha}=\rho w^{\alpha}$ for each $\alpha$; we want to show that this implies the identity $\rho e^{\zeta}f=\rho f$ for arbitrary $f\in\Cinf\da(X)$. Let us expand $\zeta=\zeta^{\alpha}\frac{\partial}{\partial w^{\alpha}}$, noting that the component functions $\zeta^{\alpha}$ are elements of $\Cinf\dm(U)$. For every function $f\in\Cinf\da(X)$, we have $$\rho(\zeta(f))=\rho(\zeta^{\alpha})\rho(\frac{\partial}{\partial w^{\alpha}}f);$$ in particular, if we can show that each component function $\zeta^{\alpha}$ satisfies $\rho(\zeta^{\alpha})=0$, then $\rho(\zeta(f))=0$ for every $f\in\Cinf\da(X)$; in this case we say that the *restriction of $\zeta$ to $Z$ vanishes*. Inductively, for each $k\geq 1$ we then have $\rho(\zeta^k(f))=0$ for every $f\in\Cinf\da(X)$, which in turn implies that $\rho e^{\zeta}=\rho$. Therefore, it is sufficient to show that, if $\zeta$ is any element of $\g\da(X)$ such that $\rho e^{\zeta}w^{\alpha}=\rho w^{\alpha}$ holds for each $\alpha$, then each of the component functions $\zeta^{\alpha}$ satisfies $\rho(\zeta^{\alpha})=0$ (equivalently, the restriction of $\zeta$ to $Z$ vanishes).
Inductively, assume that this result holds for some $A'\in\Art$, and let $\mu:A\to A'$ is a small extension with kernel $I$. Define $\zeta'=\mu(\zeta)\in\g_{A'}(X)$. Since $\mu(w^{\alpha})=w^{\alpha}$ and $\mu(\rho)=\rho$, it follows that $\rho e^{\zeta'}w^{\alpha}=\rho w^{\alpha}$ for each $\alpha$, so by the inductive hypothesis the restriction of $\zeta'$ to $Z$ vanishes. Let $\sigma:A'\to A$ be a linear splitting of $\mu$, and define $\lambda=\zeta-\sigma(\zeta')$, which by construction is an element of $I\otimes_{\R}\Cinf(TX)$. It is easy to see that the vanishing of $\zeta'$ along $Z$ implies that $\sigma(\zeta')$ vanishes along $Z$ as well, so that $\rho e^{\sigma(\zeta')}=\rho$. For each $f\in\Cinf\da(X)$ we therefore have $$\rho e^{\zeta}(f)=\rho(e^{\sigma(\zeta')}f+\lambda(f))=\rho f+\rho\lambda (f).$$ Expanding $\lambda=\lambda^{\alpha}\frac{\partial}{\partial w^{\alpha}}$, and noting that the component functions satisfy $\lambda^{\alpha}=\lambda(w^{\alpha})$, we then have $$\rho(\lambda^{\alpha})=\rho e^{\zeta}w^{\alpha}-\rho w^{\alpha}=0.$$ Therefore $\lambda$ vanishes along $Z$, so that $\zeta=\sigma(\zeta')+\lambda$ does as well.
Let us write $\hL=\L+u=e^{-u}\L$ for $u\in \Omega^1\dm(Z)$. Define $w=\pi^*u\in \Omega^1\dm(X)$, and $\xx=(\xi,w)$. The proof of Theorem \[essentially surjective\] then follows from the following lemma.
\[useful lemma\]
1. $\hB=\B\cdot e^{\xx},$ and
2. $\xx\in \T\da(X)$.
As in the proof of Theorem \[LWL normal form\], we introduce $R\subset \TTX$ as the sub-bundle spanned by $\{(\frac{\partial}{\partial y^I},0),(0,dx^i)\}_{i,I}$ and $S\subset \TTX$ as the sub-bundle spanned by $\{(\frac{\partial}{\partial x^i},0),(0,dy^I)\}_{i,I}$. In that proof, we saw that
1. $\TTX=R\oplus S$. Furthermore, both $R$ and $S$ are maximal isotropic so the pairing gives an identification $S\cong R\uv$ and $R\cong S\uv$.
2. $\J(R)=R$ and $\J(S)=S$.
3. $S|_{Z}=\TT \B$.
We also introduced the space $\Y$ of vector fields on $X_0$ of the form $\sum_{I}c^I\frac{\partial}{\partial y^I}$ for constants $\{c^I\}_I$, and defined $\RR\subset \Cinf(R)$ and $\SS\subset \Cinf(S)$ to be the subspaces of elements $\xx\in \Cinf(R),\Cinf(S)$ satisfying $(\xi,0)\cdot\xx=0$ for all $\xi\in \Y$. Via straightforward calculation, we proved the following lemma, which we state here again for convenience.
\[RSlemma\]
1. $\ll\RR,\SS\rr\subset \RR,$
2. $\ll\RR,\RR\rr=0.$
3. with respect to the isomorphism $\End(\TTX)\cong (R\oplus S)\otimes (R\oplus S)$ induced by the pairing, we have $$\J\in \RR\otimes\SS\oplus\SS\otimes\RR.$$
By construction, both $(\xi,0)$ and $(0,u)$ are elements of $\RR$, so that we have $$0=\ll(\xi,0),(0,u)\rr=(0,\pounds(\xi)u).$$ Using Proposition \[exponential\], we see that $$e^{\xx}=e^{(0,u)}e^{(\xi,0)},$$ from which the first part of Lemma \[useful lemma\] easily follows.
It remains to show that $$\xx\cdot \J=0.$$
Since $\xx\in \RR$, it follows from Lemma \[RSlemma\] that $\xx\cdot\J\in \RR\otimes\RR$, and also that $e^{-\xx}\cdot\J=\J-\xx\cdot\J$. Since the pairing identifies $R$ with $S\uv$, we see that $\xx\cdot \J$ vanishes if and only if $$\la(\xx\cdot\J)\SS,\SS\ra = 0;$$ this is equivalent to $$\la(e^{-\xx}\cdot\J)\SS,\SS\ra =0,$$ which is in turn equivalent to $$\la \J e^{\xx}(\SS),e^{\xx}\SS)\ra =0.$$ Since $\SS\subset K^{\B}$, it follows from the discussion in Remark \[rem1\] that $$\la \J e^{\xx}(\SS),e^{\xx}\SS)\ra\subset I^{Z}.$$ On the other hand, for each $x,y\in\SS$ the function $\la \J e^{\xx}(x),e^{\xx}y)\ra\subset I^{Z}$ is independent of the $y^I$ coordinates, so if it vanishes on $Z$ it vanishes on all of $X$. We therefore conclude that $$\la(e^{-\xx}\cdot\J)\SS,\SS\ra =0,$$ which finishes the proof of the lemma.
Induced Deformations {#induced deformations}
====================
In this section, we continue the study of induced (trivializable) deformations.
Fix a GC brane $\B\in\Br(X,\J)$. For each $A\in\Art$ and $x\in e^{\T\da(X)}$, we introduce the notation
$$\Sigma\da\uB(x)=\B\cdot x\in\Dep.$$ In this section we will define a formal groupoid $\De\uB(X,\J)^{tr}$, whose objects for $A\in\Art$ are the elements of $e^{\T\da(X)}$. We then extend the map $x\mapsto \Sigma\da\uB(x)$ to a functor of formal groupoids $$\Sigma\uB:\De\uB(X,\J)^{tr}\to \De\uB(X,\J),$$ which we prove is fully faithful; by construction, it is essentially surjective (and hence an equivalence) precisely when $\B$ has trivializable deformations.
In order to define the groupoid $\De\uB(X,\J)^{tr}$ we will need some preliminary definitions.
Let $\rrr(Z)$ be the following Lie algebra: as a vector space $\rrr(Z)=\Cinf(TZ)\oplus \Cinf(Z)$, and the bracket is given by $$\label{r bracket}[(\xi,f),(\eta,g)]=([\xi,\eta],\xi(g)-\eta(f)+\iota(\eta)\iota(\xi)F).$$
The Jacobi identity for the bracket (\[r bracket\]) follows from the fact that $F$ is closed.
Consider the formal group $e^{\rrr(Z)}$. For each $A\in\Art$, by Proposition \[exp prop\] we see that for every $\xi\in\g\da(Z)$ and every $f\in\Cinf\dm(Z):=\m\otimes_A\Cinf(Z)$ we have $$\label{r multiplication}e^{(\xi,0)}e^{(0,f)}=e^{(0,e^{\xi}f)}e^{(\xi,0)}.$$ We also have the following result; the proof is identical to the proof of Proposition \[exponential\], and is therefore omitted.
\[exponential r\] For each $A\in\Art$ and $(\xi,f)\in\rrr\da(Z)$, we have $$e^{(\xi,f)}=e^{(0,\int_0^1e^{t\xi}fdt)}e^{(\xi,0)}.$$
The map $$\mu:\rrr(Z)\to \cg(Z)$$ given by $$(\xi,f)\mapsto (\xi,\iota(\xi)F-df)$$ is a Lie algebra homomorphism.
Along with the Cartan formula $\pounds(\xi)=d\iota(\xi)+\iota(\xi)d$, recall also the identity $\iota([\xi,\eta])F=[\pounds(\xi),\iota(\eta)]F$. Using these, we calculate $[\mu(\xi,f),\mu(\eta,g)]$ to be $$\begin{aligned}
& [(\xi,\iota(\xi)F-df),(\eta,\iota(\eta)F-dg)] \\
= & ([\xi,\eta],\pounds(\xi)\iota(\eta)F-\pounds(\xi)dg-\pounds(\eta)\iota(\xi)F+\pounds(\eta)df)\\
= & ([\xi,\eta],\iota([\xi,\eta])F+\iota(\eta)\pounds(\xi)F-d\iota(\xi)dg-\pounds(\eta)\iota(\xi)F+d\iota(\eta)df)\\
= & ([\xi,\eta],\iota([\xi,\eta])F+\iota(\eta)d\iota(\xi)F-d\iota(\eta)\iota(\xi)F-\iota(\eta)d\iota(\xi)F-d(\xi\cdot g-\eta\cdot f))\\
= & ([\xi,\eta],\iota([\xi,\eta])F-d(\xi\cdot g-\eta\cdot f+\iota(\eta)\iota(\xi)F)) \\
=& \mu(([\xi,\eta],\xi\cdot g-\eta\cdot f+\iota(\eta)\iota(\xi)F)) \\
= & \mu([(\xi,f),(\eta,g)]).\end{aligned}$$
Let $\B$ be a GC brane on $(X,\J)$, with underlying GC submanifold $(Z,F)$. Denote by $\T^{\B}(X)$ the Lie sub-algebra of $\T(X)$ consisting of elements $\xx=(\xi,u)\in\T(X)$ such that $\xi$ is tangent to $Z$. This Lie algebra comes equipped with a restriction homomorphism $$\rho:\T^{\B}(X)\to \cg(Z).$$
\[definition KKK\] Let $\KKK(X)$ be the fiber-product of Lie algebras $$\KKK(X)=\T^{\B}(X)\times_{\cg(Z)} \rrr(Z).$$ In other words, an element of $\KKK(X)$ is a pair $(\xx,\hat{\tau})$, where $\xx=(\xi,w)\in \T\uB(X)$, $\hat{\tau}=(\tau,f)\in\rrr(Z)$, such that $\rho(\xi)=\tau$ and $\rho(w)=\iota(\tau)F-df.$ The Lie bracket is defined component-wise.
By construction, $\KKK(X)$ comes equipped with three homomorphisms:
1. $\chi:\KKK(X)\to \T(X)$
2. $\rho_1:\KKK(X)\to \cg(Z)$ and
3. $\rho_2:\KKK(X)\to \rrr(Z)$,
where $\rho_1=\mu\rho_2$. We will denote the group homomorphisms induced by these Lie algebra homomorphisms by the same symbols.
\[definition tilde extended\] Let $\Dt\uB(X,\J)^{tr}$ be the following formal groupoid: For each $A\in\Art$, the objects of $\Dt\da\uB(X,\J)^{tr}$ are elements of the group $e^{\T\da(X)}$. A morphism from $x\in e^{\T\da(X)}$ to $x'\in e^{\T\da(X)}$ is an element $y\in e^{\KKK\da(X)}$ such that $$x'=\chi(y)x.$$ Given $y,y'\in e^{\KKK\da(X)}$ and $x,x',x''\in e^{\T\da(X)}$ such that $x'=\chi(y)x$ and $x''=\chi(y')x'$, the composition $$(\xymatrix{x' \ar[r]^{y'} & x'')}\circ (\xymatrix{ x \ar[r]^y & x'})$$ is defined by $$\xymatrix{x \ar[r]^{y'y} & x''},$$ where we note that $$x''=\chi(y')x'=\chi(y')\chi(y)x=\chi(y'y)x.$$ For every $x\in e^{\T\da(X)}$, the identity morphism $x\to x$ is the group identity element $1_{e^{\KKK\da(X)}}\in e^{\KKK\da(X)}$.
For each $x\in e^{\T\da(X)}$, denote $$\widetilde{\Sigma}^{\B}\da(x)=\B\cdot x\in\Dt\uB\da(X,\J).$$ Given $x,x'\in e^{\T\da(X)}$ and $y\in e^{\K\da(X)}$ such that $x'=\chi(y)x$, we will construct $$\widetilde{\Sigma}^{\B}\da(y):\B\cdot x\to \B\cdot x'.$$ Write $x=e^{u}e^{\xi}$, $x'=e^{u'}e^{\xi'}$, $y=e^{(\yy,\hat{\tau})}$, where $\yy=(\eta,v)$ and $\hat{\tau}=(\tau,h)$. By Proposition \[exponential\], we have $$\chi(y)=e^we^{\eta},$$ where $$\label{zeroth condition}w=\int_0^1e^{t\eta}vdt.$$ The equation $x'=\chi(y)x$ implies that $$\label{first condition} e^{\xi'}=e^{\eta}e^{\xi},$$ and $$\label{second condition} u'=w+e^{\eta}u.$$ Furthermore, by definition of the Lie algebra $\KKK\da(X)$, we have $$\label{third condition} \rho e^{\eta}=e^{\tau}\rho.$$ and $$\label{fourth condition}\rho(v)=\iota(\tau)F-dh.$$
\[sigma morphism lemma\] $\widetilde{\Sigma}\uB\da(y)=(e^{\tau},\{g_I\})$ with $$\label{g definition} g_I:=\int_0^1(e^{t\tau}(\iota(\tau)a_I+h|_{W_I})dt$$ is an equivalence from $\widetilde{\Sigma}\uB\da(x)$ to $\widetilde{\Sigma}\uB\da(x')$ in $\Dt\da\uB(X,\J)$.
Write $\widetilde{\Sigma}\da\uB(x)=\B\cdot x=\hB=(\hrho,\hL)$ and $\tilde{\Sigma}\uB\da(x')=\B\cdot x'=\hB'=(\hrho',\hL')$ . According to Definition \[definition par def\], we need to check that $$\label{iso condition 1} \hrho'=e^{\tau}\hrho,$$ and that $\{g_I\}$ is an isomorphism in $\Def^{\L}_A(X)$ from $\hL'\to e^{\tau}\hL$. By construction, we have $\hrho=\rho e^{\xi}$ and $\hrho'=\rho e^{\xi'}$. Combining equation (\[first condition\]) with equation (\[third condition\]), we see that (\[iso condition 1\]) is indeed satisfied.
Next, note that by construction $$\hL'=(\{c\sij\},\{a_I+\rho(u')|_{W_I}\})$$ and $$\hL=(\{c\sij\},\{a_I+\rho(u)|_{W_I}\});$$ therefore, we also have $$e^{\tau}\hL=(\{e^{\tau}c\sij\},\{e^{\tau}a_I+e^{\tau}\rho(u)|_{W_I}\}).$$ According to Definition \[Herm def\], therefore, $\{g_I\}$ is an isomorphism in $\Def^{\L}_A(Z)$ from $\hL'\to e^{\tau}\hL$ if and only if we have $$\label{iso condition 2} g_J-g_I=e^{\tau}c_{IJ}-c_{IJ}$$ and $$\label{iso condition 3} dg_I=e^{\tau}a_I+e^{\tau}\rho(u)|_{W_I}-a_I-\rho(u')|_{W_I}.$$ By equation (\[g definition\]) we have $$\begin{aligned}
g_J-g_I & = \int_0^1e^{t\tau}(\iota(\tau)(a_J-a_I))dt \\
& = \int_0^1(e^{t\tau}(\iota(\tau)dc_{IJ})dt \\
& =\int_0^1(\pounds(\tau)e^{t\tau}c_{IJ}dt \\
& =\int_0^1\frac{d}{dt}e^{t\tau}c_{IJ}dt \\
& = e^{\tau}c_{IJ}-c_{IJ},\end{aligned}$$ which verifies condition (\[iso condition 2\]). To verify condition (\[iso condition 3\]), we calculate $$\begin{aligned}
dg_I & = \int_0^1e^{t\tau}(d\iota(\tau)a_I+dh_I)dt \\
& =\int_0^1e^{t\tau}(\pounds(\tau)a_I-\iota(\tau)da_I+\iota(\tau)F-\rho(v))dt\\
& =e^{\tau}a_I-a_I-\rho(w) \\
& = e^{\tau}a_I-a_I-\rho(u'-e^{\eta}u)\\
& = e^{\tau}a_I-a_I-\rho(u')+e^{\tau}\rho(u).\end{aligned}$$ Note that, in going from the first line to the second we used the Cartan formula for the Lie derivative, together with (\[fourth condition\]); in going from the second to the third we used that $da_I=F$, $e^{t\tau}\pounds(a_I)=\frac{d}{dt}e^{t\tau}a_I$, and (\[zeroth condition\]); in going from the third to the fourth we used (\[second condition\]); and in going from the fourth line to the last we used (\[third condition\]).
\[functoriality prop\] The map sending $x\in e^{\T\da(X)}$ to $\tilde{\Sigma}\uB\da(x)=\B\cdot x$ and $y\in e^{\K\da\uB(X)}$ to $\tilde{\Sigma}\uB\da(y)$ is functorial; that is, for every $x,x',x''\in e^{\T\da(X)}$ and $y,y'\in e^{\KKK\da(X)}$ such that $x'=\chi(y)x$ and $x''=\chi(y')x'$, we have $$\label{functoriality1}\tilde{\Sigma}\uB\da(y')\circ \tilde{\Sigma}\uB\da(y)=\tilde{\Sigma}\uB\da(y'y).$$
Suppose $x,x',x''\in e^{\T\da(X)}$ and $y,y',y''\in e^{\K\da(X)}$ satisfy $x'=\chi(y)x$ and $x''=\chi(y')x'$. Defining $y''=y'y$, we may rewrite the condtion (\[functoriality1\]) in the slightly altered form $$\label{functoriality}\tilde{\Sigma}\uB\da(y')\circ \tilde{\Sigma}\uB\da(y)=\tilde{\Sigma}\uB\da(y'').$$ Writing $\tilde{\Sigma}\uB\da(y)=(e^{\tau},\{g\si\})$, $\tilde{\Sigma}\uB\da(y')=(e^{\tau'},\{g\si'\})$, and $\tilde{\Sigma}\uB\da(y'')=(e^{\tau''},\{g''_I\})$, by Definition \[definition par def\] equation (\[functoriality\]) means that $$\label{fun cond 1} e^{\tau'}e^{\tau}=e^{\tau''},$$ and that $$\label{fun cond 2} g'\si+e^{\tau'}g\si=g''\si$$ holds for each $I$. Using formula (\[r multiplication\]), we see that the conditions (\[fun cond 1\]) and (\[fun cond 2\]) may be combined into the equations $$\label{group prop}e^{(0,g_I')}e^{(\tau_I',0)}e^{(0,g_I)}e^{(\tau_I,0)}=e^{(0,g''_I)}e^{(\tau_I'',0)}$$ in the groups $e^{\rrr\da(W_I)}$, where by definition we have $\tau_I:=\tau|_{W\si},\tau'\si:=\tau'|_{W\si},$ and $\tau''\si:=\tau''|_{W\si}$ .
Define $h\si:=h|_{W\si}$, $h'\si:=h'|_{W\si}$, and $h''\si:=h''|_{W\si}$, and also $f_I=\iota(\tau\si)a_I+h\si$, $f'_I=\iota(\tau'\si)a_I+h'\si$, and $f''_I=\iota(\tau''\si)a_I+h''\si$. Using the definition (\[g definition\]) together with Lemma \[exponential r\], we see that (\[group prop\]) may be rewritten as $$\label{group prop 2}e^{(\tau'\si,f'_I)}e^{(\tau\si,f_I)}=e^{(\tau''\si,f''_I)}.$$ By hypothesis, we have $y''=y'y$; applying the homomorphism $\rho_2:e^{\KKK\da(X)}\to e^{\rrr\da(Z)}$ and restricting to $W_I$, this implies that $$\label{group prop 3}e^{(\tau'\si,h\si')}e^{(\tau\si,h\si)}=e^{(\tau\si'',h\si'')}$$ holds in $e^{\rrr\da(W_I)}$. On the other hand, if we define $\sigma_I:\rrr\da(W_I)\to \rrr\da(W_I)$ by $$(\zeta,l)\mapsto (\zeta,l+\iota(\zeta)a_I),$$ we see that $(\tau\si,f_I)=\sigma_I(\tau\si,h\si)$, $(\tau',f'_I)=\sigma_I(\tau'\si,h'\si)$ and $(\tau'',f''_I)=\sigma_I(\tau'',h''\si)$. It is easy to check that $\sigma_I$ is a homomorphism of Lie algebras; therefore equation (\[group prop 2\]) follows from (\[group prop 2\]) by applying the group homomorphism $e^{\sigma_I}$.
Using Lemma \[sigma morphism lemma\] and Proposition \[functoriality prop\] in hand, we may now give the following definition.
\[definition sigma tilde\] The functor $\widetilde{\Sigma}\uB:\De\uB(X,\J)^{tr}\to\Dep$ is defined as follows. For each $A\in\Art$ and each $x\in e^{\T\da(X)}$, we have $$\widetilde{\Sigma}\uB\da(x)=\B\cdot x.$$ On morphisms $\widetilde{\Sigma}\uB\da$ is defined as described in Lemma \[sigma morphism lemma\].
\[fully faithful\] The functor $\widetilde{\Sigma}\uB$ is fully faithful.
We first prove that $\widetilde{\Sigma}\uB$ is faithful. Given $A\in\Art$, let $x,x'$ be elements of $e^{\T\da(X)}$, and let $y,y'\in e^{\KKK\da(X)}$ be elements satisfying $x'=\chi(y)x$ and $x'=\chi(y')x$. In particular, this implies that $\chi(y)=\chi(y')$. Let us write $y=e^{((\eta,v),(\tau,h))}$, where $(\eta,v)\in\T\da(X)$ and $(\tau,h)\in \KKK\da$, such that $\tau=\rho(\eta)$ and $\rho(v)=\iota(\tau)F-dh$. Then $y'$ is of the form $e^{((\eta,v),(\tau,h'))}$, where $\rho(v)=\iota(\tau)F-dh'$. By construction, we have $\widetilde{\Sigma}(y)=\widetilde{\Sigma}(y')$ if and only if, on each $W_I$ we have $$\int_0^1e^{t\tau}(\iota(\tau)a_I+h|_{W_I})dt=\int_0^1e^{t\tau}(\iota(\tau)a_I+h'|_{W_I})dt.$$ By Lemma \[exp formula cor\], this holds if and only if $h|_{W_I}=h'|_{W_I}$ for each $I$, or equivalently if $h=h'$. This in turn holds if and only if $y=y'$. Therefore we see that $\widetilde{\Sigma}\uB\da$ is faithful.
To show that $\widetilde{\Sigma}\uB\da$ is full, suppose we are given $x,x'\in e^{\T\da(X)}$ and a morphism $(e^{\tau},\{g\si\})$ from $\widetilde{\Sigma}\uB\da(x)\to \widetilde{\Sigma}\uB\da(x')$. Define $\tilde{y}=x'x^{-1}.$ Writing $x=e^{u}e^{\xi}$, $x'=e^{u'}e^{\xi'}$, and $\tilde{y}=e^{w}e^{\eta}$, we have $e^{\eta}=e^{\xi'}e^{-\xi}$, and $w=u'-e^{\eta}u$. Writing $\B=(Z,(\{c_{IJ}\},\{a_I\}))$, we have $$\widetilde{\Sigma}\uB\da(x)=(\rho e^{\xi},(\{c_{IJ}\},\{a_I+\rho(u)|_{W_I}\})$$ and $$\widetilde{\Sigma}\uB\da(x')=(\rho e^{\xi'},(\{c_{IJ}\},\{a_I+\rho(u')|_{W_I}\}).$$ The fact that $(e^{\tau},\{g\si\})$ is an equivalence from $\widetilde{\Sigma}\uB\da(x)$ to $\widetilde{\Sigma}\uB\da(x')$ is then equivalent to the following three conditions:
(I) $\rho e^{\xi'}=e^{\tau}\rho e^{\xi}$, or equivalently $e^{\tau}\rho=\rho e^{\xi'}e^{-\xi}=\rho e^{\eta}$, which implies that $\eta$ is tangent to $Z$ and $\rho(\eta)=\tau$.
(II) $g_J-g_I=e^{\tau}c_{IJ}-c_{IJ}$ and
(III) $dg_I = e^{\tau}(a_I+\rho(u)|_{W_I})-(a_I+\rho(u')|_{W\si})
= e^{\tau}a_I-a_I-\rho(w)|_{W_I}.$
Write $\tilde{y}=e^{(\eta,v)}$, so that $$w=\int_0^1e^{t\eta}vdt$$ and therefore $$\rho(w)=\int_0^1e^{t\tau}\rho(v)dt,$$ we see from condition (III) that $$dg\si=\int_0^1e^{t\tau}(\pounds(\tau)a_I-\rho(v)|_{W_I})dt,$$ or equivalently $$\label{int equat} dg\si-d\int_0^1e^{t\tau}\iota(\tau)a_Idt=\int_0^1e^{t\tau}(\iota(\tau)F|_{W\si}-\rho(v)|_{W\si})dt.$$ Also write $e^{(0,g_I)}e^{\tau|_{W_I}}=e^{(\tau|_{W_I},k_I)}\in e^{\rrr\da(W_I)}$, so that $$g_I=\int_0^1e^{t\tau}k_Idt.$$ Defining $h_I=k_I-\iota(\tau)a_I$, equation (\[int equat\]) implies that $$d\int_0^1 e^{t\tau}h_Idt= \int_0^1e^{t\tau}(\iota(\tau)F|_{W_I}-\rho(v)|_{W_I})dt.$$ Therefore, by Lemma \[exp formula cor\] we have $$\label{Lie equation}dh_I=(\iota(\tau)F-\rho(v))|_{W_I}.$$ On the other hand, using condition (II), we see that on each overlap $W_{IJ}$ we have $$\begin{aligned}
\int_0^1e^{t\tau}(k_J-k_I)dt & = \int_0^1e^{t\tau}\iota(\tau)dc_{IJ}dt \\
& = \int_0^1 e^{t\tau}(\iota(\tau)a_J-\iota(\tau)a_I)dt,\end{aligned}$$ which implies (again using Lemma \[exp formula cor\]) that $h_J=h_I$, so there is a unique function $h\in\Cinf\dm(Z)$ such that $h_I=h|_{W_I}$. By equation (\[Lie equation\]), it follows that $((\eta,v),(\tau,h))$ is an element of $\KKK\da(X)$, and by construction $$\widetilde{\Sigma}\uB\da(e^{((\eta,v),(\tau,h))})=(e^{\tau},\{g_I\}).$$ This completes the proof that $\widetilde{\Sigma}\uB\da$ is full.
Given an open set $U\subset X$, denote $\Dt\uB(U,\J):=\Dt^{\B|_U}(U,\J|_U)$ and $\Dt\uB(U,\J)^{tr}:=\Dt^{\B|_U}(U,\J|_U)^{tr}$, as well as $\widetilde{\Sigma}\uB(U):=\widetilde{\Sigma}^{\B|_U}$.
\[equivalence of induced\] Let $\B=(Z,\L)$ be a $GC$ brane with locally trivializable deformations. Then for every $z\in Z$, there exists a neighborhood $U\subset X$ of $z$ such that $$\widetilde{\Sigma}\uB(U):\Dt^{\B}(U,\J)^{ex}\to \Dt^{\B}(U,\J)$$ is an equivalence. In particular, this applies to LWL branes (by Proposition \[essentially surjective\]).
Given $\B=(Z,\L)\in\Br(X,\J)$ with locally trivializable deformations and $z\in Z$, by definition there exists a neighborhood $U\subset X$ of $z$ such that for every $A\in\Art$ the functor $$\widetilde{\Sigma}\da^{\B}(U):\Dt^{\B}\da(U,\J)^{ex}\to \Dt^{\B}\da(U,\J)$$ is essentially surjective. By Proposition \[fully faithful\], it is also fully faithful, and therefore an equivalence.
The proof of the following Proposition, which we omit, is a straightforward consequence of Definition \[group action on category\] and Definition \[definition tilde extended\]
There is a strict right action of $e^{\H(X)}$ on $\Dt\uB(X,\J)^{tr}$ defined as follows: given $x\in e^{\T\da(X)}$ and $z\in e^{\H\da(X)}$, we define $x\cdot z$ to be the product $xz$ (recall that $e^{\H\da(X)}$ is a subgroup of $e^{\T\da(X)}$). Given a morphism $y:x\to x'$, we define $y\cdot z=y$, now regarded as a morphism from $xz\to xz$.
For the following definition, recall Definition \[definition action groupoid\].
Let $\De\uB(X,\J)^{tr}$ be the action groupoid $\Dt\uB(X,\J)^{tr}//e^{\H(X)}$. Explicitly, for $A\in\Art$ an object of $\Dee$ is an element of $e^{\T\da(X)}$. Given $x,x'\in e^{\T\da(X)}$, a morphism from $x$ to $x'$ is a pair $(y,z)\in e^{\KKK\da(X)}\times e^{\H\da(X)}$ such that $x'z=\chi(y)x$. Composition is given by group multiplication, i.e. $$(y',z')\circ (y,z)=(y'y,z'z).$$
It is clear that the functor $\widetilde{\Sigma}\uB:\Dt\uB(X,\J)^{tr}\to\Dt\uB(X,\J)$ is compatible with the right actions of $e^{\H\da(X)}$ on both formal groupoids. Therefore, it follows from Proposition \[induced equivalence\] that $\widetilde{\Sigma}^{\B}$ naturally extends to a functor $\Sigma\uB:\De\uB\da(X,\J)^{tr}\to\De\uB\da(X,\J)$. Furthermore, $\Sigma\uB$ is an equivalence if and only if $\widetilde{\Sigma}^{\B}$ is an equivalence. This leads to the following corollary.
Let $\B=(Z,\L)$ be a leaf-wise Lagrangian brane on a GC manifold $(X,\J)$ (or more generally a brane with locally trivializable deformations). Then for every $z\in Z$, there exists a neighborhood $U\subset X$ of $z$ such that $$\Sigma\uB(U):\De\uB(U,\J)^{ex}\to \De\uB(U,\J)$$ is an equivalence of formal groupoids.
Deformations via gluing
=======================
Let $\B$ be a GC brane on $(X,\J)$. As discussed in Remark \[brane def sheaf\], the formal groupoid $\Dt\uB(X,\J)$ extends in a natural way to a presheaf of formal groupoids on $X$. In particular, for any open cover $\U=\{U\sa\}$ of $X$, we may define the groupoid of *descent data* for $\Dt\uB$ with respect to $\U$. This is described explicitly in the following definition.
\[glued deformations\] Let $\Dt\uB(\U,\J)$ be the following formal groupoid. For each $A\in\Art$, an object of $\Dt\uB\da(\U,\U)$ is pair $\hB=(\{\hB\sa\},\{\Psi\sab\})$, where
1. each $\hB\sa$ is an object of $\Dt\uB\da(U\sa,\J)$,
2. each $\Psi\sab:\hB\sb{\xymatrix@C=1.5pc{ \ar[r]^-{\cong} & }}\hB\sa$ is an isomorphism, and
3. on triple overlaps we have $\Psi_{\alpha\gamma}=\Psi_{\alpha\beta}\Psi_{\beta\gamma}:\B_{\gamma}\to \B_{\alpha}$.
A morphism from $\hB=(\{\hB\sa\},\{\Psi\sab\})$ to $\hB'=(\{\hB'\sa\},\{\Psi'\sab\})$ is a collection $\{\Phi\sa:\hB\sa\to \hB'\sa\}$ such that on each overlap we have $\Psi'_{\alpha\beta}\Phi_{\beta}=\Phi_{\alpha}\Psi_{\alpha\beta}$.
By construction, there is a natural restriction functor $$\tilde{R}\uB:\Dt\uB(X,\J)\to \Dt\uB(\U,\J).$$ Explicitly, given $A\in|Art$ and $\hB\in\Dtp$, we define $\tR(\hB)=(\{\hB\sa\},\{\Psi\sab\})$ by setting $\hB\sa=\hB|_{U\sa}$ for each $\alpha$, and $\Psi\sab=\textrm{id}_{\hB|_{U\sab}}$ on each $U\sab$. Given a morphism $\Phi:\hB\to\hB'$, we define $$\tR(\Phi)=\{\Phi|_{U\sa}\}:\tR(\hB)\to\tR(\hB').$$
\[descent property\]The restriction functor $\tilde{R}\uB:\Dt\uB(X,\J)\to \Dt\uB(\U,\J)$ is an equivalence of formal groupoids.
Fix $A\in\Art$. By inspection, it is clear that $\tilde{R}\uB$ is faithful, i.e. for each pair of objects $\hB=(\hrho,\hL)$ and $\hB'=(\hrho',\hL')$ in $\Dt^{\B}\da(X,\J)$ the induced map $$\Hom_{\Dt^{\B}\da(X,\J)}(\hB,\hB')\to \Hom_{\Dt_A^{\B}(\U,\J)}(\tilde{R}\uB\da(\hB),\tilde{R}\uB\da(\hB'))$$ is injective. On the other hand, suppose we are given a morphism $\{\Phi\sa\}:\tilde{R}\uB\da(\hB)\to \tilde{R}\uB\da(\hB')$. By definition, on each open set $U\sa$ we have an equivalence $\Phi\sa:\hB|_{U\sa}\to \hB'|_{U\sa}$, and on each overlap we have $\Phi\sa|_{U\sab}=\Phi\sb|_{U\sab}$. It is straightforward to see from Definition \[definition par def\] that this implies the existence of $\Phi:\hB\to \hB'$ whose restriction to each $U\sa$ is equal to $\Phi\sa$. Thus we see that $\tR$ is fully faithful.
To finish the proof, we will show that $\tR$ is also essentially surjective. Given $(\{\hB\sa\},\{\Psi\sab\})\in \Dt^{\B}_A(\U,\J)$, write $\Psi\sab=(e^{\tau\sab},\psi\sab)$. In particular, part (3) of Definition \[glued deformations\], implies that on each $Z\cap U_{\alpha\beta\gamma}$ we have $$e^{\tau_{\alpha\beta}}e^{\tau_{\beta\gamma}}=e^{\tau_{\alpha\gamma}}.$$
\[nonabelian coboundary\] There exist $\{\sigma\sa\in \Cinf\dm(Z\cap U\sa)\}$ such that on $Z\cap U\sab$ we have $$e^{\tau\sab}=e^{-\sigma\sa}e^{\sigma\sb}.$$
By Proposition \[induct on small extensions\], we may proceed by induction on small extensions. Thus, suppose the result holds for some $A'\in\Art$ with unique maximal ideal $\m'$, and let $\mu:A\to A'$ be a small extension. Denote the kernel of $\mu$ by $I$. Choose a linear splitting $\nu:A\to A'$ of $\mu$. Since $e^{\tau_{\alpha\beta}}e^{\tau_{\beta\gamma}}=e^{\tau_{\alpha\gamma}}$ holds in $e^{\g_{\m}(U_{\alpha\beta\gamma})}$, we also have $e^{\mu(\tau_{\alpha\beta})}e^{\mu(\tau_{\beta\gamma})}=e^{\mu(\tau_{\alpha\gamma})}$ in $e^{\g_{\m'}(U_{\alpha\beta\gamma})}$. By the inductive hypothesis, we can find $\{\tilde{\sigma}\sa\in \g_{\m'}(U_{\alpha})\}$ such that $$e^{\mu(\tau_{\alpha\beta})}=e^{-\tilde{\sigma}\sa}e^{\tilde{\sigma}\sb}.$$ We must then have $$\label{almost}e^{-\nu(\tilde{\sigma}\sa)}e^{\nu(\tilde{\sigma}\sb)}=e^{\tau\sab+\zeta_{\alpha\beta}}$$ for some $\zeta_{\alpha\beta}\in I\otimes \Cinf(T(Z\cap U_{\alpha\beta})).$ Since $I\otimes \Cinf(T(Z\cap U_{\alpha\beta})$ lies in the center of $\g_{\m}$, it is easy to see from equation (\[almost\]) that on triple overlaps $\zeta_{\beta\gamma}-\zeta_{\alpha\gamma}+\zeta_{\alpha\beta}=0$, so that we can choose $\eta_{\alpha}\in I\otimes \Cinf(T(Z\cap U_{\alpha}))$ satisfying $\eta_{\beta}-\eta_{\alpha}=-\zeta_{\alpha\beta}$. Setting $\sigma_{\alpha}=s(\tilde{\sigma}\sa)+\eta\sa$, it follows from equation (\[almost\]) that $$e^{-\sigma\sa}e^{\sigma\sb}=e^{\tau\sab}.$$
Returning to the proof of Proposition \[descent property\], choose $\{\sigma\sa\}$ as in Lemma \[nonabelian coboundary\], and define $\hrho'\sa=e^{\sigma\sa} \hrho\sa$, $\hL'\sa=e^{\sigma\sa}\hL\sa$, and
$$\hB'\sa=(\hrho'\sa,\hL'\sa).$$ Note that, by Lemma \[compatibility under isomorphism\], for each $\alpha$ we have $\hB'\sa\in \Dt^{\B}_A(U\sa,\J)$.
By construction, we have equivalences $$\Phi\sa:=(e^{\sigma\sa},id_{\hL'\sa}):\hB\sa\to \hB'\sa.$$
If we define $\Psi'_{\alpha\beta}=\Phi\sa \Psi_{\alpha\beta}\Phi^{-1}\sb$, then by construction $\hB'=(\{\hB'\sa\},\{\Psi'\sab\})\in \Dt^{\B}_A(\U,\J)$, and $\Phi=(\{\Phi\sa\})$ defines an isomorphism in $\Dt^{\B}_A(\U,\J)$ from $\hB\to \hB'$. Furthermore, by construction we have $$\begin{aligned}
\Psi'_{\alpha\beta} & =(e^{\sigma\sa},id_{e^{\sigma\sa\cdot} \hL\sa})(e^{\tau\sab},\psi\sab)(e^{-\sigma\sb},id_{e^{-\sigma\sb}\cdot \hL\sb})\\
& = (e^{\sigma\sa}e^{\tau\sab}e^{-\sigma\sb},\psi'\sab)\\
& = (1,\psi'\sab), \end{aligned}$$ where $$\psi'\sab=(e^{\sigma\sa}\cdot\psi\sab):\hL'\sa\to\hL'\sb.$$ In particular, on the overlaps $Z\cap U\sab$ we have $\hrho'\sa=\hrho'\sb$, so there exists a unique $\hrho:\Omega\ub\da(X)\to \Omega\ub\da(Z)$ such that each $\hrho'\sa$ is given by restricting $\hrho$.
For each $\alpha$ write $\hB'_{\alpha}=(\hrho\sa,\hL\sa)$. Using Proposition \[pi zero bundle\], we may assume without loss of generality that each $\hL\sa$ is of the form $(\{(c\sa)_{IJ}\},\{(\ha\sa)\si\})$, i.e. has undeformed transition functions $\{(\hc\sa)\sij=c_{IJ}|_{U\sa}\}$. Therefore, if we write $\Psi'\sab=(1,\{(g\sab)\si\})$, it follows from Definition \[definition par def\] and Definition \[Herm def\] that on each $(U\sab\cap W_I)\cap (U\sab\cap W_J) $ we have $(g\sab)_I=(g\sab)_J$, so that there is a well-defined function $g\sab:Z\cap U\sab\to \R$ whose restriction to each $U\sab\cap W\si$ is equal to $(g\sab)_I$. Furthermore, we see that on $U_{\alpha\beta\gamma}\cap Z$ we must have $$g_{\alpha\beta}+g_{\beta\gamma}=g_{\alpha\gamma},$$ so we may choose $h_{\alpha}\in\Cinf\dm(Z\cap U\sa)$ such that $g\sab=h_{\beta}-h_{\alpha}$. If we define $\{(\ha''\sa)_I=(\ha'\sa)_I-d(h\sa)|_{W\si}\}$, $\hL''\sa = (\{c_{IJ}\},\{(\ha''\sa)\si\}),$ and $$\hB''\sa=(\hrho\sa,\hL''\sa),$$ then $\hB''=(\{\hB''\sa\},\{\Psi''\sab=id\})$ defines an element of $\Dt^{\B}\da(\U,\J)$, which by construction is isomorphic to $\hB'$, and hence also to $\hB$. Also by construction, there exists a unique $\hB_0\in \Dt^{\B}_A(X)$ such that $\tR(\hB_0)=\hB''$. This completes the proof that $\tR$ is essentially surjective.
We saw in Proposition \[right action par\] that there is a strict right action of the formal group $e^{\T(X)}$ on $\Dt\uB(X,\J)$. Similarly, for each open set $U\subset X$, we have a strict right action of $e^{\T(X)}$ on $\Dt\uB(U,\J)$ constructed using the restriction homomorphism $e^{\T(X)}\to e^{\T(U)}$ together with the action of $e^{\T(U)}$ on $\Dt\uB(U,\J)$. We then have the following easy result, the proof of which is omitted.
There is a strict right action of $e^{\T(X)}$ on $\Dt\uB(\U,\J)$ defined as follows: for each $A\in\Art$, given $g\in e^{\T\da(X)}$ and $\hB=(\{\hB\sa\},\{\Psi\sab\})\in \Dt\uB\da(\U,\J)$, we define $$\hB\cdot g=(\{\hB\sa\cdot g|_{U\sa}\},\{\Psi\sab\cdot g|_{U\sab}\}).$$ Similarly, given an isomorphism $\{\Phi\sa\}:\hB\to\hB'$ in $\Dt\uB\da(\U,\J)$, we define $$\{\Phi\sa\}\cdot g=\{\Phi\sa\cdot g\}.$$
Let $\De^{\B}(\U,\J)$ be the formal groupoid $$\Dt\uB(\U,\J)//e^{\H(X)}.$$ Explicitly, for each $A\in\Art$ $\Dt\uB\da(\U,\J)$ has the same objects as $\Dt\uB\da(\U,\J)$. A morphism from $\hB\to \hB'$ in $\De\uB\da(\U,\J)$ is a pair $(\Phi,z)$, where $z\in e^{\H\da(X)}$, and $\Phi$ is a morphism in $\Dt\uB\da(\U,\J)$ from $\hB$ to $\hB'$. Composition is given by $(\Phi',z')(\Phi,z)=(\Phi'\Phi,z'z)$.
Clearly, the restriction functor $$\tilde{R}\uB:\Dt\uB(X,\J)\to \Dt\uB(\U,\J)$$ is compatible with the actions of $e^{\H(X)}$ on both formal groupoids. Using Proposition \[induced equivalence\], we may then extend $\tilde{R}\uB$ to a functor $$R\uB:\De\uB(X,\J)\to \De\uB(\U,\J).$$ Furthermore, combining Proposition \[induced equivalence\] and Proposition \[descent property\], we arrive at the following result.
\[restriction equivalence\] The restriction functor $R\uB:\De\uB(X,\J)\to \De\uB(\U,\J)$ is an equivalence of formal groupoids.
Semicosimplicial groupoids and descent {#cosimplicial groupoids}
======================================
The remainder of the paper will be devoted to constructing a DGLA governing the deformation theory of branes with locally trivializable deformations. To do so, we will adapt techniques developed in [@I], as well as [@FMM][@BM]. As a first step, in this section we review the framework of semicosimplicial groupoids and the descent groupoid construction. We will use this to reformulate the definitions introduced in the previous section more systematically. This will enable us to formulate and prove some results which would be much more cumbersome otherwise.
Most of the following discussion–including the notation–is taken from [@BM]. Let $\Delta_{mon}$ be the category whose objects are the finite ordinal sets $[n]=\{0,1,\cdots,n\}$ for $n=0,1,\cdots$, and whose morphisms are order-preserving injective maps among them. Recall that a *semicosimplicial object* in a category $\CC$ is a functor $A\ut:\Delta_{mon}\to \CC$. This may be pictured as a diagram $$\label{simplicial diagram}A\ut=\xymatrix @C=.3in{A_0 \ar@<+.4ex>[r]\ar@<-.4ex>[r] & A_1 \ar@<+.6ex>[r]\ar@<-.6ex>[r] \ar[r]& A_2\ar@<+1ex>[r]\ar@<-1ex>[r]\ar@<+.3ex>[r]\ar@<-.3ex>[r] & \dots},$$ where each $A_n:=A\ut([n])$ is an object of $\CC$. The *coface maps* $\partial^i_n:A_n\to A_{n+1}\}_{i=0}^{n+1}$ (for $i=0,\cdots, n+1$) satisfy a number of relations determined by the combinatorial structure of $\Delta_{mon}$.
More generally, given a (strict) 2-category $\CC$, we may similarly define a (strict) semicosimplicial object in $\CC$: this consists of a diagram of the form (\[simplicial diagram\]), where the entries are objects of $\CC$ and the arrows are 1-morphisms, which we require to satisfy cosimplicial relations on the nose. The example we will need is when $\CC$ is the 2-category of formal groupoids (over $\Art$).
\[example nerve groupoid\] Let $\F$ be a sheaf on a topological space $X$, valued in a category (or strict 2-category) $\CC$. Given an open cover $\U=\{U\sa\}$ of $X$, we may define a semicosimplicial object $A\ut$ in $\CC$, called the *nerve* of $\F$. For each $n=0,1,\cdots$ we take $$A\ut_{n}=\Pi_{\alpha_{0},\cdots\alpha_{n}}\F(U_{\alpha_0,\cdots\alpha_n}).$$ The coface maps $\partial^i_n:A\ut_n\to A\ut_{n+1}$ are constructed using the restriction maps of the sheaf $\F$: for each $n=0,1,\cdots$ and each $i=0,\cdots, n+1$, we have $$\partial_n^i=\Pi_{\alpha_0,\cdots,\alpha_{n+1}}\F(U_{\alpha_0\cdots\alpha_{n+1}},U_{\alpha_0\cdots \hat{\alpha}_i\cdots\alpha_{k+1}}),$$ where for each inclusion of open subset $U\subset V$ the notation $\F(U,V):\F(V)\to\F(U)$ denotes the restriction map.
\[definition descent groupoid\] [@BM Thm. 2.6] Given a semicosimplicial groupoid $\G_{\bullet}$, the *groupoid of descent data* $\Desc(\G\db)$ is defined as follows:
1. An object of $\Desc(\G\db)$ is a pair $(l,m)$ with $l$ an object of $\G_0$, and $m$ a morphism in $\G_1$ from $\partial_0l\to\partial_1l$, such that the equation $$(\partial_0m)(\partial_1m)^{-1}(\partial_2m)=1$$ holds in $\G_2$.
2. A morphism from $(l,m)$ to $(l',m')$ is a morphism $a:l\to l'$ in $\G_0$, such that the following diagram commutes: $$\xymatrix{ \partial_0l \ar[r]^{m} \ar[d]_{\partial_0a} & \partial_1l \ar[d]^{\partial_1a} \\ \partial_0l' \ar[r]^{m'} & \partial_1l'.}$$
\[nerve groupoid 2\] Continuing Example \[example nerve groupoid\], let us describe explicitly the groupoid $\Desc(\G\db)$ in the case that $\G\db$ is the nerve of a strict presheaf of groupoids on $X$. An object is a pair $(\{x\sa\},\{\varphi\sab\}\}$, where $x\sa\in \CC(U\sa)$, and the $\varphi\sab$ are morphisms $$\varphi\sab:x_{\beta}\to x_{\alpha}$$ (where both objects have implicitly been restricted to $U_{\alpha\beta}$). The morphisms must satisfy $$\varphi_{\gamma\beta}\varphi_{\beta\alpha}=\varphi_{\gamma\alpha}$$ on the triple intersections $U_{\alpha\beta\gamma}$. A morphism from $x=(\{x\sa\},\{\varphi\sab\})$ to $x'=(\{x'\sa\},\{\varphi'\sab\})$ is a collection $\{\psi\sa\}$ of morphisms $\psi\sa:x\sa\to x'\sa$ that satisfy $$\psi\sa\varphi_{\alpha\beta}=\varphi'_{\alpha\beta}\psi\sb$$ on overlaps $U\sab$.
Definition \[definition descent groupoid\] and Remark \[nerve groupoid 2\] extend immediately to the case of formal groupoids. For example, in the case of the sheaf of formal groupoids $\De\uB(X,\J)$ associated to a GC brane $\B$, the formal groupoid in Definition \[glued deformations\] with respect to an open cover $\U$ is precisely the descent groupoid for the nerve of $\De\uB(X,\J)$.
Given a GC brane $\B\in\Br(X,\J)$, recall from §\[induced deformations\] the formal groupoid $\Dt\uB\da(X,\J)^{tr}$ described in Definition \[definition tilde extended\]. This groupoid may also be refined to a sheaf of formal groupoids over $X$, which assigns to each open subset $U\subset X$ the formal groupoid $\Dt\uB(U,\J)^{tr}:=\Dt^{\B|_U}\da(U,\J|_U)^{tr}$.
Let $\Dt\uB(\U,\J)^{tr}$ be the groupoid formal of descent data associated to the nerve of the sheaf of groupoids $U\mapsto \Dt(U,\J)^{tr}$ with respect to an open cover $\U$. Explicitly, for each $A\in\Art$ an object of $\Dt\uB\da(\U,\J)^{tr}$ is a pair $(\{x\sa\},\{y\sab\})$ where
1. $x\sa$ is an element of $e^{\T\da(U\sa)}$,
2. $y\sab$ is an element of $e^{\KKK\da(U\sab)}$,
3. on each overlap $U\sab$ we have $x\sa=\chi(y\sab)x\sb$,
4. on each triple overlap $U_{\alpha\beta\gamma}$ we have $y_{\beta\gamma}y_{\alpha\gamma}^{-1}y_{\alpha\beta}=1.$
A morphism from $(\{x\sa\},\{y\sab\})$ to $(\{x'\sa\},\{y'\sab\})$ in $\Dt\uB\da(\U,\J)^{tr}$ is a collection $\{w\sa\}$, where
1. $w\sa\in e^{\KKK\da(U\sa)}$,
2. on each $U\sa$ we have $x'\sa =\chi(w\sa) x\sa$,
3. on each double overlap $U\sab$ we have $y'\sab w\sb=w\sa y\sab$.
The composition of morphisms is given by group multiplication, i.e. $$\{w\sa'\}\circ\{w\sa\}=\{w'\sa w\sa\}.$$ The identity morphisms is given by the identity group elements.
We then have the following easy result, whose proof is omitted.
For each $A\in\Art$, each object $(\{x\sa\},\{y\sab\})\in \Dt\uB\da(\U,\J)^{tr}$, and each $g\in e^{\H\da(X)}$, let $$(\{x\sa\},\{y\sab\})\cdot g=(\{x\sa\cdot g|_{U\sa}\},\{y\sab\}.$$ For each morphism $$\{w\sa\}: (\{x\sa\},\{y\sab\})\to (\{x'\sa\},\{y'\sab\}),$$ let $\{w\sa\}\cdot g=\{w\sa\}$, regarded as a morphism from $(\{x\sa\},\{y\sab\})\cdot g$ to $(\{x'\sa\},\{y'\sab\})\cdot g$. Then this defines a strict right action of $e^{\H(X)}$ on $\Dt\uB(\U,\J)^{tr}$.
\[definition of extended glued\] Let $$\De\uB(\U,\J)^{tr}:=\Dt\uB(\U,\J)^{tr}//e^{\H(X)}$$ be the formal groupoid associated to the right action of $e^{\H(X)}$ on $\Dt\uB(\U,\J)^{tr}$, as described in Definition \[definition action groupoid\].
The map taking a formal semicosimplicial groupoid to its formal groupoid of descent data is functorial in a natural sense [@BM]. Namely, there is a (strict) 2-category of semicosimplicial formal groupoids, and the descent construction defines a 2-functor from this category to the two-category of formal groupoids. For example, a map from a semicosimplicial formalgroupoid $\G\db$ to another one $\H\db$ by definition consists a collection of functors $\Psi_k:\G_k\to \H_k$ that are compatible with the boundary maps. Such a map induces a functor $\textrm{Desc}(\Psi\db):\textrm{Desc}(\G\db)\to \textrm{Desc}(\H\db)$.
\[equivalence of descent groupoids\][@BM] Let $\Psi\db:\G\db\to\H\db$ be a map of semicosimplicial formal groupoids, such that for each $k$ the functor $\Psi_k:\G_k\to \H_k$ is an equivalence. Then $\textrm{Desc}(\Psi\db):\textrm{Desc}(\G\db)\to \textrm{Desc}(\H\db)$ is also an equivalence.
Let $\CC$ and $\DD$ be (strict) sheaves of formal groupoids on $X$, and let $\Sigma:\CC\to \DD$ be a strict map of such sheaves, i.e. a collection of functors $\Sigma(U):\CC(U)\to\DD(U)$ that are strictly compatible with the restriction functors. Then $\Sigma$ induces a map of semicosimplicial formal groupoids from the nerve of $\CC$ to the nerve of $\DD$, which in turn induces a functor $\Sigma(\U)$ of the associated descent groupoids. Furthermore, if for each open subset $U\subset X$ the functor $\Sigma(U):\CC(U)\to\DD(U)$ is an equivalence of groupoids, then then induced functor $\Sigma(\U)$ is also an equivalence.
Let $\widetilde{\Sigma}\uB(\U):\Dt\uB(\U,\J)^{tr}\to \Dt\uB\da(\U,\J)$ be the functor induced by the map of sheaves described in Definition \[definition sigma tilde\].
By construction, the functor $\widetilde{\Sigma}\uB(\U):\Dt\uB(\U,\J)^{tr}\to \Dt\uB(\U,\J)$ is compatible with the actions of $e^{\H(X)}$ on both formal groupoids. Therefore, using Proposition \[induced equivalence\], we may extend $\widetilde{\Sigma}\uB$ to a functor from $\Def^{\B}(\U,\J)^{tr}$ to $\Def^{\B}(\U,\J)$.
\[definition extended nerve\] $\Sigma\uB(\U): \De\uB(\U,\J)^{tr}\to \De\uB(\U,\J)$ is the functor $\widetilde{\Sigma}\uB(\U)//e^{\H(X)}$ induced from $\widetilde{\Sigma}\uB(\U)$ using Proposition \[induced equivalence\].
\[theorem extended and LWL\] Let $\B$ be a leaf-wise Lagrangian GC brane on $(X,\J)$ (or more generally a brane with locally trivializable deformations). Then there exists an open cover $\U$ of $X$ such that $$\Sigma\uB(\U): \De\uB(\U,\J)^{tr}\to \De\uB(\U,\J)$$ is an equivalence of formal groupoids. Furthermore, by Theorem \[restriction equivalence\], the formal groupoids $\De\uB(\U,\J)^{tr}$ and $\De\uB(X,\J)$ are also equivalent.
By Corollary \[equivalence of induced\], we there exists an open cover $\U=\{U\sa\}$ of $X$ such that, for each $\alpha$, the functor $$\widetilde{\Sigma}\uB(U\sa):\Dt\uB(U\sa,\J)^{tr}\to \Dt\uB(U\sa,\J)$$ is an equivalence. It then easily follows from Proposition \[equivalence of descent groupoids\] that $$\widetilde{\Sigma}\uB(\U):\Dt\uB(\U,\J)^{tr}\to \Dt\uB(\U,\J)$$ is an equivalence as well. The proof then follows by applying Proposition \[induced equivalence\].
Construction of the DGLA for branes with locally trivializable deformations {#DGLA theory}
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DGLA’s and the Deligne functor {#DGLA obstructions}
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For simplicity, in the following we work over the field $\R$. A differential graded Lie algebra (DGLA) consists of a cochain complex $\g\ub$, together with a cochain map (the “bracket") $$[\cdot,\cdot]:\g\ub\otimes \g\ub\to \g\ub$$ of degree zero, which is skew-symmetric (in the graded sense), and satisfies a graded version of the Jacobi identity [@I].
\[Lie algebra as DGLA\] Any Lie algebra may be regarded as a DGLA concentrated in degree zero. Conversely, given an arbitrary DGLA $\g\ub$, the restriction of the bracket to $\g^0$ gives it the structure of a Lie algebra.
For any DGLA $\g\ub$, the set of *Maurer-Cartan elements* of $\g\ub$ are defined as $$\textrm{MC}(\g\ub)=\{x\in\g^1:dx+\frac{1}{2}[x,x]=0\}.$$ If $\g\ub$ is nilpotent, the group $e^{\g^0}$ is well-defined (sometimes called the *gauge group*). This group has a left action on the set of Maurer-Cartan elements of $\g\ub$ given by the formula $$\label{gauge action}e^y\cdot x=x+\sum_{n=0}^{\infty}\frac{[y,-]^n}{(n+1)!}([y,x]-dy).$$ Given an arbitrary DGLA $\g\ub$ (not necessarily nilpotent), one may define a formal groupoid $\Del_{\g\ub}$ over $\Art$, known as the *Deligne groupoid*. This is defined for each $A\in\Art$ (with unique maximal ideal $\m\subset A$) as the action groupoid $$\label{deligne groupoid} \Del_{\g\ub}(A):=\textrm{MC}(\g\ub\otimes\m)//e^{\g^0\otimes\m}.$$ Passing to equivalence classes, we obtain a functor $\df_{\g\ub}:\Art\to\textrm{Set}$, which assigns to every $A\in\Art$ the set of Maurer-Cartan elements of $\g\otimes\m$ modulo the action of $e^{\g^0\otimes\m}$.
As in Example \[Lie algebra as DGLA\], let $\g$ be a Lie algebra, regarded as a DGLA concentrated in degree $0$. Then for each $A\in\Art$, the Deligne groupoid $\Del_{\g}(A)$ may be identified with the groupoid $$\label{Deligne for Lie algebra} *//e^{\g\otimes\m}.$$ By definition, the groupoid (\[Deligne for Lie algebra\]) has a unique object $*$, and the set of morphisms from $*$ to itself are the elements of the group $e^{\g\otimes\m}$, with composition given by group multiplication.
Consider a functor $F:\Art\to \Set$. Let $\mu:A'\to A$ be a surjective map in $\Art$, and $x\in F(A)$. One says that $x$ can be *extended* to an element of $F(A')$ if there exists $x'\in F(A')$ such that $F(\mu)(x')=x$. In general there will be an obstruction to the existence of such an extension. The functor $F$ is called *unobstructed*, however, if for every surjective map $A'\to A$ the induced map $F(A')\to F(A)$ is also surjective. The fact that every surjective map in $\Art$ can be factored through a sequence of small extensions (Proposition \[induct on small extensions\]) implies that $F$ is unobstructed if and only if $F(A')\to F(A)$ is surjective for every small extension $A'\to A$.
\[H2 and obstructions\][@M2 §3]Let $\g$ be a DGLA with $H^2(\g)=0.$ Then the functor $\df_{\g}:\Art\to\Set$ is unobstructed.
Bisemicosimplicial DGLAs, totalization, and descent {#bisemi}
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The notation and terminology in this section (mostly) follows [@I]. Let $$V\ut=\xymatrix @C=.3in{V_0 \ar@<+.4ex>[r]\ar@<-.4ex>[r] & V_1 \ar@<+.6ex>[r]\ar@<-.6ex>[r] \ar[r]& V_2\ar@<+1ex>[r]\ar@<-1ex>[r]\ar@<+.3ex>[r]\ar@<-.3ex>[r] & \dots}$$ be a semicosimplicial cochain complex, i.e. a semicosimplicial object in the category of cochain complexes (differential graded vector spaces). As in §\[cosimplicial groupoids\], for each $n=0,1\cdots$, we denote by $d_n:V_n\to V_n$ the differential on the cochain complex $V_n$, and we denote by $\partial_n^i:V_n\to V_{n+1}$ for $i=0,\cdots n+1$ the coface maps. From $V\ut$ we may construct a cochain complex $\Tot(V\ut)$, called the *total complex* or *totalization* of $V\ut$. As a graded vector space, $\Tot(V\ut)$ is equal to the direct sum $\bigoplus_{n\geq 0} V_n[-n]$, where for any graded vector space $W=\oplus_i W^i$ and any integer $k$, the shifted space $W[k]$ is defined by $(W[k])^i=W^{i+k}$. For each $n$, let $$\partial_n=\sum_{i=0}^{n+1}(-1)^i\partial_n^i:V_n\to V_{n+1}.$$ Define $$d=\sum_{n=0}^{\infty}(-1)^nd_n:\bigoplus_{n\geq0} V_n[-n]\to \bigoplus_{n>0} V_n[-n]$$ and $$\partial=\sum_{n=0}^{\infty}\partial_n: \bigoplus_{n\geq0} V_n[-n]\to \bigoplus_{n\geq0} V_n[-n].$$ The differential on $\Tot(V\ut)$ is then defined to be the sum $$D=d+\delta.$$
For any category $\CC$, the collection of semicosimplicial objects in $\CC$ themselves form the objects of a category: a morphism $A\ub\to B\ub$ of semicosimplicial objects in $\CC$ is by definition a collection of morphisms $\{\psi_n:A_n\to B_n\}$ in $\CC$ that commute with the coface maps. The totalization construction sending a semicosimplicial cochain complex $V\ut$ to its total complex $\Tot(V\ut)$ extends in an natural way to a functor (from the category of semicosimplicial cochain complexes to the category of cochain complexes).
Next, let $\g\ut$ be a semicosimplicial DGLA. If we apply the above construction to the semicosimplicial cochain complex underlying $\g\ut$, there is no natural way to give the resulting cochain complex $\Tot(\g\ut)$ a DGLA structure. There is an alternate (functorial) construction, however, that takes as input a semicosimplicial DGLA $\g\ut$, and gives as output a DGLA $\Tot_{TW}(\g\ut)$, known as the *Thom-Whitney totalization of $\g\ut$*. We will not need the explicit form of this construction, only its existence and the properties summarized in Propositions \[cochain iso1\], \[descent for Deligne\], \[bi chain iso\], and \[double descent\] below.
\[cochain iso1\][@I Lemma 2.5] Let $\g\ut$ be a semicosimplicial DGLA. Then the cochain complexes $\Tot_{TW}(\g\ut)$ and $\Tot(\g\ut)$ are quasi-isomorphic.
Since $\Ttot(\g\ut)$ is a DGLA, we may apply the Deligne construction described above to construct the formal groupoid $\Del_{\Ttot(\g\ut)}$. On the other hand, applying the Deligne construction term by term to $\g\ut$ produces a semicosimplicial formal groupoid $$\G\ut(A)=\xymatrix @C=.3in{\Del_{\g_0} \ar@<+.4ex>[r]\ar@<-.4ex>[r] & \Del_{\g_1} \ar@<+.6ex>[r]\ar@<-.6ex>[r] \ar[r]& \Del_{\g_2}\ar@<+1ex>[r]\ar@<-1ex>[r]\ar@<+.3ex>[r]\ar@<-.3ex>[r] & \dots}$$ We may then form its descent groupoid $\Desc(\G\ut)$.
\[descent for Deligne\][@BM Thm. 2.6] There is a natural equivalence of formal groupoids $$\Del_{\Ttot(\g\ut)}{\xymatrix@C=1.5pc{ \ar[r]^-{\cong} & }}\Desc(\G\ut).$$
Next, let $V\utt$ be a bisemicosimplicial cochain complex, which is a diagram $$\label{square diagram}\xymatrix @C=.3in { \vdots & \vdots & \vdots \\ V_{0,1} \ar@<+.6ex>[u]\ar@<-.6ex>[u] \ar[u] \ar@<+.4ex>[r]\ar@<-.4ex>[r] &V_{1,1} \ar@<+.6ex>[u]\ar@<-.6ex>[u] \ar[u] \ar@<+.6ex>[r]\ar@<-.6ex>[r] \ar[r]& V_{2,1} \ar@<+.6ex>[u]\ar@<-.6ex>[u] \ar[u]\ar@<+1ex>[r]\ar@<-1ex>[r]\ar@<+.3ex>[r]\ar@<-.3ex>[r] & \cdots \\ V_{0,0} \ar@<+.4ex>[u]\ar@<-.4ex>[u] \ar@<+.4ex>[r]\ar@<-.4ex>[r] &V_{1,0} \ar@<+.4ex>[u]\ar@<-.4ex>[u]\ar@<+.6ex>[r]\ar@<-.6ex>[r] \ar[r]& V_{2,0} \ar@<+.4ex>[u]\ar@<-.4ex>[u]\ar@<+1ex>[r]\ar@<-1ex>[r]\ar@<+.3ex>[r]\ar@<-.3ex>[r] & \cdots}$$ We may succinctly define $V\utt$ as a functor from the category $\Delta_{mon}\times\Delta_{mon}$ to the category of cochain complexes. Each individual row $V_{\bullet,n}$ in the above diagram itself forms a semicosimplicial cochain complex, and we alternatively view $V\utt$ as a semicomisimplical object in the category of semicosimplicial cochain complexes: $$V\utt=\xymatrix @C=.3in{V_{\bullet,0} \ar@<+.4ex>[r]\ar@<-.4ex>[r] & V_{\bullet,1} \ar@<+.6ex>[r]\ar@<-.6ex>[r] \ar[r]& V_{\bullet,2} \ar@<+1ex>[r]\ar@<-1ex>[r]\ar@<+.3ex>[r]\ar@<-.3ex>[r] & \dots}$$ From this point of view, it is clear how to generalize the totalization functor to the bisemicosimplicial case: first, form the total complex of each row, yielding a semicosimplicial cochain complex $$\Tot^{\triangle}(V\utt):=\xymatrix @C=.3in{\Tot(V_{\bullet,0}) \ar@<+.4ex>[r]\ar@<-.4ex>[r] & \Tot(V_{\bullet},1) \ar@<+.6ex>[r]\ar@<-.6ex>[r] \ar[r]& \Tot(V_{\bullet},2) \ar@<+1ex>[r]\ar@<-1ex>[r]\ar@<+.3ex>[r]\ar@<-.3ex>[r] & \dots}$$ We may then applying the totalization procedure again to form the cochain complex $\Tot(\Tot^{\triangle}(V\utt))$, which we will simply denote by $\Tot(V\utt)$. Similarly, if $\g\utt$ is a bisemicosimplicial DGLA, we may iterate the Thom-Whitney totalization procedure to construct a DGLA $\Tot_{TW}(\g\utt)$
\[bi chain iso\] For any bisemicosimplicial DGLA $\g\utt$, the cochain complexes $\Tot(\g\utt)$ and $\Tot_{TW}(\g\utt))$ are quasi-isomorphic.
One may similarly define a strict bisemicosimplicial formal groupoid $\G\utt$ to be a diagram of the form \[square diagram\], where the entries are groupoids and coface maps are required to satsify the same compatibility conditions (on the nose). To define the descent groupoid of $\G\utt$, we first form the descent groupoid $\Desc(\G_{\bullet,n})$ of each row, resulting in a semicosimplicial groupoid $\Desc\ut(\G\utt)$. We may then form its formal groupoid of descent data $\Desc(\Desc\ut(\G\utt))$, which we denote simply by $\Desc(\G\utt)$.
\[double descent\] Let $\g\utt$ be a bisemicosimplicial DGLA, and let $\Del_{\g\utt}$ be the bisemicosimplicial formal groupoid formed by applying the Deligne construction term by term to $\g\utt$. Then there is a natural equivalence of formal groupoids between $\Del_{\Tot(\g\utt)}$ and $\Desc(\Del_{\g\utt})$.
Construction of the DGLA {#construction of the DGLA}
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We now apply the theory described in the previous section to the deformation theory of generalized complex branes. Specifically, for every brane $\B\in\Br(X,\J)$ and open cover $\U$ of $X$, we will construct a DGLA $L_{\B,\U}$. Using results from the previous section, we then prove that the formal groupoid $\Del_{L_{\B,\U}}$ is equivalent to the formal $\De\uB(\U,\J)$ introduced in Definition \[definition extended nerve\]. For $\B$ a LWL brane, it follows from Theorem \[theorem extended and LWL\] that $\Del_{L_{\B,\U}}$ is also equivalent to $\De\uB(X,\J)$. In particular, there is a natural isomorphism of functors $$\df_{L_{\B,\U}}{\xymatrix@C=1.5pc{ \ar[r]^-{\cong} & }}\df_{\B}.$$
As a concrete application of this construction we prove Theorem \[brane obstructions\], stated in the introduction. This generalizes the well-known result in complex geometry that the obstructions to deforming a complex submanifold $Z$ of a complex manifold $X$ are contained in the sheaf cohomology group $H^2(Z;\mathcal{O}_{NZ})$.
The construction
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Let $\B$ be a GC brane on a GC manifold $(X,\J)$, and let $\U=\{U\sa\}$ be an open cover of $X$. Consider the following diagram of Lie algebras $$\label{V diagram} \xymatrix @C=.15in {\prod \T(U\sa) \ar@<+.4ex>[r]\ar@<-.4ex>[r] & \prod\T(U_{\alpha\beta}) \ar@<+.6ex>[r]\ar@<-.6ex>[r] \ar[r]& \prod\T(U_{\alpha\beta\gamma}) \ar@<+1ex>[r]\ar@<-1ex>[r]\ar@<+.3ex>[r]\ar@<-.3ex>[r] & \cdots \\ \H(X)\oplus\prod\KKK(U\sa) \ar@<+.4ex>[u]\ar@<-.4ex>[u] \ar@<+.4ex>[r]\ar@<-.4ex>[r] & \H(X)\oplus\prod\KKK(U\sab) \ar@<+.4ex>[u]\ar@<-.4ex>[u]\ar@<+.6ex>[r]\ar@<-.6ex>[r] \ar[r]& \H(X)\oplus\prod\KKK(U\sabc) \ar@<+.4ex>[u]\ar@<-.4ex>[u]\ar@<+1ex>[r]\ar@<-1ex>[r]\ar@<+.3ex>[r]\ar@<-.3ex>[r] & \cdots}$$ The maps in the diagram are defined as follows: The top row is the semicosimplicial DGLA which is the nerve (with respect to the open cover $\U$) of the sheaf $\T$ of DGLAs on $X$, as described in Example \[example nerve groupoid\]. The bottom row is the direct sum of the nerve (with respect to $\U$) of the sheaf $\KKK$, and the semicosimplicial DGLA $$\xymatrix @C=.3in{\H(X) \ar@<+.4ex>[r]\ar@<-.4ex>[r] & \H(X) \ar@<+.6ex>[r]\ar@<-.6ex>[r] \ar[r]& \H(X)\ar@<+1ex>[r]\ar@<-1ex>[r]\ar@<+.3ex>[r]\ar@<-.3ex>[r] & \dots}$$ with all maps equal to the identity. Next we describe the vertical maps, which we denote by $\partial_V^i$ for $i=0,1$. Given $v=(\xx_f,\{y_{\alpha_0\alpha_1\cdots\alpha_k}\})\in \H(X)\oplus\prod\KKK(U_{\alpha_0\alpha_1\cdots\alpha_k})$, we have $$\partial_V^0(v)=\{(\xx_f)|_{U_{\alpha_0\alpha_1\cdots\alpha_k}}\}$$ and $$\partial_V^1(v)=\{\chi(y_{\alpha_0\alpha_1\cdots\alpha_k})\}.$$ It is easy to see that the diagram (\[V diagram\]), extended upwards by zero, defines a bisemicosimplicial DGLA, which we denote by $V\utt_{\B,\U}$.
\[concrete description\] The formal groupoid $\Desc(\Del_{V_{\B,\U}\utt})$ is equivalent to $\De\uB(\U,\J)^{tr}.$
For each $A\in\Art$, the bisemicosimplicial groupoid $\Del_{V_{\B,\U}\utt}(A)$ may be identified with $$\xymatrix @C=.1in {{*}//\prod e^{\T\da(U\sa)} \ar@<+.4ex>[r]\ar@<-.4ex>[r] & {*}//\prod e^{\T\da(U_{\alpha\beta})} \ar@<+.6ex>[r]\ar@<-.6ex>[r] \ar[r]& {*}//\prod e^{\T\da(U_{\alpha\beta\gamma})} \ar@<+1ex>[r]\ar@<-1ex>[r]\ar@<+.3ex>[r]\ar@<-.3ex>[r] & \cdots \\ {*}//e^{\H\da(X)}\times\prod e^{\KKK\da(U\sa)} \ar@<+.4ex>[u]\ar@<-.4ex>[u] \ar@<+.4ex>[r]\ar@<-.4ex>[r] & {*}//e^{\H\da(X)}\times\prod e^{\KKK\da(U\sab)} \ar@<+.4ex>[u]\ar@<-.4ex>[u]\ar@<+.6ex>[r]\ar@<-.6ex>[r] \ar[r]& {*}//e^{\H\da(X)}\times\prod e^{\KKK\da(U\sabc)} \ar@<+.4ex>[u]\ar@<-.4ex>[u]\ar@<+1ex>[r]\ar@<-1ex>[r]\ar@<+.3ex>[r]\ar@<-.3ex>[r] & \cdots}$$
Let us label these groupoids as $\GG_{ij}$, where $i=1,2$ is the row index, and $j=0,1,\cdots$ is the column index. Consider first the descent groupoid $\CC_0:=\Desc(\GG_{0,\bullet})$ of the bottom row. Recall from Definition \[definition descent groupoid\], that an object of $\CC$ is a pair $(l,m)$ where $l$ is an object of $\GG_{00}$ and $m$ is a morphism in $\GG_{01}$ from $\partial^0l$ to $\partial^1l$, which satisfies the associativity condition $$\label{assoc1} \partial^0(\partial^1)^{-1}\partial^2m=id.$$ We must have $l=*$ (since this is the only object in $\GG_{00}$), and $m$ must be an element $(z,\{y_{\alpha\beta}\}))\in e^{\H\da(X)}\times e^{\K\da(X)}$; the associativity condition (\[assoc1\]) is equivalent to $$1=(z,y_{\beta\gamma}y_{\alpha\gamma}^{-1}y_{\alpha\gamma}),$$ so we see that $z=1$ must be trivial, and the collection $\{y_{\alpha\beta}\}$ must satisfy the nonabelian cocycle condition $y_{\beta\gamma}y_{\alpha\gamma}^{-1}y_{\alpha\gamma}=1$. To simplify the notation, we will denote the object $(*,(1,\{y_{\alpha\beta}\}))$ of $\CC_0$ by $\{y_{\alpha\beta}\}$. Spelling out part (2) of Definition \[definition descent groupoid\], we see that a morphism in $\CC^0$ from $\{y_{\alpha\beta}\}$ to $\{y'_{\alpha\beta}\}$ consists of a pair $(z,\{w_{\alpha}\})$, where $z$ is an arbitrary element of $e^{\H\da(X)}$, and $\{w_{\alpha}\in e^{\KKK\da(U_{\alpha})}\}$ are subject to the condition $y'_{\alpha\beta}w_{\beta}=w_{\alpha}y_{\alpha\beta}$.
Similarly, an object of $\CC_1=\Desc(\GG_{1,\bullet})$ consists of $\{x_{\alpha\beta}\in e^{\T\da(U_{\alpha\beta})}\}$ satisfying $x_{\beta\gamma}x_{\alpha\gamma}^{-1}x_{\alpha\beta} =1$. A morphism in $\CC^1$ from $\{x_{\alpha\beta}\}$ to $\{x'_{\alpha\beta}\}$ consists of $\{v_{\alpha}\in e^{\T\da(U_{\alpha})}\}$ satisfying $x'_{\alpha\beta}v_{\beta}=v_{\alpha}x_{\alpha\beta}$.
To complete the construction of $\Desc(\Del_{V_{\B,\U}\utt}(A))$, we then then apply the descent groupoid construction in the vertical direction: $$\Desc(\Del_{V_{\B,\U}\utt}(A))=\Desc(\xymatrix @C=.3in{\CC_0 \ar@<+.4ex>[r]\ar@<-.4ex>[r] & \CC_1}).$$ Here, $\partial_V^0:\CC_0\to\CC_1$ is the functor taking and object $\{y_{\alpha\beta}\}$ in $\CC_0$ to the trivial object $\{x_{\alpha\beta}=1\}$, and a morphism $(z,\{w_{\alpha}\}):\{y_{\alpha\beta}\} \to \{y'_{\alpha\beta}\}$ to the morphism $\{z|_{U_{\alpha}}\}$ from the trivial object to itself. We also see that $\delta^1_V:\CC_0\to\CC_1$ takes $\{y_{\alpha\beta}\}$ to $\{\chi(y_{\alpha\beta})\}$, and morphism $\{(z,w_{\alpha}\}:\{y_{\alpha\beta}\}\to\{y'_{\alpha\beta}\}$ to the morphism $\{\chi(w)_{\alpha}\}:\{\chi(y_{\alpha\beta})\}\to \{\chi(y'_{\alpha\beta})\}$. Therefore, an object of $\Desc(\Del_{V_{\B,\U}\utt}(A))$ is a pair $(\{y_{\alpha\beta}\},\{x_{\alpha}\})$ with $y_{\alpha\beta}\in e^{\KKK\da(U_{\alpha\beta})}$ satisfying $y_{\beta\gamma}y^{-1}_{\alpha\gamma}y_{\alpha\beta}=1$, and $x_{\alpha}\in e^{\T\da(U_{\alpha})}$ satisfying $\chi(y_{\alpha\beta})x_{\beta}=x_{\alpha}$. A morphism in $\Desc(\Del_{V_{\B,\U}\utt}(A))$ from $(\{y_{\alpha\beta}\},\{x_{\alpha}\})$ to $(\{y'_{\alpha\beta}\},\{x'_{\alpha}\})$ is a pair $(z,\{w_{\alpha}\})$ with $z\in e^{\T\da(X)}$, $w_{\alpha}\in e^{\KKK\da(U_{\alpha})}$ such that $y'_{\alpha\beta}w_{\beta}=w_{\alpha}y_{\alpha\beta}$ and $\chi(w_{\alpha})x_{\alpha}=x'_{\alpha}z$. Comparing this with the Definition \[definition of extended glued\] of $\Def^{\B}_A(\U,\J)^{tr}$, we arrive at the desired result.
Introduce the notation $L_{\B,\U}:=\Tot_{TW}(V_{\B,\U}\utt)$. Combining Proposition \[concrete description\] and Proposition \[double descent\], we have the following result.
\[DGLA for extended\] There is an equivalence of formal groupoids $$\De\uB(\U,\J)^{tr}\cong \Del_{L_{\B,\U}}.$$
Finally, Corollary \[DGLA for extended\] with Theorem \[theorem extended and LWL\] we arrive at the following result.
\[DGLA for LWL\] Let $\B$ be a leaf-wise Lagrangian GC brane on $(X,\J)$ (or more generally a brane with locally trivializable deformations). Then there exists an open cover $\U$ of $X$ such that $\De\uB(X,\J)$ is equivalent to $\Del_{L_{\B,\U}}$. In particular, the deformation functor $\df_{L_{\B,\U}}$ is isomorphic to the functor $\df_{\B}$ associated to $\B$ as described in Definition \[brane deformation functor\].
Lie algebroid cohomology and obstructions {#cohomology}
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Let $\B\in\Br(X,\J)$ be a LWL brane (or more generally, a brane with locally trivializable deformations), and let fix an open cover $\U$ of $X$, chosen as in Theorem \[DGLA for LWL\]. Let $V\utt:=V\utt_{\B,\U}$ be the bisemicosimplicial DGLA constructed in §\[construction of the DGLA\], and $L:=L_{\B,\U}$ the DGLA $\Ttot(V_{\B,\U}\utt)$. We may also form the cochain complex $C:=\Tot(V\utt)$, which by Proposition \[bi chain iso\] is quasi-isomorphic to (the underlying cochain complex of) $L$; in particular the cohomology groups of $C$ and $L$ are isomorphic. As state in §\[DGLA obstructions\], the deformation functor $\df_L$ is unobstructed if the cohomology group $H^2(L)\cong H^2(C)$ vanishes. By Theorem \[DGLA for LWL\], the condition $H^2(C)=0$ also implies that the functor $\df_{\B}$ is unobstructed. The following theorem makes use of this fact by relating the group $H^2(C)$ to the more familiar Lie algebroid cohomology group $H^2(\B)$ associated to $\B$, which were described in §\[Lie alg cohomology\].
There is an injective linear map $$\Phi:H^2(C)\hookrightarrow H^2(\B).$$
In particular, we obtain Theorem \[brane obstructions\] as a corollary.
Using the explicit construction of the total complex $C=\Tot(V\utt)$ given in §\[bisemi\], we see that $$C^1=\prod_{\alpha}\T(U_{\alpha})\oplus \prod_{\alpha\beta}\KKK(U_{\alpha\beta})\oplus \H(X),$$ $$C^2=\prod_{\alpha\beta}\T(U_{\alpha\beta})\oplus \prod_{\alpha\beta\gamma}\KKK(U_{\alpha\beta\gamma})\oplus \H(X),$$ and $$C^3=\prod_{\alpha\beta\gamma}\T(U_{\alpha\beta\gamma})\oplus \prod_{\alpha\beta\gamma\delta}\KKK(U_{\alpha\beta\gamma\delta})\oplus \H(X).$$ The differential $D$ from $C^1$ to $ C^2$ is given by $$\label{D1}(\{x\sa\},\{y\sab\},\{z\})\mapsto (\{x_{\alpha}-x\sb-\chi(y\sab)\}+z|_{U_{\alpha\beta\gamma}},\{y_{\beta\gamma}-y_{\alpha\gamma}+y_{\alpha\beta}\},z),$$ and from $C^2$ to $C^3$ by $$\label{D2} (\{x_{\alpha\beta}\},\{y_{\alpha\beta\gamma}\},z)\mapsto (\{-x_{\beta\gamma}+x_{\alpha\gamma}-x_{\alpha\beta}-\chi(y_{\alpha\beta\gamma})+z|_{U_{\alpha\beta\gamma}},\{\delta(y)_{\alpha\beta\gamma\delta}\},0),$$ with $$(\delta y)_{\alpha\beta\gamma\delta}=y_{\beta\gamma\delta}-y_{\alpha\gamma\delta}+y_{\alpha\beta\delta}-y_{\alpha\beta\gamma}.$$
We wish to construct a map $\Phi:H^2(C\ub)\to H^2(\B)$. Let $c=(\{x_{\alpha\beta}\},\{y_{\alpha\beta\gamma}\},z)\in C^2$ be a cocycle representing a class $[c]\in H^2(C\ub)$. From (\[D1\]), we see that every element of $C^2$ is cohomologous to one with $z=0$, so without loss of generality we may assume $c$ is of this form. Write $y_{\alpha\beta\gamma}=(\tilde{y}_{\alpha\beta\gamma},f_{\alpha\beta\gamma})$, with $\tilde{y}_{\alpha\beta\gamma}\in \T\da(U_{\alpha\beta\gamma})$ and $f_{\alpha\beta\gamma}\in \Cinf(Z\cap U_{\alpha\beta\gamma})$. We claim that $c\in C^2$ is cohomologous to a cocycle of the form $(\{x'\sab\},\{(\tilde{y}'_{\alpha\beta\gamma},0)\},0)$. To see this, note using (\[D1\]) that that the condition $D(c)=0$ implies that $$f_{\beta\gamma\delta}-f_{\alpha\gamma\delta}+f_{\alpha\beta\delta}-f_{\alpha\beta\gamma}=0.$$ Therefore, using a partition of unity we may find functions $g_{\alpha\beta}\in \Cinf(Z\cap U_{\alpha\beta})$ such that $g_{\beta}-g_{\alpha}=f_{\alpha\beta}$. Let $\{\tilde{g}\sa\in\Cinf(U\sa)\}$ be a choice of functions extending $\{g_{\alpha}\}$. Note that $(0,d\tilde{g}\sa)$ is the generalized Hamiltonian vector field associated to the complex-function $i\tilde{g}\sa$, so in particular $(0,d\tilde{g}\sa)\in \T\da(U\sa)$. Furthermore, by construction $((0,d\tilde{g}\sa),-g\sa)$ is an element of $\KKK(U\sa)$. Writing $$c+D(\{0\},\{((0,d\tilde{g}\sa),-g\sa)\},0)=(\{x'\sab\},\{(\tilde{y}'_{\alpha\beta\gamma},f'_{\alpha\beta\gamma})\},0),$$ using (\[D1\]) we see that $f'_{\alpha\beta\gamma}=0$. Therefore, without loss of generality we may assume that $c$ is of the form $$c=(\{x\sab\},\{(\tilde{y}_{\alpha\beta\gamma},0\},0).$$ By Definition \[definition KKK\], we see that $\tilde{y}_{\alpha\beta\gamma}\in \KK\uB(U\sab)$, so that $$qr(\tilde{y}_{\alpha\beta\gamma})=0.$$ Therefore, if we define $\eta_{\alpha\beta}=qr(x\sab)\in \Cinf(\N|_{Z\cap U\sab})$, then using (\[D2\]) we see that $$\eta_{\beta\gamma}-\eta_{\alpha\gamma}+\eta_{\alpha\beta}=0.$$ Choose $\{\sigma_{\alpha}\in \Cinf(\N|_{Z\cap U\sa})\}$ such that $\sigma_{\beta}-\sigma_{\alpha}=\eta_{\alpha\beta}$. By Proposition \[induced holomorphic\], the fact $x\sab\in \T(\sab)$ implies that $\delta_l\mu(\eta\sab)=0$. It follows there exists a unique section $\zeta\in\Cinf(\lambda^2\l\ub)$ such that on each open set $Z\cap U\alpha$ we have $\zeta|_{Z\cap U\sa}=\delta_l\mu(\sigma\sa)$. By construction we have $\delta_l\zeta=0$, and we define $\Phi([c])=[\zeta]\in H^2(l)$.
To show that $\Phi$ is well-defined, we must check that the class $[\zeta]$ does not depend on the choice of $\{\sigma\sa\}$ or the representative $c$ for the class $[c]$. First, suppose that we have another collection $\{\sigma'\sa\in \Cinf(\N|_{Z\cap U\sa})\}$ that also satisfy $\sigma'_{\beta}-\sigma'_{\alpha}=\eta_{\alpha\beta}$, and that $\zeta'\in\Cinf(\Lambda^2l\uv)$ such that the restriction of $\zeta'$ to each $Z\cap U\sa$ is equal to $\delta_l\mu(\sigma'\sa)$. Then there is a unique section $\tau\in\Cinf(\N \B)$ such that $\sigma'_{\alpha}-\sigma\sa=\tau|_{Z\cap U\sa}$. We then have $\zeta'-\zeta=\delta_l\tau$, which implies that $[\zeta']=[\zeta]\in H^2(\B)$. This verifies that $[\zeta]$ does not depend on the choice of $\{\sigma\sa\}$. To see that $[\zeta]$ is also independent of the choice of $c$, suppose that $$c'=(\{x'\sab\},\{(\tilde{y'}_{\alpha\beta\gamma},0\},0)$$ is different cocycle such that $[c']=[c]$ in $H^2(C\ub)$. Given $b\in C^1$ such that $c'-c=Db$, by a nearly identical argument to that used above, we may assume that $b$ is of the form $b=(\{v\sa\},\{(w_{\alpha\beta},0)\},0$. We have $$x'\sab-x\sab=v_{\alpha}-v_{\beta}-w_{\alpha\beta}.$$ Since $w_{\alpha\beta}\in\K(U_{\alpha\beta})$, it follows that $$\label{diff1}qr(x'\sab)-qr(x\sab)=qr(v_{\alpha})-qr(v_{\beta}).$$ Given $\{\sigma\sa\in \Cinf(\N\B|_{Z\cap U\sa})\}$ satisfying $\sigma_\beta-\sigma_{\alpha}=x_{\alpha\beta}$, define $\sigma'\sa=\sigma\sa+qr(v\sa)$. It follows from (\[diff1\]) that $\sigma'\sb-\sigma'\sa=qr(x'\sab)$. Furthermore, since $v\sa$ is a generalized holomorphic vector field, it follows that $\delta_l\mu qr(v\sa)=0$, so we see that $\delta_l\mu(\sigma'\sa)=\delta_l\mu(\sigma\sa)$. This finishes the demonstration that $\Phi$ is well-defined.
To show that $\Phi$ is injective, suppose that $\Phi([c])=[\zeta]=0$ in $H^2(\B)$, say $\zeta=\delta_l\mu(\gamma)$ for some $\gamma\in \Cinf(\N\B)$, where $c=(\{x\sab\},\{(\tilde{y}_{\alpha\beta\gamma},0\},0)$ and $\{\sigma\sa\}$ are as above. Then on each $U\sa\cap Z$ we have $$\delta_l\mu(\sigma\sa-\gamma|_{Z\cap U\sa})=0.$$ It is not hard to show (using, for example, the arguments in §\[LWL branes example\] in the special case $A=\R[\epsilon]/(\epsilon^2$)) that there exists $\lambda\sa\in \T(U\sa)$ with $qr(\lambda\sa)=\sigma\sa-\gamma|_{Z\cap U\sa}$. Therefore we also have $qr(\lambda_{\beta})-qr(\lambda_{\alpha})=qr(x_{\alpha\beta})$, which implies that $x_{\alpha\beta}-(\lambda\sb-\lambda\sa)\in K^{\B}(U\sab)$. Setting $$b=(\{-\lambda\sa\},\{(\lambda_{\beta}-\lambda\sa-x_{\alpha\beta},0)\},0)\in C^1,$$ using the formula (\[D1\]) we see that $D(b)=c$. We therefore conclude that $[c]=0$ in $H^2(C\ub)$.
Appendix
========
*Proof of Proposition \[action\]*:
Consider the following general situation: $\g$ is a nilpotent Lie algebra acting on vector spaces $V,W,T$, we have a bilinear map $V\times W\to T$ sending $$x,y\mapsto x*y,$$ and for each $\xi\in \g$ we have $$\xi(x*y)=\xi(x)*y+x*\xi(y).$$
\[little lemma\] With these assumptions, the exponentiated action satisfies $$e^{\xi}(x*y)=e^{\xi}(x)*e^{\xi}(y).$$
Note that $$\xi^2(x*y)=\xi^2(x)*y+2\xi(x)*\xi(y)+x*\xi^2(y),$$ and more generally $$\xi^k(x*y)=\sum_{j=0}^k{k \choose j}\xi^j(x)*\xi^{k-j}(y).$$ Therefore $$\begin{aligned}
e^{\xi}([x*y])&= \sum_{k=0}^{\infty}\frac{1}{k!}\sum_{j=0}^k\frac{k!}{j!(k-j)!}\xi^j(x)*\xi^{k-j}(y) \nonumber \\
&= \sum_{j,l=0}^{\infty} \frac{1}{j!l!}\xi^j(x)*\xi^l(y) \nonumber \\
&= \sum_{j=0}^{\infty}\frac{1}{j!}\xi^j(x)*\sum_{l=0}^{\infty}\frac{1}{l!}\xi^l(y) \nonumber \\
&=e^{\xi}(x)*e^{\xi}(y).\end{aligned}$$
We note that these assumptions hold in each of the following situations (with $\g=\g\da(X)$):
1. In the case that $V=W=\Cinf\da(X)$, the action of $\g\da(X)$ is by the Lie bracket, and $*$ the Lie bracket.
2. In the case that $V=W=\Omega^{\bullet}\da(X)$, the action of $\g\da(X)$ is by the Lie derivative, and $*$ is the wedge-product.
3. In the case where $V=\Cinf\da(X)$ with $\g\da(X)$ acting by the Lie bracket, $W=\Omega^{\bullet}\da(X)$ with $\g\da(X)$ acting by the Lie derivative, and $*$ is the contraction operation of vector fields with differential forms.
To see that $e^{\xi}$ is compatible with the exterior derivative, note that (using the Cartan formula for the Lie derivative), we have $$\pounds(\xi)(da)=(\iota(\xi)d+d\iota(\xi))da=d\iota(\xi)da=d\pounds(\xi)a.$$ Iterating, we see that for each $k\geq 0$ we have $\pounds(\xi)^k(da)=d\pounds(\xi)^k(a)$, and therefore $e^{\xi}(da)=de^{\xi}(a).$
[99]{}
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[^1]: More generally, one can consider flows generated by time-dependent Hamiltonian vector fields corresponding to smooth functions $X\times [0,1]\to \R$.
[^2]: In [@G3], Gualtieri uses the term “generalized complex brane" to refer to a different type of object compared to the ones studied in this paper; our terminology is adopted from [@KL].
[^3]: For simplicity, we define the functor on $\Art$ in this paper, but an extension to $\textrm{Art}_{\C}$ is also possible.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We report the detection of large flux changes in the persistent X-ray flux of SGR 1900$+$14 during its burst active episode in 1998. Most notably, we find a factor $\sim$700 increase in the non-burst X-ray flux following the August 27$^{\rm th}$ flare, which decayed in time as a power-law. Our measurements indicate that the pulse fraction remains constant throughout this decay. This suggests a global flux enhancement as a consequence of the August 27$^{\rm th}$ flare rather than localized heating. While the persistent flux has since recovered to the pre-outburst level, the pulse profile has not. The pulse shape changed to a near sinusoidal profile within the tail of the August 27$^{\rm th}$ flare (in $\gamma$-rays) and this effect has persisted for more than 1.5 years (in X-rays). The results presented here suggest the magnetic field of the neutron star in SGR 1900$+$14 was significantly altered (perhaps globally) [*during*]{} the giant flare of August 27.'
author:
- 'Peter M. Woods, Chryssa Kouveliotou, Ersin [Göğüş]{}, Mark H. Finger, Jean Swank, Don A. Smith, Kevin Hurley, and Christopher Thompson'
title: Evidence for a Sudden Magnetic Field Reconfiguration in SGR 1900+14
---
Introduction
============
Soft Gamma Repeaters (SGRs) are believed to be a rare class of magnetized neutron stars whose collective behavior challenges physical models to explain their observed properties. There are currently four known SGRs and one candidate source (see Hurley 2000 for a review). They emit brief (durations $\sim$0.1 s), intense (super-Eddington luminosities $\lesssim$10$^{42}$ ergs s$^{-1}$) bursts of hard X-rays and soft $\gamma$-rays which repeat on timescales of seconds to years. On two exceptional occasions over the last 20 years, giant flares that reached $\sim$10$^{45}$ ergs s$^{-1}$ were observed. These two events were associated with SGR 0526$-$66 (Mazets et al. 1979) and SGR 1900$+$14 (Hurley et al. 1999a). Three of the four SGRs are positionally coincident with young supernova remnants (SNRs; SGR 1900$+$14 lies 2$^{\prime}$ outside G42.0$+$0.8 and its association is more suspect \[Vrba et al. 2000; Lorimer & Xilouris 2000\]) and all are associated with persistent X-ray counterparts. These counterparts have luminosities in quiescence of $\sim10^{34}~-~10^{35}$ ergs s$^{-1}$ and spectra that can be roughly characterized by a simple power-law ($\alpha \sim-$2.2) attenuated by interstellar absorption. Coherent pulsations have recently been detected in the persistent emission from two of these sources (SGR 1806$-$20 at 7.5 s \[Kouveliotou et al. 1998\] and SGR 1900$+$14 at 5.2 s \[Hurley et al. 1999b\]). Both sources are spinning down rapidly at rates $\sim$10$^{-10}$ s s$^{-1}$ (Kouveliotou et al. 1998 and Kouveliotou et al. 1999, respectively).
Most physical models for SGRs agree that they are young, isolated, magnetized neutron stars. The theories differ on the proposed strength of the stellar magnetic field, and hence, the energy source of the emitted radiation. Models that suggest the SGRs have field strengths $\sim10^{11}-10^{12}$ G (e.g.Marsden et al. 2000) requires accretion to account for the persistent X-ray emission. Such models have not yet offered any explanations for the hyper-Eddington burst emissions. On the other hand, Thompson & Duncan (1995, 1996) have put forward a model for the SGRs as highly magnetized neutron stars ($\sim10^{14}-10^{15}$ G), i.e. magnetars, which can account for both the persistent and burst emission properties. In the magnetar model, the decay of the superstrong magnetic field powers the persistent X-ray emission through low-level seismic activity and heating of the stellar interior (Thompson & Duncan 1996). The bursts are due to large-scale crust fractures that are driven by the evolving magnetic field (Thompson & Duncan 1995). The super-Eddington burst fluxes can be achieved in the presence of such a strong field due to the suppression of the electron scattering cross-section for some polarizations (Paczyński 1992).
SGR 1900$+$14 was discovered after only three bursts were recorded from the source in 1979 (Mazets & Golenetskii 1981). Thirteen years later, four more events were detected from this SGR (Kouveliotou et al. 1993). The pulsed signal at 5.2 s from SGR 1900$+$14 was discovered (Hurley et al. 1999b) during an April 1998 observation of the source with the [*Advanced Satellite for Cosmology and Astrophysics*]{} (ASCA). This observation coincidentally took place just three weeks prior to burst reactivation of the SGR (Hurley et al.1999c). Subsequent observations with the Proportional Counter Array (PCA) aboard the [*Rossi X-ray Timing Explorer*]{} (RXTE) of SGR 1900$+$14 confirmed the pulsations and established that the source was spinning down rapidly, having a period derivative $\sim10^{-10}$ s s$^{-1}$ (Kouveliotou et al.1999).
The peak of the burst active phase for SGR 1900$+$14 was reached on 1998 August 27 when a giant flare was recorded by numerous instruments. This flare started with a short ($\sim$0.07 s), soft spike (often referred to as the “pre-cursor”) that was followed by a much brighter, hard pulse (duration $\sim$1 s) that approached $\sim$10$^{45}$ ergs s$^{-1}$, and a soft $\gamma$-ray tail modulated at 5.2 s (Hurley et al. 1999a; Feroci et al.1999; Mazets et al. 1999). The 5.2 s oscillating tail decayed in a quasi-exponential manner over the next $\sim$6 min (Feroci et al. 2000). Integrating over the entire flare (assuming isotropic emission), at least $\sim$10$^{44}$ ergs were released in $\gamma$-rays greater than 15 keV (Mazets et al. 1999). Shortly after this flare, the All-Sky Monitor (ASM) aboard RXTE detected SGR 1900$+$14 for the first time in its four year mission at a flux level of $\sim$0.1 Crab (Remillard, Smith & Levine 1998). Approximately one week following the bright August 27$^{\rm th}$ flare, a transient radio flare lasting $\sim$10 days was recorded with the Very Large Array (Frail, Kulkarni & Bloom 1999). The temporal decay of this radio flare is consistent with a power-law having an exponent of $-$2.6 $\pm$ 1.5 (Frail et al. 1999). On August 29, another bright SGR burst was detected which resembled the August 27$^{\rm th}$ flare in many ways (Ibrahim et al. 2000). Like the giant flare, this burst had a well-defined, relatively weaker pre-cursor and was followed by a long ($\gtrsim$1000 s), oscillitory tail. However, this burst was scaled down in both peak luminosity and duration in $\gamma$-rays by a factor $\sim$100.
Sixteen days after the August 27$^{\rm th}$ flare, imaged X-ray observations with ASCA and the [*Satellite per Astronomia X*]{} (BeppoSAX) of the SGR recorded the persistent flux level of the source, which had grown by a factor $\sim$2.5 above the value prior to burst activity (Murakami et al. 1999; Woods et al. 1999a). From an earlier observation of SGR 1900$+$14 with BeppoSAX in 1997 May, while the source was in quiescence, we found that the spectrum could not be fit by a simple power-law model; however, the sum of a blackbody and a power-law provided an adequate fit to the 0.1$-$10 keV source spectrum (Woods et al. 1999a). A recent analysis of the ASCA observation in 1998 April, preceding reactivation by $\sim$3 weeks, shows that the two-component model (blackbody $+$ power-law) yields a significantly better fit to the data (D. Marsden, private communication). By including a low temperature ($kT$ $\sim$0.5 keV) blackbody component in the spectral model, the power-law index becomes flatter to accomodate the blackbody flux contribution which is most prominent at the low end of the observed energy range. Consequently, the photon index measured using the two-component model ($\alpha=1.11\pm0.19$ in 1997 May) differs significantly from the index as measured using a simple power-law model for the same data ($\alpha=1.9$ \[Woods et al. 1999a\]). When the source became more luminous during the burst active phase, the X-ray energy spectrum could be adequately modeled with a simple power-law. The thermal component was no longer significant, hence, the flux enhancement was attributed to a rise in the power-law component of the spectrum (Woods et al. 1999a). Note that this does not imply the blackbody component “disappeared” or even faded, but was rather overwhelmed by the much brighter power-law component.
Here, we have analyzed a large set of X-ray observations of the source in order to construct a more complete flux history for SGR 1900$+$14. We compare changes observed in the pulse profile during the tail of the August 27$^{\rm
th}$ event with changes in the pulse profile of the persistent emission. We discuss these observations and the constraints they place on different models for the SGRs. Finally, we suggest that these new observations support the idea of a long-lasting reconfiguration of the stellar magnetic field that took place [*during*]{} the August 27$^{\rm th}$ flare.
The X-ray Data Set
==================
Since 1992, there have been only 8 pointed observations with X-ray imaging instruments of the persistent counterpart to SGR 1900$+$14 (not including surveys). Two observations were performed with the [*Roentgen Satellite*]{} (ROSAT) High Resolution Imager (HRI), two with ASCA, and four with the BeppoSAX Narrow Field Instruments (NFI). To supplement this sparse data set, we have included more than 20 other observations using the PCA and ASM aboard RXTE. The PCA observations were particularly well-sampled during the burst active phase of the source in 1998. This time period is of key interest since it contains both the BeppoSAX and ASCA measurements of a $\sim$2.5 factor intensity increase in the persistent flux above the quiescent level and the ASM detection at 0.1 Crab.
With the exception of the two ROSAT HRI (0.1$-$2.4 keV) observations in 1994 and 1995, the instruments that observed SGR 1900$+$14 had good sensitivity within the energy range 2$-$10 keV. To make an adequate comparison of the source flux between different epochs, we have chosen to use this nominal energy range (2$-$10 keV) for flux measurements. For this reason, we have excluded the ROSAT observations from our sample. For the first two BeppoSAX NFI observations (1997 May and 1998 September) as well as for the two ASCA observations, we used the flux measurements reported earlier (Woods et al.1999a, Hurley et al. 1999b and Murakami et al. 1999, respectively). We report here on more recent NFI observations made in March and April 2000.
BeppoSAX NFI Observations
-------------------------
The BeppoSAX NFI observed the source on 2000 March 30 and April 26, approximately one year after the last detected burst activity from SGR 1900$+$14. The source exposure times were 40 ks and 40 ks for the Medium-Energy Concentrator Spectrometers (MECS) and 15 ks and 18 ks for the Low-Energy Concentrator Spectrometer (LECS) during the respective observations. In each observation, the SGR was aligned with the optical axis of the instruments. We used source extraction regions of 4$^{\prime}$ and 8$^{\prime}$ for the MECS and LECS, respectively. Due to the low Galactic latitude and dim intensity of SGR 1900$+$14, concentric rings of 6.4$^{\prime}-$9.6$^{\prime}$ and 9$^{\prime}-$13$^{\prime}$ were chosen from each pointing for background subtraction with the MECS and LECS, respectively. The resulting spectra were then analyzed using XSPEC v10.00 along with the most recent response matrices and effective area corrections.
We fit the data (0.12$-$10.5) to four different models, a simple power-law (PL), a blackbody, a thermal bremsstrahlung, and a power-law plus a blackbody (PL$+$BB), all modified by interstellar absorption. Both the blackbody and thermal bremsstrahlung models yielded poor fits to the data, while the PL and PL$+$BB models returned reduced $\chi^2$ values below 2. We find no significant difference in the spectral parameters (including source flux) between the two observations. For this reason, we decided to fit the two data sets simultaneously in order to better constrain the model parameters. We did allow for independent normalization factors for the LECS and MECS instruments as there tends to be a 5$-$10% systematic difference between the detector normalizations. As with the pre-outburst BeppoSAX observation from 1997 May, the PL$+$BB model provided the best fit to the data (see Table 1). We quantified the preference of this model over the PL model using the F-test. The probability that the measured $\chi^2$ difference (31) between the PL$+$BB model and the PL model would occur by chance is 1 $\times$ 10$^{-6}$. The unabsorbed 2$-$10 keV X-ray flux measured in 2000 March/April is 1.03(5) $\times$ 10$^{-11}$ ergs cm$^{-2}$ s$^{-1}$ with $\sim$20% of this flux coming from the thermal blackbody component. We note that the flux level in these recent observations is consistent with the flux measured in 1997 May (F$_x$ = 0.99(4) $\times$ 10$^{-11}$ ergs cm$^{-2}$ s$^{-1}$ \[Woods et al. 1999a\]), hence, we conclude that the source has returned to its quiescent flux level. Furthermore, all other spectral parameters measured in the 2000 observations are consistent with the 1997 BeppoSAX measurements with the exception of the power-law photon index ($\alpha$) which has steepened slightly (3.5$\sigma$) between the two observations.
The event times for the combined MECS units were corrected to the Solar-system barycenter and we then performed an epoch-fold period search within each observation between 5.15 and 5.25 s. We detect the pulsed signal in each observation and measure significant spin down since the last reported X-ray measurements in early 1999 (Woods et al. 1999b) as well as between the two BeppoSAX observations. The details of the timing of SGR 1900$+$14 will be discussed elsewhere (Woods et al. 2001). From the earlier BeppoSAX observations, we noted that despite a large change in source flux and pulse profile of the SGR, the root-mean-square (RMS) pulsed fraction remained constant. For the observations of 2000 March and April, we measure RMS pulse fractions of 9.4% $\pm$ 1.7% and 10.4% $\pm$ 1.7%, respectively. These values are consistent with the measurements from 1998 September (11.4% $\pm$ 1.5%) and 1997 May (12.2% $\pm$ 1.1%). This result shows the pulse fraction is constant at least within the four BeppoSAX observations which span $\sim$3 years and varying levels of source intensity and activity.
ASM Observations
----------------
Remillard et al. (1998) reported that analysis of the real-time data stream from the RXTE All-Sky Monitor (ASM) revealed a detection of SGR 1900$+$14 at $\sim$100 mCrab (2$-$12 keV) approximately two hours after the giant flare of August 27. A reexamination of the complete production light curve shows that an additional 90-s observation was performed at only $\sim$24 minutes after the flare, and this observation found SGR 1900$+$14 to be 270 $\pm$ 17 mCrab. By $\sim$7 h after the flare, the intensity of the source had fallen below the ASM detection threshhold of $\sim$20 mCrab, and has not been detected reliably since. An observation just $\sim$20 minutes before the flare shows no detectable emission from SGR 1900$+$14.
The ASM performed two passes over SGR 1900$+$14 during the 2.5 h after the flare. We have checked the Burst and Transient Source Experiment (BATSE) data for the five 90-s ASM observations performed during these passes. We find that the SGR was not occulted by the Earth for BATSE and there is no indication of significant burst activity in the large-area detector discriminator data (1.024 s time resolution). Therefore, the following ASM measurements reflect only the persistent, non-burst flux level. During each of the five 90-s ASM observations, or dwells, the best-fit average count rates for the SGR were calculated in three energy channels (1.5$-$3, 3$-$5, and 5$-$12 keV), and these count rates were averaged over each pass to yield two intensity measurements. To convert these intensities into fluxes, we took a simple power-law spectrum with a fixed slope of $-$2.2, as derived from the BeppoSAX observations of 1998 September 15, integrated over an estimated effective area for the ASM, and fit the spectral normalizations to the observed intensties. The effective area table for the ASM is a theoretical construct, compiled during the design phase of the experiment, and may differ from the actual response of the instrument. The unabsorbed flux (2$-$10 keV) at 0.4 and 2.1 hours following the beginning of the flare decayed from (7.2 $\pm$ 0.5) to (2.6 $\pm$ 0.5) $\times$ 10$^{-9}$ ergs cm$^{-2}$ s$^{-1}$. The errors here represent the statistical errors convolved with a 3% systematic error as is standard for the ASM data. There are likely further systematic errors that we are not able to accurately account for (e.g. uncertainties in the ASM effective area, spectral evolution between the ASM observations and the BeppoSAX observations).
PCA Observations
----------------
Due to the fact that the PCA is not an imaging instrument and there exist at least two bright and variable X-ray sources (4U 1907$+$09 and XTE J1906$+$09) close in angle to SGR 1900$+$14, extracting the SGR flux from these measurements required an alternative approach. From the four BeppoSAX NFI observations of SGR 1900$+$14, we know that despite the observed changes in source intensity and pulse profile, the RMS pulsed fraction remained constant. If this property of the SGR holds at all times, then a measure of the pulsed intensity would relate directly to the net source intensity. A pulsed intensity measurement with the PCA is much “cleaner” in that there is no ambiguity with where the signal originates. Under the premise that the pulsed fraction remains constant, we set out to measure the RMS pulsed intensity of SGR 1900$+$14 and indirectly, the net source flux using the PCA observations.
First, the data were grouped into segments where the PCA instrument configuration was constant, that is, observations with the same set of PCUs on and a fixed pointing. One $\sim$3 day segment near 1998 September 8 was eliminated from our sample due to the occurrence of a bright outburst from XTE J1906$+$09 (Takeshima, Corbet & Swank 1998) which dominated the count rate in the PCA. Next, the 2$-$10 keV data for each selected interval were folded on the phase connected solutions reported in Woods et al. (1999b) for the data prior to 1999 and on ephemerides reported in Woods et al. (2001) for subsequent observations. For each detector configuration, the collimator efficiency[^1] in units of counts s$^{-1}$ PCU3$^{-1}$ was calculated and the count rates in each phase bin of the folded profile were corrected accordingly. Finally, the RMS pulsed intensity for each folded light curve was measured.
The Flux History
================
The flux history of SGR 1900$+$14 over $\sim$3.8 years is given in the middle panel of Figure 1. The scale on the left vertical axis corresponds to the BeppoSAX and ASCA measurements. The right axis applies to the PCA pulsed flux measurements. The relative normalization between the pulsed flux measurements and the net source flux measurements was calculated in the following way. Prior to burst reactivation of the source in 1998 May, there was one observation each with RXTE, BeppoSAX, and ASCA. The two flux measurements during quiescence with the two imaging instruments, BeppoSAX in 1997 May and ASCA in 1998 April, are consistent with one another. Since our only measurements of a change in source flux are during a burst active interval, we assume the SGR flux is constant in quiescence. Therefore, we defined the normalization to be the ratio of the source flux as measured by BeppoSAX prior to burst activity in 1997 May to the PCA pulsed intensity measurement from 1996 September. We use this normalization to define the source flux from the PCA measurements during the burst active period. Note the good agreement of the pulsed intensity measurements (PCA) to nearly simultaneous net source intensity measurements (BeppoSAX and ASCA) following reactivation (inset of Figure 2). This strengthens our initial hypothesis that the pulsed fraction remains constant, even during burst active phases of the source. The pulsed intensity observations made with the PCA suggest the net source intensity of SGR 1900$+$14 increased by a factor $\sim$20 over the quiescent level approximately 1 day after the giant flare.
We next compared the changes observed in source flux with the change in burst activity. We quantified the burst activity using the burst rate as observed with BATSE (top panel of Figure 1). Note the good correlation between the rise and decay of the burst activity and the rise and decay of the SGR flux. We find a significant increase of the source flux during the initial burst activity of 1998 May/June, which shows the flux enhancement correlates not only with the giant flare, but also with the more common recurrent bursts. The next measurements were made starting $\sim$1 day after the giant flare and found the SGR flux was more than an order of magnitude brighter than the level during quiescence. This transient flux enhancement was an artifact of the August 27$^{\rm th}$ flare, lasting $\sim$40 days, and is discussed in greater detail below. The next sequence of pulsed flux measurements was acquired over the first seven months of 1999. During these observations, we observe a gradual decline of the X-ray flux as well as the burst occurrence rate of the SGR. It is not clear from these observations whether the flux enhancement within this portion of the X-ray lightcurve is connected with a slow decay component associated with the August 27$^{\rm th}$ flare (Kouveliotou et al. 2001), or instead, the sum of multiple smaller enhancements from burst activity in the last few months of 1998 and the first half of 1999. The next observations of SGR 1900$+$14 were performed in 2000 March$-$July, starting 11 months after the last recorded burst emission from the source. These observations found the source flux had returned to the pre-outburst level (see also section 2).
Focusing now on the large flux change observed following the giant flare, we find the net/pulsed flux decayed according to a power-law relative to the onset of the flare. Using both the net and the scaled pulsed flux measurements, we fit the data to a power-law and find a decay constant of $-$0.713 $\pm$ 0.025. This value is consistent with the power-law flux decay (F $\propto$ t$^{-0.8
\pm 0.1}$) found following the burst of August 29 (Ibrahim et al. 2000). Extrapolating the fit to the August 27$^{\rm th}$ X-ray light curve back towards the flare itself, we find fair agreement between the flux level we would expect and the ASM flux measurements (Figure 2). We note, however, that each of the ASM measurements are significantly higher than the expectation from the fit. This discrepancy may be the result of assumptions made on the source spectrum and/or the instrumental response (see section 2.2), non-negligible flux enhancements following the numerous bursts during late September, a second component related to the flare itself, and/or a slightly later reference epoch. We know from the 1998 May/June PCA measurement that the recurrent burst activity increased the persistent SGR flux by a factor $\sim$2. The bursts following the giant flare in late August and early September do not likely contribute significantly to the already large enhancement present from the August 27$^{\rm th}$ flare, however, the cumulative burst activity in mid/late September may provide comparable flux enhancements. This effect would tend to flatten the observed power-law decay slope. Another possibility is there are two components present in the ASM measurements directly after the flare; one associated with the long X-ray tail we observe for many days after the flare and one directly connected to the flare itself. Finally, the ASM measurements move closer to the fit when we push forward our reference epoch to $\sim$14 min after the [*onset*]{} of the flare. The observed duration of the August 27$^{\rm th}$ flare in $\gamma$-rays ($>$15 keV) is $\sim$6 min, so it is certainly conceivable that the SGR “afterglow” did not begin until somewhere near the end of the flare.
Pulse Profile Changes
=====================
In addition to undergoing large changes in source intensity, the pulse profile of SGR 1900$+$14 was altered dramatically during the burst active interval of 1998. Variation of the folded profile of the persistent X-ray source (2$-$10 keV) had been noted previously (Kouveliotou et al. 1999; Murakami et al.1999; Woods et al. 1999b) as well as the pulse-to-pulse evolution ($>$15 keV) found within the tail of the August 27$^{\rm th}$ flare (Mazets et al. 1999; Feroci et al. 2000). Here, we combine the observations we have analyzed with the archival data in order to show the evolution of the folded profile over the last 3.8 years (Figure 3). The light curves in the top row were all recorded prior to the August 27$^{\rm th}$ flare while the panels in the bottom row are all following. Both top and bottom rows are folded over the energy range 2$-$10 keV. The last panel on the top row was taken during/after significant burst activity in 1998 May/June, yet it shows very little change from previous observations. The two middle panels were generated from Ulysses data (25$-$150 keV) of the giant flare. The left panel was created by folding the data from 40$-$100 s after the onset of the flare, while the right panel displays the folded profile from 280$-$330 s post-trigger. The time resolution of the Ulysses data is 0.5 s, therefore, these folded profiles are under-resolved. There are certainly finer structures in these profiles that are artificially removed due to this effect. The difference between the two light curves, however, is evident: the complexity of the pulse profile diminishes with time through the burst. This is in agreement with what has been reported elsewhere (Mazets et al. 1999; Feroci et al. 2000). The same qualitative behavior can be seen in the persistent emission by comparing the top and bottom rows of Figure 3, despite being generated over a different energy range (2$-$10 keV). From this observation, we conclude that the bulk of the pulse profile change observed in the persistent emission is [*independent*]{} of source flux and appears to depend solely on the temporal relation of the observations to the August 27${\rm th}$ flare. This suggests that whatever caused the evolution of the pulse profile through the giant flare, also produced the change in the persistent X-ray light curve.
Discussion
==========
We have shown that the X-ray counterpart to SGR 1900$+$14 underwent large changes in source intensity correlated with the burst activity of the source during 1998. Following the August 27${\rm th}$ flare, the persistent SGR flux reached a maximum. This flux enhancement decayed over the next $\sim$40 days as a power-law in time with an exponent $-$0.71. All measurements of the RMS pulse fraction suggest this parameter remains constant despite the large excursion in the X-ray intensity of the SGR. Finally, we found the dramatic pulse profile change is independent of the change in luminosity of the source, but appears to be a direct consequence of the August 27$^{\rm th}$ flare.
Pulse profile changes in a fixed energy band are not uncommon in accreting X-ray binaries. However, these changes are typically correlated with variations in mass accretion rate (see e.g. White, Nagase & Parmar 1995). For a constant mass accretion rate (i.e. X-ray luminosity), the pulse profiles tend to remain unchanged. This is not the case for SGR 1900$+$14. The persistent source flux has now (March$-$July 2000) recovered to its quiescent level, yet the pulse profile has remained invariant since 1998 August 28. This result severely constrains all accretion models for SGRs.
The giant flare of August 27 released a large amount of energy in $\gamma$-rays ($E_{\rm Aug~27} \sim 10^{44}$ ergs \[Mazets et al. 1999\]) within a relatively short time period ($\sim$6 min). A fraction of a percent of this energy was stored by the star, and slowly re-radiated over the next $\sim$40 days to form the observed power-law decay of the persistent X-ray flux (Figure 2). Given the high luminosity associated with the giant flare, the radiative momentum released in the burst would excavate any hypothetical accretion disk out to a large radius where the diffusion time is of the order of months to years (Thompson et al. 2000). Therefore, accretion could not be re-established in such a short time ($\sim$10$^3$ s) as to account for the giant flare afterglow. Furthermore, kinetic energy loss (rotation) is insufficient by orders of magnitude as an energy source. We conclude that the observed power-law flux decay is clear evidence for a fundamentally new type of energy dissipation in neutron stars.
The constant pulse fraction through the long X-ray tail, and the non-recovery of the pulse profile, argues strongly against the possibility of the afterglow from the August 27$^{\rm th}$ flare being a strictly localized phenomenon. If the persistent, declining output were concentrated in a small fraction of the stellar surface or magnetosphere without reprocessing at larger radius, then the pulse fraction would be expected to change dramatically as the persistent flux returned to its baseline value. Taken at face value, this suggests that energy was released globally throughout the crust and magnetosphere of the star. The magnetosphere is the more likely location for the enhanced persistent emission because the BeppoSAX observations show the flux increase occured in the non-thermal power-law component of the persistent spectrum, not in the thermal blackbody.
We now address how the observed pulse profile change can be accommodated within the framework of the magnetar model. In the case of an isolated magnetar, the pulse shape of the persistent X-ray source is governed by the distribution of magnetic field and magnetospheric currents. Since the flux has recovered to the pre-outburst (i.e. quiescent) level, but the pulse profile has not, we conclude that either the X-ray flux is being reprocessed far outside the region where the non-thermal continuum is generated, or the distribution of currents within the magnetosphere was severely altered [*during*]{} the August 27$^{\rm
th}$ flare.
There is good evidence that the surface magnetic field of SGR 1900$+$14 had a multi-polar structure during the August 27$^{\rm th}$ giant flare, based upon the complex profile observed in the large amplitude pulsations (Feroci et al.2000). The hyper-Eddington flux of X-rays is easily channeled along (partially) open magnetic field lines in the presence of a small amount of matter. The reduction of the pulse profile from four sub-pulses to a single sub-pulse at the end of the giant flare has a simple explanation in terms of the diminishing volume of a trapped fireball (Feroci et al. 2000). Because the persistent X-ray flux is channeled passively along the field near the star, the higher multipoles are expected to remain following the giant flare. However, a giant flare may involve a global twist of the stellar magnetic field, which drives a long-lived electrical current to re-scatter X-rays at the electron cyclotron resonance near 10 stellar radii. In this model, the transient decay of the persistent emission requires the added presence of sharper field gradients, which could be created by a fracture of the rigid crust during the giant flare (Thompson et al. 2000). In light of these new observations, relevant concerns regarding this model would be the reappearance of the blackbody (presumably surface) emission in the presence of the optically thick scattering screen (which limits the amount of dissipation taking place within the screen), as well as the constancy of the pulse fraction before and after the giant flare.
We now consider an alternative explanation for the observed flux and pulse profile changes in which the magnetospheric currents of SGR 1900$+$14 are modified substantially through a global reconfiguration of the magnetic field. The reemergence of the blackbody component following the decay of the flux enhancement suggests that we are again observing radiation directly from the stellar surface. The observation of the simple pulse profile in the presence of the thermal component implies that the stellar magnetic field is now predominantly dipolar. Since the pulse profile was observed to change from complex (multi-polar field structure) to simple (di-polar field structure) within the August 27$^{\rm th}$ flare, this line of reasoning suggests that the stellar magnetic field underwent a [*global*]{} reconfiguration at the time of the giant flare. This scenario requires a reorganization of the magnetic field both inside and outside the star, which would proceed on a hydromagnetic timescale rather than the much shorter dynamical time of the stellar magnetosphere, and could plausibly take as long as $\sim$300 s. A model suggesting a global field reconfiguration during the giant flare was proposed independently by Ioka (2000). Ioka’s model was developed to explain earlier observations related to the spin period evolution of this SGR near the time of the August 27$^{\rm th}$ flare (Woods et al. 1999b). In Ioka’s model, gravitational energy was proposed to power the flare through a substantial change in the moment of inertia of the star and consequently, the total amount of [*magnetic*]{} energy released ($\sim10^{49}$ ergs) would exceed by several orders of magnitude the $\sim10^{44}$ ergs observed in the $\gamma$-ray band. One may also consider a global field reconfiguration scenario with magnetic energy as the power source for the giant flare. If the higher multipoles carry a significant fraction of the external magnetic energy, then removing them would require expending some 100$-$1000 times more energy than the observed output of the flare. In any model involving a global field reconfiguration of a magnetar, the total internal/external field energy [*must*]{} be nearly conserved during the transition, and currently, we find no compelling reason for this to be the case. In addition, if the magnetospheric currents were significantly modified during the giant flare, then the near equality between the “baseline” X-ray flux before and after the giant flare would be coincidental.
A strong, evolving magnetic field is the foundation of the magnetar model for the SGRs. The dramatic change in the pulse profile of SGR 1900$+$14 following the August 27$^{\rm th}$ flare provides strong evidence for some type of large-scale reconfiguration of the magnetic field. The two models involving stellar field dynamics outlined above each have their own merits when compared to the observations, however, both involve some level of coincidence. Moreover, the global field reconfiguration model posesses a fundamental flaw in that the energetics of the model do not match the observations. Without a comprehensive model to explain the cause and long-lasting effects of the August 27$^{\rm th}$ flare, the giant flares, and to a lesser extent SGRs in general, remain quite enigmatic.
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[cccccc]{}
BB$+$PL & 2.7 $\pm$ 0.4 & 0.43 $\pm$ 0.05 & 4 $\pm$ 1 & 1.98 $\pm$ 0.16 & 116.1/107
PL & 2.4 $\pm$ 0.2 & ... & ... & 2.45 $\pm$ 0.06 & 146.8/110
[^1]: The collimator efficiency files for each PCU can be found at the following www address: http://lheawww.gsfc.nasa.gov/users/keith/fieldofview/collimator.html
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'As a parent population to long gamma-ray bursts (LGRBs), energetic core-collapse supernovae (CC-SNe) are leading candidates as multi-messenger sources of electromagnetic and gravitational-wave emission for LIGO-Virgo and KAGRA. While their central engines are currently unknown, this outlook derives from a general association with newly born neutron stars, black holes and high-density accretion disks that may extend down to the Inner Most Stable Circular Orbit (ISCO) of the latter. We here highlight the capability of heterogeneous computing for deep searches for [*broadband extended gravitational-wave emission*]{} (BEGE) from non-axisymmetric accretion flows onto rotating black holes with durations of tens of seconds similar to Extended Emission in LGRBs and SGRBEEs. Specific attention is paid to electromagnetic priors derived from BATSE, [*BeppoSAX*]{} and [*Swift*]{} and data-analysis by GPU-accelerated butterfly filtering with over one million chirp templates per second. In deep searches using banks of up to 8 million chirp templates, the challenge is to identify signals of astrophysical origin in a background of pronounced correlations between the LIGO detectors H1 and L1. As the parent population of normal LGRBs, relatively more frequent supernovae of type Ib/c are of particular interest to blind all-sky searches, in archive LIGO S6 or real-time observation runs concurrently with electromagnetic observations covering the Local Universe up to about 100Mpc at upcoming Advanced LIGO sensitivity. Detection of their output in gravitational waves is expected to unambiguously determine the nature of their central engines and, by implication, that of GRBs.'
author:
- 'Maurice H.P.M. van Putten'
- Amir Levinson
- Filippo Frontera
- Cristiano Guidorzi
- Lorenzo Amati
- Massimo Della Valle
title: 'GPU-searches for broadband extended emission in gravitational waves in nearby energetic core-collapse supernovae'
---
\
[LIST OF SYMBOLS]{}\
[lll]{}\
$c$ & velocity of light ($3\times 10^{10}$ cm s$^{-1}$)\
$c_s$ & sound speed\
$D$ & source distance\
$E_B$ & energy in poloidal magnetic field\
$E_\gamma$ & true energy in gamma-rays\
$E_{iso}$ & isotropic equivalent energy\
$E_c$ & maximal spin energy PNS $(3\times 10^{52}$ erg)\
$E_{res}$ & energy in reservoir\
$E_{rot}$ & energy in rotation\
$\eta$ & efficiency\
$h$ & dimensionless gravitational strain\
$\xi$ & dimensionless mass-inhomogeneity\
$L_j$ & luminosity in baryon poor jet (BPJ)\
$M, M_T$ & black hole and torus mass\
$\dot{m}$ & accretion rate\
$M_\odot$ & solar mass ($2\times 10^{33}$ g)\
$\dot{N}$ & event rate\
$\nu$ & kinematic viscosity\
$\nu_H$ & black hole rotation frequency in Hz\
$\omega$ & frame dragging angular velocity\
$\Omega_{ISCO}$ & angular velocity at ISCO\
$\Omega_H$, $\Omega_T$ & black hole and torus angular velocity\
$q$ & index of rotation in accretion disk\
$R$ & branching ratio\
$R_g$ & gravitational radius $(GM/c^2)$\
$R_S$ & Schwarzschild radius ($2R_g$)\
$r_{ISCO}$ & radius of ISCO\
$r_c$ & transition radius to fragmentation by cooling\
$r_d$ & Roche radius in accretion flows\
$r_b$ & viscosity-to-radiation driven transition radius\
$T_{spin}$ & lifetime of black hole spin\
$\tau$ & coherence time scale\
$t_{ff}$ & free fall time scale\
$z$ & $r_{ISCO}/R_g$\
$\theta_H$ & half-opening angle on horizon
Introduction
============
The recent LIGO detection GW150914 poses a dramatic opening to a whole new window to the Universe [@LIG16]. Together with the upcoming commissioning of Virgo and KAGRA, we will now be in a position to pursue observations well beyond the limits of electromagnetic radiation, neutrinos and (ultra-)high-energy cosmic rays. As broadband detectors covering 30 - 2000 kHz, LIGO-Virgo and KAGRA offer unprecedented power of discovery relevant to an exceptionally broad class of astrophysical sources [@sat09; @cut02]. While the black hole merger event GW150914 was unexpected and left no conclusive signature in the electromagnetic spectrum [@kal17], it nevertheless offered new results of direct astronomical interest with estimates of mass and spin of the black hole progenitor, $$\begin{aligned}
M_1=35.7_{-3.8}^{5.4}M_\odot,~~a_1/M_1=0.31^{+0.48}_{-0.28},\\
M_2=29.1_{-4.4}^{3.8}M_\odot,~~a_2/M_2=0.46^{+0.48}_{-0.42},
\label{EQN_GW15}\end{aligned}$$ that run counter to familiar observations on stellar mass black holes in X-ray binaries in the Milky Way. The inferred high mass $M_i$ of the black holes may originate from core-collapse of Population III stars [@kin14; @ina17] and their slow dimensionless spin $a_i/M_i$ may indicate a process of spin down soon after birth such events [@van17a].
Gravitational radiation has long since been known to be important in binary evolution of compact stars, as may be seen by long time observations of binary evolution in the electromagnetic spectrum [@ver97]. Notable examples are Hulse-Taylor pulsar PSR B1913+16 [@tay89; @tay94; @wei10], the double pulsar PSR J0737-3039 [@lyn04], and ultra-short period cataclysmic variables with He mass transfer from a degenerate dwarf directly onto a companion (low mass) white dwarf [@sma67; @pac67; @fau71; @fau72; @nel05; @pos06; @bil06]. For instance, AM CVn ES Cet ($d\simeq 350$ pc) has an orbital period of about 10 min, a mass ratio $q\simeq 0.094$ of the binary with a white dwarf of mass $M\simeq0.7M_\odot$ and a luminosity $L_{EM}\simeq 10^{34}$ erg s$^{-1}$ [@wou03; @esp05]. Its gravitational wave-to-electromagnetic luminosity satisfies $L_{GW}/L_{EM}\simeq0.3$. At an orbital period of about 5 min, current data on RX J0806 [@bil06] suggest that it conceivably satisfies $$\begin{aligned}
L_{GW} \simeq {L_{EM}}.
\label{EQN_i1}\end{aligned}$$ These exceptional cases, therefore, demonstrate genuinely relativistic evolution that ultimately terminates in a binary merger. GW150914 is the most extreme example to-date. In terms of the unit of luminosity $$\begin{aligned}
L_0 = \frac{c^5}{G} = 3.6\times 10^{59} \,\mbox{erg s}^{-1},
\label{EQN_g5}\end{aligned}$$ where $G$ is Newton’s constant and $c$ is the velocity of light $c$, it featured a peak luminosity of about $0.1\%L_0$. In this light, PSR B1913+16 is remarkably gentle with $L_{GW}\simeq 10^{-29}L_0$.
Multi-messenger output in electromagnetic and gravitational radiation is expected from neutron star-neutron star (NS-NS) or neutron star-black hole (NS-BH) mergers. These mergers are widely considered to explain the most relativistic transients in the sky: cosmological gamma-ray bursts (GRBs), discovered serendipitously by nuclear treaty monitoring satellites [@kle73]. Observationally, we infer their origin in a compact relativistic inner engines from the dimensionless parameter [@van00] $$\begin{aligned}
\alpha_E = \frac{G E}{c^5\delta t}=2.75\times10^{-5} \,\left( \frac{E_{52}}{\delta t_{-3}}\right)
\label{EQN_alpha1}\end{aligned}$$ for burst energies $E= E_{52} 10^{52}$ erg and variability times $\delta t=\delta t_{-3}$ ms, where $G$ denotes Newton’s constant and $c$ is the velocity of light. The observed isotropic equivalent energies $E_{iso}=10^{48}-10^{54}$ erg s$^{-1}$ and variability times $\delta t$ down to 0.1 ms show $\alpha_E$ up to $10^{-4}$. Such values are extremely large compared to those of other transients, including GRB 980425 associated with SN 1998bw [@gal98] and galactic sources such as GRS 1915+105 [@mir94]. It implies inner engines in the form of neutron stars or stellar mass black holes, more likely so than aforementioned white dwarfs in CVs. Neutron star masses tend to cluster around $1.4M_\odot$ [@tho99], from 1.25 $M_\odot$ of PSR J0737-303B [@lyn04] to 2.1 $M_\odot$ in the NS-WD binary PSR J075+1807 [@nic04]; masses of black hole candidates in X-ray novae are broadly distributed between about 5-20 $M_\odot$ [@bai98].
GRBs show anomalous Eddington luminosities of $L_\gamma \simeq 10^8-10^{14}L_{Edd}$, given their limited durations of typically less than one minute (Fig. \[fig4B1\]). These super-Eddington luminosities defy an origin in electromagnetic interactions in a baryonic energy source. The only physical processes known that might circumvent these limitations are neutrino emissions and gravitational interactions allowed by the theory of general relativity. In anisotropic emission, the true energy in gamma-rays $E_\gamma < E_{iso}$, e.g., when GRBs are produced in jet-like outflows at finite opening angles. Even thus, some events have $E_\gamma\simeq 10^{52}$ erg. Typical values of events that reveal collimation show a relatively narrow distribution around [@fra01; @ghi06; @ghi13] $$\begin{aligned}
E_\gamma\simeq 9\times 10^{50}\,\mbox{ erg}.
\label{EQN_frail}\end{aligned}$$
Normal long GRBs have an accompanying supernova explosion with kinetic energies $E_k$ typically greater than $E_\gamma$. In exceptionally energetic events, $E_k$ points to a required energy reservoir $E_{res}$ that exceeds the maximal spin energy $E_c$ of a rapidly rotating neutron star [@van11b]. These events probably mark the birth of a black hole, rather than a neutron star. Following the Burst and Transient Source Experiment (BATSE) classification of short (SGRB) and long GRBs (LGRB) with durations $T_{90}<2$ s and, respectively, $T_{90}>2$ s (Fig. \[fig4B1\]), [*Swift*]{} discovered short GRBs with Extended Emission (SGRBEE) lasting tens of seconds to well over a minute. Their soft EE is very similar to long GRBs with accompanying supernova. In attributing SGRBs to mergers, SGRBEEs defy the dynamical time scale $T_{merger}\simeq 10$ ms of NS-NS or NS-BH mergers by a large factor.
![(Left.) The bimodal distribution of durations in the BATSE 4B Catalog, showing a population of short GRBs (less than 2 s) and long GRBs (over 2 s) (Reprinted from [@bat01].) (Right.) Core-collapse supernovae form a heterogeneous class of events, broadly partitioned in normal (narrow line) and relatively more energetic (broad line) events. (Reprinted from [@van11b], data from [@mau10]).[]{data-label="figSN1987A3"}](f2 "fig:") ![(Left.) The bimodal distribution of durations in the BATSE 4B Catalog, showing a population of short GRBs (less than 2 s) and long GRBs (over 2 s) (Reprinted from [@bat01].) (Right.) Core-collapse supernovae form a heterogeneous class of events, broadly partitioned in normal (narrow line) and relatively more energetic (broad line) events. (Reprinted from [@van11b], data from [@mau10]).[]{data-label="figSN1987A3"}](f1 "fig:")
\[fig4B1\]
These electromagnetic observations introduce the mystery of the physical nature of GRBs by extreme values of (\[EQN\_alpha1\]) and $$\begin{aligned}
%\begin{array}{ll}
L_\gamma >> L_{Edd},~ ~
E_{res} >E_c ~\mbox{(in some cases)},~ ~
T_{90}^{SGRBEE} >>T_{merger}.
%\end{array}
\label{EQN_i2}\end{aligned}$$ GRB inner engines hereby should be ultra-relativistic, conceivably operating by [*strong gravitational interactions with high density matter on the scale of their Schwarzschild radius*]{} $R_S$, defined as twice the gravitational radius $$\begin{aligned}
R_g = \frac{GM}{c^2}
\label{EQN_g1}\end{aligned}$$ for a mass $M$. If so, their inner engines may well be luminous in gravitational waves over the lifetime of the inner engine, i.e., up to tens of seconds indicated by long GRBs [@van01b]. Consequently, GRBs are of considerable interest as candidate sources of gravitational radiation that may be probed by upcoming detectors LIGO-Virgo in the US and Europe and KAGRA in Japan. “[*If gravitational waves are detected from one or more gamma-burst triggers, the waves will almost certainly reveal the physical nature of the trigger*]{}” [@cut02].
However, GRB triggers within the sensitivity distance of LIGO-Virgo and KAGRA are rare and they are difficult to detect due to beaming. Even corrected for beaming, the true GRB event rate is about one per year within a distance of 100 Mpc. Practically, it is rather similar to the event rate of double neutron star coalescence. To within the distance to Virgo $(D\simeq 20$ Mpc), it implies about one event per century. Beaming is less severe in afterglow emission that follows the prompt phase decreases as the blast wave slows down and, at late times, the emission is ultimately roughly isotropic. In such cases, an observer might detect an orphan afterglow emission at radio wavelengths [@lev02], a few months after the explosion. Identifying GRBs by afterglow emissions leaves uncertain the true time-of-onset of the trigger, however, hampering efficient search for an accompany gravitational wave burst. For SGRBs from mergers, considerable improvement in sensitivity will be realised in the upcoming advanced generation of LIGO-Virgo and KAGRA at frequencies up to a few hundred Hz by advanced seismic suspension. However, at higher frequencies expected to be relevant to LGRBs from CC-SNe, improvement in sensitivity requires high laser power expected in next generation detectors. For instance, a sensitivity distance out to 35 Mpc by advanced VIRGO-LIGO and KAGRA may include the LLGRB event GRB 980425/SN1998bw [@gal98]. Our perspectives hereby improve but not substantially with an anticipated rate of one detectable GRB-SN every few years.
To circumvent the above mentioned observational limitations to detect long GRBs, we propose a focus on type Ib/c supernovae [@mae02; @mae08; @fru06], that are far more numerous given their relatively small branching ratio of about 1% into long GRBs. Core-collapse supernovae (CC-SNe) form a remarkably heterogenous class of events (Fig. \[figSN1987A3\]), and the most energetic events of type Ib/c stand out as the parent population of normal LGRBs.
The small branching ratio of CC-SNe into successful GRBs is commonly attributed to the challenge of creating an energetic inner engine sufficiently long lived, perhaps intermittently, for its ultra-relativistic outflows to successfully break out of the progenitor remnant envelope. Unsuccessful jet breakout from the stellar envelope in a CC-SN event [@maz08; @cou11; @bro12] will lead to so called “choked GRB." Such supernovae may appear as a low-luminosity long GRB or, more broadly, as a class of X-ray transients [@sod08]. Conceivably, therefore, the formation of energetic inner engines is more frequent than successful GRB-SNe. In this review, we shall therefore focus on the outlook on gravitational wave emissions from such energetic inner engines with or without a successful long GRB. Similar considerations might apply to their emission in neutrinos [@mes01].
For energetic type Ib/c supernovae, we set out to develop an outlook and search for [*broadband extended gravitational-wave emission*]{} (BEGE) by the nature of black hole formation and evolution in CC-SNe.
Our outlook is modelled based on current phenomenology of GRBs and accompanying hyper-energetic supernovae. Detailed spectral and temporal analysis of GRBs from BATSE, [*BeppoSAX*]{} and [*Swift*]{} combined points to long-lived inner engines comprising rotating black holes, that appear common endpoints to energetic core-collapse events and mergers of neutron stars with neutron stars or stellar mass black holes alike. They hereby define a leading candidate as a universal inner engine to LGRBs and the [*Swift*]{} class of SGRBEE and LGRBNs [@van14b]. In interaction with high density accretion flows, potentially powerful gravitational wave emission may ensue, powered by accretion or the angular momentum of the black hole mediated by relativistic frame-dragging. The existence of frame dragging is not in doubt: recent measurements of non-relativistic frame dragging around the Earth are in excellent agreement with general relativity [@ciu04; @ciu07; @ciu09; @eve11]. (Gravity Probe-B measurement is equivalent to that at 5 million Schwarzschild radii of a black hole at extremal spin with the same angular momentum as the Earth.) Specifically, accretion flows onto rotating black holes offer a window to broadband extended gravitational-wave emission, from non-axisymmetric accretion flows and high density matter accumulated at the Inner Most Stable Circular Orbit (ISCO), contemporaneously with two-component relativistic outflows that may drive an accompanying supernova explosion and GRB. Some of these model considerations can be confronted with data from GRB catalogues of BATSE, [*BeppoSAX*]{} and [*Swift.*]{} The resulting outlook on long duration ascending and descending chirps from accretion flows onto rotating black holes suggests searches for BEGE, accelerated by recent developments in high performance computing.
The prospect for [*long duration*]{} gravitational-wave bursts from accretion flows onto rotating black holes core-collapse of massive stars highligted here is aimed at broadening our outlook beyond various existing discussions on gravitational waves from CC-SNe, much of which focused on complex short duration bursts from core-collapse and core-bounce in the first one or two seconds producing neutron stars [@ott09] with a LIGO sensitivity distance limited to a few Mpc [@rov09]. Their connection to GRB-supernovae and extremely energetic supernovae, however, is not obvious [@bur07; @des08]. In contrast, the universal appearance of Extended Emission in LGRBs and SGREEs indicates central engine lifetimes of tens of seconds that may be at work in the more frequent group of energetic type Ib/c supernovae. It therefore appears opportune to search for gravitational-wave bursts with extended emission of potentially similar durations from, broadly speaking, nearby energetic core-collapse supernovae and to develop the required near-optimal search algorithms to achieve maximal sensitivity distance.
The energetic output in BEGE may be large by the ample energy reservoir in angular momentum of rotating black holes, exceeding that of rotating neutron stars by some two orders of magnitude opens a radically new window to energetic bursts in gravitational waves with durations up to minutes. [If detected, LIGO-Virgo and KAGRA probes may reveal rotating black holes by calorimetry on their output in gravitational waves]{} [@van02b]. In general terms, gravitational radiation from non-axisymmetries associated with core-collapse of high angular momentum progenitors and non-axisymmetric collapse has been well appreciated [@bek73; @thu74; @nov75; @eps76; @det81], see further [e.g. @det81; @kot06; @ott09; @fry11]. Here, we emphasise potentially extreme luminosities from non-axisymmetric accretion flows down to ISCO powered by the angular momentum of the central black hole [@van01b], and hence the need for a general search method for both ascending chirps and descending chirps from accretion flows onto rotating black holes [@lev15].
Energetic type Ib/c supernova have an event rate of about 100 per year within a distance of 100 Mpc, and they are readily found in optical surveys using moderately sized telescopes. As targets of opportunity for gravitational wave bursts, they are hereby competitive with mergers, whenever the fraction successfully producing a gravitational wave burst exceeds 1% [@heo15]. Additionally, nearby galaxies such as M51 ($D\simeq 8$ Mpc) and M82 $(D\simeq 4$ Mpc) each with an event rate of over one core-collapse supernova per decade. By their proximity, these events appear of interest as well, independently of any association with type Ib/c supernovae or GRBs [@and13; @aas14]. While CC-SNe define the most energetic transients in the Universe, the dimensionless strain amplitude of any accompanying gravitational waves will be small by the time it reaches the detector [*and*]{} by a possibly prolonged duration of emission. To extract signals deeply within the detector noise, it is desirable to take full advantage of high performance computing on [*Graphics Processor Units*]{} (GPUs), to search for essentially un-modelled emission with near-optimal detection sensitivity by matched filtering against a large bank of chirp templates.
Given the current quest for a multi-messenger source of gravitational radiation, we believe it to be opportune to highlight prospects and search for gravitational radiation from energetic type Ib/c supernovae as a parent population of long GRBs, of interest to LIGO-Virgo and KAGRA up to distances of about 100 Mpc at advanced detector sensitivity.
Quadrupole gravitational radiation
----------------------------------
Normal long GRBs, SGRBs with Extended Emission, energetic core-collapse SNe and possibly superluminous SNe are all likely powered by neutrons stars or stellar mass black holes. From their generally aspherical output in electromagnetic radiation, it may be inferred that their putative inner engine should be rich in angular momentum. Angular momentum serves as a reservoir of energy that, in collapse, points to the formation of an accretion disk. In the present context, the density of any such disk will be high. Any non-axisymmetry introduces a multipole mass moment, that will inevitably luminous in gravitational waves.
Non-axisymmetries in mass-flow may result from instabilities [@van02; @kob03; @van03; @pir07] due to cooling in self-gravitating disks (e.g. [@gam01; @ric05; @mej05; @lov14; @had14]), magnetic stresses [@tag90; @tag99; @tag01; @lov14], that may account for high frequency QPOs in mciro-quasars [@tag06] or flaring in SgrA\* [@tag06b], or enhanced pressure by heating or magnetic fields due to feedback by a rotating black hole [@van03; @bro06]. In addition, intermittent accretion onto the black hole may lead once more to aforementioned excitation of QNM ringing.
Gravitational radiation is essentially [*inevitable*]{} from extreme transient events forming neutron stars and black holes, arising from non-axisymmetric mass-motion on the Schwarzschild radius $R_S$ of the system, e.g., a wobbling neutron star or a non-axisymmetric accretion flow onto the black holes. Its basic premises derive from dimensional analysis and, with no small parameters, the gravitational wave luminosity $L_{GW}$ will be a fraction of $L_0$ in (\[EQN\_g5b\]).
Gravitational radiation is a key prediction of general relativity as a mixed elliptic-hyperbolic theory of gravitation described by a metric with associated Riemann tensor [@pir57; @pir09; @tra09] coupled to matter. In what follows, we shall change to geometrical units and denote the gravitational radius (\[EQN\_g1\]) by $M$. Equivalently, we put $G=c=1$ in (\[EQN\_g1\]). Thus, $M$ parametrizes perturbations in space-time at a distance $D$ in terms of a [*dimensionless strain*]{} $$\begin{aligned}
h = \frac{M}{D}+h_{GW},
\label{EQN_g2}\end{aligned}$$ where $h_{GW}$ is the strain amplitude in gravitational radiation. At large distances, $h_{GW}$ satisfies the linearized Einstein equations in vacuo, given by a second order wave equation for small amplitude perturbations that satisfies the same dispersion relation as electromagnetic waves (Appendix A).
At the lowest frequency, gravitational radiation is described by the quadruple gravitational-wave formula, that may be derived from the rotating tidal field in a binary system. This time harmonic excitation acts as a source terms to gravitational wave emission. More generally, tidal fields arise from multipole mass moments $I_{lm}$, where $l$ and $m$ refer to the poloidal and azimuthal quantum numbers of spherical harmonics. Thorne [@tho80] gives a comprehensive overview of gravitational wave luminosity in $h_{GW}$ above from multipole mass moments defined by projections on the spherical harmonics $Y_{lm}$ $(l\ge m\ge2)$, $$\begin{aligned}
L_{lm} = \frac{1}{32\pi} \frac{G}{c^{2l+1}} \left( \frac{d^{l+1}}{dt^{l+1}} I_{lm}\right)^2
\label{EQN_Tho1a}\end{aligned}$$ by $$\begin{aligned}
I_{lm} = \frac{16\pi}{(2l+1)!!} \left[ \frac{(l+1)(l+2)}{2(l-1)l} \right]^\frac{1}{2} \int_V Y_{lm}^* r^l dm,
\label{EQN_Tho1b}\end{aligned}$$ where $dm=\rho\, d^3x$ over the source region $V$ expressed in spherical coordinates $(r,\theta,\varphi)$ as before.
In contrast, radial motion introduces time-dependence with $m=0$ which, by (\[EQN\_Tho1a\]-\[EQN\_Tho1b\]), does not tap into the angular momentum of the source. Gravitational wave emission from axisymmetric sources tends to be remarkably inefficient. Illustrative is the gravitational wave output of about 0.2% from head-on collisions of two black holes [@ann93] (cf. [@gib72]). This low efficiency reflects the effective regularization by black hole event horizons of the singular behavior of Newton’s law between point particles [@van12c]. In contrast, the $m\ne0$ tidal fields in binary mergers shows appreciable efficiency up to about 2% (e.g. [@kyu13; @szi15]) and slightly more in neutron star-neutron star coalescence [@ber15].
In a binary with binary separation $a$ much larger than the Schwarzschild radius of the system, the gravitational potential at the root of the metric is the Newtonian potential $U=M/a$ along with an orbital frequency $\omega$, $\omega^2=M/a^3$. The total luminosity $L_{GW}$ in gravitational radiation, dimensionless in geometrical units, hereby satisfies $L_{GW} \sim U^n \times (M\omega)^2 = (M/a)^{3+n}$ for some $n$, taking into account scaling with dimensionless orbital frequency $M\omega$. In the distant radiation field, $L_{GW}=4\pi D^2 w$ in terms of the radiation intensity $w=k\dot{h}^2$ for some constant $k$. In geometrical units, $w$ is of dimension cm$^{-2}$. The angular frequency of a tidal field is twice the angular velocity $\omega$ of the binary motion, i.e., $\dot{h}=2\omega h$. Since $h_{GW}$ and $L_{GW}=4\pi D^2w$ are both dimensionless, $h_{GW}$ factors over the Newtonian scale factor $M/D$ and $U$ (\[EQN\_g2\]), i.e., $L_{GW} = 4\pi D^2 (2 \omega (M/D) U)^2$. Hence $n=2$ with the familiar result [e.g. @wal84] $$\begin{aligned}
L_{GW} \propto M^2 a^4 \omega^6 \propto \left(M\omega\right)^\frac{10}{3}.
\label{EQN_g4}\end{aligned}$$ $L_{GW}$ in cgs units obtains by multiplying (\[EQN\_g4\]) with the unit of gravitational wave luminosity (\[EQN\_g5b\]).
For a binary of two masses $M_i$ $(i=1,2)$ in circular motion, a detailed derivation obtains the quadrupole formula of gravitational radiation (Appendix A) $$\begin{aligned}
L_{GW} = \frac{32}{5} \left(\mu\Omega\right)^\frac{10}{3},
\label{EQN_g5b}\end{aligned}$$ further replacing $M$ with the [*chirp mass*]{} $\mu = {M_1^\frac{3}{5}M_2^\frac{3}{5}}{(M_1+M_2)^{-\frac{1}{5}}}.$ An extension to non-circular orbits by incorporating enhanced emission at higher frequency harmonics obtains by including a factor $F(e)=(1+(73/24)e^2+(37/96)e^4)/(1-e^2)^{7/2}$ as a function of ellipticity [@pet63; @pos06]. It has been verified experimentally in long-term radio observations of the orbital decay of the Hulse-Taylor binary PSR 1913+16 to better than 0.1% [@tay94] by the additional factor of $F(e)= 11.8568$ for the observed ellipticity $e=0.6171334$. At the distance of 6.4 kpc, its $L_{GW}\simeq 8\times 10^{31}$ erg s$^{-1}$ produces an instantaneous dimensionless strain at the Earth that, as such, may be evaluated directly in geometrical units as $$\begin{aligned}
h\simeq \frac{L_{GW}^{1/2}}{\Omega D} \simeq 1.38 \times 10^{-22},
\label{EQN_h1}\end{aligned}$$ based on previous arguments with $k=1/16\pi$ (Appendix A). As a relatively compact binary, the Hulse-Taylor binary coalesces in about 310 Myr [@pos06].
Coincidentally, (\[EQN\_h1\]) is very similar to the scale for the maximal strain produced in the final merger of a circular binary of two neutron stars of total mass $M=M_1+M_2$ in the Local Universe. In this event, $L_{GW}=(2/5)\left({M}/{a}\right)^5$ defines the observed strain amplitude for an equal mass binary. Following averaging over the orientation of the source (e.g., [@pos06] for a more general discussion) $$\begin{aligned}
h = \sqrt{\frac{2}{5}}\frac{M^2}{aD} = \sqrt{\frac{2}{5}} \frac{M}{D} \left(\pi Mf_{GW}\right)^\frac{2}{3},\end{aligned}$$ where we dropped the subscript $GW$. It explicitly shows that $h$ is the product of the Newtonian specific binding energy $U=M/a$ and the scale factor $M/D$ (see also [@sat09]). Here, we expand the result to the gravitational wave frequency $f=2f_{orb}$ in terms of the orbital frequency $f_{orb}$. That is (e.g. [@tho92; @ju00; @pos06]), $$\begin{aligned}
h= 6.3 \times 10^{-23} \left(\frac{M}{3M_\odot}\right)^2 \left(\frac{D}{100\,\mbox{Mpc}}\right)^{-1} \left(\frac{f}{1000\,\mbox{Hz}}\right)^\frac{2}{3}.
\label{EQN_h2}\end{aligned}$$ In NS-NS coalescence, the chirp (\[EQN\_h2\]) holds true up to the instant when $h$ peaks at $f\simeq 800$ Hz [@bai08]. Numerical simulations show that the neutron stars subsequently break up, and merge into a hyper massive object (e.g. [@ber15a]) followed by collapse into a stellar mass black hole accompanied by a burst of quasi-normal mode (QNM) ringing. The result is a rapidly rotating low-mass black hole of close to mass $M$ with an accretion disk of about 0.01-0.1 $M_\odot$ [@bai08]. It may give rise to a short GRB, but perhaps also a SGRB with Extended Emission (SGRBEE) [@van14b].
Multi-messenger emission from SN1987A
-------------------------------------
SN1987A in the Large Magellanic Cloud (LMC, $D\simeq 50$ kpc (Figs. \[figSN1987A1\]-\[figSN1987A3\]) stands out as the first genuine multi-messenger event by a luminous output in electromagnetic radiation and MeV neutrinos. Characteristic for a core-collapse supernova, SN 1987A was radio-loud [@tur87] and aspherical [@pap89]. It also featuring relativistic jets [@nis99] with possible black hole remnant, based on a lack of detection of a neutron star and on evidence for a black hole in the rather similar type IIL event SN1979C [@mat79; @pat11]. Collectively, core-collapse supernova form a rather heterogeneous group [@fil97], that may be broadly partitioned in narrow line and broad line events (Fig. \[figSN1987A3\]), where the latter tend to be relatively more energetic featuring relativistic ejection velocities. SN1987A belongs to a class with a relatively massive progenitor [@gil87; @kir87] powered by a probably angular momentum rich inner engine based on the dramatically aspherical supernova remnant (Fig. 1).
Its output $E_\nu\simeq 10^{53}$ erg in $>10$MeV neutrinos offered our most direct view yet on the inner-most workings of a CC-SN. It may, in fact, have produced a stellar mass rotating black hole, as may be inferred from the aspherical remnant seen today. Spectroscopic observations of SNe-Ibc [@maz05; @tau09; @mod14] reveal that the geometry of ejecta of stripped envelope supernovae is, in about 50% of the observed events, is strongly asymmetric. Any non-axisymmetric angular momentum rich explosion mechanism inevitably produce gravitational waves, which may be generic to energetic supernovae.
$E_\nu$ is particularly relevant as evidence of the formation of high density matter that, combined with ample angular momentum in the progenitor as may be inferred from the aspherical supernova remnant, are just the kind of conditions leading up to an additional output in gravitational radiation, provided this collapse event developed canonical non-axisymmetric mass-motion at its core. Such output will be especially luminous, whenever such mass motion takes place on the Schwarzschild scale of the system, e.g., the Inner Most Stable Circular Orbit (ISCO) around a newly formed black hole. By virtue of the large value of $L_0$ in (\[EQN\_g5\]), the quadrupole formula (\[EQN\_g4\]) predicts an appreciable luminosity even when $L_{GW}$ in (\[EQN\_g5b\]) is small, e.g., $$\begin{aligned}
L_{GW}\simeq 10^{50}-10^{53}~\mbox{ erg~ s}^{-1}
\label{EQN_LGW0}\end{aligned}$$ by $10^{-9}-10^{-6}$ in (\[EQN\_g5b\]) is on par with the observed luminosities in the electromagnetic radiation and neutrino emissions of SN1987A. This outlook opens a broad window to observationally relevant gravitational wave luminosities, even in the face of considerable model uncertainties and chirp masses small relative to the central object.
To be specific, consider a mass-inhomogeneity with gravitation radius $\delta m$ about a mass $M$ with the aforementioned chirp mass in the limit of $M_2=\delta m$ much smaller than $M_1=M$. By virtue of (\[EQN\_g5\]), $L_{GW}$ reaches luminosities on par with SN1987A’s neutrino luminosity already for $\delta m/M\simeq 0.1\%$ orbiting at a few times the Schwarzschild radius $R_S=2R_g$. As a mass perturbation in a torus or inner disk of mass $M_T\simeq 0.01 \,M$, the gravitational wave luminosity (\[EQN\_g5b\]) of $\delta m$ satisfies $L_{GW} =\frac{32}{5} \left({\delta m}/M\right)^2\left({M}/{a}\right)^5 L_0$ in the limit of a small chirp mass. Expressed in terms of the dimensionless inhomogeneity $\xi=\delta m/M_T$ and mass $\sigma=M_T/M$, we have $$\begin{aligned}
L_{GW} = 2\times 10^{51} \left(\frac{\xi}{0.1}\right)^2 \left(\frac{\sigma}{0.01}\right)^2\left(\frac{4M}{a}\right)^{5}~\mbox{erg~s}^{-1},
\label{EQN_g4b}\end{aligned}$$ where $a$ denotes the orbital separation. The observed dimensionless strain at a source distance $D$ satisfies $$\begin{aligned}
h = \frac{L_{GW}^\frac{1}{2}}{\Omega D} = 4\sqrt{\frac{2}{5}} \xi \sigma \frac{M}{D} \left(\pi M f\right)^\frac{2}{3}.
\label{EQN_h1987AA}\end{aligned}$$ Generalized to a similar event in the Local Universe, e.g., SN1979C [@pat11], scaled to a distance of 20 Mpc, we have $$\begin{aligned}
h=3.4\times 10^{-23}\,M_1 \frac{\xi}{0.1}\,\frac{\sigma}{0.01}\left(\frac{D}{20\,\mbox{Mpc}}\right)^{-1} \left(\frac{f}{600\,\mbox{Hz}}\right)^{\frac{2}{3}},
\label{EQN_h1987A}\end{aligned}$$ where $f=2f_{orb}$ in the Newtonian approximation $2 \pi f_{orb} = M^{-1} (M/a)^{3/2}$ and $M=M_1\,10M_\odot$.
![(Left.) SN 1987A is a Type II supernovae produced by core-collapse of the supergiant Sanduleak -69$^o$ 202 in the Large Magellanic Cloud at a distance of about 50 kpc [@gil87; @kir87]. Shown is the neutrino light curve compiled from Kamiokande (stars) and IMB (circles) listed in [@bur87] associated with the optical identification of SN 1987A [@gar87; @her87; @kun87]. (Right.) The SN1987A neutrino light curve, showing an initial energy of $>$10 MeV representative for the formation of high density matter, possibly through continuing collapse of a protoneutron star. The final remnant is conceivably a stellar mass black hole, though undetected at present. (Reprinted from [@van05]).[]{data-label="figSN1987A1"}](f3 "fig:")![(Left.) SN 1987A is a Type II supernovae produced by core-collapse of the supergiant Sanduleak -69$^o$ 202 in the Large Magellanic Cloud at a distance of about 50 kpc [@gil87; @kir87]. Shown is the neutrino light curve compiled from Kamiokande (stars) and IMB (circles) listed in [@bur87] associated with the optical identification of SN 1987A [@gar87; @her87; @kun87]. (Right.) The SN1987A neutrino light curve, showing an initial energy of $>$10 MeV representative for the formation of high density matter, possibly through continuing collapse of a protoneutron star. The final remnant is conceivably a stellar mass black hole, though undetected at present. (Reprinted from [@van05]).[]{data-label="figSN1987A1"}](f4 "fig:")
Roadmap
-------
With a focus on multi-messenger emission from energetic type Ib/c supernovae powered by black hole inner engines, our roadmap is as follows.
§2 discusses evidence for black holes as a common inner engine to LGRBs and SGRBEEs from detailed analysis of BATSE data of long GRBs. In particular, LGRBs and SGRBEEs may share a common central engine in the form of a rotating black hole, and normal LGRBs may be associated with the formation of near-extremal black holes.
§3 highlights the likely complex process of birth and evolution of rotating black holes in core-collapse supernovae, various stages of accretion therein each with their own outlook on gravitational radiation with extended emission in the form of ascending and descending chirps.
§4 gives a general framework for BEGE from non-axisymmetric accretion onto rotating black holes. Specifically, we identify extended emission currently observed in GRBs with the lifetime of black hole spin, $T_{spin}$, as a secular time scale extending to tens of seconds relevant to normal long GRBs and SRBEEs. We identify ascending and descending chirps with non-axisymmetric waves in accretion flows. In the Kerr metric [@ker63], we model the latter by ISCO waves, extended by feedback over an inner torus magnetosphere with an expanding ISCO during black hole spin-down. Thus, a central engine conceivably emits simultaneously descending and [ascending chirps]{} from accretion onto rotating black holes [@van03a; @lev15].
§5 Given the outlook on extended emission in gravitational waves, we introduce a new GPU-accelerated pipeline of [butterfly filtering]{} enabling deep searches for BEGE by matched filtering against over banks of millions of chirp templates in real-time, the results of which would appears as tracks in a chirp-based spectrograms. This approach differs from Fourier-based spectrograms [@sut10; @pre12; @thr13; @thr14; @cou15; @abb15; @gos15] by bandpass filtering signals with finite slope $\left|df(t)/dt\right|\ge\delta > 0$ for some $\delta >0$ [@van14] (Fig. \[figKol\], Fig. \[figmf\]).
§6 summarises our proposed search strategy to probe nearby energetic core-collapse supernovae for BEGE, by LIGO-Virgo and KAGRA in blind all-sky searches or by follow-up of triggers from optical-radio transients surveys. The latter may be obtained from any of the existing [@dro11; @li11a] or upcoming all sky optical surveys such as Pan-STARRs [@pan11] or the planned Caltech Zwicky Transient Facility [@kul14; @bel15]. Searches for their contribution to the stochastic background in gravitational waves may be pursued by multi-year correlations between two or more gravitational wave detectors (e.g. [@sat09]). Existing observations of LGRBs and SGRBEEs justify a vigorous probe of the inner most workings of energetic CC-SNe which, if successful, may identify rotating black holes by true calorimetry on their evolution over times scales of tens of seconds.
Long GRB-supernovae and SGRBEEs
===============================
The association of normal LGRBs with supernovae and shared spectral properties in prompt GRB-emission with Extended Emission to SGRBEEs discovered by [*Swift*]{}, poses novel questions on a common inner engine to both, even as the latter derives from mergers. In what follows, we review some of the observational highlights on these two classes of GRBs.
Hyper-energetic GRB-supernovae
------------------------------
Aspherical CC-SNe [@pap89; @hof99; @mae08] such as SN1987A derive from relatively massive progenitors that are unlikely produced by their associated MeV neutrino burst. Instead, they may be powered by an internal magnetic wind [@bis70] or an internal relativistic jet [@mac99] derived from a central engine. In core-collapse following a drop in thermal pressure at the end of nuclear burning or associated with pair-instability if the mass of the star is exceptionally large [@bis66; @bar67; @gal09; @cha10], the central object thus produced is either a (rotating) neutron star or black hole.
The energy $E_k$ in a supernova explosion powered by angular momentum is constrained by the maximal rotational energy of the engine, i.e., a (proto-)neutron star or black hole, and the efficiency in expulsion of the remnant stellar envelope by a putative internal wind or jet. The maximal rotational energy $E_c$ of a neutron star is attained at its break up frequency whereas the same of a black hole of mass $M$ is attained when its angular momentum reaches $GM^2/c$. Canonical bounds for the rotational energy of the inner engine of a CC-SN are $$\begin{aligned}
%\begin{array}{ll}
E_{rot}^{NS}\le E_c & = 3\times 10^{52} \,\mbox{erg},~~
E^H_{rot} & \le 6\times 10^{54}\left(\frac{M}{10M_\odot}\right)\,\mbox{erg}
%\end{array}
\label{EQN_EcEH}\end{aligned}$$ for a rotating neutron star, respectively, black hole. Here, there is some uncertainty in the bound $E_c$ due to the equation of state of neutron stars. The efficiency $\eta=E_k/E_w$ in the expulsion of the envelope with kinetic energy $E_k$ by a wind or jet with energy $E_w$ depends on the baryon loading of the wind, i.e., [@van11b] $$\begin{aligned}
\frac{1}{2}\beta_{ej} < \eta < 1,
\label{EQN_eta}\end{aligned}$$ where $\beta_{ej} = v_{ej}/c$ denotes the observed velocity of the ejected envelope relative to the velocity of light $c$. The efficiency increases with baryon loading, as the outflow velocity becomes moderately relativistic. The efficiency reduces to $(1/2)\beta_{ej}$ in the limit of baryon-poor jets, whose velocity approaches $c$. By the above, $E_k$ is bounded by $\eta E_c$ or $\eta E^H_{rot}$.
Jet powered supernovae are of particular interest to the diversity in CC-SNe in long GRBs (Fig. \[figSN1987A3\]) as well as a diversity in their associated GRBs. Certain types of supernova explosions may lead to a relativistic shock breakout that may explain (nearby) low-luminosity GRBs (LLGRBs) but not the prompt GRB emission of normal long GRBs [@nak12]. The latter class is thought to be produced by collimated ultra-relativistic baryon-poor jets (BPJ) that penetrate through the stellar envelope. Failure to breakout the stellar envelope leads to a “chocked GRB,” which is considered the leading candidate for LLGRBs [@nak12].
For long GRBs, the association with massive stars is now supported by four pieces of evidence:
- supernovae (SNe) accompanying a few nearby events [@hjo11];
- detection of SN features in the spectra of “rebrightenings" during GRB afterglow decay, at intermediate redshifts, most recently GRB 130427A ($z$=0.34, [@mel13]) up to $z\simeq1$ [@del03];
- the host galaxies are spiral and irregular with active star formation typical for environments hosting core-collapse SN-Ic’s [@kel08; @ras08]; and
- a cosmological distribution of redshifts of long GRBs, consistent with the cosmic star formation rate [@wan10; @gri12] (Fig. \[figSwift\]).
![The distribution of observed redshifts of 230 LGRBs in the [*Swift*]{} catalogue shows a mean redshift $\mu=2.11$ with standard deviation $\sigma=1.35$. This distribution is significantly biased towards low redshifts. (Reprinted from [@van12].)[]{data-label="figSwift"}](f5)
**Table 1.**$^\dagger$ References refer to SNe except for GRB 070125. $E_*$ in units of $10^{51}$ erg.\
GRB Supernova $z$ $E_\gamma$ $E_{tot}$ $E_{SN}$ $\eta$ $E_{rot}/E_c$ Ref.
--------- ----------- -------- ------------ ----------- ---------- -------- --------------- ------ --
SN2005ap 0.283 $>10$ 1 $>0.3$ 1
SN2007bi 0.1279 $>10$ 1 $>0.3$ 1
980425 Sn1998bw 0.008 $<0.001$ 50 1 1.7 2
031203 SN2003lw 0.1055 $<0.17$ 60 0.25 10 3
060218 SN2006aj 0.033 $<0.04$ 2 0.25 0.25 4
100316D SN2006aj 0.0591 0.037-0.06 10 0.25 1.3 5
030329 SN2003dh 0.1685 0.07-0.46 40 0.25 5.3 6
050820A 2.607 42 1.4 7
050904 6.295 12.9 0.43 7
070125 1.55 25.3 0.84 7
080319B 0.937 30 1.0 7
080916C 4.25 10.2 0.34 7
090926A 2.1062 14.5 0.48 8
070125 1.55 25.3 0.84 9
\[TABLE\_1\]\
0.08in $^\dagger$ [@van11b]; 1. [@gal09; @qui11]; 2. [@gal98]; 3. [@mal04]; 4. [@mas06; @mod06; @cam06; @sol06; @mir06; @pia06; @cob06b]; 5. [@cho10; @buf11]; 6. [@sta03; @hjo03; @mat03]; 7. [@cen10]; 8. [@deu11]; 9. [@cha08].
The fact that $E^H_{rot}$ is about two orders of magnitude larger than $E_c$ in (\[EQN\_EcEH\]) allows for $E_k$ to be considerably larger than $\eta E_c$ even at modest efficiency. Additionally, black hole-disk systems can produce two-component outflows comprising an ultra-relativistic jet along their spin axis surrounding by a baryon-rich and possibly collimating disk wind. The first offers the potential for high energy emissions upon breakout from the stellar envelope, whereas the second offers the potential for an efficient explosion. In contrast, neutron stars may produce a one-component magnetic outflow, that may facilitate either one but not both.
Exceptional $E_k$’s are observed in GRB 031203/SN2003lw and 030329/SN2003dh with $E_{SN}\simeq 6\times 10^{52}$ erg and $E_{SN}\simeq 4\times 10^{52}$ erg, respectively (Table 1). Like SN1987A, supernovae accompanying LGRBs are aspherical and radio-loud. Hyper-energetic events with $E_k>E_c$ (see (\[EQN\_EcEH\])) are no exception. In the model of [@bis70], these explosive events are attributed to magnetic winds powered by angular momentum extraction of a compact object, i.e., a PNS or magnetar (e.g. [@woo10; @kas10] in SN2007bi [@nic13]) or a rotating black hole-disk system (BHS, [@van03]). Taking into account a finite efficiency for the conversion of angular momentum to a (largely radial) explosion, $\eta<1$, we determined that aforementioned two events require a central energy $E_{rot}$ in angular momentum exceeding the maximal spin energy of a rapidly rotating neutron star by a factor of 10 and, respectively, 5.3 (Table 1). With a total output of about $10^{52}$ erg in optical emission alone, SN2015L [@don15] likewise defies the limit $\eta E_c$ in the face of reasonable efficiencies $\eta$ and finite efficiency in dissipating kinetic energy to electromagnetic radiation.
Given uncertainties in the observed explosion energy by a factor of no more than a few, it appears that, in light of (\[EQN\_EcEH\]), these two events cannot be attributed to spin down of PNS, unless the efficiency is at least 100 %. For this reason, the central engine of these two events stand out as candidate BHS, powering the explosion by the spin energy of a black hole. If so, the required explosion efficiency is brought back to a reasonable few %. In particular, we have a wind energy $E_w$ that may derive from disk winds or black hole spin energy $E_{rot}^H$ satisfying $$\begin{aligned}
E_w = 6\times 10^{52} \left(\frac{\Omega_T}{0.1\Omega_H}\right)^2 \left( \frac{M}{10M_\odot}\right) \,\mbox{erg}
\label{EQN_Ew}\end{aligned}$$ for rapidly spinning black hole of mass $M$ parameterized by the ratio of the angular velocity $\Omega_T$ of a torus about the ISCO to the angular velocity $\Omega_H$ of the black hole. The fiducial scale of 0.1 in (\[EQN\_Ew\]) is rather conservative.
Local event rates of energetic CC-SNe
-------------------------------------
As a parent population of long GRBs, nearby energetic core-collapse supernovae define targets of opportunity (TOOs) to LIGO-Virgo [@abr92; @ace06; @ace07] and KAGRA [@som12; @kag14]. Current GRB and SN rates measurements point to a branching ratio $$\begin{aligned}
{R}=\frac{N(\mbox{GRB-SNe})}{N(\mbox{Type Ib/c})} \simeq 0.2-3 \%,
\label{EQN_R1}\end{aligned}$$ covering a conservative [@van04; @ghi13] to a more optimistic estimate [@gue07]. As targets of opportunity for LIGO-Virgo and KAGRA, the event rate of SN Ib/c is larger than GRB-SNe by $R^{-1}$.
The origin of the small value of $R$ in (\[EQN\_R1\]) is not well understood. Evidently, an observable GRB event requires the formation of an inner engine sufficiently powerful for its energetic outflows to overcome various adverse conditions out of which it emerged, namely a high density environment formed in core-collapse of a massive progenitor star, perhaps in a short period binary [@pac98]. A possible additional factor is the time of residence of the newly formed black hole in the center of the star, that may be sufficiently long only when the kick velocity is low. The latter may be rare. Alternatively, it may reflect the small probability of forming nearly extreme Kerr black holes, i.e., about the Thorne limit reached along a modified Bardeen trajectory as the initial condition for long GRBs [@van15].
We estimate the event rate of Type Ib/c supernovae within a distance $D_S$ in the Local Universe to be $$\begin{aligned}
\dot{N}(\mbox{Type Ib/c}, D<D_S) = 109 \,\mbox{yr}^{-1} \left(\frac{D_S}{100\,\mbox{Mpc}}\right)^3,
\label{EQN_R2}\end{aligned}$$ based on a weighted average of the volumetric rates derived from Asiago [@cap99; @bar99] and Lick surveys [@li11a; @li11b]. In a 5 year observational window, we expect about 4, 15 and 35 asymmetric Type Ib/c explosions in within a distance of 20, 30 and 40 Mpc respectively. These numbers offer a realistic perspective on simultaneous detections of GWBs and electromagnetic radiation.
The event rate (\[EQN\_R2\]) refers to regular Type Ib/c supernovae. Broad line events (cf. Fig. \[figSN1987A3\]) associated with normal LGRBs are more rare by an order of magnitude. Even so, the event rate of these energetic events is a few per year within 100 Mpc, and hence more numerous than the true event rate of LGRBs (corrected for beaming) by up to one order of magnitude.
The relatively high event rate (\[EQN\_R2\]) should be viewed in light of the anticipated high frequency in gravitational waves, where LIGO-Virgo and KAGRA sensitivity is limited by photon shot-noise. The shot-noise dominated output of these detectors is above the frequency around 100-200 Hz of minimal (thermal) noise, below which sensitivity is limited by seismic noise. By frequency alone, GWBs from Type Ib/c supernovae are less suitable as gravitational wave sources for these detectors than mergers of double neutron star binaries (§2). This drawback is considerably ameliorated by the relatively high event rate (\[EQN\_R2\]), that brings Type Ib/c supernovae statistically more nearby by a factor of about 4.8.
Detecting energetic supernovae such as Type Ib/c events is very easy, especially in the Local Universe within 100 Mpc. In contrast, electromagnetic counterparts to mergers of double neutron star binaries appear to be quite challenging [@kas13; @bar13; @tan13a] and may be seen only in the most fortuitous of cases [@tan13].
GRBs in the [*Swift*]{} era
---------------------------
Calorimetry on the kinetic energy in supernovae and the prompt GRB emission offers our most prominent views yet on the inner engine of GRB-SNe, in addition to the MeV neutrino burst from SN1987A. We next review the present classification, spectral and temporal properties of GRBs relevant to the question whether their inner engine is a black hole or neutron star. Particularly relevant are the catalogues of BATSE, [*BeppoSAX*]{} and [*Swift*]{} of long and short GRBs.
\
\
$^\dagger$ [*Swift*]{} SGRB, SGRBEE$^a$ and LGRBNs. $E_{iso}$ in $10^{52}$ erg, $E_p$ in keV.\
$T_{90}$ $z$ host$^b$ $E_{iso}$$^c$ $E_p$$^c$
----------------------------- ---------- -------- -- ----------------------------------- ---------------- ----------- -- --
[[**SGRB**]{}]{}
061201 0.760 0.111 galaxy cluster$^1$ 0.013 969
050509B 0.073 0.225 elliptical galaxy$^2$ 0.00027$^3$ -
060502B 0.131 0.287 massive red galaxy$^4$ 0.022 193
130603B 0.18 0.356 SFR$^5$ 0.2$^6$ 90$^6$
070724A 0.4 0.457 moderate SF galaxy$^7$ -
051221A 1.400 0.547 SF, late type galaxy$^8$ 0.25$^9$
131004A 1.54 0.717 low mass galaxy$^{10}$ -
101219A 0.6 0.718 faint object$^{11}$ 0.48 842
061217 0.210 0.827 faint galaxy$^{12}$ 0.008$^{12}$
090510 0.3 0.903 field galaxy$^{13}$ 3.8$^{13}$
070429B 0.47 0.904 SFR$^{14}$ 1.1$M_\odot$ yr$^{-1}$ - -
060801 0.49 1.131 - 0.027$^{15}$
100724A 1.4 1.288 probably LGRB$^{16}$ - -
050813 0.45 1.8 galaxy cluster$^{17,18}$ 0.017$^{18}$ -
090426 1.2 2.609 irreg. SF galaxy$^{19}$ - -
[**SGRBEE**]{}
060614 $^d$ $^e$ $^f$ $^g$ 108.7 0.125 faint SFR$^{20,21}$ 0.21$^{20}$ 55$^{20}$
050724 $^d$ $^e$ $^f$ $^g$ 69 0.258 elliptical, weak S$^{22}$ 0.0099$^{23}$ -
071227A $^e$ $^f$ 1.8 0.384 edge-on S$^{24}$ 0.008$^{25}$ -
061210 $^d$ $^e$ $^f$ $^g$ 85.3 0.41 bulge dominated$^{26}$ 0.046$^{26}$ -
061006 $^d$ $^e$ $^f$ $^g$ 129.9 0.438 exp. disk profile$^{27}$ 0.18 955
070714B $^d$ $^e$ $^f$ $^g$ 64 0.92 SF galaxy$^{28}$ 0.16$^{28,29}$ -
050911 $^d$ $^e$ 16.2 1.165 EDCC493 cluster$^{30}$ 0.0019$^{30}$ -
[**LGRBN**]{}
060505 4 0.089 spiral, H$^+$, no SN$^{31}$ 0.0012$^{21}$ 120
060614 $^d$ $^e$ $^f$ $^g$ 108.7 0.125 faint SFR, no SN$^{20}$ 0.21$^{21}$ -
061021 46 0.3462 no SN$^{32}$ 0.68 630
\[TABE2\]
\
0.08in [$^\dagger$ [@van14b]; $^a$ From [@HEA]; $^b$ galaxy type, SN association; $^c$ Isotropic-equivalent energy and peak energy for events with reliable estimates of the bolometric $E_{iso}$ across a large enough energy band, under the assumption $\Omega_m=0.3$ and a Hubble constant $H_0=70$ km s$^{-1}$ Mpc$^{-1}$; $^d$ [@per09]; $^e$ [@nor10]; $^f$ [@cow12]; $^g$ [@gom14]; 1. [@ber07c] ;2. [@fon10; @pag06; @per09] ; 3. [@blo06; @blo07]; 4. [@blo07]; 5. [@cuc13]; 6.[@fre13]; 7. [@koc10]; 8. [@ber05; @ber07]; 9. [@gol05]; 10. [@per13]; 11. [@per10]; 12. [@ber06b; @deu06]; 13. [@rau09; @gue12]; 14. [@cen08]; 15. [@cuc06; @ber07a]; 16. [@ukw10]; 17. [@blo07; @pro06; @ber06a; @fer07];18. [@ber05a]; 19. [@ant09]; 20. [@del06]; 21. [@fyn06a; @cob06] ; 22. [@ber05c; @pag06; @ber07a; @fon10]; 23. [@pro05]; 24. [@ber07b]; 25. [@ber07d]; 26. [@cen06]; 27. [@fon10; @ber07a]; 28. [@gra09]; 29. [@gra07]; 30. [@ber07]; 31. [@jak07]; 32. [@mor06] ]{}
![The power law index of the average PDS in the frequency range $10^{-2}<f/{\rm Hz}<1$ obtained by Fourier analysis from different data sets as a function of the observed energy. Dashed line ($\alpha_2\propto E^{-0.09}$) illustrates the $\alpha_2$ dependence on energy as estimated from Fermi data. (Reprinted from [@dic13a].)[]{data-label="fig:alpha"}](f8)
![Shown are the smoothed light curves of 72 bright long GRBs in the BeppoSAX catalog sampled at 2 kHz for the first 8-10 seconds. 42 have a pronounced autocorrelation (“red") with mean photon counts of 1.26 per 500 $\mu$ s bin, while 30 have essentially no autocorrelation (“white") with mean photon counts of 0.59 per 500 $\mu$ s bin. (Reprinted from [@van14].)[]{data-label="fig72lc"}](f9)
Immediately following the serendipitous discovery of GRBs, Stirling Colgate suggested an association to supernovae - gamma-ray flashes from type II supernova shocks [@col68; @col70; @col74] - now seen by the association of normal long GRBs with core-collapse of massive stars [@woo06]. Indeed, shock breakout in regular CC-SNe is likely to produce high energy emission in UV light [@gez08], X-rays [@cam06] up to gamma-rays [@wea76; @hof09; @kat10; @nak10; @svi12]. While conceivably relevant to low luminosity (long) GRBs (LLGRBs), the prompt GRB emission from normal long GRBs is now understood to derive, instead, from dissipation in ultra-relativistic BPJs (below).
BATSE identified short and long GRBs by the observed bimodal distribution in durations $T_{90}$ in a large number of events (Fig. \[fig4B1\]). They are now associated with, respectively, mergers of the NS-NS [@eic89] or NS-BH [@pac91] variety, albeit with a large overlap between these two populations [@bro12; @bro13]. $T_{90}$ is defined by the time interval covering a 90 percentile in total photon count [@kov93]. BATSE identified a mostly non-thermal spectrum, which is typically well described by a smoothly broken power-law (Band spectrum [@ban93]).
[*BeppoSAX*]{} seminal discovery of X-ray afterglow emission to GRB 970228 by [@cos97] allowed rapid follow-up by optical observations [@par97], providing a first cosmological redshift ($z = 0.835$ of GRB 970508) in optical absorption lines of FeII and MgII [@met97; @ama98]. When detected, afterglow emission of short GRBs tends to be very weak compared to those of LGRBs, consistent with less energy output and burst locations outside star forming regions. Weak X-ray afterglow emission discovered by [*Swift*]{} in the High Energy Transient Explorer-2 (HETE II) event GRB 050507 [@fox05] and the [*Swift*]{} event GRB 050509B [@geh05] were anticipated for GRBs from rotating black holes [@van01]. Following GRB 970508, [*BeppoSAX*]{}, HETE II and [*Swift*]{} provided a growing list of GRBs with measured redshifts. Presently, the total number of redshifts identified is about 350 with 287 due to [*Swift*]{} alone. Fig. \[figSwift\] shows the distribution of the latter.
[*Swift*]{} identified the new class of short GRBs with Extended Emission (SGRBEEs). GRB060614 ($T_{90}$=102 s) has no detectable supernova [@del06; @fyn06a; @gal06] and GRB 050724 is a SGRB with Extended Emission (SGRBEE) with an overall emission time $T_{90}$=69 s in an elliptical host galaxy [@ber05c; @ber07]. Neither is readily associated with a massive star. Since then, the list of SGRBEEs has grown considerably (Table 2). Table 2 further shows a few long GRBs with no apparent association to SNe (LGRBNs). SGRBEE and LGRBNs challenge the BATSE classification into short and long events. Though both show an initial hard pulse, characteristic of short GRBs, a subsequent long duration soft tail features a spectral peak energy ($E_{p,i}$)-radiated energy ($E_{iso}$) correlation that satisfies the Amati-correlation holding for normal long GRBs. This “hybrid" structure of observational properties of SGRBEE and LGRBNs suggests that they share the [*same astronomical origin as short GRBs with the same physics in the central engine as normal long GRBs,*]{} albeit with somewhat smaller values of $E_{iso}$.
The prompt GRB emission has a characteristic peak energy $E_{p,i}$ at which the $\nu F_\nu$ photon spectrum peaks in the cosmological rest-frame. It typically ranges from tens of keV to thousands of keV. If the isotropic-equivalent energy $E_{iso}$ is the radiation output by a GRB during its whole duration (assuming spherical symmetry), it is found that $E_{iso}$, commonly used in the absence of reliable information on the degree of collimation in individual GRB events, correlates with $E_{p,i}$ (see Fig. \[figAm1\]). This correlation, now known as the Amati relation, is well established for long GRBs, while short GRBs do not appear to satisfy this Amati-correlation ([@ama06] and references therein).
To quantify our level of confidence in the merger origin of SGRBEE and LGRBNs, we recently considered the mean values $\mu$ of the observed redshifts (Fig. \[figAm2\]), i.e., $\mu_{L}^{N}$ of LGRBNs, $\mu_{EE}$ of SGRBEEs, $\mu_{S}$ of SGRBs and $\mu_{L}$ of LGRBs, and concluded that they satisfy $$\begin{aligned}
\mu_{L}^{N}< \mu_{EE} <\mu_{S}<\mu_{L},
\label{EQN_z}\end{aligned}$$ where $\mu_S^{EE}=0.5286$, $\mu_S=0.8587$, $\mu_L^N=0.1870$, and $\mu_L=2.1069$ based on the redshifts shown in Table 2.
By a Monte Carlo test, we determined the probability that, from the mean redshift, the [*Swift*]{} samples of SGRBEE ($n_1=7$), SGRB ($n_2=15$) and LGRBNs ($n_3=3$) are drawn from the observed distribution of LGRBs ($n=230$). Because of the small $n$ samples and the broad distribution of redshifts of LGRBs (with an observational bias towards low $z$), we proceed with Monte Carlo test by drawing samples of size $n_i$ ($i=1,2,3)$ from the distribution of the $n=230$ redshifts of the latter. Doing so $N$ times for large $N$ obtains distributions of averages $\mu_i$ of the redshifts in these small $n$ samples under the Bayesian null-hypothesis of coming from the distribution of redshifts of LGRBs. We find [@van14b] $$\begin{aligned}
\begin{array}{rll}
\mbox{SGRBEE} & \not\subset \mbox{LGRB} &: ~~\sigma = 4.6700 \\
\mbox{SGRB} & \not\subset \mbox{LGRB} &: ~~\sigma = 4.7520 \\
\mbox{LGRBN} & \not\subset \mbox{LGRB} & :~~\sigma = 4.3140\\
\end{array}
\label{EQN_s}\end{aligned}$$ For SGRBs, (\[EQN\_s\]) is consistent with a relatively low redshift origin inferred from identification of host galaxies in the local Universe [@tan05]. At a level of confidence exceeding $4\sigma$, SGRBEE and LGRBNs have inner engines originating in mergers in common with normal long GRBs originating in CC-SNe, given that both share the Amati-correlation in the long/soft tail.
Our results (\[EQN\_s\]) show with relatively high confidence that the enigmatic LGRBN GRB060614 is a merger event, suggested earlier based on other arguments [@van08; @cai09], whose long durations in soft extended emission can be identified with the lifetime of spin of a rotating black hole (§2); see [@zha06; @zha07] for various other explanations of extended emissions from mergers. Baryon-rich jets such as shown in Fig. \[fig:jet\] from accretion disks produced in naked inner engines formed in mergers may dissipate into lower energy emissions, perhaps including a radio burst [@van09].
Prompt GRB emission
-------------------
Three main stages are involved in the generation of the prompt GRB emission: (i) extraction of energy and formation of outflows, (ii) dissipation of the outflow bulk energy, and (iii) conversion by dissipation into electromagnetic radiation. These processes are most likely interrelated. Successful breakout of the jet from the stellar envelope is a necessary condition for producing a GRB. Due to the compactness of the energy source, the largely non-thermal electromagnetic emission originates from large radii and, therefore, does not provide a direct probe of the central engine. Nevertheless, [*some*]{} of the temporal properties of the activity of the engine may be imprinted in the light curve of the prompt emission [@kum08; @van09a].
The conventional wisdom has been that GRB jets are powered by magnetic extraction of the rotational energy of a magnetar [@uso94; @met11] or a hyper-accreting black hole, where the latter is a particularly attractive alternative to account for low baryon-loading [@lev93; @eic11]. In some cases [@van11b], evidence is tilting towards an association to black holes rather than neutron stars. At sufficiently high accretion rates, annihilation of neutrinos that originate from the hot matter surrounding a Kerr black hole can also power a GRB outflow [@zal11; @lev13b], although magnetic extraction seems favorable. In the latter case, it is believed to form an outgoing Poynting flux which, on large enough scales, is converted somehow into kinetic energy in baryonic contaminants. This conversion process has not been identified yet, but it is generally believed to involve gradual acceleration of the flow (e.g., [@bog95; @chi91; @hey89; @lyu09]), impulsive acceleration [@gra11; @lyu11], magnetic reconnection [@gia07; @lev97; @lyu10; @lyu03; @zha11; @mck12], and/or current driven instabilities [@lev13].
The production of high energy emission requires substantial dissipation above, or just below the photosphere. It most likely results from the formation of internal [@mes93; @ree92] and/or collimation [@bro07; @laz09] shocks in cases where the flow is hydrodynamic in the vicinity of the photosphere, or magnetic reconnection [@gia07; @mck12] if the flow remains highly magnetized at large radii. Dissipation at very large optical depths will merely lead to re-acceleration of the flow, or in case of magnetic extraction to a transition to kinetic energy dominated outflows [@gra11; @lev13]. It can, nonetheless, help increasing the specific entropy, which seems to be required by the observed SED peaks.
The nature of the prompt emission mechanism is yet an open issue. The emitted spectrum, although exhibiting notable variations from source to source, can generally be described by a broken power law (Band function [@ban93]), with some exceptions, e.g., GRB 090902B. It has been originally proposed that the observed spectrum is produced by synchrotron emission of non-thermal electrons accelerated at internal collisionless shocks (for reviews see [@pir99; @pir04]). However, subsequent analysis (e.g., [@bel13; @cri97; @eic00; @pre98]) indicated that the synchrotron model has difficulties accounting for some common properties exhibited by the GRB population, specifically, the clustering of peak energies around 1MeV, the hardness of the spectrum below the peak, and the high efficiencies inferred from the observations. At the same time, it has been argued [@ryd04; @ryd05] that a thermal component appears to be present in some bursts, which may be transient as in the BeppoSAX event GRB 990712 [@fro01]. These developments, and the recent detection of some GRBs with a prominent thermal component (e.g., GRB 090902B) or multiple peaks (e.g., GRB 110721A, GRB 120323A) have motivated a reconsideration of photospheric emission [@bel13; @eic00; @gia12; @pee06; @ryd09; @vur13].
On theoretical grounds, one naively anticipates a significant dissipation of the bulk energy of a GRB outflow just below the photosphere, either by internal [@eic94; @bro11b; @mor10] or collimation shocks [@bro07; @laz09]. They are mediated by radiation and their typical size is on the order of a few Thomson depths [@bud10; @kat10; @lev08; @lev12], larger than any kinetic scale by many orders of magnitudes. Their structure and emission are, therefore, vastly different than those of collisionless shocks that can only form above the photosphere, where the Thomson optical depth is well below unity. The large shock width strongly suppresses particle acceleration [@lev08; @kat10], yet a non-thermal spectrum can be produced inside the shock via bulk Comptonization [@bud10] and formation of a Band-like spectrum is conceivable [@ker14]. Alternatively, sub-photospheric dissipation may be accomplished through magnetic reconnection if the flow remains highly magnetized at mild optical depths, in which case particle acceleration may ensue. Under such conditions, formation of a Band spectrum is also plausible [@bel13; @gia12; @vur13].
Observationally, a photospheric model (black body plus power law) or a Band function provides a satisfactory fit to most BATSE GRB light curves. However, extending spectra to the low energy range of 2-28 keV (of the BeppoSAX Wide Field Cameras) poses challenges in a number of cases. For the extended energy range of 2-2000 keV, a Comptonization model appears more robust in producing satisfactory fits [@fro13]. The low energy window is consistent with Comptonization of black body background photons by an initially non-relativistic expanding outflow, perhaps representative of the initial launch at stellar breakout of the outflow creating the GRB. A particular example, the breakout of a striped MHD jet, computed using numerical simulations, is here presented in §5.2.
The prompt GRB emission is often followed by afterglow emissions at lower energies, especially in X-rays down to radio in some cases. Afterglow emission was anticipated based on the GRB association with ultra-relativistic outflows, further enabling identifying host properties and, in some cases, calorimetry on the total energy output. We refer the reader to existing reviews on this subject [@pir99; @pir04]. [*Swift*]{} made a key discovery with the identification of long duration X-ray tails, that appear to represent latent activity of the remnant inner engine (e.g. [@chi10; @ber11]). Remarkably, these X-ray tails, when observed, are very similar to short and long GRBs, lending some support for a common engine remnant.
Kolmogorov spectra in [*BeppoSAX*]{}
------------------------------------
To probe for high frequency signatures in the prompt GRB emission and, possibly, any intermittency or quasi-periodic behavior in the inner engine, the [*BeppoSAX*]{} catalogue of 2 kHz light curves offers a unique window to broadband spectral analysis. As mentioned above, prompt GRB emission probably originates from ultra-relativistic, baryon poor jets, launched from a compact stellar mass object [@sar97; @pir97] (see also [@kob97; @nak02]), commonly believed to be black holes or neutron stars [@tho94; @met11].
Shock induced emission predicts a spectrum that is a power law in energy (e.g. [@sar98; @tho07]) and turbulent in temporal behavior. Indeed, a Fourier analysis reveals a Kolmogorov spectrum in the BATSE and BeppoSAX catalogs of light curves of long GRBs [@bel98; @bel00; @gui12; @dic13a; @dic13b]. The index of power law behavior in the observed PSD spectra is broadly distributed about the Kolmogorov value of 5/3 with a negative gradient as a function of energy (Fig. \[fig:alpha\]). Conceivably, this spectral-energy gradient might be due to scale dependent dissipation in turbulent flows, which is only beginning to be explored by high resolution numerical simulations in the approximation of relativistic hydrodynamics [@zra13] (see also [@cal14]).
However, Fourier spectra are limited to tens of Hz due to strong Poisson noise in high frequency sampled gamma-ray light curves. The Kolmogorov spectrum is expected to continue to higher frequencies. This has recently been identified in broadband spectra obtained by a chirp search method. The method uses matched filtering using chirp templates with frequencies slowly moving up or down of 1 second duration (Appendix D). Applied to 2 kHz sampled [*BeppoSAX*]{} light curves of long GRBs (\[fig72lc\]), an approximately Kolmogorov spectrum is found to extend to 1 kHz in the observer’s frame or, equivalently, a few kHz in the source frame (Fig. \[figKol\]).
The above gives considerable evidence for LGRBs produced by successful breakout of ultra-relativistic jets emerging from an energetic CC-SNe (and LLGRBs as their failed-to-breakout counterparts). While the latter may be powered by an associated baryon-rich collimating wind, the GRB is attributed to dissipation in internal and/or collimation shocks in the former, that may be the result of intermittency at the source.
The preceding data point to a common origin in mergers of SGRBEE, LGRBNs and normal LGRBs: their soft/long tail sharing the same Amati-relation (Fig. \[figAm1\]) and the Kolmogorov spectrum of gamma-ray fluctuations is smooth with no evidence for a bump that might indicate the formation of proto-pulsars. These hints point to a [*common engine producing soft extended emission*]{}, and probably so from black holes in mergers and CC-SNe alike. Additionally, rotating black holes have ample energy to account for the most energetic GRB-SNe (Table 1). Apart from the low number counts, the only reservation would be extremely sub-luminous CC-SNe (cf. [@pas07]), that would be undetectable in our sample LGRBNs (Fig. \[figAm2\]).
On the premise of black holes unifying the soft extended emission in normal LGRBs (in core-collapse of massive stars) and SGRBEEs (from mergers), we next turn to a model for extended emission from rotating black holes, parameterised largely by black hole mass, defined by a secular time scale of spin down against surrounding high density matter rather than any time scale of accretion or free fall.
Extended emission from rotating black holes
===========================================
The exact [@ker63] solution of rotating black holes is parameterised by mass $M$, angular momentum $J$ and electric charge $Q$. Rotation is commonly expressed by the dimensionless parameter $a/J$, where $a=J/M$ denotes the specific angular momentum. According to the Kerr solution, $-1\le a/M \le 1$, allowing $\sin\lambda = a/M$ in terms of $|\lambda|\le\pi/2$ [@van99]. Spacetime about a rotating black holes is dragged into rotation at an angular velocity $\omega$, that may be observed as the angular velocity of test particles (at constant radial distance and poloidal angle) with vanishing angular momentum as measured at infinity. The angular velocity $\Omega_H$ of the black hole is defined as the limit of $\omega$ as one approaches the black hole that, by the no-hair theorem, reduces to a constant $\Omega_H = \tan(\lambda/2)/2M$ on the event horizon. At a corresponding spin frequency $\nu_H=\Omega_H/2\pi$, $$\begin{aligned}
\nu_H = 1.6\,\mbox{kHz}\tan(\lambda/2)\left(\frac{10M_\odot}{M}\right),
\label{EQN_k1}\end{aligned}$$ the total energy in angular momentum satisfies $E_{rot} = 2M\sin^2(\lambda/4)$, i.e., $$\begin{aligned}
E_{rot}^H \simeq 6\times 10^{54} \,\mbox{erg} \left(\frac{M}{10M_\odot}\right)\left(\frac{\sin(\lambda/4)}{\sin(\pi/8)}\right)^2.
\label{EQN_k2}\end{aligned}$$
In an astrophysical environment, rotating black holes can naturally account for extended emission with characteristic features (\[EQN\_alpha1\]-\[EQN\_i2\]) upon identifying durations with the lifetime of their spin.
Secular time scale of black hole spin
-------------------------------------
To be specific, we propose that BATSE durations $T_{90}$ of their prompt emission (Fig. \[fig4B1\]) represents a proxy for the lifetime of the engine [@van99; @van01; @van03], i.e., $$\begin{aligned}
T_{engine} \simeq T_{90}
\label{EQN_TeT90}\end{aligned}$$ while $\Omega_H> \Omega_{ISCO}$, where $\Omega_{ISCO}= ( M (z^{3/2}\pm\sin\lambda))^{-1}$ is the angular velocity of matter at the ISCO for prograde (+) and retrograde (-) orbital motion [e.g. @sha83]. The inequality $\Omega_H>\Omega_{ISCO}$ is readily satisfied. According to the Kerr metric, it holds whenever their rotational energy exceeds about 5.3% of their maximal spin energy (for a given black hole mass-at-infinity $M$).
In the proposed identification (\[EQN\_TeT90\]), $T_{90}$ provides a unique signature of the inner engine of LGRBs and SGRBEEs, where the latter derives from the merger of a neutron star with another neutron star or rotating companion black hole. We herein identify SGRBs (without EE) with hyper-accretion onto slowly rotating black holes, produced in mergers of neutron stars with a slowly spinning black hole companion [@van01] (Fig. \[fig:origin\]).
In (\[EQN\_TeT90\]), black holes are envision to be losing angular momentum primarily to surrounding matter about the ISCO, leaving a minor release about the spin axis in open outflows powering the baryon-poor ultra-relativistic jets (BPJ) seen in gamma-rays with luminosity [@van03; @van09] $$\begin{aligned}
L_j \simeq 1.4\times 10^{51}\,\mbox{erg}\,\mbox{s}^{-1}\,\left(\frac{M}{7M_\odot}\right)
\left(\frac{T}{20\,\mbox{s}}\right) \left(\frac{\theta_H}{0.5}\right)^4.
\label{EQN_Lj}\end{aligned}$$ For rapidly rotating black holes, $L_j$ along open magnetic flux tubes is supported by Carter’s magnetic moment $\mu_e^H$ of the black hole [@car68; @coh73; @wal74], in equilibrium with the external magnetic field supported by the surrounding inner disk or torus in a state of suspended accretion [@van01; @van03].
![An extremal black hole with vanishing Carter’s magnetic moment is out-of-equilibrium with vanishing horizon flux (a), which rapidly settles down to an equilibrium state with essentially maximal horizon flux. An open magnetic flux tube forms supported by the equilibrium value ${\mu }_{H}^{e}$ (b). In (a), the black hole does not evolve. At sub-critical accretion rates, frame dragging may produce BPJs while accretion from the ISCO is suspended by feedback from the black hole by Alvén waves via an inner torus magnetosphere (b). Major dissipation ($E_D > 0$) by forced MHD turbulence about the ISCO implies ${R}_{{jD}}={E}_{j}/{E}_{D}<< 1$. By slip and no-slip boundary conditions on the event horizon and, respectively, matter at the ISCO, the black hole gradually loses angular momentum, giving rise to a finite lifetime $T_s$ of rapid spin. (Reprinted from [@van15].) []{data-label="figapj15"}](apj15)
Fig. \[figapj15\] illustrates in poloidal cross-section the structure of open outflows in a suspended accretion state, allowing the black hole to lose angular momentum $J$ to surrounding matter. Like their supermassive counter parts [e.g. @mac97; @wal13] or galactic stellar mass black holes in micro-quasars such as GRS 1915+105 [@mir94; @gre01], accretion flows in catastrophic events are believed to be likewise magnetised, exposing the central black hole to a finite magnetic flux by accretion [@ruf75; @bis76; @bla77] or the formation of a torus magnetosphere [@van99]. Strong magnetic fields may derive from the magneto-rotational instability (MRI, [@bal91; @lub94; @glo14]), whose $m=0$ component of the infrared spectrum of MHD turbulence represent a net poloidal flux. In turbulent accretion [@bis76] or by forcing [@van99], $\mu_H$ will follow changes in sign in the $m=0$ part of the infrared MHD spectrum on an Alfvén crossing time scale. Exposed to a finite variance in poloidal magnetic flux, a rapidly rotating black hole develops a finite magnetic moment that preserves maximal magnetic flux through the event horizon at all spin rates. This holds true especially at maximal spin [@wal74; @dok86; @van01p]. In the absence of a small angular parameter in the connection of magnetic flux from the latter to the former, the result is an Alvén wave over an inner torus magnetosphere, allowing black holes losing angular momentum to a torus about the ISCO. In this process, the latter is heated and expected to be driven non-axisymmetric instabilities balanced by cooling in gravitational radiation [@van12]. The strength in poloidal magnetic flux may be derived from a magnetic stability limit $E_{B}/E_k\simeq 1/15$ of energy $E_B$ of the poloidal magnetic field to the kinetic energy $E_k$ in the inner disk or torus [@van03]. Conceivably, this bound may be circumvented by strongly intermittent inner engines [@van15a], see also [@mck12]. As a result, a key parameter setting the lifetime of rapid spin of the black hole is the mass $M_T$ in the torus.
The output (\[EQN\_Lj\]) is robust in being a consequence of differential frame-dragging. ($\omega$ decays to zero at infinity from $\Omega_H$ on the black hole.) Locally, the effect is manifest in Papapetrou forces [@pap51; @pir56] acting on the canonical angular momentum of charged particles supporting a Poynting flux-dominated outflow. The line-integral thereof is a potential energy [@van09a] $E=\omega J_p$. For charged particles, $J_p$ is defined by total magnetic flux on the flux-surface at hand (which is an adiabatic invariant). In super-strong magnetic fields typically considered in models of GRB inner engines, $E$ assumes energies on the scale of Ultra-High Energy Cosmic Rays (UHECRs). This may be processed downstream to gamma-ray emission in relativistic shocks or, for intermittent sources, into UHECRs by acceleration of ionic contaminants ahead of outgoing Alfvén fronts [@van09a].
The jet luminosity (\[EQN\_Lj\]) differs from the open model of force-free flux surfaces rotating at one-half the angular velocity of the black hole envisioned in [with $\theta_H=\pi/2$ in @bla77], in that only a minor fraction of the black hole luminosity channeled in an open outflow whenever $\theta_H<<\pi/2$. In [@van15], we consider $\theta_H=\theta_H(t)$ positively correlated to the ISCO: $L_j\propto \Omega_H^2$ and, since $L_j$ is dimensionless in geometrical units, $L_j\propto (r_{ISCO}\Omega_H)^2$ for a correlation to the ISCO. Consequently, $L_j\propto z^2\Omega_H^2$ in a Taylor series expansion in $z=r_{ISCO}/M$, where $r_{ISCO}$ denotes the ISCO radius around the black hole with angular velocity $\Omega_H$. Numerical integration of the equations of suspended accretion (with fixed points at extremal and slow spin) obtains a light curve $L_j(t)$ from (\[EQN\_Lj\]), providing a template suitable for normalising light curves by matched filtering [@van12; @van17a]. If $L_j$ represents an opening outflow supported by horizon flux over the [*full hemisphere*]{} of the black hole event horizon, $E_j$ will be a substantial fraction of $E_r$. For GRBs, however, this runs counter to a generically small fraction of true energy in prompt GRB outflows relative to $E_{rot}^H$ of a black hole [@van03; @van15].
Immediately after birth, a black hole in core-collapse grows by Bondi accretion [@bon52; @sha83], up to the moment that an accretion disk first forms by angular momentum hang-up about the ISCO of the newly formed rotating black hole. Subsequently, [@bar70] accretion is expected to ensue [@mck05; @kum08b; @kin06] - modified by open outflows [@van15] - driving the black hole to a near-extremal state, provided there is sufficient mass infall to reach this state. As accretion rates become sub-critical in the sense of [@glo14], such near-extremal black hole may experience angular momentum loss to surrounding matter at the ISCO through Alfvén waves in an inner torus magnetosphere, until its angular velocity $\Omega_H$ drops to $\Omega_{ISCO}$ of matter orbiting at the ISCO - a stable fixed point in the equations of suspended accretion [@van08].
![Evolution of a rotating black hole following birth in a progenitor of mass $M_0$ in three phases of accretion: surge in direct accretion ([*dashed*]{}), growth by Bardeen accretion ([*continuous*]{}) followed by spin down against matter at the ISCO when accretion becomes subcritical (top and middle panels). Shown is further the associated evolution of any quadrupole gravitational wave signature from matter at the ISCO, marked by frequencies $f_{GW_0}$ at birth, $f_{GW1,}$ at the onset of Bardeen accretion, $f_{GW,2}$ at the onset of spin down and $f_{GW,3}$ at late times, when the black hole is slowly rotating in approximate corotation with matter at the ISCO (lower panel.)[]{data-label="figGrowthe"}](BHGrowthe)
Fig. \[figGrowthe\] illustrates the complex sequence of different types of accretion, over the course of which the black grows in mass and spins down (Bondi accretion), grows in mass and spins up (Bardeen accretion), and loses mass-energy and spins down (suspended accretion). The latter introduces a radically new secular time-scale [@van03; @van17a] $$\begin{aligned}
T_s \simeq 30 \left(\frac{\sigma}{10^{-2}}\right)^{-1}\left(\frac{M}{7M_\odot}\right)\left(\frac{z}{6}\right)^4\,\mbox{s},
\label{EQN_TsA}\end{aligned}$$ defined by the ratio $\sigma = M_T/M$ of the mass $M_T$ of a torus at the ISCO to $M$ and its normalised radius $z=R/M$. According to (\[EQN\_TsA\]), the process of losing angular momentum to matter at the ISCO can extend to ultra-long durations when $\sigma$ is small, e.g., $$\begin{aligned}
T_s \simeq 100 \left(\frac{\sigma}{10^{-7}}\right)^{-1}\,\mbox{d},
\label{EQN_TsB}\end{aligned}$$ which time scale can be found in superluminous supernovae such as SN2015L [@van17a].
The process of black holes losing angular momentum to matter at the ISCO is expected especially from a fully developed turbulent disk. Exposed to a finite variance in poloidal magnetic flux, a rapidly rotating black hole develops a finite magnetic moment that preserves maximal magnetic flux through the event horizon at all spin rates. This holds true especially at maximal spin [@wal74; @dok86; @van01p]. Fig. \[fig-T\] shows the overall efficiency of radiation thus catalytically converted from black hole spin.
Observational evidence of black hole spin down
----------------------------------------------
Normalized light curves (nLC) extracted from the BATSE catalogue of LGRBs allow a confrontation with model templates of spindown, of black holes against high density matter at the ISCO and (proto-)neutron stars by magnetic winds. In making the connection to the observed GRB emission, we consider a linear correlation between ultra-relativistic baryon-poor outflows and the observed prompt GRB emission. Fig. \[figmf\] shows match between nLC and model templates. The results favour the first, especially so for very long duration events with $T_{90}$ exceeding tens of seconds. Similar results obtain for nLC extract from [*Swift*]{} [@gup12].
Noticeable also is an improvement in the fit for relatively long duration events with $T_{90}> 20$ s [@van09]. We attribute this to the time scale of jet breakout in a remnant stellar envelope for the majority of long GRBs originating in CC-SNe [@van12], possibly further in association with the most rapidly spinning black holes [@van09]. Based on different observations related to the relatively flat distribution of $T_{90}$ below 20 s, a similar conclusion obtains [@bro13].
Eq.(\[EQN\_TeT90\]) may be contrasted with various time scales of accretion, generally associated with growth and spin-up of the central black hole. In [@kum08b], a distinction is outlined between fall-back at high and low (down to zero) accretion rates. The first implies an initial $\dot{m}\propto t^{-1/2}$ for the first ten seconds or so in transition to $\dot{m}\propto t^{-3}$ on the time scale of one hundred seconds. The second implies $\dot{m}\propto t^{-n}$ with $4/3\le n \le 2$, depending on the detailed radial profile of mass-loss in winds. Assuming a linear correlation between black hole luminosity $L_H$ powering the prompt GRB emission and accretion rate [@kum08b], the same procedure applied to these accretion models shows matches vastly sub-par to those shown in Fig. \[figmf\] [@van17a]. For power law accretion profiles, discrepant behavior appears notably in a spike between the nLC and the model light curves, characterized by a prompt switch-on (cf. short GRBs). In contast, Fig. \[figmf\], shows a satisfactory match the nLC with the model light curve of black holes losing angular momentum against matter at the ISCO across the full duration of the bursts. The results for $T_{90}>20\,$s provide some support for very long GRBs commencing from near-extremal rotating black holes, perhaps in the Thorne limit of the Bardeen trajectory of evolution [@van15]. Furthermore, extreme luminosities in GRBs can derive from a nonlinear response to intermittent accretion about the ISCO [@van15a], perhaps stimulated by feedback from the black hole starting from aforementioned nearly extremal spin.
Slow rotating remnants
----------------------
![Rotating black holes surrounded by a high density disk are a common outcome of the coalescence of neutron stars with another neutron star or black hole and core-collapse of relatively massive stars. If the black hole spins rapidly, it may loose angular momentum to the surrounding matter leading to catalytic conversion of its spin energy into gravitational waves [@van01b]. The result is a descending chirp for the lifetime of rapid spin. In the relaxation of spaceÐtime to that of a slowly rotating black hole, the ISCO expands the frequency asymptotes to (\[EQN\_APB5\]). For the merger of two neutron stars, this asymptotic result is tightly constraint by the narrow distribution of their masses and spin [@bai08], here emphasized by $H$ and $L$ for high- and low-mass neutron stars. Absent a remnant stellar envelope, such binary merger creates a naked inner engine, whose magnetic winds may produce an observable radioburst. (Reprinted from [@van09].)[]{data-label="fAPC1"}](f24)
A principle outcome of black hole evolution shown in Fig. \[figGrowthe\] is a [slowly rotating remnant]{}, whose angular velocity has reached the fixed point $\Omega_H=\Omega_{ISCO}$ in the equations of suspended accretion, satisfying $$\begin{aligned}
a/M \simeq 0.36
\label{EQN_FP}\end{aligned}$$ as a black hole gradually lost most of its angular momentum, and the surrounding Kerr space time relaxed to the space time of a slowly spinning black hole. Just such slow spin appears to be present in the progenitor binary estimates (\[EQN\_GW15\]) of GW150914.
As a stable fixed point, the black hole luminosity will reach a plateau with finite luminosity in open outflows, provided there is a continuing (latent) accretion. In attributing SN2015L to black hole spin down following (\[EQN\_TsB\]), just such plateau is seen at late times in the optical light curve. Following (\[EQN\_TsA\]), it may signal X-ray tails (XRT) over time scales of thousands of seconds, discovered by [*Swift*]{}, that are remarkably universal to LGRBs and SGRBs alike, pointing to a common remnant [@eic09]. We here identify this remnant with slowly rotating black holes about the stable fixed point (\[EQN\_FP\]), as a sure outcome regardless of prior formation and evolution history and progenitor (Fig. \[fAPC1\]). In the present context, XRT’s may possibly be accompanied by long lasting low luminosity gravitational wave emission. As a common endpoint, this may appear both to normal LGRBs and SGRBs originating in mergers, following messy break-up of neutron stars in the tidal field of a companion black hole [@lee98; @lee99] or in the merger of two neutron stars [@ros07].
We estimate the resulting release in X-rays to have a luminosity $L_X\simeq 0.25\,\dot{m}$ for an accretion rate $\dot{m}$. Accompanying gravitational wave emissions should be very weak with negligible increase in black hole mass and angular momentum in view of the observed X-ray luminosities, e.g., $L_X\simeq 10^{41}$ erg s$^{-1}$ in GRB060614 [@man07]. If unsteady, large amplitude flaring may occur in, e.g., GRB050502B [@geh09] by fluctuations between feedback of the black hole $(\Omega_H> \Omega_T)$ or accretion $(\Omega_H< \Omega_T)$. See also [@lei08]. In this broad outlook on “multi-phase multi-messenger" emissions in core-collapse events, we shall focus on a general framework for BEGE from accretion onto rotating black holes and a specific connection to the amply energy in angular momentum of rapidly spinning black holes following Fig. \[figGrowthe\].
Ascending and descending chirps in accretion flows
===================================================
The relatively high densities anticipated in accretion flows in catastrophic events such as mergers and core-collapse of massive stars forms a promising starting point for broadband extended gravitational-wave emission. In essence, we expect gravitational radiation derived from accretion flows and, possibly, ISCO waves excited by input from the black hole, converting angular momentum in orbital motion and, respectively, spin of the central black hole. A key pre-requisite for this outlook is the onset of non-axisymmetric waves.
Alpha-disk model
----------------
The [*alpha-disk*]{} model allows us to give a general frame work for mass-inhomogeneities in accretion flows. We consider a disk with the following properties:
1. A kinematic viscosity $\nu$ expressed in terms of a dimensionless $\alpha$ parameter given by $$\nu=\alpha c_s H=\frac{\alpha \Omega H^2}{\sqrt{2} }=\frac{\alpha H^2}{\sqrt{2}M}\left(\frac{M}{r}\right)^{3/2},
\label{EQN_alpha}$$ using $c_s=\Omega H/\sqrt{2} $ for the sound speed in terms of orbital angular velocity $\Omega$. Here, $r$ is the radial distance to the black hole of mass $M$ and, typically, $0.001 < \alpha < 0.1$. Where accretion flows are governed by viscous torques, the surface density of the disk satisfies $\Sigma(r)=\dot{m}/(3\pi\nu)$ [@pri81] for an accretion rate $\dot{m}$ with asymptotic radial migration velocity $v_r^\nu$, i.e., $$\begin{aligned}
%\begin{array}{ll}
\Sigma = \frac{M\dot{m}\sqrt{2}}{3\pi \alpha H^2}\left(\frac{r}{M}\right)^\frac{3}{2},~~
v_r^\nu = \frac{3\nu}{2r} = \frac{3\alpha H^2}{2\sqrt{2}M}\left(\frac{M}{r}\right)^\frac{5}{2}.
%\end{array}
\label{EQN_sigma}\end{aligned}$$ As shown below, under certain conditions, there exists a critical radius $r_\nu$ in (\[EQN\_rnu2\]) within which angular momentum loss is dominated by gravitational radiation;
2. A Lagrangian disk partition, given by annular rings of radius $r$, radial width $l(r)$ and mass $\Delta m(r)\equiv \sigma(r) M$, here in the approximation that $l(r)$ is similar to the vertical scale height $H(r)$ of the disk. A ring is parametrised by its mass inhomogeneity, total energy and gravitational wave luminosity $$\begin{aligned}
%\begin{array}{ll}
\delta m = \xi \Delta m,~~
\Delta E = \frac{\sigma M}{2} \left(\frac{M}{r}\right),~~
\Delta L_{GW} =\frac{32}{5}\xi^2\sigma^2 \left( \frac{M}{r}\right)^5.
%\end{array}
\label{EQN_dm}\end{aligned}$$ in terms of the dimensionless parameter $0\le \xi<1$ (Appendix A), where $\xi$ is not necessarily small, in a local Keplerian approximation $q=3/2$ in the angular velocity distribution $$\begin{aligned}
\Omega(r)=M^{-1}\left(\frac{M}{r}\right)^q.
\label{EQN_Omq}\end{aligned}$$
We shall cover gravitational radiation from mass-inhomogeneities in accretion flows down to the ISCO. By self-gravity, these may appear as instabilities driven by cooling, wave-like or as fragments, when cooling times are on the order of the orbital period [@gam01; @ric05]. In accretion flows onto rotating black holes, a crucial condition is that such instabilities set in at a radius outside the ISCO. In this event, accretion may be driven by angular momentum loss in gravitational radiation rather than viscous torques across some critical radius greater than $r_{ISCO}$. Although the detailed origin and structure of mass-inhomogeneities formed is uncertain, we shall, for illustrative purposes, discuss these in the quadrupole approximation. In this approximation, migration of mass-inhomogeneities is described by (\[EQN\_hchar0\]-\[EQN\_hchar2\]) of Appendix A.
Spiral in of inhomogeneities by gravitational radiation dominated angular momentum loss may commence at radii large compared to $r_{ISCO}$. In this event, the luminosity in gravitational waves at a given mass accretion rate $$\begin{aligned}
\dot{M} = \dot{m} \dot{m}_0,~~\dot{m}_0 = \frac{c^3}{G}=4\times 10^{38}\,\mbox{g}\,\mbox{s}^{-1},\end{aligned}$$ satisfies $$\frac{L_{GW}}{\dot{m}}\simeq \frac{M}{2r_{ISCO}}.
\label{EQN_AISCO2}$$ For illustrative purposes, we express (\[EQN\_AISCO2\]) in a Newtonian approximation of the gravitational binding energy at the ISCO to the central black hole. A more precise estimate for the latter is given by $1-e$, where $e$ denotes the specific energy of orbiting matter in the Kerr metric given. Our aim here is to develop leading order estimates within a factor of a few. Accretion onto the ISCO may be followed by a plunge into the black hole or mass ejection in the form of a disk wind.
The scale (\[EQN\_AISCO2\]) points to a potentially substantial energy output $E_{GW}$ in gravitational waves, provided that a window $r_c> r_{ISCO}$ or $r_b > r_{ISCO}$ for gravitational radiation dominated angular momentum transport exists. As the following two sections show, this depends on cooling, viscous transport by random walks of large scale eddies describes by the alpha disk model (2) above and mass-inhomogeneities parameterized by $\xi$. The $\alpha$ and $\xi$ are probably inversely correlated, although a detailed description thereof is not known. For instance, small $\alpha$ disks have relatively high density and/or low temperature, by which they are prone to a variety of self-gravity and wave-like instabilities that may produce $\xi$.
We introduce $$\begin{aligned}
P = \frac{\xi}{\alpha}
\label{EQN_P}\end{aligned}$$ to reflect the ratio of gravitational radiation-to-viscous mediated angular momentum transport. $P$ effectively acts as an efficiency parameter in the gravitational wave output from the extended disk $r>r_b$ in the alpha model, assumed to hold for $r\ge r_b> r_{ISCO}$. In this region, the efficiency in gravitational radiation is relatively low and the approximation $l\simeq H$ in (\[EQN\_sigma\]) gives the mass density profile $$\sigma =2\pi r\, l \Sigma =\frac{2\sqrt{2}\,M}{3\alpha H}\left(\frac{r}{M}\right)^\frac{5}{2}\dot{m}~~(r> r_b).
\label{EQN_sig}$$ Accordingly, (\[EQN\_dm\]) implies $$\begin{aligned}
\Delta L_{GW} = \frac{256}{45h^2} P^2\left(\frac{M}{r}\right)^2 \,\dot{m}~~(r>r_\nu).
\label{EQN_DLGW}\end{aligned}$$ Adopting a scaling $H=h_0 r$, the total disk luminosity $L_{GW} = \int_{r_b}^\infty \Delta L_{GW} dn$, $dn=dr/l$, satisfies $$\frac{L_{GW}}{\dot{m}}=\frac{256}{90h^3}\frac{P^2}{z^{2}_b} \,\dot{m}\simeq
1.4\, \frac{P_1^2}{z^{2}_\nu}\left(\frac{h_0}{0.1}\right)^{-3}\, \left(\frac{\dot{M}}{M_\odot\,\mbox{s}^{-1}}\right),
\label{Lgw-ineffic}$$ where $z_b= r_b/M$ and $P = 10\,P_1$ associated with a fiducial value $\alpha=0.1$.
Based on these preliminaries, we next turn to some specific estimates of $r_c$ and $r_b$.
Fragmentation chirps
--------------------
In self-gravitating accretion flows, a possible origin of mass inhomogeneities is fragmentation when the cooling time of the accreted matter in the instability zone is on the order of or shorter than the orbital time (e.g., [@gam01; @ric05; @mej05]). The disk is unstable to axisymmetric perturbations if [@too64; @gol65] $$Q= \frac{c_s\Omega}{\pi G \Sigma}<1,\label{Qtoomr}$$ and to non-axisymmetric perturbations at slightly larger values, $Q\lesssim 2$ (e.g. [@gri11]).
For our $\alpha$-disk model, (\[Qtoomr\]) yields a characteristic radius beyond which the instability may be generated [@pir07]: $$\begin{aligned}
\frac{r_c}{M} > \left(\frac{3\alpha h_1^3}{\sqrt{2}\dot{m}}\right)^{2/3} \simeq
300 \alpha_{-1}^{2/3} \left( \frac{h_1}{0.5}\right)^2 \left(\frac{\dot{M}}{M_\odot \,\mbox{s}^{-1}}\right)^{-\frac{2}{3}}
\label{EQN_rc}\end{aligned}$$ adopting $H=h_1r$ at this radius [@pop99; @che07] with the fiducial scale $h_1=0.5$ for the relatively colder disk flow further out. The characteristic wavelength of the fastest growing mode is of the order of $QH$, and its mass is $(QH)^2\Sigma$.
Cooling may derive from several channels. Among electron-positron pair annihilation to neutrinos, URCA process, and photo-disintegration of $^4$He, it has been argued that the latter may be most effective one in the instability zone [@pir07]. Rapid cooling may thus lead to fragmentation into a gravitationally bound clumps of mass $m_f$ up to a few percent of the mass of the black hole, i.e. [@pir07]: $$m_f = M \sigma_f\simeq M\Sigma(QH)^2\simeq \frac{M}{\pi}h_1^3\simeq 0.04\, M.\label{sig_frag}$$ It is unclear how many fragments are produced in this process. In [@pir07], it is suggested that if multiple fragments form, they may merge into a mass of $0.1-1\,M_\odot$. Fragments thus produced will subsequently migrate inwards, initially so by viscous stresses. Any gravitational wave emission hereby derives its energy from the accretion flow. The characteristic strain amplitude hereby scales with the instantaneous strain amplitude, i.e., $h_{char}(f)\propto f^{2/3}$ (cf. \[EQN\_h1987A\]). When it reaches small enough radius with associated transition frequency $f_{e}$ in gravitational waves, angular momentum loss may be overtaken by gravitational wave emission. In this event, the gravitational wave luminosity effectively derives from gravitational binding energy of the inhomogeneities to the central object, as opposed to the accretion flow, until complete disruption by tidal forces.
At the fragment’s Roche radius $r_d\simeq 1.26 \eta M (m_f/M)^\frac{1}{3}$ for a black hole size $\eta M$, where $1\le \eta\le 2$ parametrizes uncertainty in black hole spin [@fis72; @lat74; @lat76], the orbital frequency is roughly $\Omega_{d}\simeq 1/M (M/r_d)^\frac{2}{3}$. With (\[sig\_frag\]) adopted for $m_f$, the corresponding gravitational wave frequency window is $$\begin{aligned}
W_c:~~ f_c< f< \min\{f_d,2f_{ISCO}\},
\label{EQN_WC}
\end{aligned}$$ where where $f_{ISCO}$ denotes the orbital frequency at the ISCO and $$\begin{aligned}
f_c\simeq 1.2 \,M_1^{-1} \frac{1}{\alpha_{-1}} \left( \frac{h_1}{0.5}\right)^{-3}
\left(\frac{\dot{M}}{M_\odot \,\mbox{s}^{-1}}\right) \,\mbox{Hz},~~f_{d}\simeq 300-900 \,M_{1}^{-1}\left(\frac{h_1}{0.5}\right)^\frac{3}{2}\,\mbox{ Hz},
\label{EQN_fc1}\end{aligned}$$ where $M=M_{1}\,10M_\odot$ and the range in $f_d$ refers to the uncertainty in $\eta$. The broad bandwidth in $W_c$ essentially covers the full operational bandwidth of sensitivity of LIGO-Virgo and KAGRA. The associated characteristic strain amplitude satisfies the canonical scaling of binary coalescence. For a mass fragment (\[sig\_frag\]), we have, adapted from (\[EQN\_hchar2\]) of Appendix A: $$\begin{aligned}
%\begin{array}{lll}
h_{char}(f>f_e) =
1.7 \times 10^{-22} D_{100}^{-1} \,M_1{^\frac{1}{3}} \left(\frac{\sigma_f}{0.04}\right)^\frac{1}{2} \left(\frac{f}{100\,\mbox{Hz}}\right)^{-\frac{1}{6}}~(f\epsilon W_{c})
%\end{array}
\label{EQN_hchar2b2}\end{aligned}$$ with $\sigma_f = m_f/M$, $D=D_{100} \,100$ Mpc and $W_{c}=[f_{e},\min\{f_d,2f_{ISCO}\}]$. A plot of the fragmentation chirps is exhibited in Fig. \[fig:0\].
Wave patterns in accretion disks
--------------------------------
We shall consider continuous accretion in a non-fragmented disk with deformations $\xi>0$ originate at large radii in wave motion, such as spiral waves by self-gravity in Lin-Shu type wave instabilities [@gri11]. These deformations may evolve as matter accretes inwards, though this is poorly understood at present.
In what follows, we consider a general framework of steady-state accretion and assume, for simplicity, that $\xi$ is a constant. Given the definition of $\delta m$ in (\[EQN\_dm\]), this means that $\delta m$ varies in proportion to $\sigma=\sigma(r)$. This model approach is hereby distinct from ordinary constant mass inhomogeneities.
Once more we focus on quadrupole emission, although any Jeans type self-gravitating instability, as in fragmentation, is inherently local and need not couple to the large scale structure of the disk to ensure that the lowest order modes are the most unstable. For illustrative purposes, we nevertheless focus on the $l=m=2$ instabilities.
Fig. \[fig:0\] illustrates our identification of quadrupole mass-moments in a spiral wave, in rings following discretization in polar coordinates. Each ring is has finite annular width. By the underlying spiral wave, each ring has a quadrupole mass-moment of over-dense regions, here represented by a pair of mass-inhomogeneities $\delta m$. Due to rotation, each pair of $\delta m$ emits quadrupole gravitational wave emission at twice the Keplerian frequency, defined by the radius of each ring. In accretion flows, the spiral wave gradually tightens. As $\delta m$ migrates inwards, their gravitational radiation broadens in frequency, each imprinting their own stamp in the gravitational wave spectrum. The total luminosity is the sum of the luminosity in each ring by Parseval’s theorem.
Our focus on quadrupole wave emissions gives a leading order approximation, that ignores possibly further emissions by higher order mass moments. The full spectrum of gravitational wave emission from all multiple mass moments can be calculated, but likely so the total luminosity is dominant in $m=2$ [@van02; @bro06].
![(a) A spiral density wave pattern in a disk. (b) Over-dense regions (thick black) in an annular region $4.5 < r < 5$ (thin black circles) of finite angular extend $\delta \varphi/2\pi<<1$. (a) Approximation of the over-dense regions in (b) by local mass-inhomogeneities $\delta m$, here $2\times 32$ on a grid with 32 annular rings. (d) Keplerian rotation implies distinct quadrupole emission spectra of each $\delta m$ in (c) that are non-overlapping in frequencies. The Fourier coefficients $|c_k|^2$ show broadening due to accretion, shown in (d) for two accretion rates, corresponding to radial migrations $\delta r_2<\delta r_1<0$. (Reprinted from [@lev15].)[]{data-label="fig:0"}](f21)
We consider a transition radius $r_b$, beyond which accretion is by viscous angular momentum transport following standard thin disk accretion theory. Within $r_b$, accretion is driven by gravitational radiation losses in the inspiral in region $$\begin{aligned}
r_{ISCO} < r < r_b.
\label{EQN_AISCO1}\end{aligned}$$ Provided that $r_b$ is sufficiently larger than $r_{ISCO}$, (\[EQN\_AISCO1\]) implies maximal efficiency in gravitational radiation with a luminosity satisfying (\[EQN\_AISCO2\]). We now show that $r_b>r_{ISCO}$ is satisfied at hyper-accretion rates.
By $\Delta L_{GW}$ in (\[EQN\_dm\]), the ring shrinks, thereby reducing its total energy by $d\Delta E = M^2\sigma/2r^2dr = -\Delta L_{GW}dt_{GW}$ $(dr<0)$, that is $$dt_{GW} = \frac{5}{64} \left(\frac{M}{r}\right)^{-3} \xi^{-2}\sigma^{-1}dr.
\label{EQN_tgw}$$ The time for a radial drift $dr$ by viscous torques alone is $$\begin{aligned}
dt_b = \frac{2r}{3\nu} dr = \frac{\sqrt{8}M^2}{3\alpha H^2} \left(\frac{M}{r}\right)^{-\frac{5}{2}}dr.
\label{EQN_tnu}\end{aligned}$$ We define the transition radius $r_b$ below which gravitational driven migration dominates over viscous transport by equating (\[EQN\_tgw\]) and (\[EQN\_tnu\]), i.e., $dt_{GW}=dt_b$ at the transition radius $$\begin{aligned}
\frac{r_b}{M} =\left( \frac{128\sqrt{2}}{15} \frac{\xi^2 \sigma M^2}{\alpha H^2}\right)^2.
\label{EQN_rnu}\end{aligned}$$
To be specific, consider a vertical scale height $H(r)/r$ slowly increasing with $r$ from about 0.1 within a few gravitational radii to about 0.5 at about 100 $M$, based on numerical results for neutrino cooled disk models [@pop99; @che07]. These numerical results show that $H/r$ depends relatively weakly on $\alpha$. For illustrative purposes, we shall therefore adopt $H(r)=h_0 r$, $h_0\simeq0.1$, in the inner region. Our $(\alpha,\xi)$ model (\[EQN\_alpha\]-\[EQN\_dm\]) thus obtains $$\frac{r_b}{M}=\frac{512\xi^2}{45\alpha^2 h_0^3}\dot{m}\simeq 5.7\,P_1^2\,\left(\frac{h_0}{0.1}\right)^{-3}\left(\frac{\dot{M}}{M_\odot\,\mbox{s}^{-1}}\right),
\label{EQN_rnu2}$$ where $P=10\,P_1$, $P_1=\xi / \alpha_{-1}$, $\alpha=0.1\alpha_{-1}$. Integration of (\[EQN\_DLGW\]) then yields $$\begin{aligned}
%\begin{array}{ll}
L_{GW}(r) = \frac{\dot{m}Mr_\nu}{4r^2}~~(r>r_b),~~
L_{GW}(r_b) = 2.2\times 10^{-7} P_1^{-2}\left(\frac{h_0}{0.1}\right)^3.
%\end{array}
\label{Lgw-ineffic2}\end{aligned}$$
In the region $r_{ISCO}<r<r_b$ the accretion flow is driven by gravitational radiation losses, where the mass profile of the enclosed number of $dn=dr/l$ rings is determined from the relation $(\sigma M)dn=\Delta m \,dn = \dot{m} \,dt_{GW}$. Using (\[EQN\_tgw\]) with $l=H$, $H=h_0r$, we obtain $$\sigma =\frac{1}{5.7\xi} \sqrt{{5h_0\dot{m}}} \left(\frac{r}{M}\right)^2.
\label{sig-gw}$$ Note the relatively steep radial dependence compared to (\[EQN\_sig\]) due to a modified surface density $\Sigma$. Following substitution of (\[sig-gw\]) into (\[EQN\_dm\]), it is seen that $\Delta L_{GW}\propto \dot{m}\, r^{-1}$ is strongest at the smallest radii. The total luminosity is roughly the sum over the inner rings, $L_{GW}\simeq (r_{ISCO}/l)\Delta L_{GW}(r_{ISCO})\simeq h_0^{-1}\Delta L_{GW}(r_{ISCO})$. In the high efficiency regime $r_b>r_{ISCO}$, we thus obtain the total disk luminosity (\[EQN\_AISCO2\]). It shows that a major fraction of the accretion flow is converted to gravitational radiation, [*independent*]{} of $P$.
According to (\[EQN\_rnu2\]), $r_{b}>r_{ISCO}$ holds at hyper-accretion rates $$\begin{aligned}
\frac{\dot{M}}{M_\odot\,\mbox{s}^{-1}} > \frac{1}{5.7} z\,P_1^{-2}
\label{EQN_hz}\end{aligned}$$ with luminosity (\[EQN\_AISCO2\]) provided that mass inhomogeneities originating at $r>r_b>r_{ISCO}$ survive all the way to $r_{ISCO}=zM$. If they dissipate at $r_{ISCO}<r_{diss}<r_b$, then $r_{ISCO}$ should be replaced by $r_{diss}$ in (\[EQN\_AISCO2\]) , and the net result is a relatively lower $L_{GW}$ emitted at lower frequencies. It will also be appreciated that for (\[EQN\_hz\]) to proceed on a long duration time scale of tens of seconds, a large progenitor remnant stellar envelope mass is required or $\alpha$ is small.
We express the outlook above in terms of the frequency window associated with quadrupole emissions at $r_b$ and $r_{ISCO}$: $$\begin{aligned}
W_b:~~ f_b < f< \max\{f_{diss},2f_{ISCO}\},
\label{EQN_Wnu}
\end{aligned}$$ where $$f_b\simeq 468\,M_1^{-1} P_1^{-3}\left(\frac{h_0}{0.1}\right)^{\frac{9}{2}} \left(\frac{\dot{M}}{M_\odot\,\mbox{s}^{-1}}\right)^{-\frac{3}{2} }\,\mbox{Hz}.
\label{EQN_fnu}$$
The observable relevant to $W_b$ is the associated characteristic strain amplitude. Over a time period $\tau$, an accretion rate $\dot{m}$ implies a mass migration $\dot{m}\tau$ in the annular region down to $r$ from $r_b$ by the associated energy $\Delta E_{rad}=({1}/{2})\left({M}/{r}\right)\dot{m}\tau$ in gravitational radiation. For $r_{ISCO}\le r << r_b$, it is emitted over a bandwidth given by the difference between the gravitational wave frequency at $r$ and $f_b$, $$\begin{aligned}
\Delta f = f(r) - f_b \simeq f(r) = (\pi M)^{-1} \left(\frac{M}{r}\right)^\frac{3}{2}
\label{EQN_A4}\end{aligned}$$ in the present Keplerian approximation. Consequently, $$\begin{aligned}
\frac{\Delta {E}_{rad}}{\Delta f} = \frac{\pi\dot{M}\tau M}{2}\left(\frac{M}{r}\right)^{-\frac{1}{2}},
\label{EQN_A5}\end{aligned}$$ and hence the characteristic strain for a source at a distance $D$ is $$\begin{aligned}
%\begin{array}{ll}
h_{char}(f) = \frac{\sqrt{2}}{\pi D}\sqrt{\frac{\Delta { E_{rad}}}{\Delta f}} \simeq
\frac{M}{\sqrt{\pi} D}\, \sqrt{\frac{\dot{M}\tau}{M }} \left(\pi M f \right)^{-1/6}=\sqrt{2}\kappa \left(f/f_b\right)^{-\frac{1}{6}}
%\end{array}
\label{EQN_A6A}\end{aligned}$$ for $f\epsilon W_b$ with $$\begin{aligned}
\kappa = \kappa_0 \left(\frac{f_b}{f_0}\right)^{-\frac{1}{6}},~~\kappa_0= \frac{M}{\sqrt{2\pi }D}\, \sqrt{\frac{\dot{M}\tau}{M}} \left(\pi M f_0 \right)^{-1/6}.
\label{EQN_A6B}\end{aligned}$$ Numerically, we have $$\begin{aligned}
\kappa_0 = 8.3 \times 10^{-22}\,\frac{M_1^\frac{1}{3}}{D_{100}} \sqrt{\frac{M_{acc}}{M_\odot}} \left(\frac{f_0}{1000\,\mbox{Hz}}\right)^{-\frac{1}{6}},
\label{EQN_A6C}\end{aligned}$$ where we put $M_{acc}=\dot{M}\tau$. It shows that $h_{char}(f)$ is maximal in $W_b$ at $f\simeq f_b$. Observationally, $f_b$ is the most relevant frequency in $W_\nu$ provided that it falls in the thermal or shot noise dominated region of the LIGO-Virgo and KAGRA detectors.
By the strong dependence of $f_b$ on $\dot{M}$ in (\[EQN\_fnu\]), it will be appreciated that $W_b$ opens up a window $f_b< 2f_{ISCO}$ only at large hyper-accretion rates, when $$\begin{aligned}
%\begin{array}{ll}
\frac{\dot{M}}{M_\odot\,\mbox{s}^{-1}} > \left( \frac{M_1f_{ISCO} }{216 \,\mbox{Hz}}\right)^{-\frac{2}{3}} \alpha_{-1}^2\simeq
\frac{1}{6} \left(z^\frac{3}{2}+\hat{a}\right)^\frac{2}{3}\alpha_{-1}^2\ge 0.26\,\alpha_{-1}^2,
%\end{array}
\label{EQN_m0}\end{aligned}$$ where $z=r_{ISCO}/M$, $\hat{a}=a/M$ denotes the dimensionless spin rate of the black hole and the lower bound on the right hand side refers to an extremal Kerr black hole $(z=\hat{a}=1)$. The inequality on the right hand side of (\[EQN\_m0\]) provides a necessary but not sufficient condition. Around non-extremal black holes, the required $\dot{M}$ is larger. Furthermore, the frequency windows $W_c$ and $W_b$ satisfy different scalings with accretion rate $\dot{M}$. Around a nearly extremal Kerr black hole, we note that $f_c = f_b$ only when $$\begin{aligned}
\dot{m} \simeq 10 \,\left(\frac{h_1}{0.5}\right)^\frac{6}{5} \alpha_{-1}^\frac{8}{5}.\end{aligned}$$ Unless $\alpha_{-1}$ is small, i.e., $0.001 < \alpha < 0.01$, we expect $$\begin{aligned}
W_b \subset W_c.
\label{EQN_WW}\end{aligned}$$
In the external region $r>r_b$, the emission in gravitational waves is at frequencies $f<f_b$. In this region, the emission is relatively inefficient and satisfies (\[Lgw-ineffic2\]). We can estimate the effective strain $h_{eff}=h(f) \sqrt{f\tau}$ from the instantaneous strain $h=L_{GW}^{1/2}/\Omega D$, $\Omega = \pi f$, and the number of wave periods $n=f\tau$. By $h_{char}\sqrt{f^{-1} \Delta f}=\sqrt{2}h_{eff}$, we thus estimate the characteristic strain associated with a one-sided frequency spectrum as $$\begin{aligned}
h_{char}(f) = \frac{M}{\sqrt{2 \pi} D} \sqrt{\frac{M_{acc}}{M}} \left(\pi M f_b\right)^{-\frac{1}{6}} (f/f_b)^{\frac{1}{6}}
\,(f < f_\nu).
\label{EQN_inefficD}\end{aligned}$$
A matching of the expressions (\[EQN\_A6A\]) and (\[Lgw-ineffic2\]) obtains by noting that the luminosity in the inner radiatively efficient region $[r,r_b]$, $r_{ISCO}\le r << r_b$, satisfies $$\begin{aligned}
L_{GW}^i([r,r_\nu])= \frac{M\dot{m}}{2r} - \frac{M\dot{m}}{4r_b} = \frac{M\dot{m}}{4r}\left[2-(f_b/f)^\frac{2}{3}\right] .
\label{EQN_LGWi}\end{aligned}$$ With (\[EQN\_A6C\]), we therefore have [@lev15] $$\begin{aligned}
h_{char}(f) = \kappa \left\{
\begin{array}{lr}
(f/f_b)^\frac{1}{6} & (f<f_b)\\
(f/f_b)^{-\frac{1}{6}}\left[ 2 - (f_b/f)^\frac{2}{3}\right]^\frac{1}{2} & (f\ge f_b).
\end{array}\right.
\label{EQN_A6D}\end{aligned}$$ This broadband spectrum increases with $\dot{m}$. Due to (\[Lgw-ineffic2\]), the increase in $f<f_b$ is entirely due a shrinking of the bandwidth, as $r_\nu$ increases and $f_b$ decreases with $\dot{M}$. In contrast, the increase in $f>f_b$ is due to an increase in the luminosity (\[EQN\_LGWi\]). A plot of the proposed broadband emission (\[EQN\_A6D\]) is exhibited in Fig. \[fig:hchar\] for various choices of $M_{acc} = \dot{m}\tau$ and $f_b$.
ISCO waves
----------
Following (\[EQN\_TeT90\]), we consider sub-critical accretion allowing for the formation of a torus in suspended accretion, which catalytically converts most of the input from the black hole into gravitational radiation at frequencies about $$\begin{aligned}
f_{GW} \simeq \pi^{-1} \Omega_{ISCO}
\label{EQN_fgw2}\end{aligned}$$ with lesser emission at higher frequencies [@bro06], accompanied by a minor output (\[EQN\_Lj\]) in BPJs. The former features a distinctive descending chirp, due to expansion of the ISCO as the black hole spins down [@van08]. This emission is subsequent to any gravitational burst associated with the initial formation of the black hole and continuing accretion [@ree74; @due04; @lip83; @leu97; @fon01; @nag07; @lip09; @fry02] and separate from any quasi-mode ringing (QNR) of the event horizon [@tho86; @kok99] with relatively high frequencies ($l=m=2$) [@ech98] $$\begin{aligned}
f_{QNM}=3200 \,\mbox{Hz}\,\left[1-0.63\left(1-\frac{a}{M_H}\right)\right]^\frac{3}{10} \left(\frac{M_H}{10M_\odot}\right)^{-1}.
\label{EQN_fQNR}\end{aligned}$$
Quite generally, gravitational radiation from ISCO waves is described by mass moments $I_{lm}$ in the quantum numbers $l$ and $m$ of spherical harmonics has a luminosity (Appendix A) $$\begin{aligned}
L_{GW} \propto \Omega_T^{2l+2}I_{lm}^2.
\label{EQN_LQ1}\end{aligned}$$ In practice, it appears that most of the emission comes from $m=l$ [@bro06]. To leading order $\Omega_T^2\propto r^{-3}$, whereby (\[EQN\_LQ1\]) satisfies $$\begin{aligned}
L_{GW} \propto \frac{1}{r^{m+3}}.
\label{EQN_Q2}\end{aligned}$$ Consequently, gravitational wave luminosity tends to be maximal at the ISCO, such as exemplified in the previous section.
Susceptability to non-axisymmetric instabilities at the ISCO is due to a variety of processes. In particular, energetic feedback by black hole sets in at $E_{rot}^H$ in excess of 5.3% of $E_{rot}^{max}$, when $\Omega_H>\Omega_{ISCO}$. Black holes losing angular momentum to surrounding matter enforce turbulence with associated heating and enhanced (thermal and magnetic) pressure [@van02; @van12]. An improved criterion for the black hole to be rapidly spinning, is when its forcing onto the surrounding matter exceeds the luminosity in magnetic winds from the latter, taking advantage of $\Omega_H/\Omega_{ISCO}>1$ (the ratio extends up to 1.4396 in the Kerr metric) and the substantial surface area of the event horizon. Using the Shakura-Sunyaev solution [@sha73] as a leading order approximation of the inner disk, the time integrated luminosity of the black hole $E_H$ onto the inner disk exceeds the energy loss $E_w^*$ of the latter in magnetic winds whenever $a/M\ge 0.4433$ [@van12]. Equivalently, $E_{rot}^H$ initially exceeds $E_{rot}^{max}$ by about 9%, which is still small. The excess $E_H - E_w^*$ can be radiated off in gravitational waves and MeV neutrinos. Here, we attribute the excitation of non-axisymmetric wave instabilities to enhanced thermal and magnetic pressures induced by such feedback.
### Mass moments about the ISCO
Papaloizou-Pringle [@pap84] considered (\[EQN\_Omq\]) with non-Keplerian rotation index $q>3/2$. Applied to a torus, the associated surface gravity allows surface waves to be excited at the inner and outer surface. In the approximation of an incompressible fluid, a detailed analysis shows that the coupling between waves on these two opposite faces allows for angular momentum transport outwards on a dynamical time scale. In the singular limit of an infinitesimally slender torus considered in [@pap84], these surface waves become unstable to all of $m\ge 1$ modes at the same critical rotation index $q_c=\sqrt{3}$.
Here, we envision $q>3/2$ induced by feedback from the central rotating black hole. Extending the Papaloizou-Pringle instability to tori of arbitrary width [@van02], non-axisymmetric instabilities are found to set in consecutively, starting at $m=1$, as a function of the rotation index of a torus (Appendix B). When $q>q_{cm}(\delta)\ge \sqrt{3}$ exceeds a critical value, its azimuthal modes $m=1,2,\cdots$ become successively unstable (as a function of $q$), where [@van02] $$\begin{aligned}
%\begin{array}{ll}
q_{c1}(\delta)= \sqrt{3} + 0.27 \left(\frac{\delta}{0.7506}\right)^2,~~
q_{c2}=\sqrt{3} +0.27\left(\frac{\delta}{0.3260}\right)^2,\cdots
%\end{array}
\label{EQN_APB4}\end{aligned}$$ as a function of the ratio $\delta=b/R$ of minor-to-major radius of the torus. Here, $\sqrt{3}$ refers to the singular limit $\delta=0$ of infinitesimally slender tori in [@pap84]. The curves (\[EQN\_APB4\]) are analytic approximations to semi-analytical results shown in Fig. \[fig:p6a\]. This instability has also been observed in recent numerical simulations [@kiu11].
From heating alone, a minimum value of $q$ can be associated with the temperature $T=T_{10}10^{10}$ K [@van03] $$\begin{aligned}
T_{10} \simeq 2L_{\nu,52} \left(\frac{M_T}{0.1M_\odot}\right)^{-\frac{1}{6}}.
\label{EQN_APB2}\end{aligned}$$ For a torus of minor-to-major radius $\delta = b/R$, the resulting thermal pressure enhances $q$ according to $$\begin{aligned}
q = 1.5 + 0.15 \left(\frac{R}{M}\right) \left(\frac{\delta}{0.2} \right)^2 T_{10}.
\label{EQN_APB3}\end{aligned}$$ Thermal pressures and magnetic pressures will be similar at temperatures of about 2 MeV [@van09a]. When $q>q_{cm}(\delta)\ge \sqrt{3}$ exceeds a critical value, its azimuthal modes $m=1,2,\cdots$ become unstable as shown in Fig. \[fig:p6a\].
A mode that becomes unstable is generally strengthened by gravitational radiation back reaction [@van02]. As a result, inducing gravitational wave emissions by (\[EQN\_APB4\]) is reminiscent of a Hopf bifurcation [@van12]. The result is a characteristic descending chirp during black hole spin down (Fig. \[fig:p6a\]).
Ultimately, the luminosity is determined by a nonlinear saturation amplitude, balancing [*heating*]{} in dissipation and [*cooling*]{} in gravitational waves, MeV neutrino emission and magnetic winds [@van12], unless aforementioned feedback is intermittent or the torus as a whole is unstable. The first allows for an analytic estimate in case of a flat infrared spectrum of MHD turbulence [@van01b]. The latter is possibly of interest to the most luminous sources, perhaps resulting from hyper-accretion onto the ISCO in core-collapse supernovae [@van15a].
![(Left panel.) The neutral stability curves for buckling modes expressed in terms of the critical rotating index $q_c$ as a function of minor-to-major radius $\delta=b/a$. Labels refer to azimuthal quantum numbers $m=1, 2, \cdots$, where instability sets in above and stability sets in below. Of particular interest is the range $q \le 2$, where $m=0$ is Rayleigh-stable. For $q = 2$, instability sets in for $b/a < 0.7385$ ($m = 1$), 0.3225 ($m = 2$) and, asymptotically, for $b/a< 0.56/m$ ($m \ge 3$). Included are the curves of $q_c$ as a function of $\delta$ at various temperatures $T=T_{10}\,10^{10}$ K. (Reprinted from [@van02; @van12].) (Middle and right panels.) Theoretical wave form of ISCO waves induced around an initially extremal black hole obtained by integration of the equations of suspended accretion in the strong interaction limit $L_{GW}\simeq L_H\simeq -\dot{M}$. Shown is the orientation averaged strain amplitude $h(t)/\sqrt{5}$. Due to a turbulent background accretion flow, phase-incoherence is anticipated to be limited intermediate time scales. The wave form is sliced into chirp templates of intermediate duration $\tau=1$s for use in matched filtering. (Reprinted from [@van08].)[]{data-label="fig-fsc1"}](f22 "fig:")![(Left panel.) The neutral stability curves for buckling modes expressed in terms of the critical rotating index $q_c$ as a function of minor-to-major radius $\delta=b/a$. Labels refer to azimuthal quantum numbers $m=1, 2, \cdots$, where instability sets in above and stability sets in below. Of particular interest is the range $q \le 2$, where $m=0$ is Rayleigh-stable. For $q = 2$, instability sets in for $b/a < 0.7385$ ($m = 1$), 0.3225 ($m = 2$) and, asymptotically, for $b/a< 0.56/m$ ($m \ge 3$). Included are the curves of $q_c$ as a function of $\delta$ at various temperatures $T=T_{10}\,10^{10}$ K. (Reprinted from [@van02; @van12].) (Middle and right panels.) Theoretical wave form of ISCO waves induced around an initially extremal black hole obtained by integration of the equations of suspended accretion in the strong interaction limit $L_{GW}\simeq L_H\simeq -\dot{M}$. Shown is the orientation averaged strain amplitude $h(t)/\sqrt{5}$. Due to a turbulent background accretion flow, phase-incoherence is anticipated to be limited intermediate time scales. The wave form is sliced into chirp templates of intermediate duration $\tau=1$s for use in matched filtering. (Reprinted from [@van08].)[]{data-label="fig-fsc1"}](f23a "fig:")![(Left panel.) The neutral stability curves for buckling modes expressed in terms of the critical rotating index $q_c$ as a function of minor-to-major radius $\delta=b/a$. Labels refer to azimuthal quantum numbers $m=1, 2, \cdots$, where instability sets in above and stability sets in below. Of particular interest is the range $q \le 2$, where $m=0$ is Rayleigh-stable. For $q = 2$, instability sets in for $b/a < 0.7385$ ($m = 1$), 0.3225 ($m = 2$) and, asymptotically, for $b/a< 0.56/m$ ($m \ge 3$). Included are the curves of $q_c$ as a function of $\delta$ at various temperatures $T=T_{10}\,10^{10}$ K. (Reprinted from [@van02; @van12].) (Middle and right panels.) Theoretical wave form of ISCO waves induced around an initially extremal black hole obtained by integration of the equations of suspended accretion in the strong interaction limit $L_{GW}\simeq L_H\simeq -\dot{M}$. Shown is the orientation averaged strain amplitude $h(t)/\sqrt{5}$. Due to a turbulent background accretion flow, phase-incoherence is anticipated to be limited intermediate time scales. The wave form is sliced into chirp templates of intermediate duration $\tau=1$s for use in matched filtering. (Reprinted from [@van08].)[]{data-label="fig-fsc1"}](f23b "fig:")
\[fig:p6a\]
### Dimensionless strain amplitude
The wave instability (\[EQN\_APB4\]) has the desirable property that the associated mass-moments are predominantly at the lowest quantum numbers, which ensures that most of the gravitational wave output is at the lowest quadrupole emission frequency. Around a stellar mass black hole of $10M_\odot$, the resulting frequency is broadly in the range of 500-3000 Hz for emissions from quadruple mass moments, which falls within the sensitivity wave band of the upcoming advanced ground based detectors. Emissions from $m=3$ and higher are unlikely to be detectable by these detectors. (Emissions from $m=1,2$ in a disk or torus produce quadrupole emissions at the same frequency from the combined black hole plus disk or torus system). Even so, their output may be of interest to future, next generation detectors. The same conclusion holds for gravitational wave spectra produced by magnetic pressure induced multipole mass moments based on a numerical simulation [@bro06]. [*Accordingly, the frequency of gravitational wave emissions is fixed by the Kerr metric for a given mass and angular momentum of the black hole with those of quadrupole emission satisfying*]{} $$\begin{aligned}
f_{GW} = 2f_{ISCO},
\label{EQN_fgw}\end{aligned}$$ where $f_{ISCO}=\Omega_T/2\pi$, where $\Omega_T$ denotes the angular velocity of the torus formed at the ISCO. For stellar mass black hole systems, this ensures that frequencies are within the LIGO-Virgo and KAGRA bandwidth of sensitivity. At late times, when the angular velocity of the black hole approaches that of the ISCO, we have for an initial black hole mass $M$ the frequency range [@van11] $$\begin{aligned}
f_{GW} = 595 - 704 \,\mbox{Hz}\,\left(\frac{M}{10M_\odot} \right)^{-1},
\label{EQN_APB5}\end{aligned}$$ where the 15% frequency range 595-704 Hz refers to different choices of black hole initial spin. (The minimum value of 595 Hz corresponds to an initially maximally spinning black hole.)
![Schematic overview of quadrupole gravitational radiation derived from accretion flows onto rotating black holes in non-axisymmetric wave patterns ($W_b$) or spiral in of fragments ($W_c$), or ISCO waves (red curve). The first produces one or more ascending chirps at sufficiently high accretion rates, the second descending chirps. Fragmentation may occur at sufficient cooling in the extended accretion disk. Wave instabilities from a torus about the ISCO arise from heating and magnetic pressure by feedback from the central black hole if $\Omega_H > \Omega_{ISCO}$. High frequency radiation can be produced by Quasi-Normal Mode ringing of the event horizon that may be exciting by matter plunging in (black curve). The spectrum may contain additional radiation from higher order modes (not shown), e.g., by high $m$ multipole mass moments in the disk or torus, as well as fragments in elliptical orbits (not shown).[]{data-label="fig:fisco"}](f25)
The instantaneous strain implies an [*effective*]{} strain $h_{eff}=\sqrt{n}h$, where $n\simeq fT$ at a characteristic frequency $f$ for a burst duration $T$. Thus, $h_{eff} \simeq \pi^{-1} D^{-1} \sqrt{E/f}$ represents the strain that can be recovered by matched filtering in the ideal limit of matching the entire signal with a model. Similar to the derivation of (\[EQN\_inefficD\]), we have, in the frequency domain [@fla98a; @fla98b], the corresponding characteristic amplitude satisfies $$\begin{aligned}
h_{char}(f) \simeq \frac{\sqrt{2}}{\pi D} \sqrt{ \frac{dE}{df} } \simeq 3.5\times 10^{-21}\,D_{100}^{-1} \,M_1
\label{EQN_heff}\end{aligned}$$ based on $dE/df \simeq 4.5\times 10^{12} M_1^2$ cm$^{2}$ across a frequency range of about 600 - 1500 Hz $(10M_\odot/M)$ calculated in the next section.
Fig. \[fAPC1\] summarizes our perspective on gravitational wave chirps associated with mergers and black holes losing angular momentum following Fig. \[fig:origin\]. For the suspended accretion state, the chirps shown have a characteristic late time frequency around 600-700 Hz for a $M=10M_\odot$ black hole following (\[EQN\_APB5\]). It should be mentioned that Fig. \[fAPC1\] represents trajectories when matter about the ISCO orbits in the equatorial plane. This assumption need not hold. Any misalignment between its angular momentum and that of the black hole would lead to Lense-Thirring precession and hence line-broadening of the trajectories shown [@van04].
For a low mass black hole formed in a double neutron star merger producing a SGRBEE, the late frequency (\[EQN\_APB5\]) of about 2.3 kHz is particularly well defined, given the narrow range of neutron star masses around $1.5 M_\odot$. The latter outlook may be of interest to future narrow band detector modes [@cut02].
Figs. \[fig:fisco\]-\[fig:hchar\] summarize the frequencies and strain amplitude of quadrupole emissions from accretion flows and a torus about the ISCO. Concluding this section, we anticipate, further to (\[EQN\_TeT90\]), that for the gravitational wave emissions most likely observable by LIGO-Virgo and KAGRA with durations $T_{GW}\simeq T_{engine}$, where $T_{GW}$ represents either the time scale of hyper-accretion or the lifetime of spin of the central black hole.
Stochastic background from CC-SNe
---------------------------------
In a homogeneous isotropic universe, the contribution to the stochastic background in gravitational waves from a source population locked to the cosmic star formation rate (SFR) can be expressed in terms of the spectral energy density $\epsilon_B^\prime$ per unit volume per unit frequency. It is common to express the same per unit logarithmic frequency relative to the closure density, i.e., $f\epsilon_B^\prime$ relative to $\rho_c$, where $f$ denotes the observed frequency in gravitational waves and $\rho_c = 3H_0^2/8\pi$ in geometrical units for a present-day Hubble constant $H_0$. It can be calculated [@van05b] based on (a) Einstein’s adiabatic relationship $E=hf$ for the energy $E$ of a graviton of frequency $f$, where $h$ is Planck’s constant; (b) conservation of radiation energy within a co-moving volume during cosmological evolution; and (c) a scaling of the SFR over the cosmic evolution described by the Hubble constant $H(z)=H_0h(z)$ [e.g. @por01]. In three-flat $\Lambda$CDM at late times, $h(z)=\sqrt{1-\omega_m+\omega_m(1+z)^3}$ with present-day matter (dark and baryonic) density $\omega_m$ expressed relative to aforementioned $\rho_c$.
The event number density per unit redshift $N(z)$ and the event rate volume density $R(z)$ satisfy $N(z)dz=R(z)dt_e$ where $dz$ refers to the redshift interval with a corresponding locally measured time interval $dt_e$ (measured in the source frame). Consequently, $$\begin{aligned}
\begin{array}{l}
N(z) = R(z) \frac{dt_e}{dz} dz = R(z) \frac{dt_e}{dt} \frac{dt}{dr} \frac{dr}{dz} = \frac{R(z) dz}{(1+z)H_0h(z)},
\end{array}
\label{EQN_B1}\end{aligned}$$ where $dt_e/dt = 1/(1+z)$ is the cosmological dilation in time, $dr/dt=1$ is the velocity of light measured at $z=0$ and $dr/dz=(H_0h(z))^{-1}$ denotes the change in proper distance with respect to redshift. According to (c) derived in [@por01], the cosmic SFR expressed in terms of a rate per unit volume $R_{SF2}(z,\Omega_\Lambda)$ in a three-flat cosmology parametrized by $\Omega_\Lambda$ satisfies $$\begin{aligned}
R_{SF2}(z,\Omega_\Lambda) = R_{SF2}(z,0)\frac{h(z)}{h_0(z)},~~h_0=(1+z)^\frac{3}{2}.
\label{EQN_B2}\end{aligned}$$
By Einstein’s adiabatic relationship (a), $dE/df$ is redshift invariant as a function of $(1+z)f$ for a given $f$. Hence, by conservation of radiation (b), we have (cf. [@phi01] for a closely related expression) $$\begin{aligned}
\epsilon_B^\prime(f) = \int_0^{\infty} \frac{dE}{df}((1+z)f) N(z) dz,
\label{EQN_B3}\end{aligned}$$ where $z_{max}$ denotes the maximal redshift in the cosmic SFR model rate (\[EQN\_B2\]). For a gravitational wave source locked to the cosmic SFR, $$\begin{aligned}
N(z)=N_0 \frac{R_{SF2}(z,\Omega_\Lambda) }{ R_{SF2}(0,\Omega_\Lambda)},\end{aligned}$$ where $N_0$ denotes the the observed rate volume density at $z=0$. By (\[EQN\_B2\]), it follows that $$\begin{aligned}
\begin{array}{l}
\epsilon_B^\prime(f) = n_0 \int_0^{\infty} \frac{dE}{df}\left[(1+z)f\right] \frac{\hat{R}_{SF2}(z)}{(1+z)^\frac{3}{2}} \,dz, n_0=\frac{R_0}{H_0},
\end{array}
\label{EQN_B4}\end{aligned}$$ where $\hat{R}_{SF2}(z)=R_{SR2}(z,0)/R_{SF2}(0,0)$ denotes the normalized cosmic SFR satisfying $\hat{R}_{SF2}(0,0)=1$.
A concrete example of a cosmic SFR model rate is [@por01] $$\begin{aligned}
R_{SF2}(z,0) = \frac{0.16 h_{73} }{1+660 e^{-3.4(1+z)}}\,M_\odot \,\mbox{yr}^{-1}\,\mbox{Mpc}^{-3},
\label{EQN_B5}\end{aligned}$$ where $H_0=h_{73} \times 73$ km s$^{-1}$ Mpc$^{-1}$, whereby $$\begin{aligned}
\begin{array}{l}
\hat{R}_{SF2}(z) = \frac{23}{1+660 e^{-3.4(1+z)}},~~
N(z)= \frac{23N_0}{(1+660 e^{-3.4(1+z)})(1+z)^\frac{3}{2}}
\end{array}
\label{EQN_B5b}\end{aligned}$$ is an approximation to the observed cosmic SFR over the redshift range $0\le z \le 5$. For a given observational parameter $N_0$ and source model $dE/df$, $\epsilon_B^\prime(f)$ can thus be evaluated by numerical integration. In what follows, we shall write $E_f = dE/df$. Thus, Schwarz’ inequality provides an a priori bound on $\epsilon_B^\prime(f)$, given by $$\begin{aligned}
\begin{array}{l}
\epsilon_B^\prime(f) \le n_0A_2E_2,~E_2= \sqrt{ \int_0^\infty E_f^2(x) \frac{dx}{x} },~~
A_2=\sqrt{ \int_0^{\infty} \frac{\hat{R}^2_{SF2}(z)}{(1+z)^2} \,dz}
\end{array}
\label{EQN_B6}\end{aligned}$$ with $A_2=12.72$. For a source effectively described by a constant $E_f=E_f^0$ over a finite bandwidth $B=f_2-f_1$ between two cut-off frequencies $f_{1,2}$, i.e., we have $$\begin{aligned}
\begin{array}{l}
\epsilon_B^\prime(f) = n_0E^0_f \int_{1+z=f_1/f}^{1+z=f_2/f} \frac{\hat{R}(z)}{(1+z)^\frac{3}{2}}\,dz \le n_0E^0_f A_\frac{3}{2},~~
A_\frac{3}{2}=\int_0^\infty \frac{\hat{R}(z)}{(1+z)^\frac{3}{2}}\,dz,
\end{array}
\label{EQN_B7a}\end{aligned}$$ where $A_\frac{3}{2}=5.8$. The maximum of $\epsilon_B^\prime(f)$ attains at $\epsilon_B^{\prime\prime}(f) = 0$, i.e., $$\begin{aligned}
\hat{R}\left(\frac{f_1}{f}-1\right) = \sqrt{\frac{f_1}{f_2}} \hat{R}\left(\frac{f_2}{f}-1\right).
\label{EQN_B7b}\end{aligned}$$ For a cosmic SFR that is asymptotically constant, i.e., $\hat{R}(z)\simeq \hat{R}_*$ at large $z$, e.g., $\hat{R}_*=23$ in (\[EQN\_B5\]), (\[EQN\_B7b\]) reduces to the implicit equation $$\begin{aligned}
\hat{R}\left(\frac{f_1}{f}-1\right) = \sqrt{\frac{f_1}{f_2}}\hat{R}_*
\label{EQN_B7c}\end{aligned}$$ whenever $f_2>>f_1$. To exemplify, $f_1=600$ Hz and $f_2=3000$ Hz associated with a black hole mass $M=10\,M_\odot$ imply a maximum at $$\begin{aligned}
f_{B,peak}=272\,\mbox{Hz}
\label{EQN_B7d}\end{aligned}$$ as the root of (\[EQN\_B7c\]) for the model rate (\[EQN\_B5\]) (with a corresponding $z_c=0.84$). Note that $f_2/f_c=11.0$, which is still within the redshift range of star formation. At this frequency, (\[EQN\_B7a\]) gives the maximum $$\begin{aligned}
\epsilon_B^\prime(f) = n_0E^0_f A_*,
\label{EQN_B7e}\end{aligned}$$ where $A_*=3.52$. In this approximation, we have, consequently, $$\begin{aligned}
\Omega_B = \frac{B\epsilon_B^\prime(f)}{\rho_c} \simeq 10^{-8}\, \left(\frac{k}{10\%}\right)
\label{EQN_B7f}\end{aligned}$$ for a bandwidth $B\simeq 1000$ Hz in gravitational waves from SN Ib/c. Here, we consider a branching ratio $k$ of SN Ib/c into broad line events that may successfully produce long gravitational wave bursts.
GPU-accelerated searches for broadband extended gravitational-wave emission
===========================================================================
![Butterfly filtering is a bandpass filter of trajectories of long-duration chirps with finite slope $0<\delta \le df(t)/dt$ in frequency $f(t)$, suppressing signals with essentially constant frequencies. Butterfly filtering is realized by matched filtering against a bank of chirp templates, here of intermediate duration $\tau = 1$s and covering of bandwidth of 350-2000 Hz in frequency. (Reprinted from [@van16].)[]{data-label="figWb"}](butterfly)
[Deep searches]{} for BEGE from multi-phase central engines in core-collapse supernovae with durations (\[EQN\_TsA\]), possibly extended to (\[EQN\_TsB\]), can be pursued by [*butterfly filtering*]{}, to extract trajectories in the $(f(t),\dot{f}(t))$-plane with finite slope $0<\delta \le df(t)/dt$ for some $\delta>0$. These trajectories typify sources exhausting a finite reservoir in energy and angular momentum. From (magneto-)hydrodynamic sources such as accretion flows onto compact objects, phase coherence in gravitational wave emission is expected to be limited to intermediate time scales $\tau << T_{90}$, associated with possibly turbulent accretion flows.
Butterfly filtering is realized by matched filtering against a bank of chirp templates of duration $\tau$, here $\tau=1\,$s, covering 350-2000 Hz permitted by LIGO S6 data down-sampled to 4096 Hz. In this frequency range, LIGO noise is essentially Gaussian over the finite bandwidth of a given chirp template. The output of correlations against such chirp templates is again Gaussian in the absence of any signal or anomaly in the detector output. This high frequency range fortuitously covers our outlook on BEGE discussed in the previous section. Butterfly filtering hereby attains essentially maximal sensitivity based on the theory of matched filtering [@van16]. Butterfly filtering output is a [*chirp-based*]{} spectrogram, distinct from the more common Fourier-based spectrograms.
Illustrative for the power of butterfly filtering is the identification of a Kolmogorov spectrum in high frequency [*BeppoSAX*]{} data, extracted from a mean of 1.26 photons per 0.5ms bin (Fig. \[figKol\]). Realization of near-optimal sensitivity requires a large bank of chirp templates, that densely covers the two-dimensional parameter space inherent to butterfly filtering. In covering frequencies $O(10^3)$Hz and chirps with $\Delta f=O(10^2)$Hz, the theoretical minimum size of a bank will be on the order of $O(10^{5})$ templates. The results of Fig. \[figKol\] derive from a bank of 8.64 million templates.
For LIGO, an effective gravitational wave strain noise amplitude is $h_n = \sqrt{fS_h(f)} =3.6\times 10^{-22}$ about 1000 Hz at advanced detector sensitivity [@cut02]. Butterfly filtering by chirps of $\tau=1\,$s duration recovers (\[EQN\_heff\]) with a signal-to-noise ratio $S/N= ({\pi r h_n})^{-1} \sqrt{ { 2 dE }/{5 df} }$ as an average over detector orientations, i.e., $$\begin{aligned}
S/N \simeq 3.7 \, D_{100}^{-1}\, M_1 \left(\frac{\sqrt{fS_h[1000\,\mbox{Hz}]}}{3.6\times 10^{-22}}\right)^{-1}.
\label{EQN_SN}\end{aligned}$$ This result obtains for a single gravitational wave detector. Detection in all three LIGO-Virgo detectors would imply an enhanced signal-to-noise ratio $\rho\simeq 6.4$, apart from additional significance gained from an associated supernova signature (e.g. [@van02]). Here, we use a dimensionless strain $h$ amplitude, satisfying $$\begin{aligned}
h = \left(\frac{M}{D}\right) \,h_S
\label{EQN_10a}\end{aligned}$$ for a source of mass $M$ at a distance $D$. For rotating systems, $h_S$ effectively scales with the Newtonian tidal field of a mass-inhomogeneity. The experimental and data-analysis challenge is defined by the fact that, for stellar mass systems at astrophysical distances, $M/D$ is exceptionally small, whereby $h$ is, at best, on the order of $10^{-23}-10^{-22}$ for the most extreme sources in the Local Universe.
While LIGO data in the shot-noise dominated frequency range 350-2000 Hz is largely Gaussian over intermediate bandwidths of chirps over $\tau = 1\,$s, it features frequent spurious signals, commonly referred to as [*glitches*]{}. This necessitates going beyond single detector analysis, seeking [*correlated detections*]{} in both the H1 and L1 detectors. Furthermore, LIGO data sets are large. LIGO S6 alone covers well over one year of observations in 1 TByte of data. Deep searches in LIGO data by butterfly filtering similar to that in (\[figKol\]) requires acceleration on [*Graphics Processor Units*]{} (GPU). On about a dozen modern GPUs, heterogeneous GPU-CPU analysis enables butterfly filtering against banks of up to 16 million templates in size at over one million correlations per second, i.e., deep searches in all of LIGO S6 or better than real-time analysis of LIGO observation runs. Specifically, it enables blind all-sky searches without triggers from electromagnetic or neutrinos.
![Pseudo-spectra of H1 and L1 in simultaneous hits $(\rho>5.5\sigma)$ in LIGO S6 butterfly filtering against a bank of 8M chirp templates, shown as an average over 20 blocks (182 hr of H1$\land$L1 data) with control by H1$_0\land$L1$_0$ following time randomization.[]{data-label="fPER1"}](plot_PERd_Aug)
For H1L1-correlated searches, chirp-based spectrograms of H1 and L1 each can be combined to a new spectrogram of correlations $C_l$ and $C_f$ in H1L1 butterfly output $\rho$ and, respectively, $f$, that we shall refer to as correlations in [*loudness*]{} and, respectively, [*pitch*]{}. H1L1 correlations in loudness and pitch are unambiguous, by the relatively small standard errors in the mean (below). Overall, they are of known and unknown origin. For instance, H1 and L1 are frequently subject to LIGO injections for calibration purposes. However, the observed correlations appear to be quite consistent with time, though strongly non-uniform over frequency. While a detailed analysis of this H1L1 correlations is beyond the scope of this discussion, the results given serve to show the challenges encountered in unraveling detections of astrophysical origin obtained by the present deep searches.
LIGO S6 correlations in loudness and pitch
------------------------------------------
In searching for astrophysical signals in H1L1 correlations, we recently applied GPU-accelerated butterfly filtering to LIGO S6 [@van17b]. In this approach, butterfly output to the CPU from the GPU is restricted to tails in matched filtering correlations $\rho$ with standard deviation $\sigma$, satisfying $$\begin{aligned}
\rho > \kappa \sigma,
\label{EQN_kappa}\end{aligned}$$ where we put $\kappa=5.5$. Given our focus on signals with finite slope $df(t)/dt)\ge\delta>0$, we are at liberty to down select “hits" (\[EQN\_kappa\]) at the same GPS time of the LIGO detectors, as any time-delay $\Delta t$ in signal arrival time has an equivalent frequency shift $\Delta f \ge \delta \Delta t$.
Fig. \[fPER1\] shows H1L1 correlations in simultaneous hits in butterfly filtering against a bank of 8 million chirp templates, along with control obtained by time randomization of H1 and L1 data. Somewhat unexpected, we observe a pronounced [*excess probability ratio*]{} $$\begin{aligned}
EPR = \frac{p_{12}}{p_1p_2} = 5.11 \pm 0.0268
\label{EQN_EPR}\end{aligned}$$ H1L1 correlations, where $p_i$ refer to the probability of hits (\[EQN\_kappa\]) by each detector individually ($i=1$ for H1, $i=2$ for L1). The result is beyond EPR=1, as expected and measured by our control, defined by the same analysis following time randomization of H1 and L1. (Effectively the same control obtains following time randomization of either H1 or L1.) Going one step deeper, the results unambiguously show correlations $C_l$ in loudness and $C_f$ in pitch with mean values $\mu_l$ and, respectively, $\mu_f$, satisfying $$\begin{aligned}
\mbox{loudness:}~\mu_l =0.063 \pm 0.0014,~~\mbox{pitch:}~\mu_f = 0.108 \pm 0.00172.
\label{EQN_sf}\end{aligned}$$
It should be mentioned that H1 and L1 are frequently given hardware injections, but over much less than 1% of of total measurement time. As such, injections are unlikely to account for (\[EQN\_EPR\]-\[EQN\_sf\]).
Correlations (\[EQN\_EPR\]) and, respectively, (\[EQN\_sf\]) obtained by butterfly filtering with a massive bank of 8 million templates seems to justify further analysis. This requiring novel algorithms, to search for evidence of signals of astrophysical origin in correlations that are highly dispersed over frequency and unsteady in time.
Summary and future prospects
============================
While the central engine of GRBs remains to be identified, diverse evidence from electromagnetic observations (Table 3) points to an association with black holes featuring ultra-relativistic outflows marked by (\[EQN\_alpha1\]-\[EQN\_i2\]), that defy an explanation in terms of Newtonian processes from baryonic inner engines. In particular, extended emission in LGRBs and SGRBEEs points to rotating black holes described by the Kerr metric. While these events are somewhat rare, the parent population of supernovae of type Ib/c is far more numerous by about two orders of magnitude. The formation and evolution of accreting black holes in core-collapse events gives an outlook on multi-messenger emission over short and long timescales, the latter including the lifetime of black holes spin associated with relativistic frame dragging induced interactions with matter at the ISCO. Thus, rapidly rotating black holes may account for (\[EQN\_frail\]-\[EQN\_i2\]), that are otherwise challenging to explain by magnetic winds from rotating neuntron stars. Specifically, we mention
- [*Universality of soft extended emission*]{} in [*Swift*]{}’s SGRBEE and LGRBs alike (Table 2), satisfying the same Amati relation (Fig. \[figAm1\]), whose durations can be identified with the lifetime of black hole spin (Fig. \[figmf\]);
- [*Large energy reservoir in angular momentum*]{} (\[EQN\_EcEH\]), that can account for hyper-energetic events whose central engines defy the limitations of (proto)-neutron stars (Table 1);
- [*Prominent intermittency*]{} in GRB light curves with a positive correlation to luminosity [@rei01], consistent with intermittent feedback by rotating black holes [@van15a];
- [*Smooth broadband Kolmogorov spectra*]{} in [*BeppoSAX*]{} light curves of LGRBs (Fig. \[figKol\]) up to the kHz range with no bump at high frequency, i.e., any signature of proto-pulsars is strikingly absent (Fig. \[fig72lc\]);
- [*Slowly spinning remnants*]{} common to LGRBs from core-collapse and SGRBEEs from mergers (Fig. \[fig:origin\]) with $a/M\simeq0.3$, that appears both in (\[EQN\_GW15\]) and the plateau of the superluminous event SN2015L [@van17a].
**Table 3.** Observational evidence for LGRBs from rotating black holes. (Adapted from [@van16].)
[Instrument]{} [Observation/Discovery]{} [Result]{} [Ref.]{}
--------------------- ---------------------------------- --------------------------------------- ---------- -- --
LGRBs with no SN, SGRBEE Extended Emission to mergers (1)
0.2in Amati-relation Universal to LGRBs and EEs to SGRBs (1)
0.2in X-ray afterglows SGRBs SGRB 050509B (2)
0.in [*HETE-II*]{} X-ray afterglows SGRBs SGRB050709 (3)
0.in [*BeppoSAX*]{} X-ray afterglows LGRBs GRB970228, common to LGRBs, SGRB(EE)s (1,4)
0.2in Broadband Kolmogorov spectrum No signature (proto-)pulsars (5)
0.in BATSE Bi-modal distribution durations. short-hard and long-soft GRBs (6)
0.2in ms variability Compact relativistic central engine (7)
0.2in Normalized light curves LGRBs BH spin-down against ISCO (8)
0.in Optical LGRB association to SNe Ib/c Branching ratio $<1\%$ (9)
0.2in Calorimetry, $E_k$ SNe $E_{rot}>E_c[NS]$ in some GRB-SNe (10)
\[TABLE\_3\]\
0.08in [**Note.**]{} (1) Revisited in [@van14b]; (2) [@geh05]; (3) [@vil05; @fox05; @hjo05]; (4) [@cos97]; (5) [@van14b]; (6) [@kov93]; (7) [@sar97; @pir97; @kob97], and [@van00; @nak02]; (8) [@van09; @van12; @nat15], see further [@van08b; @sha15]; (9) [@van04; @gue07]; (10) in the model of [@bis70], $E_{rot}$ exceeds the maximal spin energy $E_c[NS]$ of a (proto-)neutron star in some hyper-energetic events [@van11b].
For non-axisymmetric accreting flows onto rotating black holes formed in core-collapse events, broadband extended gravitational-wave emission may derive from orbital angular momentum and (catalytic) conversion of spin of the black hole, losing angular momentum to matter at the ISCO. Long duration times scales derive from in fall of the stellar remnant envelope and, respectively, the lifetime of black hole spin. An overview of this broad outlook on gravitational radiation from core-collapse events is shown in Fig. \[fig:hchar\] and summarized in Table 4. In particular, we mention
1. [*Ascending chirps*]{} from accretion flows with fragments or non-axisymmetric wave patterns, formed within a critical radius where cooling conspires with self-gravity or where angular momentum loss by gravitational radiation is dominant over angular momentum loss by viscous transport;
2. [*Descending chirps*]{} from ISCO waves induced by energetic feedback of the central black hole via an inner torus magnetosphere. Non-axisymmetric waves are expected from heating and enhanced magnetic pressure, balanced by cooling in gravitational radiation, MeV neutrino emission and magnetic winds. In this process, the ISCO expands as the black hole loses angular momentum;
3. [*Broadband turbulence*]{} appears to be a prevalent in LGRBs, deep within about the ISCO of the central engine [@van01b; @van12] and in the prompt gamma-ray emission process (Figs. \[fig:alpha\] and \[figmf\]).
These emissions may be preceded by QNM ringing and the random black hole kicks in black hole formation, during a surge in black hole mass prior to the formation of an accretion disk, possibly continued during further growth to a nearly extremal black hole leading up to an accompanying LGRB.
**Table 4.** Broadband extended gravitational-wave emission from accretion flows onto rotating black holes.
[Quantity]{} [Scale]{} [Comment]{} [Ref.]{}
----------------------------- --------------------------------------------- --------------------------- ----------------------------- -- --
[BH DISK-TORUS]{}
0.2in $M$ $10^1$ $M_\odot$ [@bai98; @woo06]
0.2in $E_{rot}^H$ $1\, M_\odot c^2$ [@ker63]
0.2in $f_{ISCO}$ $10^{2-3}$ Hz Fig. 11
0.2in $M_{acc}$ $ 10^{-1}-10^0$ $M_\odot$ $\dot{M}\tau$ (\[EQN\_A6C\])
0.2in $M_{T}$ $10^{-2}$ $M$ (\[EQN\_g4b\])
[LONG GW BURST]{}
0.2in efficiency $< 50\%$ Fig. \[fig-T\]
0.2in $E_{rad}$ $< 1\,M_\odot c^2$ Fig. \[fig-T\]
0.2in $T$ $10^1$ s $T_{90},t_{ff}, T_{spin}$ Figs. 11-12, (\[EQN\_TsA\])
0.2in $L$ $10^{-1}$ $M_\odot c^2$ s$^{-1}$ Fig. \[fig-T\]
[ACCRETION]{}
[0.08in Fragments]{} [@pir07]
0.2in $f_c < f < 2f_{ISCO}$ $10^0-10^3$ Hz (\[EQN\_WC\])
0.2in $h_{char}(f)$ $1.7\times10^{-22}(f/f_{e})^{\frac{2}{3}}$ $f_c<f<f_{e}$ cf. (\[EQN\_h1987A\])
0.2in $h_{char}(f)$ $1.7\times10^{-22}(f/f_{e})^{-\frac{1}{6}}$ $f_{e} < f < 2f_{ISCO}$ (\[EQN\_hchar2b2\])
0.08in [ Disk waves]{} [@lev15]
0.2in $f<2f_{ISCO}$ $10^0-10^3$ Hz
0.2in $h_{char}(f)$ $1.2\times10^{-21}(f/f_b)^{\frac{1}{6}}$ $f<f_b$ (\[EQN\_A6D\])
0.2in $h_{char}(f)$ $1.2\times10^{-21} (f/f_b)^{-\frac{1}{6}}$ $f_b < f < 2 f_{ISCO}$ (\[EQN\_A6D\])
0.08in [ ISCO waves]{} [@van02; @van03]
0.2in $f=2f_{ISCO}$ $10^{2-3}$ Hz
0.2in $h_{char}(f)$ $3.4\times10^{-21}$ (\[EQN\_heff\]),[@van01b]
[DETECTION]{}
0.2in $D[\rm{source}]$ $10^2$ Mpc spectral bump (\[EQN\_SN\])
0.2in $D[\rm{chirp}]$ $10^1$ Mpc track in $tf$-plane [@van11]
0.2in $\Omega_B$ $10^{-9}-10^{-8}$ stochastic (\[EQN\_B7e\])
0.2in $f_{B,peak}$ $10^2$ Hz spectral bump (\[EQN\_B7d\])
\[TABLE\_4\]\
0.08in
Deep searches by GPU-accelerated butterfly filtering
----------------------------------------------------
Probes of BEGE (Fig. 18) with complex time and frequency behavior due to non-axisymmetric mass motion and turbulence requires a broadband detection algorithm, seeking to capture phase coherence on intermediate time scales of chirps, that may be ascending and descending. To this end, we consider extracting chirp-based spectrograms by butterfly filtering. Banks of large size allow deep searches in noisy time-series, that successfully identified broadband Kolmogorov spectrum in LGRBs (Fig. \[figKol\]) and broadband H1L1 correlations by a factor of 5.11 (Fig. 20). The latter poses a challenge, to identify signals of astrophysical origin in a background of detector correlations that is dispersed over a broad frequency range with unsteady behavior (in time).
Overall, we distinguish three types of searches:
1. [*Source detection*]{} by detecting excess hit counts of ascending and descending chirps below, respectively, above a few hundred Hz. In the frequency range of 350-2000 Hz, essentially maximal sensitivity is attained when the bank of templates is large, allowing for a distances up to about 100 Mpc for energetic CC-SNe and LGRBs for Advanced LIGO-Virgo. Source detection ignores time ordering in matches to individual chirps templates, however;
2. [*Chirp detection*]{} of long duration comprising a large number of hits. This is expected to reach a distance sensitivity of tens of Mpc for Advanced LIGO-Virgo.
3. [*Stochastic background detection*]{} from an astrophysical source population. In particular, Type Ib/c supernovae may collectively produce a bump at a few hundred Hz (\[EQN\_B7d\]), derived from their cosmological distribution locked to the cosmic star formation rate.
Scientific objective and observational strategy
-----------------------------------------------
For upcoming searches by LIGO-Virgo and KAGRA, it appears advantageous to focus on the progenitor population of energetic CC-SNe rather than GRBs themselves, to circumvent the relatively limited event rates of GRBs and the challenge in finding them due to beaming. Supernovae are much more numerous and quite easy to find using modest sized robot telescopes. If a fraction of 1% or more is luminous in gravitational waves, they will be competitive with neutron star mergers as suitable targets of opportunity.
True calorimetry will be essential to identifying the true nature of the inner engine of energetic CC-SNe, that should include gravitational radiation following the identification of high density matter - and possibly high-density mass motion - in the pivotal event SN1987A. A detailed probe may identify, for instance, different engines to narrow- and broad line events (Fig. \[figSN1987A3\]), provide first principle constraints on total mass accretion (ascending chirps) and the total rotational energy of the central compact object (descending chirps). The latter offers a window to identification of Kerr black holes, including their initial state and evolution, as the most luminous objects in Nature [@van02b]. For calorimetry, ignoring temporal evolution, the above mentioned source detection is sufficient.
Sensitivity of LIGO-Virgo and KAGRA detectors is limited by instrumental strain noise. They are broadband covering a window of tens of Hz to a few kHz, susceptable to seismic noise at low frequency, thermal noise at intermediate frequencies and shot noise at higher frequencies. Below a few hundred Hz, they are ideally suited to probe mergers of stellar mass compact objects. While these are amply highlighted in existing development programs preparing for upcoming gravitational wave searches, we here highlight a window on broadband extended emission from CC-SNe that extends to the high frequency bandwidth of sensitivity. Our outlook is guided by evidence for black holes in energetic events based on their association with LGRBs, aforementioned SN1987A and some general considerations of accretion flows onto rotating black holes.
In detecting nearby Type Ib/c supernovae in optical surveys, and the main challenges are sufficient competeness of the survey and cadence to resolve the supernova light curve [@heo15]. A narrow window of the time-of-onset to within one day will be crucial to minimizing computational effort in source detection by application of TSMF. The most nearby events may be captured by surveys focused on galaxies most prolific in core-collapse supernovae, e.g., M51 and M82, each featuring over one CC-SN per decade.
[**Acknowledgments.**]{} M.H.P.M. van Putten acknowledges support from the National Research Foundation of Korea and faculty research fund of Sejong University in 2012. A. Levinson acknowledges supported by a grant from the Israel Science Foundation no. 1277/13. The BeppoSAX mission was an effort of the Italian Space Agency ASI with the participation of The Netherlands Space Agency NIVR. M.H.P.M. van Putten acknowledges computational support in part by the National Science Foundation through TeraGrid (now XSEDE) resources provided by Purdue University under grant No. TG-DMS100033. Some calculations were performed at CAC/KIAS and KISTI. F. Frontera, C. Guidorzi and L. Amati acknowledge financial support from the Italian Ministry of Education, University, and Research through the PRIN-MIUR 2009 project on Gamma-Ray Bursts (Prot. 2009 ERC3HT). This work was partially supported by the National Research Foundation of Korea under grants 2015R1D1A1A01059793 and 2016R1A5A1013277 and made use of LIGO S6 data from the LIGO Open Science Center (losc.ligo. org), provided by the LIGO Laboratory and LIGO Scientific Collaboration. LIGO is funded by the U.S. National Science Foundation. Additional support is acknowledged from MEXT, JSPS Leading- edge Research Infrastructure Program, JSPS Grant-in- Aid for Specially Promoted Research 26000005, MEXT Grant-in-Aid for Scientific Research on Innovative Areas 24103005, JSPS Core-to-Core Program, A. Advanced Re- search Networks, and the joint research program of the Institute for Cosmic Ray Research.
[**Appendix A.**]{} Gravitational radiation
===========================================
The Einstein equations are a hyperbolic-elliptic system of equations for a four metric $g_{ab}$ with signature $(-,+,+,+)$ in the line element $$\begin{aligned}
ds^2 = g_{ab}dx^adx^b.
\label{EQN_line1}\end{aligned}$$ This can be made explicit following a foliation of space-time in Cauchy surfaces [@arn62] or in a Lorentz gauge on SO(3,1) connections in the Riemann-Cartan formalism [@van96]. The resulting gravitational wave motion is subject to elliptic constraints given by conservation of energy and momentum.
Hyperbolicity
-------------
The linearized equations of motion about Minkowski space-time reveal the two modes of transverse gravitational waves of spin two. Traditionally, the derivation is given in 3+1 in a special choice of coordinates, that exploits gauge covariance in the choice of coordinates. Here, we note a derivation utilizing the constraints of energy-momentum conservation.
To start, consider slicing of space-time into space-like hypersurfaces of constant coordinate time $t=x^0$. Let $h_{ij}$ denote the three-metric intrinsic to these hypersurfaces, that are coordinated by the remaining $x^i$ $(i=1,2,3)$. The line-element can be equivalently expressed as (e.g. [@tho86]) $$\begin{aligned}
ds^2 = -\alpha^2 dt^2 + h_{ij}\left( dx^i + \beta^i dt\right) \left( dx^j + \beta^j dt\right),
\label{EQN_AP1}\end{aligned}$$ where $(\alpha,\beta^i)$ denote the lapse function and, respectively, shift functions. In foliating space-time into three-dimensional hypersurfaces, the $(\alpha,\beta^i)$ are a gauge, and are not dynamical variables. The three-metric $h_{ij}$ defines parallel transport of vectors in the hypersurfaces of constant coordinate time $t$ and, as such, comes with a covariant three-derivative $D_i$ and associated Christoffel symbols $\Gamma_{ij}^k$.
Following (\[EQN\_AP1\]), the Ricci tensor $^{(4)}R$ of $g_{ab}$ expands to $^{(3)}R$ of $h_{ij}$ and quadratic terms of the extrinsic curvature tensor $K_{ij} = -1/(2\alpha) L_t h_{ij}$, where $L_t$ denotes the Lie derivative of $h_{ij}$ with respect to the coordinate time $t$. The Hilbert action for $g_{ab}$ hereby reveals explicit contributions from “potential" and “kinetic" energies in $$\begin{aligned}
S= \frac{1}{16\pi} \int \left( ^{(3)}R + K_{ij}K^{ij} - K^2\right) \alpha \sqrt{h} dx^3 dt.
\label{EQN_AP2}\end{aligned}$$ Following [@arn62], variations with respect to the non-dynamical variables $(\alpha, \beta^i)$ obtain the conservation laws of energy and momentum, given by the constraints $$\begin{aligned}
R - K_{ij}K^{ij} + K^2 = 0,~~D^iK_{ij} - D_j K = 0,
\label{EQN_AP3c}\end{aligned}$$ where $R=R_{i}^i$ denotes the trace of the Ricci tensor of $h_{ij}$. Variation with respect to $h_{ij}$ gives the first-order evolution equation $$\begin{aligned}
L_t K_{ij} =\left(R_{ij} - D_iD_j\right) \alpha + \left( K K_{ij} - 2K_i^m K_{jm}\right) \alpha,
\label{EQN_AP3a}\end{aligned}$$ where $L_t K_{ij} = \partial_t K_{ij} - \left( K_i^m D_m\beta_j + K_j^m D_m \beta_i + \beta^m D_m K_{ij} \right)$. Similar to the latter, we may expand $L_t h_{ij}$ to obtain the first-order evolution equation $$\begin{aligned}
\partial_t h_{ij} = D_i \beta_j + D_j \beta_i - {2\alpha} K_{ij}.
\label{EQN_AP3b}\end{aligned}$$ Combined, (\[EQN\_AP3c\]-\[EQN\_AP3b\]) defines a constraint Hamiltonian system of equations for the dynamical variables $(h_{ij},K_{ij})$.
The existence of gravitational waves represents the hyperbolic structure of (\[EQN\_AP3c\]-\[EQN\_AP3b\]). This becomes explicit by analysis of harmonic perturbations in the curvature driven gauge [@van10; @van12b] $$\begin{aligned}
\partial_t \alpha = -K, ~~\beta^i = 0.
\label{EQN_AP4a}\end{aligned}$$ We next consider small perturbations about Minkowski space-time $(h_{ij} = \delta_{ij},\alpha=1,K_{ij}=0$), where $\delta_{ij}$ denotes the Kronecker delta symbol, that is, $h_{ij} = \delta_{ij} + \delta h_{ij}$, $\alpha = 1 + \delta \alpha$ and $K_{ij} = \delta K_{ij}$. In the gauge (\[EQN\_AP4a\]), (\[EQN\_AP3a\]-\[EQN\_AP3b\]) imply $$\begin{aligned}
\partial_t^2 h_{ij} = - 2 \left( R_{ij} - D_iD_j \delta \alpha\right) ,~~\partial_t^2 K = \Delta K,
\label{EQN_AP4b}\end{aligned}$$ where we used that $^{(3)}R$ is second order according to the Hamiltonian energy constraint in (\[EQN\_AP3c\]), Here, we recall the perturbative expansion [@wal84] $$\begin{aligned}
R_{ij} = - \frac{1}{2} \Delta \delta h_{ij} + \frac{1}{2} \partial_i \partial^e \delta \bar{h}_{ej} + \frac{1}{2} \partial_j \delta \bar{h}_{ej},
\label{EQN_AP4c}\end{aligned}$$ where $\bar{h}_{ij} = \delta h_{ij} - \frac{1}{2} \delta_{ij} \delta h$, $\delta h = \delta^{ij} \delta h_{ij}$. A harmonic plane wave $\delta h_{ij} = \hat{h}_{ij} e^{-i\omega t} e^{ik_ix^i}$ (similarly for $K_{ij}$) of angular frequency $\omega$ with wave vector $k_i$ can be applied to (\[EQN\_AP3a\]) and (\[EQN\_AP4a\]), giving $k_i\hat{K}_{ij} = k_j \hat{K}$, $-i\omega \hat{\alpha} = - \hat{K}$, $\delta \hat{h}_{ij}=-2i\omega^{-1} \hat{K}_{ij}$ and, for $\partial_i\partial^e\bar{h}_{ej}$, $$\begin{aligned}
%\begin{array}{ll}
k_i k^e \hat{h}_{ej} - \frac{1}{2} k_i k_j \delta \hat{h} =
i \omega^{-1} \left( - 2k_i k^e \hat{K}_{ej} + k_i k_j \hat{K}\right) = - i\omega^{-1} k_i k_j \hat{K}.
%\end{array}
\label{EQN_5a}\end{aligned}$$ By (\[EQN\_AP4c\]), it follows that $\hat{R}_{ij} - \partial_i\partial_j \hat{\alpha} = \frac{1}{2} k^2 \hat{h}_{ij} - i\omega^{-1} k_i k_j \hat{K} + i\omega^{-1} k_i k_j \hat{K},$ whereby the first evolution equation in (\[EQN\_AP4b\]) reduces to the dispersion relation $$\begin{aligned}
\omega^2 = k^2
\label{EQN_5b}\end{aligned}$$ of propagation along light cones in a local Minkowski background space-time.
The Einstein equations, $G_{ab} = 16\pi T_{ab}$ describe the response of space-time curvature to a stress-energy tensor $T_{ab}$ of matter and fields, where $G_{ab} = ^{(4)}R_{ab} - \frac{1}{2}g_{ab}^{(4)}R$ is the Einstein tensor. Following (\[EQN\_AP4c\]) and (\[EQN\_5b\]), the covariant wave equation for perturbations $g_{ab} = \eta_{ab} + \delta g_{ab}$ in the four-metric on a fixed background space-time with metric $\eta_{ab}$ is $$\begin{aligned}
\Box_\eta \delta g_{ab} = -16\pi T_{ab},
\label{EQN_Gab}\end{aligned}$$ where $\Box$ denotes the d’Alembertian associated with $\eta_{ab}$. The coefficient $-16\pi$ in (\[EQN\_Gab\]) results from the factor $-\frac{1}{2}$ in (\[EQN\_AP4c\]).
Searches for contemporaneous emission in electromagnetic and gravitational from cosmological GRBs have been suggested to test for gravitons to be massless as described by (\[EQN\_Gab\]). However, these tests only serve to identify differences in masses of gravitons and photons, as may be seen by expressing wave motion of both in terms of four vector fields. Let $\omega_{a\mu\nu}$ denote the Riemann-Cartan connection of four-dimensional space-time in the SO(3,1) tetrad formalism and $A_a$ denote the vector potential of the electromagnetic field. In the Lorentz gauge to both [@van96], propagation in vacuum satisfies $$\begin{aligned}
%\begin{array}{ll}
\hat{\Box} \omega_{a\mu\mu} - R_a^c\omega_{c\mu\nu} - [\omega^c,\nabla_a\omega_c]_{\mu\nu} = 0,~~
\Box A_a - R_a^cA_c =0,
%\end{array}
\label{EQN_APWE}\end{aligned}$$ where $\hat{\Box}$ denotes the d’Alembertian associated with the SO(3,1) gauge covariant derivative $\hat{\nabla}_a = \nabla_a + [\omega_a,\cdot]$. In the presence of a cosmological constant $\Lambda>0$ [@rie98; @per99], $R_{ab} = \Lambda g_{ab}$, whereby (\[EQN\_APWE\]) becomes $$\begin{aligned}
%\begin{array}{ll}
\hat{\Box} \omega_{a\mu\mu} - \Lambda\omega_{c\mu\nu} - [\omega^c,\nabla_a\omega_c]_{\mu\nu} = 0,~~
\Box A_a - \Lambda A_c =0,
%\end{array}
\label{EQN_APWEb}\end{aligned}$$ showing gravitons and photons of the same effective mass $m=\sqrt{\Lambda}\simeq 10^{-29}$ cm$^{-1}$ in geometrical units, as defined by the dispersion relation of (\[EQN\_APWEb\]).
Luminosity in gravitational radiation
-------------------------------------
Consider the transverse traceless perturbations [@mis73] $$\begin{aligned}
\delta h^{TT}_{ij} = \left( \begin{array}{ccc} h_+ & h_\times & 0 \\ h_\times & -h_+ & 0 \\ 0 & 0 & 0 \end{array} \right) =
h_+ e_{ij}^+ + h_\times e_{ij}^\times,
\label{EQN_AP6a}\end{aligned}$$ decomposed in the two linear polarization tensors $e_{ij}^+$ and $e_{ij}^\times$ of gravitational waves in the $(x,y)$ plane orthogonal to the direction of propagation along the $z-$axis. The perturbed line-element (\[EQN\_line1\]) now assumes the form $$\begin{aligned}
ds^2=\eta_{ab} dx^a dx^b + h_+(dx^2-dy^2) + 2h_\times dx dy.
\label{EQN_line2}\end{aligned}$$ With rotational symmetry over an angle $\pi$ about the $z-$axis, gravitational waves are of spin-2 [@fie39]. In the far field region away from a source region, the Hilbert action (\[EQN\_AP2\]) reduces to the kinetic term $K_{ij}K^{ij}$, and hence by (\[EQN\_AP3b\]) to $$\begin{aligned}
S = \frac{1}{16\pi} \int \frac{1}{2}\left[ \left(\partial_a h_+\right)^2 + \left( \partial_a h_\times\right)^2 \right] dx^3 dt.
\label{EQN_AP2r}\end{aligned}$$ We can now read off the stress-energy tensor of gravitational wave motion: $$\begin{aligned}
t^{00} = t^{0z} = t^{zz} = \frac{1}{16\pi} \left < \dot{h}_+^2 + \dot{h}_\times^2 \right>
\label{EQN_AP2s}\end{aligned}$$ The two polarization wave modes (\[EQN\_AP6a\]) and the associated gravitational wave stress-energy tensor (\[EQN\_AP2s\]) are characteristic properties of general general relativity. Alternative theories may have additional degrees of freedom [@sat09].
For a source described by a stress-energy tensor $T_{ab}$, the gravitational wave emission results from the associated time harmonic perturbations in the tidal gravitational field. Following (\[EQN\_Gab\]), we have in response to a distance source over a region $V$ $$\begin{aligned}
\delta h_{ij}(r,t)= \frac{4}{r} \int_V T_{ij}(t-r,x^i) d^3x.
\label{EQN_AP2w}\end{aligned}$$ Explicit evaluation for a circular binary of point masses $M_1$ and $M_2$, orbital frequency $\Omega$ and orbital separation $a$ shows the quadrupole gravitational wave formula [@wal84; @tho02] $$\begin{aligned}
L_{GW} = \frac{32}{5} \Omega^6 a^4 \mu^2 = \frac{32}{5} \left(\Omega \mu\right)^{\frac{10}{3}}
\label{EQN_AP2gw}\end{aligned}$$ in units of $L_0 = c^5/G$, where $\mu = M_1^{\frac{3}{5}}M_2^\frac{3}{5}/(M_1+M_2)^\frac{1}{5}$ denotes the chirp mass, $c$ is the velocity of light and $G$ is Newton’s constant.
To see (\[EQN\_AP2gw\]), we first recall the following identity for a mass distribution with velocity four-vector $u^b$ (e.g. [@tho02]) $$\begin{aligned}
\int_V T^{ij}d^3x = \frac{1}{2}\partial_0^2 I_0^{ij}
\label{EQN_idT}\end{aligned}$$ between $\int_V T^{ij} d^3x = \int_V u^iu^jdm$, $dm=\rho d^3x$ and the moment of inertia tensor $$\begin{aligned}
I^{ij}_0 = \int_VT^{00}x^ix^j d^3x \simeq \int_V x^ix^j dm,
\label{EQN_Iij}\end{aligned}$$ where the latter refers to the non-relativistic limit $u^0\simeq 1$. The identity (\[EQN\_idT\]) follows from the conservation of energy-momentum, $\nabla_aT^{ab}=0$. About a flat space-time background, $\nabla_a\nabla_bT^{ab}=0$ implies $\partial_0^2T^{00} + 2\partial_0\partial_i T^{0i} + \partial_i\partial_jT^{ij}=0$, i.e., $\partial_0^2T^{00}-\partial_i\partial_jT^{ij}=0$ using $\partial_0T^{00}+\partial_iT^{0i}=0$. Integration by parts twice of $\partial_0^2 \int_V x^ix^j T^{00} d^3x$ = $\int_V x^ix^j \partial_k\partial_l T^{kl}$ obtains (\[EQN\_idT\]).
Consequently, (\[EQN\_AP2w\]) gives for the [*traceless*]{} metric perturbations $$h_{ij}^{T}(t,r)=\frac{2}{r}\frac{d^2I_{ij}(t-r)}{dt^2},~~{I}^{jk}=I^{jk}_0-\frac{1}{3}\delta^{jk}\delta_{lm}I^{lm}_0.
\label{qmom}$$ By (\[EQN\_AP2r\]), we arrive at the gravitational wave luminosity $$L_{GW}=\frac{dE_{GW}}{dt}=\frac{1}{5}
\langle{ \frac{d^3I_{jk}}{dt^3} \frac{d^3I^{jk}}{dt^3}} \rangle,
\label{dEgw/dt}$$ taking into account and reduction factor 2/5 as only two of the five degrees of freedom in the traceless metric perturbation $\delta h^T_{ij}$ are physical degrees of freedom representing outgoing gravitational radiation [@tho02].
Radiation from multipole mass moments
-------------------------------------
Consider a ring having cross-sectional radius $b$ and density $\rho$, rotating around a central object in the $(x,y)$ plan at angular velocity $\Omega$ in a circular orbit of radius $r$. Let $m=\int_V{\rho d^3x}$ denotes the total mass of the ring, where the integration is over the ring’s volume $V$. We restrict the analysis to a thin ring, $b<<r$, and compute $I^{jk}$ to order $O(b^2/r^2)$. Let $(x',y')$ denotes a Cartesian coordinate system rotating with the ring. One can always choose the axis such that $$\begin{aligned}
I^{x'x'}_0=\frac{1}{2}mr^2(1+\xi),~~
I^{y'y'}_0=\frac{1}{2}mr^2(1-\xi),~~
I^{x'y'}_0=0.\end{aligned}$$ to order $O(b^2/r^2)$. Here, $\xi$ quantifies the mass quadrupole inhomogeneity, with $\xi=0$ for an axi-symmetric ring. For example, for a ring having a density $\rho=\rho_0+\rho_2\cos^2\theta$ in cylindrical coordinates, with $\rho_0$ and $\rho_2$ being constants, one obtains $\xi=m_2 /4m$, where $m=\int_Vd^3x\rho$ is the total mass, and $m_2=\int_Vd^3x\rho_2$. Now, transforming to the non-rotating frame, $$\left(\begin{array}{c} x\\ y
\end{array}\right)
=\left(\begin{array}{cc}
x'\cos\Omega t & y'\sin\Omega t\\
-x'\sin\Omega t & +y'\cos\Omega t
\end{array}\right)$$ yields $$\begin{aligned}
\begin{array}{l}
I^{xx}_0=I^{x'x'}_0\cos^2\Omega t+I^{y'y'}_0\sin^2\Omega t=\frac{1}{2}mr^2(1+\xi\cos2\Omega t),\nonumber\\
I^{yy}_0=I^{x'x'}_0\sin^2\Omega t+I^{y'y'}_0\cos^2\Omega t=\frac{1}{2}mr^2(1-\xi\cos2\Omega t),\label{Ijk}\\
I^{xy}_0=\frac{1}{2}(I^{x'x'}_0-I^{y'y'}_0)\sin2\Omega t=-\frac{1}{2}mr^2\xi\sin2\Omega t.\nonumber
\end{array}\end{aligned}$$ By employing (\[qmom\]) and (\[Ijk\]) one has $${\frac{d^3I^{ij}}{dt^3}} =4\xi mr^2\Omega^3
\left(\begin{array}{cc}
\sin\Omega t & \cos\Omega t\\
\cos\Omega t & -\sin\Omega t
\end{array}\right).$$ Substituting into Equation (\[dEgw/dt\]) finally gives $$L_{GW}=\frac{32}{5}\xi^2m^2 r^4\Omega^6.
\label{EQN_ALGW}$$ The quadrupole formula for a circular binary of point masses is obtained upon taking $\xi=1$, $m=\mu$ and $r=a$, here $\mu$ is the reduced mass and $a$ is the binary separation.
In (\[EQN\_ALGW\]), the limit $\xi=1$ obtains the canonical formula of quadrupole gravitational wave emission. A circular binary of masses $M_i$ $(i=1,2)$ with chirp mass $\mu = (M_1M_2)^{5/3}(M_1+M_2)^{-1/5}$. For a matched filtering detection method, the relevant quantity is the amplitude that takes into account the square root of the associated number of wave periods. In the frequency domain, the corresponding quantity is the [*characteristic strain amplitude*]{}, given by the square root of the energy per unit logarithmic frequency interval [@fla98a], $$\begin{aligned}
h_{char}(f) = \frac{\sqrt{2}}{\pi D}\sqrt{\left|\frac{dE}{df}\right|}.
\label{EQN_hchar0}\end{aligned}$$ Consider a circular binary with small mass-ratio $\sigma=M_2/M_1<<1$, so that $M=M_1+M_2\simeq M_2$. At an orbital separation $a$, it emits quadrupole gravitational radiation at a frequency $\pi M f = (M/a)^\frac{2}{3}$. The total energy $E=-\frac{1}{2}\sigma M^2/a=-\frac{1}{2}M\sigma (M\pi f)^\frac{2}{3}$ hereby shrinks, whereby (cf. [@mis73]) $$\begin{aligned}
\frac{dE}{df} = - \frac{\pi \sigma M^2}{3 (\pi M f)^\frac{1}{3}}
\label{EQN_hchar1}\end{aligned}$$ and hence (cf. [@tho98; @ju00]) $$\begin{aligned}
\begin{array}{l}
h_{char}(f) = 8.6 \times 10^{-22} \,\sigma^\frac{1}{2} M_1{^\frac{1}{3}} \left(\frac{D}{100\,\mbox{Mpc}}\right)^{-1} \left(\frac{f}{100\,\mbox{Hz}}\right)^{-\frac{1}{6}}
\end{array}
\label{EQN_hchar2}\end{aligned}$$ for a central mass $M=M_1\times 10\,M_\odot$. It shows that the low frequency emission at early in spiral is particularly important for detection.
In the presence of an ellipticity $e$, the luminosity in gravitational waves is greater than (\[EQN\_ALGW\]) by additional radiation at frequency harmonics $m>2$ [@pet63]. The result can be expressed by an enhancement factor $F(e)\ge1$. The time rate of change in orbital frequency satisfies $$\begin{aligned}
\dot{f}_{orb} = \frac{96}{5} (2\pi)^{\frac{8}{3}} f_{orb}^{\frac{11}{3}} F(e), F(e)=\frac{1+\frac{73}{24}e^2+\frac{37}{96}e^4}{(1-e^2)^\frac{7}{2}}.
\label{EQN_Aforb1}\end{aligned}$$ Given an initial ellipticity $e_0$, the orbital separation $a=a(t)$ hereby satisfies $$\begin{aligned}
a(t) = a_0 \left( 1 - \frac{t}{\tau_0}\right)^\frac{1}{4}, \tau_0 = \frac{5a^4_0}{256 M_1M_2(M_1+M_2) }f(e_0),
\label{EQN_Aforb2}\end{aligned}$$ where $f(e_0)\le 1$ $(e_0\ge 0)$ obtains as an integral over $0\le e \le e_0$ (see, e.g., [@pos06] for a detailed expression).
It should be mentioned that (\[EQN\_Tho1a\]-\[EQN\_Tho1b\]) is derived for sources about a Minskowski background space-time. In an applying to the multipole mass-moments of a strongly magnetized torus about the ISCO of a rotating black hole [@bro06], the radiation is emitted in a strongly curved space-time. It requires extending (\[EQN\_Tho1a\]) by an additional grey body factor, that represents suppression of radiation at low $l$ for a given $I_{lm}$. The grey body factor derives from scattering of relatively low frequency gravitational waves in the curved space-time around black holes, that results in partial absorption by the black hole. However, in a suspended accretion state which balances heating by input from the black hole and cooling in gravitational radiation [@van12], $I_{lm}$ is self-regulated such that $L_{lm}$ in (\[EQN\_Tho1a\]) times such grey body factor balances with the energetic input from the black hole. Emissions from $I_{lm}$ beyond the ISCO are relatively less affected by space-time curvature. At large distances away from the black hole, the grey body factor is effectively one. A detailed derivation of the grey body factor is beyond the scope of this review.
[**Appendix B.**]{} Relativistic frame dragging
===============================================
The Kerr metric in Boyer-Lindquist coordinates $(t,r,\theta,\phi)$ explicitly brings about the Killing vectors $k^b=(\partial_t)^b$ and $m^b=(\partial_\phi)^b$ of time slices of constant coordinate time $t$. It gives an exact solution of frame dragging in terms of the angular velocity $\omega$ of particles of zero-angular momentum. In Boyer-Lindquist coordinates $(t,r,\theta,\phi)$ of the Kerr metric [@tho86], the world line of zero-angular momentum observers (ZAMOs, [@tho86] are orthogonal to slices of constant time-at-infinity. The angular velocity $\omega=d\phi/dt$ decays with the cube of the distance to the black hole at large distances.
By frame dragging, the ISCO of corotating orbits shrinks to the event horizon from $6M$ around a non-rotating Schwarzschild black hole. This appears in X-ray spectroscopy of MCG 6-30-15 [@iwa96]. These results are time variable over a time scale of a year, that may reflect intermittency in the inner radius of the disk [@fab95] or, alternatively, in circumnuclear clouds intermittently absorbing disk emissions.
The complete gravitational field induced by the angular momentum and mass of a rotating black hole is described by the Riemann tensor. For completeness, we here include a brief summary of earlier derivations on the associated energetic interactions [@van05; @van12].
Gravitational spin-orbit energy
-------------------------------
Consider the tetrad 1-forms $$\begin{aligned}
%\begin{array}{l}
e_{(0)}= \alpha dt,~~
e_{(1)} =\frac{\Sigma}{\rho}(d\phi - \omega dt)\sin\theta,~~
e_{(2)}= \frac{\rho}{\sqrt{\Delta}} dr,~~
e_{(3)} = \rho d\theta,
%\end{array}\end{aligned}$$ where $\alpha={\rho}{\Sigma}^{-1}\sqrt{\Delta}$ is the redshift factor, $\Sigma^2=(r^2+a^2)^2-a^2\Delta\sin\theta$, $\rho=r^2+a^2\cos^2\theta$, $\Delta=r^2-2Mr+a^2$ and $\omega={2aMr}{\Sigma}^{-2}$ is the angular velocity of frame dragging. The Riemann tensor has the following non-zero components [@cha83] $$\begin{aligned}
\begin{array}{rcl}
R_{0123} &=& A,~~R_{1230} = AC,~~R_{1302} = AD\\
-R_{3002} &=& R_{1213} = -3aA\sqrt{\Delta}\Sigma^{-2}(r^2 + a^2)\sin\theta\\
-R_{1220} &=& R_{1330} = -3aB\sqrt{\Delta}\Sigma^{-2}(r^2 + a^2)\sin\theta\\
-R_{1010} &=& R_{2323} = B = R_{0202} + R_{0303}\\
-R_{1313} &=& R_{0202} = BD,~~-R_{1212} = R_{0303} = -BC,
\end{array}
\label{EQN_R}\end{aligned}$$ where $$\begin{aligned}
\begin{array}{lcl}
A = aM\rho^{-6}(3r^2 - a^2 \cos^2\theta),~~
B = Mr\rho^{-6}(r^2 - 3a^2 \cos^2\theta),\\
C = \Sigma^{-2}[(r^2 + a^2)^2 + 2a^2\Delta\sin^2\theta], ~~
D = \Sigma^{-2}[2(r^2 + a^2)2 + a^2\Delta\sin^2\theta].
\end{array}\end{aligned}$$ About the black hole spin axis ($\theta=0$), $2A = -\partial_r\omega = {2aM}{\rho^{-6}}(3r^2 - a^2)$, $C = 1$, $D = 2,$ $J$ induced components appear in the first three of (\[EQN\_R\]). Integrating the Papapetrou force on a test particle with velocity four-vector $u^b$ satisfies [@pap51] $$\begin{aligned}
F_2 =\frac{1}{2}\epsilon_{abef}R^{cf}_{cd}J_p^au^bu^d = J_pR_{3120} = J_pAD = -\partial_2\omega J_p\end{aligned}$$ gives $$\begin{aligned}
{E}=\int_r^\infty F_2 ds.
\label{EQN_E2}\end{aligned}$$
Alternatively, consider the angular velocity $\Omega=u^{\phi}/u^t$. The normalization $-1 = u^cu_c = \left[g_{tt} + g_{\phi\phi}\Omega(\Omega-2\omega)\right](u^t)^2$ gives two roots $$\begin{aligned}
\Omega_{\pm} = \omega \pm \sqrt{\omega^2 - (g_{tt} + (u^t)^{-2})/g_{\phi\phi}}.
\label{EQN_ompm}\end{aligned}$$ Two particles with the same angular momentum in absolute value, $$\begin{aligned}
%\begin{array}{ll}
J_{p,\pm} = g_{\phi\phi}u^t(\Omega_{\pm}+\omega) =
g_{\phi\phi}u^t \sqrt{\omega^2-(g_{tt}+(u^t)^{-2})/g_{\phi\phi}}=\pm J_p
% \end{array}\end{aligned}$$ hereby have (with the same $u^t$) the total energies $E_{\pm} = (u^t)^{-1} + \Omega_{\pm}J_{\pm}$. One-half the difference satisfies $$\begin{aligned}
{E} =\frac{1}{2}(E_+ - E_-) = \omega J_p.
\label{EQN_E3}\end{aligned}$$
As a gravitational interaction, the curvature-spin coupling (\[EQN\_E3\]) acts universally on angular momentum, whether mechanical or electromagnetic in origin.
The result of (\[EQN\_E2\]) combined with canonical pair-creation processes will be a possibly force-free outflow along open magnetic field lines along the black hole spin axis, such as envisioned in [@bla77]. Intermittent inner engines hereby produce outgoing Alfvén fronts, that communicate the raw Faraday induced potential within the inner engine (roughly, on the event horizon of the black hole) out to large distance. The result may thus produce a linear accelerator ahead of the Alfvén front in regions of relatively low opacity, facilitating the production of UHECRs [@van09].
The structure of force-free outflows is a limit of ideal MHD [@glo14], which neglects inertia (and hence Reynolds stresses) in addition to being free of dissipation of the electromagnetic field. Originally, this limit was motivated to model extragalactic outflows, e.g., [@fan74], but increasingly this limit appears to be relevant also to extreme sources such as GRBs (e.g. [@lyu03; @lyu03b]).
Let $p_B=B^2/8\pi$ and $e_B=B^2/8/pi$ denote the magnetic pressure and energy density. In a magnetic flux tube of radius $R$, the dissipationless limit implies adiabatic compression: $p_B(2\pi RdR)=d(\pi e_BR^2)$, i.e., the magnetic flux $\Phi=\pi BR^2$ is [*frozen*]{} into the fluid. In contrast, a torsional perturbation mediating angular momentum outflow creates an [*Alfvén wave*]{} with velocity [@lic67] $$\begin{aligned}
v_A=\frac{B}{\sqrt{4\pi\rho+B^2}},
\label{EQN_VA}\end{aligned}$$ where $\rho$ denotes the fluid density as seen in the comoving frame. The Alfvén wave is purely rotational, leaving density (and magnetic flux) invariant. It should be mentioned that (\[EQN\_VA\]) is unique to MHD in U(1). It does not generalize to colored MHD [@van94].
Neglecting inertia, the Alfvén velocity reaches the velocity of light. Neglecting Reynolds stresses, $$\begin{aligned}
F_{ab}j^b=0,
\label{EQN_FF}\end{aligned}$$ which reduces the number of degrees of freedom in the electromagnetic field to two. For an electric current $j^b=\rho_e v^b$ associated with a charge density $\rho_e$ with four-velocity $v^b$, (\[EQN\_FF\]) implies $v^i\partial_iA_\phi=0$ and $v^i\partial_iA_0=0$ for a time-independent tube $A_\phi=$const. along the polar axis $\theta=0$. The electric potential hereby satisfies $A_0=A_0(A_\phi)$, and the electric field $\partial_iA_0=A_0^\prime\partial_iA_\phi$, in the Boyer-Lindquist frame of reference, is normal to the flux surfaces. Force-free flux surfaces are equipotential surfaces ([@gol69; @bla77; @tho86]).
Alfvén surfaces in force-free outflows from Intermittent inner engines can thus transmit Faraday induced potentials outwards. They can produce linear accelerators upstream at large distances from the source, providing a suitable condition for the creation of UHCRs from ionic contaminants by, e.g., UV-irradiation from a surrounding torus in AGN.
Alfvén waves in a torus magnetosphere
-------------------------------------
Consider the electromagnetic two-tensor $F_{ab}$ [@lic67] $$\begin{aligned}
{\bf F} = {\bf u}\wedge {\bf e} + * {\bf u} \wedge {\bf h}\end{aligned}$$ in the four-vector representation $(u^b,e^b,h^b)$ associated with a time like unit tangent $u^b$, $u^cu_c=-1$, of ZAMOs. Following [@bar72; @tho86], we have the one-form ${\bf u} = -\alpha {\bf d}t$ with redshift $\alpha$. Then ${\bf u}=\alpha^{-1}({\bf k}+\omega {\bf m})$ is linear combination of the Killing vectors, satisfying $\nabla_cu^c=0$. ZAMOs measure an electric field $e^b$ and a magnetic field $h^b$, $e^b=u_cF^{ac}$ and $h^b=u_c*F^{cb},$ each with three degrees of freedom given $u^ce_c=u^ch_c=0$. The same ZAMOs observe ${\bf e}=(0,E^i)$ and ${\bf h}=(0,B^i)$, where $i=1,2,3$ refers to the coordinates of the surfaces of constant $t$. The star $*$ denotes the Hodge dual, satisfying $*^2=-1$ in four dimensions.
Faraday’s equation $$\begin{aligned}
\nabla_a*F^{ab}=0\end{aligned}$$ can be expanded by considering $\nabla_a (u^ah^b-u^bh^a) = {L}_u h^b + (\nabla_cu^c)h^b-(\nabla_ch^c)u^b,$ where ${L}_u h^b = (u^c\nabla_c)h^b - (h^c\nabla_c)u^b$ denotes the Lie-derivative of $h^b$ with respect to $u^b$. Projected onto surfaces of constant $t$ (orthogonal to $u^b$), we have $$\begin{aligned}
\left({L}_{\bf u} {\bf h}\right)_\perp = \alpha^{-1}\left(\partial_t {\bf B} + {L}_\omega {\bf B}\right)\end{aligned}$$ evaluated in the frame of ZAMOs, where $L_\omega$ is the Lie-derivative with respect to $\omega^i\equiv\omega m^i$ ($m^i$ is not a unit three-vector). Next, $\nabla_a = D_a- u_a(u^c\nabla_c)$ and $(*{\bf u}\wedge {\bf h})_{abcd} = \epsilon_{abcd}u^ce^d$. With acceleration $(u^c\nabla_c)u_b=\alpha^{-1}\nabla_b\alpha$, consider $\nabla^b(\epsilon_{abcd}u^ce^d)=\epsilon_{abcd}(D^bu^c)e^d-\epsilon_{abcd}u^ba^ce^d
+\epsilon_{abcd}u^c\nabla^be^d.$ Projection of the right hand side onto the space like coordinates $i=(r,\theta,\phi)$ normal to $u^b$ satisfies $$\begin{aligned}
%\begin{array}{ll}
\epsilon_{ibcd}(D^bu^c)e^d+\tilde{\epsilon}_{ijk}a^je^k+\tilde{\epsilon}_{ijk}\nabla^je^k=
\epsilon_{ibcd}(D^bu^c)e^d+\alpha^{-1}\tilde{\epsilon}_{ijk}\nabla^j(\alpha e^k),
%\end{array}\end{aligned}$$ where $\epsilon_{aijk}u^a=\tilde{\epsilon}_{ijk}=\sqrt{h}\Delta_{ijk}$ with $\sqrt{-g}=\alpha\sqrt{h}$ over the space like volume element $\sqrt{h}$, where $\Delta_{ijk}$, $\Delta_{123}=1$. The first term on the right hand side vanishes, since $D_bu_c$ is spacelike: $u^b(D_bu_c)=0$ by construction and $u^cD_bu_c=0$ by $u^2=-1$. Consequently, Faraday’s law includes an additional term (derived alternatively in [@tho86] and references therein) $$\begin{aligned}
\tilde{\nabla}\times \alpha {\bf E} = -\partial_t {\bf B} + 4\pi {J}_m,
\label{EQN_FAR}\end{aligned}$$ where $\tilde{\nabla}_i=D_i$ and $$\begin{aligned}
{J}_m = -\frac{1}{4\pi} { L}_\omega {\bf B}.
\label{EQN_JM}\end{aligned}$$ $J_m$ appears analogously to a current of virtual magnetic monopoles.
Applied to a torus magnetosphere, (\[EQN\_JM\]) satisfies $$\begin{aligned}
%\begin{array}{ll}
\omega_i { J}_m^i \simeq \frac{1}{8\pi} {\bf B}\cdot \tilde{\nabla}(\omega_i\omega^i) > 0, ~~
\omega_i\omega^i = 4\frac{z^2\sin^2\lambda}{(z^2+\sin^2\lambda)^3} ~(\theta=\frac{\pi}{2}),
%\end{array}
\label{EQN_LOM}\end{aligned}$$ where the inequality refers to a poloidally ingoing magnetic field.
By (\[EQN\_LOM\]), frame dragging induced poloidal current loops in the inner torus magnetosphere. The resulting poloidal Afvén waves produces Maxwell stresses by which rotating black holes lose angular momentum to surrounding matter. The black hole hereby spins down, which should have an imprint on any light curve derived from high energy emissions derived from (\[EQN\_E2\]), while surrounding matter is brought into a state of forced turbulence by competing torques acting on the inner and outer faces [@van99], possibly related to forced turbulence in Taylor-Couette flows [@ste09].
The Alfvén waves effectively transport angular momentum out an onto the torus, provided they are not canceled by Reynolds stresses from a baryon-rich torus wind back into the black hole. In this event, the inner face of the torus will be spun up, whereby it assumes a state of super-Keplerian motion. The resulting differential rotation can induce non-axisymmetric wave instabilities (Appendix C). The associated surface gravity suppresses baryon-rich outflows from the inner face of the torus. A detailed description of this suppression of Reynolds stresses falls outside the scope of the present discussion, however.
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A reformulation of the problem of the Interpretation of Quantum Mechanics
=========================================================================
In this paper, I discuss a novel view of quantum mechanics. This point of view is not antagonistic to current ones, as the Copenhagen \[Heisenberg 1927, Bohr 1935\], consistent histories \[Griffiths 1984, Griffiths 1996, Omnes 1988, Gell-Mann and Hartle 1990\], many-worlds \[Everett 1957, Wheeler 1957, DeWitt 1970\], quantum event \[Huges 1989\], many minds \[Albert and Lower 1988, 1989, Lockwood 1986, Donald 1990\] or modal \[Shimony 1969, van Fraassen 1991, Fleming 1992\] interpretations, but rather combines and complements aspects of them. This paper is based on a critique of a notion generally assumed uncritically. As such, it bears a vague resemblance with Einstein’s discussion of special relativity, which is based on the critique of the notion of absolute simultaneity. The notion rejected here is the notion of absolute, or observer-independent, state of a system; equivalently, the notion of observer-independent values of physical quantities. The thesis of the present work is that by abandoning such a notion (in favor of the weaker notion of state –and values of physical quantities– [ *relative*]{} to something), quantum mechanics makes much more sense. This conclusion derives from the observation that the experimental evidence at the basis of quantum mechanics forces us to accept that distinct observers give different descriptions of the same events. From this, I shall argue that the notion of observer-independent state of a system is inadequate to describe the physical world beyond the $\hbar \rightarrow 0$ limit, in the same way in which the notion of observer-independent time is inadequate to describe the physical world beyond the $c \rightarrow\infty$ limit. I then consider the possibility of replacing the notion of absolute state with a notion that refers to the relation between physical systems.
The motivation for the present work is the commonplace observation that in spite of the 70 years-lapse from the discovery of quantum mechanics, and in spite of the variety of approaches developed with the aim of clarifying its content and improving the original formulation, quantum mechanics still maintains a remarkable level of obscurity. It is even accused of being unreasonable and unacceptable, even inconsistent, by world-class physicists (For example \[Newman 1993\]). My point of view in this regard is that quantum mechanics synthesizes most of what we have learned so far about the physical world: The issue is thus not to replace or fix it, but rather to understand [*what*]{} precisely it says about the world; or, equivalently: what precisely we have learned from experimental micro-physics.
It [*is*]{} difficult to overcome the sense of unease that quantum mechanics communicates. The troubling aspect of the theory assumes different faces within different interpretations, and a complete description of the problem can only be based on a survey of the solutions proposed. Here, I do not attempt such a survey; for a classical review see \[d’Espagnat 1971\], a more modern survey is in the first chapters of \[Albert 1992\], or, in compact form, see \[Butterfield 1995\]. The unease is expressed, for instance, in the objections the supporters of each interpretation raise against other interpretations. Some of these objections are perhaps naive or ill-posed, but the fact that that no interpretation has so far succeeded in convincing the majority of the physicists, indicates, I believe, that the problem of the interpretation of quantum mechanics has not been fully disentangled yet. This unease, and the variety of interpretations of quantum mechanics that it has generated is sometimes denoted as the measurement problem. In this paper, I address this problem and consider a way out.
The paper is based on two ideas:
- That the unease may derive from the use of a concept which is inappropriate to describe the physical world at the quantum level. I shall argue that this concept is the concept of observer-independent state of a system, or, equivalently, the concept of observer-independent values of physical quantities.
- That quantum mechanics will cease to look puzzling only when we will be able to [*derive*]{} the formalism of the theory from a set of simple physical assertions (“postulates", “principles") about the world. Therefore, we should not try to [*append* ]{} a reasonable interpretation to the quantum mechanics [ *formalism*]{}, but rather to [*derive*]{} the formalism from a set of experimentally motivated postulates.
The reasons for exploring such a strategy are illuminated by an obvious historical precedent: special relativity. I shall make use of this analogy for explanatory purposes, in spite of the evident limits of the simile.
Special relativity is a well understood physical theory, appropriately credited to Einstein’s 1905 celebrated paper. The formal content of special relativity, however, is coded into the Lorentz transformations, written by Lorentz, not by Einstein, and before 1905. So, what was Einstein’s contribution? It was to understand the physical meaning of the Lorentz transformations. (And more, but this is what is of interest here). We could say –admittedly in a provocative manner– that Einstein’s contribution to special relativity has been the interpretation of the theory, not its formalism: the formalism already existed.
Lorentz transformations were discussed at the beginning of the century, and their [*interpretation*]{} was debated. In spite of the recognized fact that they represent an extension of the Galilean group compatible with Maxwell theory, the Lorentz transformation were perceived as “unreasonable” and “unacceptable as a fundamental spacetime symmetry”, even “inconsistent”, before 1905; words that recall nowadays comments on quantum mechanics. The physical interpretation proposed by Lorentz himself (and defended by Lorentz long after 1905) was a physical contraction of moving bodies, caused by a complex and unknown electromagnetic interaction between the atoms of the bodies and the ether. It was quite an unattractive interpretation, remarkably similar to certain interpretations of the wave function collapse presently investigated! Einstein’s 1905 paper suddenly clarified the matter by pointing out the reason for the unease in taking Lorentz transformations “seriously": the implicit use of a concept (observer-independent time) inappropriate to describe reality when velocities are high. Equivalently: a common deep assumption about reality (simultaneity is observer-independent) which is physically untenable. The unease with the Lorentz transformations derived from a conceptual scheme in which an incorrect notion –absolute simultaneity– was assumed, yielding any sort of paradoxical consequences. Once this notion was removed the physical interpretation of the Lorentz transformations stood clear, and special relativity is now considered rather uncontroversial.
Here I consider the hypothesis that all “paradoxical" situations associated with quantum mechanics –as the famous and unfortunate half-dead Schrödinger cat \[Schrödinger 1935\]– may derive from some analogous incorrect notion that we use in thinking about quantum mechanics. (Not in *using* quantum mechanics, since we seem to have learned to use it in a remarkably effective way.) The aim of this paper is to hunt for this incorrect notion, with the hope that by exposing it clearly to public contempt, we could free ourselves from the present unease with our best present theory of motion, and fully understand what does the theory assert about the world.
Furthermore, Einstein was so persuasive with his interpretation of the Lorentz equations because he did not append an interpretation to them: rather, he re-derivedÊ them, starting from two postulates with terse physical content –equivalence of inertial observers and universality of the speed of light– taken as facts of experience. It was this re-derivation that unraveled the physical content of the Lorentz transformations and provided them a convincing interpretation. I would like to suggest here that in order to clarify the physical meaning of quantum mechanics, a similar result should be searched: Finding a small number of simple statements about nature –which may perhaps seem contradictory, as the two postulates of special relativity do– with clear physical content, from which the formalism of quantum mechanics could be derived. In other words, I have a methodological suggestion for the problem of the interpretation of quantum mechanics: Finding the set of physical facts from which the quantum mechanics’s formalism can be derived. To my knowledge, such a derivation has not been achieved yet. In this paper, I do not achieve such a result in a satisfactory manner, but I discuss a possible reconstruction scheme.
The program outlined is thus to do for the formalism of quantum mechanics what Einstein did for the Lorentz transformations: i. Find a set of simple assertions about the world, with clear physical meaning, that we know are experimentally true (postulates); ii. Analyze these postulates, and show that from their conjunction it follows that certain common assumptions about the world are incorrect; iii. Derive the full formalism of quantum mechanics from these postulates. I expect that if this program could be completed, we would at long last begin to agree that we have understood quantum mechanics.
In section 2, I analyze the measurement process as described by two distinct observers. This analysis leads to the main idea: the observer dependence of state and physical quantities, and to recognize a few key concepts in terms of which, I would like to suggest, the quantum mechanical description of reality “makes sense". Prominent among these is the concept of information \[Shannon 1949, Wheeler 1988, 1989, 1992\]. In section 3, I switch from an inductive to a (very mildly) deductive mode, and put forward a set of notions, and a set of simple physical statements, from which the formalism of quantum mechanics can be reconstructed. I denote these statements as postulates, at the risk of misunderstanding: I do not claim any mathematical nor philosophical rigor, nor completeness –supplementary assumptions are made along the way. I am not interested here in a formalization of the subject, but only in better grasping its physics. Ideas and techniques for the reconstruction are borrowed from quantum logic research, but motivations and spirit are different. Finally, in section 4, I discuss the picture of the physical world that has emerged, and attempt an evaluation. In particular, I compare the approach I have developed with some currently popular interpretations of quantum mechanics, and argue that the differences between these disappear, if the results presented here are taken into account.
In order to prevent the reader from channeling his/her thoughts in the wrong direction, let me anticipate a few terminological remarks. By using the word “observer" I do not make any reference to conscious, animate, or computing, or in any other manner special, system. I use the word “observer" in the sense in which it is conventionally used in Galilean relativity when we say that an object has a velocity “with respect to a certain observer". The observer can be any physical object having a definite state of motion. For instance, I say that my hand moves at a velocity $v$ with respect to the lamp on my table. Velocity is a relational notion (in Galilean as well as in special relativistic physics), and thus it is always (explicitly or implicitly) referred to something; it is traditional to denote this something as the observer, but it is important in the following discussion to keep in mind that the observer can be a table lamp. Also, I use information theory in its information-theory meaning (Shannon): information is a measure of the number of states in which a system can be –or in which several systems whose states are physically constrained (correlated) can be. Thus, a pen on my table has information because it points in this or that direction. We do not need a human being, a cat, or a computer, to make use of this notion of information.
Quantum mechanics is a theory about information
===============================================
In this section, a preliminary analysis of the process of measurement is presented, and the main ideas are introduced. Throughout this section, standard quantum mechanics and standard interpretation –by which I mean for instance: formalism and interpretation in \[Dirac 1930\] or \[Messiah 1958\]– are assumed.
The third person problem
------------------------
Consider an observer $O$ (Observer) that makes a measurement on a system $S$ (System). For the moment we may think of $O$ as a classical macroscopic measuring apparatus, including or not including a human being. Assume that the quantity being measured, say $q$, takes two values, 1 and 2; and let the states of the system $S$ be described by vectors (rays) in a two (complex) dimensional Hilbert space $H_S$. Let the two eigenstates of the operator corresponding to the measurement of $q$ be $|1\rangle$ and $|2\rangle$. As it is well known: if $S$ is in a generic normalized state $|\psi\rangle =\alpha |1\rangle + \beta
|2\rangle$, where $\alpha$ and $\beta$ are complex numbers and $|\alpha|^2 + |\beta|^2 = 1$, then $O$ can measure either one of the two values 1 and 2 – with respective probabilities $|\alpha|^2$ and $|\beta|^2$.
Assume that in a [*given specific measurement*]{} the outcome of the measurement is 1. From now on, we concentrate on describing this specific experiment, which we denote as $\cal E$. The system $S$ is affected by the measurement, and at a time $t=t_2$ after the measurement, the state of the system is $|1\rangle$. In the physical sequence of events $\cal E$, the states of the system at $t_1$ and $t_2$ are thus $$\begin{aligned}
t_1 & \longrightarrow & t_2 \nonumber \\
\alpha |1\rangle + \beta |2\rangle & \longrightarrow
&|1\rangle
\label{uno}\end{aligned}$$
Let us now consider this same sequence of events $\cal E$, as described by a second observer, which we refer to as $P$. I shall refer to $O$ as “he" and to $P$ as “she". $P$ describes the interacting system formed by $S$ and $O$. Again, assume $P$ uses conventional quantum mechanics. Also, assume that $P$ does not perform any measurement on the $S-O$ system during the $t_1-t_2$ interval, but that she knows the initial states of both $S$ and $O$, and is thus able to give a quantum mechanical description of the set of events $\cal E$. She describes the system $S$ by means of the Hilbert space $H_S$ considered above, and $O$ by means of a Hilbert space $H_O$. The $S-O$ system is then described by the tensor product $H_{SO} = H_S\otimes H_O$. As it has become conventional, let us denote the vector in $H_O$ that describes the state of the observer $O$ at $t=t_1$ (prior to the measurement) as $|init\rangle$. The physical process during which $O$ measures the quantity $q$ of the system $S$ implies a physical interaction between $O$ and $S$. In the process of this interaction, the state of $O$ changes. If the initial state of $S$ is $|1\rangle$ (resp $|2\rangle$) (and the initial state of $O$ is $|init\rangle$), then $|init\rangle$ evolves to a state that we denote as $|O1\rangle$ (resp $|O2\rangle$). Think of $|O1\rangle$ (resp $|O2\rangle$) as a state in which “the position of the hand of a measuring apparatus points towards the mark ‘1’ (resp ‘2’)". It is not difficult to construct model Hamiltonians that produce evolutions of this kind, and that can be taken as models for the physical interactions that produce a measurement. Let us consider the actual case of the experiment $\cal E$, in which the initial state of $S$ is $|\psi\rangle = \alpha |1\rangle+\beta |2\rangle$. The initial full state of the $S-O$ system is then $|\psi\rangle\otimes |init\rangle
=
(\alpha |1\rangle+\beta |2\rangle)\otimes|init\rangle$. As well known, the linearity of quantum mechanics implies $$\begin{aligned}
t_1 & \longrightarrow & t_2 \nonumber \\
(\alpha |1\rangle + \beta |2\rangle)\otimes |init\rangle
& \longrightarrow &
\alpha |1\rangle\otimes |O1\rangle + \beta |2\rangle\otimes
|O2\rangle \label{due}\end{aligned}$$ Thus, at $t=t_2$Ê the system $S-O$ is in the state $(\alpha
|1\rangle\otimes
|O1\rangle+\beta|2\rangle\otimes|O2\rangle)$. This is the conventional description of a measurement as a physical process \[von Neumann 1932\].
I have described an actual physical process $\cal E$ taking place in a real laboratory. Standard quantum mechanics requires us to distinguish system from observer, but it allows us freedom in drawing the line that distinguishes the two. In the above analysis this freedom has been exploited in order to describe the same sequence of physical events in terms of two different descriptions. In the first description, equation (1), the line that distinguishes system from observer is set between $S$ and O. In the second, equation (2), between $S-O$ and $P$. Recall that we have assumed that $P$ is not making a measurement on the $S-O$ system; there is no physical interaction between $S-O$ and $P$ during the $t_1-t_2$ interval. $P$ may make measurements at a later time $t_{3}$: if she measures the value of $q$ on $S$ and the position of the hand on $O$, she finds that the two agree, because the first measurement collapses the state into one of the two factors of (2), leaving the second measurement fully determined to be the consistent value. Thus, we have two descriptions of the physical sequence of events $\cal E$: The description (1) given by the observer $O$ and the description (2) given by the observer $P$. These are two distinct [*correct*]{} descriptions of the same sequence of events $\cal E$. At time $t_2$, in the $O$ description, the system $S$ is in the state $|1\rangle$ and the quantity $q$ has value 1. According to the $P$ description, $S$ is not in the state $|1\rangle$ and the hand of the measuring apparatus does not indicate ‘1’.
Thus, I come to the observation on which the rest of the paper relies.
> [**Main observation:**]{} In quantum mechanics different observers may give different accounts of the same sequence of events.
For a very similar conclusion, see \[Zurek 1982\] and \[Kochen 1979\]. In the rest of the work, I explore the consequences of taking this observation fully into account. Since this observation is crucial, I now pause to discuss and refute various objections to the main observation. The reader who finds the above observation plausible may skip this rather long list of objections and jump to section II.C.
Objections to the main observation
----------------------------------
[*Objection 1. Whether the account (1) or the account (2) is correct depends on which kind of system $O$ happens to be. There are systems that induce the collapse of the wave function. For instance, if $O$ is macroscopic (1) is correct, if $O$ is microscopic (2) is correct.* ]{}
The derivation of (2) does not rely on any assumption on the systems, but only from the basics of quantum mechanics (linearity). Therefore a particular $O$ system yielding (1) instead of (2) via Schrödinger evolution must behave in a way that contradicts the formalism of quantum mechanics as we know it. This implies that $O$ cannot be described as a genuine quantum system. Namely that there are special systems that do not obey conventional quantum mechanics, but are intrinsically classical in that they produce collapse of the wave functions –or the actualization of quantities’ values. This idea underlies a variety of old and recent attempts to unravel the quantum puzzle. The special systems being for instance gravity \[Penrose 1989\], or minds \[Albert and Loewer 1988\], or macroscopic systems \[Bohr 1949\]. If we accept this idea, we have to separate reality into two kinds of systems: quantum mechanical systems on the one hand, and special systems on the other. Bohr declares explicitly that we must renounce giving a quantum description of the classical world \[Bohr 1949\]. This is echoed in texts as \[Landau and Lifschit 1977\]. Wigner pushes this view to the extreme consequences and distinguishes material systems (observed) from consciousness (observer) \[Wigner 1961\]. Here, on the contrary, I wish to assume
> [**Hypothesis 1**]{}: All systems are equivalent: Nothing distinguishes a priori macroscopic systems from quantum systems. If the observer $O$ can give a quantum description of the system $S$, then it is also legitimate for an observer $P$ to give a quantum description of the system formed by the observer $O$.
Of course, I have no proof of hypothesis 1, only plausibility arguments. I am suspicious toward attempts to introduce special non-quantum and not-yet-understood new physics, in order to alleviate the strangeness of quantum mechanics: they look to me very much like Lorentz’ attempt to postulate a mysterious interaction that Lorentz-contracts physical bodies “for real" –something that we now see was very much off of the point, in the light of Einstein’s clarity. Virtually all those views modify quantum mechanical predictions, in spite of statements of the contrary: if at $t_2$, the state is as in (1), then $P$ can never detect interference terms between the two branches in (2), contrary to quantum theory predictions. These discrepancies are likely to be minute, as shown by the beautiful discovery of the physical mechanism of decoherence \[Zurek 1981, Joos and Zeh 1985\], which “saves the phenomena". But they are nevertheless different from zero, and thus observable (more on this later). I am inclined to trust that a sophisticated experiment able to detect those minute discrepancies will fully vindicate quantum mechanics against distortions due to postulated intrinsic classicality of specific systems. In any case, the question is experimentally decidable; and we shall see. Second, I do not like the idea that the present over-successful theory of motion can only be understood in terms of its failures yet-to-be-detected. Finally, I think it is reasonable to remain committed, up to compelling disproof, to the rule that all physical systems are equivalent with respect to mechanics: this rule has proven so successful, that I would not dismiss it as far as there is another way out.
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[*Objection 2. What the discussion indicates is that the quantum state is different in the two accounts, but the quantum state is a fictitious non-physical mental construction; the physical content of the theory is given by the outcomes of the measurements.* ]{}
Indeed, one can take the view that outcomes of measurements are the physical content of the theory, and the quantum state is a secondary theoretical construction. This is the way I read \[Heisenberg 1927\] and \[van Fraassen 1991\]. According to this view, anything in between two measurement outcomes is like the “non-existing" trajectory of the electron, to use Heisenberg’s vivid expression, of which there is nothing to say. I am very sympathetic with this view, which plays an important role in section III. This view, however, does not circumvent the main observation for the following reason. The account (2) states that there is nothing to be said about the value of the quantity $q$ of $S$ at time $t_2$: for $P$, at $t= t_2$ the quantity $q$ does not have a determined value. On the other hand, for $O$, at $t= t_2$, $q$ has value 1. From which the main observation follows again.
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[*Objection 3. As before (only outcomes of measurements are physical), but the truth of the matter is that $P$ is right and $O$ is wrong.* ]{}
This is indefensible. Since all physical experiments of which we know can be seen as instances of the $S-O$ measurement, this would imply that not a single outcome of measurement has ever being obtained yet. If so, how could have we learned quantum theory?
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[*Objection 4. As before (only outcomes of measurements are physical), but the truth of the matter is that $O$ is right and P is wrong.*]{}
If $P$ is wrong, quantum mechanics cannot be applied to the $S-O$ system (because her account is a straightforward implementation of textbook quantum mechanics). Thus this objection predicts discrepancies, so far never observed, with quantum mechanical predictions, which include observable interference effects between the two terms of (2).
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[*Objection 5. As before, but under the assumption that $O$ is macroscopic. Then interference terms become extremely small because of decoherence effects. If they are small enough, they are unobservable, and thus $q=1$ becomes an absolute property of $S$, which is true and absolutely determined, albeit unknown to $P$, who could measure it anytime, and would not see interference effects.*]{}
Strictly speaking this is wrong, because decoherence depends on which observation $P$ will make. Therefore, the property $q=1$ of $S$ would become an absolute property at time $t_2$, or not, according to which subsequent properties of $S$ the observer $P$ considers. This is the reason for which the idea of exploiting physical decoherence for the basic interpretation of quantum mechanics problem has evolved into the consistent histories interpretations, where probabilities are (consistently) assigned to histories, and not to single outcomes of measurements within a history. (See, however, the discussion on the no-histories slogan in \[Butterfield 1995\].)
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[*Objection 6. There is no collapse. The description (1) is not correct, because “the wave function never really collapses". The account (2) is the correct one. There are no values assigned to classical properties of system; there are only quantum states.*]{}
If so, then the observer $P$ cannot measure the value of the property $q$ either, since (by assumption) there are no values assigned to classical properties, but only quantum states; thus the quantity $q$ doesn’t ever have a value. But we do describe the world in terms of “properties" that the systems have and values assumed by various quantities, not in terms of states in Hilbert space. In a description of the world purely in terms of quantum states, the systems never have definite properties and I do not see how to match such a description with [*any*]{} observation. For a detailed elaboration of this point, which is too often neglected, but I think is very strong, see \[Albert 1992\].
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[*Objection 7. There is no collapse. The description (1) is not correct, because “the wave function never really collapses". The account (2) is the correct one. The values assigned to classical properties are different from branch to branch.*]{}
This is a form of Everett’s view \[Everett 1957\], which entails the idea that when we measure the electron’s spin being up, the electron spin is also and simultaneously down “in some other branch" –or “world", hence the many world denomination of this view. The property of the electron of having spin up is not absolutely true, but only true relative to “this" branch. We have a new “parameter" for expressing contingency: “which branch" is a new “dimension" of indexicality, in addition to the familiar ones “which time" and “which place". Thus, the state of affairs of the example is that, at $t_2$, $q$ has value 1 in one branch and has value 2 in the other; the two branches being theoretically described by the two terms in (2). This is a fascinating idea that has recently been implemented in a variety of diverse incarnations. Traditionally, the idea has been discussed in the context of a notion of apparatus, namely a distinguished set of subsystems of the universe, and a distinguished quantity of such an apparatus –the preferred basis. Such a (collection of) preferred apparatus and preferred basis are needed in order to define branching, and thus in order to have assignment of values \[Butterfield 1995\]; the view has recently branched into the many mind interpretations, where the distinguished subsystems are related to various aspect of the human brain. (See \[Butterfield 1995\] for a recent discussion). These versions of Everett’s idea violate hypothesis 1, and thus I am not concerned with them. Alternatively, there are versions of Everett’s idea that reject the specification of preferred apparatus and preferred basis, and in which the branching itself is indexed by an arbitrarily chosen system playing the role of apparatus and an arbitrarily chosen basis. To my knowledge, the only elaborated versions of this view which avoids the difficulties mentioned in objection 5 have evolved into the histories formalisms considered below.
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[*Objection 8. What is absolute and observer independent is the probability of a sequence $A_1, ... A_n$ of property ascriptions (such that the interference terms mentioned above are extremely small - decoherence); this probability is independent from the existence of any observer measuring these properties.* ]{}
This is certainly correct. In fact, this observation is at the root of the consistent histories (CH) interpretations of quantum mechanics \[Griffiths 1984, 1996, Omnes 1988, Gell-Mann and Hartle 1990\]. However, in my understanding, CH confirms the observation above that different observers give different accounts of the same sequence of events, for the following reason. The beauty of the histories interpretations is the fact that the probability of a sequence of events in a consistent family of sequences does not depends on the observer, precisely as it doesn’t in classical mechanics. One can be content with this powerful result of the theory and stop here. However, probabilities depends on the choice of the consistent [ *family*]{} of histories, which is chosen (to avoid misunderstanding: [*whether or not*]{} a physical occurrence can be assigned a probability depends on the family chosen). One (who?) makes a choice in picking up a family of alternative histories in terms of which he chooses to describe the system. Griffiths has introduced the vivid expression “framework” to indicate a consistent family of histories \[Griffiths 1996\]. There exist funny cases in which one framework is: “Is the value of the physical quantity $Q$ equal to 1 at time $t$, or not?"; and a second framework is “Is the value of the physical quantity $Q$ equal to 2 at time $t$, or not?", and in each framework the answer is yes with probability 1! \[Kent 1995, Kent and Dowker 1995, Griffiths 1996.\] Therefore the description of what has happened at time $t$, that we can give on the basis of a fixed set of data, is: At $t$, $Q$ was equal to 1, if we ask whether $Q$ was 1 or not. Or: At $t$, $Q$ was equal to 2, if we ask whether $Q$ was 2 or not. There is no contradiction here (in Copenhagen terms, the same mathematics would indicate that the outcome depends on which apparatus is present), but it is difficult to deny that a large majority of physicists still want to understand more about this strangeness of quantum mechanics. In the Copenhagen view, the choice that corresponds to the choice between frameworks is determined by which classical apparatus is present. Namely, the framework is determined by the interaction of the quantum system with a classical object. In CH, one claims that property ascription does not need a classical interaction; the price to pay is that (probabilistic) predictions, rather than being uniquely determined, are framework dependent. In the example of the previous section, the observers $O$ and $P$ may choose two distinct frameworks, and the corresponding two descriptions are both valid: each one in its own framework. However, observer $O$ does not have the choice of using the framework that the observer $P$ uses, because he has “seen” $q=1$. After having seen that $q=1$, $O$ has no option anymore of allowing a framework in which $q$ is not 1. The fact that $q=1$ has become one of his “dataÕÕ; and data [ *determine*]{} which frameworks are consistent. Therefore, the two observers $O$ and $P$ have different sets of frameworks at their disposal for describing the same events, because they have different data (for the same set of events). The framework in which (2) makes sense is available to $P$, but not to $O$, because $O$ has data that include that fact that $Q=1$ at $t_2$. What is [*data*]{} for $O$ is not data for $P$, who considers the full $S-O$ system: $P$ is still allowed to choose a framework which does not include the value 1 of $q$ at $t_2$. Once the data are specified, all predictions are well defined in CH, but the characterization of what may count as data, and therefore which frameworks are available, is different for the two observers of the above example. Once more, we have that two different observers give different descriptions of the same set of events: what is data for $O$ is only a possible choice of framework for $P$. I will return on this delicate point in the last section.
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In conclusion, it seems to me that whatever view of quantum theory (consistent with hypothesis 1) one holds, the main observation is inescapable. I may thus proceed to the main point of this work.
Main discussion
---------------
If different observers give different accounts of the same sequence of events, then each quantum mechanical description has to be understood as relative to a particular observer. Thus, a quantum mechanical description of a certain system (state and/or values of physical quantities) cannot be taken as an “absolute" (observer independent) description of reality, but rather as a formalization, or codification, of properties of a system [*relative*]{} to a given observer. Quantum mechanics can therefore be viewed as a theory about the states of systems and values of physical quantities relative to other systems.
A quantum description of the state of a system $S$ exists only if some system $O$ (considered as an observer) is actually “describing" $S$, or, more precisely, has interacted with $S$. The quantum state of a system is always a state of that system with respect to a certain other system. More precisely: when we say that a physical quantity takes the value $v$, we should always (explicitly or implicitly) qualify this statement as: the physical quantity takes the value $v$ with respect to the so and so observer. Thus, in the example considered in section 2.1, $q$ has value 1 [*with respect to* ]{} $O$, but not with respect to $P$.
Therefore, I suggest that in quantum mechanics “state" as well as “value of a variable" –or “outcome of a measurement–" are relational notions in the same sense in which velocity is relational in classical mechanics. We say “the object $S$ has velocity $v$” meaning “with respect to a reference object $O$". Similarly, I maintain that “the system is in such a quantum state" or “$q=1$" are always to be understood “with respect to the reference $O$." In quantum mechanics [*all*]{} physical variables are relational, as is velocity.
If quantum mechanics describes relative information only, one could consider the possibility that there is a deeper underlying theory that describes what happens “in reality". This is the thesis of the incompleteness of quantum mechanics (first suggested in \[Born 1926\]!). Examples of hypothetical underlying theories are hidden variables theories \[Bohm 1951, Belifante 1973\]. Alternatively, the “wave-function-collapse-producing" systems can be “special" because of some non-yet-understood physics, which becomes relevant due to large number of degrees of freedom \[Ghirardi Rimini and Weber 1986, Bell 1987\], complexity \[Hughes 1989\], quantum gravity \[Penrose 1989\] or other.
As is well known, there are no indications on [*physical*]{} grounds that quantum mechanics is incomplete. Indeed, the [ *practice*]{} of quantum mechanics supports the view that quantum mechanics represents the best we can say about the world at the present state of experimentation, and suggests that the structure of the world grasped by quantum mechanics is deeper, and not shallower, than the scheme of description of the world of classical mechanics. On the other hand, one could consider motivations on [ *metaphysical*]{} grounds, in support of the incompleteness of quantum mechanics. One could argue: “Since reality has to be real and universal, and the same for everybody, then a theory in which the description of reality is observer-dependent is certainly an incomplete theory". If such a theory were complete, our concept of reality would be disturbed.
But the way I reformulated the problem of the interpretation of quantum mechanics in section I. should make us suspicious and attentive precisely to such kinds of arguments, I believe. Indeed, what we are looking for is precisely some “wrong general assumption" that we suspect to have, and that could be at the origin of the unease with quantum mechanics. Thus, I discard the thesis of the incompleteness of quantum mechanics and assume
> [**Hypothesis 2**]{} (Completeness): Quantum mechanics provides a complete and self-consistent scheme of description of the physical world, appropriate to our present level of experimental observations.
The conjunction of this hypothesis 2 with the main observation of section II.A and the discussion above leads to the following idea:
> [*Quantum mechanics is a theory about the physical description of physical systems relative to other systems, and this is a complete description of the world.*]{}
The thesis of this paper is that this conclusion is not self-contradictory. If this conclusion is valid, then the incorrect notion at the source of our unease with quantum theory has been uncovered: it is the notion of true, universal, observer-independent description of the state of the world. If the notion of observer-independent description of the world is unphysical, a complete description of the world is exhausted by the relevant information that systems have about each other. Namely, there is neither an absolute state of the system, nor absolute properties that the system has at a certain time. Physics is fully relational, not just as far as the notions of rest and motion are considered, but with respect to all physical quantities. Accounts (1) and (2) of the sequence of events $\cal E$ are both correct, even if distinct: any time we talk about a state or property of a system, we have to refer these notions to a specific observing, or reference system. Thus, I propose the idea that quantum mechanics indicates that the notion of a universal description of the state of the world, shared by all observers, is a concept which is physically untenable, on experimental ground.[^1]
Thus, the hypothesis on which I base this paper is that accounts (1) and (2) are both fully correct. They refer to different observers. I propose to reinterpret every contingent statement about nature (“the electron has spin up", “the atom is in the so and so excited state", the “spring is compressed", “the chair is here and not there") as elliptic expressions for relational assertions (“the electron has spin up [*with respect to the Stern Gerlac apparatus*]{}" ... “the chair is here and not there [*with respect to my eyes*]{}", and so on). A general physical theory is a theory about the state that physical systems have, relative to each other. I explore and elaborate this possibility in this paper.
Relation between descriptions
-----------------------------
The multiplication of points of view induced by the relational notion of state and physical quantities’ values considered above raises the problem of the relation between distinct descriptions of the same events. What is the relation between the value of a variable $q$ relative to an observer $O$, and the value of the same variable relative to a different observer? This problem is subtle. Consider the example of section II.A. We expect some relation between the description of the world illustrated in (1) and in (2).
First of all, one may ask what is the “actual”, “absolute” relation between the description of the world relative to $O$ and the one relative to $P$. This is a question debated in the context of “perspectival” interpretations of quantum mechanics. I think that the question is ill-posed. The absolute state of affairs of the world is a meaningless notion; asking about the absolute relation between two descriptions is precisely asking about such an absolute state of affairs of the world. Therefore there is no meaning in the “absolute” relation between the views of different observers. In particular, there is no way of deducing the view of one from the view of the other.
Does this mean that there is no relation whatsoever between views of different observers? Certainly not; it means that the relation itself must be understood quantum mechanically rather than classically. Namely the issue of the relation between views must be addressed within the view of one of the two observers (or of a third one). In other words, we may investigate the view of the world of $O$, as seen by $P$. Still in other words: the fact that a certain quantity $q$ has a value with respect to $O$ is a physical fact; as a physical fact, its being true, or not true, must be understood as relative to an observer, say $P$. Thus, the relation between $O$’s and $P$’s views is not absolute either, but it can be described in the framework of, say, $P$’s view.
There is an important physical reason behind this fact: It [*is*]{} possible to compare different views, but the process of comparison is always a physical interaction, and all physical interactions are quantum mechanical in nature. I think that this simple fact is forgotten in most discussions on quantum mechanics, yielding serious conceptual errors. Suppose a physical quantity $q$ has value with respect to you, as well as with respect to me. Can we compare these values? Yes we can, by communicating among us. But communication is a physical interaction and therefore is quantum mechanical. In particular, it is intrinsically probabilistic. Therefore you can inquire about the value of $q$ with respect to me, but this is (in principle) a quantum measurement as well.
Next, one must distinguish between two different questions: (i) Does $P$ “know" that $S$ “knows" the value of $q$? (ii) Does $P$ know what is the value of $q$ relative to $O$? (I know [*that*]{} you know the amount of your salary, but I do not know [*what*]{} you know about the amount of your salary).
\(i) Can $P$ “know" that $O$ has made a measurement on $S$ at time $t_2$? The answer is yes. $P$ has a full account of the events $\cal
E$. Description (2) expresses the fact that $O$ has measured $S$. The key observation is that in the state at $t_2$ in (2), the variables $q$ (with eigenstates $|1\rangle$ and $|2\rangle$) and the pointer variable (with eigenstates $|O1\rangle$ and $|O2\rangle$) are correlated. From this fact, $P$ understands that the pointer variable in $O$ has information about $q$. In fact, the state of $S-O$ is the quantum superposition of two states: in the first, ($|1\rangle\otimes |O1\rangle$), $S$ is in the $|1\rangle $ state and the hand of the observer is correctly on the ‘1’ mark. In the second, ($|2\rangle\otimes |O2\rangle$), $S$ is in the $|2\rangle$ state and the hand of the observer is, correctly again, on the ‘2’ mark. In both cases, the hand of $O$ is on the mark that correctly represents the state of the system. More formally, there is an operator $M$ on the Hilbert space of the $S-O$ system whose physical interpretation is “Is the pointer correctly correlated to $q$?” If $P$ measures $M$, then the outcome of this measurement would be yes with certainty, when the state of the $S-O$ system is as in (2). The operator $M$ is given by $$\begin{aligned}
M \ (|1\rangle \otimes |O1\rangle)& = &|1\rangle\otimes
|O1\rangle
\nonumber \\
M \ (|1\rangle\otimes |O2\rangle)& = & 0
\nonumber \\
M \ (|2\rangle\otimes |O2\rangle)& = & |2\rangle\otimes
|O2\rangle
\nonumber \\
M \ ( |2\rangle\otimes |O1\rangle ) & = & 0 \end{aligned}$$ where the eigenvalue 1 means “yes, the hand of $O$ indicates the correct state of S" and the eigenvalue 0 means “no, the hand of $O$ does not indicate the correct state of S". At time $t_2$, the $S-O$ system is in an eigenstate of $M$ with eigenvalue 1; therefore $P$ can predict with certainty that $O$ “knows” the value of $q$. Thus, it is meaningful to say that, according to the $P$ description of the events $\cal E$, $O$ “knows" the quantity $q$ of $S$, or that he “has measured" the quantity $q$ of $S$, and the pointer variable embodies the information.
A side remark is important. In general, the state of the $S-O$ system will not be an eigenvalue of $M$. In particular, the physical interaction between $S$ and $O$ which establishes the correlation will take time. Therefore the correlation between the $q$ variable of $S$ and the pointer variable of $O$ will be established gradually. Does this mean that, in $P$ views, the measurement is made “gradually”, namely that, according to $P$, $q$ will have value with respect to $O$ only partially? This is a much debated question: “Half the way through the measurement, has a measurement being done?”. By realizing that $P$’s knowledge about $O$ is also quantum mechanical, we find –I believe– the solution of the puzzle: If the state of the $S-O$ system is not an eigenstate of $M$, then, following standard quantum mechanics rule, this means that any eventual attempt of $P$ to verify whether or not a measurement has happened will have outcome “yes” or “no” with a certain respective probability. In other words: there is no half-a-measurement; there is probability one-half that the measurement has been made! We never see quantum superpositions of physical values, we only see physical values, but we can predict which one we are going to see only probabilistically. Similarly, I can say only probabilistically whether or not a physical quantity has taken value for you; but I should not say that you “half-see” a physical quantity! Thus, by representing the fact that (for $P$) “the pointer variable of $O$ has information about the $q$ variable in $S$" by means of the operator $M$ resolves the well- known and formidable problem of defining the “precise moment" in which the measurement is performed, or the precise “amount of correlation" needed for a measurement to be established –see for instance \[Bacciagaluppi and Hemmo 1995\]. Such questions are not classical questions, but quantum mechanical questions, because whether or not $O$ has measured $S$ is not an absolute property of the $S-O$ state, but a quantum property of the quantum $S-O$ system, that can be investigated by $P$, and whose yes/no answers are, in general, determined only probabilistically. In other words: [*imperfect correlation does not imply no measurement performed, but only a smaller than 1 probability that the measurement has been completed.* ]{}
A second remark in this regard is that, due to the well-known bi-orthogonal decomposition theorem, there are always correlated variables in any coupled system (in a pure state). Therefore there is always “some” operator $M$ for which the $S-O$ system is an eigenstate. Much emphasis has been given to this fact in the literature. I do not think this fact is very relevant. Imagine we have a quantum particle in a box, with a finite probability to tunnel out of it (say this models a nuclear decay). At some initial time we describe the state with a wave function concentrated in the box. At some later time a Geiger counter detects the particle outside the box, and we describe the particle as a position eigenstate at the Geiger counter position. During the time in between, we can describe the state of the particle by giving the wave form of its Schrödinger wave $\psi(x)$ as it leaks out of the box. Now, in principle, we know that there is an operator $A$ in the Hilbert space of the particle such that $\psi(x)$ is an eigenstate of $A$. Therefore we know that “some” quantity is uniquely defined at any moment. But what is the interest of such observation? Very little, I would say. $A$ will correspond to some totally uninteresting and practically non measurable quantity. Similarly, given an arbitrary state of the coupled $S-O$ system, there will always be a basis in each of the two Hilbert spaces which gives the bi-orthogonal decomposition, and therefore which defines an $M$ for which the coupled system is an eigenstate. But this is of null practical nor theoretical significance. We are interested in [*certain*]{} self-adjoint operators only, representing observables that we know how to measure; for this same reason, we are only interested in correlations between [*certain*]{} quantities: the ones we know how to measure.
The second question $P$ may ask is: (ii.) What is the outcome of the measurement performed by $O$? It is important not to confuse the statement “$P$ knows [*that*]{} $O$ knows the value of $q$" with the statement “$P$ knows [*what*]{} $O$ knows about $q$". In general, the observer $P$ does not know “what is the value of the observable $q$ that $O$ has measured" (unless $\alpha$ or $\beta$ in (2) vanish). An observer with sufficient initial information may predict which variable the other observer has measured, but not the outcome of the measurement. Communication of measurements results is however possible (and fairly common!). $P$ can measure the outcome of the measurement performed by $O$. She can, indeed, measure whether $O$ is in $|O1\rangle$, or in $|O2\rangle$.
Notice that there is a consistency condition to be fulfilled, which is the following: if $P$ knows that $O$ has measured $q$, then she measures $q$, and then she measures what $O$ has obtained in measuring $q$ (namely she measures the pointer variable), then consistency requires that the results obtained by $P$ on the $q$ variable and on the pointer variable be correlated. Indeed, they are! as was first noticed by von Neumann, and as is clear from (2). Thus, there is a satisfied consistency requirement in the notion of relative description discussed. This can be expressed in terms of standard quantum mechanical language: From the point of view of the $P$ description:
> [*The fact that the pointer variable in $O$ has information about $S$ (has measured $q$) is expressed by the existence of a correlation between the $q$ variable of $S$ and the pointer variable of $O$. The existence of this correlation is a measurable property of the $S-O$ state.*]{}
Information
-----------
It is time to introduce the main concept in terms of which I propose to interpret quantum mechanics: information.
What is the precise nature of the relation between the variable $q$ and the system $O$ expressed in the statement “$q=1$ relative to $O$"? Does this relation have a comprehensible physical meaning? Can we analyze it in physical terms? The answer has emerged in the previous subsection. Let me recapitulate the main idea: The statement “$q$ has a value relative to $O$" refers to the contingent state of the $S-O$ system. But the contingent state of the $S-O$ system has no observer-independent meaning. We can make statements about the state of the $S-O$ system only provided that we interpret these statements as relative to a third physical system $P$. Therefore, it should be possible to understand what is the physical meaning of “$q$ has a value relative to $O$” by considering the description that $P$ gives (or could give) of the $S-O$ system. This description is not in terms of classical physics, but in quantum mechanical terms; it is the one given in detail above. The result is that “$q$ has value with respect to $S$ means that there is a correlation between the variable $q$ and the pointer variable in $O$, namely that $P$ is able to predict that subsequent measurements she will make on $q$ and on the pointer variable will produce correlated outcomes.
Correlation is “information” in the sense of information theory \[Shannon 1949\]. If the state of the $S-O$ system is in an eigenstate of $M$ with eigenvalue 1, then the four possible configurations that the $q$ variable and the pointer variable can take are reduced to two. Therefore (by definition) the pointer variable has information about $q$. Let me then take a lexical move. I will from now on express the fact that $q$ has a certain value with respect to $O$ by saying: $O$ has the “information” that $q=1$.
The notion of information I employ here should not be confused with other notions of information used in other contexts. I use here a notion of information that does not require distinction between human and non-human observers, systems that understand meaning or don’t, very-complicated or simple systems, and so on. As it is well known, the problem of defining such a notion was brilliantly solved by Shannon: in the technical sense of information-theory, the amount of information is the number of the elements of a set of alternatives out of which a configuration is chosen. Information expresses the fact that a system is in a certain configuration, which is correlated to the configuration of another system (information source). The relation between this notion of information and more elaborate notions of information is given by the fact that the information-theoretical information is a minimal condition for more elaborate notions. In a physical theory it is sufficient to deal with this basic information-theoretical notion of information. This is very weak; it does not require us to consider information storage, thermodynamics, complex systems, meaning, or anything of the sort. In particular: (i.) information can be lost dynamically (correlated state may become uncorrelated); (ii.) we do not distinguish between correlation obtained on purpose and accidental correlation; Most important: (iii.) any physical system may contain information about another physical system. For instance if we have two spin-1/2 particles that have the same value of the spin in the same direction, we say that one has information about the other one. Thus observer system in this paper is any possible physical system (with more than one state). If there is any hope of understanding how a system may behave as observer without renouncing the postulate that all systems are equivalent, then the same kind of processes –“collapse”– that happens between an electron and a CERN machine, may also happen between an electron and another electron. Observers are not “physically special systems" in any sense. The relevance of information theory for understanding quantum physics has been advocated by John Wheeler \[Wheeler 1988, 1989, 1992\].
Thus, the physical nature of the relation between $S$ and $O$ expressed in the fact that $q$ has a value relative to $O$ is captured by the fact that $O$ has information (in the sense of information theory) about $q$. By “$q$ has a value relative to $O$", we mean “relative to $P$, there is a certain correlation in the $S$ and $O$ states", or, equivalently, “O has information about $q$".
Notice that this is, in a sense, only a partial answer to the question formulated at the beginning of this section. First, it is a quantum mechanical answer, because $P$’s information about the $S-O$ system is probabilistic. Second, it is an answer that only shifts the problem by one step, because the information possessed by $O$ is explained in terms of the information possessed by $P$. Thus, the notion of information I use has a double valence. On the one hand, I want to weaken all physical statements that we make: not “the spin is up", but “we have information that the spin is up" –which leaves the possibility open to the fact that somebody other has different information. Thus, [*information*]{} indicates the usual ascription of values to quantities that founds physics, but emphasizes their relational aspect. On the other hand, this ascription can be described within the theory itself, as information-theoretical [ *information*]{}, namely correlation. But such a description, in turn, is quantum mechanical and observer dependent, because a universal observer-independent description of the state of affairs of the world does not exist. Finally, there is a key irreducible distinction between $P$’s knowledge that $O$ has information about $q$ and $O$’s knowledge of $q$. Physics is the theory of the relative information that systems have about each other. This information exhausts everything we can say about the world.
At this point, the main ideas and concepts have been formulated. In the next section, I consider a certain number of postulates expressed in terms of these concepts, and derive quantum mechanics from these postulates.
On the reconstruction of Quantum Mechanics
==========================================
Basic concepts
---------------
Physics is concerned with relations between physical systems. In particular, it is concerned with the description that physical systems give of other physical systems. Following hypothesis 1, I reject any fundamental distinctions as: system/observer, quantum/ classical system, physical system/consciousness. Assume that the world can be decomposed (possibly in a variety of ways) in a collection of systems, each of which can be equivalently considered as an observing system or as an observed system. A system (observing system) may have information about another system (observed system). Information is exchanged via physical interactions. The actual process through which information is collected and perhaps stored is not of particular interest here, but can be physically described in any specific instance.
Information is a discrete quantity: there is a minimum amount of information exchangeable (a single bit, or the information that distinguishes between just two alternatives.) I will denote a process of acquisition of information (a measurement) as a “question" that a system (observing system) asks another system (observed system). Since information is discrete, any process of acquisition of information can be decomposed into acquisitions of elementary bits of information. I refer to an elementary question that collects a single bit of information as a “yes/no question", and I denote these questions as $Q_1, Q_2, \ldots $.
Any system $S$, viewed as an observed system, is characterized by the family of yes/no questions that can be asked to it. These correspond to the physical variables of classical mechanics and to the observables of conventional quantum mechanics. I denote the set of these questions as $W(S) = \{Q_i, i \in I\}$, where the index $i$ belongs to an index set $I$ characteristic of $S$. The general kinematical features of $S$ are representable as relations between the questions $Q_i$ in $W(S)$, that is, structures over $W(S)$. For instance, meaningful questions that can be asked to an electron are whether the particle is in a certain region of space, whether its spin along a certain direction is positive, and so on.
The result of a sequence of questions $( Q_1, Q_2 , Q_3, \ldots )$ to $S$, from an observer system $O$, can be represented by a string $$(e_1, e_2, e_3, \ldots )$$ where each $e_i$ is either 0 or 1 (no or yes) and represents the response of the system to the question $Q_i$. Thus the information that $O$ has about $S$ can be represented as a binary string. It is a basic fact about nature that knowledge of a portion $(e_1, \ldots ,
e_n)$ of this string provides indications about the subsequent outcomes $(e_{n+1}, e_{n+2}, \ldots )$. It is in this sense that a string (4) contains the information that $O$ has about $S$.
Trivially repeating the same question (experiment) and obtaining always the same outcome does not increase the information on $S$. The [*relevant*]{} information (from now on, simply information) that $O$ has about $S$ is defined as the non-trivial content of the (potentially infinite) string (4), that is the part of (4) relevant for predicting future answers of possible future questions. The relevant information is the subset of the string (4), obtained discarding the $e_i$’s that do not affect the outcomes of future questions.
The relation between the notions introduced and traditional notions used in quantum mechanics is transparent: A question is a version of a measurement. The idea that quantum measurements can be reduced to yes/no measurements is old. A yes/no measurement is represented by a projection operator onto a linear subset of the Hilbert space, or by the linear subset of the Hilbert space itself. Here this idea is not derived from the quantum mechanical formalism, but is justified in information-theoretical terms. The notions of observing system and observed system reflect the traditional notions of observer and system (but any system can play both roles here). $W(S)$ corresponds to the set of the observables. Recall that in algebraic approaches a system is characterized by the (algebraic) structure of the family of its observables.
A notion does not appear here: the state of the system. The absence of this notion is the prime feature of the interpretation considered here. In place of the notion of state, which refers solely to the system, the notion of the information that a system has about another system has been introduced. I view this notion very concretely: a piece of paper on which outcomes of measurements are written, hands of measuring apparatus, memory of scientists, or a two-value variable which is up or down after an interaction.
For simplicity, in the following I focus on systems that in conventional quantum mechanics are described by a finite dimensional Hilbert space. This choice simplifies the mathematical treatment of the theory, avoiding continuum spectrum and other infinitary issues.
The two main postulates
-----------------------
> [**Postulate 1**]{} (Limited information). There is a maximum amount of relevant information that can be extracted from a system.
The physical meaning of postulate 1 is that it is possible to exhaust, or give a complete description of the system. In other words, any future prediction that can be inferred about the system out of an infinite string (4), can also be inferred from a finite subset $$s = [e_1, \ldots , e_N]$$ of (4), where $N$ is a number that characterizes the system $S$. The finite string (5) represents the maximal knowledge that $O$ has about $S$.[^2] One may say that any system $S$ has a maximal “information capacity" $N$, where $N$, an amount of information, is expressed in bits. This means that $N$ bits of information exhaust everything we can say about $S$. Thus, each system is characterized by a number $N$. In terms of traditional notions, we can view $N$ as the smallest integer such that $N \geq \log_2 k$, where $k$ is the dimension of the Hilbert space of the system $S$. Recall that the outcomes of the measurement of a complete set of commuting observables, characterizes the state, and in a system described by a $k = 2^N$ dimensional Hilbert space such measurements distinguish one outcome out of $2^N$ alternative (the number of orthogonal basis vectors): this means that one gains information $N$ on the system. Postulate 1 is confirmed by our experience about the world (within the assumption above, that we restrict to finite dimensional Hilbert space systems. Generalization to infinite systems should not be difficult.)
Notice that postulate 1 already adds the Planck’s constant to classical physics. Consider a classical system described by a variable $q$ that takes bounded but continuous values; for instance, the position of a particle. Classically, the amount of information we can gather about it is infinite: we can locate its state in the system’s phase space with arbitrary precision. Quantum mechanically, this infinite localization is impossible because of postulate 1. Thus, maximum available information can localize the state only within a finite region of the phase space. Since the dimensions of the classical phase space of any system are $(L^{2}T^{-1}M)^n$, there must be a universal constant with dimension $L^{2}T^{-1}M$, that determines the minimal localizability of objects in phase space. This constant is of course Planck’s constant. Thus we can view Planck’s constant just as the transformation coefficient between physical units (position $\times$ momentum) and information theoretical units (bits).
What happens if, after having asked the $N$ questions such that the maximal information about $S$ has been gathered, the system $O$ asks a further question $Q_{N+1}$?
> [**Postulate 2**]{} (Unlimited information). It is always possible to acquire new information about a system.
If, after having gathered the maximal information about $S$, the system $O$ asks a further question $Q$, to the observed system $S$, there are two extreme possibilities: either the question $Q$ is fully determined by previous questions, or not. In the first case, no new information is gained. However, the second postulate asserts that there is always a way to acquire new information. This postulate implies therefore that the sequence of responses we obtain from observing a system cannot be fully deterministic.
The motivation for the second postulate is fully experimental. We know that all quantum systems (and all systems are quantum systems) have the property that even if we know their quantum state $|\psi\rangle$ exactly, we can still “learn" something new about them by performing a measurement of a quantity $O$ such that $|\psi\rangle$ is not an eigenstate of $O$. This is an [*experimental*]{} result about the world, coded in quantum mechanics. Postulate 2 expresses this result.
Since the amount of information that $O$ can have about $S$ is limited by postulate 1, when new information is acquired, part of the old relevant-information becomes irrelevant. In particular, if a new question $Q$ (not determined by the previous information gathered), is asked, then $O$ looses (at least) one bit of the previous information. So that, after asking the question $Q$, new information is available, but the total amount of relevant information about the system does not exceed $N$ bits.
Rather surprisingly, those two postulates are (almost) sufficient to reconstruct the full formalism of quantum mechanics. Namely, one may assert that the physical content of the general formalism of quantum mechanics is (almost) nothing but a sequence of consequences of two physical facts expressed in postulates 1 and 2. This is illustrated in the next section.
Reconstruction of the formalism, and the third postulate
--------------------------------------------------------
In this section, I discuss the possibility of deriving the formalism of quantum mechanics from the physical assertions contained in the postulates 1 and 2. This section is technical, and the uninterested reader may skip it and jump to section III.D. The technical machinery I employ has been developed (with different motivations) in quantum logic analyses. See for example \[Beltrametti and Cassinelli 1981\]. As I mentioned in the introduction, the reconstruction attempt is not fully successful. I will be forced to introduce a third postulate (besides various relative minor assumptions). I will speculate on the possibility of giving this postulate a simple physical meaning, but I do not have any clear result. This difficulty reflects parallel difficulties in the quantum logic reconstruction attempts.
Let me begin by analyzing the consequences of the first postulate. The number of questions in $W(S)$ can be much larger than $N$. Some of these questions may not be independent. In particular, one may find (experimentally) that they can be related by implication $(Q_1\Rightarrow Q_2)$, union $(Q_3=Q_1\vee Q_2)$ and intersection $(Q_3=Q_1 \wedge Q_2)$. One can define an always false $(Q_0)$ and an always true question $(Q_\infty)$, the negation of a question $(\neg
Q)$, and a notion of orthogonality as follows: if $Q_1 \Rightarrow
\neg Q_2$, then $Q_1$ and $Q_2$ are orthogonal (we indicate this as $Q_1 \bot Q_2$). Equipped with these structures, and under the (non-trivial) additional assumption that $\vee$ and $\wedge$ are defined for every pair of questions, $W(S)$ is an orthomodular lattice \[Beltrametti and Cassinelli 1981, Huges 1989\].
If there is a maximal amount of information that can be extracted from the system, we may assume that one can select in $W(S)$ an ensemble of $N$ questions $Q_i$, which we denote as $c = \{Q_i, i=1, N\}$, that are independent from each other. There is nothing canonical in this choice, so there may be many distinct families $c, b, d, ...$ of $N$ independent questions in $W(S)$. If a system $O$ asks the $N$ questions in the family $c$ to a system $S$, then the answers obtained can be represented as a string that we denote as $$s_cÊ = [e_1, ...... , e_N]_c$$ The string $s_c$ represents the information that $O$ has about $S$, as a result of the interaction that allowed it to ask the questions in $c$. The string $s_c$ Êcan take $2^N = K$ values; we denote these values as $s_c^{(1)}, s_c^{(2)}, ... , s_c^{(K)}$. So that $$\begin{aligned}
s_c^{(1)} &=& [0, 0, \ldots , 0]_c
\nonumber \\
s_c^{(2)} &=& [0, 0, \ldots , 1]_c
\nonumber \\
&\ldots&, \nonumber \\
s_c^{(K)} &=& [1, 1, \ldots , 1]_c \end{aligned}$$ Since the $2^N$ possible outcomes $s_c^{(1)},Ês_c^{(2)}, ... ,s_c^{(K)}$ of the $N$ yes/no questions are (by construction) mutually exclusive, we can define $2^N$ new questions $Q_c^{(1)}... Q_c^{(K)}$ such that the yes answer to $Q_c^{(i)}$ corresponds to the string of answers $s_c^{(i)}$: $$\begin{aligned}
Q_c^{(1)} &=& \neg Q_1 \wedge \neg Q_2 \wedge ....
\wedge
\neg Q_N \nonumber \\
Q_c^{(2)} &=& \neg Q_1 \wedge \neg Q_2 \wedge .... \wedge Q_N
\nonumber \\ &...& \nonumber \\
Q_c^{(k)} &=& Q_1 \wedge Q_2 \wedge .... \wedge
Q_N \end{aligned}$$ We refer to questions of this kind as “complete questions". By taking all possible unions of sets of complete questions $Q_c^{(i)}$ (of the same family $c$), we construct a Boolean algebra that has $Q_c^{(i)} $ as atoms.
Alternatively, the observer $O$ could use a different family of $N$ independent yes-no questions, in order to gather information about $S$. Denote an alternative set as $b$. Then, he will still have a maximal amount of relevant information about $S$ formed by an $N$-bit string $s_bÊ = [e_1, ...... , e_N]_b$. Thus, $O$ can give different kinds of descriptions of $S$, by asking different questions. Correspondingly, denote as $s_b^{(1)}... s_b^{(K)}$ the $2^N$ values that $s_b$ can take, and consider the corresponding complete questions $Q_b^{(1)}... Q_b^{(K)}$ and the Boolean algebra they generate. Thus, it follows from the first postulate that the set of the questions $W(S)$ that can be asked to a system $S$ has a natural structure of an orthomodular lattice containing subsets that form Boolean algebras. This is precisely the algebraic structure formed by the family of the linear subsets of a Hilbert space, which represent the yes/no measurements in ordinary quantum mechanics! \[Jauch 1968, Finkelstein 1969, Piron, 1972, Beltrametti and Cassinelli 1981.\]
The next question is the extent to which the information (6) about the set of questions c determines the outcome of an additional question $Q$. There are two extreme possibilities: that $Q$ is fully determined by (6), or that it is fully independent, namely that the probability of getting a yes answer is 1/2. In addition, there is a range of intermediate possibilities: The outcome of $Q$ may be determined probabilistically by $s_c$. The second postulate states explicitly that there are questions that are non-determined. Define, in general, as $p(Q, Q_c^{(i)})$ the probability that a yes answer to $Q$ will follow the string $s_c^{(i)}$. Given two complete families of information $s_c$ and $s_b$, we can then consider the probabilities[^3] $$p^{ij} = p(Q_b^{(i)}, Q_c^{(j)})$$ From the way it is defined, the $2^N \times 2^N$ matrix $p^{ij}$ cannot be fully arbitrary. First, we must have $$0 \geq p^{ij} \geq 1$$ Then, if the information $s_c^{(j)}$, is available about the system, one and only one of the outcomes $s_b^{(i)}$, may result. Therefore $$\sum_i p^{ij} = 1$$ We also assume that $p(Q_b^{(i)}, Q_c^{(j)}) = p(Q_c^{(j)},Q_b^{(i)})$ (this is a new assumption! There is a relation with time reversal, but I leave it here as an unjustified assumption at this stage), from which we must have $$\sum_j p^{ij} = 1$$ The conditions (10-11-12) are strong constraints on the matrix $p^{ij}$. They are satisfied if $$p^{ij} = | U^{ij} |^2$$ where $U$ is a unitary matrix, and $p^{ij}$ can always be written in this form for some unitary matrix $U$ (which, however, is not fully determined by $p^{ij}$).
Consider a question in the Boolean algebra generated by a family $s_c$, for instance $$Q_c^{(jk)} = Q_c^{(j)} \vee Q_c^{(k)}$$ In order to take this question into account, we cannot consider probabilities of the form $p(Q_b^{(i)},
Q_c^{(jk)})$, because a yes answer to $Q_c^{(jk)}$ is less than the maximum amount of relevant information. But we may consider probabilities of the form, say, $$p^{i(jk)i} = p(Q_b^{(i)}, Q_c^{(jk)} Q_b^{(i)})$$ defined as the probability that a yes answer to $Q_b^{(i)}$ will follow a yes answer to $Q_b^{(i)}$ ($N$ bits of information) and a subsequent yes answer to $Q_c^{(jk)}$ ($N-1$ bits of information). As is well known, we have (experimentally!) that $$\begin{aligned}
p^{i(jk)i} &\neq &
p(Q_b^{(i)}, Q_c^{(j)})\ p(Q_c^{(j)},Q_b^{(i)})
\nonumber \\
&& + p(Q_b{(i)},Q_c^{(k)})\ p(Q_c^{(k)},Q_b^{(i)}) \nonumber \\
& = & (p^{ij})^2 + (p^{ik})^2 \end{aligned}$$ Accordingly, we can determine the missing phases of $U$ in (13) by means of the correct relation, which is $$p^{i(jk)i} = | U^{ij}U^{ji} + U^{ik}U^{ki} |^2$$ It would be extremely interesting to study the constraints that the probabilistic nature of all quantities $p$ implies, and to investigate to which extent the structure of quantum mechanics can be derived in full from these constraints. One could conjecture that eqs.(13-17) could be derived solely by the properties of conditional probabilities –or find exactly the weakest formulation of the superposition principle directly in terms of probabilities: this would be a strong result. Alternatively, it would be even more interesting to investigate the extent to which the noticed consistency between different observers’ descriptions, which I believe characterizes quantum mechanics so marvelously, could be taken as the missing input for reconstructing the full formalism. I have a suspicion this could work, but have no definite result. Here, I content myself with the more modest step of introducing a third postulate. For strictly related attempts to reconstruct the quantum mechanical formalism from the algebraic structure of the measurement outcomes, see \[Mackey 1963, Maczinski 1967, Finkelstein 1969, Jauch 1968, Piron 1972\].
> [**Postulate 3**]{} (Superposition principle). If $c$ and $b$ define two complete families of questions, then the unitary matrix $U_{cb}$ in $$p( Q_c^{(i)}, Q_b^{(j)} ) = |U_{cb}^{ij}|^{2}$$ can be chosen in such a way that for every $c$, $b$ and $d$, we have $U_{cd} = U_{cb} U_{bd}$ and the effect of composite questions is given by eq.(17).
It follows that we may consider any question as a vector in a complex Hilbert space, fix a basis $|Q_c^{(i)} \rangle $ in this space and represent any other question $|Q_b^{(j)} \rangle$ as a linear combination of these: $$|Q_b^{(j)} \rangle = \sum_i U_{bc}^{ji}\ \
|Q_c^{(i)}
\rangle$$ The matrices $U_{bc}^{ij}$ are then a unitary change of basis from the $|Q_c^{(i)} \rangle$ to the $|Q_b^{(j)} \rangle$ basis. Recall now the conventional quantum mechanical probability rule: if $|v^{(i)}
\rangle$ are a set of basis vectors and $|w^{(j)} \rangle$ a second set of basis vectors related to the first ones by $$|w^{(j)} \rangle = \sum_i U^{ji}\ |v^{(i)} \rangle$$ then the probability of measuring the state $|w^{(j)} \rangle$ if the system is in the state $|w^{(i)} \rangle$ is $$p^{ij} = | \langle v^{(i)} | w^{(j)} \rangle |^2$$ (20) and (21) yield $p^{ij} = |U^{ij}|^2$, which is equation (18). Therefore the conventional formalism of quantum mechanics as well as the standard probability rules follow completely from the three postulates. The set $W(S)$ has the structure of a set of linear subspaces in the Hilbert space. For any yes/no question $Q_i$, let $L_i$ be the corresponding linear subset of $H$. The relations $\{ \Rightarrow,
\vee, \wedge, \neg, \bot \}$ between questions $Q_i$ correspond to the relations $\{$inclusion, orthogonal sum, intersection, orthogonal-complement, orthogonality$\}$ between the corresponding linear subspaces $L_i$.
The inclusion of dynamics in the above scheme is straightforward. Two questions can be considered as distinct if defined by the same operations but performed at different times. Thus, any question can be labeled by the time variable $t$, indicating the time at which it is asked: denote as $t \rightarrow Q(t)$ the one-parameter family of questions defined by the same procedure performed at different times. In this way we have naturally the Heisenberg picture. As we have seen, the set $W(S)$ has the structure of a set of linear subspaces in the Hilbert space. Assuming that time evolution is a symmetry in the theory, the set of all the questions at time $t_2$ must be isomorphic to the set of all the questions at time $t_1$. Therefore the corresponding family of linear subspaces must have the same structure; therefore there should be a unitary transformation $U( t_2 - t_1 )$ such that $$Q( t_2 ) = U( t_2 - t_1 ) Q( t_1 ) U^{-1}( t_2 - t_1 )$$ By conventional arguments, these unitary matrices form an abelian group and $U( t_2 - t_1 ) = exp\{-\imath(t_2-t_1)H\}$, where $H$ is a self-adjoint operator on the Hilbert space, the Hamiltonian. The Schrödinger equation follows immediately if we transform from the Heisenberg to the Schrödinger picture.
The observer observed
----------------------
We now have the full formal machinery of quantum mechanics, with an interpretative novelty: the absence of the notion of state the system. I now return to the issue of the relation between information of distinct observers. How can a system $P$ have information about the fact that $O$ has information about $S$? The information possessed by distinct observers cannot be compared directly. This is the key point of the construction. A statement [*about*]{} the information possessed by $O$ is a statement about the physical state of $O$; the observer $O$ is a regular physical system. Since there is no absolute meaning to the state of a system, any statement regarding the state of $O$, including the information it possess, is to be referred to some other system observing $O$. A second observer $P$ can have information about the fact that $O$ has information about $S$, but any acquisition of information implies a physical interaction. $P$ can get new information about the information that $O$ has about $S$ only by physically interacting with the $S-O$ system.
At the cost of repeating myself, let me stress again that I believe that the common mistake in analyzing measurement issues in quantum mechanics is to forget that two observers can compare their information (their measurement outcomes) only by physically interacting with each other. This means that there is no way to compare “the information possessed by $O$” with “the information possessed by $P$”, without considering a [*quantum*]{} physical interaction, or a quantum measurement, between the two.
The relation between the information possessed by distinct observers is thus given by the following: Viewed by $O$, information about $S$ is the primary concept in terms of which one describes the world; viewed by $P$, the information that $O$ has about $S$ information is just a property of some degrees of freedom in $O$ being correlated with some property of $S$. This can be taken as an additional ingredient to the structure defined by the three postulates; it ties the distinct observers to each other.
Again, the direct question “Do observers $O$ and $P$ have the same information on a system $S$?” is meaningless, because it is a question about the absolute state of $O$ and $P$. What is meaningful is to rephrase the question in terms of some observer. For instance, we could ask it in terms of the information possessed by a further observer, or by $P$ herself. Consider this last case. At time $t_1$, $O$ gets information about $S$. $P$ has information about the initial state, and therefore has the information that the measurement has been performed. The meaning of this is that she knows that the states of the $S-O$ systems are correlated, or more precisely she knows that if at a later time $t_3$ she asks a question to $S$ concerning property $A$, and a question to $O$ concerning his knowledge about $A$ (or, equivalently, concerning the position of a pointer), she will get consistent results.
From the dynamical point of view, knowledge of the structure of the family of questions $W(S)$ implies the knowledge of the dynamics of $S$ (because $W(S)$ includes all Heisenberg observables at all times). In Hilbert space terms, this means knowing the Hamiltonian of the evolution of the observed system. If $P$ knows the dynamics of the $S-O$ system, she knows the two Hamiltonians of $O$ and $S$ [ *and*]{} the interaction Hamiltonian. The interaction Hamiltonian cannot be vanishing because a measurement ($O$ measuring $S$) implies an interaction: this is the only way in which a correlation can be dynamically established. From the point of view of $P$, the measurement is therefore a fully unitary evolution, determined by a peculiar interaction Hamiltonian between $O$ and $S$. The interaction is a measurement if it brings the states (relative $P$) to a correlated configuration. On the other hand, $O$ gives a dynamical description of $S$ alone. Therefore he can only use the $S$ Hamiltonian. Since between times $t_1$ and $t_2$ the evolution of $S$ is affected by its interaction with $O$, the description of the unitary evolution of $S$ given by $O$ breaks down. The unitary evolution does not break down for mysterious physical quantum jumps, or due to unknown effects, but simply because $O$ is not giving a full dynamical description of the interaction. $O$ cannot have a full description of the interaction of $S$ with himself ($O$), because his information is correlation, and there is no meaning in being correlated with oneself.
The reader may convince himself that even if we take into account several observers observing each other, there is no way in which contradiction may develop, provided that one does not violate the two rules:
- \(i) There is no meaning to the state of a system or the information that a system has, except within the information of a further observer.
- \(ii) There is no way a system $P$ may get information about a system $O$ without physically interacting with it, and therefore without breaking down (at the time of the interaction) the unitary evolution description of $O$.
For instance, there is no way two observers $P$ and $O$ can get information about a system $S$ independently from each other: one of two (say $O$) will have to obtain the information first. In doing so, he will interact with $S$ at a certain time $t$. This interaction implies that there is a non vanishing interaction Hamiltonian between $S$ and $O$. If $P$ asks a question to $O$ at a later time $t'$, she will either have to consider the interacting correlated $S-O$ system, or to realize that the unitary evolution of the $O$ dynamics has broken down, due to the physical interaction she was not taking into account.
Critique of the concept of state
================================
“Any observation requires an observer”: summary of the ideas presented
----------------------------------------------------------------------
Let me summarize the path covered. I started from the distinction between observer and observed-system. I assumed (hypothesis 1) that all systems are equivalent, so that any observer can be described by the same physics as any other system. In particular, I assumed that an observer that measures a system can be described by quantum mechanics. I have analyzed a fixed physical sequence of events $\cal
E$, from two different points of observations, the one of the observer and the one of a third system, external to the measurement. I have concluded that two observers give different accounts of the same physical set of events (main observation).
Rather than backtracking in front of this observation, and giving up the commitment to the belief that all systems are equivalent, I have decided to take this experimental fact at its face value, and consider it as a starting point for understanding the world. [*If different observers give different descriptions of the state of the same system, this means that the notion of state is observer dependent*]{}. I have taken this deduction seriously, and have considered a conceptual scheme in which the notion of absolute observer-independent state of a system is replaced by the notion of information about a system that a physical system may possess.
I have considered three postulates that this information must satisfy, which summarize present experimental evidence about the world. The first limits the amount of relevant information that a system can have; the second summarizes the novelty revealed by the experiments from which quantum mechanics derives, by asserting that whatever the information we have about a system we can always get new information. The third limits the structure of the set of questions; this third postulate can probably be sharpened. Out of these postulates the conventional Hilbert space formalism of quantum mechanics and the corresponding rules for calculating probabilities (and therefore any other equivalent formalism) can be rederived.
A physical system is characterized by the structure on the set $W(S)$ of questions that can be asked to the system. This set has the structure of the non-Boolean algebra of a family of linear subspaces of a complex $k$-dimensional Hilbert space. The information about $S$ that any observer $O$ can possess can be represented as a string $s$, containing an amount of information $N$.
I have investigated the meaning of this information out of which the theory is constructed. I have shown that the fact that a variable in a system $O$ has information about a variable in a system $S$ means that the variables of $S$ and $O$ are correlated, meaning that a third observer $P$ has information about the coupled $S-O$ system that allows her to predict correlated outcomes between questions to $S$ and questions to $O$. Thus correlation has no absolute meaning, because states have no absolute meaning, and must be interpreted as the content of the information that a third system has about the $S-O$ couple.
Finally, since we take quantum mechanics as a complete description of the world at the present level of experimental knowledge (hypothesis 2), we are forced to accept the result that there is no objective, or more precisely observer-independent meaning to the ascription of a property to a system. Thus, the properties of the systems are to be described by an interrelated net of observations and information collected from observations. Any complex situation can be described [*in toto*]{} by a further additional observer, and the interrelation is consistent. However, such [*in toto*]{} description is deficient in two directions: upward, because an even more general observer is needed to describe the global observer itself, and –more importantly– downward, because the [*in toto*]{} observer knows the content of the information that the single component systems possess about each other only probabilistically.
There is no way to “exit" from the observer-observed global system: “Any observation requires an observer” (The expression is freely taken from \[Maturana and Varela 1920\]). In other words, I suggest that it is a matter of natural science whether or not the descriptions that different observers give of the same ensemble of events is universal or not:
> [*Quantum mechanics is the theoretical formalization of the experimental discovery that the descriptions that different observers give of the same events are not universal.*]{}
The concept that quantum mechanics forces us to give up is the concept of a description of a system independent from the observer providing such a description; that is, the concept of absolute state of a system. The structure of the classical scientific description of the world in terms of [*systems*]{} that are in certain states is perhaps incorrect, and inappropriate to describe the world beyond the $\hbar \rightarrow 0$ limit.
Relation with other interpretations
-----------------------------------
I conclude with a brief discussion on the relation between the view presented here and some popular views of quantum mechanics. I follow \[Butterfield 1995\] to organize current strategies on the quantum puzzle. The first strategy (Dynamics) is to reject the quantum postulate that an isolated system evolves according to the linear Schrödinger equation, and consider additional mechanisms that modify this evolution –in a sense physically producing the wave function collapse. Examples are the interpretations in which the measurement process is replaced by some hypothetical process that violates the linear Schrödinger equation \[Ghirardi Rimini and Weber 1986, Penrose 1989\]. These interpretation are radically different from the present approach, since they violate hypothesis 2. My effort here is not to modify quantum mechanics to make it consistent with my view of the world, but to modify my view of the world to make it consistent with quantum mechanics.[^4]
The second and third strategies maintain the idea that probabilistic expectations of values of any isolated physical system are given by the linear Schrödinger evolution. They must then face the problem of reconciling (in the example of section II.A) the probabilities expressed by the state at time $t_2$ in equation (2), –$q=1$ with probability 1/2 and $q=2$ with probability 1/2– with the assertion that the the observer $O$ assigns the value $q=1$ to the variable $q$ at the same time $t_2$. As Butterfield emphasizes, if this value assignment coexists with the probability distribution expressed by (2), then the eigenstate-eigenvalue link must be in some sense weakened, and the possibility of assigning values to variables in addition to the eigenstate case (extra values) allowed. The second and the third strategy in Butterfield’s classification differ on whether these extra values are “wholly a matter of physics" (Physics Values), or are “somehow mental or perspectival" (Perspectival Values). In the first case, the assignment is (in every sense) observer-independent. In the second case, it is (in some sense) observer-dependent.
A prime example of the second strategy (Physics Values) is Bohr’s, or Copenhagen, interpretation –at least in one possible reading. Bohr assumes a classical world. In Bohr’s view, this classical world is physically distinct from the microsystems described by quantum mechanics, and it is precisely the classical nature of the apparatus that gives measurement interactions a special status \[Bohr 1949; for a clear discussion of this point, see Landau Lifshitz 1977\].
Within the point of view developed in this paper, one can fix once and for all a privileged system $S_o$ as “The Observer" (capitalized). This system $S_o$ can be formed for instance by all the macroscopic objects around us. In this way we recover Bohr’s view. The quantum mechanical “state" of a system $S$ is then the information that the privileged system $S_o$ has about $S$. Bohr’s choice is simply the assumption of a set of systems (the classical systems) as privileged observers. This is consistent with the view presented here.[^5] By taking Bohr’s step, one becomes blind to the net of interrelations that are at the foundation of the theory, and puzzled about the fact that the theory treats one system, $S_o$, the classical world, in a way which is different from the other systems. The disturbing aspect of Bohr’s view is the inapplicability of quantum theory to macrophysics. This disturbing aspect vanishes, I believe, at the light of the discussion in this paper.
Therefore, the considerations in this paper do not suggest any modification to the conventional [*use*]{} of quantum mechanics: there is nothing incorrect in fixing the preferred observer $S_o$ once and for all. If we adopt the point of view suggested here, we continue to use quantum mechanics precisely as is it is currently used. On the other hand, this point of view (I hope) brings clarity about the physical significance of the strange theoretical procedure adopted in Bohr’s quantum mechanics: treating a portion of the world in a different manner than the rest of it. This different treatment is, I believe, the origin of the unease with quantum mechanics.
The strident aspect of Bohr’s quantum mechanics is cleanly characterized by von Neumann’s introduction of the “projection postulate", according to which systems have two different kinds of evolutions: the unitary and deterministic Schrödinger evolution, and the instantaneous, probabilistic measurement collapse \[von Neumann 1932\]. According to the point of view described here, the Schrödinger unitary evolution of the system $S$ breaks down simply because the system interacts with something which is not taken into account by the evolution equations. Unitary evolution requires the system to be isolated, which is exactly what ceases to be true during the measurement, because of the interaction with the observer. If we include the observer into the system, then the evolution is still unitary, but we are now dealing with the description of a different observer. As suggested by Ashtekar, the point of view presented here can then be described as a fundamental assumption prohibiting an observer to be able to give a full description of “itself" \[Ashtekar 1993\]. In this respect, these ideas are related to earlier suggestions that quantum mechanics is a theory that necessarily excludes the observer \[Peres and Zurek 1982, Roessler 1987, Finkelstein 1988, Primas 1990\]. A recent result in this regard is a general theorem proven by Breuer \[Breuer 1994\], according to which no system (quantum nor classical) can perform a complete self-measurement. The relation between the point of view presented here and Breuer’s result deserves to be explored.
Other views within Butterfield’s second strategy (Physical Values) are Bohm’s hidden variables theory, which violates hypothesis 2 (completeness), and modal interpretations, which deny the collapse but assume the existence of physical quantities’ values. Of these, I am familiar with \[van Fraassen 1991\], or the idea of actualization of potentialities in \[Shimony 1969, Fleming 1992\]. The assumed values must be consistent with the standard theory’s predictions, be probabilistically determined by a unitary evolving wave function, but they are not constrained by the eigenstate-eigenvalue link. One may doubt these acrobatics could work \[See Bacciagaluppi 1995, Bacciagaluppi and Hemmo 1995\]. I am very sympathetic with the idea that the object of quantum mechanics is a set of quantities’ values and their distribution. Here, I have assumed value assignment as in these interpretations, but with two crucial differences. First, this value assignment is observer-dependent. Second, it need not be consistent with a unitary Schrödinger evolution, because the evolution is not unitary when the observed system interacts with the observer. Namely, there is collapse in each observer-dependent evolution of expected probabilities. These two differences allow values to be assigned to physical quantities without any of the consistency worries that plague modal interpretations. The point is that the break of the eigenstate-eigenvalue link is bypassed by that fact that the eigenvalue refers to one observer, and the state to a different observer. For a fixed observer, the eigenstate-eigenvalue link is maintained. Consistency should only be recovered between different observers, but consistency is only quantum mechanical as discussed in section III.D. Actuality is observer dependent. The fact that the values of physical quantities are relational and their consistency is only probabilistically required circumvents the potential difficulties of the modal interpretations.
A class of interpretations of quantum mechanics that Butterfield does not include in his classification, but which are presently very popular among physicists, is the consistent histories (CH) interpretations \[Griffiths 1984, Omnes 1988, Gell-Mann and Hartle 1990\]. These interpretations reduce the description of a system to the prediction of temporal sequences of values of physical variables. The key novelties are three: (i) probabilities are assigned to sequences of values, as opposed to single values; (ii) only certain sequences can be considered; (iii) probability is interpreted as probability of the given sequence of values within a chosen family of sequences, or framework. The restriction (ii) incorporates the quantum mechanical prohibition of giving value, say, to position and momentum at the same time. More precisely, in combination with (iii) it excludes all the instances in which observable interference effects would make probability assignments inconsistent. In a sense, CH represent a sophisticated implementation of the program of discovering a minimum consistent value attribution scheme. The price paid for consistency is that a single value attribution is meaningless: whether or not a variable has a value may very well depend on whether we are asking or not if at a later time another variable has a value.
There is a key subtlety in the CH scheme that has rarely being emphasized[^6]: this is the distinction between “properties that hold with probability one” and “data”. The interesting question to ask is: What is it that determines which frameworks are allowed? The standard answer in CH is that we have certain information on the system, call it [*data*]{}. In particular, we know that certain physical quantities of the system have certain values. The fact that certain quantities have certain values determines which framework are allowed to describe a phenomenon. Thus: [*we rely on quantities having values (data) for selecting the allowed framework*]{}. Then we derive probabilistic predictions about value attribution. These probabilistic predictions are framework dependent. It is often stated that in CH [*all*]{} value attributions are framework dependent; this is the way Nature is. [*But if all value attributions are framework dependent, what are the data?*]{} In CH there seem to be two distinct kind of value attributions: a weak value attribution: the framework dependent values, and a strong value attribution: the data, or facts. In CH the focus is totally on predictions about weak value attributions. But everyday life and scientific practice are about (possibly probabilistic) predictions of facts, namely facts that could be later used as data. To put it pictorially (and a bit imprecisely): I do not care about a science that tells me that my airplane will not crash “in one framework”; I want a science that will tell me that my airplane will just not crash!
Consider the following situation (situation A): a set (D) of data on a system is given. From these data, it follows that there is a framework (call it F1) in which the question “Is $Q$ equal to 1 at time t?” can be asked, and the answer is yes with probability one. Then consider the following other situation (situation B): we have a set of data on the system, which consist in the set (D) [*plus the data that $Q=1$ at time t*]{}. Is situation A physically the same as situation B in the consistent histories approach? The answer is no. In fact, situation A still allows for $Q=2$ (strange but true) in a different framework, say F2, while situation B is incompatible with $Q=2$, because F2 is not an admitted framework. [*Therefore there should exist a physical way in which we pass from situation A to situation B*]{}. Namely there should be way for a probability 1 prediction (in a framework) to become a data. The key question that, as far as I can see, the consistent histories approaches does not address is: how do we concretely pass (in a lab) from situation A to situation B? What is it that transforms a probability-one event (framework dependent) into something that we can use as data (framework independent)? If the transformation of a “probability-one-in-a-framework" situation into [*data* ]{} is an actual occurrence in Nature, then I believe CH fails to tell me how this can happen (when does a framework becomes realized as data). If, on the other hand, what is data for me may fail to be data for somebody other, then one falls precisely into the scheme presented in this paper.
In the Copenhagen view, it is the interaction with a classical object that actualizes properties. A different solution has been suggested in this paper: interaction with any object, but then actualization of properties is only relative to that object. I do not see the solution of this problem within the history views. This is not to say that there is anything wrong in the CH approaches. To the opposite, I believe that the CH views are correct and precise. Still, there is a question that they leave open: the physical meaning of the framework dependence of the value assignments; more precisely, the understanding of how there can be facts, or data, if property ascriptions are only framework dependent. I think that the answer is simply that there are no (observer independent) data at all: the data that I have, and therefore the family of frameworks that I can use is different from your data, and therefore the family of framework that you can use. The histories interpretations are not inconsistent with the analysis developed here. What I try to add here is attention to the process through which the observer-independent, but framework-dependent probabilities attached to histories, may be related to actual observer-dependent descriptions of the facts of the world.
Finally, let me come to the third strategy (Perspectival Values), whose prime example is the many worlds interpretation \[Everett 1957, Wheeler 1957, DeWitt 1970\], and its variants. If the “branching" of the wave function in the many worlds interpretation is considered as a physical process, it raises the very same sort of difficulties as the von Neumann “collapse" does. When does it happen? Which systems are measuring systems that make the world branch? These difficulties of the many-world interpretation have been discussed in the literature \[See Earman 1986\]. Alternatively, we may forget branching as a physical process, and keep evolving the wave function under unitary evolution. The problem is then to interpret the observation of the “internal" observers. As discussed in \[Butterfield 1985\] and \[Albert 1992\], this can be done by giving preferred status to special observers (apparatus) whose values determine a (perspectival) branching. See Objection 7 in section II.B. A variant is to take brains –“Minds"– as the preferred systems that determine this perspectival branching, and thus whose state determines the new “dimension" of indexicality. Preferred apparatus, or bringing Minds into the game, violates hypothesis 1.
There is a way of having (perspectival) branching keeping all systems on the same footing: the way followed in this paper, namely to assume that all values assignments are completely relational, not just relational with respect to apparatus or Minds. Notice, however, that from this perspective Everett’s wave function is a very misleading notion, not only because it represents the perspective of a non-existent observer, but because it even fails to contain any relevant information about the values observed by each single observer! There is no description of the universe [*in-toto*]{}, only a quantum-interrelated net of partial descriptions.
With respect to Butterfield’s classification, the interpretation proposed here is thus in the second, as well as in the third, group: the extra values assigned are “somehow perspectival" (but definitely not mental!), in that they are observer-dependent, but at the same time “wholly a matter of physics", in the sense in which the “perspectival" aspect of simultaneity is “wholly a matter of physics" in relativity. In one word: value assignment in a measurement is not inconsistent with unitary evolution of the apparatus+system ensemble, because value assignment refers to the properties of the system with respect to the apparatus, while the unitary evolution refers to properties with respect to an external system.
From the point of view discussed here, Bohr’s interpretation, consistent histories interpretations, as well as the many worlds interpretation, are all correct. The point of view closest to the one presented here is perhaps Heisenberg’s. Heisenberg’s insistence on the fact that the lesson to be taken from the atomic experiments is that we should stop thinking of the “state of the system", has been obscured by the subsequent terse definition of the theory in terms of states given by Dirac. Here, I have taken Heisenberg’s lesson to extreme consequences.[^7]
Crane is developing a point of view similar to the one discussed here and has attempted an ambitious extension of these ideas to the cosmological general-covariant gravitational case \[Crane 1995\]. It was recently brought to my attention that Zurek ends his paper \[Zurek 1982\] with conclusions that are identical to the ones developed here: “Properties of quantum systems have no absolute meaning. Rather, they must be always characterized with respect to other physical systems" and “correlations between the properties of quantum systems are more basic that the properties themselves" \[Zurek 1982\]. Finally, Rob Clifton has brought to my attention an unpublished preprint by Kochen \[Kochen 1979\], with ideas extremely similar to the ones presented here.
.5cm
[**Acknowledgments**]{} .5cm
The ideas in this work emerged from: i. Conversations with Abhay Ashtekar, Julian Barbour, Alain Connes, Jürgen Ehlers, Brigitte Faulknburg, Gordon Flemming, Jonathan Halliwell, Jim Hartle, Chris Isham, Al Janis, Ted Newman, Roger Penrose, Lee Smolin, John Wheeler and HD Zeh; ii. A seminar run by Bob Griffiths at Carnegie Mellon University (1993); iii. A seminar run by John Earman at Pittsburgh University (1992); iv. Louis Crane’s ideas on quantum cosmology; v. The teachings of Paola Cesari on the importance of taking the observer into account. It is a pleasure to thank them all. I also thank Gordon Belot, Jeremy Butterfield, Bob Clifton, John Earman, Simon Saunders and Euan Squires for discussions and comments on the first version of this work.
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[^1]: To counter objections based on instinct alone, it is perhaps worthwhile recalling the great resistance that the idea of fully relational notions of “rest" and “motion" encountered at the beginning of the scientific revolution. I think that quantum mechanics (and general relativity) could well be in the course of triggering a –not yet developed– revision of world views as far reaching as the seventeenth century’s one (on this, see \[Rovelli 1995\]).
[^2]: The string (5) is essentially the state. The novelty here is not the fact that the state is defined as the response of the system to a set of yes/no experiments: this is the traditional reading of the state as a preparation procedure. The novelty is that this notion of state is relative to the observer that has asked the questions.
[^3]: I do not wish to enter here the debate on the meaning of probability in quantum mechanics. I think that the shift of perspective I am suggesting is meaningful in the framework of an objective definition of probability, tied to the notion of repeated measurements, as well as in the context of subjective probability, or any variant of this, if one does not accept Jayne’s criticisms of the last.
[^4]: Note added. I have recently become aware of an idea to circumvent this problem by exploiting the infinite-number-of-degrees-of-freedom nature of the observing system. This could generate an apparently non-linear evolution from conventional Schrödinger evolution, via a symmetry-breaking instability generating effective superselection rules. See in particular \[Jona-Lasinio [*et. al.*]{} 81, 86\] and \[Wightman 95\].
[^5]: A separate problem is why the observing system chosen –$S_o$, or the macroscopic world– admits, in turn, a description in which expectations probabilities evolve classically, namely are virtually always concentrated on values 0 and 1, and interference terms are invisible. It is to this question that the physical decoherence mechanism \[Joos Zeh 1985, Zurek 1981\] provides an answer. Namely, [*after*]{} having an answer on what determines extra- values ascriptions (the observer-observed structure, in the view proposed here), the physical decoherence mechanism helps explaining why those ascriptions are consistent with classical physics in macroscopic systems.
[^6]: The only discussion on this point I heard was by Isham.
[^7]: With a large number of exceptions, most physicists hold some version of naive realism, or some version of naive empiricism. I am aware of the “philosophical qualm" that the ideas presented here may then generate. The conventional reply, which I reiterate, is that Galileo’s relational notion of velocity used to produce analogous qualms, and that physics seems to have the remarkable capacity of challenging even its own conceptual premises, in the course of its evolution. Historically, the discovery of quantum mechanics has had a strong impact on the philosophical credo of many physicists, as well as on part of contemporary philosophy. It is possible that this process is not concluded. But I certainly do not want to venture into philosophical terrains, and I leave this aspect of the discussion to competent thinkers. Just a few observations: The relational aspect of knowledge is one of the themes around which large part of western philosophy has developed. In Kantian terms, only to mention a characteristic voice, any phenomenal substance which may be object of possible experience is “entirely made up of mere relations" \[Kant 1787\]. In recent years, the idea that the notion of observer- independent description of a system is meaningless has become almost a commonplace in disparate areas of the contemporary culture, from anthropology to certain biology and neuro-physiology, from the post-neopositivist tradition to (much more radically) continental philosophy \[Gadamer 1989\], all the way to theoretical physical education \[Bragagnolo Cesari and Facci 1993\]. I find the fact that quantum mechanics, which has directly contributed to inspire many of these views, has then remained unconnected to these conceptual development, quite curious.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present a series of observations of the nearby obscured Seyfert galaxies ESO103-G35, IC5063, NGC4507 and NGC7172. The period of monitoring ranges from seven days for NGC7172 up to about seven months for ESO103-G035. The spectra of all galaxies are well fit with a highly obscured ($N_H>10^{23}$ ) power-law and an Fe line at 6.4 keV. We find strong evidence for the presence of a reflection component in ESO103-G35 and NGC4507. The observed flux presents strong variability on day timescales in all objects. Spectral variability is also detected in the sense that the spectrum steepens with increasing flux similar to the behaviour witnessed in some Seyfert-1 galaxies.'
author:
- |
I. Georgantopoulos$^1$, I. E. Papadakis $^2$\
$^1$ Institute of Astronomy & Astrophysics, National Observatory of Athens, Lofos Koufou, Palaia Penteli, 15236, Athens, Greece\
$^2$ Physics Department, University of Crete, 71003, Heraklion, Greece\
title: '**[*RXTE*]{} observations of Seyfert-2 galaxies: evidence for spectral variability**'
---
2[$\chi^{2}$]{} 2[$\chi^{2}$]{}
= ==1=1=0pt =2=2=0pt =2=2=0pt
. galaxies:active – X-ray:galaxies – galaxies:Seyfert – galaxies:individual: ESO103-G35; IC5063; NGC4507; NGC7172
INTRODUCTION
============
Recent X-ray missions have brought a rapid progress in our understanding of the X-ray properties of Seyfert galaxies (for a review see Mushotzky, Done & Pounds 1993). The X-ray spectra of Seyfert 2 galaxies as observed by , and recently [*BeppoSAX*]{} have proved to be very complex (Smith & Done 1996; Turner et al. 1997). In broad terms most Seyfert 2 X-ray spectra can be well fitted by a power-law continuum (typically $\Gamma \sim 1.8$), plus an Fe-K emission line at 6.4 keV and a reflection component (e.g. Lightman & White 1988, George & Fabian 1991). This latter component, which suggests the presence of a large amount of cold material in the vicinity of the nucleus, flattens the observed continuum and can dominate the spectrum above $\sim 10$ keV. In most Seyfert 2s the above emission components are viewed through a large absorbing column density, typically $N_H>10^{23}$ . This screen, which suppresses the soft X-ray emission through the process of photoelectric absorption in cool atomic and molecular gas, is possibly associated with a large (pc scale) molecular torus. In the limit of very high column densities, Thomson scattering will also diminish the more penetrating hard X-ray emission. In some sources additional emission in the form of a soft X-ray excess is observed below $\sim 3$ keV probably as a result of scattering of the intrinsic power-law continuum by a strongly photoionised medium (eg Griffiths et al. 1998 in the case of Markarian 3). In order to observe such soft X-ray emission it is clearly a requirement that the scattering medium should extend beyond the bounds of the obscuration of the molecular torus. The X-ray spectra of Seyfert-2 galaxies gave wide support to the standard AGN unification model (Antonucci & Miller 1985). According to this paradigm, the nucleus (supermassive black hole, accretion disc and broad line region) has basically the same structure in both type 1 and type 2 Seyferts, but depending on the circumstances, can be hidden from view by the molecular torus (Krolik & Begelman 1986). Specifically if the source is observed at a sufficiently high inclination angle, and thus the line of sight intersects the torus, it would be classified as a Seyfert 2, whereas for all other orientations it would be deemed to be a Seyfert 1.
In contrast to the recent progress in understanding the X-ray spectral characteristics of Seyfert 2 galaxies, our knowledge of their X-ray variability properties remains limited. According to the standard unification scenario, the hard X-ray continuum should vary with large amplitude in a similar way to that observed in Seyfert 1 galaxies (Mushotzky, Done & Pounds 1993). However, in Seyfert-2s a large fraction of the emission must come from reprocessed radiation. Since a large fraction of the Fe-K line, the reflection emission and the soft excess are likely to originate from regions of parsec scale-size, it follows that significantly less variability might be expected in Seyfert 2 objects, at least in those parts of the spectrum where the reprocessing makes a substantial contribution to the overall flux. Time variability studies can provide strong constraints on the geometry of the nucleus and the surounding region. For example, the time lag between the variation of the power-law continuum and the Fe line flux or the reflection component would give information on the location and the size of the reflecting material. In particular if a large amount of the Fe line originates from the accretion disk, instead of the torus, we would expect variations of the line flux in timescales of days.
Here, we present the results from several archival observations on four well known Seyfert-2 galaxies: ESO103-G35, IC5063, NGC4507 and NGC7172. In another paper (Georgantopoulos et al. 1999) we presented monitoring observations of Markarian 3. The extended energy range of the [*RXTE*]{} PCA detectors and the large number of observations for each object give us the opportunity to investigate in detail the properties of the spectral components present in these galaxies such as the intrinsic power-law and the reflection component. Our main objective is to use the variability exhibited in the 3–24 keV band to place constraints on the geometry of the nucleus and its circumnuclear matter.
THE SAMPLE
==========
ESO 103-G35
-----------
The HEAO-1 hard X-ray source 1H 1832-653 has been identified with the galaxy ESO 103-G35 at a redshift z=0.013. Optical spectroscopy revealed a high excitation forbidden line spectrum with weak broad emission line wings and therefore classified this object as a Seyfert-1.9 galaxy (Phillips et al. 1979). obtained the first X-ray spectrum showing a power-law spectrum absorbed by a column of $N_H\sim 10^{23}$ (Warwick, Pounds & Turner 1988). The observations also showed a variation in the column density over a period of 90 days. Smith & Done (1996) present observations detecting an Fe line around 6.4 keV with an equivalent width of $\sim$ 350 eV. The spectrum of the source could be characterised by a heavily absorbed power-law continuum of slope $\Gamma \sim 1.8$ and $N_{H}\sim 2\times 10^{23}$ . Smith & Done (1996) also found evidence for the presence of a reflection component. Turner et al. (1997) present the first data for this object. They resolve the Fe line into possibly three components (6.4, 6.68 and 6.96 keV). Recently, Forster, Leighly & Kay (1999) presented three observations separated by $\sim$ 2 years. They find some marginal evidence for a double Fe line in their first observation. Their best fit $\Gamma$ and $N_{H}$ values agree with the results. They also find that the power-law continuum flux increased by a factor of two while the Fe line equivalent width has decreased by about a similar amount. Two observations of ESO103-G35 with [*BeppoSAX*]{} separated by a year, from $\sim$2 to 60 keV (Akylas et al. 2000), show a variation of the power-law flux by a factor of two, no variation of the Fe line flux while there is no strong evidence for the presence of a reflection component. Comparison of the historical data (Poletta et al. 1996) from HEAO-1 up to ASCA show a variation of the flux of a factor of about four.
IC5063
------
IC5063 is an S0 galaxy (z=0.011) presenting a typical Seyfert-2 galaxy spectrum (Colina, Sparks & Macchetto 1991). Scattered broad $H_\alpha$ emission has been detected by Inglis et al. (1993). [*HST*]{} NICMOS observations show a very red unresolved point source (Kulkarni et al. 1998). IC5063 has a radio power at 1.4 GHz two orders of magnitude greater than that of typical Seyfert galaxies and thus it can be classified as a Narrow-Line Radio galaxy as well (Ulvestad & Wilson 1984). A observation showed a power-law photon index of $\Gamma=1.5$ and $N_{H}\sim 2.5\times 10^{23}$ (Smith & Done 1996). Turner et al. (1997) presented two observations of IC5063. The spectrum is in good agreement with the observations. IC5063 shows short-term X-ray flux variability at the 90 per cent confidence level (Turner et al. 1997). However, the and the observation obtained about four years later give similar fluxes.
NGC4507
-------
NGC4507 is a nearby (z=0.012) barred spiral galaxy. Optical spectra present high excitation, narrow emission lines classifying it as a Seyfert-2 galaxy (Durret & Bergeron 1986). observations showed a flat X-ray spectrum ($\Gamma \sim 1.3$), with $N_{H}\sim 4\times
10^{23}$ and a strong Fe line (equivalent width, EW $\sim 800$ eV) (Smith & Done 1996). Instead, OSSE observations (Bassani et al. 1995) showed a steeper photon index ($\Gamma\sim 2.1 \pm 0.3$) in the 50-200 keV energy range suggesting the presence of a strong reflection component. observations (Turner et al. 1997, Comastri et al. 1998) showed a flat power-law index, with a possible variation of the iron line intensity and the absorption column density. The limited spectral bandpass of did not allow to constrain the properties of a reflection component. Comparison between the and observations show a flux variability of a factor of about two.
NGC7172
-------
NGC7172 is an S0 galaxy (z=0.0087) belonging to a compact group of galaxies (Hickson90). It is classified as a Seyfert-2 galaxy on the basis of its optical spectrum (Sharples et al. 1984). obtained the first X-ray spectrum of the source yielding a photon index of $\Gamma\sim 1.8$ (Turner & Pounds 1989). The same photon index value was also found in a observation of the source (Smith & Done 1996). The data analysis yielded a column density of $\sim 1\times
10^{23}$ , and a rather weak Fe line (EW $\sim 50$ eV). Turner et al. (1997) find an EW of $\sim 80$ eV. The data as well as combination with [*GRO/OSSE*]{} observations (Ryde et al. 1997) show a flat spectrum of $\Gamma\sim 1.5$ and therefore provided evidence for spectral variability between the and epoch. The 2-10 keV flux has been fairly constant during the 1977-1989 period at a level of about $3-4\times
10^{-11}$ . However, Guainazzi et al. (1998) presented evidence for significant short term (hours) and long term (months) variability using two observations separated by a year. They find a flux decrease by a factor of 3 over a year. The Fe line flux decreased by a similar amount providing important constraints for the size of the reflecting material.
DATA ANALYSIS
=============
In this work, we present the results from the analysis of the PCA (Proportional Counter Array) data only. The PCA consists of five collimated (1$^{\circ}$ FWHM) Xenon proportional counter units (PCU). The PCU are sensitive to energies between 2 and 60 keV. However, the effective area drops very rapidly below 3 and above 20 keV. The energy resolution is 18 per cent at 6.6 keV (Glasser, Odell & Seufert 1994). The collecting area of each PCU is 1300 $\rm cm^2$. We use only 3 PCUs (0 to 2); the data from the other two PCU were discarded as these detectors were turned off on some occasions. We extracted PCU light curves and spectra from only the top Xenon layer in order to maximize the signal-to-noise ratio. The data were selected using standard screening criteria: we exclude data taken at an Earth elevation angle of less than 10$^{\circ}$, pointing offset larger than 0.02$^{\circ}$, during and 30 minutes after the satellite passage through the South Atlantic Anomaly (SAA), and when the particle counts [electron0,1,2 ]{} are higher than 0.1.
We use the [PCABACKEST v2]{} routine of [FTOOLS v 4.1.1]{} to generate the background models which take into account both the cosmic and the internal background. The internal background is estimated by matching the conditions of the observations with those in various model files. Most of the internal background is correlated with the L7 rate, the sum of 7 rates from pairs of adjacent anode wires. However, there is a residual background component correlated with recent passages from the SAA. Therefore, the use of a second, activation component is also employed. Comparison with blank fields observations shows that the above background model represents the real background accurately (within 1.5 per cent in the 3-20 keV band).
Due to the large field-of-view of the PCA detector, contamination of the observed spectrum by nearby sources is likely. We have therefore checked the GIS 2-10 keV images of our sources. These show no nearby sources (the GIS field-of-view is 40 arcmin diameter). However, Turner et al. (1997) point out that even the spectrum of NGC4507 may be somewhat contaminated by an adjacent point source (present in the soft PSPC 0.1-2 keV band). This source cannot be resolved within the limited GIS spatial resolution (about 3 arcmin half-power-diameter). Possible contamination at radii larger than the GIS field-of-view can be investigated using the PSPC images of our sources. Indeed, in NGC7172 and ESO103-G35 we find bright sources at distances from 30 to 60 arcmin from our targets. Inspection of the NASA extragalactic database (NED) reveals no coincidences. Therefore, these are most probably Galactic stellar sources and thus with soft X-ray spectra whose contamination in the band is expected to be small.
THE OBSERVATIONS
================
In total 43 observations with exposure time of 151.0, 46.3, 118.6, and 78.2 ksec have been obtained with for ESO103-G35, IC5063, NGC4507 and NGC7172 respectively. The period of observations ranges from about 7 days for NGC7172, and 15 days for NGC4507, up to about 5 months for IC5063 and 7 months for ESO103-G35. The observation dates for each dataset together with the exposure time as well as the observed background-subtracted count rate in the full 2-60 keV PCA energy band are given in Tables 1 to 4.
Date Exp (ksec) Ctr
------------- ------------ ----------------
11-Apr-1997 14.4 15.38$\pm$0.11
12-Apr-1997 10.7 14.94$\pm$0.11
12-Apr-1997 15.7 14.68$\pm$0.10
13-Apr-1997 6.3 15.76$\pm$0.15
18-Jul-1997 6.2 12.08$\pm$0.16
20-Jul-1997 8.6 13.10$\pm$0.14
21-Jul-1997 11.7 14.12$\pm$0.11
22-Jul-1997 11.4 15.92$\pm$0.12
23-Jul-1997 10.4 15.36$\pm$0.11
24-Jul-1997 11.5 15.32$\pm$0.11
26-Jul-1997 12.4 14.30$\pm$0.11
13-Nov-1997 1.9 13.69$\pm$0.27
14-Nov-1997 8.7 12.84$\pm$0.13
14-Nov-1997 19.4 13.20$\pm$0.09
14-Nov-1997 1.4 16.10$\pm$0.32
: The ESO103-G35 observations[]{data-label="eso103_obsn"}
Date Exp (ksec) Ctr
------------- ------------ ----------------
22-Jul-1996 18.0 6.34$\pm$0.07
01-Sep-1996 9.8 4.45$\pm$0.10
01-Sep-1996 1.4 3.42$\pm$ 0.24
19-Dec-1996 11.1 4.61$\pm$0.09
20-Dec-1996 6.0 4.73$\pm$0.12
: The IC5063 observations[]{data-label="ic5063_obsn"}
Date Exp (ksec) Ctr
------------- ------------ ---------------
24-Feb-1996 12.6 7.26$\pm$0.08
24-Feb-1996 3.3 6.96$\pm$0.16
26-Feb-1996 12.7 8.33$\pm$0.08
26-Feb-1996 14.0 6.99$\pm$0.07
27-Feb-1996 3.6 7.46$\pm$0.15
28-Feb-1996 14.5 6.75$\pm$0.08
28-Feb-1996 15.2 5.39$\pm$0.09
29-Feb-1996 0.6 5.83$\pm$0.35
29-Feb-1996 3.2 6.84$\pm$0.07
29-Feb-1996 3.2 6.73$\pm$0.16
03-Mar-1996 12.5 7.70$\pm$0.08
07-Mar-1996 16.3 9.58$\pm$0.07
07-Mar-1996 0.8 9.04$\pm$0.32
10-Mar-1996 6.1 7.69$\pm$0.11
: The NGC4507 observations[]{data-label="ngc4507_obsn"}
Date Exp (ksec) Ctr
------------- ------------ ----------------
18-Dec-1996 12.8 7.32$\pm$0.09
18-Dec-1996 4.4 5.42$\pm$0.14
18-Dec-1996 5.5 7.61$\pm$0.12
19-Dec-1996 10.0 8.13$\pm$0.10
19-Dec-1996 3.2 8.38$\pm$0.17
23-Dec-1996 14.3 6.12$\pm$0.08
23-Dec-1996 9.5 4.84$\pm$0.09
25-Dec-1996 11.7 2.25$\pm$ 0.09
25-Dec-1996 6.6 2.47$\pm$ 0.11
: The NGC7172 observations[]{data-label="ngc7172_obsn"}
In Fig. \[lcurve\] we present the 2-10 keV light curves for all sources using a bin size of 25ksec (time in this figure is measured from the beginning of the first observation of each source). The light curves show variations on all sampled timescales (days/months). For example, ESO103-G35 shows a $\sim 30$ per cent variation within a few days, while IC5063 shows a decline by a factor of $\sim 2$ within 4 months. The maximum amplitude of the observed variability is $\sim 3$ in the case of NGC7172, $\sim 2$ for NGC4507 and IC5063, and $\sim 60$ per cent in the case of ESO103-G35 light curve. Consequently, on timescales larger than $\sim 1$ day, these four Seyfert 2 galaxies show $2-10$ keV flux variability similar to that observed in Seyfert 1. In fact, the light curves show significant variations on time scales as short as $\sim$ a few hundred seconds as well (these results will be presented elsewhere). This is not an unexpected result, since for these Seyfert 2 galaxies we believe that we can detect the nuclear emission above a few keV directly.
{height="10.0cm"}
Hardness Ratios
---------------
Next, we investigate if there is any spectral variability during the flux variations. For this reason we extracted light curves at the following energy bands: $2-5$, $5-7$, $7-10$ and $10-20$ keV, using a bin size of 5400 sec (this is roughly equal to the orbital period of ). Variations of the emission/absorption components in the spectrum of Seyfert 2 galaxies are expected to contribute in a different way in these energy bands. For example, any absorbing column density changes around $1\times
10^{23}$ should affect mainly the $2-5$ keV band, while the Fe-K emission line and the reflection component variations should affect the $5-7$ keV and $10-20$ keV bands respectively. On the other hand, the $7-10$ keV band light curve should be representative of the power-law continuum variations only. Using these light curves we calculated three hardness ratios: $HR1=CR_{7-10keV}/CR_{2-5keV},
HR2=CR_{10-20keV}/CR_{7-10keV}$, and $HR3=CR_{7-10keV}/CR_{5-7keV}$, where $CR_{E1-E2}$ is the count rate of the $E1-E2$ band light curve. Under the hypothesis that there are no spectral changes, the $HR$s should remain constant.
Fig. \[ratio\] shows the $HR$ plot for the sources as a function of the normalised $CR_{7-10keV}$ (the normalisation is necessary in order to compare the sources since they have different luminosities). NGC7172 shows the largest amplitude variations in the $7-10$ keV band. Within the common range of the normalised $CR_{7-10keV}$ values, the $HR$s have similar values for all objects except for NGC4507 (open diamonds in Fig. \[ratio\]). Both $HR2$ and $HR3$ for this source have a large value. In fact we could not calculate $HR1$ as its $2-5$ keV light curve has a mean value of almost zero. The similarity of the $HR$ values for ESO103-G35, IC5063 and NGC7172 suggests that they have similar spectra, while the NGC4507 spectrum appears to be much “harder". This is probably due to the fact that NGC4507 has a larger absorbing column density, which reduces mainly the count rate in the “soft" bands, and hence results in larger $HR$ values.
In order to investigate the presence of spectral variations we fitted the data for each galaxy with a model of the form: $\log(HR)=a+b*log(CR_{7-10 keV})$ (in fitting the data we took account of the errors in both $HR$ and $CR_{7-10 keV}$). In all cases, a line in the log-log space fits the data well (ie the best fit 2 values can be accepted with confidence larger than $5$ per cent). If the $HR$ values remain constant during the flux variations, we expect $b$ (ie the slope of the line) to be consistent with zero. We discuss below the best fit results for each galaxy.
ESO103-G5 shows no $HR1$ variations ($b_{HR1}=0.05\pm0.06$). On the other hand, both $HR2$ and $HR3$ are variable: $b_{HR2}=-0.23\pm0.06$ and $b_{HR3}=-0.10\pm0.04$. The $HR2$ variations suggest that the spectrum of the source above $7$ keV steepens as the source brightens. The $HR3$ ratio shows the opposite behaviour. It implies a “flattening" of the $5-10$ keV spectrum as the source brightens. In the case of IC5063, both $HR1$ and $HR2$ are variable ($b_{HR1}=-1.0\pm0.2$, and $b_{HR2}=-0.40\pm0.10$) while $HR3$ remains constant ($b_{HR3}=0.06\pm
0.07$). Both $HR1$ and $HR2$ indicate that the source’s spectrum becomes softer as it brightens. NGC4507 shows a different behaviour. $HR2$ remains constant ($b_{HR2}=-0.04\pm0.03$), while $HR3$ is significantly variable ($b_{HR3}=0.45\pm 0.08$). The spectrum below $10$ keV appears to become softer as the source flux increases. NGC7172 shows a similar behaviour. $HR2$ remains constant ($b_{HR2}=-0.04\pm 0.03$), $HR3$ shows marginal changes ($b_{HR3}=0.06\pm0.03$) while $HR1$ shows more significant variations ($b_{HR1}=-0.12\pm0.4$). These variations indicate again a steepening of the spectrum below $10$ keV as the source flux increases.
SPECTRAL ANALYSIS
=================
In our spectral analysis, we used only data between 3 and 20 keV, where the effective area of PCA, and hence the signal-to-noise ratio, is highest. By discarding the data below 3 keV we also avoid the complications associated with the soft X-ray excess in these sources (see Turner et al. 1997). The PCA data from each observation were binned to give a minimum of 20 counts per channel (source plus background). The spectral fitting analysis was carried out using the [XSPEC v.10]{} software package on the basis of “joint fits” to our [*RXTE*]{} observations. All errors correspond to the 90 per cent confidence level for one interesting parameter. All energies quoted refer to the emitter’s rest-frame.
We have employed two group of models (see below) in order to characterize the spectra of the objects at each observation. Within each group, in order to assess the significance of new parameters added to the initial model of the group we have adopted the 95 per cent confidence level using the F-test for additional terms. To compare models with different number of parameters within each group and models between the two groups we have used the “likelihood ratio" (Mushotzky 1982) and accepted values of the ratio larger than 100 as showing significant improvement in the model fits.
Note that the background estimation software calculates errors on the simulated background spectrum assuming Poisson statistics. This is an overestimate of the statistical error on the background spectrum, which is estimated using a large amount of data. In fact, for faint objects (such as AGN) the error that the standard software calculates for the background is comparable to the error in the total observed rate. In practice, the background spectrum errors should be negligible compared to the errors on the observed spectrum. Since the [XSPEC]{} software estimates the error in the net spectrum (source - background) by combinning the errors in the observed spectrum with those of the simulated background, the reduced 2 values in many cases are very small. However, the F and “likelihood ratio” tests are not affected by this. Finally, due to the large errors, the confidence regions that we estimate for the best fit parameter values are almost certainly enlarged.
Power Law and Gaussian Line Models.
-----------------------------------
Following previous and results, we first employ a simple spectral model consisting of a power-law continuum, with photon spectral index $\Gamma$, modified by absorption in a column density, $N_H$, of cool neutral material. A Gaussian line was also included to account for Fe-K emission. The line energy and the width were allowed to vary freely. However, when the width was found to be smaller than the spectral resolution, we fixed it at $\sigma_{line}=0.1$ keV. The values of the $N_H$, the photon index and the normalization of the Fe line are free parameters but tied to the same value for all observations. The normalization of the power law is allowed to vary freely. The results of fitting this spectral model are presented as model A in Tables 5, 6, 7 and 8. Next, we allow for the column density and the power-law index to vary freely (models B and C in the same tables). Finally, in model D we allow for the normalization of the Gaussian line to vary freely while now both the power-law index and the column density are tied to the same value between all the observations. We discuss the results for each galaxy in turn.
[**ESO103-G35.**]{} It shows a large column density ($N_{H}\sim 1.4\times
10^{23}$ ) in excellent agreement with the observations. However, the spectral index is flat ($\Gamma\sim 1.6$, model A) in contrast with the results. Moreover, the best fit energy line is very low ($E \sim$ 6 keV) and marginally resolved, ie broader than the energy resolution of PCA, having a $\sigma$ of $\sim
0.7$ keV. The EW of the line does not remain constant. It ranges from $490^{+41}_{-38}$ to $661^{+56}_{-51}$ eV. The data show strong evidence for spectral variability, since models B,C and D give a significantly better fit to the data when compared to model A. When the line normalisation is untied (model D), we obtain a $\Delta \chi^{2}=68$ for 14 addtitional parameters. A larger $\Delta \chi^{2}$ value is obtained when the column density is untied and an even larger $\Delta \chi^{2}$ when the power law index is untied ($\Delta \chi^{2}=107$ and 162 respectively for 14 additional parameters). Based on the likelihood ratios, model C gives the best fit to the data.
Following Forster et al. (1997), we tried an additional model. We added an absorption edge component (model [EDGE]{} in XSPEC) to model A and kept the Fe line energy fixed at 6.4 keV. We obtain a highly statistically significant improvement in 2 compared to model A ($\Delta\chi^2\sim 143$ for one additional parameter). The photon index becomes now steeper ($\Gamma=
1.68^{+0.03}_{-0.02}$ and $N_H= 1.7^{+0.5}_{-0.5}\times 10^{23}$ ). The best-fit energy for the edge is $6.8^{+0.10}_{-0.10}$ keV and the optical depth is $\tau= 0.2 \pm 0.02$.
[**IC5063.**]{} Model A does not provide a good fit in this case. The column density is $2\times 10^{23}$ while the power-law is flat ($\Gamma \sim 1.6$) both in good agreement with the results. The EW of the Fe line varies between $298^{+26}_{-45}$ and $462^{+55}_{-70}$ eV. As before with ESO 103-G35, we find highly significant evidence for spectral variability. Models B and C give a significant 2 improvement compared to model A (in this case model D does not improve the fit significantly). The best fit is obtained when the photon index is untied (model C): $\Gamma = 1.15 - 1.70$. However, the Fe line energy is low ($E\sim 6.12$ keV with $\sigma \sim 0.5$). We therefore included an absorption edge in model C. The line energy is marginally improved, $E=6.19^{+0.10}_{-0.12}$ keV, with $\sigma=0.48$. The edge energy was fixed at 6.9 keV while the optical depth found was $\tau=0.07$. However, the inclusion of the edge did not improve the fit at a statistically significant level ($\chi^2=175.2/235$, as compared to $\chi^2=177.2/236$ for model C).
[**NGC4507.**]{} Model A results in a high absorbing column ($\sim 4.5\times
10^{23}$ ) and a flat spectrum ($\Gamma \sim 1.4$), in agreement with the observations. The line energy is $\sim 6.16$ keV with the line width being unresolved ($\sigma \sim 0.33$). The EW ranges from $353^{+47}_{-34}$ up to $655^{+89}_{-62}$ eV. All models B,C and D give a significant 2 improvement when compared to model A, indicating again the presence of significant spectral variability. Based on the likelihood ratios, it is model B this time that provides the best fit to the data.
[**NGC7172.**]{} Model A shows a large column density ($N_{H} \sim 1.1\times
10^{23}$ ) in agreement with previous results. The power-law has a value of $\Gamma\sim 1.8$, also in good agreement with the results, and typical of the intrinsic AGN spectral index. The Fe line has an energy of $\sim 6.24$ keV. The line width is consistent with the line being narrow. The resulting EW varies from $96^{+13}_{-22}$ to $360^{+49}_{-28}$ eV in the nine observations. Model D does not improve the fit, while models B and C give a statistically significant improvement to the 2 value of the fit. When the column density becomes a free parameter (model B) we obtain a $\Delta \chi^2 \sim 30$ for 12 additional parameters. When the photon index becomes untied (model C), we obtain an equally good 2 with $\Gamma$ varying between $1.7$ and $2.0$. This suggests that the two parameters (photon index and column density) are degenerate and it is therefore difficult to disentangle the origin of the observed spectral variation. We plot the variations of column density and of the photon index as a function of the $3-20$ keV flux in Fig. \[nh\] and \[gamma\] respectively. We can see that the changes in the spectrum of this source can be explained by a decrease of $N_{H}$ by $\sim 30$ per cent as the source flux increases by a factor of $\sim 3.5$. Alternatively, a $\Gamma$ increase by $\sim 0.3$ (from $\sim
1.65$ up to $\sim 1.95$) can explain the spectral changes equally well.
Power Law, Gaussian Line and Reflection Component Models.
---------------------------------------------------------
Although the power-law plus gaussian line prescription defined above gives an acceptable fit in terms of the $\chi^{2}$ statistic, there is evidence from the flat power-laws derived for NGC4507, IC5063 and ESO103-G35 for the presence of a strong Compton reflection component. The next step in the current analysis was therefore to include a reflection component in the spectral modeling. Specifically we use the [PEXRAV]{} model (Magdziarz & Zdziarski 1995) in [XSPEC]{}. This model calculates the expected X-ray spectrum when a source of X-rays is incident on optically thick, neutral (except hydrogen and helium) material. We assume that both the reflection component and the power-law are absorbed by the cold gas column density. The strength of the reflection component is governed by the parameter R, representing the strength of the reflected signal relative to the level of the incident power-law continuum. We set R=1 (which corresponds to a $2\pi$ solid angle subtended by the optically thick material). We fix the inclination angle for the disk at $i=60^{\circ}$, since the shape of the reflection spectrum below 20 keV is relatively independent of the inclination angle. We also fix the energy of the exponential cutoff at 300 keV. Both the iron and light element abundances were kept fixed at the solar abundance values.
We added this reflection component to all models A,B,C and D, creating a new set of models (A’,B’,C’ and D’). In all new models, the normalisation of the reflection component is tied to a single value across the set of observations. We also set the spectral index of the power-law continuum to $\Gamma=1.9$ (in all models apart from model C’). As before, in model A’ the power-law normalisation is allowed to vary in each observation while the column density and the Fe line normalization are tied between the different observations. In model B’ we untie the column density, in model C’ we untie the photon index and in model D’ we let the gaussian line normalisation free. Finally, we let the reflection component normalisation free (model E’). The results are presented in tables 9 to 12.
[**ESO103-G35.**]{} In the case of ESO103-G35, the inclusion of a reflection component gives a statistically significant improvement to the data fit. For example, we obtain a $\Delta \chi^2 = 163$ (with the addition of one extra parameter) when we include the reflection component in model A. This is in agreement with the findings of Smith & Done (1996) using data. Furthermore, all models (B’,C’,D’ and E’) significantly improve the fit when compared to model A’. The best fit to the data is obtained when the photon-index is allowed to vary with time (model C’, $\Gamma\sim 1.8-2$). An equally good fit is also obtained when the reflection component normalisation is allowed to vary with time (model E’). Both these models give a better fit than model C, showing again that the inclusion of the reflection component significantly improves the fit.
In Fig. \[eso103\_pexrav\] we present the variations of the power-law and the reflection flux between the different observations (model E’; time in this figure and in Fig. \[ic5063\_pexrav\] as well is measured in seconds from the mission launch date). We see that the power-law component shows rapid flux variations (up to a factor of 50 per cent) within days. Similar variations, but not well correlated with the power-law variations, are observed in the reflection component flux as well. In Fig. \[gamma\] we plot model’s C’ best fit $\Gamma$ values as a function of the $3-20$ keV flux. A change of $\sim 0.15$ (from $\Gamma \sim 1.8$ up to $\Gamma \sim 2$) can explain the spectral variability in this case.
However, the best fit line energy in models C’ and E’ is abnormally low ($E\sim 6.15$ keV). We therefore introduced an absorption edge to model E’. The resulting 2 is 505/714, which indicates that, compared to model E’, the improvement to the fit is statistically significant. The best fit line energy now becomes $6.23\pm 0.10$ keV, roughly consistent with the line energy of neutral Fe. The best fit edge energy is $E_{edge}=7.69
\pm 0.14$ keV (ionised Fe) while the optical depth is $\tau=0.1\pm 0.01$. Similar results are obtained when we add a warm edge component to model C’.
[**IC5063.**]{} In IC5063 the inclusion of a reflection component does not represent a statistically significant improvement. Models C’, E’ and C (which does not include a reflection component) give an equally good fit to the data. However, the inclusion of the reflection component (for example model E’) yields a line with an energy of $E=6.25\pm0.10$ keV, which is roughly consistent with the energy of line from neutral Fe. In Fig. \[ic5063\_pexrav\] the variations of the power-law and the reflection flux are presented (model E’). The power-law component flux decreases by a factor of two with time. In contrast, the reflection signal, although variable as well, does not correlate with the power-law flux. Alternatively, a change of $\Gamma$ between roughly $1.6$ and $2.2$ (model C’, see Fig. \[gamma\]) yields a similarly good fit to the data, and results in the same best fit line energy value. However, the photon index does not correlate well with the continuum flux.
[**NGC4507.**]{} In this case, the use of a reflection component improves the fit dramatically. Models B’ and C’ give a much better fit to the data than model B. These two models (untied column densities or photon indices) fit equally well the data: in the first case the $N_H$ varies roughly between $4.8$ and $5.3\times 10^{23}$ , while in the second case $\Gamma$ varies between $1.9$ and $2.0$.
The line energy is again low ($E\sim 6.2$ keV) and therefore we included an absorption edge component in both models B’ and C’ above. We obtain a statistically significant improvement in the fit in both cases. However, the best fit is now clearly obtained in the case of a column density variation (2=643/794 as compared to 2=655/794 for an untied $\Gamma$). The edge energy was frozen at 6.9 keV while its optical depth was found to be $0.16\pm0.02$. The line energy was kept fixed at 6.4 keV ($\sigma$=0.1 keV). In Fig. \[nh\] we plot the variations of $N_{H}$ as a function of the 3-20 keV flux (model B’ with edge). A $\sim 10$ per cent decrease in the absorpting column while the source flux increases by a factor of 2 can explain the observed spectral changes.
[**NGC7172.**]{} In this case, the inclusion of a reflection component does not improve the fit at a statistically significant level. Models B’ and C’ present the best fit to the data, however the goodness of fit is comparable to that of the models B and C. The lack of a reflection component is rather expected given the steep power-law index found earlier in the case of the simple power-law and a Gaussian line model. Note that the results required the presence of a reflection component only marginally.
![ESO103-G35: The 3-20 keV flux of the power-law (upper panel) and of the reflection component (lower panel) as a function of time from the mission’s launch (90 per cent errors are plotted)[]{data-label="eso103_pexrav"}](eso103.ps){height="7.0cm"}
![IC5063: the 3-20 keV flux of the power-law (upper panel) and of the reflection component (lower panel) as a function of time from the mission’s launch (90 per cent errors are plotted).[]{data-label="ic5063_pexrav"}](ic5063.ps){height="7.0cm"}
{height="8.0cm"}
{height="8.0cm"}
Soft excess models
------------------
The X-ray spectra of Seyfert-2 galaxies usually show some level of soft X-ray emission despite the strong photoelectric absorption (eg Turner et al. 1997). This soft excess could be due to either a strong star-forming component or scattered emission from the nucleus. The contribution of the soft excess in our case (ie above 3 keV) should be negligible. Nevertheless, we tried to investigate whether a constant soft excess combined with a power-law component with varying normalization may explain the strong spectral variability observed in our data. Therefore, we include a power-law component with the same photon index as the hard X-ray power-law and with a free normalization (but tied between all the observations) in the models above.
In NGC7172 we add the soft power-law to model A since a reflection component is not needed by the data. We obtain 2=303.8 ie identical to that of model A, despite the addition of one extra parameter. We conclude that an extra soft component is not needed by the data. In the other three cases we add the soft power-law component to model A’. In the case of ESO103-G35 we obtain 2=568.5/729 which represents a statistically significant improvement to model A’. However, the fit is worse when compared to models C’ and E’. In NGC4507 we obtain $\chi^2=738/806$ similar to model A’ with the soft power-law normalization being zero. Finally, in the case of IC5063 we obtain an identical $\chi^2$ to model A’ (269/239) while the soft power-law normalization is again zero. The above confirm that the presence of any soft excesses do not affect our data above 3 keV, and they cannot account for the observed spectral variability.
SUMMARY
=======
[**ESO103-G35**]{} presents a complex spectrum. Models C’and E’, with the addition of a warm edge, give the best fit to our data. Our best fit edge energy is $7.69\pm0.14$ keV with $\tau=0.1\pm 0.01$. The best fit edge energy matches the results from the 1988 and 1991 observation (Warwick et al 1993, Smith and Done 1996), and also the 1996 observation (Forster et al 1999). The energy of the edge implies the presence a significant amount of ionised iron. The optical depth of the iron edge was observed to increase between the 1994 and 1996 observations (Forster et al 1999), and the edge line in the former observation was consistent with that from neutral iron. Our best fit edge optical depth is significantly smaller than the respective result (Forster et al. 1999). However, the quality of the present data set is not good enough to investigate if there exist any changes in the optical depth and/or the energy of the edge during the observations. The spectral variability can be explained by a variable spectral index (model C’, Fig. \[gamma\]). Although these variations are subtle, they are highly statistically significant. The photon index is correlated with the 3-20 keV flux in the sense that the spectrum steepens with increasing flux. The same variations can be equally well explained by a variable reflection component (model E’, Fig. \[eso103\_pexrav\]). Interestingly, Akylas et al. (2000) find no reflection component in the observations despite the large passband ($\sim$2-60 keV) and the good signal-to-noise ratio of their observations. This may suggest that the relative normalization of the power-law and the reflection component must change in different epochs, and supports the interpratation that the spectral variations seen by are caused by a variable reflection component. Furthermore, in this case we can also explain naturally the fact that although the $HR2$ ratio of ESO103-G35 is variable, the $HR1$ ratio remains constant.
[**IC5063**]{} can be well fit with a simple spectrum consisting of a power-law and a Gaussian line but the energy of the line is abnormally low ($\sim6.12$ keV). The addition of an edge does not yield a reasonable value for the energy of the Fe line. However, the addition of a reflection component (models C’ and E’), although it does not improve the 2, yields an Fe line with an energy of $6.25\pm0.1$ keV compatible within the errors with the energy of cold Fe line. IC5063 shows strong evidence for spectral variability (Fig. \[ratio\]). The spectral variability can be explained either by the reflection component being variable between the observations (model E’) or by the photon index being variable (model C’, $\Gamma \sim$ 1.6 - 2.2). According to model E’ best fit results there is no correlation between the reflection component and the intrinsic power law flux variability (Fig. \[ic5063\_pexrav\]). Similarly, in the model C’ case, there is no clear correlation between the photon index and the continuum flux (Fig. \[gamma\]).
[**NGC4507**]{} presents a similar spectrum to ESO103-G35. We find strong evidence for a reflection component (note that such a component was not evident in neither the nor the data). We also find evidence for a cold Fe edge (with an energy fixed at 6.9 keV), in excess of the absorption edge associated with the reflection component. Assuming the iron cross section given by Leahy and Creighton (1993), the optical depth of the edge ($\tau=0.16\pm 0.02$) implies an equivalent hydrogen column density of $\sim 2\times 10^{23}$ . This is smaller than our best fit $N_{H}$ value of $5\pm1.0 \times 10^{23}$. The edge component is $not$ required by the data, but with its addition we can get a good fit to the data with the iron line energy fixed at 6.4 keV. However, there is a possibility that residuals in the PCA calibration matrix around 5 keV due to Xe may still be present. In such a case it is possible that the line energy obtains somewhat low values due to these residuals. Therefore, we believe that the detection of a cold Fe edge in this object remains uncertain. We find strong evidence for spectral variability (Fig. \[ratio\]). The best model fit to the data (model B’) suggests flux correlated column density changes: the column becomes lower with increasing flux (Fig. \[nh\]). This model can explain naturally the $HR3$ variations observed for this object, and at the same time the lack of $HR2$ variations.
[**NGC7172**]{} is the only object which shows no evidence for a reflection component. found marginal evidence for such a component while in data a reflection component is needed only if the spectral index is fixed to the canonical $\Gamma=1.9$ value (Akylas et al. 2000). Similarly, if $\Gamma$ is fixed to 1.9, models with reflection (B’ and C’) can fit the present data set well but simple power law models (B and C) can fit the data equally well. There is no evidence for an edge, and spectral variability is observed again. The spectral fitting cannot discriminate between variations in the photon index or the column density: as the flux increases, either the column drops ($14 - 11\times 10^{23}$) or the photon index steepens ($1.66 -1.96$) in a similar fashion to ESO103-G35 and NGC4507.
DISCUSSION AND CONCLUSIONS
==========================
The observations of ESO103-G35, IC5063, NGC4507 and NGC7172 show that, in all objects, the X-ray flux is significantly variable on time scales larger than $\sim$ a day, with an amplitude similar to that observed in Seyfert 1 galaxies. This result on its own suggests that, in these objects, we are seeing directly the nuclear emission. This emission is transmitted through a heavily absorbing material with a density of $\sim 1-5 \times 10^{23}$ . Apart from the flux variability, we searched for the presence of spectral variations, taking advantage of the fact that the present data set consists of many observations for each object all made with the same instrument. In the past, spectral variability studies in Seyfert 2 galaxies were not conclusive mainly because it is difficult to compare the results from different spectral fits performed across various instruments with different bandpasses. We detected significant, flux related variations in the hardness ratios that we calculated for all objects (Fig. \[ratio\]). This result suggests that spectral variability may be a ubiquitous feature in Seyfert-2 galaxies.
In principle, the observed hardness ratio variations could be an artefact of poor background subtraction. However, the $HR$ variations do [*not*]{} show the same behaviour in all sources, and are well correlated with the source flux. Therefore, if these variations are due to background subtraction errors, those errors should be different at different epochs and should operate in such a way so that the resulting ratios should mimic the flux correlated variations seen in Fig. \[ratio\]. We consider this possibility unlikely to be the cause of the spectral variations that we observe.
Still, the explanation for the observed spectral variability is model dependent and is based on the spectral fitting results. In some cases, these results are somehow uncertain. For example, in NGC7172 and NGC4507, significantly different $N_{H}$ values are observed in just one observation, namely the one with the highest flux (Fig. \[nh\]). Although the hardness ratios clearly show that the variability in both NGC4507 and NGC7172 is attributed to more than one observation, we investigated whether the spectral fitting results could be affected by problems associated with the background subtraction in the highest flux observations. Hence, we examined the data below 3 keV in layer 1 where the signal is mostly due to particles as the effective area of the PCA is negligible. We also examined the signal in the full energy band of layers 2 and 3 which are again sensitive to particles only. We find that the particle spectrum was consistent with the background model predictions, especially below 5 keV. In conclusion, we believe that the spectral variability is real in all four objects, and that our model fit results can be used in order to explain its origin.
The origin of the spectral variability appears to be different in each object. We discuss below briefly the implications of our results on the processes that operate in the nuclei of Seyfert 2 galaxies.
Photon index changes
--------------------
In three cases, namely ESO103-G35, IC5063 and NGC7172, models with a variable photon index (model C’ for the first two sources and model C for NGC7172) can fit the data well. Turner et al (1998) have found significant photon index variability in many Seyfert 2 galaxies by comparing historical X-ray observations. Significant photon index variability has also been observed in many Type 1 objects (eg MCG 6-30-15, Lee et al. 1999; NGC4051, Guainazzi et al. 1996; Mrk 766, Leighly et al. 1996; NGC4151, Warwick et al. 1996). In all these objects, the power law becomes steeper with increasing flux. The same pattern is followed by the photon index with flux in our case as well. In ESO103-G35 and NGC7172 we observe $\Gamma$ increasing by $\sim 0.2$ and $\sim 0.3$ for a $\sim 1.6$ and $\sim3$ flux increase respectively. In IC5063, $\Gamma$ increased by $\sim
0.6$ while the source doubled its flux, but is far from clear in this case that the $\Gamma$ variations are correlated with the source flux.
According to standard Comptonization models, hard X-rays in AGN are produced through the scattering of the soft UV photons from the accretion disk on a hot ($>$ 40 keV) corona above the accretion disk. Opacity changes in the corona can give rise to significant spectral variability, even if the total luminosity of the hard X-rays remains constant (Haardt et al 1997). If the corona opacity is not dominated by pairs, as the optical depth of the corona, $\tau$, increases the intrinsic spectrum becomes steeper (as long as $\tau < 1$). As a result, as long as $\Gamma <
2$, the the photon index correlates with the photon flux: a $\sim 2.5$ increase in the $2-10$ keV flux can result in a $\Delta\Gamma\sim 0.4$ increase in the photon index (Fig. 7 in Haardt et al 1997). The best fit $\Gamma$ values that we find are less than $2$ in almost all cases and the $\Delta \Gamma$ changes that we detect for the observed source flux variations are in broad agreement with the predictions of this model.
We note in passing that Seyfert 2 galaxies with “flat spectrum" (ie Smith & Done 1996) may not be intrinsically flat. If those sources also exhibit flux related photon index variations, then values as flat as $\sim 1.5 -
1.6$ can be expected according to Haardt et al. (1997). Monitoring observations of these sources could resolve the issue.
Absorption column changes
-------------------------
In NGC4507 and NGC7172, models with a variable absorption column density (models B’ and B respectively) give a good fit to the data. In fact, model B’ gives the best fit to the NGC4507 data. In both cases, the column density is anticorrelated with the X-ray flux: a $\sim 5\times 10^{22}$ decrease in $N_{H}$ is observed while the source flux increases by a factor of $\sim 2$ and $\sim 3$ respectively (Fig. \[nh\]). Absorbing column variations in Seyfert 2 galaxies have been observed in the past as well (eg NGC526A, Turner & Pounds 1989; NGC7582, Warwick et al. 1993; ESO103-G35, Warwick, Pounds and Turner 1988). In the first two cases significant column changes on a time scale of years were detected, while two observations of ESO103-G35 revealed a significant drop in the column density within 90 days.
The flux correlated $N_{H}$ variations can be explained if the obscuration is not due to a uniform torus, but due to individual clouds rotating around the nucleus instead. We may expect clouds of different column density to intercept the emission from the nucleus at different epochs. In this case, a small increase of the column density will be associated with a decrease in the total flux. This model requires both high velocities for the covering clouds and a small X-ray source size. For example, in the case of NGC7172, the lowest flux observations (last two points in Fig. \[lcurve\]) have similar $N_{H}$ values and are $\sim 4.5$ and $\sim 5.5$ days apart from the highest flux observation (third point in Fig. \[lcurve\]). Assuming that a cloud of $N_{H} \sim 3-5\times 10^{22}$ and velocity of $\sim
10^{4}$ km/sec covered the X-ray source in $4.5-5.5$ days, then the size of the X-ray source should be smaller than a few ($\sim 3.5-4.5$) light hours.
The $N_{H}$ variations could also be explained by partial photo-ionisation of the obscuring screen. Varying degrees of photo-ionisation caused by intrinsic luminosity variations of the source can result in column density changes. This model can explain succesfully the variable X-ray absorption in MR2251-178 (Halpern 1984, Pan et al. 1990). Recently, Comastri et al. (1998), presented strong evidence for the existence of photonionised material with a column density of $\sim 8\times 10^{22}$ in NGC4507. In order for the absorbing column to adjust itself instantaneous to the continuum flux variations (and hence explain the rather “continuous" variations shown by its $HR2$ plot in Fig. \[ratio\]), the absorbing material should be close to the central source and in a thin shell (so that its density will be larger than $\sim$ a few $10^{8}$ cm$^{-3}$, Netzer 1993). Interestingly, Turner et al. (1998) have recently presented evidence for the existence of absorbing material close to the central source in Seyfert-2 galaxies. Neither the nor the data show evidence for a warm edge in the spectrum of NGC4507. For an ionised material with a column density of $\sim$ a few $\times 10^{23}$ the respective edge should have an optical depth of less than 0.1, and hence difficult to detect. On the other hand, ESO103-G35 shows a warm iron edge in its spectrum but no indication of absorption column variations. Perhaps in this case, the small amplitude continuum variability together with possible variability of other spectral components (ie $\Gamma$, reflection component normalisation) render the detection of any absorption variations difficult. Finally, in the case of NGC7172, ASCA observations did not show any evidence of photo-ionised material (Guainazzi et al 1998). The $2-10$ keV flux of the source during the ASCA observation was $\sim 1.3\times
10^{-11}$ erg cm$^{-2}$s$^{-1}$, which is similar to the flux during the last observation. Their best fit $\Gamma$ value $1.5\pm0.15$, which is consistent with our estimate ($1.66\pm 0.15$, Table 8). This result supports the idea that in NGC7172 flux related $\Gamma$ changes (ie model C) rather than $N_{H}$ variations (model B) are responsible for the spectral variability.
The detection of $N_{H}$ variations in just one source indicates that distribution of the absorbing material in Seyfert 2 galaxies may not be the same in all objects (eg Turner et al. 1998). Material away from the nucleus is not expected to be ionised and will not respond to the continuum variability. As a result, $N_{H}$ in these objects will remain constant with time.
Reflection component and Iron Line
----------------------------------
Model E’ (a variable reflection component) can explain the spectral variability observed in ESO103-G35 and IC5063. If true, this would imply that the cold material responsible for reflection is close to the nucleus eg an accretion disk as suggested by Nandra et al. (1997) for Seyfert 1 galaxies. In this case we expect a positive correlation and no appreciable lags in the reflection response to the continuum flux variations. Contrary to this, we find that the reflection variations are not correlated with the continuum variations, especially in the case of IC5063 (Fig. \[eso103\_pexrav\], \[ic5063\_pexrav\]). One can assume, rather arbitrarily, that there is after all a lag and therefore the reflection component is responding to past, unobserved continuum variations. Further evidence for the existence and possible variability of the reflection component can be provided by the iron line results.
All sources show a strong, unresolved emission line at $\sim 6.4$ keV. The only exception is ESO103-G35 where the line may be marginally resolved($\sigma \sim 0.5$ keV, models C’ and E’). This is consistent with the results of Turner et al. (1997) for this source. The intensity of the line though remains constant. This is indicated by the fact that model D’ (a variable iron line) does not improve the fit when compared with model A’ in IC5063 and NGC4507. The same holds for model D when compared with model A for NGC7172. Fig. \[ew\], shows a plot of the line EW as a function of the source $2-10$ keV flux. The equivalent widths were estimated using results from the best fit of model B’ for NGC4507 and NGC7172, model C’ for IC5063 and the same model plus a warm iron edge for ESO103-G35. Since all models require a constant line, in all cases the EW decreases as the source flux increases. The errors on the EW are small enough to see that the EW variations are significant.
The fact that the lines are narrow (except for ESO103-G35) and constant implies that they are produced by the obscuring torus. For IC5063, NGC4507 and NGC7172, the EW of the iron line can be as large as $\sim 300$ eV in the lowest flux observations (Fig. \[ew\]). The measured column densities in the line of sight toward those sources is $\sim 3\times 10^{23}$, $\sim 5\times 10^{23}$ and $\sim 1\times 10^{23}$ respectively. For these $N_{H}$ values, we expect an emission line with equivalent width of $\sim 120$ eV, $\sim 200$ eV and $\sim 40$ eV respectively (Ghisellini et al. 1994). The larger than expected EW suggests that there exists a source of line photons other than from the line-of-sight absorbing material. A 6.4 keV iron line can also be produced by fluorescence from an acrretion disk. The expected angle-averaged EW in this case is $\sim 110$ eV. Consequently, in NGC4507 and IC5063 the iron line that we observed is probably the sum of two lines - most of the line is produced by the torus and there is additional contribution by the accretion disk itself. Any variations of the accretion disk line would be diluted by the presence of the torus line and hence difficult to detect. Furthermore, the lack of line variability indicates that the torus should be in a distance larger than approximately two light weeks and approximately six light months in NGC4507 and IC5063 respectively.
For NGC7172, the discrepancy between the EW observed in the two lowest flux observations of this source ($\sim 300$ eV), and the EW expected in the case of a torus plus an accretion disk line ($\sim 150-200$ eV) remains. Furthermore, the and results (Smith & Done 1996, Turner et al. 1997) showed a cold iron line with EW $\sim 50$ and $\sim
70$ eV respectively (these estimates had large errors associated with them). The continuum flux was $\sim 4 \times 10^{-11}$ erg s $^{-1}$ cm$^{-2}$ in both cases. Despite the large errors, for this source flux level, the equivalent widths are in agreement with these expected according to EW vs flux plot for this source in Fig 7. Guainazzi et al. (1998) report the same EW when the source was $\sim 4$ times fainter. Their estimate appears to be inconsistent with our results, and suggests that the line follows the continuum on time scales of $\sim$ a year. The observations were made within a week. One explanation could then be that the line is mainly produced by the torus which is located at a distance larger than $\sim$ a light week and smaller than $\sim$ a light year.
ESO103-G35 was observed three times by (Forster et al. 1999). In all cases, the source flux was lower than during the observations. The EW of the iron line was higher ($\sim 400-200$ eV), in agreement with the trend shown in Fig. \[ew\]. As in the case of NGC7172, the large EW observed by requires an extra source of line photons apart from the torus and the accretion disk. Forster et al. (1999) suggest various possibilities in order to explain this. The fact that the iron line appears broad in the observations is a direct indication that a part of the line is produced by fluorescence from the accretion disk. In fact, ESO103-G35 is the only source were a model with a variable iron line (model D’) improves significantly the goodness fit when compared to model A’. This indication of line variability supports the idea that the spectral variability in this object is due to a variable reflection component (model E’). Such a model could also explain naturally the fact that $HR2$ in the case of ESO103-G35 is variable while $HR1$ remains constant. The possible detection of line and reflection component variability in this object only indicates that the orientation of the central source may not be the same in Seyfert 2 galaxies (despite the predictions of the unification models in their simplest form). Our results are consistent with the idea that the central source in ESO103-G35 is seen close to face on and at a larger inclination in the other objects. In the latter case, the contribution from the reflection component and the disk iron line emission will be supressed and it will be more difficult to detect any variations associated with them.
The main result from the observations of the four Seyfert 2 galaxies ESO103-G35, IC5063, NGC4507 and NGC7172 is that they all show significant spectral variability in the sense that the spectrum steepens with increasing flux. This is a model independent result and is based on the analysis of the hardness ratios (see Section 4.1, Fig. 2). The origin of this variability may not be the same for all of them. In NGC4507 and NGC7172 we observe flux related $N_{H}$ and $\Gamma$ variations respectively. In ESO103-G35 the spectral variability could be due either to photon index variations (like NGC7172) or to reflection component changes. The results for IC5063 are puzzling. The stability of the iron line flux does not support reflection component variations as the reason of the observed spectral variations. On the other hand, the photon index variations are not well correlated with the continuum flux. Since this object is a radio galaxy, perhaps the normal “Seyfert - type" X-ray source in this case is “contaminated" by emission from the non-thermal source which is responsible for the radio emission.
The differences in the origin of the spectral variability observed in these objects suggest that the orientation of the nucleus and/or the distribution of the absorbing matterial around the nucleus is not the same in all Seyfert 2 galaxies. We believe that future monitoring observations of these objects with XMM mission which has both a large effective area and good spectral resolution are necessary in order to clarify the issue of spectral variability in Seyfert 2 galaxies and provide stringent constraints on the geometry of the nucleus and the location of the cirumnuclear matter.
Acknowledgements
================
We are grateful to D.A. Smith for numerous conversations and suggestions regarding the background. This research has made use of data obtained from the High Energy Astrophysics Science Archive Research Center (HEASARC), provided by NASA’s Goddard Space Flight Center.
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Model $\rm N_H^5$ $\Gamma$ Energy (keV) $\sigma$ (keV) 2/dof
------- ----------------------------------------- --------------------------------------------- ------------------------ ------------------------ -----------
A$^1$ $14.1^{+0.5}_{-0.4}$ $1.57^{+0.02}_{-0.03}$ $5.93^{+0.01}_{-0.01}$ $0.69^{+0.05}_{-0.04}$ 791.3/730
B$^2$ $12.6^{+0.6}_{-0.6}-16.0^{+1.0}_{-1.0}$ $1.55^{+0.03}_{-0.03}$ $5.91^{+0.04}_{-0.04}$ $0.72^{+0.05}_{-0.05}$ 684.3/716
C$^3$ $13.9^{+0.4}_{-0.4}$ $1.47^{+0.17}_{-0.10}-1.61^{+0.04}_{-0.04}$ $5.90^{+0.04}_{-0.04}$ $0.73^{+0.05}_{-0.05}$ 629.4/716
D$^4$ $13.8^{+0.5}_{-0.5}$ $1.54^{+0.02}_{-0.02}$ $5.90^{+0.05}_{-0.05}$ 0.1 723.7/716
Notes: $^1$ $N_H$, $\Gamma$ tied; $^2$ $N_H$ untied, $\Gamma$ tied; $^3$ $N_H$ tied, $\Gamma$ untied;\
$^4$ gaussian line normalizations untied ; $^5$ in units of $10^{22}$
----------------------------------------------------------------------------------------------------------------------------------------
Model $\rm N_H^5$ $\Gamma$ Energy (keV) $\sigma$ (keV) 2/dof
------- ----------------------------------------- ------------------------ ------------------------ ------------------------ -----------
A$^1$ $19.2^{+1.7}_{-1.7}$ $1.57^{+0.07}_{-0.07}$ $6.11^{+0.08}_{-0.09}$ $0.37^{+0.13}_{-0.17}$ 310.0/240
B$^2$ $15.0^{+4.0}_{-4.0}-28.0^{+2.0}_{-3.0}$ $1.52^{+0.10}_{-0.09}$ $6.02^{+0.10}_{-0.03}$ $0.51^{+0.14}_{-0.06}$ 191.9/236
C$^3$ $17.9^{+1.6}_{-1.6}$ $1.15^{+0.10}_{-0.13}- $6.05^{+0.09}_{-0.10}$ $0.50^{+0.15}_{-0.14}$ 177.2/236
1.70^{+0.20}_{-0.19}$
D$^4$ $17.8^{+2.5}_{-1.3}$ $1.47^{+0.15}_{-0.06}$ $6.04^{+0.13}_{-0.14}$ $0.53^{+0.09}_{-0.27}$ 302.6/236
----------------------------------------------------------------------------------------------------------------------------------------
Notes: $^1$ $N_H$, $\Gamma$ tied; $^2$ $N_H$ untied, $\Gamma$ tied; $^3$ $N_H$ tied, $\Gamma$ untied;\
$^4$ gaussian line normalizations untied ; $^5$ in units of $10^{22}$
Model $\rm N_H^5$ $\Gamma$ Energy (keV) $\sigma$ (keV) 2/dof
------- ----------------------------------------- ---------------------------------------------- ------------------------ ------------------------ -----------
A$^1$ $45.0^{+1.5}_{-1.5}$ $1.40^{+0.05}_{-0.01}$ $6.09^{+0.04}_{-0.06}$ $0.33^{+0.04}_{-0.06}$ 887.2/807
B$^2$ $40.3^{+1.0}_{-1.0}-50.9^{+2.0}_{-3.0}$ $1.40^{+0.05}_{-0.03}$ $6.03^{+0.05}_{-0.15}$ $0.42^{+0.05}_{-0.05}$ 742.5/794
C$^3$ $45.2^{+2}_{-2}$ $1.27^{+0.06}_{-0.01}- 1.53^{+0.05}_{-0.04}$ $6.05^{+0.05}_{-0.04}$ $0.40^{+0.04}_{-0.03}$ 763.7/794
D$^4$ $46.7^{+1}_{-1}$ $1.42^{+0.03}_{-0.03}$ $5.96^{+0.03}_{-0.03}$ $0.52^{+0.04}_{-0.04}$ 815.8/794
Notes: $^1$ $N_H$, $\Gamma$ tied; $^2$ $N_H$ untied, $\Gamma$ tied; $^3$ $N_H$ tied, $\Gamma$ untied;\
$^4$ gaussian line normalizations untied ; $^5$ in units of $10^{22}$
Model $\rm N_H^5$ $\Gamma$ Energy (keV) $\sigma$ (keV) 2/dof
------- ----------------------------------------- --------------------------------------------- ------------------------ ---------------- -----------
A$^1$ $11.5^{+0.3}_{-0.8}$ $1.81^{+0.03}_{-0.05}$ $6.19^{+0.09}_{-0.09}$ 0.1 304.2/437
B$^2$ $11.0^{+1.0}_{-1.0}-14.0^{+1.0}_{-1.0}$ $1.81^{+0.06}_{-0.05}$ $6.18^{+0.08}_{-0.10}$ 0.1 274.6/429
C$^3$ $11.5^{+0.7}_{-0.8}$ $1.66^{+0.14}_{-0.16}-1.96^{+0.03}_{-0.08}$ $6.19^{+0.09}_{-0.09}$ 0.1 274.7/429
D$^4$ $11.3^{+0.5}_{-0.5}$ $1.80^{+0.01}_{-0.04}$ $6.17^{+0.08}_{-0.08}$ 0.1 299.8/429
Notes: $^1$ $N_H$, $\Gamma$ tied; $^2$ $N_H$ untied, $\Gamma$ tied; $^3$ $N_H$ tied, $\Gamma$ untied;\
$^4$ gaussian line normalizations untied ; $^5$ in units of $10^{22}$
Model $\rm N_H^1$ $\Gamma$ Energy (keV) $\sigma$ (keV) 2/dof
-------- ----------------------------------------- --------------------------------------------- ------------------------- ------------------------ -----------
A’$^2$ $17.2^{+0.3}_{-0.2}$ 1.9 $6.07^{+0.03}_{-0.03}$ $0.5^{+0.10}_{-0.10}$ 627.5/730
B’$^3$ $15.9^{+1.0}_{-1.0}-19.0^{+1.0}_{-1.0}$ 1.9 $6.06^{+0.05}_{-0.05}$ $0.5^{+0.09}_{-0.03}$ 556.2/716
C’$^4$ $17.5 ^{+0.2}_{-0.2}$ $1.84^{+0.08}_{-0.08}-2.02^{+0.05}_{-0.05}$ $6.07^{+0.06}_{-0.06} $ $0.47^{+0.08}_{-0.04}$ 523.5/715
D’$^5$ $17.2^{+0.2}_{-0.2}$ 1.9 $6.06^{+0.05}_{-0.05}$ $0.51^{+0.08}_{-0.04}$ 604.7/716
E’$^6$ $17.3^{+0.2}_{-0.2}$ 1.9 $6.06^{+0.04}_{-0.05}$ $0.51^{+0.08}_{-0.04}$ 525.8/716
Notes: $^1$ in units of $10^{22}$ ; $^2$ $N_H$, $\Gamma$ tied; $^3$ $N_H$ untied, $\Gamma$ tied; $^4$ $N_H$ tied, $\Gamma$ untied;\
$^5$ gaussian line normalizations untied ; $^6$ reflection component normalisation untied
Model $\rm N_H^1$ $\Gamma$ Energy (keV) $\sigma$ (keV) 2/dof
-------- ----------------------------------------- --------------------------------------------- ------------------------- ---------------- -----------
A’$^2$ $21.9^{+1.0}_{-1.0}$ 1.9 $6.18^{+0.10}_{-0.10}$ 0.1 269.3/241
B’$^3$ $18.4^{+4.0}_{-4.0}-31.0^{+2.5}_{-3.5}$ 1.9 $6.17^{+0.05}_{-0.05}$ 0.1 186.8/237
C’$^4$ $22.7^{+0.8}_{-0.8}$ $1.57^{+0.10}_{-0.10}-2.20^{+0.15}_{-0.15}$ $6.18^{+0.05}_{-0.05} $ 0.1 181.8/236
D’$^5$ $21.9^{+1.0}_{-1.0}$ 1.9 $6.20^{+0.07}_{-0.09}$ 0.1 264.3/237
E’$^6$ $22.3^{+1.0}_{-1.0}$ 1.9 $6.18^{+0.10}_{-0.10}$ 0.1 176.7/237
Notes: $^1$ in units of $10^{22}$ ; $^2$ $N_H$, $\Gamma$ tied; $^3$ $N_H$ untied, $\Gamma$ tied; $^4$ $N_H$ tied, $\Gamma$ untied;\
$^5$ gaussian line normalizations untied ; $^6$ reflection component normalisation untied
Model $\rm N_H^1$ $\Gamma$ Energy (keV) $\sigma$ (keV) 2/dof
-------- ----------------------------------------- --------------------------------------------- ------------------------- ---------------- -----------
A’$^2$ $50.4^{+1.1}_{-0.6}$ 1.9 $6.14^{+0.03}_{-0.04}$ 0.1 737.2/807
B’$^3$ $47.9^{+0.8}_{-0.8}-53.4^{+1.2}_{-1.2}$ 1.9 $6.13^{+0.04}_{-0.04}$ 0.1 701.5/795
C’$^4$ $51.9^{+1.0}_{-1.0}$ $1.90^{+0.05}_{-0.05}-2.01^{+0.07}_{-0.07}$ $6.13^{+0.04}_{-0.04} $ 0.1 700.7/794
D’$^5$ $50.8^{+1.0}_{-1.0}$ 1.9 $6.13^{+0.04}_{-0.04}$ 0.1 732.0/795
E’$^6$ $50.8^{+1.0}_{-1.0}$ 1.9 $6.14^{+0.03}_{-0.04}$ 0.1 717.8/795
Notes: $^1$ in units of $10^{22}$ ; $^2$ $N_H$, $\Gamma$ tied; $^3$ $N_H$ untied, $\Gamma$ tied; $^4$ $N_H$ tied, $\Gamma$ untied;\
$^5$ gaussian line normalizations untied ; $^6$ reflection component normalisation untied
Model $\rm N_H^1$ $\Gamma$ Energy (keV) $\sigma$ (keV) 2/dof
-------- ---------------------------------------- --------------------------------------------- ------------------------ ---------------- -----------
A’$^2$ $12.2^{+0.4}_{-0.4}$ 1.9 $6.21^{+0.09}_{-0.09}$ 0.1 299.7/436
B’$^3$ $9.9^{+1.0}_{-1.0}-14.0^{+3.0}_{-3.0}$ 1.9 $6.21^{+0.10}_{-0.10}$ 0.1 273.5/428
C’$^4$ $11.7^{+0.8}_{-0.4}$ $1.74^{+0.15}_{-0.22}-1.99^{+0.08}_{-0.07}$ $6.20^{+0.10}_{-0.10}$ 0.1 274.1/428
D’$^5$ $12.2^{+0.4}_{-0.2}$ 1.9 $6.20^{+0.10}_{-0.10}$ 0.1 297.1/428
E’$^6$ $12.1^{+0.5}_{-0.5}$ 1.9 $6.20^{+0.10}_{-0.08}$ 0.1 280.7/428
Notes: $^1$ in units of $10^{22}$ ; $^2$ $N_H$, $\Gamma$ tied; $^3$ $N_H$ untied, $\Gamma$ tied; $^4$ $N_H$ tied, $\Gamma$ untied;\
$^5$ gaussian line normalizations untied ; $^6$ reflection component normalisation untied
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We study the finite temperature dynamical structure factor $S(k,\omega)$ of a 1D Bose gas using numerical simulations of the Gross–Pitaevskii equation appropriate to a weakly interacting system. The lineshape of the phonon peaks in $S(k,\omega)$ has a width $\propto |k|^{3/2}$ at low wavevectors. This anomalous width arises from resonant three-phonon interactions, and reveals a remarkable connection to the Kardar–Parisi–Zhang universality class of dynamical critical phenomena.'
author:
- Manas Kulkarni
- Austen Lamacraft
date:
-
-
title: 'From GPE to KPZ: finite temperature dynamical structure factor of the 1D Bose gas'
---
The statistical mechanics of low dimensional fluids, both quantum and classical, has long been a source of theoretical surprises. To give just two examples:
1. The long-time tail $\propto t^{-d/2}$ in the velocity autocorrelation function of a $d$-dimensional classical fluid invalidates hydrodynamics for $d\leq 2$ [@Alder:1970; @Ernst:1970; @Dorfman:1970].
2. The Luttinger liquid description [@Haldane:1981] provides a universal language for 1D quantum liquids, with a panoply of phases arising upon perturbation [@Giamarchi:2004].
Despite these achievements recent developments in the theory of 1D quantum liquids away from the low energy limit make it clear that our understanding of these systems is still rather limited (for a review, see Ref. [@Imambekov:2011]). To take a simple example, consider the dynamical structure factor $S(k,\omega)$ that gives the cross section for inelastic scattering from the liquid as a function of momentum $\hbar k$ and energy $\hbar\omega$ transferred. The Luttinger liquid theory predicts that $S(k,\omega)$ consists only of a pair of delta function peaks $\omega = \pm c|k|$, with $c$ the velocity of sound, corresponding to an undamped phonon oscillation. At finite $k$, however, one expects this delta function to broaden due to interactions between phonons. Attempts to find the resulting lineshape using perturbation theory within the Luttinger framework are plagued by divergences [@Aristov:2007], whose origin we will describe below. As a result, the possibility of capturing the relevant physics within the Luttinger or hydrodynamic formalism is now viewed with a degree of pessimism [@Cheianov:2009].
Almost all of the developments reviewed in Ref. [@Imambekov:2011] pertain to zero temperature. In this work we study the dynamical structure factor of a 1D Bose gas at finite temperature. Aside from being of paramount importance in real systems, we will show that finite temperature brings qualitatively new features that cannot be interpreted simply as a smearing of the zero temperature lineshape. Using analytical arguments and simulations of the Gross–Pitaevskii equation (GPE) appropriate to weak interactions and finite temperature $T$, we find that the lineshape of the phonon peak in $S(k,\omega)$ has a width $\Gamma_{k}\propto |k|^{3/2}$ at low $k$ (see Fig. \[fig:Lineshape\]). As well as dominating any zero temperature structure (generally $\propto k^{2}$) at low wavevectors, the $3/2$ power is *anomalous* relative to the $k^{2}$ scaling that follows from linearized hydrodynamics [@Forster:1975].
![Dynamical structure factor $S(k,\omega)$ of a 1D Bose gas described by the Gross–Pitaevskii equation for wavevectors $k=2\pi p/L$, $p=64,32,16$ (right to left). $L=5\times 2^{13}$ and temperature $T=0.005$, with length being measured in units of the healing length, and energy in units of the chemical potential. Inset: Scaling collapse of the phonon peaks using the ansatz Eq. .[]{data-label="fig:Lineshape"}](RawAndScaled.png){width="\columnwidth"}
This unusual scaling points to a very rich phenomenology. According to a remarkable recent conjecture [@beijeren2011], the long wavelength dynamics of a classical 1D fluid at finite temperature is in the Kardar–Parisi–Zhang (KPZ) universality class describing interface growth [@Kardar:1986; @kriecherbauer2010; @sasamoto2010]. Specifically, the phonon (Brillouin) peaks in $S(k,\omega)$ have the scaling form at low wavenumber $$\label{GPEtoKPZ_PhononScaling}
S^{(\pm)}_{\text{phonon}}(k,\omega) \propto \frac{1}{\Gamma_{k}}f_{\text{PS}}\left(\frac{\omega\pm c|k|}{\Gamma_{k}}\right)$$ where $f_{\text{PS}}(x)$ is given in Eq. (5.7) of Ref. [@prahofer2004]. The meaning of Eq. is that in a frame moving at the speed of sound the density fluctuations moving in the same direction behave exactly as the fluctuations of the interface slope in the KPZ problem.
There are very few experiments confirming KPZ scaling to date [@wakita1997; @maunuksela1997; @takeuchi2010]. Our hope is that the results of this Letter will lead to its observation in new systems. For example, the structure factor of 1D Bose gases of $^{87}$Rb was recently measured using Bragg spectroscopy [@Fabbri:2011], while in Ref. [@savard2011] the hydrodynamics of superfluid Helium in a single nanohole was investigated. In the latter case the sound absorption coefficient is presumably more accessible than the structure factor, and being $\propto\Gamma_{k}$ displays the same anomalous scaling.
*Hydrodynamic description*. Our starting point is the *classical* Hamiltonian describing a 1D gas of bosons of mass $m$ and interaction parameter $g$ $$\label{GPEtoKPZ_GPHam}
H = \int dx\,\left[\frac{|\partial_{x}\Psi|^{2}}{2m}+\frac{g}{2}|\Psi|^{4}\right],$$ where the complex field $\Psi(x)$ obeys the Poisson bracket $\left\{\Psi^{\dagger}(x),\Psi(y)\right\}=i\delta(x-y)$, and we have set $\hbar=1$. The dynamics of $\Psi(x,t)$ is described by the familiar Gross–Pitaevskii equation. The mean field description embodied by the GPE is appropriate when the number of particles in a healing length $\xi\equiv\left(g\rho_{0}m\right)^{-1/2}$ is large, where $\rho_{0}$ denotes the mean density. This corresponds to ‘Luttinger parameter’ $K\equiv\frac{\pi \rho_{0}}{mc}\gg 1$, with $c=\sqrt{g\rho_{0}/m}$ the speed of sound in the uniform state.
After writing the condensate order parameter as $\Psi(x)=\sqrt{\rho(x)}e^{i\theta(x)}$ in terms of the canonically conjugate density $\rho(x)$ and phase $\theta(x)$, the Hamiltonian takes the form $$\label{GPEtoKPZ_HydroHam}
H = \int dx \left[\frac{\rho\left(\partial_{x} \theta\right)^{2}}{2m}+\frac{(\partial_{x} \sqrt{\rho})^{2}}{2m}+\frac{g}{2}\rho^{2}\right].$$ Dynamics near a state of uniform density with $\rho(x,t)=\rho_{0}$, $\theta(x,t)=0$ can be described in the first approximation by writing $\rho=\rho_{0}+\varrho$, retaining only terms quadratic in $\varrho$ and $\theta$ from Eq. $$\label{GPEtoKPZ_QuadHam}
H_2 = \int dx \left[\frac{\rho_{0}\left(\partial_{x} \theta\right)^{2}}{2m}+\frac{(\partial_{x} \varrho)^{2}}{8m\rho_{0}}+\frac{g}{2}\varrho^{2}\right].$$ $H_{2}$ is solved by introducing the mode expansions for $\varrho(x)$ and $\theta(x)$ for a system of length $L$ $$\label{Chris_notes_genmodes}
\begin{split}
\varrho(x)&=\sqrt{\frac{\rho_{0}}{2L}}\sum_{k\neq 0}e^{-\kappa_{k}}\left(b_{k}e^{ikx}+\text{c.c}\right) \\
\theta(x)&= \frac{i}{\sqrt{2\rho_{0}L}} \sum_{k\neq 0}e^{\kappa_{k}}\left(b_{k}e^{ikx}-\text{c.c}\right).
\end{split}$$ After substitution in Eq. , $e^{\kappa_{k}}$ is chosen to diagonalize $H_{2}=\sum_{k} {{\cal E}}_{k}|{b^{\vphantom{\dagger}}}_{k}|^{2}$ with ${{\cal E}}_{k}=\left[\frac{k^{2}}{2m}\left(\frac{k^{2}}{2m}+2g \rho_{0} \right)\right]^{1/2}$ the Bogoliubov dispersion relation. At low $k$ ${{\cal E}}_{k}\to c|k|+O(k^{3})$. The deviation from the linear dispersion is due to the second term of Eqs. and , sometimes called the ‘quantum pressure’.
Interactions between the modes are described by the anharmonic parts of Eq. . The most important interaction arises from the first term, and has the form $$\label{GPEtoKPZ_CubicVertex}
\begin{split}
H_{3} &= \int dx\, \frac{\varrho(\partial_{x}\theta)^{2}}{2m}\\
&=\sum_{k_{1}+k_{2}+k_{3}=0}\sqrt{\frac{c|k_{1}k_{2}k_{3}|}{32L\rho_{0}m}}\left({b^{\vphantom{\dagger}}}_{k_{1}}{b^{\vphantom{\dagger}}}_{k_{2}}{b^{\vphantom{\dagger}}}_{k_{3}}\right.\\
&\qquad\qquad\left.-{b^{\vphantom{\dagger}}}_{k_{1}}{b^\dagger}_{-k_{2}}{b^\dagger}_{-k_{3}}-{b^\dagger}_{k_{1}}{b^{\vphantom{\dagger}}}_{-k_{2}}{b^{\vphantom{\dagger}}}_{-k_{3}}\right).
\end{split}$$ where for simplicity we have assumed the low $k$ limit for $e^{\kappa_{k}}$. The difficulty associated with a perturbative treatment of this interaction is now apparent. Substituting the time dependence ${b^{\vphantom{\dagger}}}_{k}\to{b^{\vphantom{\dagger}}}_{k}e^{-i{{\cal E}}_{k}t}$ associated with $H_{2}$, we see that the second and third terms of Eq. are *resonant* for purely linear dispersion when all three modes move in the same direction (in quantum mechanical language energy and momentum conservation are simultaneously satisfied). One may object that the $O(k^{3})$ deviation from linearity at low $k$ due to the quantum pressure term removes this difficulty, but the $|k|^{3/2}$ broadening that we find dominates this effect at low $k$. The other nonlinearities arising from Eq. are likewise irrelevant in this limit.
The need for a non-perturbative approach was recognized long ago in Ref. [@andreev:1980], where a self-consistent mode-coupling (SCMC) treatment of the cubic interaction was given, ignoring vertex corrections, and yielded $\Gamma_{k}\propto \sqrt{T |k|^{3}}$, where $T$ is the temperature (see also Ref. [@Samokhin:1998]). The same result was independently rederived much later [@delfini2006]. In contrast, a renormalization group (RG) argument based on Galilean invariance (first appearing in the related context of the noisy Burgers equation [@forster1977]) predicts a dynamical critical exponent $z=1+d/2$ for $d<2$ [@narayan:2002]. This suggests that Galilean invariance lies behind the success of the SCMC approach, an idea that finds support in the analysis of vertex corrections for the case of Burgers equation [@frey1996].
We wish to emphasize that the SCMC theory of Refs. [@andreev:1980; @Samokhin:1998; @delfini2006] is an uncontrolled approximation, while the RG analysis of Ref. [@narayan:2002] was based on the equations of viscous 1D hydrodynamics with thermal fluctuations appearing as noise sources. It is therefore desirable to study the purely Hamiltonian dynamics described by Eq. .
*Equations of motion.* To gain some intuition regarding the connection of the GPE to the noisy Burgers equation and thence to the KPZ universality class, we discuss the equations of motion of the Hamiltonian Eq. . Ignoring the quantum pressure term, these are $$\label{GPEtoKPZ_HydroEq}
\begin{split}
\partial_{t} \rho+\partial_{x}\left(v\rho\right)=0 \\
\partial_{t}v + v\partial_{x}v+(g/m)\partial_{x}\rho=0
\end{split}$$ where $v=\partial_{x}\theta/m$ is the superfluid velocity. These are the continuity and Euler equations for the 1D Bose gas, and may be put in the Riemann form [@Menikoff:1989]. $$\label{GPEtoKPZ_RHproblem}
\partial_{t}(v\pm 2c_{\rho})+v_{\pm}\partial_{x}(v\pm 2c_{\rho})=0,$$ where $v_{\pm}=v\pm c_{\rho}$, and $c_{\rho}=c\sqrt{\rho/\rho_{0}}$ is the speed of sound in a frame in which the fluid is locally at rest. Eq. tells us that $v\pm 2c_{\rho}$ are constant along their respective characteristic curves $X_{\pm}(t)$ defined by $\dot X_{\pm}(t)=v_{\pm}(X_{\pm}(t),t)$. Alternatively, we may write Eq. as $$\label{GPEtoKPZ_2Burgers}
\partial_{t}v_{\pm}+v_{\pm}\partial_{x}v_{\pm}=\frac{1}{3}(\partial_{t}+v_{\pm}\partial_{x})v_{\mp}.$$ The interpretation of Eq. is as follows. Right and left moving sound waves propagate through the fluid at the velocities $v_{\pm}$, but their motion is perturbed by the variation in the comoving frame of the velocity of the counterpropagating wave. As a result, the characteristic curves are not straight lines (see Fig. \[fig:characteristics\]). We can think of Eq. as a pair of driven Burgers equations, in which the left moving waves act as a noise term on the propagation of the right moving waves, and vice versa. The viscous $\nu\partial_x^2v_{\pm}$ term is absent for this Hamiltonian system but will be generated upon coarse graining.
![The characteristic curve $X_{+}(t)$ giving the path of a right-moving phonon wavepacket is not straight due to the influence of the counterpropoagating waves[]{data-label="fig:characteristics"}](characteristics.png){width="0.8\columnwidth"}
It is natural to ask how this situation changes for a *Fermi* gas, which has the hydrodynamic description $$\label{GPEtoKPZ_FermiHydro}
H_{\text{Fermi}}= \int dx \left[\frac{\rho\left(\partial_{x} \theta\right)^{2}}{2m}+\frac{\pi^{2}\rho^{3}}{6m}\right],$$ with the second term representing the Fermi pressure. The same analysis now yields the *uncoupled* Burgers equations $$\label{GPEtoKPZ_FermiBurgers}
\partial_{t}v_{\pm}+v_{\pm}\partial_{x}v_{\pm} =0$$ where $v_{\pm}=v\pm \pi \rho/m$ are the right and left moving Fermi velocities. The characteristics $X_{\pm}(t)$ are now straight lines, and the free Fermi gas therefore represents an exceptional fluid in which we expect no anomalous broadening of the type discussed here.
*Numerical simulations*. The GPE is solved using the splitting method, whereby $\Psi(x,t)$ is evolved for a timestep $\tau$ alternately by the kinetic $T=\frac{1}{2}\int dx\, |\partial_{x}\Psi|^{2}$ and potential $V=\frac{1}{2}\int\ dx\,|\Psi|^{4}$ terms of the Hamiltonian [@McLachlan:1993] $$\label{GPEtoKPZ_TVsplitting}
\begin{split}
&{{\cal T}}_{\tau}:\tilde\Psi(k,t)\to e^{-ik^{2}\tau/2}\tilde\Psi(k,t)\\
&{{\cal V}}_{\tau}:\Psi(x,t)\to e^{-i\tau|\Psi(x,t)|^{2}}\Psi(x,t),
\end{split}$$ where $\tilde\Psi(k,t)$ denotes the Fourier transform of $\Psi(x,t)$, and we now switch to measuring distance in units of the healing length $\xi$, time in units of (inverse) chemical potential $\mu\equiv g\rho_{0}$, and $\Psi$ in units of $\sqrt{\rho_{0}}$. Algorithms of this type are *symplectic*. This means that the method exactly simulates a Hamiltonian $H_{\tau}$ with $H_{\tau}-H$ a power series in $\tau$. The lowest power of $\tau$ in the series determines the *order* of the method. Two benefits of symplectic integrators for statistical mechanical simulations are: i) exact conservation of phase space volume (i.e. Liouville’s theorem is satisfied) and ii) no drift in the energy due to exact conservation of $H_{\tau}$.
We use the method ${{\cal V}}_{\tau/2}\cdot{{\cal T}}_{\tau}\cdot{{\cal V}}_{\tau/2}$ – often called ‘Leapfrog’ – which is second order with [@McLachlan:1993] $$\label{GPEtoKPZ_Htau}
\begin{split}
H_{\tau}-H &= \frac{\tau^{2}}{24}\left(2\{T,\{V,T\}\}+\{V,\{T,V\}\}\right)+O(\tau^{4})\\
&=\frac{\tau^{2}}{24}\int dx\left[\rho^{2}\left(\partial_{x}\theta\partial_{x}^{3}\theta-(\partial_{x}^{2}\theta)^{2}\right)-\rho(\partial_{x}\rho)^{2}\right]\\
&\qquad\qquad+O(\tau^{4})
\end{split}$$ We display the explicit form of the first correction term in $H_{\tau}$ to demonstrate that the additional terms generated by discretizing time are higher order in spatial gradients than Eq. , and are therefore not expected to change the low $k$ behavior.
The harmonic modes are initially populated according to equipartition, so that the ${b^{\vphantom{\dagger}}}_{k}$ are taken as complex Gaussian random variables with $\langle|{b^{\vphantom{\dagger}}}_{k}|^{2}\rangle=\frac{T}{{{\cal E}}_{k}}$ for temperature $T$. This assumes sufficiently weak nonlinearity, which is confirmed by the absence of transient behavior for the parameter ranges explored, indicating that the initial state is close to thermal.
We choose a spatial discretization scale $a=5$ to ensure all wavevectors are in the regime of linear dispersion $k\ll \xi^{-1}$. The time step of $\tau=2$ is then at the limit of stability of the algorithm. Most simulations use systems of length $L=2^{13}a$, but we check that the results are not significantly altered for $L=2^{14}a$ (see Fig. \[fig:widths\]). Periodic boundary conditions are used throughout.
At each time step we compute the Fourier components of the density $\rho(x,t)$ $$\label{GPEtoKPZ_rhoq}
\rho_{k}(t) = \sum_{n=0}^{N-1} |\Psi(na,t)|^{2}e^{-2\pi i kna}\qquad k = 0,\frac{2\pi}{L},\ldots, \frac{\pi}{a}.$$ The resulting time series is then Fourier transformed to give the dynamical structure factor $$\label{GPEtoKPZ_StructureFactor}
S(k,\omega)=\langle |\rho_{k,\omega}|^{2}\rangle, \qquad \omega = 0,\pm\frac{2\pi}{N_{\text{bin}}\tau},\ldots$$ where $N_{\text{bin}}$ is the bin size used for the computation of the power spectra (at least $2^{20}$), and the angular brackets denotes an average over $\sim 128$ runs of length $N_{\text{steps}}\tau$ with different random initial conditions. Thus for $N_{\text{bin}}=2^{20}$, $N_{\text{steps}}=2^{21}$ corresponds to an effective average over $\sim 256$ runs, assuming no correlation between the two halves of a given run.
We gather data for wavevectors $k=\frac{2\pi p}{L}$, with $p=1,2, 4,\ldots 4096$. Typical results, displaying good data collapse (assuming a width scaling as $|k|^{3/2}$) are shown in Fig. \[fig:Lineshape\]. For an unbiased test, power spectra are folded assuming symmetry between positive and negative frequencies, and the phonon peaks fitted to a Lorentzian to extract the amplitude, peak frequency, and width. Data are shown in Fig. \[fig:widths\]. A good fit to the scaling form $\Gamma_{k}\propto |k|^{z}$ is obtained over 1.5 decades and yields $z = 1.510 \pm 0.018$. Significant deviations from scaling are obtained at higher wavevectors.
![Dependence of linewidth on wavevector, for system sizes $N=2^{13}$ (red dots) and $N=2^{14}$ (blue crosses), with $T=0.005$. The data are almost perfectly coincident. The blue line is a fit for wavevectors $2\pi p / L$, $p=2,4,\ldots 64$ giving $z = 1.510 \pm 0.018$. Higher wavevectors show a marked deviation from $3/2$ scaling, and $k=2\pi/L$ is omitted because the linewidth lies below resolution. []{data-label="fig:widths"}](WidthFit.png){width="\columnwidth"}
*Conclusion*. We have provided convincing analytical and experimental evidence of the relationship between the finite temperature dynamics of the 1D Bose gas and the KPZ universality class. Galilean invariance is of paramount importance: simulations of wave equations with cubic nonlinearity but without Galilean invariance show $z\sim 1$ [@Kulkarni:]. Numerous extensions of the results of this work to multicomponent (or spinor) quantum fluids, and to transient rather than equilibrium dynamics, may be envisaged. In addition, the challenging problem of describing – within a single framework – the finite temperature phenomena described here, and the zero temperature results reviewed in Ref. [@Imambekov:2011], remains to be solved.
*Acknowledgements* Our thanks are due to Robert McLachlan and Uwe Täuber for helpful discussions, and to the University of Virginia Alliance for Computational Science and Engineering, especially Katherine Holcomb, for their assistance. A.L. gratefully acknowledges the support of the Research Corporation and the NSF through award DMR-0846788. M.K thanks Saul Lapidus for useful discussions.
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---
abstract: 'The temperature dependence of the London penetration depth $\lambda $ was measured for an untwined single crystal of YBa$_2$Cu$_3$O$_{7-\delta}$ along the three principal crystallographic directions ($a$, $b$, and $c$). Both in-plane components ($\lambda_a$ and $\lambda_b$) show an inflection point in their temperature dependence which is absent in the component along the $c$ direction ($\lambda_c$). The data provide convincing evidence that the in-plane superconducting order parameter is a mixture of $s+d-$wave symmetry whereas it is exclusively $s-$wave along the $c$ direction. In conjunction with previous results it is concluded that coupled $s+d-$order parameters are universal and intrinsic to cuprate superconductors.'
author:
- 'R. Khasanov'
- 'S. Strässle'
- 'D. Di Castro'
- 'T. Masui'
- 'S. Miyasaka'
- 'S. Tajima'
- 'A. Bussmann-Holder'
- 'H. Keller'
title: Universal observation of multiple order parameters in cuprate superconductors
---
It is believed that the CuO$_2$ planes are the essential building units in cuprate high-temperature superconductors (HTS’s) where superconductivity occurs. Even though either static or dynamic distortions of these planes destroy the cubic symmetry, many theoretical approaches ignore the observed orthorhombicity and idealize the planar structure, mostly in order to justify a pure $d-$wave order parameter. Early on it was, however, emphasized that cuprates must have a more complex order parameter than just $d$-wave [@Muller95; @Muller97], supported by many experiments like nuclear magnetic resonance (NMR) [@Martindale98], Raman scattering [@Masui03; @Friedl90], angle-resolved electron tunneling [@Smilde05], Andreev reflection [@Deutscher05], angular-resolved photoemission (ARPES) [@Lu01], muon-spin rotation ($\mu$SR) [@Khasanov07_La214; @Khasanov07_Y124; @Khasano07_La214_book], and neutron crystal-field spectroscopy [@Furrer07]. In addition, experiments along the $c$ axis like, [*e.g.*]{}, tunneling [@Sun94], bi-crystal twist Josephson junctions [@Li99], optical pulsed probe [@Kabanov99], and optical reflectivity [@Muller04] suggest that a pure $s-$wave order parameter is realized here.
Multiple order parameter scenarios were proposed shortly after the BCS theory in order to account for a complex band structure and interband scattering [@Suhl59; @Moskalenko59; @Kresin73]. This approach has the advantage that high-temperature superconductivity can easily be realized even within weak coupling theories since interband scattering provides a pairwise exchange between different bands which strongly enhances the transition temperature ($T_c$) as compared to a single band model. The first realization of two-band superconductivity has been made in Nb doped SrTiO$_3$ [@Binnig80] and has not attracted very much attention. With the discovery of high-temperature superconductivity in MgB$_2$ two-gap superconductivity became more prominent, and meanwhile many more systems exhibiting multi-band superconductivity have been discovered (see, e.g., Refs. [@Binnig80; @Giubileo01; @Boaknin03; @Shulga98; @Seyfarth05]). Interestingly, in all these systems the coupled superconducting order parameters are of the same symmetry, i.e., $s+s$, $d+d$. In this respect HTS’s are novel since here mixed order parameter symmetries are realized, namely, $s+d$. Theoretically it has been shown that mixed order parameters support even high values of $T_c$ and lead to an almost doubling of the transition temperature as compared to coupled order parameters of the same symmetry [@Bussmann-Holder03a].
In order to prove that complex order parameters are intrinsic and universal to HTS, previous $\mu$SR measurements [@Khasanov07_La214; @Khasanov07_Y124; @Khasano07_La214_book] were continued for another HTS family, namely YBa$_2$Cu$_3$O$_{7-\delta}$. The $\mu$SR technique has the advantage that it is bulk sensitive and a direct probe of the London penetration depth which is highly anisotropic in HTS’s. Recent results for La$_{1-x}$Sr$_x$CuO$_4$ and YBa$_2$Cu$_4$O$_8$ [@Khasanov07_La214; @Khasanov07_Y124; @Khasano07_La214_book] clearly demonstrate the existence of two coupled $s+d-$wave gaps in the CuO$_2$ planes and an $s-$wave gap along the $c$ axis in YBa$_2$Cu$_4$O$_8$. While these findings already suggest that a complex gap structure is intrinsic to HTS, the new results on YBa$_2$Cu$_3$O$_{7-\delta}$, presented below, support this conclusion consistently. We are thus reasoning that $s+d-$wave superconductivity in the planes and $s-$wave superconductivity along the $c$ direction are intrinsic and universal to this complex material class. In addition, this finding imposes serious constraints for theoretical models, [*e.g.*]{}, neither purely electronic nor 2D based approaches capture this complicated picture.
The crystal was grown by a crystal pulling technique [@Y123_sample_preparation] and exhibited a rectangular shape of approximate size of 4x4x1 mm$^3$. The sample was detwinned by annealing it under stress for 2 months at 400$^{\rm o}$C. The total fraction, where $a$ and $b$ axis are exchanged, occupies approximately 8 to 10% of the entire crystal, as confirmed by measurements with a polarized microscope. $T_c$ and the transition width were determined by DC-magnetization measurements and found to be 91.2 K and 2 K, respectively.
The transverse-field $\mu$SR experiments were carried out at the $\pi$M3 beam line at the Paul Scherrer Institute (Villigen, Switzerland). The samples were field cooled from above $T_c$ to 1.7 K in magnetic fields ranging from 0.012 T to 0.64 T. The typical counting statistics were $\sim$20-25 million muon detections per experimental point. The experimental data were analyzed within the same scheme as described in Refs. . This is based on a four component Gaussian fit of the $\mu$SR time spectra where one component describes the background signal stemming from muons stopped outside the sample, and the three other components describe the asymmetric local magnetic field $P(B)$ distribution in the superconductor in the mixed state. The magnetic field penetration depth $\lambda$ was derived from the second moment of $P(B)$ since $\lambda^{-4}\propto\langle\Delta
B^2\rangle\propto\sigma_{sc}^2$ [@Brandt88]. The superconducting part of the square root of the second moment ($\sigma_{sc}\propto\lambda^{-2}$ ) was obtained by subtracting the normal state nuclear moment contribution ($\sigma_{nm}$) from the measured $\sigma$, as $\sigma_{sc}^2=\sigma^2-\sigma_{nm }^2$ (see Ref. for details).
![(Color online) Temperature dependences of $\sigma_{ab}\propto \lambda^{-2}_{ab}$ of YBa$_2$Cu$_3$O$_{7-\delta}$ measured after field cooling in $\mu_0H=$0.1 T (a) and 0.64 T (b). The red lines represent results of a numerical calculation using the two-gap model [@Khasanov07_La214; @Khasanov07_Y124; @Khasano07_La214_book] with parameters as summarized in Table \[Table:two-gap\]. The contributions of the small $s-$wave and the large $d-$wave gap to the in-plane superfluid density are shown by the blue and the black lines, respectively.[]{data-label="fig:sigma_vs_field"}](figure1){width="0.8\linewidth"}
Since cuprates are highly anisotropic, the relation $\sigma_{sc}\propto\lambda^{-2}$ has to be extended to account for magnetic fields applied along the three crystallographic directions ($i,j,k=a,b,c$). For the field applied along the $i-$th principal axis the penetration depth is determined from the second moment like $\lambda_{jk}^{-2}=(\lambda_j\lambda_k)^{-1}\propto\sigma_{jk} =
\sqrt{\sigma_j\sigma_k}$ [@Thiemann89]. Here the index [*sc*]{} was omitted for clarity. The magnetic field dependence of the in-plane penetration depth has first been measured for different fields (0.05 T, 0.1 T, 0.2 T, and 0.64 T). The field was applied along the crystallographic $c$ axis and, subsequently, the sample was cooled down from above $T_c$ to 1.7 K. For this field configuration the in-plane component of the superfluid density $\sigma_{ab}\propto\lambda^{-2}_{ab}=(\lambda_a\lambda_b)^{-1}$ was obtained. In Fig. \[fig:sigma\_vs\_field\] $\sigma_{ab}$ is shown as a function of temperature for two representative fields of 0.1 T and 0.64 T. In analogy to previous results [@Khasanov07_La214; @Khasanov07_Y124; @Khasano07_La214_book], an inflection point in $\sigma_{ab}(T)$ is observed at $T\simeq 10$ K, which is a typical signature for the coexistence of a small $s-$wave and a large $d-$wave gap. Accordingly, the analysis of the data was performed by assuming that $\sigma(T)$ can be decomposed into two components having $d-$wave and $s-$wave symmetry as $\sigma(T)=\sigma^s(T)+\sigma^d(T)$ [@Khasanov07_La214; @Khasanov07_Y124; @Khasano07_La214_book]. The temperature dependences of the individual components were obtained within the same framework as presented in Refs. . The comparison of experimental and theoretical results is made in Fig. \[fig:sigma\_vs\_field\], where the red lines refer to the sum of the two components, whereas the blue and the black lines display the individual $s-$ and $d-$wave contributions, respectively. For all magnetic fields the analysis was based on common zero-temperature gap values ($\Delta^s_0$ and $\Delta^d_0$) but field dependent second moments \[$\sigma^s(0)$, $\sigma^d(0)$\]. The zero-temperature gap values obtained in this way are $\Delta^s_0=0.707(11)$ meV, $\Delta_0^d=22.92(9)$ meV, and are in good agreement with results from tunneling experiments for the $d-$wave gap (see, [*e.g.*]{}, Ref. ). The parameters obtained from the analysis are summarized in Table \[Table:two-gap\].
The $d-$wave contribution to the total superfluid density $\omega=\sigma^d(0)/[\sigma^s(0)+\sigma^d(0)]$ increases with increasing field (see Fig. \[fig:omega\_vs\_field\]), similar to the field dependence observed for La$_{1.83}$Sr$_{0.17}$CuO$_4$ [@Khasanov07_La214]. This dependence can be understood by the fact that superconductivity is suppressed stronger in the $s-$wave band with increasing field than in the $d-$wave band [@Khasanov07_La214].
![(Color online) The $d-$wave contribution to the in-plane superfluid density $\omega=\sigma^d(0)/[\sigma^s(0)+\sigma^d(0)]$ as a function of the magnetic field of YBa$_2$Cu$_3$O$_{7-\delta}$. The line is a guide to the eye. []{data-label="fig:omega_vs_field"}](figure2){width="0.8\linewidth"}
---------- ----------------- ----------------- ----------- -------------- -------------- -- -- --
$\mu_0H$ $\sigma^d(0)$ $\sigma^s(0)$ $\omega$ $\Delta^d_0$ $\Delta^s_0$
(T) ($\mu$s$^{-1}$) ($\mu$s$^{-1}$) (meV) (meV)
0.05 1.78(2) 4.80(7) 0.729(12)
0.1 2.01(2) 5.53(6) 0.734(11)
0.2 1.87(2) 5.63(7) 0.751(13) 22.92(9) 0.707(11)
0.64 1.49(2) 5.01(7) 0.771(16)
---------- ----------------- ----------------- ----------- -------------- -------------- -- -- --
: \[Table:two-gap\] Summary of the two-gap analysis for untwined single-crystal YBa$_2$Cu$_3$O$_{7-\delta}$ for the magnetic field applied along the $c$ direction. The meaning of the parameters is – $\mu_0H$: external magnetic field, $\sigma^d(0)$ and $\sigma^s(0)$: $d-$wave and $s-$wave contribution to the zero-temperature $\mu$SR relaxation rate $\sigma(0)$, $\omega=\sigma^d(0)/[\sigma^s(0)+\sigma^d(0)]$: the contribution of the large $d-$wave gap to the total in-plane superfluid density, $\Delta^d_0$: $d-$wave gap at $T=0$ K, $\Delta^s_0$: $s-$wave gap at $T=0$ K.
The individual components of the inverse squared penetration depth ($\lambda_a^{-2}$, $\lambda_b^{-2}$, and $\lambda_c^{-2}$) as functions of temperature were obtained by applying the magnetic field along the three crystallographic axes (0.012 T along $a$ and $b$, and $0.1$ T along $c$). This yields the axis-related superfluid densities according to [@Khasanov07_Y124]: $$\sigma_i=\sigma_{ij}\cdot\sigma_{ik}/\sigma_{jk}
\propto\lambda_i^{-2}.
\label{eq:lambda_i}$$ The results are shown in Fig. \[fig:sigma\_i\] (a). Obviously $\sigma_a$ and $\sigma_b$ have a very similar temperature dependence, in particular, the inflection point at $T\simeq 10$ K and the linear increase in an intermediate range of temperatures ($60$ K$\gtrsim T \gtrsim 10$ K). The temperature dependence of $\sigma_c$ is qualitatively very different from the one of $\sigma_a$ and $\sigma_b$. Here a saturation is observed at $T\lesssim40$ K. $\sigma_a(T)$ and $\sigma_b(T)$ can be well described by the two-component approach mentioned above where the same zero-temperature gap values were used. From this analysis the following individual contributions from the $s-$ and the $d-$wave components along the $a$ and $b$ axis are obtained: $\sigma_a^s(0)=1.19$ $\mu$s$^{-1}$, $\sigma_a^d(0)=3.83$ $\mu$s$^{-1}$ and $\sigma_b^s(0)=2.51$ $\mu$s$^{-1}$, $\sigma_b^d(0)=5.95$ $\mu$s$^{-1}$. Since the relative contributions of the large $d-$wave component are almost the same along $a$ and $b$ directions, namely, $\omega_a=0.70$, $\omega_b=0.76$, it is plausible to conclude that not the CuO chains are the cause of the two-component behavior [@Atkinson95], but that this is an intrinsic property of cuprates. The same conclusions were reached from different experimental techniques as, e.g., NMR [@Martindale98], Raman scattering [@Masui03; @Friedl90], and ARPES [@Lu01]. In particular, Masui [*et al.*]{} [@Masui03] showed that the $s+d$ symmetry of the order parameter is required in order to describe the Raman data, even for [*tetragonal*]{} Tl$_2$Ba$_2$CuO$_{6+\delta}$ HTS’s.
![(Color online) (a) Temperature dependences of $\sigma_{a}\propto \lambda^{-2}_{a}$, $\sigma_{b}\propto
\lambda^{-2}_{b}$, and $\sigma_{c}\propto \lambda^{-2}_{c}$ of YBa$_2$Cu$_3$O$_{7-\delta}$ obtained from $\sigma(T)$ measured along the crystallographic $a$, $b$, and $c$ directions by using Eq. (\[eq:lambda\_i\]). Lines represent results of the analysis within the two-component (black and red lines) and one-component (blue line) models [@Khasanov07_Y124]. (b) Temperature dependences of the anisotropy parameters $\gamma_{ab}$, $\gamma_{ca}$, and $\gamma_{cb}$ obtained as $\gamma_{ij}=\lambda_i/\lambda_j=\sqrt{\sigma_j/\sigma_i}$ (see text for details). []{data-label="fig:sigma_i"}](figure3){width="0.9\linewidth"}
The temperature dependence of the $c$ axis related superfluid density $\sigma_{sc}\propto\lambda_c^{-2}$ resembles very much the one observed for YBa$_2$Cu$_4$O$_8$ [@Khasanov07_Y124] and follows closely the one expected for a single $s-$wave gap. The full blue curve in Fig. \[fig:sigma\_i\] (a) corresponds to an analysis based on an isotropic $s-$wave gap with $\Delta^s_0=17.52$ meV and $\sigma_c(0)=0.183$ $\mu$s$^{-1}$. The present results are in good agreement with previous findings of tunnelling experiments, where an $s-$wave gap along the $c$ direction was reported [@Sun94]. The $s-$wave component along the $c$ axis is not easily detectable by most experimental methods because either very well oriented films should be prepared or bulk methods have to be used. Due to the fact that many experiments are surface sensitive only, and $ab$ oriented samples and films are hardly available, these techniques are unable to see the $s-$wave component along the $c$ axis. Its observation is, however, important since the coupling of a major $d-$wave component in the $ab$ plane to the $s-$wave component along the $c$ axis mixes both symmetries in the planes already. In addition, theory predicts for this scenario, that a pure $d-$wave order parameter is never observable. On the other hand, also along the $c$ direction an admixture of the $d-$wave order parameter has to be present [@Bussmann-Holder03a]. This latter statement could provide an explanation for the optical conductivity spectra along the $c$ direction where strongly anisotropic gap like features have been observed together with a finite density of states at the Fermi energy [@Schutzmann94; @Tajima97].
Finally, the anisotropy along all three crystallographic directions is addressed. This can be calculated by using Eq. (\[eq:lambda\_i\]) and defining the anisotropy as $\gamma_{ij}=\lambda_i/\lambda_j=\sqrt{\sigma_j/\sigma_i}$. The results are shown in Fig. \[fig:sigma\_i\] (b). While the in-plane anisotropy ($\gamma_{ab}$) is almost constant for all temperatures, both out-of-plane components ($\gamma_{ca}$ and $\gamma_{cb}$) exhibit a sharp increase, as expected from Fig. \[fig:sigma\_i\] (a). In addition, it is seen that while $\gamma_{ab}$ is almost always close to 1.2, $\gamma_{ca}$, $\gamma_{cb}$ are substantially larger and reach values between 5 and 6.5 in the low-$T$ limit. This high anisotropy is in very good agreement with previously reported values [@Ishida97; @Ager00]. It reflects the fact that the CuO$_2$ planes have nearly Fermi liquid-like metallic properties whereas along the $c$ direction mostly insulating behavior is observed.
Our conclusions from the above presented data are manifold. Since $s+d-$wave symmetries of the superconducting order parameter were observed previously in various cuprate families by various different techniques [@Martindale98; @Masui03; @Friedl90; @Smilde05; @Deutscher05; @Lu01; @Khasanov07_La214; @Khasanov07_Y124; @Khasano07_La214_book; @Furrer07], the new $\mu$SR data together with earlier results on structurally different compounds [@Khasanov07_La214; @Khasanov07_Y124; @Khasano07_La214_book] support the idea that this behavior is [*intrinsic*]{} and [*universal*]{}. Similarly, the observation of an $s-$wave order parameter along the crystallographic $c$ axis is proposed to be intrinsic as well. Specifically, this latter point emphasizes the importance of the [*third*]{} dimension for HTS’s which was neglected in most of the theoretical models. In this context it is worth mentioning that the importance of the $c$ axis has already been emphasized from ab-initio band structure calculations, where trends in $T_c$ were correlated with CuO$_2$ apical oxygen distances [@Pavarini01]. Also, from first principles doping dependent computations of ARPES intensities, it was concluded that contributions from the $c$ axis are of crucial importance in understanding the physics of HTS’s [@Sahrahorpi05]. On the other hand, the observation of mixed order parameters and more specifically, the additional $s-$wave component, require that the lattice must be considered in the physics of HTS’s.
It is a pleasure to acknowledge many stimulating and supporting discussions with K. A. Müller. This work was partly performed at the Swiss Muon Source (S$\mu$S), Paul Scherrer Institute (PSI, Switzerland). The authors are grateful to A. Amato and D. Herlach for assistance during the $\mu$SR measurements. This work was supported by the Swiss National Science Foundation, by the K. Alex Müller Foundation, and the EU Project CoMePhS.
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{
"pile_set_name": "ArXiv"
}
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---
author:
- 'S. Hesselbach[^1]'
- 'S. Moretti[^2]'
- 'S. Munir[^3]'
- 'P. Poulose[^4]'
title: 'Di-Photon Higgs Decay in SUSY with CP Violation'
---
Introduction {#intro}
============
One of the main objectives of the upcoming Large Hadron Collider (LHC) is to investigate ElectroWeak Symmetry Breaking (EWSB). While the Standard (SM) Higgs mechanism predicts one physical scalar particle, many models beyond it have more than one scalar states as well as pseudo-scalar (and charged) ones. In the presence of CP-violation scalar/pseudo-scalar mixing occurs to give the physical Higgs states. Studying CP properties of the Higgs particles thus becomes an important feature in distinguishing different models. Among the new-physics scenarios Supersymmetry (SUSY) is one of the favourites of particle physicists. LHC will investigate various aspects of SUSY with special attention to the Minimal Supersymmetric Standard Model (MSSM). While the phenomenology of the CP-conserved MSSM has been thoroughly studied, many issues of the MSSM with CP violation are yet to be investigated. Many parameters of the MSSM can well be complex and thus explicitly break CP invariance inducing CP violation also in the Higgs sector beyond Born approximation [@Pilaftsis:1998dd]. After elimination of unphysical phases and imposing universality conditions at the unification scale two independent phases remain, the phase $\phi_\mu$ of the higgsino mass term $\mu$ and a common phase $\phi_{A_f}$ of the soft trilinear Yukawa couplings $A_f$ in the sfermion sector [@Pilaftsis:1999qt]. Experimental searches of Electric Dipole Moments (EDMs) of electron and neutron put constraints on the CP phases of any model. In the MSSM with CP violation these constraints can be avoided by taking the sfermions belonging to the first two generations to be very heavy (see [@Olive:2005ru] for a review).
We consider the di-photon decay mode, $H_1\rightarrow \gamma\gamma$, of the lightest neutral Higgs boson $H_1$, which involves direct, i.e. leading, effects of the SUSY phases through couplings of the $H_1$ to SUSY particles in the loops as well as indirect, i.e. sub-leading, effects through the scalar/pseudo-scalar mixing yielding the Higgs mass-eigenstate $H_1$. In scenarios with heavy SUSY particles, where CP violation enters solely through the scalar/pseudo-scalar mixing, the SUSY CP phases can result in a strong suppression of the BR of the decay $H_1\rightarrow \gamma\gamma$ as well as of the rate of the combined production and decay process $gg \to H_1 \to \gamma\gamma$ [@Choi:2001iu]. Here, we summarize the results of [@Moretti:2007th; @Hesselbach:2007en] focusing especially on the effects of light SUSY particles in the decay $H_1\rightarrow \gamma\gamma$. The analysis of the full production and decay process at the LHC is postponed to a forthcoming publication [@fullprocess].
The $H_1\rightarrow \gamma\gamma$ decay
=======================================
As mentioned in the Introduction we consider explicit CP violation (and assume that the Higgs vacuum expectation values are real). Thus, in this particular scenario under study with common phases for the trilinear couplings and separately for the gaugino masses, we are left with two independent phases after symmetry considerations. As intimated, we take these to be $\phi_\mu$ and $\phi_{A_f}$. The leading terms in the CP-violating scalar/pseudo-scalar mixing in the Higgs sector are proportional to $\mathrm{Im}(\mu A_f)$, hence we assume $\phi_{A_f} = 0$ and analyse the effects of a non-zero $\phi_\mu$ in the following. With the help of the publicly available <span style="font-variant:small-caps;">Fortran</span> code <span style="font-variant:small-caps;">CPSuperH</span> [@Lee:2003nta], version 2, which calculates the mass spectrum and decay widths of all Higgs bosons along with their couplings to SM and SUSY particles, we analyse the Higgs decay into the di-photon channel in the CP-violating MSSM and compare it with that of the CP-conserving MSSM. A Higgs boson in the MSSM decays at one-loop level into two photons through loops of fermions, sfermions, $W^\pm$ bosons, charged Higgs bosons and charginos, see Fig. \[fig:HiggsPho\]. A random parameter space scan over about 100,000 parameter space points to study the general behaviour of the $\mathrm{BR}(H_1 \to \gamma\gamma)$ for non-zero $\phi_\mu$ has revealed that about 50% deviations are possible for $M_{H_1}$ around 104 GeV for $\phi_\mu=100^\circ$. In the considered mass range of 90–130 GeV an average of 30% deviation is found to occur. Furthermore, this study of the average behaviour with and without a light stop clearly establishes the strong impact of a $\tilde{t}_1$ with a mass around 200 GeV on the deviations of the BR [@Moretti:2007th]. Fig. \[fig:ssp\] illustrates this fact for a particular parameter set except for the stop mass and $\phi_\mu$, where $BR(H_1\rightarrow \gamma\gamma)$ is plotted against $M_{H_1}$ for different values of $\phi_\mu$ in the two cases of light and heavy stop.
![Diagrams for Higgs decay into $\gamma\gamma$ pairs in the CP-violating MSSM: $f\equiv t,\: b,\:\tau;\;\;\tilde f\equiv \tilde t_{1,2}, \tilde b_{1,2},
\tilde \tau_{1,2}$.[]{data-label="fig:HiggsPho"}](fdiag.ps){width="8cm"}
![BR($H_1\rightarrow \gamma\gamma$) plotted against $M_{H_1}$. Parameters used are: $\tan\beta=20,\;\; M_1=100\;{\rm GeV},\;\;M_2=M_3=1\;{\rm TeV},\;\;
M_{(\tilde Q_3, \tilde D_3, \tilde L_3, \tilde E_3)}=1\;{\rm TeV},\;\;
|\mu|=1\;{\rm TeV},\;\;
|A_f|= 1.5\;{\rm TeV}$. The upper plot is for $M_{\tilde U_3}=1$ TeV while the lower one is for $M_{\tilde U_3}=250$ GeV (the latter giving a rather light stop, ${m}_{\tilde t_1} = 200$ GeV).[]{data-label="fig:ssp"}](Width_msusy1_A1.5_mu1_tb20_pA0.ps "fig:"){width="8cm"} ![BR($H_1\rightarrow \gamma\gamma$) plotted against $M_{H_1}$. Parameters used are: $\tan\beta=20,\;\; M_1=100\;{\rm GeV},\;\;M_2=M_3=1\;{\rm TeV},\;\;
M_{(\tilde Q_3, \tilde D_3, \tilde L_3, \tilde E_3)}=1\;{\rm TeV},\;\;
|\mu|=1\;{\rm TeV},\;\;
|A_f|= 1.5\;{\rm TeV}$. The upper plot is for $M_{\tilde U_3}=1$ TeV while the lower one is for $M_{\tilde U_3}=250$ GeV (the latter giving a rather light stop, ${m}_{\tilde t_1} = 200$ GeV).[]{data-label="fig:ssp"}](Width_msusy1_mt.25_A1.5_mu1_tb20_pA0.ps "fig:"){width="8cm"}
A detailed analysis at the matrix element level was undertaken in [@Hesselbach:2007en], which consolidated the above observations. Since the mass of the Higgs particle itself changes by changing $\phi_\mu$ (and keeping all other parameters the same), the difference in the BR read out from Fig. \[fig:ssp\] will have to be corrected for this change in $M_{H_1}$. (For the parameter set considered, the change in $M_{H_1}$ going from CP-conserving MSSM to CP-violating MSSM, by changing the value of $\phi_\mu$, is within the typical experimental uncertainty expected at LHC.) In Fig. \[fig:BR\] we plot the BR($H_1\rightarrow \gamma\gamma$) for five representative $\phi_\mu$ values between $0^\circ$ and $180^\circ$ as a function of $M_{H^+}$ for the two cases $M_{\tilde U_3}=1$ TeV (all SUSY particles heavy) and $M_{\tilde U_3}=250$ GeV (light $\tilde{t}_1$). The respective values of $M_{H_1}$ are indicated separately on the horizontal lines for each $\phi_\mu$ value. The cross over point in the Higgs mass eigenstates at $M_{H^+} \sim 150$ GeV is clearly visible. This corresponds to the sharp rise of the BR at around $M_{H_1}\sim 120$ GeV in Fig. \[fig:ssp\]. Below this point the BRs are very small and there is a strong $\phi_\mu$ dependence of $M_{H_1}$, hence our analysis is not relevant in this parameter region. Above $M_{H^+} \sim 150$ GeV and $M_{H_1} \gtrsim 115$ GeV, the $\phi_\mu$ dependence of $M_{H_1}$ is within the expected experimental uncertainty and the BR is large enough to be important for the LHC Higgs search. In scenarios with heavy SUSY particles (upper plot) the BR increases with increasing $\phi_\mu$ leading to a 50% increase for $\phi_\mu=90^\circ$ at $M_{H^+}\sim 200$ GeV. This $\phi_\mu$ dependence is caused mainly by the $\phi_\mu$ dependence of the $H_1$ couplings to $W^\pm$ bosons and $t$ and $b$ quarks, which appear in the loop-induced decay $H_1\rightarrow \gamma\gamma$. When a light $\tilde{t}_1$ is present (lower plot) the additional $\phi_\mu$ dependence in the stop sector causes a considerable change of the $\phi_\mu$ dependence of the BR. In fact, the BR increases again with increasing $\phi_\mu$ up to a maximum for some value of $\phi_\mu$ around $40^\circ$, beyond which, however, the BR decreases to about 50% at $\phi_\mu = 180^\circ$.
![\[fig:BR\] BR of $H_1\rightarrow \gamma\gamma$ for $|A_f|=1.5$ TeV, $|\mu|=1$ TeV and $\tan\beta=20$. Values of $M_{H_1}$ corresponding to representative points on the $M_{H^+}$ axis are indicated on the horizontal lines above separately for the values of $\phi_\mu$ used. The upper plot corresponds to the case with $M_{\tilde U_3}=1$ TeV (no light SUSY particles), while the lower plot corresponds to the case with $M_{\tilde U_3}=250$ GeV (a light stop is present).](BR_none0250_A1500_mu1000_tb20_phimu_log.eps "fig:"){width="18pc"} ![\[fig:BR\] BR of $H_1\rightarrow \gamma\gamma$ for $|A_f|=1.5$ TeV, $|\mu|=1$ TeV and $\tan\beta=20$. Values of $M_{H_1}$ corresponding to representative points on the $M_{H^+}$ axis are indicated on the horizontal lines above separately for the values of $\phi_\mu$ used. The upper plot corresponds to the case with $M_{\tilde U_3}=1$ TeV (no light SUSY particles), while the lower plot corresponds to the case with $M_{\tilde U_3}=250$ GeV (a light stop is present).](BR___mt0250_A1500_mu1000_tb20_phimu_log.eps "fig:"){width="18pc"}
![\[fig:BR\_A500\] The same as Fig. \[fig:BR\] but with $|A_f|=0.5$ TeV, $|\mu|=1$ TeV and $\tan\beta=20$.](BR_none0250_A0500_mu1000_tb20_phimu_log.eps "fig:"){width="18pc"} ![\[fig:BR\_A500\] The same as Fig. \[fig:BR\] but with $|A_f|=0.5$ TeV, $|\mu|=1$ TeV and $\tan\beta=20$.](BR___mt0250_A0500_mu1000_tb20_phimu_log.eps "fig:"){width="18pc"}
![\[fig:BR\_mu500\] The same as Fig. \[fig:BR\] but with $|A_f|=1.5$ TeV, $|\mu|=0.5$ TeV and $\tan\beta=20$.](BR_none0250_A1500_mu0500_tb20_phimu_log.eps "fig:"){width="18pc"} ![\[fig:BR\_mu500\] The same as Fig. \[fig:BR\] but with $|A_f|=1.5$ TeV, $|\mu|=0.5$ TeV and $\tan\beta=20$.](BR___mt0250_A1500_mu0500_tb20_phimu_log.eps "fig:"){width="18pc"}
Our analysis with other relevant sparticles like sbottom and stau being light shows that they do not play any major role in the $H_1\rightarrow
\gamma\gamma$ decay, even for $\tan\beta$ values as large as 50. We have also taken care that the changes in the masses of sparticles are not too large (i.e., again within expected experimental errors) when going from $\phi_\mu=0$ to non-zero values of $\phi_\mu$ while keeping the other parameters constant. Concerning the dependence on other SUSY parameters, we have found that a smaller value of $|A_f|$ considerably changes the $\phi_\mu$ dependence of the BR in scenarios with light a $\tilde{t}_1$ (see lower plot in Fig. \[fig:BR\_A500\]) whereas a smaller $|\mu|$ value leads generally to a smaller $\phi_\mu$ dependence (Fig. \[fig:BR\_mu500\]).
Summary
=======
We have analysed the BR of the di-photon decay of the lightest Higgs boson in the CP-violating MSSM with a complex $\mu$ parameter. The presence of a light scalar top is found to influence the $\phi_\mu$ dependence of the BR considerably while other sparticles have only negligible effect. In general, the BR may be increased or decreased for a non-zero $\phi_\mu$ depending on the SUSY parameter point. In scenarios where the relevant SUSY spectrum is already established by the LHC and consistent with both the CP-conserving and CP-violating scenario, our analyses of $H_1\rightarrow \gamma\gamma$ will be able to distinguish between these two cases.
[999]{}
A. Pilaftsis, Phys. Lett. B [**435**]{} (1998) 88, hep-ph/9805373. A. Pilaftsis and C. E. M. Wagner, Nucl. Phys. B [**553**]{} (1999) 3, hep-ph/9902371. K. A. Olive, M. Pospelov, A. Ritz and Y. Santoso, Phys. Rev. D [**72**]{} (2005) 075001, hep-ph/0506106. S. Y. Choi, K. Hagiwara and J. S. Lee, Phys. Lett. B [**529**]{} (2002) 212, hep-ph/0110138. S. Moretti, S. Munir and P. Poulose, Phys. Lett. B [**649**]{} (2007) 206, hep-ph/0702242. S. Hesselbach, S. Moretti, S. Munir and P. Poulose, arXiv:0706.4269 \[hep-ph\]. S. Hesselbach, S. Moretti, S. Munir and P. Poulose, in preparation.
J. S. Lee [*et al.*]{}, Comput. Phys. Commun. [**156**]{} (2004) 283, hep-ph/0307377.
[^1]: *Email:* [email protected]
[^2]: *Email:* [email protected]
[^3]: *Email:* [email protected]
[^4]: *Email:* [email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In recent times, neural networks have become a powerful tool for the analysis of complex and abstract data models. However, their introduction intrinsically increases our uncertainty about which features of the analysis are model-related and which are due to the neural network. This means that predictions by neural networks have biases which cannot be trivially distinguished from being due to the true nature of the creation and observation of data or not. In order to attempt to address such issues we discuss Bayesian neural networks: neural networks where the uncertainty due to the network can be characterised. In particular, we present the Bayesian statistical framework which allows us to categorise uncertainty in terms of the ingrained randomness of observing certain data and the uncertainty from our *lack of knowledge* about how data can be created and observed. In presenting such techniques we show how errors in prediction by neural networks can be obtained in principle, and provide the two favoured methods for characterising these errors. We will also describe how both of these methods have substantial pitfalls when put into practice, highlighting the need for other statistical techniques to truly be able to do inference when using neural networks.'
address:
- |
$^1$Sorbonne Université, CNRS, UMR 7095, Institut d’Astrophysique de Paris,\
98 bis bd Arago, 75014 Paris, France,\
[email protected]
- |
$^2$Department of Physics, Univesité de Montréal and Mila,\
Montréal,\
[email protected]
- |
$^3$AIM, CEA, CNRS, Université Paris-Saclay, Université Paris Diderot, Sorbonne Paris Cité, F-91191 Gif-sur-Yvette, France\
[email protected]
author:
- 'Tom Charnock$^1$, Laurence Perreault-Levasseur$^2$, François Lanusse$^3$'
bibliography:
- 'references.bib'
---
Bayesian Neural Networks {#ra_ch1}
========================
Introduction {#sec:intro}
------------
In recent times we have seen the power and ability that neural networks and deep learning methods can provide for fitting abstractly complex data models. However, any prediction from a neural network is necessarily and unknowably biased due to factors such as: choices in network architecture; methods for fitting networks; cuts in sets of training data; uncertainty in the distribution of realistic data; and lack of knowledge about the physical processes which generate such data. In this chapter we elucidate ways in which can learn how to separate, as much as possible, the sources of error which are intrinsic to the data model and those that we have introduced by choosing to add neural networks to that model.
### Aleatoric and epistemic uncertainty
Uncertainty can be categorised into two classes: aleatoric and epistemic. These two uncertainties explain, respectively, scatter from what we cannot know and error due to lack of knowledge. For example, we might observe some particular set of observations, but we do not know how likely it would be to observe that exact set and not a different set. This is an intrinsic uncertainty due to the random nature of the way the observed objects are created and the way we make observations. As such, we call this uncertainty *aleatoric* since it cannot be reduced through greater understanding. On the other hand, when we are trying to understand a particular set of observations there are things we do not know but could, in principle, learn about: what are the properties of a physical processes which are necessary to create such data? what types of distribution could describe how likely were we to see such an observation? and how certain are we that such a model is supported by our data? These are questions prescient to the statistical study of science. By analysing the observed data, we can find answers to these questions, and so any uncertainty due to the lack of knowledge is reducible. This uncertainty is called *epistemic*.
When adding a neural network for the analysis of data, we are modifying our data model to include any effects that are introduced by the network. There is, therefore, intrinsic (aleatoric) uncertainty due to the stochastic nature of the data, and epistemic uncertainty now due to both the lack of knowledge about underlying model of the data as well as the form of the neural network, the way it is trained, the choice of cost function used to characterise how well the network performs, etc. It is only by understanding the epistemic uncertainty introduced by the neural networks that we can attempt to make understandable predictions using them.
For the most part, estimates of how well a neural network generalises are obtained using large sets of validation and testing data and relative agreement then suggests that a neural network “works”. However, these neural networks do not address the probability that any prediction coincides with the truth. There is no separation between aleatoric and epistemic uncertainty and no knowledge of how likely (or well) a new example of data is to provide a realistic prediction. It is *possible* to quantify this epistemic error, though, caused by our lack of knowledge about the properties of a neural network, and characterising this uncertainty can allow us to perform reasoned inference. In this chapter we will lay down the formalism for Bayesian neural networks: treating neural networks as statistical models whose parameters are attributed probabilities as a degree of belief which can be logically updated under the support from data. In such a form, neural networks can be used to make statements of inference about how likely we are to believe the outputs of neural networks, reducing the lack of trust that is inherent in the standard deep learning setup. We will also show some ways of practically implementing this Bayesian formalism and indicating where some of these implementations have been used in astronomy and cosmology.
Bayesian neural networks {#sec:bnn}
------------------------
In this section we will show how one can use a Bayesian statistical framework to assess both aleatoric and epistemic uncertainty in a model which includes neural networks, and describe how epistemic uncertainty can be reduced under the evidence of supporting data using Bayesian inference.
### Bayesian statistics
When speaking of *uncertainty*, we are really describing our lack of knowledge about the truth. This uncertainty is subjective, in that it is not an inherent property of a problem but rather the way we construct the problem. If we are uncertain about the results of a particular experiment, we do not know exactly what the result of that experiment will be. The Bayesian (or subjective) statistical framework is a scientific viewpoint in which we admit that we do not (and are not able to) know the truth about any particular hypothesis. Our uncertainty, or our degree of belief in the truth, are attributed probabilities, i.e. hypotheses we believe more strongly are described as being more likely. Of course, in this construction, probabilities can vary from person to person, since different beliefs can be held by different people. Without any *prior* knowledge, we are free to believe what we will. However, by using Bayesian inference, we are able to reduce epistemic uncertainty and update our *a priori* knowledge by obtaining evidence, *a posteriori*. It is important to realise that, whilst our *a priori* beliefs describe the epistemic uncertainty, this quantification can be artificially small without the support of observations. If our beliefs are not supported by the evidence then the *a posteriori* probability describing the state of our belief after obtaining evidence will become more uncertain, which is a better characterisation of the state of our knowledge. Under repeated application of new evidence, we can update our beliefs to hone in on the best supported result.
#### Statistical models
A Bayesian statistical framework is a natural setting to build models with which we can infer the most likely distributions and underlying processes that generate some observable events. Such a statistical model can be thought of as a measurable space of possible events, $({\ensuremath{\mathcal{S}}},{\ensuremath{\mathcal{E}}},\mathcal{P})$. The first element is the sampling space, ${\ensuremath{\mathcal{S}}}$, which describes the set of all possible outcomes for a given problem. Each outcome is a random variable, ${\ensuremath{\mathcal{d}}}\in{\ensuremath{\mathcal{S}}}$, whose value is a single measured observation or result. An event, ${\ensuremath{\mathcal{D}}}\subset{\ensuremath{\mathcal{S}}}$, is defined as a subset of possible outcomes. The set of all events that can possibly occur is ${\ensuremath{\mathcal{E}}}$.We can impose a measure on the space of possible events, $\mathcal{P}: {\ensuremath{\mathcal{D}}}\in {\ensuremath{\mathcal{E}}}\mapsto \mathcal{P}({\ensuremath{\mathcal{D}}}) \in [0,1]$, which is a function that assigns a value between 0 and 1 to every event, ${\ensuremath{\mathcal{D}}}\in{\ensuremath{\mathcal{E}}}$, describing how likely it is for such an event to occur. This probability indicates that an event, ${\ensuremath{\mathcal{D}}}$, is impossible when $\mathcal{P}({\ensuremath{\mathcal{D}}})=0$ and is certain when $\mathcal{P}({\ensuremath{\mathcal{D}}})=1$. The measure, $\mathcal{P}$, of the measurable space of a statistical model is additive, so that it is certain that any *possible* event *can* occur, $\mathcal{P}({\ensuremath{\mathcal{E}}})=1$.
Whilst we can observe some subset of all possible outcomes from this probability space we do not necessarily know *which* particular outcomes we will observe from the random processes generating any $\mathcal{D}$. That is, even if we knew, exactly, the statistical model, $({\ensuremath{\mathcal{S}}}, {\ensuremath{\mathcal{E}}}, \mathcal{P})$, we would not know which event would occur from the distribution, $\mathcal{P}$. This is akin to not having access to the state (or seed) of a random process and is the true source of *aleatoric uncertainty*.
In practice, we also do not know the form of $\mathcal{P}$ either and as such we attempt to model the probability measure using another statistical model $({\ensuremath{\mathcal{S}}}_\alpha,{\ensuremath{\mathcal{E}}}_\alpha,{\ensuremath{\mathcal{p}}})$. In a Bayesian context, ${\ensuremath{\mathcal{S}}}_\alpha$ is another sampling space of possible parameterised distributions with an outcome, $\alpha\in\mathcal{S}_\alpha$, representing all properties of a particular distribution, i.e. functional form, shape, as well as the possible values of some unobservable random variables, $\omega\in\Omega_\alpha$, which generate ${\ensuremath{\mathcal{d}}}\in{\ensuremath{\mathcal{S}}}$, etc. Any possible set of $\alpha\in{\ensuremath{\mathcal{S}}}_\alpha$ is an event in the space of possible distributions, ${\ensuremath{\mathcal{a}}}\in{\ensuremath{\mathcal{E}}}_\alpha$, which can model $({\ensuremath{\mathcal{S}}},{\ensuremath{\mathcal{E}}},\mathcal{P})$. We assign a probabilistic degree of belief from our *prior* knowledge, ${\ensuremath{\mathcal{p}}}({\ensuremath{\mathcal{a}}})$, that any set of possible distributions, ${\ensuremath{\mathcal{a}}}\in{\ensuremath{\mathcal{E}}}_\alpha$, encapsulates the true probability measure, $\mathcal{P}$, describing the probability of events, ${\ensuremath{\mathcal{D}}}$, occurring. The lack of knowledge about the possible values of ${\ensuremath{\mathcal{a}}}$ is the source of *epistemic* uncertainty. Whilst we might never know the exact statistical model, $({\ensuremath{\mathcal{S}}},{\ensuremath{\mathcal{E}}},\mathcal{P})$, we can increase our knowledge about how to model it with ${\ensuremath{\mathcal{a}}}\in{\ensuremath{\mathcal{E}}}_\alpha$ under the evidence of observed events, ${\ensuremath{\mathcal{D}}}\in{\ensuremath{\mathcal{E}}}$, thereby reducing the epistemic uncertainty.
Effectively, any ${\ensuremath{\mathcal{a}}}\in{\ensuremath{\mathcal{E}}}_\alpha$ defines a model $(({\ensuremath{\mathcal{S}}},\Omega_{\ensuremath{\mathcal{a}}}), ({\ensuremath{\mathcal{E}}},{\ensuremath{\mathcal{E}}}_\omega), {\ensuremath{\mathcal{p}_\mathcal{a}}})$ of $({\ensuremath{\mathcal{S}}},{\ensuremath{\mathcal{E}}},\mathcal{P})$. That is, any ${\ensuremath{\mathcal{a}}}\in{\ensuremath{\mathcal{E}}}_\alpha$ introduces a sampling space of unobservable random variables, $\Omega_{\ensuremath{\mathcal{a}}}$, whose values, $\omega\in\Omega_{\ensuremath{\mathcal{a}}}$, can generate outcomes, ${\ensuremath{\mathcal{d}}}\in{\ensuremath{\mathcal{S}}}$. The set of all possible unobservable random variables, ${\ensuremath{\mathcal{w}}}\subset\Omega_\alpha$, which can generate events, ${\ensuremath{\mathcal{D}}}\in{\ensuremath{\mathcal{E}}}$, is then called ${\ensuremath{\mathcal{E}}}_\omega$. Any ${\ensuremath{\mathcal{a}}}\in{\ensuremath{\mathcal{E}}}_\alpha$ also defines a probability measure, ${\ensuremath{\mathcal{p}_\mathcal{a}}}: ({\ensuremath{\mathcal{D}}},{\ensuremath{\mathcal{w}}}) \in ({\ensuremath{\mathcal{E}}},{\ensuremath{\mathcal{E}}}_\omega) \mapsto {\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{D}}},{\ensuremath{\mathcal{w}}}) \in [0,1]$, describing how likely any observable-event-and-unobservable-parameter pair are. The possible parameterised statistical model characterises what physical processes generate an observable outcome, our belief in the values of the parameters of those physical processes, and how likely we are to obtain any set of outcomes *and* physical parameters. We call ${\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{D}}},{\ensuremath{\mathcal{w}}})$ the joint probability of observables and parameters. Of course, the choice in the way that ${\ensuremath{\mathcal{p}_\mathcal{a}}}$ attributes probabilities to $({\ensuremath{\mathcal{D}}},{\ensuremath{\mathcal{w}}})$ depends on our prior belief, ${\ensuremath{\mathcal{p}}}({\ensuremath{\mathcal{a}}})$, in the ability for any particular parameterised distribution, ${\ensuremath{\mathcal{a}}}\in{\ensuremath{\mathcal{E}}}_\alpha$, to represent $({\ensuremath{\mathcal{S}}},{\ensuremath{\mathcal{E}}},\mathcal{P})$.
Since the unobservable parameters, ${\ensuremath{\mathcal{w}}}\in{\ensuremath{\mathcal{E}}}_\omega$, generate possible sets of observable outcomes, ${\ensuremath{\mathcal{D}}}\in{\ensuremath{\mathcal{E}}}$, we can write down how likely we are to observe some event, ${\ensuremath{\mathcal{D}}}$, *given* that the unobservable parameters, ${\ensuremath{\mathcal{w}}}$, have a particular value, $${\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{D}}},{\ensuremath{\mathcal{w}}}) = {\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{D}}}|{\ensuremath{\mathcal{w}}}){\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{w}}}).$$ We call ${\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{D}}}|{\ensuremath{\mathcal{w}}})$ the *likelihood* of some values of observables ${\ensuremath{\mathcal{D}}}$, given the values of parameters, ${\ensuremath{\mathcal{w}}}$, and ${\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{w}}})$ is the *a priori* (or prior) distribution of parameters describing what we believe the values of ${\ensuremath{\mathcal{w}}}$ to be based on our current knowledge. Therefore, some (but not all) of the epistemic uncertainty is encapsulated by ${\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{w}}})$. The prior distribution, ${\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{w}}})$, does not, however, describe the form of the parameterised distribution, ${\ensuremath{\mathcal{p}_\mathcal{a}}}$, modelling, $({\ensuremath{\mathcal{S}}},{\ensuremath{\mathcal{E}}},\mathcal{P})$, and so we must also consider our prior knowledge of the possible distributions, ${\ensuremath{\mathcal{p}}}({\ensuremath{\mathcal{a}}})$, to properly characterise the epistemic uncertainty.
#### Bayesian inference
By observing events, $\mathcal{D}\in{\ensuremath{\mathcal{E}}}$, we can update our belief that values of any set of unobservable random variables, ${\ensuremath{\mathcal{w}}}\in{\ensuremath{\mathcal{E}}}_\omega$, and distributions, ${\ensuremath{\mathcal{a}}}\in{\ensuremath{\mathcal{E}}}_\alpha$, correctly model the statistical model, $({\ensuremath{\mathcal{S}}},{\ensuremath{\mathcal{E}}},{\ensuremath{\mathcal{p}}})$. This is how we can reduce our epistemic uncertainty. The probability describing our belief in the possible values of ${\ensuremath{\mathcal{w}}}$ and ${\ensuremath{\mathcal{a}}}$ obtained *after* we have observed an event, $\mathcal{D}$, is called the *a posteriori* (or posterior) distribution and can be derived by expanding the joint distribution $$\begin{aligned}
{\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{D}}},{\ensuremath{\mathcal{w}}}){\ensuremath{\mathcal{p}}}({\ensuremath{\mathcal{a}}})&={\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{D}}}|{\ensuremath{\mathcal{w}}}){\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{w}}}){\ensuremath{\mathcal{p}}}({\ensuremath{\mathcal{a}}})\nonumber\\
&={\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{w}}}|{\ensuremath{\mathcal{D}}}){\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{D}}}){\ensuremath{\mathcal{p}}}({\ensuremath{\mathcal{a}}}),\end{aligned}$$ and equating both sides to get Bayes’ theorem $${\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{w}}}|{\ensuremath{\mathcal{D}}}) = \frac{{\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{D}}}|{\ensuremath{\mathcal{w}}}){\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{w}}})}{{\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{D}}})}.\label{eq:bayes}$$ This equation tells us that, given a particular parameterised model, ${\ensuremath{\mathcal{a}}}$, the probability that some parameters, ${\ensuremath{\mathcal{w}}}$, have a particular value when some event, ${\ensuremath{\mathcal{D}}}$, is observed is proportional to the likelihood of the observation of such an event given a particular value of the parameters, ${\ensuremath{\mathcal{w}}}$, generating the event. The probability of those parameter values is described by our belief in their value, ${\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{w}}})$. The *evidence* that the parameterised distribution accurately describes the distribution of some event is $${\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{D}}})=\int_{{\ensuremath{\mathcal{E}}}_\omega}d{\ensuremath{\mathcal{w}}}\,{\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{D}}}|{\ensuremath{\mathcal{w}}}){\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{w}}}).\label{eq:evidence}$$ If the probability of ${\ensuremath{\mathcal{D}}}$ is small when the likelihood is integrated over all possible sets of parameter values, ${\ensuremath{\mathcal{w}}}\in{\ensuremath{\mathcal{E}}}_\omega$, both of which are defined by ${\ensuremath{\mathcal{a}}}$, then there is little support for that choice of a value of ${\ensuremath{\mathcal{a}}}\in{\ensuremath{\mathcal{E}}}_\alpha$. This would suggest that we need to update our belief in the chosen parameterised distribution, ${\ensuremath{\mathcal{p}}}({\ensuremath{\mathcal{a}}})$, in being able to represent the true model, $({\ensuremath{\mathcal{S}}},{\ensuremath{\mathcal{E}}},\mathcal{P})$.
##### Maximum likelihood estimation
In classical statistics, the unobserved random variables, ${\ensuremath{\mathcal{w}}}\in{\ensuremath{\mathcal{E}}}_\omega$, are considered to be fixed parameters of a particular statistical model, ${\ensuremath{\mathcal{a}}}\in{\ensuremath{\mathcal{E}}}_\alpha$. The parameters which best describes some event, $\mathcal{D}$, can be found maximising the likelihood function $$\widehat{{\ensuremath{\mathcal{w}}}}=\underset{{\ensuremath{\mathcal{w}}}\in{\ensuremath{\mathcal{E}}}_\omega}{\operatorname*{arg\,max}}\,\mathcal{p}_{\ensuremath{\mathcal{a}}}({\ensuremath{\mathcal{D}}}|{\ensuremath{\mathcal{w}}}).$$ Although this point in parameter space maximises the likelihood and can be found fairly easily by various optimisation schemes, it is completely ignorant about both the shape of the distribution, ${\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{D}}}|{\ensuremath{\mathcal{w}}})$, and how likely we think any particular value of ${\ensuremath{\mathcal{w}}}$ (and ${\ensuremath{\mathcal{a}}}$) are. This means that the possible parameters values are degenerated to one point and absolute certainty is ascribed to a choice of model and its parameters. Furthermore, for skewed distributions, the mode of the likelihood can be far away from the expectation value (or mean) of the distribution and therefore the maximum likelihood estimate might not even be representative. Any epistemic uncertainty in the model is ignored since we do not consider our belief in ${\ensuremath{\mathcal{w}}}$, nevermind how likely ${\ensuremath{\mathcal{a}}}$ is.
##### Maximum *a posteriori* estimation
The simplest form of Bayesian inference is finding the maximum *a posteriori* (MAP) estimate, i.e. the mode of the posterior distribution for a given model, ${\ensuremath{\mathcal{a}}}$, as $$\begin{aligned}
\widehat{{\ensuremath{\mathcal{w}}}}&=\underset{{\ensuremath{\mathcal{w}}}\in{\ensuremath{\mathcal{E}}}_{\ensuremath{\mathcal{a}}}}{\operatorname*{arg\,max}}\,{\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{w}}}|{\ensuremath{\mathcal{D}}})\nonumber\\
&=\underset{{\ensuremath{\mathcal{w}}}\in{\ensuremath{\mathcal{E}}}_{\ensuremath{\mathcal{a}}}}{\operatorname*{arg\,max}}\,{\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{D}}}|{\ensuremath{\mathcal{w}}}){\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{w}}}).\end{aligned}$$ Note that, when we think that any values of the model parameters are equally likely, i.e. the prior distribution, ${\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{w}}})$, is uniform, then ${\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{D}}}|{\ensuremath{\mathcal{w}}})\propto{\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{w}}}|{\ensuremath{\mathcal{D}}})$ and MAP estimation is equivalent to maximum likelihood estimation. So, whilst MAP estimation is Bayesian due to the addition of our belief in possible parameter values, ${\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{w}}})$, this form of inference suffers in exactly the same way that maximum likelihood estimation does : the mode of the posterior might also be far from the expectation value and not be representative, and all information about the epistemic uncertainty is underestimated because knowledge about the distribution of parameters is ignored.
##### Bayesian posterior inference
To effectively characterise the epistemic uncertainty, not only should we consider Bayes’ theorem , one should work with the marginal distribution over the prior probability of parameterised models $$\begin{aligned}
{\ensuremath{\mathcal{p}}}({\ensuremath{\mathcal{D}}})&=\int_{{\ensuremath{\mathcal{E}}}_\alpha}d{\ensuremath{\mathcal{a}}}\,{\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{D}}}){\ensuremath{\mathcal{p}}}({\ensuremath{\mathcal{a}}}),\nonumber\\
&=\int_{{\ensuremath{\mathcal{E}}}_\alpha}\int_{{\ensuremath{\mathcal{E}}}_\omega}d{\ensuremath{\mathcal{a}}}d{\ensuremath{\mathcal{w}}}\,{\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{D}}}|{\ensuremath{\mathcal{w}}}){\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{w}}}){\ensuremath{\mathcal{p}}}({\ensuremath{\mathcal{a}}}).\end{aligned}$$ Practically, the space of possible models, ${\ensuremath{\mathcal{E}}}_\alpha$, can be infinitely large, although our belief in possible models, ${\ensuremath{\mathcal{p}}}({\ensuremath{\mathcal{a}}})$, does not have to be. Still, the integration over all possible models often makes the calculation of ${\ensuremath{\mathcal{p}}}({\ensuremath{\mathcal{D}}})$ effectively intractable. In practice, we tend to choose a particular model and, in the best case (where we have lots of time and computational power) use empirical Bayes to calculate the mode of the possible marginal distributions $$\begin{aligned}
\hat{{\ensuremath{\mathcal{a}}}}&=\underset{{\ensuremath{\mathcal{a}}}\in{\ensuremath{\mathcal{E}}}_\alpha}{\operatorname*{arg\,max}}\,{\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{D}}}){\ensuremath{\mathcal{p}}}({\ensuremath{\mathcal{a}}}),\nonumber\\
&=\underset{{\ensuremath{\mathcal{a}}}\in{\ensuremath{\mathcal{E}}}_\alpha}{\operatorname*{arg\,max}}\,\int_{{\ensuremath{\mathcal{E}}}_\omega}d{\ensuremath{\mathcal{w}}}\,{\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{D}}}|{\ensuremath{\mathcal{w}}}){\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{w}}}){\ensuremath{\mathcal{p}}}({\ensuremath{\mathcal{a}}}).\label{eq:EB}\end{aligned}$$ As with the MAP estimate of the parameters, $\hat{{\ensuremath{\mathcal{a}}}}$ describes the most likely believed model that supports an event, $\mathcal{D}$. However, again as with the MAP estimate of the parameters, a model, ${\ensuremath{\mathcal{a}}}=\hat{{\ensuremath{\mathcal{a}}}}$, might have artificially small epistemic uncertainty due to discarding the rest of the knowledge of the distribution. To be able to correctly estimate this epistemic uncertainty, one must update, logically, the probability of any possible models and parameters based on the acquisition of knowledge.
### Neural networks formulated as statistical models {#sec:smnn}
We can consider neural networks as part of a statistical model. In this case, we usually think of an observable outcome as a pair of input and target random variable pairs, ${\ensuremath{\mathcal{d}}}=(x,y)\in{\ensuremath{\mathcal{S}}}\,$[^1]. An event is then a subset of pairs ${\ensuremath{\mathcal{D}}}=({\ensuremath{\mathcal{x}}},{\ensuremath{\mathcal{y}}})\in{\ensuremath{\mathcal{E}}}$ with probability $\mathcal{P}({\ensuremath{\mathcal{x}}},{\ensuremath{\mathcal{y}}})$. We can then use a neural network as a parameterised, non-linear function $${\ensuremath{\mathcal{r}}}=\mathcal{f}_{{\ensuremath{\mathcal{w}}},{\ensuremath{\mathcal{a}}}}({\ensuremath{\mathcal{x}}})\label{eq:neural_network}$$ where ${\ensuremath{\mathcal{r}}}$ are considered the parameters of a distribution which models the likelihood of targets given inputs, ${\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{y}}}|{\ensuremath{\mathcal{x}}},{\ensuremath{\mathcal{w}}})$. The form of the function, i.e. the architecture, the number, value and distribution of parameters ${\ensuremath{\mathcal{w}}}\in{\ensuremath{\mathcal{E}}}_\omega$, initialisation of the network, etc. is described by some hyperparameters, ${\ensuremath{\mathcal{a}}}\in{\ensuremath{\mathcal{E}}}_\alpha$. The prescription for the likelihood ${\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{y}}}|{\ensuremath{\mathcal{x}}},{\ensuremath{\mathcal{w}}})$ can range from being defined as ${\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{y}}}|{\ensuremath{\mathcal{x}}},{\ensuremath{\mathcal{w}}})\propto\exp[-\mathcal{L}({\ensuremath{\mathcal{y}}},{\ensuremath{\mathcal{r}}})]$, where $\mathcal{L}({\ensuremath{\mathcal{y}}},{\ensuremath{\mathcal{r}}})$ is an unregularised loss function measuring the similarity of the output of a neural network, ${\ensuremath{\mathcal{r}}}$, to some target, ${\ensuremath{\mathcal{y}}}$[^2] , to parametric distributions such as a mixture of distributions or neural density estimators.
When considering a neural network as an abstract function, it can be possible to obtain virtually any value of ${\ensuremath{\mathcal{r}}}$ for a given input ${\ensuremath{\mathcal{x}}}$ at any values of the network parameters, ${\ensuremath{\mathcal{w}}}$, since the network parameters are often unidentifiable [@Muller1998] and the functional form of the possible values of ${\ensuremath{\mathcal{r}}}$ is very likely infinite in extent and no statement about convexity can be made. Therefore, evaluating the neural network and interpreting its output as the value of the parameters which correctly defines the likelihood of a target will be misleading[^3]. This statement is true for *any* value of the network parameters, ${\ensuremath{\mathcal{w}}}\in{\ensuremath{\mathcal{E}}}_\omega$, since most values of ${\ensuremath{\mathcal{w}}}$ do not correspond to neural networks which perform the desired function. This is essentially why we use neural networks, we can carve out parts of useful parameter space which provides the function which describes how to best fit some known data, $({\ensuremath{\mathcal{x}}},{\ensuremath{\mathcal{y}}})$, using the likelihood, ${\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{y}}}|{\ensuremath{\mathcal{x}}},{\ensuremath{\mathcal{w}}})$, as defined by the data itself. We normally describe this set of known data which ascribes acceptable regions of parameter space where the likelihood makes sense as a *training* set, $({\ensuremath{\mathcal{x}}},{\ensuremath{\mathcal{y}}})_\textrm{train}\in{\ensuremath{\mathcal{E}}}$.
Having described neural networks as statistical models we can, further, place them in a Bayesian context by associating a quantification of belief, ${\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{w}}})$, to the values of the network parameters, ${\ensuremath{\mathcal{w}}}\in{\ensuremath{\mathcal{E}}}_\omega$, for a network ${\ensuremath{\mathcal{a}}}\in{\ensuremath{\mathcal{E}}}_\alpha$, which we believe to be able to represent the the true distribution of observed events, $\mathcal{P}({\ensuremath{\mathcal{x}}}, {\ensuremath{\mathcal{y}}})$, with probability ${\ensuremath{\mathcal{p}}}({\ensuremath{\mathcal{a}}})$. ${\ensuremath{\mathcal{p}}}({\ensuremath{\mathcal{a}}})$ (and the associated ${\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{w}}})$) represent the epistemic uncertainty due to the neural network, whilst the aleatoric uncertainty arises due to the fact that it is not known exactly which $({\ensuremath{\mathcal{x}}}, {\ensuremath{\mathcal{y}}})$ would arise from the statistical model $({\ensuremath{\mathcal{S}}},{\ensuremath{\mathcal{E}}},\mathcal{P})$. We can use Bayesian statistics to update our beliefs and obtain posterior predictive estimates of targets, ${\ensuremath{\mathcal{y}}}$, based on this information via the posterior predictive distribution $${\ensuremath{\mathcal{p}}}({\ensuremath{\mathcal{y}}}| {\ensuremath{\mathcal{x}}})=\int_{{\ensuremath{\mathcal{E}}}_\alpha}\int_{{\ensuremath{\mathcal{E}}}_\omega} d{\ensuremath{\mathcal{a}}}d{\ensuremath{\mathcal{w}}}\,{\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{y}}}|{\ensuremath{\mathcal{x}}},{\ensuremath{\mathcal{w}}}){\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{w}}}){\ensuremath{\mathcal{p}}}({\ensuremath{\mathcal{a}}}).\label{eq:post_pred}$$ By integrating over all possible parameters for all possible network choices, we obtain a distribution describing how probable different values of ${\ensuremath{\mathcal{y}}}$ are, from our model, which incorporates our lack of knowledge.
The region where we believe that the parameters allow the network to perform its intended purpose is described by, ${\ensuremath{\mathcal{p}_\mathcal{a}}}\left({\ensuremath{\mathcal{w}}}|({\ensuremath{\mathcal{x}}},{\ensuremath{\mathcal{y}}})_\textrm{train}\right)$. This is our first step in the Bayesian inference. Bayes’ theorem tells us $${\ensuremath{\mathcal{p}_\mathcal{a}}}\left({\ensuremath{\mathcal{w}}}|({\ensuremath{\mathcal{x}}},{\ensuremath{\mathcal{y}}})_\textrm{train}\right)=\frac{{\ensuremath{\mathcal{p}_\mathcal{a}}}\left({\ensuremath{\mathcal{y}}}_\textrm{train}|{\ensuremath{\mathcal{x}}}_\textrm{train},{\ensuremath{\mathcal{w}}}\right){\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{w}}})}{{\ensuremath{\mathcal{p}_\mathcal{a}}}\left(({\ensuremath{\mathcal{x}}},{\ensuremath{\mathcal{y}}})_\textrm{train}\right)},\label{eq:post_w}$$ so that updating our knowledge of the parameters given the presence of a training set allows us to better characterise the probability of obtaining ${\ensuremath{\mathcal{y}}}$ from ${\ensuremath{\mathcal{x}}}$ with a particular neural network $$\mathcal{p}\left({\ensuremath{\mathcal{y}}}|{\ensuremath{\mathcal{x}}},({\ensuremath{\mathcal{x}}},{\ensuremath{\mathcal{y}}})_\textrm{train}\right)=\int_{{\ensuremath{\mathcal{E}}}_\alpha}\int_{{\ensuremath{\mathcal{E}}}_\omega}d{\ensuremath{\mathcal{a}}}d{\ensuremath{\mathcal{w}}}\,{\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{y}}}|{\ensuremath{\mathcal{x}}},{\ensuremath{\mathcal{w}}}){\ensuremath{\mathcal{p}_\mathcal{a}}}\left({\ensuremath{\mathcal{w}}}|({\ensuremath{\mathcal{x}}},{\ensuremath{\mathcal{y}}})_\textrm{train}\right){\ensuremath{\mathcal{p}}}({\ensuremath{\mathcal{a}}}).\label{eq:post_pred_post_w}$$ To encapsulate the uncertainty in the network we need to calculate the posterior distribution of network parameters, ${\ensuremath{\mathcal{w}}}$, as in , which we can then use to calculate the distribution of possible ${\ensuremath{\mathcal{y}}}$ as described by the predicted ${\ensuremath{\mathcal{r}}}$ from the network, as in . Attention must be paid to the initial choice of ${\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{w}}})$ which still occurs in [^4].
This description of Bayesian neural networks, therefore, refers solely to networks which are part of a Bayesian model[^5], i.e. networks where the epistemic uncertainty in the network parameters are characterised by probability distributions, ${\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{w}}}|{\ensuremath{\mathcal{x}}},{\ensuremath{\mathcal{y}}})$, and thus we are interested in the inference of ${\ensuremath{\mathcal{w}}}$. There are several approaches which are effective for characterising distributions, but each of them have their pros and cons. In section \[sec:PI\], we present some numerically approximate schemes using the exact distributions and some exact schemes using approximate distributions, these fall under the realms of Monte Carlo methods and variational inference.
#### Limitations of the Bayesian neural network formulation
The goal of a Bayesian neural network is to capture epistemic uncertainties. In the absence of any data, the behaviour of the model is only controlled by the prior, and should produce large epistemic uncertainties (high variance of the model outputs) for any given input. We then expect that as we update the posterior of network parameters with training data, the epistemic uncertainties should decrease in the vicinity of these training points, as the model is now at least somewhat constrained, but the variance should remain large for Out-Of-Distribution (OOD) regions far from the training set. This is the behaviour that one would expect, however, we want to highlight that nothing in the BNN derivation presented in this section necessarily implies this behaviour in practice.
As in any Bayesian model, the behaviour of a Bayesian neural network when data is not constraining is tightly coupled to the choice of prior. However the priors typically used in BNNs are chosen based on practicality and empirical observation rather than principled considerations on the functional space spanned by the neural network. There is indeed little guarantee that a Gaussian prior on the weights of a deep dense neural network implies any meaningful uncertainties away from the training distribution. In fact, it is easily shown[@Hafner2018] that putting priors on weights can fail at properly capturing epistemic uncertainties, even on very simple examples.
#### Relation to classical neural networks {#sec:classicalnn}
Since neural networks are, in general, able to fit arbitrarily complex models when large enough, we might be able to justify a relatively narrow prior on the hyperparameters, ${\ensuremath{\mathcal{p}}}({\ensuremath{\mathcal{a}}})\approx\delta({\ensuremath{\mathcal{a}}}-\hat{{\ensuremath{\mathcal{a}}}})$, meaning that we think that an arbitrarily complex network can encapsulate the statistical model $({\ensuremath{\mathcal{S}}},{\ensuremath{\mathcal{E}}},\mathcal{P})$[^6]. Marginalising over the possible hyperparameters gives us $$\begin{aligned}
{\ensuremath{\mathcal{p}}}({\ensuremath{\mathcal{x}}},{\ensuremath{\mathcal{y}}},{\ensuremath{\mathcal{w}}})&=\int_{{\ensuremath{\mathcal{E}}}_\alpha}d{\ensuremath{\mathcal{a}}}\,{\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{x}}},{\ensuremath{\mathcal{y}}},{\ensuremath{\mathcal{w}}}){\ensuremath{\mathcal{p}}}({\ensuremath{\mathcal{a}}})\nonumber\\
&=\int_{{\ensuremath{\mathcal{E}}}_\alpha}d{\ensuremath{\mathcal{a}}}\,{\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{x}}},{\ensuremath{\mathcal{y}}},{\ensuremath{\mathcal{w}}})\delta({\ensuremath{\mathcal{a}}}-\hat{{\ensuremath{\mathcal{a}}}})\nonumber\\
&={\ensuremath{\mathcal{p}}}_{\hat{{\ensuremath{\mathcal{a}}}}}({\ensuremath{\mathcal{x}}},{\ensuremath{\mathcal{y}}},{\ensuremath{\mathcal{w}}}).\end{aligned}$$ This describes the probability of possible input-target pairs and network parameters for any given choice of hyperparameters, from which we can write $${\ensuremath{\mathcal{p}}}\left({\ensuremath{\mathcal{y}}}|{\ensuremath{\mathcal{x}}}, ({\ensuremath{\mathcal{x}}},{\ensuremath{\mathcal{y}}})_\textrm{train}\right)=\int_{{\ensuremath{\mathcal{E}}}_\omega}d{\ensuremath{\mathcal{w}}}\,{\ensuremath{\mathcal{p}}}_{\hat{a}}({\ensuremath{\mathcal{y}}}|{\ensuremath{\mathcal{x}}}, {\ensuremath{\mathcal{w}}}){\ensuremath{\mathcal{p}}}_{\hat{a}}\left({\ensuremath{\mathcal{w}}}|({\ensuremath{\mathcal{x}}},{\ensuremath{\mathcal{y}}})_\textrm{train}\right).\label{eq:post_pred_w}$$
In a non-Bayesian context, having restricted the possible forms of neural networks via fixing ${\ensuremath{\mathcal{a}}}=\hat{{\ensuremath{\mathcal{a}}}}$, it is common to find the mode of the distribution of neural network parameters, ${\ensuremath{\mathcal{w}}}$, by maximising the likelihood[^7] of observing some training set $({\ensuremath{\mathcal{x}}},{\ensuremath{\mathcal{y}}})_\textrm{train}\in{\ensuremath{\mathcal{E}}}$ when given those parameters $$\begin{aligned}
\widehat{{\ensuremath{\mathcal{w}}}}&=\underset{{\ensuremath{\mathcal{w}}}\in{\ensuremath{\mathcal{E}}}_\omega}{\operatorname*{arg\,max}}\,{\ensuremath{\mathcal{p}}}_{\hat{{\ensuremath{\mathcal{a}}}}}({\ensuremath{\mathcal{y}}}_\textrm{train}|{\ensuremath{\mathcal{x}}}_\textrm{train},{\ensuremath{\mathcal{w}}}).\end{aligned}$$
Once an estimate for the network parameters is made, the posterior distribution of parameter values, ${\ensuremath{\mathcal{p}}}_{\hat{{\ensuremath{\mathcal{a}}}}}\left({\ensuremath{\mathcal{w}}}|({\ensuremath{\mathcal{x}}},{\ensuremath{\mathcal{y}}})_\textrm{train}\right)$, is usually degenerated to a delta function at the maximum likelihood estimate of the network parameters, ${\ensuremath{\mathcal{p}}}_{\hat{{\ensuremath{\mathcal{a}}}}}\left({\ensuremath{\mathcal{w}}}|({\ensuremath{\mathcal{x}}},{\ensuremath{\mathcal{y}}})_\textrm{train}\right)\Rightarrow\delta({\ensuremath{\mathcal{w}}}-\widehat{{\ensuremath{\mathcal{w}}}})$. The prediction of a target, ${\ensuremath{\mathcal{y}}}$, from an input, ${\ensuremath{\mathcal{x}}}$, then occurs with a probability equal to the likelihood evaluated at the maximum likelihood estimate of the value of the network parameters $$\begin{aligned}
{\ensuremath{\mathcal{p}}}\left({\ensuremath{\mathcal{y}}}|{\ensuremath{\mathcal{x}}}, ({\ensuremath{\mathcal{x}}},{\ensuremath{\mathcal{y}}})_\textrm{train}\right)&=\int_{{\ensuremath{\mathcal{E}}}_\omega}d{\ensuremath{\mathcal{w}}}\,{\ensuremath{\mathcal{p}}}_{\hat{{\ensuremath{\mathcal{a}}}}}({\ensuremath{\mathcal{y}}}|{\ensuremath{\mathcal{x}}},{\ensuremath{\mathcal{w}}}){\ensuremath{\mathcal{p}}}_{\hat{{\ensuremath{\mathcal{a}}}}}\left({\ensuremath{\mathcal{w}}}|({\ensuremath{\mathcal{x}}},{\ensuremath{\mathcal{y}}})_\textrm{train}\right)\nonumber\\
&=\int_{{\ensuremath{\mathcal{E}}}_\omega}d{\ensuremath{\mathcal{w}}}\,{\ensuremath{\mathcal{p}}}_{\hat{{\ensuremath{\mathcal{a}}}}}({\ensuremath{\mathcal{y}}}|{\ensuremath{\mathcal{x}}},{\ensuremath{\mathcal{w}}})\delta({\ensuremath{\mathcal{w}}}-\widehat{{\ensuremath{\mathcal{w}}}})\nonumber\\
&={\ensuremath{\mathcal{p}}}_{\hat{{\ensuremath{\mathcal{a}}}}}({\ensuremath{\mathcal{y}}}|{\ensuremath{\mathcal{x}}},\widehat{{\ensuremath{\mathcal{w}}}}).\label{eq:classical_network}\end{aligned}$$ Once optimised, the form of the distribution chosen to evaluate the training samples, i.e. the loss function, is often ignored and the network output, ${\ensuremath{\mathcal{r}}}$, is assumed to coincide with the truth, ${\ensuremath{\mathcal{y}}}$. Note, however, that the result of is actually a distribution, characterised by the loss function or a variational distribution, at ${\ensuremath{\mathcal{w}}}=\widehat{{\ensuremath{\mathcal{w}}}}$, peaked at whatever is dictated by the output of the neural network (and not necessarily the true value of ${\ensuremath{\mathcal{y}}}$). Therefore, even in the classical case, we can make an estimation of how likely targets are by evaluating the loss function for different ${\ensuremath{\mathcal{y}}}$ using frameworks such as Markov methods (described in section \[sec:mcmc\]) or fitting the variational distribution for ${\ensuremath{\mathcal{p}}}({\ensuremath{\mathcal{y}}}|{\ensuremath{\mathcal{x}}},{\ensuremath{\mathcal{w}}})$ (described in section \[sec:VI\]).
However, this form of Bayesian inference does not characterise the uncertainties due to the neural network. Using the maximum likelihood of the network parameters (and hyperparameters) as degenerated prior distributions for calculating the posterior predictive distribution, ${\ensuremath{\mathcal{p}}}\left({\ensuremath{\mathcal{y}}}|{\ensuremath{\mathcal{x}}}, ({\ensuremath{\mathcal{x}}}, {\ensuremath{\mathcal{y}}})_\textrm{train}\right)$ completely ignores the epistemic uncertainty introduced by the network by assuming that the likelihood with such parameters exactly describes the distribution of ${\ensuremath{\mathcal{y}}}$ given a value of ${\ensuremath{\mathcal{x}}}$. Again, even though the value of ${\ensuremath{\mathcal{w}}}=\widehat{{\ensuremath{\mathcal{w}}}}$ that maximises the likelihood can be found fairly easily by various optimisation schemes, information about the shape of the likelihood is discarded and therefore may not be supported by the bulk of the probability. To incorporate our lack of knowledge and build true Bayesian neural networks, we have to revert back to .
Practical implementations {#sec:PI}
-------------------------
The methods laid out in this chapter showcase some practical ways for characterising distributions, be it the posterior distribution of network parameters, ${\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{w}}}|{\ensuremath{\mathcal{x}}},{\ensuremath{\mathcal{y}}})$, necessary for performing inference with Bayesian neural networks, the posterior distribution of targets given inputs, ${\ensuremath{\mathcal{p}}}({\ensuremath{\mathcal{y}}}|{\ensuremath{\mathcal{x}}})$, normally considered in model inference or, indeed, any other distribution. For simplicity we will refer, abstractly, to the target distribution as $${\ensuremath{\mathcal{p}}}(\lambda|\chi)=\frac{{\ensuremath{\mathcal{p}}}(\chi|\lambda){\ensuremath{\mathcal{p}}}(\lambda)}{{\ensuremath{\mathcal{p}}}(\chi)}$$ for variables $\lambda\in{\ensuremath{\mathcal{E}}}_\Lambda$ and observables $\chi\in{\ensuremath{\mathcal{E}}}_X$.
### Numerically approximate inference:Monte Carlo methods {#sec:mcmc}
Monte Carlo methods define a class of solutions to probabilistic problems. One particularly important method is Markov chain Monte Carlo (MCMC) in which a Markov chain of samples is constructed with such properties that the samples can be attributed as belonging to a target distribution.
A Markov chain is a stochastic model of events where each event depends on *only* one previous event. For example, labelling an event as $\lambda_i\in{\ensuremath{\mathcal{E}}}_\Lambda$, the probability of transitioning to another event, $\lambda_{i+1}\in{\ensuremath{\mathcal{E}}}_\Lambda$ is given by a transition probability, ${\ensuremath{\mathcal{t}}}(\lambda_{i+1}|\lambda_i)$. A chain consists of a set of events, called samples, of the state, $\{\lambda_i|\,i\in[1, n]\}$, in which each consecutive sample is correlated with the next. Although the transition probability is only conditional on the previous state, the chains are correlated over long distances. Only states that are physically uncorrelated can be kept as samples from some target distribution, ${\ensuremath{\mathcal{p}}}(\lambda|\chi)$.
One property that a Markov chain must have to represent a set of samples from a target distribution, is *ergodicity*. This means that it is possible to move from any possible state to another in some finite number of transitions from one state to the next and that no long term repeating cycles occur in the chain. The stationary distribution of the chain, in the asymptotic limit of infinite samples, can be denoted $\pi(\lambda)$. Since an infinite number of samples are needed to prove the stationary condition, MCMC techniques can only be considered numerical approximations to the target distribution. It should be noted that the initial steps in any Markov chain tend to be out of equilibrium and as such those samples can be out of distribution. All the samples until the stationary distribution is reached are considered *burn-in* samples and need to be discarded in order not to skew the approximated target distribution.
#### Metropolis-Hastings algorithm {#sec:MH}
The Metropolis-Hastings algorithm is a methods which allows states to be generated from a target distribution, ${\ensuremath{\mathcal{p}}}(\lambda|\chi)$, by defining transition probabilities between states such that the distribution of samples, $\pi(\lambda)$, in a Markov chain is stationary and ergodic. This can be ensured easily by invoking *detailed balance*, i.e. making the transition probability from state $\lambda_i$ to $\lambda_{i+1}$ reversible such that the Markov chain is necessarily in a steady state. Detailed balance can be written as $$\pi(\lambda_i){\ensuremath{\mathcal{t}}}(\lambda_{i+1}|\lambda_i)=\pi(\lambda_{i+1}){\ensuremath{\mathcal{t}}}(\lambda_i|\lambda_{i+1}),$$ which is the probability of being in state $\lambda_i$ and transitioning to state $\lambda_{i+1}$ is equal to the probability of being in state $\lambda_{i+1}$ and transitioning to state $\lambda_i$.
As described in section \[eq:bayes\], it can be effectively impossible to characterise a distribution, ${\ensuremath{\mathcal{p}}}(\lambda|\chi)$, since the integral necessary for calculating the marginal, ${\ensuremath{\mathcal{p}}}(\chi)$, can often be intractable. This isn’t a problem when using the Metropolis-Hastings algorithm, thanks to detailed balance. First, substituting the target distribution, $\pi(\lambda)\approx{\ensuremath{\mathcal{p}}}(\lambda|\chi)$, into the detailed balance equation and rearranging gives $$\begin{aligned}
\frac{{\ensuremath{\mathcal{t}}}(\lambda_{i+1}|\lambda_i)}{{\ensuremath{\mathcal{t}}}(\lambda_i|\lambda_{i+1})} & = \frac{{\ensuremath{\mathcal{p}}}(\lambda_{i+1}|\chi)}{{\ensuremath{\mathcal{p}}}(\lambda_i|\chi)}\nonumber\\
&=\frac{{\ensuremath{\mathcal{p}}}(\chi|\lambda_{i+1}){\ensuremath{\mathcal{p}}}(\lambda_{i+1})/{\ensuremath{\mathcal{p}}}(\chi)}{{\ensuremath{\mathcal{p}}}(\chi|\lambda_i){\ensuremath{\mathcal{p}}}(\lambda_i)/{\ensuremath{\mathcal{p}}}(\chi)}\nonumber\\
&=\frac{{\ensuremath{\mathcal{p}}}(\chi|\lambda_{i+1}){\ensuremath{\mathcal{p}}}(\lambda_{i+1})}{{\ensuremath{\mathcal{p}}}(\chi|\lambda_i){\ensuremath{\mathcal{p}}}(\lambda_i)}.\end{aligned}$$ The intractable integral cancels out and as such we can work with the unnormalised posterior, $$\begin{aligned}
\varrho(\lambda|\chi)&\equiv{\ensuremath{\mathcal{p}}}(\chi|\lambda){\ensuremath{\mathcal{p}}}(\lambda)\nonumber\\
&={\ensuremath{\mathcal{p}}}(\lambda|\chi){\ensuremath{\mathcal{p}}}(\chi)\nonumber\\
&={\ensuremath{\mathcal{p}}}(\chi,\lambda),\end{aligned}$$ such that $$\frac{{\ensuremath{\mathcal{t}}}(\lambda_{i+1}|\lambda_i)}{{\ensuremath{\mathcal{t}}}(\lambda_i|\lambda_{i+1})}=\frac{\varrho(\lambda_{i+1}|\chi)}{\varrho(\lambda_i|\chi)}.$$ The Metropolis-Hastings algorithm involves breaking the transition probability into two steps, ${\ensuremath{\mathcal{t}}}(\lambda_{i+1}|\lambda_i)=a(\lambda_{i+1}, \lambda_i)s(\lambda_{i+1}|\lambda_i)$, with a conditional distribution, $s(\lambda_{i+1}|\lambda_i)$, proposing a new sample and a probability, $a(\lambda_{i+1}, \lambda_i)$, describing whether the new sample is accepted as a valid proposal or not. Substituting these into the detailed balance equations gives $$\frac{a(\lambda_{i+1},\lambda_i)}{a(\lambda_i,\lambda_{i+1})}=\frac{\varrho(\lambda_{i+1}|\chi)s(\lambda_i|\lambda_{i+1})}{\varrho(\lambda_i|\chi)s(\lambda_{i+1}|\lambda_i)}.$$ A reversible acceptance probability can then be identified as $$a(\lambda_{i+1},\lambda_i)=\min\left[1,\frac{\varrho(\lambda_{i+1}|\chi)s(\lambda_i|\lambda_{i+1})}{\varrho(\lambda_i|\chi)s(\lambda_{i+1}|\lambda_i)}\right],$$ such that either $a(\lambda_{i+1},\lambda_i)=1$ or $a(\lambda_i,\lambda_{i+1})=1$[^8].
The algorithm itself has two free choices, the first is the number of iterations needed to overcome the correlation of states in the chain and properly approximate the target distribution, but in principle it should approach infinity. The second is the choice of the proposal distribution, $s(\lambda_{i+1}|\lambda_i)$. It is often chosen to be a multivariate Gaussian whose covariance can be optimised during burn-in to properly represent useful step sizes in the direction of each element of a state. This ensures that the Markov chain is a random walk. A poor choice of proposal distribution can cause extremely inefficient sampling and as such it should be chosen carefully.
Whilst, in principle, Metropolis-Hastings MCMC will work in high dimensions, the rejection rate can be high and the correlation length very long. Above a handful of parameters the computational time of Metropolis-Hastings becomes a limitation, meaning that it is not efficient for sampling high dimensional distributions such as the posterior distribution of neural network parameters.
#### Hamiltonian Monte Carlo
One way of dealing with the large correlation between samples, high rejection rate and small step sizes which occur in Metropolis-Hastings is to introduce a new sampling proposal procedure based on a Gibbs sampling step and a Metropolis-Hastings acceptance step. In Hamiltonian Monte Carlo (HMC), we introduce an arbitrary *momentum* vector, $\nu$, with as many elements as $\lambda$ has. We describe the Markov process as a classical mechanical system with a total energy (Hamiltonian) $$\begin{aligned}
\mathcal{H}(\lambda,\nu) &= \mathcal{K}(\nu)+\mathcal{V}(\lambda)\nonumber\\
&=\frac{1}{2}\nu^T{\bf M}^{-1}\nu-\log \varrho(\lambda|\chi).\end{aligned}$$ $\mathcal{K}(\nu)$ is a *kinetic energy* with a “mass” matrix, ${\bf M}$, describing the strength of correlation between parameters. $\mathcal{V}(\lambda)$ is a *potential energy* equal to the negative logarithm of the target distribution. A state, $\mathcal{z}=(\lambda, \nu)$, in the stationary distribution of the Markov chain, $\pi(\lambda,\nu)$, is a sample from the distribution ${\ensuremath{\mathcal{p}}}(\lambda,\nu|\chi)=\exp[-\mathcal{H}(\lambda,\nu)]$, found by solving the ordinary differential equation (ODE) derived from Hamiltonian dynamics $$\begin{aligned}
\dot{\lambda} & = {\bf M}^{-1}\nu\label{eq:HMCw}\\
\dot{\nu} & = -\nabla\mathcal{V}(\lambda),\label{eq:HMCp}\end{aligned}$$ where the dots are derivatives with respect to some time-like variable, which is introduced to define the dynamical system. The stationary distribution, $\pi(\lambda,\nu)\approx\mathcal{H}(\lambda,\nu)$, of the Markov chain is separable, $\exp[-\mathcal{H}(\lambda,\nu)]=\exp[-\mathcal{K}(\nu)]\exp[-\mathcal{V}(\lambda)]$, and so ${\ensuremath{\mathcal{p}}}(\lambda,\nu|\chi)\propto{\ensuremath{\mathcal{p}}}(\lambda|\chi){\ensuremath{\mathcal{p}}}(\nu)$. This means that a Gibbs sample of the $i^\textrm{th}$ momentum can be drawn, $\nu_i\sim{\ensuremath{\mathcal{p}}}(\nu)=\mathbb{N}({\bf 0}, {\bf M})$, and by evolving the state $\mathcal{z}_i=(\lambda_i,\nu_i)$ using Hamilton’s equations, a proposed sample obtained, $\mathcal{z}_{i+1}=(\lambda_{i+1}, \nu_{i+1})\sim\mathcal{p}(\lambda,\nu|\chi)$. The acceptance condition for the detailed balance is obtained by computing the difference in energies between the $i^\textrm{th}$ state and the proposed, $(i+1)^\textrm{th}$, state $$a(\mathcal{z}_{i+1},\mathcal{z}_i) = \min\left[1, \exp(\Delta\mathcal{H})\right],$$ where any loss in total energy $\Delta\mathcal{H}=\mathcal{H}(\lambda_{i+1}, \nu_{i+1})-\mathcal{H}(\lambda_i, \nu_i)$ arises from the discretisation of solving Hamilton’s equations. If the equations were solved exactly (the Hamiltonian is conserved), then every single proposal would be accepted. It is typical to use $\epsilon$-discretisation (the leapfrog method, see algorithm \[al:leapfrog\]) to solve the ODE over a number of steps, $L$, where $\epsilon$ describes the step size of the integrator. Smaller step sizes result in higher acceptance rate at the expense of longer computational times of the integrator, whilst larger step sizes result in shorter integration times, but lower acceptance. It is possible to allow for self adaptation of $\epsilon$ using properties of the chain, such as the average acceptance as a function of iteration, and a target acceptance rate, $\delta\in[0, 1]$. It has been shown that, for HMC, the optimal acceptance rate is $\delta\approx0.65$[@alex2010optimal] and so we can adapt $\epsilon$ to be of this order. Care has to be taken though, since the initial samples in the Markov chain will be out of equilibrium and so adapting $\epsilon$ in the early iterations can still lead to poor step size later on, and so this adaptation should only be attempted after the burn-in phase. A priori, it is not known how many steps to take in the integrator and so multiple examples of the HMC may need to be run to tune the value of $L$, which can be very expensive[^9].
Initial state, $\mathcal{z}=(\lambda,\nu)$; number of steps, $L$; step size, $\epsilon$; mass matrix, ${\bf M}$ Proposed state, $\mathcal{z}=(\lambda,\nu)$ Gradient of target distribution, $\nabla\mathcal{V}(\lambda)$ $\nu\leftarrow\nu-\epsilon\nabla\mathcal{V}(\lambda)/2$ $\nu\leftarrow\nu-\epsilon\nabla\mathcal{V}(\lambda)/2$
##### No U-turn sampler[@Hoffman2014]
A proposed extension to HMC to deal with the unknown number of steps in the integrator is the No U-turn sampler (NUTs). Here, the idea is to find a condition which describes whether or not running more steps in the integrator would carry on increasing the distance between the initial sample and a proposed one. A simple choice of criterion is the derivative with respect to Hamiltonian time of the half squared distance between the current proposed and initial states $$\begin{aligned}
\mathcal{s}&=\frac{d}{dt}\frac{(\lambda_{i+1}-\lambda_i)\cdot(\lambda_{i+1}-\lambda_i)}{2}\nonumber\\
&=(\lambda_{i+1}-\lambda_i)\cdot\nu.\end{aligned}$$ If $\mathcal{s}=0$ then this indicates that the dynamical system is starting to turn back on itself, i.e. making a U-turn, and further proposals can be closer to the initial state. In practice, a balanced binary tree of possible samples is created by running the leapfrog integrator either forwards or backwards for a doubling number of steps (1, 2, 4, 8, ...) where each of these steps is a leaf of the tree, $\mathcal{F}=\{(\lambda^{L\pm}, \nu^{L\pm})|L\in[1, 2, 4, ...]\}$. When the furthest distance in the trajectory, $\lambda^{\max L+}-\lambda^{\max L-}$, starts to decrease then the computation can be stopped and we can sample from $\mathcal{F}$ via a detailed-balance preserving method. Such an algorithm can greatly reduce the cost of tuning the number of steps in the integrator, $L$, in the HMC and is therefore highly beneficial when attempting to characterise a target distribution.
Thanks to the high acceptance rate and the ability to take large steps to efficiently obtain samples, HMC a is good proposition for numerically approximating the distributions such as the posterior distribution of neural network parameters. One severe limitation, though, is the choice of the mass matrix, ${\bf M}$. The mass matrix must be properly defined since it defines the direction and size of steps and correlations between parameters. It is not easy to choose its value *a priori* and a poor choice can lead to very inefficient sampling. We present below two methods which deal with the mass matrix[^10].
##### Quasi-Newtonian HMC[@Fu2016]
With quasi-Newtonian HMC (QNHMC) we make use of the second order geometric information of the target distribution as well as the gradient. The QNHMC modifies Hamilton’s equations to $$\begin{aligned}
\dot{\lambda} &={\bf B}{\bf M}^{-1} \nu \\
\dot{\nu} &=-{\bf B}\nabla\mathcal{V}(\lambda)\end{aligned}$$ where ${\bf B}$ is an approximation to the inverse Hessian derived from the L-BFGS technique (or other more computationally and memory efficient methods for large dimensions) found using quasi-Newton methods $${\bf B}_{i+1}=\left(\mathbb{I}-\frac{\mathcal{h}_i\mathcal{g}_i^T}{\mathcal{g}_i^T\mathcal{h}_i}\right){\bf B}_i\left(\mathbb{I}-\frac{\mathcal{g}_i\mathcal{h}_i^T}{\mathcal{g}_i^T\mathcal{h}_i}\right)+\frac{\mathcal{h}_i\mathcal{h}_i^T}{\mathcal{h}_i^T\mathcal{h}_i},$$ where $\mathcal{h}_i=\lambda_{i+1}-\lambda_i$, $\mathcal{g}_i=\nabla\mathcal{V}(\lambda_{i+1})-\nabla\mathcal{V}(\lambda_i)$ and $\mathbb{I}$ is the identity matrix. Obtaining ${\bf B}_{i+1}$, is extremely efficient because both $\mathcal{h}_i$ and $\mathcal{g}_i$ are calculated when solving Hamilton’s equations using leapfrog methods. Note that the approximate inverse Hessian varies with proposal, but is kept constant whilst solving Hamilton’s equations. The inverse Hessian effectively rescales the momenta and parameters such that each dimension has a similar scale and thus the movement around the target distribution is more efficient with less correlated proposals. It is easiest to begin with an initial inverse Hessian, ${\bf B}_0=\mathbb{I}$, and allow the adaptation of the Hessian to the geometry of the space. Note that the mass matrix, ${\bf M}$, is still present to set the dynamical time-like scales of Hamilton’s equations along each direction, but the rescaling of the momenta via ${\bf B}$ allows us to be fairly ambiguous about its value. The optimal mass matrix for sampling is equal to the covariance of the target distribution, but in practice, a diagonal mass matrix with approximately correct variance values for the distribution works well.
##### Example: Inference of the halo mass distribution function
To be able to extract cosmological information from the large scale structure distribution of matter in the universe, such as the mass, location and clustering of galaxies, obtained by galaxy surveys, we either have to summarise the data into statistical quantities (such as the power spectrum, etc.) or learn about the placement of all the objects in these surveys. Whilst the first method is (potentially very) lossy, the complexity of the likelihood describing the distribution of structures in the universe generally makes the second technique intractable. With the goal of maximising the cosmological information extracted from galaxy surveys the Aquila consortium has developed an algorithm for Bayesian origins reconstruction from galaxies (BORG)[@Jasche:2013; @Jasche:2015; @Lavaux:2016] which assumes a Bayesian hierarchical model to relate Gaussian initial conditions of the early universe to the complex distribution of galaxies observed today. As part of this model, one needs to relate observed galaxies to the underlying, and otherwise invisible, dark matter field through a so-called *bias* model, which is an effective description for extremely complex astrophysical effects. Finding a flexible enough and yet tractable parameterisation for this model *a priori* is a difficult task.
![The halo mass distribution function as a function of mass. The diamonds connected by a dashed line indicates the number density of haloes from an observed halo catalogue of a given mass, where the different colours represent the value of the density environment for those haloes. The lines higher in number density correspond to the more dense regions, i.e. there are more large haloes in denser environments. The solid lines show the mean halo number density from samples (taken from the Markov chain) from the neural bias model, with the shaded bands as the 68% credible intervals of these samples. There is a very good agreement between the observed halo number density and that obtained by the neural bias model.*Figure credit: Charnock et al. (2020)[@Charnock2019NeuralFunction]*[]{data-label="fig:hmdf"}](figures/1.pdf)
Using physical considerations, such as locality and radial symmetry, to reduce the numbers of degrees of freedom, a very simple mixture density network with 17 parameters was proposed to model this bias[@Charnock2019NeuralFunction]. This network, dubbed a neural physical engine due to its physical inductive biases, is small enough that each parameter is exactly identifiable, so that sensible priors could be defined for those parameters. The ability to place these meaningful priors on network parameters is well motivated for this physically motivated problem, but may be more difficult to design for problems without physical intuition.
Sampling from this model could also be integrated within the larger hierarchical model of the BORG framework using QNHMC (see figure \[fig:mdn\_diagram\] for a description of the BORG+neural bias model algorithm). Concretely, BORG was run in two blocks, first using HMC to propose samples of the dark matter density field and then using QNHMC to propose samples of the neural bias model. This allowed the exact joint posterior of both density field and network parameters to be inferred under the observation of a mock halo catalogue from the <span style="font-variant:small-caps;">velmass</span> simulations. Such inference was an example of zero-shot training since there was no *training* data necessary. Finally, and maybe most importantly, the effect of the neural network can be marginalised out to obtain the distribution of initial phases of the dark matter conditional on the observed data, agnostic of the exact form of the bias model within the posterior distribution of bias model parameters.
##### Riemannian Manifold HMC
[@Girolami2009]\[sec:RMHMC\] Whilst we have so far depended on a choice of mass matrix to set the time-like steps in the integrator, it is possible to exploit the geometry of the Hamiltonian to adaptively avoid having to choose. Samples from the Hamiltonian are effectively points in a Riemannian surface with a metric defined by the Fisher information of the target distribution, $\mathcal{I}(\lambda)=\langle\nabla\varrho(\lambda|\chi)(\nabla\varrho(\lambda|\chi))^T\rangle_\lambda$. In essence, this metric is a position-dependent equivalent to the mass matrix which we have so far considered, but since we have to calculate $\nabla\varrho(\lambda|\chi)$ in the integrator anyway, we can actually approximate the Fisher information cheaply. However, to ensure that the Hamiltonian is still the logarithm of a density it must be regularised leaving the Hamiltonian as $$\begin{aligned}
\mathcal{H}(\lambda,\nu)&=\mathcal{K}(\lambda,\nu)+\mathcal{V}(\lambda)\nonumber\\
&=\frac{1}{2}\nu^T\mathcal{I}(\lambda)^{-1}\nu-\log\varrho(\lambda|\chi)+\frac{1}{2}\log(2\pi)^{\textrm{dim }\lambda}|\mathcal{I}(\lambda)|.\end{aligned}$$ Here the kinetic term now involves a dependence on the parameters, $\lambda$, and so the Hamiltonian is not separable, i.e. the momenta are drawn from a parameter dependent mass matrix, $\nu\sim\mathbb{N}({\bf 0},\mathcal{I}(\lambda))$. The equations of motion in this case become $$\begin{aligned}
\dot{\lambda} & = \mathcal{I}(\lambda)^{-1}\nu\\
\dot{\nu} & = -\nabla\mathcal{V}(\lambda)+\frac{1}{2}\textrm{Trace}\left[\mathcal{I}(\lambda)^{-1}\nabla\mathcal{I}(\lambda)\right]\nonumber\\
&\phantom{somestuff}-\frac{1}{2}\nu^T\mathcal{I}(\lambda)^{-1}\nabla\mathcal{I}(\lambda)\mathcal{I}(\lambda)^{-1}\nu.\end{aligned}$$ With such a change, the scaling of the momenta along each parameter direction becomes automatic, but the reversibility and volume preserving evolution using the leapfrog integrator is broken and so the proposed states do not adhere to detailed balance. Instead a new symplectic integrator is required which first makes a volume preserving transformation of the momenta by calculating the Jacobian of the inverse Fisher matrix. Whilst this adds extra complexity to the equations of motion, it is equivalent to only two additional steps in the integrator since the Fisher information can be approximated cheaply from the calculation of the gradient of the potential energy. By using the RMHMC, we avoid the need to choose a mass matrix (or approximate the Hessian). $\epsilon$ can be fixed to some value as the adaptive matrix is able to overcome the step size, and $L$, i.e. the number of steps in the integrator, can be chosen to tune the acceptance rate.
##### Stochastic gradient HMC
[@Chen2014] Whilst we can sample effectively using Hamiltonian Monte Carlo, and its variants shown above, we must also address the question of the size of the data we are interested in. As so far presented, HMC requires an entire dataset, $\chi\in{\ensuremath{\mathcal{E}}}_X$, to evaluate the distribution. However, in modern times, datasets can be extremely large (big data) and it may not be possible to evaluate it all simultaneously. Furthermore, we are more and more likely to be in the regime where the data is obtained continually and streamed for inference. For this reason, most optimisation techniques rely on stochastic estimation techniques over minibatches, $\chi_i$ i.e. the union of all minibatches is the complete set, $\bigcup_i^\textrm{batches}\chi_i=\chi$. This stochastic sampling of data can be considered as being equivalent to adding a source of random noise, or scatter, around the target distribution. For clarity, the gradient of the potential energy for a minibatch becomes $$\begin{aligned}
\nabla\mathcal{V}(\lambda)&=-\log\varrho(\lambda|\chi_i)\nonumber\\
&\to-\log\varrho(\lambda|\chi)+\gamma\end{aligned}$$ where $\gamma\sim\textrm{Dist}(\lambda)$ is a random variable drawn from the distribution of noise and whose shape is described by some diffusion matrix, $\mathcal{Q}(\lambda)$. For large minibatch sizes, we can relatively safely assume this distribution is Gaussianly distributed, $\gamma\sim\mathbb{N}({\bf 0},\epsilon\mathcal{Q}(\lambda)/2)$, due to the central limit theorem where the diffusion matrix at any step in the integration can be equated to the variance of the noise, $\epsilon\mathcal{Q}(\lambda)=2\Sigma(\lambda)$. Note, that we may not, necessarily, be in the regime where we can make this assumption.
Making noisy estimates of the target distribution using minibatches brakes the Hamiltonian dynamics of the HMC and as such extremely high rejection rates can occur. In particular, the additional noise term acts as force which can push the states far from the target distribution. We can reduce this effect by taking further inspiration from mechanical systems - we can use Langevin dynamics to describe the macroscopic states of a statistical mechanical system with a stochastic noise term describing the expected effect of some ensemble of microscopic states. In particular, using second-order Langevin equations is equivalent to including a friction term which decreases the energy and, thus, counterbalances the effect of the noise. Equations \[eq:HMCw\] and \[eq:HMCp\] therefore get promoted to $$\begin{aligned}
\dot{\lambda} & = {\bf M}^{-1}\nu\\
\dot{\nu} & = -\nabla\mathcal{V}(\lambda)-\mathcal{Q}(\lambda){\bf M}^{-1}\nu+\gamma\end{aligned}$$ Solving these equations provides a stationary distribution, $\pi(\lambda,\nu)\approx\mathcal{H}(\lambda,\nu)$, with the distribution of samples, $\lambda\sim{\ensuremath{\mathcal{p}}}(\lambda|\chi)$. Of course, this method depends on knowing the distribution of the noise well, but for large minibatch sizes, this approaches Gaussian. The stochastic gradient HMC, in this case, provides a way to obtain samples from the target distribution even when not using the entire dataset and therefore vastly reducing computational expense and allowing for active collection and inference of data.
### Variational Inference {#sec:VI}
Whilst a target probability distribution can be approximately characterised by obtaining exact samples from the distribution via Monte Carlo methods, it is often a very costly process. Instead we can use variational inference, where a *variational* distribution, say ${\ensuremath{\mathcal{q}^\mathcal{m}}}(\lambda|\chi)$, is chosen to represent a very close approximation to the target distribution, ${\ensuremath{\mathcal{p}}}(\lambda|\chi)$. In general, ${\ensuremath{\mathcal{q}^\mathcal{m}}}(\lambda|\chi)$ is a tractable distribution, parameterised by some $\mathcal{m}$, and via the optimisation of these parameters ${\ensuremath{\mathcal{q}^\mathcal{m}}}(\lambda|\chi)$ can hopefully be made close to the target distribution, ${\ensuremath{\mathcal{p}}}(\lambda|\chi)$. Note, again, that if the target distribution is the posterior predictive distribution of some model, ${\ensuremath{\mathcal{p}}}({\ensuremath{\mathcal{y}}}|{\ensuremath{\mathcal{x}}})$, then fitting a variational distribution to this is not a Bayesian procedure in the same way that maximum likelihood estimation is not Bayesian.
To describe what is meant by *close* in the context of distributions we often consider the Kullback-Leibler (KL) divergence (or relative entropy). In a statistical setting, the KL-divergence is a measure of information lost when approximating a distribution ${\ensuremath{\mathcal{p}}}(\lambda|\chi)$ with some other ${\ensuremath{\mathcal{q}^\mathcal{m}}}(\lambda|\chi)$, $$\mathbb{KL}({\ensuremath{\mathcal{p}}}||{\ensuremath{\mathcal{q}^\mathcal{m}}})=\int_{{\ensuremath{\mathcal{E}}}_\Lambda}d\lambda\,{\ensuremath{\mathcal{p}}}(\lambda|\chi)\log\frac{{\ensuremath{\mathcal{p}}}(\lambda|\chi)}{{\ensuremath{\mathcal{q}^\mathcal{m}}}(\lambda|\chi)}.\label{eq:KL}$$ When $\mathbb{KL}({\ensuremath{\mathcal{p}}}||{\ensuremath{\mathcal{q}^\mathcal{m}}})=0$, there is no information loss and so ${\ensuremath{\mathcal{p}}}(\lambda|\chi)$ and ${\ensuremath{\mathcal{q}^\mathcal{m}}}(\lambda|\chi)$ are equivalent. Values $\mathbb{KL}({\ensuremath{\mathcal{p}}}||{\ensuremath{\mathcal{q}^\mathcal{m}}})>0$ indicate the degree of information lost. Note that the KL-divergence is not symmetric and as such is not a real distance metric.
The form of assumes the integral of ${\ensuremath{\mathcal{p}}}(\lambda|\chi)$ to be tractable. In fact, in the case that the expectation can be approximated well, we can use the KL-divergence to perform expectation propagation. However, if we are considering the approximation of the posterior distribution of network parameters, as stated in section \[sec:bnn\], we can expect the integral of ${\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{w}}}|{\ensuremath{\mathcal{x}}},{\ensuremath{\mathcal{y}}})$ to be intractable meaning that calculating $\mathbb{KL}({\ensuremath{\mathcal{p}_\mathcal{a}}}||{\ensuremath{\mathcal{q}^\mathcal{m}}})$ would be necessarily hard. Instead we can consider the reverse KL-divergence $$\mathbb{KL}({\ensuremath{\mathcal{q}^\mathcal{m}}}||{\ensuremath{\mathcal{p}}})=\int_{{\ensuremath{\mathcal{E}}}_\Lambda}d\lambda\,{\ensuremath{\mathcal{q}^\mathcal{m}}}(\lambda|\chi)\log\frac{{\ensuremath{\mathcal{q}^\mathcal{m}}}(\lambda|\chi)}{{\ensuremath{\mathcal{p}}}(\lambda|\chi)}.$$ The choice of ${\ensuremath{\mathcal{q}^\mathcal{m}}}(\lambda|\chi)$ is specified so that expectations are tractable. However, evaluating ${\ensuremath{\mathcal{p}}}(\lambda|\chi)$ would require calculating the evidence, ${\ensuremath{\mathcal{p}}}(\chi)$, which, although constant for different $\lambda$, remains intractable. For convenience we can consider the unnormalised distribution (as we did for detailed balance in section \[sec:MH\]) $$\begin{aligned}
\varrho(\lambda|\chi)&={\ensuremath{\mathcal{p}}}(\chi|\lambda){\ensuremath{\mathcal{p}}}(\lambda)\nonumber\\
&={\ensuremath{\mathcal{p}}}(\lambda|\chi){\ensuremath{\mathcal{p}}}(\chi)\nonumber\\
&={\ensuremath{\mathcal{p}}}(\chi,\lambda),\end{aligned}$$ and calculate a new measure $$\textrm{ELBO}({\ensuremath{\mathcal{q}^\mathcal{m}}})=-\int_{{\ensuremath{\mathcal{E}}}_\Lambda}d\lambda\,{\ensuremath{\mathcal{q}^\mathcal{m}}}(\lambda|\chi)\log\frac{{\ensuremath{\mathcal{q}^\mathcal{m}}}(\lambda|\chi)}{\varrho(\lambda|\chi)}.$$ Note that this has the form of minus the reverse KL-divergence, but is not equivalent since $\varrho(\lambda|\chi)$ is not normalised. By substitution we can see that $$\begin{aligned}
\textrm{ELBO}({\ensuremath{\mathcal{q}^\mathcal{m}}})&=-\int_{{\ensuremath{\mathcal{E}}}_\Lambda}d\lambda\,{\ensuremath{\mathcal{q}^\mathcal{m}}}(\lambda|\chi)\log\frac{{\ensuremath{\mathcal{q}^\mathcal{m}}}(\lambda|\chi)}{{\ensuremath{\mathcal{p}}}(\lambda|\chi){\ensuremath{\mathcal{p}}}(\chi)}\nonumber\\
&=-\int_{{\ensuremath{\mathcal{E}}}_\Lambda}d\lambda\,{\ensuremath{\mathcal{q}^\mathcal{m}}}(\lambda|\chi)\log\frac{{\ensuremath{\mathcal{q}^\mathcal{m}}}(\lambda|\chi)}{{\ensuremath{\mathcal{p}}}(\lambda|\chi)}+\log{\ensuremath{\mathcal{p}}}(\chi)\nonumber\\
&=-\mathbb{KL}({\ensuremath{\mathcal{q}^\mathcal{m}}}||{\ensuremath{\mathcal{p}}})+\log{\ensuremath{\mathcal{p}}}(\chi).\label{eq:ELBO}\end{aligned}$$ Since $\log{\ensuremath{\mathcal{p}}}(\chi)$ is constant with respect to the parameters, $\lambda$, maximising $\textrm{ELBO}({\ensuremath{\mathcal{q}^\mathcal{m}}})$ will force ${\ensuremath{\mathcal{q}^\mathcal{m}}}(\lambda|\chi)$ close to the target distribution ${\ensuremath{\mathcal{p}}}(\lambda|\chi)$. The term ELBO comes from the fact that the KL-divergence is non-negative and so $\textrm{ELBO}({\ensuremath{\mathcal{q}^\mathcal{m}}})$ defines a lower bound to the evidence, ${\ensuremath{\mathcal{p}}}(\chi)$.
#### Mean-field variation
One efficient way of parameterising a distribution for approximating a target, ${\ensuremath{\mathcal{p}}}(\lambda|\chi)$, is to make it factorise along each dimension of the parameters, i.e. $${\ensuremath{\mathcal{q}^\mathcal{m}}}(\lambda|\chi)=\prod_{i=1}^{\textrm{dim}\,\lambda}{\ensuremath{\mathcal{q}^\mathcal{m}}}_i(\lambda_i|\chi).$$ In doing such, the ELBO for any individual ${\ensuremath{\mathcal{q}^\mathcal{m}}}_j$ is $$\begin{aligned}
\textrm{ELBO}({\ensuremath{\mathcal{q}^\mathcal{m}}}_j) &= \underset{{\ensuremath{\mathcal{E}}}_{\Lambda,j}}{\int\cdots\int} d\lambda_j\prod_i{\ensuremath{\mathcal{q}^\mathcal{m}}}_i(\lambda_i|\chi)\times\left[\log\varrho(\lambda|\chi)-\sum_k\log{\ensuremath{\mathcal{q}^\mathcal{m}}}_k(\lambda_k|\chi)\right]\nonumber\\
&=\int_{{\ensuremath{\mathcal{E}}}_{\Lambda,j}}d\lambda_j\,{\ensuremath{\mathcal{q}^\mathcal{m}}}_j(\lambda_j|\chi)\underset{{\ensuremath{\mathcal{E}}}_{\Lambda,i\ne j}}{\int\cdots\int}d\lambda_i\prod_{i\ne j}{\ensuremath{\mathcal{q}^\mathcal{m}}}_i(\lambda_i|\chi)\nonumber\\
&\phantom{biglongline}\times\left[\log\varrho(\lambda|\chi)-\sum_k\log{\ensuremath{\mathcal{q}^\mathcal{m}}}_k(\lambda_k|\chi)\right]\nonumber\\
&=\int_{{\ensuremath{\mathcal{E}}}_{\Lambda,j}}d\lambda_j\,{\ensuremath{\mathcal{q}^\mathcal{m}}}_j(\lambda_j|\chi)\times\left[\underset{i\ne j}{\mathbb{E}}[\log\varrho(\lambda_j|\chi)]-\log{\ensuremath{\mathcal{q}^\mathcal{m}}}_j(\lambda_j|\chi)\right]\nonumber\\
&\phantom{biglongline}+ \textrm{const}\end{aligned}$$ where the constant is the expectation value of the factorised distributions, ${\ensuremath{\mathcal{q}^\mathcal{m}}}_i(\lambda_i|\chi)$, in the dimensions where $i\ne j$ and is unimportant for the optimisation of the distribution for the $j^{th}$ dimension since it is independent of $\lambda_j$. $\mathbb{E}_{i\ne j}[\log\varrho(\lambda_j|\chi)]$ is the expectation value of the logarithm of the target distribution for every ${\ensuremath{\mathcal{q}^\mathcal{m}}}_i(\lambda_i|\chi)$ where $i\ne j$, and remains due to its dependence on $\lambda_j$. The optimal $j^{th}$ distribution is the one that maximises the ELBO which is equivalent to optimising each of the factorised distributions, ${\ensuremath{\mathcal{q}^\mathcal{m}}}_j(\lambda_j|\chi)$, in turn to obtain ${\ensuremath{\mathcal{q}^\mathcal{m}}}_j(\lambda_j|\chi)=\exp\left[\mathbb{E}_{i\ne j}\left[\log\varrho(\lambda_j|\chi)\right]\right]$. This provides a mean-field approximation of the target distribution.
#### Bayes by Backprop
Bayes by Backprop[@2015arXiv150505424B; @Dikov2019BayesianArchitectures] (a form of stochastic gradient variational Bayes) provides a method for approximating a target distribution, ${\ensuremath{\mathcal{p}}}(\lambda|\chi)$, using differentiable functions such as neural networks. The basic premise of Bayes by Backprop relies on a technique known as the *reparameterisation trick*. This states that a random sample, ${\ensuremath{\mathcal{w}}}$, from a Gaussian distribution can be obtained, but also be differentiable with respect to the parameters (mean and standard deviation) of the distribution, $m=(\mu,\sigma)$, requiring only to be able to sample from a normal distribution, $\epsilon\sim\mathbb{N}(0, 1)$, i.e. $$\begin{aligned}
{\ensuremath{\mathcal{w}}}(m_i) &\sim\mathbb{N}(\mu_i, \sigma_i)\nonumber\\
&=\mu_i + \sigma_i \epsilon_i.\end{aligned}$$ Here, we can view ${\ensuremath{\mathcal{w}}}(m_i)$ as the $i^\textrm{th}$ random variable parameter of a neural network with a total of $n_\textrm{w}$ network parameters where ${\ensuremath{\mathcal{w}}}({\ensuremath{\mathcal{m}}})=\{{\ensuremath{\mathcal{w}}}(m_i)|\,i\in[1,n_\textrm{w}]\}$. Any evaluation of the neural network is a sample $\widehat{\lambda}\sim{\ensuremath{\mathcal{q}^\mathcal{m}}}(\lambda|\chi,{\ensuremath{\mathcal{w}}})$. Maximising the ELBO , between ${\ensuremath{\mathcal{q}^\mathcal{m}}}(\lambda|\chi,{\ensuremath{\mathcal{w}}})$ and an unnormalised target distribution, $\varrho(\lambda|\chi)$, can now be done via backpropagation since we can calculate $$\partial_{\ensuremath{\mathcal{m}}}\textrm{ELBO}({\ensuremath{\mathcal{q}^\mathcal{m}}})=\begin{pmatrix}\partial_{{\ensuremath{\mathcal{w}}}({\ensuremath{\mathcal{m}}})}\textrm{ELBO}({\ensuremath{\mathcal{q}^\mathcal{m}}})+\partial_{\mu}\textrm{ELBO}({\ensuremath{\mathcal{q}^\mathcal{m}}})\\(\epsilon/\sigma)\partial_{{\ensuremath{\mathcal{w}}}({\ensuremath{\mathcal{m}}})}\textrm{ELBO}({\ensuremath{\mathcal{q}^\mathcal{m}}})+\partial_{\sigma}\textrm{ELBO}({\ensuremath{\mathcal{q}^\mathcal{m}}})\end{pmatrix}$$ and update the parameters using $${\ensuremath{\mathcal{m}}}\leftarrow{\ensuremath{\mathcal{m}}}-\eta\partial_{{\ensuremath{\mathcal{m}}}}\textrm{ELBO({\ensuremath{\mathcal{q}^\mathcal{m}}})}\label{eq:BbB}$$ where $\eta$ is a learning rate. Note that the $\partial_{{\ensuremath{\mathcal{w}}}({\ensuremath{\mathcal{m}}})}\textrm{ELBO}({\ensuremath{\mathcal{q}^\mathcal{m}}})$ terms in are exactly the same as the gradients normally associated with backpropagation in neural networks.
As originally presented, Bayes by Backprop was an attempt to make Bayesian posterior predictions of targets, ${\ensuremath{\mathcal{y}}}$, from inputs, ${\ensuremath{\mathcal{x}}}$, as in , where ${\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{w}}}|{\ensuremath{\mathcal{x}}}_\textrm{train},{\ensuremath{\mathcal{y}}}_\textrm{train})\equiv\prod_{i=1}^{n_\textrm{w}}\mathbb{N}(\mu_i,\sigma_i)$. Here, the values of all $\mu_i$ and $\sigma_i$ are fit using maximum likelihood estimation (or maximum *a posterior* estimation) given data ${\ensuremath{\mathcal{x}}}_\textrm{train}$ and ${\ensuremath{\mathcal{y}}}_\textrm{train}$. As explained in section \[sec:smnn\], the distribution of weights, ${\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{w}}})$, is likely to be extremely non-trivial, since most network parameters are non-identifiable and highly degenerate with other parameters. Therefore, modelling this distribution as a Gaussian is unlikely to be very accurate. This can, therefore, incorrectly conflate the epistemic uncertainty for ${\ensuremath{\mathcal{y}}}$ from a particular network, ${\ensuremath{\mathcal{a}}}$, with parameters, ${\ensuremath{\mathcal{w}}}$, and input, ${\ensuremath{\mathcal{x}}}$, with the posterior prediction. In essence, Bayes by Backprop provides a way of sampling from a single choice of an (arbitrarily complex) approximation of a target distribution much more efficiently than using numerical schemes such as Markov methods, but there is little knowledge in how close this approximation is to the desired target. By fitting the parameters, ${\ensuremath{\mathcal{m}}}$, of the neural distribution rather than characterising their distribution, ${\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{m}}})$, the characterisation of the epistemic uncertainty is biased. As such, just using Bayes by Backprop provides a network that is Bayesian in principle, but with a limited choice of prior distribution which may fail to capture our epistemic uncertainty.
Whilst Bayes by Backprop allows us to characterise the mean and standard deviation of a Gaussian distribution from which we can draw network parameters, it can be extremely expensive to draw different parameters for each example of the data to perform the optimisation. Therefore, the data is often split into $n_\textrm{batches}$ minibatches of $n_\textrm{elems}$ elements and a single sample of each parameter drawn for all $n_\textrm{elems}$ elements in each minibatch. This clearly does not represent the variability of the distribution of parameters well and leads to artificially high variance in the stochastic gradient calculation. Furthermore, by sharing the same parameter values for all elements in a minibatch, correlations between gradients prevents the high variance from being eliminated.
##### Example: Classification of photometric light-curves
Because Bayes by Backprop can be comparatively more expensive that other practical techniques introduced below, there are a fairly limited number of examples of applications in the physics literature. One notable example however is the probabilistic classification of SuperNovae lightcurves method `SuperNNova`[@2020MNRAS.491.4277M]. The aim of that study is to analyse time-series measuring the brightness of distant galaxies as a function of time, and detect potential SuperNovae Type Ia events, of particular interest for cosmology.
[.45]{} ![Calibration of predicted class probabilities for the SuperNovae lightcurve classification problem.The BBB RNN (purple triangle) exhibits better calibration than a vanilla RNN (orange dot) of matching architecture but optimised by maximum likelihood. *Figure credit: Möller & de Boissiere (2019)[@2020MNRAS.491.4277M]*[]{data-label="fig:moller2019_calib"}](figures/moller2019_baseline_lc.png "fig:"){width="\textwidth"}
[.1]{}
[.45]{} ![Calibration of predicted class probabilities for the SuperNovae lightcurve classification problem.The BBB RNN (purple triangle) exhibits better calibration than a vanilla RNN (orange dot) of matching architecture but optimised by maximum likelihood. *Figure credit: Möller & de Boissiere (2019)[@2020MNRAS.491.4277M]*[]{data-label="fig:moller2019_calib"}](figures/moller2019_calib_multimodels.png "fig:"){width="\textwidth"}
\[fig:moller2019\]
Figure \[fig:moller2019\_lc\] illustrates the light curve classification problem on a simulated SuperNova Ia event. In that study, the authors aim to compare the performance of a vanilla recurrent neural network (RNN) classifier to a probabilistic model quantifying some uncertainties. For that purpose the authors introduce a “*Bayesian*” recurrent neural network[@Fortunato2017] (BRNN) based on a bidirectional LSTM, but using a variational Gaussian distribution for the posterior of network parameters, optimised by back-propagation. Following the approach presented in this section, the loss function for this model becomes $$\mathcal{L}({\ensuremath{\mathcal{m}}}) = -\underset{{\ensuremath{\mathcal{w}}}\sim{\ensuremath{\mathcal{q}^\mathcal{m}}}}{\mathbb{E}}\left[ \log {\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{y}}}_\textrm{train} | {\ensuremath{\mathcal{x}}}_\textrm{train}, {\ensuremath{\mathcal{w}}}) \right] + \mathbb{KL}({\ensuremath{\mathcal{q}^\mathcal{m}}}|| {\ensuremath{\mathcal{p}_\mathcal{a}}})$$ which corresponds to the ELBO introduced in , and where $\log {\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{y}}}_\textrm{train} | {\ensuremath{\mathcal{x}}}_\textrm{train}, {\ensuremath{\mathcal{w}}})$ is the log-likelihood of a categorical distribution with probabilities predicted by the neural network, and ${\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{w}}})$ is the prior on the BRNN parameters.
With this approach, the authors attempt to distinguish between aleatoric uncertainties which are uniquely determined by the categorical probabilities predicted by the model for a given set of network parameters ${\ensuremath{\mathcal{w}}}$, and the epistemic uncertainties which are, this case, characterised by the variational approximation to the posterior of network parameters ${\ensuremath{\mathcal{q}^\mathcal{m}}}({\ensuremath{\mathcal{w}}}|({\ensuremath{\mathcal{x}}},{\ensuremath{\mathcal{y}}})_\textrm{train})\approx{\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{w}}}|({\ensuremath{\mathcal{x}}},{\ensuremath{\mathcal{y}}})_\textrm{train})$. As highlighted multiple times before, one should however always be careful in interpreting these probabilities, and in that study the authors empirically check the calibration of the mean posterior probabilities using a reliability diagram[@degroot1983] . The reliability diagram shows the fraction of true positives in a binary classification problem as a function of the probabilities predicted by the model. For a perfectly calibrated classifier, only 10% of objects which received a detection probability of 0.1 are true positives.
Figure \[fig:moller2019\_calib\] shows this calibration diagram for different classifiers, but of particular interest are the curves for the *baseline RNN* and *BBB RNN*, in both cases the actual neural network architecture is identical, but the former is optimised to find the maximum likelihood estimates of the network parameters, while the later is trained by Bayes by Backprop. The *BBB RNN* predicted probabilities are closer to the diagonal representing a more *correct* calibration than the baseline RNN. In this example including a model for the epistemic uncertainties improves the model calibration.
#### Local reparameterisation trick
Although for large numbers of network parameters, $n_\textrm{w}$, characterising the global uncertainty of the parameters using the reparameterisation trick becomes computationally unfeasible for each element of data and for each parameter in the network, a local noise approximation[@KingmaVariationalTrick] can be made to transform the perturbation of parameters to a perturbation of activation values $$o^l_{jn}\sim\mathbb{N}\left(\sum_{i=1}^{\textrm{dim }l-1}\mu^l_{ji}a_{in}^{l-1},\sum_{i=1}^{\textrm{dim }l-1}\left(\sigma_{ji}^{l}\right)^2\left(a_{in}^{l-1}\right)^2\right),$$ where $a_{jn}^l=\mathcal{f}(o_{jn}^l)$ is the activated output (with possibly non-linear activation function $\mathcal{f}$) of the $j^\textrm{th}$ unit of the $l^\textrm{th}$ layer of a neural network with $n_\textrm{l}$ layers, according to the $n^\textrm{th}$ element of the input minibatch. As with the reparameterisation trick, the sampling of the activation value of any layer can be written as $$o^l_{jn}=\sum_{i=1}^{\textrm{dim }l-1}\mu^l_{ji}a_{in}^{l-1}+\epsilon^l_{jn}\sqrt{\left(\sigma_{ji}^{l}\right)^2\left(a_{in}^{l-1}\right)^2}$$ where $\epsilon^l_{jn}\sim\mathbb{N}(0, 1)$. Whilst the parameters $m_{ji}^l=(\mu_{ji}^l,\sigma_{ji}^l)$ of the Gaussian distribution describe the probabilistic model for a network parameter, ${\ensuremath{\mathcal{w}}}(m_{ji}^l)$, from unit $i$ of layer $l-1$ to unit $j$ of layer $l$, this model is never sampled, and only the activation values are sampled. The dimensionality of the probabilistic interpretation of layer outputs, i.e. the number of $\boldsymbol{\epsilon}=\{\epsilon_{in}^l|\,i\in[1,\textrm{dim }l],l\in[1,n_\textrm{l}],n\in[1,n_\textrm{elems}]\}$ needed to be stored for computation of the gradient is much lower than when considering the number of random draws needed for every single network parameter and every element of data in the minibatch. Furthermore, the variance of the gradient is much less when using the local reparameterisation trick than when assuming a single random draw for each parameter being the same for all of the elements of data in a minibatch.
#### Variational dropout
One limitation of the local reparameterisation trick is that it only applies to networks with no weight sharing, i.e. fully-connected neural networks. However, inspired by the local reparameterisation trick, a general method for approximating distributions using multiplicative noise can be implemented. Variational dropout[@2015arXiv150602142G; @Gal2017ConcreteDropout] is another way of approximating a target distribution, ${\ensuremath{\mathcal{p}}}(\lambda|\chi)$ with ${\ensuremath{\mathcal{q}^\mathcal{m}}}(\lambda|\chi,{\ensuremath{\mathcal{w}}})$. In this case, the distribution is defined by the application of random variables to the outputs of hidden layers in a neural network. Much like the local reparameterisation trick, the outputs of the $n_\textrm{l}$ layers of a neural network are draws from some multiplicative noise model, $a^l_{in}=\epsilon_{in}^l\mathcal{f}(o^{l}_{in})$, where $o_{in}^{l}$ are the non-activated outputs of the $l^\textrm{th}$ layer of a neural network at element $n$ in the minibatch. Note that $o_{in}^{l}$ can be obtained using any function, i.e. fully connected, convolutional, etc. $\mathcal{f}$ is some (possibly non-linear) activation function and $\epsilon_{in}^l\sim\textrm{Dist}(m_{in}^l)$ is a random variable drawn from some distribution parameterised by some ${\ensuremath{\mathcal{m}}}=\{m_{in}^l|\,i\in[1,\textrm{dim }l],l\in[1,n_\textrm{l}],n\in[1,n_\textrm{elems}]\}$. Selecting some form for the distribution and values for its parameters, ${\ensuremath{\mathcal{m}}}$, provides a way of obtaining samples from the neural network by running the network forward with many draws of $\epsilon$. This makes the network a model of a Bayesian neural network rather than a Bayesian neural network itself - there is no sampling of the parameters of the network, and no attempt to characterise their uncertainty. Furthermore, the value of ${\ensuremath{\mathcal{m}}}$ cannot be fit using Bayes by Backprop and it is an *a priori* choice for the sampling distribution[^11].
##### Bernoulli dropout
One method of performing variational dropout is by using $\boldsymbol{\epsilon}\sim\textrm{Bernoulli}({\ensuremath{\mathcal{m}}})$, which amounts to feeding forward an input to a network with dropout[@Hinton2012] with a keep rate ${\ensuremath{\mathcal{m}}}$ for each of the outputs of each layer of the neural network multiple times. The outputted samples can then be interpreted as the distribution of possible targets which can be obtained using that network (and the choice of the Bernoulli distribution with parameters ${\ensuremath{\mathcal{m}}}$). It is very common to set all values of $m_{in}^l\in{\ensuremath{\mathcal{m}}}$ to the same value, although it can be optimised via expectation maximisation. The ease with which this method can be implemented has made it very popular, and in the limit of large number of samples, the activated outputs approach a Gaussian distribution thanks to the central limit theorem. Note that the choice of a Bernoulli distribution changes the expected output of any activation layer as $\langle a^l_{in}\rangle=m_{in}^l(1-m_{in}^l)a^l_{in}$, therefore there is a scaling which needs to be taken into account.
##### Gaussian dropout
A second option is to draw the random variable from a unit-mean Gaussian[@KingmaVariationalTrick; @Houthooft2016VIME:Exploration], $\boldsymbol{\epsilon}\sim\mathbb{N}(\mathbb{I},\textrm{diag}({\ensuremath{\mathcal{m}}}))$, so that the expectation value of the multiplication of the output of a unit of a layer by the random variable remains, $\langle a^l_{in}\rangle=a^l_{in}$ the same since $\langle\epsilon\rangle=1$. Furthermore, by calculating the variational objective the value of ${\ensuremath{\mathcal{m}}}$ in the multiplicative noise distribution can be fit using expectation maximisation.
For both the Bernoulli, Gaussian or any other multiplicative dropout distribution, by maximising the $\textrm{ELBO}({\ensuremath{\mathcal{q}^\mathcal{m}}})$, we can get ${\ensuremath{\mathcal{q}^\mathcal{m}}}(\lambda|\chi,{\ensuremath{\mathcal{w}}})$ close to ${\ensuremath{\mathcal{p}}}(\lambda|\chi)$ allowing us to make estimates of this distribution. Again it should be stated that this is not Bayesian in the sense that if the variational distribution provided by variational dropout is approximating the posterior predictive distribution, ${\ensuremath{\mathcal{p}}}({\ensuremath{\mathcal{y}}}|{\ensuremath{\mathcal{x}}})$, there is no sense of certainty in how good that approximation is. There is no attempt to characterise our lack of knowledge of the parameters of the network *or* the parameters of the distributions, ${\ensuremath{\mathcal{m}}}$[^12].
#### Monte Carlo Dropout
Very closely related to Bernoulli variational dropout, is the MC Dropout model[@2015arXiv150602142G]. Completely similar to the previous section, MC Dropout provides a Bayesian framework to interpret the effect of traditional dropout[@Hinton2012] on neural networks. A variational distribution ${\ensuremath{\mathcal{q}^\mathcal{m}}}({\ensuremath{\mathcal{w}}}|\chi, \lambda)$ assumed for the network parameter posterior can be parameterised as ${\ensuremath{\mathcal{w}}}= \mathbf{M} \cdot \mathrm{diag}([z_j]_{j=1}^J)$ with $z_j \sim \mathrm{Bernoulli({\ensuremath{\mathcal{m}}})}$, $\mathbf{M}$ being a $K \times J$ weight matrix, and ${\ensuremath{\mathcal{m}}}$ being the dropout rate. Given this formulation for the variational distribution it can be shown that a KL divergence with respect to an implicit prior can be approximated as a simple $\ell_2$ regularisation term[@2015arXiv150602142G]. Training a neural network under dropout and with $\ell_2$ weight regularization therefore maximising the $\textrm{ELBO}({\ensuremath{\mathcal{q}^\mathcal{m}}})$ and is performing proper variational inference at no extra cost.
##### Example: Probabilistic classification of galaxy morphologies and active learning
MC Dropout is the most frequent solution adopted for probabilistic modelling using neural networks, and was the first such application in astrophysics[@2017ApJ...850L...7P], for a strong gravitational lensing parameter estimation problem. To illustrate the method and its applications on a more recent example[@2020MNRAS.491.1554W], we will consider the problem of classifying galaxy types from cutout images. In the context of modern large galaxy surveys, the challenge is to be able to automatically determine galaxy morphological types without (or with minimal) human visual inspection.
Such a study is based on the result of a large citizen science effort asking volunteers to answer a series of questions to characterise the type and morphology of a series of galaxy images. The task for the neural network is to predict volunteer responses for some galaxy types of particular interest. These answers are modeled using a binomial distribution, $\mathrm{Bin}({\ensuremath{\mathcal{r}}}, N)$, where ${\ensuremath{\mathcal{r}}}$ is the probability of a volunteer providing a positive response, and $N$ the number of volunteers asked to answer the question. Based on this model, a probabilistic prediction model can be built from a neural network estimating the parameter ${\ensuremath{\mathcal{r}}}$ from a given image ${\ensuremath{\mathcal{x}}}$: $$\mathcal{L} = - \log \mathrm{Bin}(k| {\ensuremath{\mathcal{x}}}, {\ensuremath{\mathcal{w}}}, {\ensuremath{\mathcal{a}}}, N) + \lambda \parallel {\ensuremath{\mathcal{w}}}\parallel_2^2 \;.$$
![Posterior distributions of number $k$ of positive answers to the question “Bar ?” for $N$ votes. Left: Image cutouts as presented to the citizen scientists and CNN. Center: Approximate posterior distribution predicted by one model at fixed network parameters, i.e only modeling aleatoric uncertainties, the red line represents the true observed number. Right: Approximate posterior taking into account a model for both aleatoric and epistemic uncertainties, i.e. obtained by sampling 30 realisations from the MC Dropout network. The full posterior distribution (green) is generally broader and better calibrated than individual approximate posterior samples (black). *Figure credit: Walmsley et al. (2019)[@2020MNRAS.491.1554W]*[]{data-label="fig:walmsley2019_comparison"}](figures/walmsley2019_mc_model_7){width="\textwidth"}
Figure \[fig:walmsley2019\_comparison\] illustrates the difference between posterior predictions from the fitted model with and without sampling of network outputs via MC Dropout. For any given fixed realisation (central column), the distribution of predictions is generally over confident, leading to apparent mis-calibration as two out of the seven examples appear to give very low probability to the actual value (second and fifth rows). On the contrary, after sampling from the multiplicative noise distribution (right column), the mean model approximate posterior (green) is significantly wider *and* also can exhibit more complex shapes than a simple binomial distribution. The high variance on the posterior predictions indicates that epistemic uncertainties are more significant, and the authors further measure empirically a much improved, but not perfect, calibration of the MC Dropout posterior.
Recognising that the uncertainties modelled by a choice in probabilistic network are not perfect, the authors still propose an excellent use case for them, in the form of active learning. In the active scenario, approximate and fast inference is preferred over more exact, but computationally and time expensive results. For this reason, the authors propose a strategy to identify galaxies for which the model uncertainies are largest, and preferentially ask human volunteers to label those, as a way to selectively invest human resources where they will be the most useful to help constrain the model.
In particular, the authors adopt the Bayesian Active Learning by Disagreement[@Mackay1992] (BALD) strategy which is based on selecting examples that maximize the mutual information $\mathbb{I}[k, {\ensuremath{\mathcal{w}}}]$. This quantity measures, for a given galaxy ${\ensuremath{\mathcal{x}}}$, how much information can be gained on the parameters of the neural network ${\ensuremath{\mathcal{w}}}$ from knowing the true label $k$ of that galaxy. While estimating this mutual information is in general a difficult task, a practical estimator can be derived in the case of a MC dropout. In their experiments, it is found that for some prediction tasks, as much as 60% fewer training galaxies are necessary to reach a given testing score when selected through active learning, compared to selected through a uniform random sampling.
#### Flipout
Flipout[@Wen2018] is an alternative to the local reparameterisation trick (or variational dropout) that proposes an efficient way to generate (and store) pseudo-independent perturbations to decorrelate the gradients with respect to parameters ${\ensuremath{\mathcal{m}}}=\{(\mu_i,\Delta{\ensuremath{\mathcal{w}}}_i)|i\in[1,n_\textrm{w}]\}$ according to each of the $n_\textrm{elems}$ elements of data within a minibatch. $\mu_i$ and $\Delta{\ensuremath{\mathcal{w}}}_i$ are the mean and stochastic perturbation of some network parameter ${\ensuremath{\mathcal{w}}}(m_i)$. This method therefore is more closely akin to stochastic gradient variational Bayes, where the distribution of network parameters is fitted. Note that for Flipout, the requirements on the distribution for each network parameter is that they are differentiable with respect to the parameters ${\ensuremath{\mathcal{m}}}$, that the distribution of perturbations of the network parameters, $\Delta{\ensuremath{\mathcal{w}}}_i$, is symmetric about zero (but not necessarily Gaussian) and the network parameters are independent. With $\Delta{\ensuremath{\mathcal{w}}}_i$ symmetric amount zero, the multiplication by a random matrix of signs leaves it identically distributed. This means that by choosing a single $\Delta{\ensuremath{\mathcal{w}}}_i$ for each network parameter (like $\Delta{\ensuremath{\mathcal{w}}}_i=\sigma_i\epsilon_i$ for the Gaussian case) somewhat decorrelated gradients can be obtained for each element of a minibatch by identically distributing $\Delta{\ensuremath{\mathcal{w}}}_i$ via random draws from two $n_\textrm{elems}$-length vectors, $\mathcal{j}_i=\{j_{in}=2b_{in}-1| b_{in}\sim\textrm{Bernoulli}(0.5), n\in[1,n_\textrm{elems}]\}$ and $\mathcal{k}_i=\{k_{in}=2b_{in}-1| b_{in}\sim\textrm{Bernoulli}(0.5), n\in[1,n_\textrm{elems}]\}$. Each parameter of the network is then obtained, as with the reparameterisation trick, via $${\ensuremath{\mathcal{w}}}_n(m_i)\sim\mu_{i}+\Delta{\ensuremath{\mathcal{w}}}_ij_{in}k_{in}.$$ This can be performed very quickly using matrix multiplication, affording a decrease in the variance of the stochastic gradient by a factor of $\sim1/n_\textrm{elems}$ in comparison to using shared parameter values for an entire minibatch for approximately twice the computational cost, although due to parallelisation this can be done in equal time.
##### Example: Cosmological parameter inference and uncertainty calibration
The inference of cosmological parameter values from data, such as maps of the Cosmic Microwave Background radiation, is an important task in these times of precision cosmology. It is therefore useful to consider the comparison of several of the variational inference methods to calibrate their performance[@Hortua2019ParametersNetworks] . The study uses two CNN architectures, AlexNet and VGG to predict, from an image of the CMB, a Gaussian posterior distribution on a limited set of three cosmological parameters. The outputs of these neural networks, parameterised with ${\ensuremath{\mathcal{w}}}$, are therefore chosen to be the mean, $\mu\equiv\mu({\ensuremath{\mathcal{x}}}, {\ensuremath{\mathcal{w}}})$, and covariance, $\Sigma\equiv\Sigma({\ensuremath{\mathcal{x}}}, {\ensuremath{\mathcal{w}}})$, of a multivariate Gaussian distribution. The loss function used to train these networks under a Flipout model is $$\begin{aligned}
\mathcal{L}({\ensuremath{\mathcal{m}}}) =& - \underset{{\ensuremath{\mathcal{w}}}\sim{\ensuremath{\mathcal{q}^\mathcal{m}}}}{\mathbb{E}} \left[\frac{1}{2} ({\ensuremath{\mathcal{y}}}- \mu)^T \Sigma^{-1}({\ensuremath{\mathcal{y}}}- \mu) + \frac{1}{2} \log\det \Sigma \right] + \mathbb{KL}\left( {\ensuremath{\mathcal{q}^\mathcal{m}}}\parallel {\ensuremath{\mathcal{p}_\mathcal{a}}}\right).\end{aligned}$$
![Left: Reliability diagrams for different models, before (solid) and after (dashed) post-training re-calibratrion. Right: Approximate posteriors for cosmological parameters obtained after calibration of the models. This illustrates the difficulty of obtaining well calibrated probabilistic models from neural networks directly out of the optimisation procedure, but post-hoc calibration can correct some of these biases. *Figure credit: Hortua et al. (2019)[@Hortua2019ParametersNetworks]*[]{data-label="fig:hortua2019"}](figures/hortua2019_calibration1 "fig:"){width="55.00000%"}![Left: Reliability diagrams for different models, before (solid) and after (dashed) post-training re-calibratrion. Right: Approximate posteriors for cosmological parameters obtained after calibration of the models. This illustrates the difficulty of obtaining well calibrated probabilistic models from neural networks directly out of the optimisation procedure, but post-hoc calibration can correct some of these biases. *Figure credit: Hortua et al. (2019)[@Hortua2019ParametersNetworks]*[]{data-label="fig:hortua2019"}](figures/hortua2019_triangular1 "fig:"){width="45.00000%"}
In this work, the authors perform a post-training re-calibration of the models to ensure that some coverage properties are respected. In practice, they adopt the Platt Scaling method[@Kull2017], to empirically adjust the posteriors as to make the reliability diagram of their coverage probability well calibrated. Note however that this simple scaling cannot account for all deviations from the true posterior shape.
The results of this procedure are illustrated on Figure \[fig:hortua2019\] where the left plot shows the reliability diagrams of the various models before and after calibration. The right plot illustrates the confidence contours for the approximate posterior of cosmological parameters predicted by four different re-calibrated models on the same input data. They are fairly similar in terms of sizes, but not identical, showing that this re-calibration cannot account for complex departures in posterior shapes.
One of the takeaways of this work is that overall Flipout appear to be the best performing method in terms of calibration, training speed, and accuracy, out of the four explored (reparameterisation, Flipout, MC Dropout, DropConnect).
#### Neutra
We can also use variational inference as part of a Markov chain sampling scheme. Neutra[@Hoffman2019] is a method which samples from a normal distribution and then performs a bijective transformation, $\mathcal{g}:\epsilon\sim\mathbb{N}(\bf{0},\mathbb{I})\to\lambda\sim{\ensuremath{\mathcal{p}}}(\lambda|\chi)$, to a space approximating the target distribution. In this way it can be seen as an approximation to the Riemannian manifold HMC (section \[sec:RMHMC\]) where the metric is defined by the bijective function, $\mathbb{I}(\lambda)=(\mathcal{J}\mathcal{J}^T)^{-1}$, where the Jacobian is $\mathcal{J}=\partial_\epsilon\mathcal{g}$. Using neural networks (and particularly many of the modern density estimators such as inverse autoregressive flows, etc.) this Jacobian is very easy to evaluate and therefore $\mathcal{g}$ can be arbitrarily complex, fittable to the desired function by maximising the ELBO and quick to evaluate. Samples, using HMC can be obtained very easily from the normal distribution and the bijected forward to get samples from an approximation to the target distribution much more efficiently than samples can be obtained by directly evaluating the target distribution. It should be noted that is, again, is not Bayesian in nature, since there is no quantification in how well the bijection is really performing, and therefore no way to tell if the samples, $\mathcal{g}(\epsilon)$, actually coincide with samples $\lambda\sim{\ensuremath{\mathcal{p}}}(\lambda|\chi)$.
Concluding Remarks and Outlook
------------------------------
Despite impressive accuracy in supervised learning benchmarks, current state of the art neural networks are poor at quantifying predictive uncertainty, and as such are prone to produce overconfident predictions and biases which are extremely difficult to disentangle from true properties of the data. The fact that proper uncertainty quantification is crucial for many practical applications justifies the formulation of neural networks as statistical models as a first step towards using them to make statements of inference.
While truly Bayesian neural networks have the capacity to fully characterize the epistemic uncertainty introduced by the neural network, in practice, exact Bayesian inference is intractable for neural networks, and it is common to resort to either using numerically approximated by exact samples of posterior distribution of network parameters, that is, Monte Carlo methods, or to using approximate distributions as a proxy for the true Bayesian posterior, through variational inference. The fact that, through the former method, Bayesian neural networks are often harder to train and implement than non-Bayesian neural networks means that, in the literature, variational methods have gained a lot of popularity in the recent years. However, as we have have stressed in this chapter, those latter approximate methods suffer from many pitfalls, in particular the lack of guarantee that the approximate distribution is sufficiently close to the desired target.
Because of these, other statistical tools and tests should be used in concurrence with approximate Bayesian neural networks, such as calibration and test of generalization of the predictive uncertainty to domain shifts[@NIPS2019_9547]. However, it is worth noting that Bayesian neural networks are not necessarily the most useful for doing the best reasoned inference of network outputs. For this, other methods, such as likelihood-free (simulation-based) inference, could be more efficient, powerful, and easier to implement.
[^1]: Although we discuss pairs $x$ and $y$ suggesting *inputs* and *targets*, note that we are referring generically. For example, for auto-encoders, we would consider the target to be equivalent to the input, and for generative networks we would consider the input to be some latent variables with which to generate some targets, etc.
[^2]: For example, a classical mean squared loss corresponds to modelling the negative logarithm of the likelihood as a simple standard unit variance diagonal (multivariate) Gaussian with a mean at the neural network output, ${\ensuremath{\mathcal{r}}}$.
[^3]: A sensible likelihood for network targets can be created by making the parameters of the network identifiable. One such method is to use *neural physical engines*[@Charnock2019NeuralFunction] , where neural networks are designed using physical motivation for the parameters. However, there is a trade-off with this identifiability which comes at the expense of fitting far less complex functions than are usually considered when using neural networks, but far less data and energy is needed to train such models.\[fn:npe\]
[^4]: Historically the choice of prior on the weights has normally been chosen to make the gradients of the likelihood manageable, but this may not be the best justified. Such a choice in prior could be made more meaningful by designing a model where parameters having meaning (see footnote \[fn:npe\]). Another way to solve this problem is not to consider Bayesian neural networks, but instead transfer the prior distribution of network parameters to the prior distribution of data, $\mathcal{P}({\ensuremath{\mathcal{x}}},{\ensuremath{\mathcal{y}}})$[@Hafner2018] . Note that, in any case, the prior distribution of data should be considered for a fully Bayesian analysis.
[^5]: There is a common misuse of the term Bayesian neural networks to mean networks which predict posterior distributions, say some variational distribution characterised by a neural density estimator for targets, ${\ensuremath{\mathcal{p}_\mathcal{a}}}({\ensuremath{\mathcal{y}}}|{\ensuremath{\mathcal{x}}},{\ensuremath{\mathcal{w}}})$, but these networks are not providing the true posterior distribution of the target, rather they are simply a fitted distribution approximating (to an unknown degree) the posterior (see section \[sec:classicalnn\]).
[^6]: In assuming ${\ensuremath{\mathcal{p}}}({\ensuremath{\mathcal{a}}})\approx\delta({\ensuremath{\mathcal{a}}}-\hat{{\ensuremath{\mathcal{a}}}})$ we are of course neglecting a source of epistemic uncertainty. One possible way that allows us to attempt to characterise the distribution of some subset of ${\ensuremath{\mathcal{a}}}$ is the use of Bayesian model averaging or ensemble methods[@NIPS2017_7219] . This could be used to sample randomly, for example, from the initialisation values of network parameters or the order with which minibatches of data are shuffled, all of which can affect the preferred region of network parameter space which fits the intended function.
[^7]: As described earlier, an unregularised loss function can be used to evaluate the negative logarithm of likelihood. A regularisation term on the network parameters can be added describing our belief in how the weights should behave. In this case the regularised loss is proportional to the negative logarithm of the posterior distribution and maximising the regularised loss is equivalent to MAP estimation.
[^8]: Whilst a reversible Markov chain enforces stationarity, it also leads to a probability of rejecting samples, which can be inefficient. Although we will not go into detail here, it is also possible to construct a continuous, directional Markov process which is still ergodic. In this case every sample from the state will be accepted making the algorithm more efficient for collecting samples - although the computation could be more costly. One example of such a method is the Bouncy Particle Sampler[@PhysRevE.85.026703; @Bouchard2015] in which samples are obtained from the target distribution by picking a random direction in parameter space and sampling along a piecewise-linear trajectory until the value of target distribution at that state is less than or equal to the value of the target distribution at the initial state. At this point there is a Poissonian probability of the trajectory bouncing back along another randomised trajectory, drawing samples along the way. Such methods are state-of-the-art but mostly untested in the literature on sampling neural networks.
[^9]: Recent work has been done using neural networks to approximate the gradient of target distribution, $\nabla\mathcal{V}(\lambda)$[@PhysRevD.100.104023]. Whilst this could lead to errors if trusted for the whole process, the neural gradients are only used in the leapfrog steps to propose new targets, at which point the true target distribution can be evaluated. In this case, a poorly trained estimator of the gradient of the target distribution proposes poor states, and as such the acceptance rate drops, but the samples obtained are still evaluated from the actual target distribution and therefore it is unbiased by the neural network. Furthermore, any rejected states could be rerun numerically (rather than being estimated) and added to the training set to further fit the estimator, potentially providing exponential speed up as samples are drawn. Note, the gradient of the target distribution could be fit using efficient methods described in section \[sec:VI\].
[^10]: Note we are not going to discuss relativistic HMC [@Lu2017] , where the kinetic energy is replaced with its relativistic form $\mathcal{K}(\nu_j)=\sum_{i=1}^{\textrm{dim }\lambda}m_ic_i^2\sqrt{(\nu_j/m_ic)^2+1}$. Whilst this method is valid for preventing the run-away of particles on very glassy target distributions thanks to an upper bound on the distance able to be travelled per iteration, $m$ and $c$ are (in practice) needed for every momenta in the dimension of $\lambda$. This makes it as difficult a problem as *a priori* knowing the mass matrix, ${\bf M}$, in the classical case.
[^11]: It should be noted that the parameters, ${\ensuremath{\mathcal{m}}}$, of any variational dropout distribution can be optimised via expectation maximisation.
[^12]: Of course, if considering a MAP estimate, then some characterisation of our lack of knowledge is taken into account, but the distribution is still neglected.
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{
"pile_set_name": "ArXiv"
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---
abstract: 'We present a class of Lie algebraic similarity transformations generated by exponentials of two-body on-site hermitian operators whose Hausdorff series can be summed exactly without truncation. The correlators are defined over the entire lattice and include the Gutzwiller factor $n_{i\uparrow}n_{i\downarrow}$, and two-site products of density $(n_{i\uparrow} + n_{i\downarrow})$ and spin $(n_{i\uparrow}-n_{i\downarrow})$ operators. The resulting non-hermitian many-body Hamiltonian can be solved in a biorthogonal mean-field approach with polynomial computational cost. The proposed similarity transformation generates locally weighted orbital transformations of the reference determinant. Although the energy of the model is unbound, projective equations in the spirit of coupled cluster theory lead to well-defined solutions. The theory is tested on the 1D and 2D repulsive Hubbard model where we find accurate results across all interaction strengths.'
author:
- 'Jacob M. Wahlen-Strothman'
- 'Carlos A. Jiménez-Hoyos'
- 'Thomas M. Henderson'
- 'Gustavo E. Scuseria'
date:
title: Lie Algebraic Similarity Transformed Hamiltonians for Lattice Model Systems
---
*Introduction.*—Hamiltonian similarity transformations are ubiquitous in many areas of physics, including electronic structure and condensed matter theories, and have been applied in a myriad of contexts [@Boys; @Glazek; @Wegner; @White; @Yanai; @Nooijen]. Jastrow-Gutzwiller correlation factors are also very popular as variational wave functions in quantum Monte Carlo and other applications [@Fulde; @Gutz-review; @Baeriswyl2; @Sorella; @Baeriswyl1; @Henderson; @Gutz; @Neuscamman2; @Imada]. Non-variational solutions have also been discussed in the literature. Tsuneyuki [@Tsuneyuki] presented a Hilbert space Jastrow method based on a Gutzwiller factor $\sum_{i}n_{i\uparrow
}n_{i\downarrow }$ and applied it to the 1D Hubbard model, minimizing its energy variance as in the transcorrelated method [@Handy; @Ten-no; @Tsuneyuki2]. Neuscamman *et al.* [@Chan] proposed many-body Jastrow correlators, diagonal in the lattice basis, and truncated them to a subset of sites matching a given pattern; these authors compared projective solutions with those obtained stochastically via Monte Carlo.
Here, we consider Hamiltonian transformations of the form $e^{-J}He^{J}$ based on hermitian correlators $J
$ built from general two-body products of on-site operators over the entire lattice. The transformations here are generated by density (charge), spin, and Gutzwiller factor correlators, including density-spin crossed terms. Similar Jastrow-type correlators have been extensively discussed in the literature but almost always in a variational context [@Gutz-review]. Our transformed Hamiltonian is non-hermitian but can be solved in mean-field via projective equations similar in spirit to those of coupled cluster theory [@Chan; @Bartlett]. In this sense, the model is an extension that fits under the *generalized* coupled cluster label [@Kutzelnigg; @Piecuch; @Nooijen2]. The fundamental difference is that traditional coupled cluster is formulated with particle-hole excitations out of a reference determinant via a non-hermitian cluster operator; the present model is constructed with on-site hermitian correlators.
The main result of this paper is the realization that the Hausdorff series resulting from the non-unitary similarity transformation $e^{-J}He^{J}$ can be analytically summed. This result follows from Lie algebraic arguments [@Gilmore] after recognizing that both the Hamiltonian and the correlator $J$ can be written in the basis of generators of an enveloping algebra built from on-site operators [@Jinmo]. Topologically, our transformation is non-compact and yields a non-hermitian Hamiltonian, whereas traditional canonical transformations are almost always chosen to be unitary, thus compact, and preserve hermiticity. There is a mistaken belief that quantum canonical transformations must be unitary [@Arlen]; this is not correct even in the linear case [@NUHF]. From this perspective, traditional coupled cluster exponentiates the shifts of a nilpotent algebra, whose Hausdorff series truncates at the fourth commutator (for a two-body $H$). For two-body correlators, our model leads to a renormalized $N$-body Hamiltonian that produces locally weighted orbital rotations of a reference state, leading to expectation values between non-orthogonal determinants. The general theory of enveloping algebras in electronic structure theory, upon which the present results follow, will be discussed in detail elsewhere [@Jinmo]. Here, we introduce the main mathematical results in a self-contained manner, touching upon the physical aspects of the model, and present benchmark applications to the 1D and 2D Hubbard models.
*Theory.*—Consider on-site fermion creation and annihilation operators $c_{i\sigma}^{\dag }$, $c_{i\sigma}$ and on-site number operators $n_{i\sigma }=c_{i \sigma}^{\dag }c_{i\sigma }$, where $\sigma=\uparrow,\downarrow$. The number operators are idempotent ($n_{i\sigma}^{2}=n_{i\sigma}$) and satisfy $$\left[ n_{i\sigma },n_{j\sigma'}\right] =0.$$The elemental fermion operators are eigenoperators (shifts) of the on-site number operators: $$\left[ n_{i\sigma},c_{j\sigma'}^{\dag }\right] =\delta _{i\sigma,j\sigma'}c_{j\sigma'}^{\dag }.$$Let us now define a general two-body correlator, $$J=\frac{1}{2}\sum_{i\sigma, j\sigma' }\alpha _{i\sigma, j\sigma'}n_{i\sigma}n_{j\sigma'}, \label{J_def}$$that is chosen hermitian ($J^{\dag }=J$) with real $\alpha $ amplitudes. We require $\alpha$ to be zero on the diagonal to exclude one-body operators. The main result of this paper is the realization that a *global* non-unitary similarity transformation using the correlator above yields$$\begin{aligned}
e^{-J}c_{k\sigma}^{\dag }e^{J} &=& \exp \bigg( -\sum_{j\sigma'}\alpha _{k\sigma,j\sigma'}n_{j\sigma'}\bigg)
c_{k\sigma}^{\dag } \label{Haus} \\
&=& \exp\big(-J_{k\sigma}\big)c^\dagger_{k\sigma}, \nonumber\end{aligned}$$the exponential of a Hermitian one-body operator that commutes with the fermion operator being transformed. Using this result and its adjoint, we obtain$$e^{-J}c_{k\sigma}^{\dag }c_{l\sigma' }e^{J} = e^{-J_{k\sigma}} c_{k\sigma }^{\dag }c_{l\sigma' }e^{J_{l\sigma'}}. \label{onebody}$$ Note how the exponentials carry a *local* weight $\alpha$ that depends on the transformed fermion operators. As $J_{k\sigma}$ only consists of on-site number operators, it is a diagonal one-body operator generating a local transformation. The algebraic derivation is straightforward and can be found in the Supplemental Material. When acting on a Slater determinant, these one-body exponentials produce local, non-uniform orbital rotations. An orbital rotation, $e^K$, where $K=\sum_{ij}\Lambda_{ij}c^\dagger_ic_j$, acting on a Slater determinant $|\Phi\rangle$ defined by orbital coefficients $C_{ip}$, where $p\in\{1,\dots, N_o\}$ labels the occupied orbitals for $N_o$ occupied states, produces a new unnormalized Slater determinant with transformed coefficients (see eg. [@PHF]) $$e^K|\Phi\rangle=|\Phi'\rangle,\quad C'=e^\Lambda C.$$ Using this property, expectation values of Eq. (\[onebody\]) with $|\Phi\rangle$ can be calculated as transition density matrix elements between non-orthogonal states [@blaizot] by applying the local transformations to the reference state $$\langle\Phi|e^{ -J_{p\sigma}} c_{p\sigma }^{\dag }c_{q\sigma' }e^{J_{q\sigma'}}|\Phi\rangle=\det(S)~\rho_{q\sigma',p\sigma}, \label{density}$$ where $$\begin{aligned}
S &=& (e^{-\Lambda^{p\sigma}}C)^\dagger~(e^{\Lambda^{q\sigma'}}C),\\
\quad \rho &=& (e^{\Lambda^{q\sigma'}}C)~S^{-1}(e^{-\Lambda^{p\sigma}}C)^\dagger. \nonumber\end{aligned}$$ Here, $\Lambda^{p\sigma}$ is a diagonal matrix containing the set of coefficients for the operator $J_{p\sigma}$. The evaluation of a single element of the transformed density for a general correlator of the form given in Eq. (\[J\_def\]) therefore has $O(MN_o^2)$ cost, where $M$ is the size of the basis. This cost can be reduced for some cases, as explained below. The extension of Eq. (\[onebody\]) for a two-body operator is straightforward, resulting in similar expressions with one local weight per fermion operator. This can be evaluated as a two-site dependent transformation on the Slater determinant with the same cost for the evaluation of each element.
The correlator (\[J\_def\]) includes all combinations of two-body on-site operators. The quantities of interest here are $N_{i}N_{j},S_{i}^{z}S_{j}^{z},N_{i}S_{j}^{z},S_{i}^{z}N_{j},D_{i}+D_j$, where$$\begin{aligned}
N_{i} &=&n_{i\uparrow }+n_{i\downarrow }, \label{cartan}\\
S_{i}^{z} &=&n_{i\uparrow }-n_{i\downarrow }, \nonumber \\
D_{i} &=&n_{i\uparrow }n_{i\downarrow }. \nonumber\end{aligned}$$These two-body operators acting on a reference modify correlation corresponding to density, spin, and double-occupancy providing flexibility to improve approximate wavefunctions with poor descriptions of these correlation functions. Here we seek to add these corrections in an efficient manner via similarity transformation.
The nearest-neighbor Hubbard Hamiltonian $$H=-t\sum_{\left\langle ij\right\rangle \sigma }c_{i\sigma }^{\dag
}c_{j\sigma } +U\sum_{i}n_{i\uparrow
}n_{i\downarrow } \nonumber$$contains at most two-site terms; $\langle ij\rangle$ represents nearest-neighbors, $t$ is the energy for a particle to hop from one site to a neighboring site, and $U$ is the interaction of two particles on the same site. Clearly, $J$ as defined in Eq. (\[J\_def\]) commutes with the interaction but not with hopping. The proposed similarity transformation yields the nonhermitian effective Hamiltonian $\overline H=e^{-J}He^J$, $$\begin{aligned}
\overline{H}&=&-t\sum_{\langle ij\rangle\sigma }e^{-J_{i\sigma}}c^\dagger_{i\sigma}c_{j\sigma}e^{J_{j\sigma}}+U\sum_{i}n_{i\uparrow }n_{i\downarrow }.\end{aligned}$$The correlator parameters must be optimized and a suitable non-hermitian optimization scheme is needed as the energy, $E_J=\langle\overline H\rangle$, is unbound with respect to $\alpha$.
The Hamiltonian can be treated via left projection with the component operators of $J$. Schrödinger’s equation is projected into a subspace, as in coupled cluster theory, leading to a system of equations that determine the unknowns $\alpha_{ij}$ [@Chan; @Bartlett] $$\begin{aligned}
\langle n_{i\sigma}n_{j\sigma'}(\overline{H}-\langle\overline H\rangle)\rangle&=&0,\quad \forall~i\sigma\neq j\sigma' \label{Proj}.\end{aligned}$$ This exactly solves Schrödinger’s equation projected onto the subspace $\{n_{i\sigma}n_{j\sigma'}|\Phi\rangle, |\Phi\rangle\}$. A hermitized variance can be constructed and minimized as in transcorrelation [@Tsuneyuki] and is discussed in the Supplemental Material. Quantities other than the energy can be calculated via a Lagrangian formulation analogous to that used in coupled cluster theory [@Bartlett], $$L=\langle \overline H\rangle+\sum_{i\sigma\neq j\sigma'}z_{i\sigma,j\sigma'}\langle n_{i\sigma}n_{j\sigma'}(\overline H -\langle\overline H\rangle)\rangle. \label{lagrange}$$ Requiring $\partial L/\partial z_{i\sigma,j\sigma'}=0$ results in Eq. (\[Proj\]) and $\partial L/\partial\alpha_{i\sigma,j\sigma'}=0$ gives the equations for the linear response amplitudes $z$. The expectation value of an arbitrary operator $\mathcal{O}$ can then be calculated, including the contributions from the response equations, as $$\langle\mathcal{O}\rangle_J = \langle \overline\mathcal{O}\rangle
+ \sum_{i\sigma\neq j\sigma'}z_{i\sigma,j\sigma'}\langle n_{i\sigma}n_{j\sigma'}(\overline\mathcal{O} -\langle \overline\mathcal{O}\rangle)\rangle \label{response}$$ where $\overline\mathcal{O}=e^{-J}\mathcal{O}e^J$. (See Ref. [@Bartlett] for more details on response equations.)
![Correlation energy of 8-hole-doped Hubbard chains for $U=2$ with open and closed boundaries on an RHF reference. DMRG is used to find exact energies for open systems [@ALPS1; @ALPS2].[]{data-label="fig:dope"}](./Doped_8_2.eps){width="3.4in"}
*Results.*—We present benchmark calculations for one and two-dimensional Hubbard systems with a Hartree-Fock Slater determinant reference. Unless otherwise stated, the calculations include Gutzwiller, density-density, and spin-spin terms, with energy in units of $t$. The correlation energy is measured with respect to restricted Hartree-Fock (RHF) energies.
------------- ---- --- --------- --------- --------- -------
$6\times 6$ 24 4 -1.0546 -1.1684 -1.1853 87.06
$6\times 6$ 24 8 -0.6097 -0.9845 -1.0393 87.25
$8\times 8$ 28 4 -1.0078 -1.0659 -1.0718 90.78
$8\times 8$ 44 4 -1.0542 -1.1693 -1.1858 87.75
------------- ---- --- --------- --------- --------- -------
: Energy per site and portion of the recovered correlation energy ($E_c$) for 2D, periodic lattices with $N_o$ electrons, an RHF reference wavefunction and released-constraint Monte Carlo ($E_{MC}$) [@Zhang1; @Zhang2] as the best estimate for the exact result.[]{data-label="table1"}
In Fig. \[fig:dope\] we compare the correlation energy captured for 8-hole doped systems with periodic and non-periodic boundaries. The theory is most accurate for systems with few particles and open boundary conditions, but as we increase the size of the system, finite size effects are suppressed and we begin to approach the thermodynamic limit while still recovering more than 95% of the correlation energy. We produce highly accurate results for doped systems and find some reduction in the quality as we approach the thermodynamic limit but still find significant improvements.
We have applied the method to a select set of two-dimensional Hubbard lattices where high quality reference data are available (Table \[table1\]). By screening the incorrect double-occupancy with the Gutzwiller factor and incorporating corrections to the correlations in the RHF wavefunction with the spin and density terms, most of the correlation energy is recovered, dramatically improving the results. This supports the method as a cost-effective way to treat larger systems with high accuracy. Calculations on much bigger lattices are feasible and they will be reported in due course.
We can calculate other significant quantities using Eq. (\[response\]) and results agree well with other state-of-the-art methods. Figure \[fig:spin\] shows the discrete Fourier transform of the spin-spin correlation function, $S(i)=\langle S^z_0S^z_i\rangle$, for a one-dimensional Hubbard ring. We find that the Jastrow correlator adds most of the correct correlation on an otherwise smooth background. The function is only slightly underestimated at $q=\pi$, unlike the comparatively flat reference, and has the correct long-range decay.
![The spin-spin correlation function in Fourier space calculated using Eq. (\[response\]) ($S_J$) for a 30-site Hubbard ring at half-filling and U=3 compared to the RHF reference and DMRG [@ALPS1; @ALPS2].[]{data-label="fig:spin"}](Kspace3.eps){width="3.3in"}
If the wavefunction $|\Phi\rangle$ is a right eigenstate of the transformed Hamiltonian, then $e^J|\Phi\rangle$ is a solution to the original Hamiltonian, and we expect good approximations to $|\Phi\rangle$ and $J$ to have a similar approximate relationship. In order to attest to the power of the method, we compare transformed ($E_J$) and variational energies, $$E_v=\frac{\langle e^JHe^J\rangle}{\langle e^{2J}\rangle},$$ for a 14-site system (Table \[table\]). By directly computing overlaps with the exact wavefunction, we can determine how close the correlated wavefunction is to the exact solution. There is strong agreement in the weak-coupling regime, where the results are of excellent quality, and reasonable agreement at larger interaction strengths as seen before. This is further supported by the overlap of the reference and correlated wavefunctions with the true ground state and by the variance per particle ($0.0038$, 0.0174, and 0.0481 for $U$ of $1$, $2$, and $3$ respectively). As $e^J|\Phi\rangle$ is close to the true ground state, the Schrödinger equation is nearly satisfied and the energy evaluation using the transformed Hamiltonian is close to the corresponding variational energy.
--- ---------- ---------- ---------- ---------- -------- --------
1 -14.6983 -14.7003 -14.7147 -14.4758 0.9721 0.9972
2 -11.8486 -11.8765 -11.9543 -10.9758 0.8780 0.9815
3 -9.3925 -9.5059 -9.7488 -7.4758 0.7100 0.9378
4 -7.4688 -7.4745 -8.0883 -3.9758 0.5296 0.8711
5 -5.5017 -6.0807 -6.8531 -0.4758 0.3967 0.8544
6 -3.7983 -4.8766 -5.9165 3.0242 0.3086 0.8437
--- ---------- ---------- ---------- ---------- -------- --------
: Energies and overlaps for the exact $|0\rangle$, RHF $|\Phi\rangle$, and correlated wavefunctions $|J\rangle$ for a 14-site ring, where $|J\rangle=e^J|\Phi\rangle/|\langle\Phi |e^{2J}|\Phi\rangle|^{\frac{1}{2}}$. []{data-label="table"}
---- --- --------- --------- --------- --------- ---------
14 2 -1.1172 -1.1644 -1.1634 -1.1920 -1.1982
14 4 -0.7344 -0.8808 -0.9018 -0.9595 -0.9840
14 8 0.0313 -0.5921 -0.5354 -0.6691 -0.7418
16 2 -1.0000 -1.0973 -1.0509 -1.1188 -1.1261
16 4 -0.5000 -0.7854 -0.6931 -0.8270 -0.8514
16 8 0.5000 -0.4619 -0.2235 -0.4873 -0.5293
---- --- --------- --------- --------- --------- ---------
: Energies for $4\times 4$ Hubbard lattices with RHF ($E_{RJ}$) and UHF ($E_{UJ}$) references including spin-density correlators ($S^z_iN_j$), compared to exact energies ($E_{ED}$) [@Zhang1; @Fano].[]{data-label="table3"}
For the treatment of larger systems, the cost can be moderated by restricting the correlation amplitudes to include only local interactions. For sufficiently weak $U$, the correlations can be limited to short range without significant impact on the quality of the results. Figure \[fig:range\] illustrates this effect. As is clear from the plot, weaker interactions benefit little from correlation beyond second-nearest neighbors. Truncation at range $R$ results in $O(MR)$ equations ($O(R)$ for translationally invariant systems) instead of $O(M^2)$, greatly reducing the computational effort required. Additionally the cost for construction, inversion, and the determinant evaluation of the overlap matrix in Eq. (\[onebody\]) can be reduced by a factor of $M$ via an update of the overlap for each new iteration due to the simple diagonal structure of the transformations. Truncating the range of the transformation in this manner will restrict the range at which correlation functions calculated with Eq. (\[lagrange\]) will be accurate, and we believe this approximation is most appropriate in systems where correlations decay rapidly.
![The cumulative fraction of correlation energy captured by limiting the range $R$ of the correlators compared to the full $R=15$ set for a 30-site ring at half-filling with an RHF reference. $R=0$ includes the Gutzwiller factor.[]{data-label="fig:range"}](./corrRange3.eps){width="3.4in"}
There is some reduction in accuracy near half-filling. This can be addressed using a spin-broken reference (Table \[table3\]). Whereas the RHF reference has large ionic contributions (zero or double occupancies), the UHF wavefunction possesses the correct qualitative antiferromagnetic character near half-filling. As all two-body on-site correlators are included in our model (Eq. \[cartan\]) it is sufficiently flexible to accommodate the necessary correlations depending on the nature of the reference. In the case of RHF, the largest contribution to the correlation energy is typically due to the Gutzwiller factor. Unlike the RHF case, we find a non-zero contribution from spin-density cross terms with a symmetry-broken reference as the up and down orbitals are no longer identical. Results improve significantly with the UHF reference, particularly for large values of U, and we typically recover 80% or more of the correlation energy. Additional results are available in the Supplemental Material.
*Conclusions.*—We have presented similarity transformations generated by exponentials of hermitian on-site operators resulting in a Hausdorff series which can be resummed and leads to expressions that can be easily evaluated with polynomial cost. Results from this model are in very good agreement with the variational energies, indicating it is a cost effective way of treating wavefunctions of the form $e^J|\Phi\rangle$. Results for 1D and 2D systems are of high quality with little computational effort. Our method is size extensive, preserves symmetries that commute with $J$, and is an alternative to variational Monte Carlo sampling with no stochastic error. The strategy here adopted represents a reasonable approach to optimizing wavefunctions of the form considered in this work without the need to evaluate the variational energy, which is combinatorial in cost if computed exactly or gains statistical error if calculated via Monte Carlo.
In extending this idea to non-lattice Hamiltonians, it will be necessary to determine an “on-site” basis for the correlators. In lattice models, we have an obvious choice. The atomic orbital basis may be a good starting point, but better choices might exist.
There are many possible extensions to improve the quality of the results. The theory can incorporate more flexible references such as Hartree-Fock-Bogoliubov or projected BCS wavefunctions with reference optimization. The marriage of the current on-site correlators and pair coupled cluster doubles (non-hermitian pairing excitation operators in the particle-hole basis) [@pCCD; @pqCCD; @pCCD2] is promising as they separately address weak and strong correlation, and is a topic of further study.
*Acknowledgments.*—This work was supported by the National Science Foundation (CHE-1110884). GES is a Welch Foundation Chair (C-0036).
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{
"pile_set_name": "ArXiv"
}
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abstract: |
We formulate the saturation property for vector measures in lcHs as a nonseparability condition on the derived Boolean $\sigma$-algebras by drawing on the topological structure of vector measure algebras. We exploit a Pettis-like notion of vector integration in lcHs, the Bourbaki–Kluvánek–Lewis integral, to derive an exact version of the Lyapunov convexity theorem in lcHs without the BDS property. We apply our Lyapunov convexity theorem to the bang-bang principle in Lyapunov control systems in lcHs to provide a further characterization of the saturation property.
[**Key words:**]{} vector measure; locally convex Hausdorff space; Maharam-type; saturation; Lyapunov convexity theorem; multifunction; control system; bang-bang principle.
[**MSC 2000:**]{} Primary: 28B05, 46G10; Secondary: 28B20, 49J30.
author:
- |
M. Ali Khan\
[Department of Economics, The Johns Hopkins University]{}\
[Baltimore, MD 21218, United States]{}\
[e-mail: [email protected]]{}
- |
\
Nobusumi Sagara[^1]\
[Department of Economics, Hosei University]{}\
[4342, Aihara, Machida, Tokyo 194–0298, Japan]{}\
[e-mail: [email protected]]{}
title: 'Maharam-Types and Lyapunov’s Theorem for Vector Measures on Locally Convex Spaces without Control Measures[^2]'
---
Introduction
============
Lyapunov’s convexity theorem on the range of an atomless vector measure has proved to be of tremendous use and significance in applied work in the theories of optimal control, of statistical decisions, of Nash equilibria in large games, and of Walrasian equilibria in mathematical economics; see the references of the extended introduction in [@ks14]. As explored in [@ks13; @ks14; @ks15] at length, the so-called saturation property is an indispensable structure on measure spaces for the Lyapunov convexity theorem to be valid without the closure operation for measures taking values in infinite-dimensional spaces. This property of measure spaces is by now a well established notion in measure theory, as formulated in the scalar measure case (see [@fr12; @hk84; @ma42]). For vector measures with values in locally convex Hausdorff spaces (lcHs), the notion of saturation is easily adapted under the Bartle–Dunford–Schwartz (BDS) property of lcHs, a property automatically satisfied in Banach spaces (see [@ks13; @ks15]), to guarantee the existence of control measures for any vector measure.
A natural question pertains to a fruitful formulation of the notion of saturation for vector measures in lcHs [*without*]{} control measures. Such a motivation stems from the recent development on the Lyapunov convexity theorem in Banach spaces, and we provide an answer to this question here. The main contribution of this answer hinges on what we see as the following threefold contribution:
(i) We formulate the saturation property for vector measures in lcHs as a nonseparability condition on the Boolean $\sigma$-algebras along the lines of [@fr12; @ks09; @ma42; @ri92] by drawing on the topological structure of vector measure algebras established in the important monograph [@kk75].
(ii) We exploit a Pettis-like notion of vector integration in lcHs, the Bourbaki–Kluvánek–Lewis integral, (see [@bo59; @kl70; @le70]) to derive an exact version of the Lyapunov convexity theorem in lcHs without the BDS property, a result that is a natural extension of those presented in [@ks13; @ks15] (necessity and sufficiency).
(iii) We apply our Lyapunov convexity theorem to the bang-bang principle established in [@ks14] and Lyapunov control systems in lcHs explored in [@kk75; @kn75] to provide a further characterization of saturation.
The organization of the paper is as follows. In Section 2 we introduce the notion of Maharam types and saturation in Boolean $\sigma$-algebras and define vector measure algebras in lcHs. Section 3 deals with the space of integrable functions with respect to a vector measure in lcHs and investigates its topological properties, especially completeness and separability. Section 4 presents the main result of the paper in which the equivalence of the saturation property of a vector measure algebra and the Lyapunov convexity theorem is established in full generality. Section 5 contains the application of the main result to the bang-bang principle and Lyapunov control systems.
Preliminaries
=============
Boolean Algebras and Maharam Types
----------------------------------
Let $\F$ be a Boolean algebra with binary operations $\vee$ and $\wedge$, and a unary operation $^c$, endowed with the order $\le$ given by $A\le B\Longleftrightarrow A=A\wedge B$, where $\O=\Omega^c$ is the smallest element in $\F$ and $\Omega=\O^c$ is the largest element in $\F$. A subset $\I$ of $\F$ is an *ideal* if $\O\in \I$, $A\vee B\in \I$ for every $A,B\in \I$ and $B\le A$ with $A\in \I$ implies $B\in \I$. The *principal ideal* $\F_E$ generated by $E\in \F$ is an ideal of $\F$ given by $\F_E=\{ A\in \F\mid A\le E \}$, which is a Boolean algebra with unit $E$. An element $A\in \F$ is an *atom* if $A\ne \O$ and $E\le A$ with $E\in \F$ implies either $E=\O$ or $E=A$; $\F$ is *nonatomic* if it has no atom.
A *subalgebra* of $\F$ is a subset of $\F$ that contains $\Omega$ and is closed under the Boolean operations $\vee$, $\wedge$ and ${}^c$. A subalgebra $\U$ of $\F$ is *order-closed* with respect to the order $\le$ if any nonempty upwards directed subsets of $\U$ with its supremum in $\F$ has the supremum in $\U$. A subset $\U\subset \F$ *completely generates* $\F$ if the smallest order closed subalgebra in $\F$ containing $\U$ is $\F$ itself. The *Maharam type* of a Boolean algebra $\F$ is the smallest cardinal of any subset $\U\subset \F$ which completely generates $\F$. By $\kappa(\F)$ we denote the Maharam type of $\F$. A Boolean algebra is *saturated* if for every $E\in \F$ with $E\ne \O$ the Maharam type $\kappa(\F_E)$ of the principal ideal $\F_E$ generated by $E$ is uncountable. A Boolean algebra $\F$ is nonatomic if and only if $\kappa(\F_E)$ is infinite for every $E\in \F$ with $E\ne \O$ (see [@ks14 Proposition 2.1]).
Vector Measure Algebras in lcHs
-------------------------------
Let $(\Omega,\F)$ be a measurable space and $X$ be a locally convex Hausdorff space (briefly, lcHs). A set function $m:\F\to X$ is *countably additive* if for every pairwise disjoint sequence $\{ A_n \}$ in $\F$, we have $m(\bigcup_{n=1}^\infty A_n)=\sum_{n=1}^\infty m(A_n)$, where the series is unconditionally convergent with respect to the locally convex topology on $X$. It is well known that if $m$ is countably additive with respect to “some" locally convex topology on $X$ that is consistent with the dual pair $\langle X,X^* \rangle$, then it is countably additive with respect to “any" locally consistent convex topology on $X$ (see [@tw70 Proposition 4]). This is a consequence of Orlicz–Pettis theorem (see [@mc67 Theorem 1]). Therefore, the countable additivity of vector measures is independent of the particular topologies lying between the weak and Mackey topologies of $X$.
For a vector measure $m:\F\to X$, a set $N\in \F$ is *$m$-null* if $m(A\cap N)=\bold{0}$ for every $A\in \F$. An equivalence relation $\sim$ on $\F$ is given by $A\sim B$ if and only if $A\triangle B$ is $m$-null, where $A\triangle B$ is the symmetric difference of $A$ and $B$ in $\F$. The collection of equivalence classes is denoted by $\widehat{\F}=\F/\sim$ and its generic element $\widehat{A}$ is the equivalence class of $A\in \F$. The lattice operations $\vee$ and $\wedge$ in $\widehat{\F}$ are given in a usual way by $\widehat{A}\vee \widehat{B}=\widehat{A\cup B}$ and $\widehat{A}\wedge \widehat{B}=\widehat{A\cap B}$. The unary operation ${}^c$ in $\widehat{\F}$ is obtained for taking complements in $\widehat{\F}$ by $\widehat{A}^{c}=\widehat{(A^c)}$. Under these operations $\widehat{\F}$ is a Boolean $\sigma$-algebra. Let $\hat{m}:\widehat{\F}\to X$ be an $X$-valued countably additive function on $\widehat{\F}$ defined by $\hat{m}(\widehat{A})=m(A)$ for $\widehat{A}\in \widehat{\F}$. Then the pair $(\widehat{\F},\hat{m})$ is called a *vector measure algebra* induced by $m$. Denote by $\F_E=\{ A\cap E\mid A\in \F \}$, a $\sigma$-algebra of $E\in \F$ inherited from $\F$, and $(\widehat{\F_E},\hat{m})$ a vector measure algebra induced by the restriction of $m$ to $\F_E$. Then $\widehat{\F_E}$ is the principal ideal of $\widehat{\F}$ generated by the element $\widehat{E}\in \widehat{\F}$. A $\sigma$-algebra $\F$ is *$m$-essentially countably generated* if there exists a countably generated sub $\sigma$-algebra $\G$ of $\F$ such that $\widehat{\G}=\widehat{\F}$.
A set $A\in \F$ is an *atom* of $m$ if $m(A)\ne \bold{0}$ and $\widehat{A}$ is an atom of $\widehat{\F}$. If $m$ has no atom, it is said to be *nonatomic*. The Maharam type of a vector measure $m:\F\to X$ is defined to be $\kappa(\widehat{\F})$. Thus, $\kappa(\widehat{\F})$ is countable if and only if $\F$ is $m$-essentially countably generated. Hence, $m$ is nonatomic if and only if $\kappa(\widehat{\F_E})$ is infinite for every $m$-nonnull $E\in \F$. A vector measure space $(\Omega,\F,m)$ (or a vector measure $m$) is *saturated* if $\kappa(\widehat{\F_E})$ is uncountable for every $m$-nonnull $E\in \F$. Let $Y$ be a lcHs. A vector measure $n:\F\to Y$ is *absolutely continuous* (or *$m$-continuous*) with respect to a vector measure $m:\F\to X$ if every $m$-null set is $n$-null. It is evident that for every lcHs $Y$ a vector measure $n:\F\to Y$ is saturated (resp. nonatomic) if and only if there exists a saturated (resp. nonatomic) vector measure $m:\F\to X$ with respect to which $n$ is absolutely continuous.
The $L^1$-Space of Vector Measures
==================================
Integrals with Respect to Vector Measures
-----------------------------------------
The following Pettis-like notion of the integral of measurable functions with respect to a vector measure was introduced in @bo59 and elaborated independently by @kl70 [@le70; @le72]. For a detailed treatment of this integral, see [@kk75; @osr08; @pa08].
Let $\langle x^*,m \rangle:\F \to \R$ be the scalar measure defined by $\langle x^*,m \rangle(A):=\langle x^*,m(A) \rangle$ with $x^*\in X^*$ and $A\in \F$. A measurable function $f:\Omega\to \R$ is *$m$-integrable* if it is integrable with respect to the scalar measure $\langle x^*,m \rangle$ for every $x^*\in X^*$ and for every $A\in \F$ there exists a vector $x_A\in X$ such that $$\left\langle x^*,x_A \right\rangle=\int_Afd\langle x^*,m \rangle \quad \text{for every $x^*\in X^*$}.$$
Since the dual space $X^*$ is a total family of linear functionals on a lcHs $X$, the vector $x_A$ is unique, which we denote by $\int_Afdm$. Unlike the definition of the integrals with respect to $m$ formulated in [@ks14] (see also the references therein), no assumption is made about the completeness of $X$ and the existence of a scalar measure with respect to which $m$ is absolutely continuous is unnecessary.
The vector space of all $m$-integrable functions is denoted by $L(m)$. Let $\P$ be a separating family of seminorms in $X$ that generates the locally convex topology $\tau$. Let $U_p^\circ$ denotes the polar of the set $U_p=\{ x\in X\mid p(x)\le 1 \}$, that is, $U_p^\circ=\{ x^*\in X^*\mid |\langle x^*,x \rangle|\le 1 \ \forall x\in U_p \}$. Each $p\in \P$ induces a seminorm $p(m)$ in $L(m)$ via the formula $$p(m)(f)=\sup_{x^*\in U_p^\circ}\int |f|d|\langle x^*,m \rangle|, \quad f\in L(m),$$ where $|\langle x^*,m \rangle|$ denotes the variation of the scalar measure $\langle x^*,m \rangle$. The above seminorms turn $L(m)$ into a locally convex space. The quotient space of $L(m)$ modulo the subspace $\bigcap_{p\in \P}p(m)^{-1}(0)$ of all $m$-null functions is denoted by $L^1(m)$, which is a lcHs with its topology denoted by $\tau(m)$.
Denote by $L^\infty(m)$ (the equivalence classes of) the space of all $m$-essentially bounded measurable functions $f$ on $\Omega$, endowed with the $m$-essentially supremum norm $$\| f \|_\infty=\inf\{ \alpha>0\mid \text{$\{ \omega\in \Omega\mid |f(\omega)|>\alpha \}$ is $m$-\hspace{0pt}null} \}.$$ Recall that $X$ is said to be *sequentially complete* if every Cauchy sequence in $X$ converges. If $X$ is a sequentially complete lcHs, then $L^\infty(m)\subset L^1(m)$ (see [@kk75 Lemma II.3.1] or [@pa08 Theorem 4.1.9$'$]) and $(L^\infty(m),\| \cdot \|_\infty)$ is a Banach space (see [@pa08 Theorem 4.5.8]).
A linear operator $T_m:L^1(m)\to X$ defined by $T_mf=\int fdm$ for $f\in L^1(m)$ is called an *integration operator* of $m$. We also denote $T_mf$ by $m(f)$. Since $L^\infty(m)\subset L^1(m)$ whenever $X$ is sequentially complete, one can restrict the integration operator $T_m$ to $L^\infty(m)$ endowed with the $m$-essential sup norm. Hence, if $X$ is a sequentially complete lcHs, then the integration operator $T_m:L^\infty(m)\to X$ is continuous (see [@kk75 Lemma II.3.1]). Moreover, the following continuity result of the integration operator is true without any completeness assumption on $X$.
\[lem1\] The integration operator $T_m:L^1(m)\to X$ is continuous for the weak topologies[^3]of $L^1(m)$ and $X$.
We first show that the finite signed measure $x^*m$ is a continuous linear functional on $L^1(m)$ for every $x^*\in X^*$ with respect to $\tau(m)$-topology. To this end, let $\{ f_\alpha \}$ be a net in $L^1(m)$ such that $p(m)(f_\alpha-f)$ for every $p\in \P$. If $x^*\in X^*$ vanishes on $U_p$, then $x^*\in U_p^\circ$, and hence, we obtain $$\left| \int f_\alpha d\langle x^*,m \rangle-\int fd\langle x^*,m \rangle \right|\le \int|f_\alpha-f|d|\langle x^*,m \rangle|\le p(m)(f_\alpha-f)\to 0.$$ If $x^*\ne 0$ on $U_p$, define $\tilde{p}(m)(x^*)=\sup_{x\in U_p}|\langle x^*,x \rangle|$. By normalization, we have $y^*:=x^*/\tilde{p}(m)(x^*)\in U_p^\circ$, and hence $$\begin{aligned}
\left| \int f_\alpha d\langle x^*,m \rangle-\int fd\langle x^*,m \rangle \right|
& \le \tilde{p}(m)(x^*)\int|f_\alpha-f|d|\langle y^*,m \rangle| \\
& \le [\tilde{p}(m)(x^*)][p(m)(f_\alpha-f)]\to 0.\end{aligned}$$ Therefore, $\int f_\alpha d\langle x^*,m \rangle\to \int f d\langle x^*,m \rangle$ for every $x^*\in X^*$. This means that $\langle x^*,m \rangle$ is an element of the dual space $(L^1(m))^*$ for every $x^*\in X^*$. Let $\{ g_\alpha \}$ be a net in $L^1(m)$ that converges weakly to $g\in L^1(m)$. Then for every $x^*\in X^*$, we have $$\begin{aligned}
\langle x^*,T_mg_\alpha \rangle=\int g_\alpha d\langle x^*,m \rangle\to \int g d\langle x^*,m \rangle=\langle x^*, T_mg \rangle.\end{aligned}$$ Hence, $T_m$ is continuous for the weak topologies of $L^1(m)$ and $X$.
Completeness of $L^1(m)$
------------------------
A measure space $(\Omega,\F,\mu)$ (possibly $\mu$ is an infinite measure) is *localizable* if for every continuous linear functional $\varphi$ on $L^1(\mu)$ there is a bounded measurable function $g:\Omega\to \R$ such that $\varphi(f)=\int fgd\mu$ for every $f\in L^1(\mu)$ (see [@kk75 Section I.3]). When $\mu$ is $\sigma$-finite, localizability is automatically satisfied because the dual of $L^1(\mu)$ is $L^\infty(\mu)$. It is well known that such duality is no longer true if $\mu$ is not $\sigma$-finite. A measure space $(\Omega,\F,\mu)$ is localizable if and only if its measure algebra is a complete Boolean algebra as a partially ordered set (see [@se50 Theorem 5.1]).
For $p\in \P$, the *$p$-semivariation* of a vector measure $m:\F\to X$ is a set function $\| m \|_p:\F\to \R$ defined by $$\| m \|_p(A)=\sup_{x^*\in U_p^\circ}|\langle x^*,m \rangle|(A), \quad A\in \F.$$ The $p$-semivariation $\| m \|_p$ of $m$ is bounded, monotone and countably subadditive with the following estimate (see [@kk75 Lemma II.1.2]): $$\sup_{E\subset A}p(m(E))\le\| m \|_p(A)\le 2\sup_{E\subset A}p(m(E)).$$ A finite (scalar) measure $\mu_p$ is a *$p$-control measure* of $m$ whenever $\mu_p(A)=0$ if and only if $\| m \|_p(A)=0$. If $X$ is a lcHs, then for every $p\in \P$ there exists a $p$-control measure of $m$ (see [@kk75 Corollaries II.1.2 on p.19 and II.1.1 on p.21]). A vector measure $m:\F\to X$ is *absolutely continuous* with respect to a scalar measure $\mu$ if $\mu(A)=0$ implies that $m(A)=\bold{0}$. A finite measure $\mu$ is a *control measure* of $m$ whenever $\mu(A)=0$ if and only if $m(A\cap E)=\bold{0}$ for every $E\in \F$.
Since each characteristic function of $\F$ is identified with an element of $L^1(m)$, we can restrict the locally convex Hausdorff topology $\tau(m)$ of $L^1(m)$ to the vector measure algebra $(\widehat{\F},\hat{m})$ of $m$. Thus, the relative $\tau(m)$-topology on $\widehat{\F}$ is generated by a family of semimetrics $\{ d_p\mid p\in \P \}$ on $\widehat{\F}$ by the formula $$d_p(\widehat{A},\widehat{B}):=p(m)(\chi_A-\chi_B)=\| m \|_p(A\triangle B), \quad A,B\in \F.$$ Denote by $L^1_E(m)=\{ f\chi_E\mid f\in L^1(m) \}$ the vector subspace consisting of $m$-integrable functions on $\Omega$ restricted to $E\in \F$ and similarly, $L^\infty_E(m)=\{ f\chi_E\mid f\in L^\infty(m) \}$ the vector subspace consisting of $m$-essentially bounded functions on $\Omega$ restricted to $E$.
Without completeness, the function space $L^1(m)$ would be useless in the limit operation of integrals. To overcome this difficulty, @kk75 introduced the notion of closed vector measures in lcHs.
A vector measure $m:\F\to X$ is *closed* if $\widehat{\F}$ is complete in the $\tau(m)$-topology.
> Closed vector measures are those for which most of the classical theory of $L^1$ spaces carries over, especially results concerning completeness. The phenomenon of non-closed measures is observable only if the range space is not metrizable. ([@kk75 p.67].)
A sufficient condition for the closedness of $m$ given by @ri84 [Proposition 1] is the metrizability of the range $m(\F)$ in a lcHs $X$. In particular, if $X$ is a Fréchet space, then $L^1(m)$ is also a Fréchet space (see [@kk75 Theorems IV.4.1 and IV.7.1] and [@fnr Theorem 1]). The significance of the notion of closed vector measures is exemplified by the next characterization of the completeness of $L^1(m)$ attributed to [@kk75], which can be slightly generalized as the current form, where the assumption of quasicompleteness is replaced by that of sequentially completeness (see [@ri83 Proposition 1] and [@fnr Proposition 3]).
\[pro1\] Let $X$ be a sequentially complete lcHs. Then a vector measure $m:\F\to X$ is closed if and only if $L^1(m)$ is $\tau(m)$-complete.
For other characterizations of closed vector measures, see [@or95; @or99; @ri84; @ri90].
It follows immediately from Proposition \[pro1\] that a vector measure is closed if it has a control measure. A lcHs $X$ has the *Bartle–Dunford–Schwartz (BDS) property* if every $X$-valued vector measure has a control measure. As demonstrated in @ks14 [Example 2.1], a lcHs $X$ has the BDS property for each of the following cases: (i) $X$ is metrizable; (ii) $X$ is Suslin; (iii) $X^*$ is weakly$^*$ separable.
A vector measure $m:\F\to X$ is *countably determined* if there exists a sequence $\{ x_n^* \}$ in $X^*$ such that a set $A\in \F$ is $m$-null if it is $\langle x_n^*,m \rangle$-null for each $n=1,2,\dots$. A countably determined vector measure $m$ in a lcHs $X$ has a control measure. Indeed, its control measure is given by $$\mu(A)=\sum_{n=1}^\infty \frac{|\langle x_n^*,m \rangle|(A)}{2^n(1+|\langle x_n^*,m \rangle|(\Omega))},\quad A\in \F.$$
At a first glance, the closedness of vector measures seems innocuous, but it demands a lot because a vector measure $m:\F\to X$ is closed if and only if there exists a localizable measure $\mu$ such that $m$ is $\mu$-continuous (see [@kk75 Theorem IV.7.3] and [@kl77 Lemma 11 and Corollary 13]). Indeed, it is “nothing" but absolute continuity in disguise! Without $\sigma$-finiteness, the localizable measures nevertheless would not play a significant role as in the control measures of a vector measure because of the lack of the duality between $L^1(\mu)$ and $L^\infty(\mu)$.
Separability of $L^1(m)$
------------------------
The following observation due to @ri92 [Proposition 1A] (see also [@pa08 Theorems 4.6.2 and 4.6.3]) plays a crucial role to develop the notion of saturation for vector measures in lcHs.
\[pro2\] If $X$ is a sequentially complete lcHs, then $L^1(m)$ is separable if and only if $\widehat{\F}$ is separable.
Indeed, for a scalar measure $\mu$, the non-separability of $L^1(\mu)$ is a defining property for the saturation of the measure space $(\Omega,\F,\mu)$ (see [@fr12 331O and 365X(p)], [@hk84 Corollary 4.5], and [@ks09 Fact 2.5]).
\[thm2\] Let $X$ be a sequentially complete lcHs and $m:\F\to X$ be a vector measure. Then the following conditions are equivalent:
1. $(\Omega,\F,m)$ is saturated;
2. $\widehat{\F_E}$ is non-separable for every $m$-nonnull $E\in \F$;
3. $L^1_E(m)$ is non-separable for every $m$-nonnull $E\in \F$.
\(ii) $\Leftrightarrow$ (iii): See Proposition \[pro2\].
\(iii) $\Rightarrow$ (i): If the Maharam type of $\widehat{\F_E}$ is countable for some $m$-nonnull $E\in \F$, then there is a countable subset $\widehat{\U}$ of $\widehat{\F_E}$ that completely generates $\widehat{\F_E}$. By virtue of the axiom of choice, there is a choice function $\varphi: \widehat{\F_E}\to \F_E$ such that $\varphi(\widehat{A})\in \widehat{A}$ for every $\widehat{A}\in \widehat{\F_E}$. Let $\G$ be the subalgebra of $\F_E$ completely generated by $\varphi(\widehat{\U})$. By construction, $\G$ is $m$-essentially countably generated satisfying $\widehat{\G}=\widehat{\F_E}$, and hence, $L^1_E(m)$ is separable (see [@ri92 Proposition 2]).
\(i) $\Rightarrow$ (iii): If the Maharam type of $\widehat{\F_E}$ is uncountable for every $m$-nonnull $E\in \F$, then the cardinality of $\widehat{\F_E}$ is uncountable. Suppose, to the contrary, that $L^1_E(m)$ is separable for some $m$-nonnull $E\in \F$. By Proposition \[pro2\], $\widehat{\F_E}$ is separable, so there exists a countable subset $\G$ of $\F_E$ such that for every $A\in \F_E$, every $\varepsilon_1,\dots,\varepsilon_k>0$ and every $p_1,\dots,p_k\in \P$ there exists $B\in \G$ satisfying $\| m \|_{p_i}(A\triangle B)<\varepsilon_i$ for each $i=1,\dots,k$. On the other hand, since $\widehat{\F_E}\setminus \widehat{\G}$ is uncountable, there exists an $m$-nonnull set $A\in \F_E\setminus \G$ with $\widehat{A}\in \widehat{\F_E}\setminus \widehat{\G}$. Since $\P$ is a separating family of seminorms, we can take a seminorm $p\in \P$ with $p(m(A))>0$. Take any $B\in \G$. Then $A\cap B=\emptyset$, and hence, $\| m \|_p(A\triangle B)=\| m \|_p(A\cup B)\ge \| m \|_p(A)\ge p(m(A))$ for every $B\in \G$, a contradiction. Therefore, $L^1_E(m)$ is non-separable.
Lyapunov Convexity Theorem in LcHs
==================================
Lyapunov Measures and Lyapunov Operators
----------------------------------------
Following [@ks13; @ks15], we characterize the Lyapunov convexity theorem in terms of the integration operator.
A vector measure $m:\F\to X$ is a *Lyapunov measure* if for every $E\in \F$ the set $m(\F_E)$ is weakly compact and convex in $X$.
The integration operator $T_m:L^\infty(m)\to X$ is said to be:
(i) a *nonatomic operator* if for every $m$-nonnull $E\in \F$ and every neighborhood $U$ of the origin in $X$ there exists $f\in L^\infty_E(m)\setminus \{ 0 \}$ with signed values $\{ -1,0,1 \}$ such that $T_mf\in U$;
(ii) a *Lyapunov operator* of $m$ if for every $m$-nonnull $E\in \F$ the restriction $T_m:L^\infty_E(m)\to X$ is not injective.
\[thm3\] Let $X$ be a sequentially complete lcHs and $m:\F\to X$ be a vector measure. Then $m$ is nonatomic if and only if $T_m:L^\infty(m)\to X$ is a nonatomic operator.
Suppose that $T_m$ is a nonatomic operator. If $m$ has an atom $E\in \F$, then $m(E)\ne \bold{0}$. Thus, for every neighborhood $U$ of $\bold{0}$ there exists $f\in L^\infty_E(m)\setminus \{ 0 \}$ with signed values $\{ -1,0,1 \}$ such that $T_mf\in U\cap (-U)$. Since measurable functions are constant on atoms of $m$, either $f=\chi_E$ or $f=-\chi_E$. We thus obtain $m(E)\in \{ \pm T_mf \}\subset U\cap(-U)$ for every neighborhood $U$ of $\bold{0}$, and hence, $m(E)=\bold{0}$, a contradiction.
Conversely, suppose that $T_m$ is not a nonatomic operator. Then there exists $E\in \F$ with $m(E)\ne \bold{0}$ and a convex, balanced, absorbing neighborhood $U$ of $\bold{0}$ such that $T_mf\not\in U$ for every $f\in L^\infty_E(m)\setminus \{ 0 \}$ with signed values $\{ -1,0,1 \}$. (Such a neighborhood can be taken as $U\cap (-U)$ as above). Since there exists $p\in \P$ such that $U=U_p$ (see [@ru73 Theorems 1.34 and 1.35]), for every $m$-nonnull $A\in \F_E$, we have $m(A)=T_m\chi_A\not\in U_p$. Thus, $\| m \|_p(A)\ge p(m(A))>1$ for every $m$-nonnull $A\in \F_E$. If $E$ is not an atom of $m$, then there exists $A\in \F_E$ such that $m(A)\ne m(E)$ and $m(A)\ne \bold{0}$. Take any $p$-control measure $\mu_p$ of $m$. By the $\mu_p$-continuity of $\| m \|_p$, we have $\mu_p(A)>0$. Hence, there exists $\delta>0$ such that for every $B\in \F$ with $\mu_p(A\cap B)<\delta$, we have $\| m \|_p(A\cap B)<1$, a contradiction. Therefore, $E$ is an atom of $m$.
The range of a Lyapunov measure $m$ is weakly compact and convex in $X$ in view of $m(\F)=m(\F_\Omega)$. If $m$ has an atom $E\in \F$, then evidently, $m(\F_E)$ is not convex in $X$. Therefore, every Lyapunov measure is nonatomic. As the next result demonstrates, the nonatomicity of vector measures is reinforced as well by the notion of Lyapunov operators (see [@ks13 Theorem 3.2]).
\[thm4\] Let $X$ be a sequentially complete lcHs and $m:\F\to X$ be a vector measure. If $T_m:L^\infty(m)\to X$ is a Lyapunov operator, then it is a nonatomic operator.
The following result is due to [@kk75 Theorem V.1.1].
\[pro3\] Let $X$ be a quasicomplete lcHs and $m:\F\to X$ be a closed vector measure. Then $m$ is a Lyapunov measure if and only if $T_m:L^\infty(m)\to X$ is a Lyapunov operator. The range of a Lyapunov measure $m$ is given by $$m(\F)=\{ m(f) \in X\mid 0\le f\le 1,\,f\in L^\infty(m) \}.$$
Saturation: A Sufficiency Theorem
---------------------------------
\[lem2\] Let $X$ be a sequentially complete lcHs and $m:\F\to X$ be a closed vector measure. Then a bounded closed convex subset of $L^\infty(m)$ is weakly compact in $L^1(m)$.
Let $K$ be a bounded, closed, convex subset of $L^\infty(m)$. By Proposition \[pro1\], $K$ is $\tau(m)$-complete in $L^1(m)$. Since the $\tau(m)$-topology of $L^1(m)$ is generated by the family of seminorms $\{q_{p,x^*}\mid p\in \P,\,x^*\in U_p^\circ \}$ defined by $q_{p,x^*}(f)=\int|f|d|\langle x^*,m \rangle|$, the quotient space $L^1(m)/q_{p,x^*}^{-1}(0)$ is a vector subspace of $L^1(|\langle x^*,m \rangle|)$ endowed with the $L^1(|\langle x^*,m \rangle|)$-norm. Let $\pi_{p,x^*}:L^1(m)\to L^1(m)/q_{p,x^*}^{-1}(0)$ be the natural projection. Then $\pi_{p,x^*}(K)$ is a closed subset of $L^1(|\langle x^*,m \rangle|)$. Since the boundedness of $K$ in $L^\infty(m)$ implies that $\pi_{p,x^*}(K)$ is uniformly integrable in the sense that $$\lim_{|\langle x^*,m \rangle|(A)\to 0}\sup_{f\in \pi_{p,x^*}(K)}\int_A|f|d|\langle x^*,m \rangle|=0$$ by the Dunford–Pettis criterion (see [@ds58 Corollary IV.8.11]), $\pi_{p,x^*}(K)$ is weakly sequentially compact, and hence, weakly compact in $L^1(|\langle x^*,m \rangle|)$ for every $x^*\in U_p^\circ$ and $p\in \P$ in view of the Eberlein–Smulian theorem (see [@ds58 Theorem V.6.1]). By [@kk75 Theorem I.1.1], the weak compactness of $\pi_{p,x^*}(K)$ in $L^1(|\langle x^*,m \rangle|)$ for every $p\in \P$ and $x^*\in X^*$ implies the weak compactness of $K$ in $L^1(m)$.
The *density* of a topological space $S$, denoted by $\mathrm{dens}\,S$, is the smallest cardinal of any dense subset of $S$. The density of a lcHs $X$ is equal to the *topological dimension* of $X$, i.e., the smallest cardinal of any set whose linear span is dense in $X$ whenever $\mathrm{dens}\,X$ is infinite.
\[lem7\] $\kappa(\widehat{\F_E})\le \mathrm{dens}\,\widehat{\F_E}$ for every $m$-nonnull $E\in \F$.
Let $E\in \F$ be an $m$-nonnull set and $\{ \widehat{A}_\alpha \}_{\alpha<\mathrm{dens}\,\widehat{\F_E}}$ be a family of elements in $\widehat{\F_E}$ such that its $\tau(m)$-closure coincides with $\widehat{\F_E}$. Denote by $\widehat{\U}$ the subalgebra of $\widehat{\F_E}$ completely generated by $\{ \widehat{A}_\alpha \}_{\alpha<\mathrm{dens}\,\widehat{\F_E}}$. Since $\{ \widehat{A}_\alpha \}_{\alpha<\mathrm{dens}\,\widehat{\F_E}}$ is contained in $\widehat{\U}$, the $\tau(m)$-closure $\mathrm{cl}\,\widehat{\U}$ of $\widehat{\U}$ coincides with $\widehat{\F_E}$. If we demonstrate that $\widehat{\U}=\widehat{\F_E}$, then we have $\kappa(\widehat{\F_E})\le \mathrm{dens}\,\widehat{\F_E}$. To this end, it suffices to show the $\tau(m)$-closedness of $\widehat{\U}$. Let $\{ \widehat{A}^\nu \}$ be a net in $\widehat{\U}$ converging to $\widehat{A}$. Since $\mathrm{cl}\,\widehat{\U}=\widehat{\F_E}$, there exists $\widehat{B}^\nu\in \widehat{\U}$ such that $\widehat{B}^\nu\le \widehat{A}\wedge \widehat{A}^\nu\in \widehat{\F_E}$ for each $\nu$ and the net $\{ \widehat{B}^\nu \}$ converges to $\widehat{A}$. Extracting a subnet from $\{ \widehat{B}^\nu \}$ (which we do not relabel), one may assume that $\{ \widehat{B}^\nu \}$ is upward directed in $\widehat{\U}$ with $\lim_\nu\widehat{B}^\nu=\sup_\nu \widehat{B}^\nu=\widehat{A}$. Since $\widehat{\U}$ is order closed, we have $\widehat{A}\in \widehat{\U}$.
\[lem3\] If $X$ is a sequentially complete lcHs and $m:\F\to X$ is a closed vector measure such that $\mathrm{dens}\,X<\kappa(\widehat{\F_E})$ for every $m$-nonnull $E\in \F$, then $T_m:L^\infty(m)\to X$ is a Lyapunov operator.
Suppose to the contrary that $T_m$ is not a Lyapunov operator. Then there exists an $m$-nonnull set $E\in \F$ such that the restriction $T_m:L^\infty_E(m)\to X$ is an injection. Let $\B$ denote the closed unit ball in $L^\infty(m)$ and set $\B_E=\B\cap L^\infty_E(m)$. Since $L^\infty_E(m)$ is a closed vector subspace of $L^1(m)$, by Lemma \[lem2\], $\B_E$ is a weakly compact subset of $L^1(m)$. Since the integration operator $T_m:L^1(m)\to X$ is continuous for the weak topologies of $L^1(m)$ and $X$ by Lemma \[lem1\], the restriction $T_m$ to $\B_E$ is a homeomorphism between $\B_E$ and $T_m(\B_E)$. It follows from the convexity and weak closedness of $T_m(\B_E)$ that $T_m(\B_E)$ is also closed in the strong topology of $X$. Let $\{ x_\alpha \}_{\alpha<\mathrm{dens}\,X}$ be a dense subset of $T_m(\B_E)$ and define $f_\alpha=T^{-1}_mx_\alpha$. Then $\{ f_\alpha \}_{\alpha<\mathrm{dens}\,X}$ is a weakly dense subset of $\B_E$ in $L^1(m)$. Since $\B_E$ is convex, it is $\tau(m)$-closed, and hence, $\{ f_\alpha \}_{\alpha<\mathrm{dens}\,X}$ is also a $\tau(m)$-dense subset of $\B_E$. Since $\widehat{\F_E}\subset \B_E$, by Lemma \[lem7\], we have $\kappa(\widehat{\F_E})\le \mathrm{dens}\,\widehat{\F_E}\le \mathrm{dens}\,\B_E\le \mathrm{dens}\,X$, a contradiction to the hypothesis.
An immediate consequence of Proposition \[pro3\] and Lemma \[lem3\] is the following version of the Lyapunov theorem, which is a further generalization of [@gp13; @ks13; @ks15].
\[thm5\] If $X$ is a quasicomplete lcHs and $m:\F\to X$ is a closed vector measure such that $\mathrm{dens}\,X<\kappa(\widehat{\F_E})$ for every $m$-nonnull $E\in \F$, then $m$ is a Lyapunov measure with its range given by $$m(\F)=\{ m(f)\in X\mid 0\le f\le 1,\,f\in L^\infty(m) \}.$$
In particular, if $X$ is separable and $m$ is saturated, then the density hypothesis of Theorem \[thm5\] is automatic because $\mathrm{dens}\,X=\aleph_0<\aleph_1\le \kappa(\widehat{\F_E})$ for every $m$-nonnull $E\in \F$.
The *algebraic dimension* of a lcHs $X$ is the cardinality of a Hamel basis in $X$. We denote by $\dim X$ the algebraic dimension of $X$. Consider the dimensionality condition: $$\dim L^\infty_E(m)>\dim X \quad \text{for every $m$-\hspace{0pt}nonnull $E\in \F$}.$$ This is obviously a sufficient condition for $T_m$ to be a Lyapunov operator. As evident from Lemma \[lem3\], the dimensionality condition is sharpened to the density condition: $$\mathrm{dens}\,L^\infty_E(m)>\mathrm{dens}\,X \quad \text{for every $m$-\hspace{0pt}nonnull $E\in \F$}.$$ For the discussion of the dimensionality and density conditions, see [@ks13 Remark 3.2].
Saturation: A Necessity Theorem
-------------------------------
Let $Y$ be a lcHs. Denote by $\mathit{ca}(\F,m,Y)$ the space of $Y$-valued closed vector measures on $\F$ which are absolutely continuous with respect to a vector measure $m:\F\to X$.
\[thm8\] Let $X$ be a quasicomplete separable lcHs, $Y$ be a quasicomplete, separable lcHs with a vector subspace that is isomorphic to an infinite-dimensional Banach space, and $m:\F\to X$ be a closed vector measure. Then the following conditions are equivalent:
1. $(\Omega,\F,m)$ is saturated;
2. $T_n:L^\infty(n)\to Y$ is a Lyapunov operator for every $n\in \mathit{ca}(\F,m,Y)$;
3. Every vector measure in $\mathit{ca}(\F,m,Y)$ is saturated;
4. Every vector measure in $\mathit{ca}(\F,m,Y)$ is a Lyapunov measure.
\(i) $\Rightarrow$ (iii): Trivial.
\(iii) $\Rightarrow$ (iv): See Lemma \[lem3\] and Theorem \[thm5\].
\(iv) $\Rightarrow$ (i): If $m$ is not saturated, then there exists an $m$-nonnull set $E\in \F$ such that $\tau(\widehat{\F_E})$ is a countable cardinal, where $(\widehat{\F_E},\hat{m})$ is the restriction of the vector measure algebra $(\widehat{\F},\hat{m})$ to $E$. This means that $\F_E$ is $m$-essentially countably generated, and hence, there exists a countably generated sub $\sigma$-algebra $\G$ of $\F_E$ such that $\widehat{\G}=\widehat{\F_E}$. Since the restriction of $m$ to $\F_E$ is an $X$-valued Lyapunov measure on $\F_E$, it is nonatomic on $\F_E$ by Theorems \[thm3\] and \[thm4\]. Take any $p$-control measure $\mu_p$ of $m$. It follows from $\widehat{\G}=\widehat{\F_E}$ that $m$ is nonatomic on $\G$, and hence, $\mu_p$ is also nonatomic on $\G$ because of the absolutely continuity of $\mu_p$ with respect to $m$. Therefore, $(E,\G,\mu_p)$ is a countably generated, nonatomic, finite measure space, which is not obviously saturated. Let $\tilde{Y}$ be a vector subspace of $Y$ that is isomorphic to an infinite-dimensional Banach space. By [@ks13 Lemma 4.1], there exists a non-Lyapunov measure $n_F\in \mathit{ca}(\G_F,\mu_p, \tilde{Y})$ for some $F\subset E$ with $\mu_p(F)>0$. Extend $n_F$ from $\G_F$ to $\F$ by $n(A)=n_F(A\cap F)$ for $A\in \F$. Indeed, $n$ is closed since it absolutely continuous with respect to $\mu_p$. Hence, $n\in \mathit{ca}(\F,m,Y)$ is not a Lyapunov measure.
\(i) $\Leftrightarrow$ (ii): See Proposition \[pro3\].
By taking $m$ as a scalar measure, we can recover obviously the necessity result explored in [@ks13; @ks15].
Further Characterization of Saturation
======================================
The Bang-Bang Principle in LcHs
-------------------------------
Let $I$ be an arbitrary subset of the set $\mathbb{N}$ of natural numbers and $X_i$ be a lcHs for each $i\in I$. Denote by $\prod_{i\in I}X_i$ the product space of $X_i$ consisting of all mappings $I\ni i\mapsto x_i\in X_i$, endowed with the product topology (given by the pointwise convergence in $X_i$ for each $i\in I$). Let $m:\F\to \prod_{i\in I}X_i$ be a vector measure with a component measure $m_i:\F\to X_i$ for $i\in I$. Given a vector measure space $(\Omega,\F,m)$, two measurable functions $f:\Omega\to \R^I$ and $g:\Omega\to \R^I$ are regarded as equivalent if $f(\omega)=g(\omega)$ except on the $m$-null set.
The infinite-dimensional control systems under scrutiny here are described by the lcHs $\prod_{i\in I}X_i$, the vector measure space $(\Omega,\F,m)$, and a multifunction $\Gamma:\Omega\twoheadrightarrow \R^I$. Denote by $\mathcal{S}^1_\Gamma$ the set of measurable selectors $f:\Omega\to \R^I$ from $\Gamma$ such that each component function $f_i:\Omega\to \R$ is $m_i$-integrable. Thus, $\mathcal{S}^1_\Gamma$ is regarded as a subset of the product space $\prod_{i\in I}L^1(m_i)$. For each $f_i\in L^1(m_i)$, let $m_i(f_i)=\int f_i dm_i\in X_i$. Then the integral of $\Gamma$ with respect to $m$ is given by $$\int \Gamma dm=\{ (m_i(f_i))_{i\in I}\in X \mid f=(f_i)_{i\in I}\in \mathcal{S}_\Gamma^1 \}.$$
A vector measure $m:\F\to \prod_{i\in I}X_i$ satisfies the *bang-bang principle* for a multifunction $\Gamma:\Omega\twoheadrightarrow \R^I$ if $\int \Gamma dm=\int \mathrm{ex}\,\Gamma dm$, where $\mathrm{ex}\,\Gamma:\Omega\twoheadrightarrow \R^I$ is the multifunction given by the extreme points of $\Gamma(\omega)$ at each $\omega\in \Omega$.
A multifunction $\Gamma:\Omega\twoheadrightarrow \R^I$ is *bounded* if there exists a bounded set $K\subset \R^I$ such that $\Gamma(\omega)\subset K$ for every $\omega\in \Omega$. Denote by $\M_{\mathit{bfc}}(\Omega,\R^I)$ the set of graph measurable, bounded multifunctions with closed convex values from $\Omega$ to $\R^I$.
The next result is a special case of [@cv77 Theorems IV.14 and IV.15].
\[lem5\] For every $\Gamma\in \M_{\mathit{bfc}}(\Omega,\R^I)$, the following conditions hold.
1. $\mathrm{ex}\,\mathcal{S}_{\Gamma}^1=\mathcal{S}_{\mathrm{ex}\,\Gamma}^1$.
2. For every $f\in \mathcal{S}_{\Gamma}^1$ there exists a measurable function $g:\Omega\to \R^I$ such that $f\pm g\in \mathcal{S}_{\Gamma}^1$ and $g(\omega)\ne \bold{0}$ whenever $f(\omega)\not\in \mathrm{ex}\,\Gamma(\omega)$.
The following result is a further extension of [@ks14 Theorem 4.3] to the lcHs setting.
\[thm7\] If $X_i$ is a quasicomplete lcHs for each $i\in I$ and $m:\F\to \prod_{i\in I}X_i$ is a closed vector measure such that $\mathrm{dens}\,\prod_{i\in I}X_i<\kappa(\widehat{\F_E})$ for every $m$-nonnull $E\in \F$, then $m$ satisfies the bang-bang principle for every $\Gamma\in \M_{\mathit{bfc}}(\Omega,\R^I)$.
It follows from the boundedness of $\Gamma$ that $\mathcal{S}_\Gamma^1$ is a bounded closed subset of $\prod_{i\in I}L^\infty(m_i)$, and hence, it is a weakly compact subset of $\prod_{i\in I}L^1(m_i)$ by Lemma \[lem2\]. Define the integration operator $T:\prod_{i\in I}L^1(m_i)\to \prod_{i\in I}X_i$ by $T((f_i)_{i\in I})=(m_i(f_i))_{i\in I}$ for $f_i\in L^1(m_i)$ with $i\in I$. Then $T$ is a linear operator that is continuous in the weak topologies for $\prod_{i\in I}L^1(m_i)$ and $\prod_{i\in I}X_i$ since the integration operator $f_i\mapsto m_i(f_i)$ is continuous in the weak topologies for $L^1(m_i)$ and $X_i$ by Lemma \[lem1\]. Let $\hat{x}\in T(\mathcal{S}_\Gamma^1)=\int \Gamma dm$ be given arbitrarily. Then the set $T^{-1}(\hat{x})\cap \mathcal{S}_\Gamma^1$ is a weakly compact, convex subset of $\prod_{i\in I}L^1(m_i)$, and hence, it has an extreme point $\hat{f}$. It suffices to show that $\hat{f}\in \mathcal{S}_{\mathrm{ex}\,\Gamma}^1$. If $\hat{f}\not\in \mathcal{S}_{\mathrm{ex}\,\Gamma}^1$, then there exists an $m$-nonnull set $E\in \F$ such that $\hat{f}(\omega)\not\in \mathrm{ex}\,\Gamma(\omega)$ for every $\omega\in E$. By the boundedness of $\Gamma$ and Lemma \[lem5\], there exists $g=(g_i)_{i\in I}\in \prod_{i\in I}L^1(m_i)$ such that $\hat{f}\pm g\in \mathcal{S}_{\Gamma}^1$ and $g\ne \bold{0}$ on $E$. Since $m$ is a Lyapunov measure, the range of $m$ is convex in $\prod_{i\in I}X_i$. Take $F\subset E$ with $m(F)=\frac{1}{2}m(E)$ and define $\hat{g}:\Omega\to X$ by $$\hat{g}(\omega)=
\begin{cases}
\hspace{0.35cm} g(\omega) & \text{if $\omega\in E\setminus F$}, \\
-g(\omega) & \text{if $\omega\in F$}, \\
\quad \bold{0} & \text{otherwise}.
\end{cases}$$ Then $\hat{f}=\frac{1}{2}(\hat{f}+\hat{g})+\frac{1}{2}(\hat{f}-\hat{g})$ and $\hat{f}\pm \hat{g}\in \mathcal{S}_{\Gamma}^1$. A simple calculation yields $\int \hat{g}dm=(m_i(\hat{g}_i))_{i\in I}=\bold{0}\in X$, and hence, $\hat{f}\pm \hat{g}\in T^{-1}(\hat{x})\cap \mathcal{S}_\Gamma^1$. This contradicts the fact that $\hat{f}$ is an extreme point of $T^{-1}(\hat{x})\cap \mathcal{S}_\Gamma^1$.
\[lem6\] Let $X_i$ be a lcHs and $m_i:\F\to X_i$ be a vector measure for each $i\in I$ and define the set by $$\mathcal{I}_E=\left\{ (f_i)_{i\in I}\in \prod_{i\in I}L^\infty_E(m_i) \mid 0\le f_i\le 1\ \forall i\in I \right\}, \quad E\in \F.$$ We then have $$\mathrm{ex}\,\mathcal{I}_E=\left\{ (\chi_{A_i})_{i\in I}\in \prod_{i\in I}L^\infty_E(m_i) \mid A_i\in \F_E \ \forall i\in I \right\}.$$
Clearly, any sequence of characteristic functions $(\chi_{A_i})_{i\in I}$ is an extreme point of $\I_E$. On the other hand, if an extreme point $(f_i)_{i\in I}$ in $\I_E$ is not a sequence of characteristic functions, then without loss of generality, we may assume that there exist $\varepsilon>0$ and $m_j$-nonnull set $A\in \F_E$ for some index $j\in I$ such that $\varepsilon\le f_j\le 1-\varepsilon$ on $A$. Then the sequence of functions $[f_j\pm \varepsilon \chi_A,(f_i)_{i\in I\setminus \{ j \}}]$ belongs to $\I_E$ and $$(f_i)_{i\in I}=\frac{1}{2}\left[ f_j+\varepsilon \chi_A,(f_i)_{i\in I\setminus \{ j \}} \right]+\frac{1}{2}\left[ f_j-\varepsilon \chi_A,(f_i)_{i\in I\setminus \{ j \}} \right],$$ a contradiction to the fact that $(f_i)_{i\in I}$ is an extreme point of $\I_E$.
\[thm6\] Let $X_i$ be a quasicomplete separable lcHs with a vector subspace that is isomorphic to an infinite-dimensional Banach space for each $i\in I$ and $m:\F\to \prod_{i\in I}X_i$ is a closed vector measure. Then $m$ is saturated if and only if every vector measure in $\mathit{ca}(\Omega,m,\prod_{i\in I}X_i)$ satisfies the bang-bang principle for every $\Gamma\in \M_\textit{bfc}(\Omega,\R^I)$.
Suppose that every $n\in \mathit{ca}(\Omega,m,\prod_{i\in I}X_i)$ satisfies the bang-bang principle for every $\Gamma\in \M_\textit{bfc}(\Omega,\R^I)$. Take any $E\in \F$ and define the multifunction $\Gamma_E:\Omega\twoheadrightarrow \R^I$ by $$\Gamma_E(t)=\{ (f_i(t))_{i\in I}\in \R^I \mid (f_i)_{i\in I}\in \I_E \}.$$ We then have $\Gamma_E\in \M_\textit{bfc}(\Omega,\R^I)$ and $\mathcal{S}^1_{\Gamma_E}=\I_E$. Since $\I_E$ is weakly compact in the product topology of $\prod_{i\in I}L^1(n_i)$ by Lemma \[lem2\] and the integration operator $T:\prod_{i\in I}L^1(n_i)\to \prod_{i\in I}X_i$ provided in the proof of Theorem \[thm7\] is continuous for the weak topologies in $\prod_{i\in I}L^1(n_i)$, the set $T(\mathcal{S}^1_{\Gamma_E})=\int \Gamma_Edn$ is weakly compact and convex in $X$. By Lemma \[lem5\], we have $\mathcal{S}^1_{\mathrm{ex}\,\Gamma_E}=\mathrm{ex}\,\mathcal{S}^1_{\Gamma_E}=\mathrm{ex}\,\I_E$, and hence, it follows from Lemma \[lem6\] that $$\int \Gamma_E dn=\int \mathrm{ex}\,\Gamma_E dn=\left\{ (n_i(A_i))_{i\in I}\in X\mid A_i\in \F_E \ \forall i\in I \right\}=n(\F_E).$$ The bang-bang principle for $n$ implies that $n(\F_E)$ is weakly compact and convex in $\prod_{i\in I}X_i$ for every $E\in \F$. Hence, every $n\in \mathit{ca}(\Omega,m,\prod_{i\in I}X_i)$ is a Lyapunov measure. The saturation of $m$ follows from Theorem \[thm8\]. The converse implication is a consequence of Theorem \[thm7\].
Lyapunov Control Systems in LcHs
--------------------------------
As an application of the bang-bang principle, we examine Lyapunov control systems developed by @kk75 [@kl73; @kk73; @kk75; @kk78; @kn75; @kn76a; @kn77].
Let $X$ be a lcHs and $X^I$ be the product space $\prod_{i\in I}X_i$ with $X_i=X$ for each $i\in I$. A vector measure $m:\F\to X^I$ is a *control system* in $X^I$ if the sum $\sum_{i\in I}x_i$ is (unconditionally) convergent in $X$ for every $x_i\in m_i(\F)$ with $i\in I$. The attainable set for a control system $m:\F\to X^I$ and a measurable correspondence $\Gamma:\Omega\twoheadrightarrow \R^I$ is defined by $$\A_\Gamma(m)=\left\{ \sum_{i\in I}m_i(f_i)\in X\mid (f_i)_{i\in I}\in \mathcal{S}^1_\Gamma, \text{ $\sum_{i\in I}m_i(f_i)$ exists} \right\}.$$
A vector measure $m:\F\to X^I$ is a *Lyapunov control system* for a multifunction $\Gamma:\Omega\twoheadrightarrow \R^I$ if $\A_\Gamma(m)=\A_{\mathrm{ex}\,\Gamma}(m)$.
We provide a necessary and sufficient condition for control systems to be Lyapunov in terms of the saturation property.
Let $X$ be a quasicomplete separable lcHs and $m:\F\to X^I$ be a closed vector measure. Then $m$ is saturated if and only if every vector measure in $\mathit{ca}(\Omega,m,X^I)$ is a Lyapunov control system for every $\Gamma\in \M_{\textit{bfc}}(\Omega,\R^I)$.
Suppose that every $n\in \mathit{ca}(\Omega,m,X^I)$ is a Lyapunov control system for every $\Gamma\in \M_{\textit{bfc}}(\Omega,\R^I)$. Let $\Gamma^j_E\in \M_\textit{bfc}(\Omega,\R^I)$ be the multifunction defined by $$\Gamma^j_E(\omega)=\{ (f(\omega),(0)_{i\in I\setminus \{ j \}})\in \R^I \mid 0\le f\le 1,\,f\in L^\infty_E(n_j) \},$$ where the nonzero entry $f(\omega)$ of the defining sequence in $\Gamma^j_E(\omega)$ appears in the $j$th component in $\R^I$. In view of Lemmas \[lem5\] and \[lem6\], we have $$\mathcal{S}_{\mathrm{ex}\,\Gamma^j_E}^1=\mathrm{ex}\,\mathcal{S}_{\Gamma^j_E}^1=\{ (\chi_A,(0)_{i\in I\setminus \{ j \}})\mid A\in \F_E \}.$$ Since $n$ is a Lyapunov control system for $\Gamma^j_E$, we obtain $$\A_{\Gamma^j_{E}}(n)=\A_{\mathrm{ex}\,\Gamma^j_E}(n)=\{ n_j(A)\in X\mid A\in \F_E \}=n_j(\F_E).$$ By Lemma \[lem2\], $\mathcal{S}_{\Gamma^j_E}^1$ is weakly compact in $\prod_{i\in I}L^1(n_i)$ and by Lemma \[lem1\], $\A_{\Gamma^j_{E}}(n)$ is weakly compact and convex in $X$. Therefore, $n_j(\F_E)$ is weakly compact and convex in $X$ for every $E\in \F$, and hence, $n$ is a Lyapunov measure. The saturation of $n$ follows from Theorem \[thm8\]. The converse implication is a consequence of Theorem \[thm7\].
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Ricker, W.J., (1990). “Completeness of the $L^1$-space of closed vector measures", *Proc. Edinb. Math. Soc. (2)* **33**, 71–78.
Ricker, W.J., (1992). “Separability of the $L^1$-space of a vector measure", *Glasg. Math. J.* **34**, 1–9.
Rudin, W., (1973). *Functional Analysis*, McGraw-Hill, New York.
Segal, I.E., (1950). “Equivalences of measure spaces", *Amer. J. Math.* **73**, 275–313.
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[^1]: Corresponding author.
[^2]: This version of the paper was completed during Sagara’s visit to the Johns Hopkins University. This research is supported by a Grant-in-Aid for Scientific Research (No.26380246) from the Ministry of Education, Culture, Sports, Science and Technology, Japan.
[^3]: See [@ok93] for the specification of the dual space of $L^1(m)$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In this paper, we propose an approach to detect incipient slip, i.e. predict slip, by using a high-resolution vision-based tactile sensor, GelSlim. The sensor dynamically captures the tactile imprints of the contact object and their changes with a soft gel pad. The method assumes the object is mostly rigid and treats the motion of object’s imprint on sensor surface as a 2D rigid-body motion. We use the deviation of the true motion field from that of a 2D planar rigid transformation as a measure of slip. The output is a dense slip field which we use to detect when small areas of the contact patch start to slip (incipient slip). The method can detect both translational and rotational incipient slip without any prior knowledge of the object at 24 Hz. We test the method on 10 objects 240 times and achieve 86.25% detection accuracy. We further show how the slip feedback can be used to monitor the gripping force to avoid slip with a closed-loop bottle-cap screwing and unscrewing experiment with incipient slip detection feedback. The method was demonstrated to be useful for the robot to apply proper gripping force and stop screwing at the right point before breaking objects. The method can be applied to many manipulation tasks in both structured and unstructured environments.'
author:
- '[^1]'
bibliography:
- 'Ref.bib'
title: '**Maintaining Grasps within Slipping Bound by Monitoring Incipient Slip** '
---
INTRODUCTION
============
Human hands can perform complex manipulations because of the various tactile sensors distributed in the skin. One important function of these tactile sensors is to detect incipient slip. People detect incipient slip by sensing the stretch of the skin [@vallbo1984properties], which enables them to naturally control forceful manipulation tasks such as adjusting grasping force, resisting external perturbances, following the contours of objects or using tools while maintaining firm grasps.
![Top: the robot arm is dropping a tilted screw-driver to the ground such that the screw driver slips rotationally when it collides with the ground. Bottom: the robot is screwing a bottle cap, and translational slip starts when the cap is screwed to the end of the threads. The second and fourth rows show the evolution of processed GelSlim images from no shear or slip, to shear with no slip, to full slip for each scenario. The arrows represent real marker displacements (in yellow) and estimated marker displacements (in red) under assumption of no slippage.[]{data-label="fig:figure1"}](figure/figure1_2.jpg){width="0.96\linewidth"}
One major limitation of current robotic systems is their inability to control gripping force – resulting largely from the lack of tactile sensors capable of providing similar incipient slip detection feedback to that of human skin. Slip detection helps to minimize wear on robotic fingers while also maintaining a firm grasp. Closing the loop with slip detection could aid in tasks such as: a water bottle slipping out of a grasp due to moving liquid, a tool dropping from the hand as a consequence of exerting too much force, or a broken object due to excessive grasping force.
In this paper, we propose a general purpose incipient slip detection method using the GelSlim tactile sensor [@GelSlim_v1] to enable more intelligent slip-aware grasp control. As shown in Figure \[fig:figure1\] top row, the robot can sense rotational slippage when the screw driver collides with the ground and then naturally decreases the grasping force to gently place it on the ground. As shown in Figure \[fig:figure1\] bottom row, when screwing the bottle cap, the robot can also determine the correct stopping point by detecting translational slip in the process.
The GelSlim sensor [@GelSlim_v1], shown in Figure \[fig:figure1\], is a vision-based tactile sensor. It uses a Raspberry Pi Spy Camera to capture a deformation of a gel-based skin on the sensor surface to measure the contact position and local texture of the object. The gel is covered by a piece of fabric, which functions as a protective layer and enhances the contact signal with its own texture. The tactile imprints of the screw driver handle and bottle cap captured by GelSlim sensor are shown in the second and fourth row (respectively) of Figure \[fig:figure1\]. On the surface of the gel, there are evenly distributed black markers. The markers move under contact force and the displacement field of the markers represents how much the gel is stretched in and around the contact region.
We define incipient slip as the condition under which parts of the contact region (generally the peripheral) start to slip. The marker displacement in slip region must be smaller than the displacement of the object. The real marker displacement field in the contact region is shown with yellow arrows in Figure \[fig:figure1\]. Assuming no slip happens in the central region of the contact patch, we model the movement of the object-gel contact region as a 2D rigid body to calculate the estimated marker displacement field (shown with red arrows in Figure \[fig:figure1\]). If the real and estimated displacement fields are similar, no slip happens (see second column of the GelSlim images in Figure \[fig:figure1\]). However, if they have quite amount of difference, incipient slip happens (see third column of the GelSlim images). Based on the principle described above, our incipient slip detector can detect both translational and rotational slippage without any prior knowledge of the object, such as coefficient of friction, mass or shape. With a light computation cost, the detector can run at around 24 Hz.
We test the detection accuracy of the method with 10 daily objects under a large number of external forces, and achieve 86.25% success rate. We also successfully implement closed-loop bottle cap screwing and unscrewing experiments with incipient slip detection feedback, which informs the robot when to increase the gripping force and when to stop. The method can also be applied to other vision-based tactile sensors with dense markers on the surface [@GelSight_Dong; @GelForce2005] and it can find applications in manipulation tasks like: in-hand manipulation [@yousef2011tactile], closed-loop grasp control [@Zeng2018], contour-following [@hellman2018functional].
RELATED WORK
============
Various tactile sensors have been developed in the past few decades specially to detect incipient slip or slip [@slipreview2013]. There are mainly two categories of working principles: vibration based and relative motion based slip detection.
**Vibration based method** Since relative motion between two rough surfaces results in high-frequency signals, many tactile sensors have been built to detect slip by sensing vibration [@veiga2015stabilizing; @li2014learning]. Sch[ö]{}pfer *et al*. [@schopfer2010_piezo] proposed a incipient slip detection method by using piezo-resistive tactile sensor, which could measure the normal force at 83 Hz within 80 mm $\times$ 80 mm sensing area. The time-sequence force data was reprocessed by using a Fourier transform first and then fed into a neural network to detect slip. Sch[ü]{}rmann *et al*. [@schurmann2012highspeed] further improved the frame rate up to 1.9k Hz and successfully implemented grasp force control with slip detection feedback. The method is able to detect slip with unknown objects, however, the configuration of the tactile sensor is hard to directly to grasp objects with. Romano *et al*. [@romano2011human] used the vibration signals from an accelerometer (running at 3 kHz) and pressure cells (running at 24.4 Hz) embedded in the palm of the gripper to detect slip. They also performed a similar grasp force control experiment with slip feedback control at around 1K Hz with a PR2 robot.
**Relative motion based method** Relative motion based slip detection is mostly utilized in vision-based tactile sensors [@ueda2005development; @ueda2005grip]. Maldonado *et al*. [@maldonado2012improving] designed a compact fingertip sensor, where a small camera with 30 $\times$ 30 pixels spatial resolution and 1 $mm^2$ sensing region is embedded inside, to detect the relative motion between the object and fingertip. Given the area of the sensor’s contact region, it is hard for this sensor to detect slip when the object has no texture. The GelSight sensor [@GelSightUSB; @GelSight_Dong], which is also a vision-based sensor but uses an camera to capture the deformation and stretch of an elastomer gel, was also used to detect slip. Yuan *et al*. [@GelSightShear] proposed a method to detect incipient slip by monitoring the entropy of the marker displacement according to the effect that the stretch of the gel surface is inhomogeneous in slip state. However, this method only works well when the contact surface has little texture. Dong *et al*. [@GelSight_Dong] used the GelSight sensor to measure the relative motion between the markers on the gel surface, object texture, and maker displacement ratio between the center contact region and the peripheral area, which achieved $71\%$ test accuracy with $37$ different daily objects. These two mechanisms are complementary for objects with & without textures, however, the relatively small sensing area of the GelSight sensor, and the relatively large computational cost makes it difficult to use in closed-loop control scenarios. Li *et al*. [@li2018slip] further improved the detection accuracy to 88% by adding an external camera beside the GelSight sensor and training a convolutional neural network. However, this technique further decreases the manipulation speed. Compared to these methods above, our approach speeds up the detecting frame rate by using a simpler algorithm, and achieves similar/better detection accuracy meanwhile. In addition, compared to other vision-based sensors, the GelSlim sensor has much larger sensing region ($3cm \times 4cm$), which also provides more buffer time to sense and react to the slip event.
Method
======
**Physical principle** When a small portion of the contact area has lost contact, we call this condition “incipient slip". Under a shear load, the gel surface will be stretched along the force direction. This stretch is easily calculated by tracking the markers on the gel surface. As the force increases, gel in the contact area will move with the object until the frictional force is not sufficient to hold the contact. Before incipient slip happens, the gel in the contact region has the same displacement as the object. When slip starts, the peripheral edge of the contact region will slip first [@andre2011effect], which means the displacement field of that region will have smaller magnitude than that at the center. This is the trigger signal for our method to detect incipient slip.
**Algebra** The method starts from the core assumption that under no slip, the contact patch will move as a planar rigid body. The motion of every point in the contact patch can then be considered as pure rotation around a point, which is called the instantaneous center of rotation (ICR). If we know the position ($x$, $y$), velocity ($\upsilon_{x}$, $\upsilon_{y}$) and angular velocity $\omega$ of a reference point in the rigid body, we can calculate the velocity ($\upsilon_{x}$, $\upsilon_{y}$) and angular velocity $\omega'$ of any other arbitrary point with known position ($x'$, $y'$) inside the rigid body according to equation \[eq:1\]. If the point ($x$, $y$) is the ICR, $\upsilon_{x}$ and $\upsilon_{y}$ are zero.
$$\label{eq:1}
\begin{gathered}
\upsilon_{x'} = \upsilon_{x} + \omega*(y-y')\\
\upsilon_{y'} = \upsilon_{y} + \omega*(x'-x)\\
\omega' = \omega
\end{gathered}$$
As explained above, generally the peripheral region with lower pressure distribution loses contact first, but the inner part of the contact region still holds when incipient slip happens. So we can use the markers in the inner contact region as the reference points of the rigid body motion to calculate the estimated displacement field of the markers in the contact region. When the real displacements of several points in the contact region are sufficiently different from the estimated displacements, we trigger incipient slip.
**Algorithm** The algorithm includes 4 steps (illustrated in Figure \[fig:steps\]). We use a translational incipient slip case as an example (Figure \[fig:steps\] left column) to explain in detail.
- **Step1: Detect contact region** since the contact region is highlighted by the fabric textures (Figure \[fig:steps\](a1)), we use the Canny filter to detect the edges and several morphological filters to group the edges together resulting in the contact region (highlighted with green color in Figure \[fig:steps\](b1)).
![The procedure to detect incipient translational slip (left column) and rotational slip (right column).[]{data-label="fig:steps"}](figure/method2.jpg){width="0.95\linewidth"}
- **Step2: Calculate real displacement field** Then the black markers are selected with a threshold and we deploy the SimpleBlobDetector function in OpenCV to detect the position of each marker in the contact region. Through matching and comparing the relative displacement of the marker positions in the reference frame and current frame, we compute the displacement of each selected marker according to the reference frame. The displacement vectors are shown in Figure \[fig:steps\](b1) with yellow arrows. The reference frame we use is captured in the beginning. However, after incipient slip signal is detected, we update the reference frame with the current frame (i.e.,update contact region and reference marker positions) to make the method work dynamically.
- **Step3: Estimate rigid motion of reference point** The next step is to find the inner region and localize the markers inside to find the reference point of the rigid body motion. A simple erosion filter is used to remove the edge region of the contact area and the markers in the remaining area (highlighted with white in Figure \[fig:steps\](c)) are used to calculate the position $(x, y)$, velocity ($\upsilon$) and angular velocity ($\omega$) of the reference point. Since the positions and displacement of these markers are already computed from the last step, we feed them into the estimateRigidTransform function in OpenCV to compute the rigid transform matrix. The angular velocity of the reference point can be extracted from the transform matrix, and the position and velocity are computed by averaging the positions and displacements of the markers inside. For small motion, we use marker displacements as velocities. The reference point and its velocity (displacement) are labeled with a red arrow in Figure \[fig:steps\](c1). We labeled the ICR with a yellow dot in Figure \[fig:steps\](c1). The black dashed lines are perpendicular to their corresponding displacements vectors.
- **Step4: Calculate estimated displacement field and slip field** we calculate the estimated marker displacement filed in the contact region (shown with red arrows in Figure \[fig:steps\](d1)) according to Equation \[eq:1\]. The slip field are the difference between the real and estimated marker displacement fields (black arrows in Figure \[fig:steps\](e1)). If multiple markers (labeled with purple dots) in the slip field are larger than a threshold, it indicates that incipient slip has happened. We notice that all of them lie in the peripheral region, which also verifies the phenomenon described in [@andre2011effect].
The right column of Figure \[fig:steps\] shows the same procedure on detecting rotational slip. Since the object rotates in the clockwise direction and the ICR is roughly in the center of the contact patch, the upper right and down left region lose contact first, which can be clearly seen from the slip field. To make the algorithm clearer, we include Figure \[fig:pipeline\] to show the algorithm pipeline.
This method can detect any type of directional slip (translational, rotational or both) as long as the contact patch spans a cluster of markers. The rotation center for rotational slip can be easily computed if needed. It does not require any prior knowledge of the object and contact, such as coefficient of friction, mass or shape. The method can run at around 24 Hz and the largest computational cost is due to on localizing the marker positions, which means smaller contact patch results in a higher speed. In addition, we can expect large improvement from parallel or GPU computation.
![Incipient slip detection algorithm pipeline[]{data-label="fig:pipeline"}](figure/pipeline2.png){width="0.8\linewidth"}
Experiment
==========
Experiment Setup
----------------
We conduct experiments in a setup based on the grasping system developed in [@Zeng2018; @hogan2018tactile] with a 6-DOF ABB-1600 robot arm equipped with a WSG-50 parallel gripper. Two GelSlim sensors are mounted in the gripper as two fingers (Figure \[fig:figure1\]). The GelSlim sensor captures the imprints of the contact surface with 640\*480 spatial resolution at 30 Hz. The sensing surface of the sensor measures $3\times4$ $cm^2$. In this experiment, for computational simplicity, only one of the two sensors is used to detect incipient slip.
Incipient Slip Detection Test
-----------------------------
We test the accuracy of detecting incipient slip on 10 daily objects (Table \[tab:objects\]) with different shapes, masses, softness and friction coefficients. The yellow duck, brain foam and scrub sponges are soft and deformable. The water bottle (full of water) is much heavier than other objects. The crayons box, scrub sponges and glue bottle have flat surfaces. The surfaces of scrub sponges, brain foam and glue bottle are smooth to the sensor surface.
Each object is initially held by the gripper and then pushed, pulled or rotated manually. For translational slip, we pushed the objects along 4 evenly spaced directions in the plane parallel to the finger phalanges. For rotational slip, we rotate the objects in clockwise and anticlockwise directions. For the non-slip case, we try to push or rotate the object without making it slip. Since it is difficult to accurately control the whether or not the object slips in this open-loop experiment, we relabel the data afterwards according to human observation. To test the algorithm in different working scenarios, we repeat the experiment with three different forces (from 5 to 30N for most objects). Therefore, each object is tested 24 times (18 slip cases and 6 non-slip cases) for a total of 240 experiments. The algorithm is running in real time, and the results are saved automatically.
Tape Crayons Scissor Total
--------------------------- ------------------------------------------------ ------------------------------------------------ ------------------------------------------------ ------------------------------------------------ ------------------------------------------------ ------------------------------------------------ ------------------------------------------------ ----------------------------------------------------- ------------------------------------------------ ------------------------------------------------- --------
{width="6.00000%"} {width="5.00000%"} {width="3.20000%"} {width="3.30000%"} {width="3.30000%"} {width="5.50000%"} {width="5.00000%"} {width="3.00000%"} {width="5.00000%"} {width="5.00000%"}
Success Rate 100.0% 49.97 % 100.0% 100.0% 95.8% 87.5% 100.0% 100% 33.33% 95.83%
Failure (False Positives) 0.0% 4.17% 0.0% 0.0% 4.17% 0.0% 0.0% 0.0% 0.0% 0.0% 0.83%
Failure (False Negatives) 0.0% 0.0% 0.0% 0.0% 12.5% 0.0% 0.0% 4.17% 12.92%
Bottle Cap Screwing and unscrewing with Incipient-Slip Feedback
---------------------------------------------------------------
![The experimental setup and steps for screwing process[]{data-label="fig:screw_experiment"}](figure/screw_cap.png){width="0.9\linewidth"}
There are many manipulation tasks in daily life that require exerting force or torque while maintaining stable grasps. We propose using incipient-slip detection to implement them. To demonstrate the function of our incipient slip detector, we performed bottle cap screwing and unscrewing experiments, where the detector informs when the robot increases the gripping force and stops screwing. The experimental setup and screwing process are shown in Figure \[fig:screw\_experiment\]. The water bottle is fixed on a platform, and the cap is initially placed on top of the bottle loosely. We assume the position of the bottle cap is known. The gripper is aligned with the central axis of the bottle, and the sensing region of the GelSlim sensor is horizontally aligned with the bottle cap (Figure \[fig:screw\_experiment\](a)). The robot then grips the cap with 10N as the initial force and starts with a low speed screwing motion. Because of the compliance of the gel, we do not need to know the exact pitch of the thread. While the robot screws the cap, our incipient slip detector gives warnings (Figure \[fig:screw\_experiment\](b)). When incipient slip is detected, we stop the robot at that point momentarily, increase the gripping force by 10N and start the screwing again (Figure \[fig:screw\_experiment\](c)). The process is repeated until the force reaches 60N, when we stop the screwing process and open the gripper (Figure \[fig:screw\_experiment\](d)). The data of the GelSlim sensor is recorded on-line for posterior analysis (Section \[Results\]).
For the unscrewing experiment, the process is reversed (Figure \[fig:unscrew\_experiment\]). The cap starts tightly screwed. We still select 10N as the initial force and 60N as the maximum force. The only difference between these two experiments replies on the time when slip happened. For unscrewing process, slip happens at the beginning and it is not necessary for the gripper to increase to the max force limit. This is in contrast to, the screwing experiment where using maximum gripping force allows the robot to firmly torque the bottle cap on. We repeat the two experiments 3 times to test the stability of the algorithm.
![The experiment setup and steps for unscrewing process[]{data-label="fig:unscrew_experiment"}](figure/unscrew_cap.png){width="0.9\linewidth"}
Experiment Results and Analysis {#Results}
===============================
Incipient Slip Detection Accuracy
---------------------------------
Table \[tab:objects\] summarizes the detection accuracy for each object. Based on the experiment type and the output of the detector, we group the results into three different classes: successful, false positives and false negatives. False positives (0.83%) means the detector gives false alarm, i.e., the detector is triggered without slip event. False negative (12.90%) means the detector fails to detect incipient slip when it happens. In general, the detector provides correct detection results for most of the objects with nearly 100% accuracy except for the crayon box and scrub sponges. On average, the detection accuracy of our method is 86.25%.
Most of the false negatives errors (12.92%) result from the experiments with the crayon box and scrub sponges. A more detailed analysis indicates that the contact signals for these two objects are too weak for the algorithm to work (see Figure \[fig:failure\]). The two objects with flat and smooth surface make little pressure and shear force on the sensor surface under small gripping force. The shallow texture and tiny displacement field shown in Figure \[fig:failure\] corroborate this point. Therefore, for objects with smooth and flat surfaces, we need to apply larger grasping force to generate a good contact signal for our method to work. We redid the same experiment for these two objects with 50N gripper force. The success rate increased to 100.0% for the crayon box and 66.67% for the scrub sponges. It is interesting to note since the quality and size of the contact patch can be measured in real time and used as an assessment if the algorithm is reliable.
{width="0.8\linewidth"}
. \[fig:failure\]
Bottle Cap Screwing
-------------------
{width="0.9\linewidth"}
In the bottle cap screwing experiment, the incipient slip detector is helpful to determine when increased gripping force is needed and when the cap is already tightly screwed. The robot was able to tightly screw the bottle cap for the three experiment trials. We recorded the gripping force change over time, the GelSlim images and external camera images at each detected slip point, which are visualized in Figure \[fig:exp1\]. For the GelSlim images, we still use yellow arrows to represent real marker displacements, red arrows to represent estimated marker displacements and the slipped markers are labeled in purple.
In the first half of the process, the bottle cap was loose and the gel surface shown in Figure \[fig:exp1\](1) was barely stretched. The red and yellow arrows nearly overlapped with each other. The first slip signal was triggered in the middle of the process, the small error in z axis made the arrows in Figure \[fig:exp1\](2) point down. After the gripping force increased to 30N, the contact patch in time (3) also expanded. As the the cap became screwed on more, the accumulated error in z axis triggered another slip, which is shown in Figure \[fig:exp1\](3). Since the cap was already in the thread, the slip actually helped to correct the error. The last three detected slippage happened closely in the last few seconds, which indicates that the cap was almost at the end of the thread. The large maker displacements in the three GelSlim images indicates that the gripper was exerting larger forces on the cap. Since the reference frame is updated right after incipient slip is detected, the displacements in the next frame will be very small. However, after we cleared the displacements in time (4) and (5), the marker displacements in time (5) and (6) were still large, which implies that slip happened continuously in this time period and was continuously detected. The unchanged bottle cap orientation in external images (5) and (6) also validate this point. It is interesting to note that the directionality of the slip field could potentially be used to distinguish between vertical slip (cap goes down, OK) and horizontal slip (lose the grasp, NOT OK).
Bottle Cap Unscrewing
---------------------
![The force change over time (top), the images taken with external camera (middle) and the GelSlim images (bottom) at the time points labelled in the plot for the bottle cap unscrewing process.[]{data-label="fig:unscrew"}](figure/unscrew.png){width="0.9\linewidth"}
For the bottle cap unscrewing process, incipient slip happened right after we started the experiment, which can be seen in Figure \[fig:unscrew\](1). The arrows point to the opposite direction compared to those in screwing process, since the two tasks have opposite force directions. The second slip was triggered closely to the first one, which implies the cap did not move with the sensor yet. The cap in the external images of Figure \[fig:unscrew\](1)&(2) also maintains the same orientation. Right after the gripping force increased to 30N, the bottle cap started to follow the gripper, which can be seen by comparing the orientations of the bottle cap in Figure \[fig:unscrew\](2)&(3). Afterwards, there was no more slip event happened and the marker displacements kept with small values (Figure \[fig:unscrew\](3)). It is worth nothing that all of the slip happens for markers located in peripheral region of the contact area. The bottle cap was successfully opened in all of the three trials.
CONCLUSIONS
===========
We have proposed an incipient slip algorithm that exploits the displacement field tracking capability of the GelSlim tactile sensor [@GelSlim_v1]. The key idea is to compare the displacement field in the contact patch with that of a perfectly rigid-body displacement. The algorithm achieves 86.25% detection accuracy in the 240 slip experimental trials with 10 daily objects, which is comparable to [@li2018slip] but at least 8 times faster (24 Hz for our method). This method works best for the objects that make large contact patches and apply large tangential forces to the sensor. For objects with flat and smooth surfaces, a large gripping force is needed to make the method reliable. The method can detect any directional slip without any prior knowledge of the grasped object. In addition, we demonstrate the method provides useful real-time incipient slip feedback in the bottle cap screwing and unscrewing experiment. It can be potentially applied to many manipulation tasks that requires slip detection feedback, such as grasp force control, in hand manipulation, and tool use.
[^1]: This work was supported by the Amazon Research Awards, and the Toyota Research Institute (TRI). This article solely reflects the opinions and conclusions of its authors and not Amazon or Toyota.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'A new engineering technique using continuous quantum measurement in conjunction with feed-forward is proposed to improve indistinguishability of a single-photon source. The technique involves continuous monitoring of the state of the emitter, processing the noisy output signal with a simple linear estimation algorithm, and feed forward to control a variable delay at the output. In the weak coupling regime, the information gained by monitoring the state of the emitter is used to reduce the time uncertainty inherent in photon emission from the source, which improves the indistinguishability of the emitted photons.'
author:
- Shesha Raghunathan
- Todd Brun
title: 'Continuous monitoring can improve indistinguishability of a single-photon source'
---
Developing a scalable model to implement quantum computation on physical systems has generated consuming interest in the quantum computing community ever since Shor [@Shor94] and Grover [@Grover97] showed that quantum computing can outperform any classical device for certain algorithms of practical interest. Knill *et al.* [@Knill01] developed a scalable model based on linear optics and single qubit measurement. This model, however, assumes the availability of good quality single photon states. This, consequently, has lead to considerable interest in the design and implementation of good quality single-photon source. The applications of a single-photon source, however, are not restricted to quantum computing alone, and would be highly useful in the areas of quantum imaging, metrology and quantum cryptography, to name a few [@Oxborrow05].
Amongst various physical implementations of a single-photon source, semiconductor quantum dot (QD or dot for short) based sources are of great interest, as they scale well upon integration, and are amenable to commercial fabrication techniques [@Oxborrow05; @QDs]. Since photon emission is isotropic in free space, spatial confinement of photons is vital to improve the collection efficiency of a single-photon source. Design and fabrication of microcavities has received much attention over the last few years, and a variety of cavities have been fabricated: micropillar, microdisk, and photonic crystal to name a few. These microcavities have different sizes and shapes, and varying quality factors [@Vahala03].
A quantum dot in a microcavity can be pumped either optically or electrically. Electrical pumping is interesting for a variety of reasons. In particular, since the electrical pumping process cannot directly insert photons into the cavity, one could pump directly into an energy level resonant with the cavity mode. Electrical pumping of a quantum dot involves the application of voltage across the source and drain; if the dot has no electron in the valence band, an electron from the source tunnels through to the dot [@electrical-pumping]. Once an electron tunnels through to the dot, no new electron can tunnel through due to Coulomb repulsion of like charges, a phenomenon widely referred to as the *Coulomb blockade* effect [@QDs; @electrical-pumping]. Thus, a new electron can tunnel through only when the electron currently in the dot ‘tunnels out.’ The electron in the quantum dot tunnels out when there is a difference in voltage between the dot and the drain; this relative voltage can be controlled by the application of gate voltage on the dot. By observing the voltage across the source and drain of a quantum dot, we can also gain information about the state of the dot. This observation can be seen as a weak quantum measurement that provides some information about the system [@Levi07].
![(Color online) Continuous monitoring with feed forward involves 3 components: ($i$) emitter $+$ cavity, ($ii$) AMMSE-based transition time estimation, and, ($iii$) variable delay element. []{data-label="fig:overview"}](QD+Cavity.pdf){width="2.75in" height="1.75in"}
In this Letter, we combine continuous quantum measurements and feed forward to reduce the uncertainty in the time at which a photon leaks out of a source (Fig. \[fig:overview\]). The idea is to continuously monitor the state of the emitter, and to use this information to make a time correction at the end, so that the time uncertainty of the single-photon source is reduced. To reduce the effect of noise on our estimate of when the emitter transitioned to its ground state, we process the signal with the simple error estimation technique of Affine Minimum Mean Squared Error estimation (AMMSE). This processed signal is fed forward to control a variable delay at the output of the single-photon source. This approach works well in the weak coupling regime, as the transition time of the emitter from an excited state to its ground state controls, fairly closely, the time of the photon leaking out of the cavity.
Figure \[fig:3LA\] shows the energy level diagram of the model under consideration [@SPS-theoretical]. The system consists of an emitter (QD) with 3 energy levels whose first excited state is resonantly coupled to the cavity mode. The energy levels are represented by $\ket{G}$, $\ket{X_1}$, $\ket{X_2}$, with $G$ standing for the ground state of the emitter while $X_1$ and $X_2$ represent the first and second excited states of the emitter, respectively. The interaction between the emitter and the cavity in the emitter $+$ cavity system is determined by the Jaynes-Cummings Hamiltonian: $ \H_I = \, i \hbar \, g \left( \adag \Sm_1 \, - \, \a \Sp_1 \right)$, where $g$ is the interaction strength, and $\Sm_1 = \ket{G}\bra{X_1}$; we operate in the interaction picture, and henceforth set the total Hamiltonian $\H$ to $\H_I$. It is also assumed that the cavity can contain at most one photon. Further, in this work, we do not model the pumping process explicitly, and we assume that the system is initially in $\ket{X_2,0}$. This assumption is reasonable if the pumping is strong.
![<span style="font-variant:small-caps;">Emitter $+$ Cavity System:</span> Energy level diagram[]{data-label="fig:3LA"}](3la-energy.pdf){width="1.75in" height="1.25in"}
Incoherent or decoherence processes considered in this model include spontaneous decay, cavity leakage and dephasing. $\Gamma_1$ and $\Gamma_2$ are spontaneous emission rates for $\ket{X_1} \rightarrow \ket{G}$ and $\ket{X_2} \rightarrow \ket{X_1}$ transitions respectively; $\kappa$ is the rate of (photon) leakage from the cavity, and $\gamma$ is the dephasing rate between $\ket{X_1,0}$ and $\ket{G,1}$. The time evolution of the system is described by a stochastic master equation (SME) [@Trajectories]: $$\begin{split}
d\rho
&= - \, \frac{i}{\hbar} \, [ \H, \rho ] \, dt \, + \, \left( \Gamma_1\, \mathcal{H}[ \Sm_{1}] \, + \, \Gamma_2 \,\mathcal{H}[ \Sm_{2}] \,\right) \rho \, dt \\
& \, + \, \left( \kappa \, \mathcal{H}[ \a] \, + \, \gamma \, \mathcal{H}[ \Op] \, \right) \rho \, dt \, + \, \sqrt{\eta \gamma} \, \mathcal{D}[\Op] \,\rho \, dW_t,
\end{split}
\label{eqn:sme}$$ where $\mathcal{H}$ and $\mathcal{D}$ are super-operators and are defined to be $\mathcal{H}[\A] \rho = \A \rho \Adag - (\Adag \A \rho + \rho \Adag \A)/2$, and $\mathcal{D}[\Op]\rho = (\Op - \expect{\Op}) \rho + \rho (\Opdag - \expect{\Opdag})$ where $\expect{\Op}=Tr\{\Op\rho\}$. Also, $\Sm_2 = \ket{X_1}\bra{X_2}$, $\a$ is the annihilation operator acting on the cavity part of the emitter $+$ cavity system, and $\Op = \id - \ket{G}\bra{G}$ is the dephasing or “observer” operator acting on the emitter part of the system. Dephasing is caused by the interaction of the system (emitter) with other degrees of freedom, for instance phonon modes in the quantum dot [@QDs]. If we can tap into these modes, we can obtain information regarding the system of interest; $\eta$ is the efficiency with which we (the “observer”) can read the information in these degrees of freedom. We assume that we operate in the bad cavity limit, so that the parameters satisfy the following condition: $$\Gamma_1 << g, \gamma < \kappa.$$ The output signal obtained from our continuous measurement is called the measurement record, and is given (in rescaled units) by $$J(t) = \, \probOp(t) \, + \, \frac{\xi(t)}{\sqrt{\eta \gamma}},$$ where $\xi(t) = dW_t/dt$ is the Gaussian white noise with zero mean, $i.e.$ $\mathbb{E}[dW_t] = 0$ where $\mathbb{E}$ is the expectation or mean of a random variable, and $dW_t^2 = dt$ [@Trajectories]. Integrating the measurement record over time, we obtain an estimate of when the emitter transitioned to its ground state: $$\nu = \int_0^T J(t) \, dt,$$ where $T$ is a sufficiently long period of time. The best estimate of the transition time of the emitter would be $$\tau = \int_0^T \probOp(t) \, dt.$$ We, however, have access to $J(t)$ and not $\tau$.
To allow for this estimation error, we pass the output signal $\nu$ through an AMMSE procedure [@Papoulis91], whose output $\hat{\tau} = G \nu + m$ represents a linear estimate of the transition time of the emitter. The idea of AMMSE is to minimize the mean squared error $\mathbb{E}[|\tau - \hat{\tau}|^2]$. We derive equations for $G$ and $m$ in terms of the mean and variance of $\tau$ such that $\mathbb{E}\{|\tau - \hat{\tau}|^2\}$ is minimized: $$\begin{aligned}
G &= &\frac{\sigma_{\tau}^2}{\sigma_{\tau}^2 \, + \, \beta^2 \, T} \ , \\
m &=& m_{\tau} \left( \frac{\beta^2 \, T}{ \sigma_{\tau}^2 \, + \, \beta^2 \, T} \right),
\label{eqn:G&m}\end{aligned}$$ where $m_{\tau}$ and $\sigma_{\tau}^2$ are mean and variance of signal $\tau$ repectively, $\beta = (\eta \gamma)^{-1/2}$, and $T$ is the total integration time. In the solution above, we have, as a first approximation, assumed that there is no correlation between the output signal ($\nu$) and noise ($\nu - \tau$) affecting the system [@Raghunathan07]. In the bad cavity limit, the the emitter displays a quasi-exponential behaviour [@Haroche06]. Since the variance of an exponential distribution is square of the mean, we have $\sigma_{\tau}^2 \approx m_{\tau}^2$. The mean, $m_{\tau}$, is calculated by numerically integrating the deterministic (Lindblad) master equation, $i.e.$ Eq. (\[eqn:sme\]) with $\eta = 0.0$.
There are two, sometimes conflicting, requirements for an emitter to be a good single-photon source: (a) that a single photon leaks out of the source (cavity) with high probability, and, (b) that the photon that leaks out is highly indistinguishable from other photons produced by the same source. Indistinguishability is most strongly affected by the uncertainty in time of when a photon leaks out of the source. This uncertainty is unavoidable, as the the process of generating a photon involves incoherent processes; in a system represented by Fig. \[fig:3LA\], for a photon to leak out of the cavity, the system has to undergo the transitions $\ket{X_2,0} \stackrel{\Gamma_2}{\longrightarrow} \ket{X_1,0} \stackrel{g}{\longleftrightarrow} \ket{G,1} \stackrel{\kappa}{\longrightarrow} \ket{G,0}$ where $\Gamma_2$ is an incoherent process. In the weak coupling regime, $\kappa$ dominates $g$, and hence the reabsorption rate of the energy in the cavity by the emitter (QD) is fairly small. Thus, the $\ket{X_1} \longrightarrow \ket{G}$ transition of the emitter largely determines when a photon leaks out of the cavity. This implies that knowing the state of the emitter, should reduce the time uncertainty.
To calculate the indistinguishability, consider a canonical Hang-Ou-Mandel-like [@HOM87] experimental setup as shown in Fig. \[fig:experimental\]. The setup has two independent, but identical, single-photon sources, $SPS_1$ and $SPS_2$; the sources have the same parameter values $(g, \kappa, \gamma, \Gamma_2, \Gamma_1)$, and the noise acting on them is independent. The sources $SPS_1$ and $SPS_2$ have 2 components: (a) an emitter $+$ cavity system, and, (b) a variable delay. The emitter $+$ cavity system is the physical photon source, while the variable delay is introduced to correct for the uncertainty in time regarding when a photon is emitted. The delay is determined by the information gained by continuous monitoring of the system.
Sources $SPS_1$ and $SPS_2$ emit photons into modes $1$ and $2$, respectively, which are then passed through a ($50:50$) beam-splitter whose output modes are labeled $3$ and $4$. *Indistinguishability* ($\Lambda$) is defined as the lack of coincidence at the output modes of the beam-splitter, $i.e.$ $\Lambda = 1 - p_c$, where $p_c$ is the coincidence probability. The coincidence probability is the normalized second-order correlation function of the output of the beam-splitter [@SPS-theoretical]: $$p_c = \frac{\int_0^T \, dt \, \int_0^{T-t} \, d\tau \, \, G^{(2)}_{3,4}(t,\tau) }{\int_0^T \, dt \, \int_0^{T-t} \, d\tau \, \expectLR{\adag_3(t) \, \a_3(t)} \, \expectLR{\adag_4(t+\tau) \, \a_4(t+\tau)}}$$ where $G^{(2)}_{3,4}(t,\tau) = \expectLR{\adag_3(t) \, \adag_4(t+\tau) \, \a_4(t+\tau) \, \a_3(t)}$ is the second order correlation function between output modes $3$ and $4$. The output modes of the beam-splitter can be expressed in terms of its input modes using simple linear equations $\a_3(t) \, = ( \a_1(t) \, - \, \a_2(t) )/\sqrt{2}$ and $\a_4(t) \, = \, ( \a_1(t) \, + \, \a_2(t) )/\sqrt{2}$. Assuming that the photons from sources $SPS_1$ and $SPS_2$ do not scatter into modes other than $1$ and $2$, modes $1$ and $2$ are proportional to the cavity mode in their respective sources ($SPS_1$ and $SPS_2$).
![(Color online) <span style="font-variant:small-caps;">Canonical experiment to calculate indistinguishability.</span> $SPS_1$ and $SPS_2$ are identical and independent sources with variable delays included at the output of their respective physical emitter $+$ cavity system. Variable delay is controlled by signal derived from continuous monitoring of the emitter. A ($50:50$) beam-splitter takes the photons from $SPS_1$ and $SPS_2$ as input, with output modes $3$ and $4$; coincidence counter records the output of the detectors present in output modes $3$ and $4$. If the photons in modes $1$ and $2$ are identical, they both will always go into mode $3$ or $4$.[]{data-label="fig:experimental"}](sps-block-diag.pdf){width="3.25in" height="2.5in"}
![(Color online) <span style="font-variant:small-caps;">Results.</span> We plot indistinguishability ($\Lambda$) as a function of $\Gamma_2$ with system parameters: $g=0.1, \kappa=1, \gamma=0.1, \Gamma_1=0.001$. The 3 cases are: ($i$) no dephasing, ($ii$) dephasing but no feed forward, and, ($iii$) dephasing and with feed forward. (Inset) We plot $\%$ improvement in $\Lambda$ of case ($iii$) with respect to case ($ii$).[]{data-label="fig:results"}](numerical.pdf){width="3.25in" height="3.25in"}
To analyze the performance of the continuous monitoring feed forward technique, we compare 3 cases: ($i$) no dephasing, ($ii$) dephasing but no feed forward correction, and, ($iii$) dephasing with feed forward correction. In case ($i$), the evolution of the system is given by a deterministic (Lindblad) master equation (Eq. (\[eqn:sme\]) with $\gamma=0$). In case ($ii$), there is dephasing in the system ($\gamma \ne 0$), but we (the “observer”) either have no access to the information in the external modes coupled to the system, or, we choose to ignore the information; the evolution of this system is deterministic as well (Eq. (\[eqn:sme\]) with $\eta=0$). Case ($iii$) is the continuous monitoring feed forward technique, which reduces uncertainty in time by introducing a variable delay at the output of the source.
We simulated this system numerically using fourth-order Runge-Kutta integrator ($rk4$) along with a pseudo-random number generator ($RNG$) [@Numerics]; the integrator was used to evolve the stochastic master equation, while $RNG$ was used to generate Gaussian white noise ($dW_t$ in Eq. (\[eqn:sme\])). Monte-Carlo simulation of the SME (Eq. (\[eqn:sme\])) was carried out by averaging $7000$ trajectories (case ($iii$) in Fig. \[fig:results\]).
In Fig. \[fig:results\], we plot indistinguishability ($\Lambda$) as a function of $\Gamma_2$ for the 3 cases considered above. The plot is for $g=0.1, \kappa=1, \gamma=0.1, \Gamma_1=0.001$ and for case ($iii$) we have set $\eta=1$. It can be seen that $\Lambda$ for case ($i$) performs the best while case ($ii$) performs the worst. However, we find that continuous monitoring based feed forward technique performs significantly better than case ($ii$). From Fig. \[fig:results\] we see that the performance improvement as compared with dephasing with no feed forward is about $25\%$ or more. The enhancement is larger for smaller $\Gamma_2$; this is due to the fact that for smaller values of $\Gamma_2$, the time uncertainty of when a photon leaks out is higher, and the impact of feed forward based time-correction (due to variable delay) is larger.
In the numerical study presented above, it was assumed that we get information with very high efficiency, $i.e.$ $\eta = 1$, and that the process of continuous monitoring does not in itself increase dephasing. Both these assumptions need not be true in reality. One of the motivations for developing this technique is that the same circuit used in pumping can provide information regarding the state of the quantum dot. In this case, the continuous monitoring should not have an adverse impact on the system. However, if other techniques have to be used for pumping and continuous monitoring, then it is possible that monitoring may increase dephasing in the system. Engineering a quantum dot as a single-photon source requires a more detailed study of such adverse effects, and a wider parameter space needs to be explored [@Raghunathan07].
That said, what this work shows is that there is potentially a lot to be gained by continuous monitoring (in conjuction with feed forward) to minimize the time uncertainty of photons from a single-photon source, and that this approach is simple enough to be implemented with current technology.
In conclusion, we have shown that continuous monitoring can be used to improve the indistinguishability of a single-photon source. The technique follows a process of (a) continuously monitoring the state of the emitter, (b) processing the noisy output, and (c) feeding forward the information gained to a variable delay at the output of the single-photon source. This simple approach led to a significant improvement in indistinguishability (about $25\%$ or more) in numerical simulations; the most encouraging fact is that this approach requires only a simple linear algorithm along with variable delay elements, which should be achievable with current technology. We believe that continuous monitoring with feed forward has the potential to be a practical engineering tool to improve performance, and similar approaches will find more applications in future.
[**Acknowledgements.**]{} We thank A.F.J. Levi for useful discussions regarding implementation of continuous monitoring in QDs. This work was supported in part by NSF Grant No. ECS-0507270 and NSF CAREER Grant No. CCF-0448658.
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We used $rk4$ from *Numerical Recipes* and $RNG$ from the GNU $C++$ library.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: '[We develop the theory of versal deformations of dialgebras and describe a method for constructing a miniversal deformation of a dialgebra.]{}'
address:
- |
Alice Fialowski, Institute of Mathematics, E$\ddot{o}$tv$\ddot{o}$s Lor$\acute{a}$nd University, 1117, Budapest, Hungary.\
E-mail: [[email protected]]{}
- |
Anita Majumdar, Dept. of Mathematics, Indian Institute of Science, Bangalore-560012, India.\
E-mail: [[email protected]]{}
author:
- Alice Fialowski and Anita Majumdar
title: Miniversal deformations of dialgebras
---
[^1] [^2] [^3]
Introduction
============
The notion of Leibniz algebras and dialgebras was discovered by J.-L. Loday while studying periodicity phenomena in algebraic K-theory [@L]. Leibniz algebras are a non-commutative variation of Lie algebras and dialgebras are a variation of associative algebras. Recall that any associative algebra gives rise to a Lie algebra by $[x,y]=xy-yx$. The notion of dialgebras was invented in order to build analogue of the couple $$\text{Lie algebras} \leftrightarrow \mbox{associative algebras},$$ where Lie algebras are replaced by Leibniz algebras. Shortly, dialgebra is to Leibniz algebra, what associative algebra is to Lie algebra. A (co)homology theory associated to dialgebras was developed by J.-L. Loday, called the dialgebra cohomology where planar binary trees play a crucial role in the construction. Dialgebra cohomology with coefficients was studied by A. Frabetti [@F1; @F2] and deformations of dialgebras were developed in [@MM1]. In the present paper, we develop a deformation theory of dialgebras over a commutative unital algebra base, following [@Fi1], and show that dialgebra cohomology is a natural candidate for the cohomology controlling the deformations. We work out a construction of a versal deformation for dialgebras, following [@FF].
The paper is organized as follows. In Section2, we recall some facts on dialgebra and its cohomology. In Section3, we introduce the definitions of deformations of dialgebras over a commutative, unital algebra base. In Section4, we produce an example of an infinitesimal deformation of a dialgebra $D$ over a field $K$ , denoted by $\eta_D$, and also show that this deformation is co-universal in the sense that, given any infinitesimal deformation $\lambda$ of a dialgebra $D$ with a finite dimensional base $A$, there exists a unique homomorphism $\phi: K\oplus HY^2(D,D)' \longrightarrow A$, where $HY^2(D,D)$ denotes the two dimensional cohomology of $D$ with coefficients in itself, such that $\lambda$ is equivalent to the push-out $\phi_*\eta_D.$ Section5 comprises results of Harrison cohomology of a commutative unital algebra $A$ with coefficients in a $A$-module $M$, which have been used in the paper. In Section6 we introduce obstructions to extending a deformation over a base $A$, to a deformation over a base $B$, where there exists an extension $0\rightarrow K\stackrel{i}{\rightarrow} B
\stackrel{p}{\rightarrow} A\rightarrow 0$ of $A$. We show that an obstruction is a cohomology class, vanishing of which is a necessary and sufficient condition for the given deformation of $D$ over base $A$ to be extended to a deformation of $D$ over base $B$. In Section7 we discuss extendible deformations. Let $\lambda$ be the deformation of $D$ over $A$ which is extendible. We state that the two dimensional cohomology group $HY^2(D,D)$ operates transitively on the set of equivalence classes of deformations $\mu$ of $D$ with base $B$ such that $p_*\mu=\lambda.$ We also state that the group of automorphisms of the extension $0\rightarrow
K\stackrel{i}{\rightarrow}B\stackrel{p}{\rightarrow}A\rightarrow
0$ operates on the set of equivalence classes of deformations $\mu$ such that $p_*\mu= \lambda$. These two actions are related by a map called the *differential* $d\lambda:
TA\rightarrow HY^2(D,D),$ where $TA$ denotes the tangent space of $A$. In the last section, we present a construction of a miniversal deformation of a dialgebra $D$.
Dialgebra and its Cohomology
============================
Throughtout this paper, $K$ will denote the ground field of characteristic zero. All tensor products shall be over $K$ unless specified. In this section, we recall the definition of a dialgebra and the construction of the dialgebra cochain complex. Since we are interested in coefficients in the dialgebra itself, we shall restrict our definition to the same.
A dialgebra $D$ over $K$ is a vector space over $K$ along with two $K$-linear maps $\dashv : D\otimes D \longrightarrow D$ called left and $\vdash : D\otimes D \longrightarrow D$ called right satisfying the following axioms : $$\begin{array}{rcl}
x\dashv (y \dashv z)&\stackrel{1}=&(x\dashv y)\dashv z \stackrel{2}= x\dashv
(y \vdash z)\\
(x \vdash y) \dashv z &\stackrel{3}=& x \vdash (y \dashv z) \\
(x \dashv y)\vdash z &\stackrel{4}=&x \vdash (y \vdash z) \stackrel{5}=
(x\vdash y)\vdash z
\end{array}$$\
for all $x,y,z\in D$.
Apart from the known algebraic examples of dialgebras, [@L], we cite an interesting example of a family of dialgebras in the context of functional analysis, [@RF].
Let ${\mathcal H}$ be a Hilbert space and $e \in \mathcal H$ with $||e||=1$. Define two linear operations $\dashv$ and $\vdash$ by $$a
\dashv b = \left\langle b, e\right\rangle a, ~~~~~ a \vdash b =
\left\langle a, e\right\rangle b,$$ for $a, b \in {\mathcal H}$. Then $({\mathcal H}, \dashv, \vdash)$ is a dialgebra, more precisely, a normed dialgebra [@RF].
A morphism $\phi : D \longrightarrow D'$ between two dialgebras is a K-linear map such that $\phi(x\dashv y)=\phi(x) \dashv \phi(y)$ and $\phi(x\vdash y)=\phi(x) \vdash \phi(y).$
A planar binary tree with $n$ vertices (in short, $n$-tree) is a planar tree with $(n+1)$ leaves, one root and each vertex trivalent. Let $Y_n$ denote the set of all $n$-trees. Let $Y_0$ be the singleton set consisting of a root only. The $n$-trees for $0\leq n\leq 3$ are given by the following diagrams:
1.9cm
For any $y\in Y_n$, the $(n+1)$ leaves are labelled by $\{0,1,\ldots
,n\}$ from left to right and the vertices are labelled $\{1,2,\ldots
,n\}$ so that the $i$th vertex is between the leaves $(i-1)$ and $i$. The only element $|$ of $Y_0$ is denoted by $[0]$ and the only element of $Y_1$ is denoted by $[1]$. The grafting of a $p$-tree $y_1$ and a $q$-tree $y_2$ is a $(p+q+1)$-tree denoted by $y_1\vee y_2$ which is obtained by joining the roots of $y_1$ and $y_2$ and creating a new root from that vertex. This is denoted by $[y_1~ p+q+1~ y_2]$ with the convention that all zeros are deleted except for the element in $Y_0$. With this notation, the trees pictured above from left to right are $[0],[1],[12],[21],[123],[213], [131], [312], [321]$.
For any $i$, $0\leq i\leq n$, there is a map, called the face map, $d_i :
Y_n \longrightarrow Y_{n-1}$, $y\mapsto d_iy$ where $d_iy$ is obtained from $y$ by deleting the $i$th leaf. The face maps satisfy the relations $d_id_j=d_{j-1}d_i$, for all $i<j$.
Let $D$ be a dialgebra over a field $K$. The cochain complex $CY^*(D,D)$ which defines the dialgebra cohomology $HY^*(D,D)$ is defined as follows. For any $n\geq 0$, let $K[Y_n]$ denote the $K$-vector space spanned by $Y_n$ and $CY^n(D,D):= \mbox{Hom}_K(K[Y_n]\otimes D^{\otimes n},D)$ be the module of $n$-cochains of $D$ with coefficients in $D$. The coboundary operator $\delta : CY^n(D,D) \longrightarrow CY^{n+1}(D,D)$ is defined as the $K$-linear map $\delta = \sum _{i=0}^{n+1}(-1)^i \delta^i$, where $$(\delta^i f)(y;a_1,a_2,\ldots,a_{n+1})=\left\{\begin{array}{ll}
a_1 \circ_0^y f(d_0y;a_2,\ldots,a_{n+1}), & i=0\\
f(d_iy;a_1,\ldots,a_i \circ_i^y a_{i+1},\ldots,a_{n+1}),
& 1 \leq i\leq n\\
f(d_{n+1}y;a_1,\ldots,a_n)\circ_{n+1}^ya_{n+1}, & i=n+1
\end{array}
\right.$$ for any $y \in Y_{n+1}$; $a_1, \ldots,a_{n+1}\in D$ and $f:K[Y_n]\otimes D^{\otimes n} \longrightarrow D$. Here, for any $i$, $0\leq i\leq n+1$. The maps $\circ_i:Y_{n+1}
\longrightarrow \{\dashv,\vdash\}$, are defined by
$$\circ_0(y)= \circ_0^y:=\left \{\begin{array}{ll} \dashv & \mbox {if
$y$ is of the form $| \vee y_1$, for some $n$-tree $y_1$}\\ \vdash &
\mbox{otherwise} \end{array} \right.$$
$$\circ_i(y)= \circ_i^y:=\left \{\begin{array}{ll} \dashv & \mbox {if the
$i^{th}$ leaf of $y$ is oriented like `$\backslash $'}\\ \vdash &
\mbox{if the $i^{th}$ leaf of $y$ is oriented like `$/$' } \end{array}
\right.$$ for $ 1\leq i\leq n$ and
$$\circ_{n+1}(y)= \circ_{n+1}^y:=\left \{\begin{array}{ll}
\vdash & \mbox{if $y$ is of the form $y_1 \vee |$, for some $n$-tree
$y_1$}\\
\dashv & \mbox{otherwise}
\end{array},
\right.$$ where the symbol $`\vee'$ stands for grafting of trees [@L].
There exists a pre-Lie algebra structure on $CY^*(D,D)$, [@MM1; @MM2], the pre-Lie product being denoted by $$\circ: CY^n(D,D) \otimes CY^m(D,D)\longrightarrow CY^{n+m-1}.$$ Also, if we modify the coboundary map $\delta$ by a sign, say $dx=(-1)^{|x|}\delta(x)$, and define a bracket product on $CY^*(D,D)$ by $[x,y]= x\circ y -(-1)^{|x||y|} y\circ x$, which is the commutator of the pre-Lie product, then $(CY^*(D,D), d)$ forms a differential graded Lie algebra, [@MM2], where $|x|= \mbox{deg}~x-1$.
Deformations of Dialgebras
==========================
Let $D$ be a dialgebra over $K$ and let $A$ be a commutative unital algebra over $K$ with a fixed augmentation $\epsilon: A\longrightarrow
K$ with $\epsilon(1)=1.$ Let $\mathsf m = \ker\,\epsilon.$ We assume $\dim\,({\mathsf m}^k/{\mathsf m}^{k+1})<\infty$, for all $k \geq 1$.
A deformation $\lambda$ of $D$ with base $(A,\mathsf m)$ is a dialgebra structure on the tensor product $A\otimes_K D$ with the products $\dashv_{\lambda}$ and $\vdash_{\lambda}$ being $A$-linear (or simply, an $A$-dialgebra structure) such that $\epsilon\otimes \mbox{id}~:
A\otimes_K D\longrightarrow K\otimes D \cong D$ is a $A$-linear dialgebra morphism. The left action of $A$ on $K\otimes D$ is given by the augmentation map.
We note that for $ x_1, x_2 \in D$, and $ a, b \in A$, $$a\otimes x_1 *_\lambda b\otimes x_2= ab ( 1\otimes x_1 * 1\otimes x_2),$$ by A-linearity of the products, where $* =\{ \dashv, \vdash\}$. Also, since $\epsilon \otimes ~\mbox{id}~: A\otimes D\longrightarrow K\otimes
D$ is a $A$-linear dialgebra homomorphism, $$\begin{array}{rcl}
(\epsilon\otimes ~\mbox{id})~\{ 1\otimes x_1 *_\lambda 1\otimes x_2\}
&=&(\epsilon\otimes ~\mbox{id})(1\otimes x_1) * (\epsilon\otimes ~\mbox{id})(1\otimes x_2)\\
&=&(1\otimes x_1) * (1\otimes x_2)\\
&=&1\otimes (x_1* x_2)\\
&=&(\epsilon \otimes ~\mbox{id})(1\otimes (x_1* x_2)).
\end{array}$$ So, $ (1\otimes x_1) *_\lambda (1\otimes x_2) - 1\otimes (x_1 * x_2)
\in \ker(\epsilon\otimes \mbox{id})$. Hence $ (1\otimes x_1) *_\lambda
(1\otimes x_2) = 1\otimes (x_1 * x_2) + \sum_i m_i \otimes d_i$, where $m_i \in \ker\,\epsilon= \mathsf m$ and $d_i \in D$ and $\sum_i m_i \otimes
d_i$ is a finite sum.
Two deformations of $D$ with the same base $A$ are called *equivalent* if there exists a $A$-linear dialgebra isomorphism between the two copies of $A\otimes D$ with the two dialgebra structures, compatible with $\epsilon \otimes \mbox{id}.$ A deformation of $D$ with base $A$ is called *local* if the algebra $A$ is local, and will be called *infinitesimal* if, in addition, $\mathsf m^2=0$, where $\mathsf m$ is the maximal ideal of $A$.
Let $A$ be a complete local algebra, that is, $A=
\overleftarrow{\displaystyle{\lim_{n\to\infty}}} (A/\mathsf m^n)$, $\mathsf m$ denoting the maximal ideal in $A$. A formal deformation of $D$ with base $A$ is a $A$-dialgebra structure on the completed tensor product $A \widehat{\otimes}D=
\overleftarrow{\displaystyle{\lim_{n\to\infty}}} (A/\mathsf m^n)
\otimes D)$, such that $\epsilon
\widehat{\otimes}\mbox{id}~: A \widehat{\otimes} D\longrightarrow
K\otimes D=D$ is a $A$-linear dialgebra morphism.
Two formal deformations of a dialgebra $D$ with the same base $A$ are called *equivalent* if there exists a dialgebra isomorphism between the two copies of $A \widehat{\otimes} D$ with the two dialgebra structures compatible with $\epsilon \widehat{\otimes}\mbox{id}.$
If $A= K[[t]]$ then a formal deformation of $D$ with base $A$ is the same as a formal one-parameter deformation of $D$, [@MM1].
Let $A'$ be a commutative algebra with identity, with a fixed augmentation $\epsilon': A' \longrightarrow K$ and let $\phi: A\longrightarrow A'$ be an algebra homomorphism, with $\phi(1)=1$ and $\epsilon' \circ \phi =
\epsilon.$ Then we can construct a deformation of $D$ with base $A'$ in the following way.
Let $\lambda$ be a deformation of the dialgebra $D$ with base $(A,\mathsf m)$. The *push*-*out* $\phi_*\lambda$ is the deformation of $D$ with base $(A',\mathsf m'=\ker\,\epsilon')$, which is the dialgebra structure given by $$\begin{aligned}
a_1'\otimes_A(a_1\otimes x_1)
\dashv_{\phi_*\lambda}\ a_2'\otimes_A(a_2\otimes x_2)
=&\ a_1'a_2' \otimes_A (a_1\otimes x_1 \dashv_\lambda a_2\otimes x_2)\\
a_1'\otimes_A(a_1\otimes x_1)
\vdash_{\phi_*\lambda}\ a_2'\otimes_A(a_2\otimes x_2)
=&\ a_1'a_2' \otimes_A (a_1\otimes x_1 \vdash_\lambda a_2\otimes x_2),\end{aligned}$$ where $a_1, a_2\in A'$, $a_1, a_2 \in A$ and $l_1, l_2 \in D.$ Here we make use of the fact that $A'\otimes D= (A' \otimes_A A)\otimes D=
A'\otimes_A(A\otimes D),$ where $A'$ is regarded as an $A$-module by the structure $a'a=a'\phi(a).$
Similarly, one can define the *push*-*out* of formal deformations.
We note that if the dialgebra structure $\lambda$ on $A\otimes D$ is given by $$(1\otimes x_1)*_\lambda (1\otimes x_2)= 1\otimes (x_1 * x_2)
+ \sum_{i=1}^n m_i \otimes d_i;~~ m_i \in\mathsf m, d_i \in D,$$ then the dialgebra structure $\phi_*\lambda$ on $A'\otimes D$ is given by $$(1\otimes x_1)*_{\phi_*\lambda} (1\otimes x_2)
= 1\otimes (x_1 * x_2)+ \sum_{i=1}^n \phi(m_i) \otimes d_i.$$
Universal Infinitesimal and Miniversal Deformations of Dialgebras
=================================================================
In [@FF], the authors have produced a fundamental example of an infinitesimal deformation of Lie algebras. Here we produce an example of an infinitesimal deformation of dialgebras, which is obtained from the aforesaid example, with slight modifications. Suppose $\dim\,HY^2(D,D) <\infty.$ This is, in particular, true if $\dim\, D <\infty$. Consider the base of the deformation to be $A= K\oplus HY^2(D,D)',$ with $'$ denoting the linear dual. Here, $A$ is local with the maximal ideal $\mathsf m=
HY^2(D,D)',$ and $\mathsf m^2=0.$
Let $$\mu: HY^2(D,D) \longrightarrow CY^2(D,D)= \mbox{Hom\,}(K[Y_2]\otimes
D^{\otimes 2},D)$$ which takes a cohomology class into a cocycle representing the class. Define a dialgebra structure on $$\begin{array}{ll}
A\otimes D&= (K\oplus HY^2(D,D)')\otimes D\\
&=(K\otimes D) \oplus (HY^2(D,D)'\otimes D)\\
&=D \oplus (HY^2(D,D)'\otimes D)\\
&= D \oplus \mbox{Hom}~(HY^2(D,D),D)
\end{array}$$ by
$$\begin{array}{ll}
(x_1,\phi_1)\dashv (x_2,\phi_2)&=(x_1\dashv x_2, \psi_{\ell})\\
(x_1,\phi_1)\vdash (x_2,\phi_2)&=(x_1\vdash x_2, \psi_r)
\end{array}$$ where $$\begin{array}{ll}
\psi_{\ell}(\alpha)&=\mu(\alpha)([21]; x_1, x_2) + \phi_1(\alpha)\dashv x_2+
x_1\dashv \phi_2(\alpha)\\
\psi_r(\alpha)&=\mu(\alpha)([12]; x_1, x_2) + \phi_1(\alpha)\vdash x_2+
x_1\vdash \phi_2(\alpha),
\end{array}$$ for $\alpha \in HY^2(D,D)$.
Using the dialgebra structure of $D$ and the fact that $\mu(\alpha)$ is a $2$-cocycle of $D$, one can check that the $\dashv$ and $\vdash$ products defined this way satisfy the dialgebra axioms. It is to be noted that this deformation does not depend on the choice of $\mu$, upto an isomorphism.
Let $\mu': HY^2(D,D) \longrightarrow CY^2(D,D)$ be another choice of $\mu.$ Define a homomorphism $$\nu: HY^2(D,D) \longrightarrow CY^1(D,D)\cong \text{Hom}(D,D)$$ by $\mu'(\alpha) - \mu (\alpha)= \delta\nu(\alpha),$ for all $\alpha \in
HY^2(D,D)$. We define a linear automorphism $\rho$ of the space $A\otimes D= D\oplus
\text{Hom} (HY^2(D,D),D)$ by $\rho(x, \phi)= (x, \psi)$ where $\psi(\alpha)= \phi(\alpha)+
\nu(\alpha)(x).$ It is straightforward to check that $\rho$ defines a dialgebra isomorphism between the two dialgebra structures induced by $\mu$ and $\mu'$ respectively. We denote the infinitesimal deformation of $D$ as constructed above by $\eta_D.$
Below we will show the *couniversality* of $\eta_D$ in the class of infinitesimal deformations:
Let $\lambda$ be an infinitesimal deformation of the dialgebra $D$, with a finite dimensional local algebra base $A$, with $\mathsf m^2=0$, where $\mathsf m$ is the maximal ideal of $A$. Let $\xi \in {\mathsf m}'= \text{Hom}_K\,(\mathsf m,K).$ This is equivalent to $\xi \in \text{Hom}_K~(A,K)$ with $\xi(1)=0.$
For $x_1, x_2 \in D,$ let us define a $2$-cochain as follows: $$\alpha_{\lambda, \xi}([21]; x_1, x_2)= (\xi \otimes \mbox{id})((1\otimes
x_1)\dashv_{\lambda}(1\otimes x_2))$$ and $$\alpha_{\lambda, \xi}([12]; x_1, x_2)= (\xi \otimes \mbox{id})((1\otimes
x_1)\vdash_{\lambda}(1\otimes x_2)).$$
We claim that $\alpha_{\lambda,\xi}\in CY^2(D,D)$ is a $2$-cocycle. This is because $$\begin{aligned}
\delta \alpha_{\lambda,\xi}([321];&x_1,x_2,x_3)\\
=\ &x_1\dashv
\alpha_{\lambda,\xi}([21];x_2,x_3)-\alpha_{\lambda,\xi}([21];x_1\dashv
x_2,x_3)\\
&+ \alpha_{\lambda,\xi}([21]; x_1, x_2\dashv x_3)
-\alpha_{\lambda,\xi}([21]; x_1, x_2) \dashv x_3\\
=\ &x_1\dashv(\xi\otimes \mbox{id}~) ((1\otimes x_2)\vdash_{\lambda}
(1\otimes x_3))- (\xi \otimes \mbox{id})(1\otimes (x_1\dashv
x_2)\dashv_{\lambda}1\otimes x_3)\\
&+ (\xi\otimes \mbox{id}) (1\otimes x_1 \dashv_{\lambda} 1\otimes
x_2\dashv x_3)- (\xi\otimes \mbox{id}~)(1\otimes x_1\dashv_{\lambda}1\otimes
x_2)\dashv x_3.\end{aligned}$$ If $\epsilon$ denotes the fixed augmentation of the algebra $A$, then $$\epsilon \otimes \mbox{id}~:(1\otimes x_1 \dashv_{\lambda} 1\otimes x_2 -
1\otimes x_1\dashv x_2)=0,$$ i.e. $1\otimes x_1 \dashv_{\lambda} 1\otimes x_2 - 1\otimes x_1\dashv x_2
\in\mathsf m\otimes D.$ So, $$\begin{aligned}
(\xi\otimes \mbox{id}~) &((1\otimes x_1 \dashv_{\lambda} 1\otimes x_2)
\dashv_{\lambda} (1\otimes x_3))\\
=& (\xi\otimes \mbox{id}~) (((1\otimes x_1 \dashv x_2)+ \sum_i m_i\otimes
y_i)\dashv_{\lambda} (1\otimes x_3)))\\
=& (\xi\otimes \mbox{id}~) ((1\otimes x_1 \dashv
x_2)\dashv_{\lambda}(1\otimes x_3))+
(\xi\otimes \mbox{id}~) (\sum_i (m_i\otimes y_i)\dashv_{\lambda} (1\otimes
x_3))\\
=& (\xi\otimes \mbox{id}~) ((1\otimes x_1 \dashv
x_2)\dashv_{\lambda}(1\otimes x_3))+
(\xi\otimes \mbox{id}~) (\sum_i m_i(1\otimes y_i)\dashv_{\lambda} (1\otimes
x_3))\\
=&\alpha_{\lambda, \xi}([21];x_1\dashv x_2, x_3) + (\xi\otimes
\mbox{id})\sum_i m_i(1\otimes y_i \dashv_\lambda 1\otimes x_3).\end{aligned}$$ Note that in the second step from the end we make use of the action of the algebra $A$ on $A\otimes D$.
Now we have $$1\otimes y_i \dashv_{\lambda} 1\otimes x_3 - 1\otimes y_i\dashv x_3 \in
m\otimes D,$$ $$1\otimes y_i \dashv_{\lambda} 1\otimes x_3 = 1\otimes y_i\dashv x_3 +h,$$ where $h \in \mathsf m\otimes D.$ Hence, $$m_i(1\otimes y_i \dashv_{\lambda} 1\otimes x_3)= m_i (1\otimes y_i\dashv
x_3 +h).$$
Since $\mathsf m^2=0$, we have $m_i h=0.$ So, $m_i(1\otimes y_i \dashv_{\lambda}
1\otimes x_3)= m_i \otimes (y_i\dashv x_3),$ making use of the action of $A$ on $A\otimes D.$ Next $$\begin{aligned}
(\xi\otimes \mbox{id}~)\sum_i m_i(1\otimes y_i\dashv_{\lambda} 1\otimes
x_3)&=\sum_i(\xi\otimes \mbox{id}~)(m_i\otimes y_i\dashv x_3)\\
&=\sum_i\xi(m_i)(y_i\dashv x_3)\\
&= \sum_i(\xi(m_i)y_i \dashv x_3)\\
&=(\xi\otimes \mbox{id}~)(\sum_im_i\otimes y_i)\dashv x_3\\
&=(\xi \otimes \mbox{id})\{(1\otimes x_1\dashv_\lambda 1\otimes
x_2)-1\otimes x_1\dashv x_2\}\dashv x_3\\
&=((\xi\otimes \mbox{id})(1\otimes x_1 \dashv_{\lambda}1\otimes x_2)\dashv
x_3\quad
\text{[using $\xi(1)=0$]}\\
&=\alpha_{\lambda, \xi}([12];x_1, x_2)\dashv x_3.\end{aligned}$$ Thus, $$\xi\otimes \mbox{id}~ ((1\otimes x_1 \dashv_{\lambda} 1\otimes x_2)
\dashv_{\lambda}(1\otimes x_3))
=\alpha_{\lambda, \xi}([21]; x_1\dashv x_2, x_3) + \alpha_{\lambda,
\xi}([21];x_1, x_2)\dashv x_3.$$ In the same way, $$\xi\otimes \mbox{id}~ (1\otimes x_1 \dashv_{\lambda} (1\otimes
x_2\dashv_{\lambda}(1\otimes x_3))\\
=x_1 \dashv \alpha_{\lambda, \xi}([21]; x_2, x_3) + \alpha_{\lambda,
\xi}([21];x_1, x_2\dashv x_3).$$
Since $$\xi\otimes \mbox{id}((1\otimes x_1 \dashv_{\lambda} 1\otimes
x_2)\dashv_{\lambda}(1\otimes x_3))-\xi\otimes\mbox{id}~(1\otimes x_1
\dashv_{\lambda} (1\otimes x_2\dashv_{\lambda}1\otimes x_3))=0,$$ we have $$\delta\alpha_{\lambda, \xi}([321]; x_1, x_2, x_3)=0,$$ and we can also show that $\delta\alpha_{\lambda, \xi}(y; x_1,
x_2, x_3)=0$ for all $y \in \{[312],[131], [213], [123]\}$.\
\
The following proposition classifies all infinitesimal deformations of $D$ over finite dimensional bases.
For any infinitesimal deformation $\lambda$ of a dialgebra $D$ with a finite dimensional base $A$ there exists a unique homomorphism $\phi: K\oplus
HY^2(D,D)' \longrightarrow A$ such that $\lambda$ is equivalent to the push-out $\phi_*\eta_D.$
[**Proof.**]{} Let $a_{\lambda, \xi} \in HY^2(D,D)$ be the cohomology class of the cocycle $\alpha_{\lambda, \xi}$, corresponding to $\xi\in\mathsf m'.$ Thus we have the following homomorphisms:
$$\begin{array}{ll}
\alpha_\lambda:\mathsf m'&\longrightarrow CY^2(D,D)\\
a_\lambda:\mathsf m'&\longrightarrow HY^2(D,D).
\end{array}$$
[**Step 1.**]{} We show that the deformations $\lambda, \lambda'$ are equivalent if and only if $a_{\lambda}= a_{\lambda'}$. Let $\lambda_1$ and $\lambda_2$ be two equivalent deformations of the dialgebra $D$, with base $A$. By definition, there exists a $A$-linear dialgebra isomorphism
$$\label{iso}
\rho: A\otimes D\longrightarrow A\otimes D,~ \text{such that}~
(\epsilon \otimes \mbox{id})\circ \rho= \epsilon \otimes \mbox{id}.$$
Since $A\otimes D= D \oplus (\mathsf m\otimes D)$, the isomorphism $\rho$ can be written as $\rho = \rho_1 + \rho_2$ where $\rho_1: D\longrightarrow
D$ and $\rho_2: D \longrightarrow\mathsf m\otimes D$.
By using equation (\[iso\]), we get $\rho_1=\mbox{id}.$ Note that by the adjunction property of tensor products, $$\mbox{Hom}(D;\mathsf m\otimes D) \cong\mathsf m\otimes \mbox{Hom} (D,D)
\cong \mbox{Hom}(\mathsf m'; \mbox{Hom}(D,D)),$$ where the isomorphisms are given by
$$\label{adiso}
\rho_2\longmapsto \sum_1^k m_i \otimes \phi_i\longmapsto \sum_i^k \chi_i.$$
Here $\phi_i = (\xi_i \otimes \mbox{id})\circ \rho_2$ and $\chi_i
(\xi_j)= \delta_{i, j}\phi_i$, where $\{m_i\}_{1\leq i \leq k}$ is a basis of $\mathsf m$ and $\{ \xi_j\}_{1\leq j\leq k}$ is a basis of $\mathsf m'$.
We have by equation (\[adiso\]), $$\begin{array}{rcl}
\rho(1\otimes x)&=& \rho_1(1\otimes x) + \rho_2(1\otimes x)\\
&=& 1\otimes x + \sum_1^k m_i \otimes \phi_i(x).
\end{array}$$
Using the notation $*=\{\dashv, \vdash\}$, the map $\rho$ is a dialgebra homomorphism iff $$\rho( 1\otimes x_1 *_{\lambda_1} 1\otimes x_2)
= \rho( 1\otimes x_1) *_{\lambda_2} \rho( 1\otimes x_2),$$
Let us set $\psi_i^r= \alpha_{\lambda_r, \xi_i}$, $i=1,2,\ldots, k$ and $r=1,2.$ Then we have
$$1\otimes x_1 \dashv_{\lambda_r} 1\otimes x_2= 1\otimes x_1\dashv x_2 + \sum_i^k m_i \otimes \psi_i^r([21]; x_1, x_2)$$
and $$1\otimes x_1 \vdash_{\lambda_r} 1\otimes x_2= 1\otimes x_1\vdash x_2 + \sum_i^k m_i \otimes \psi_i^r([1 2]; x_1, x_2).$$
Therefore, using the fact that $m_i. m_j=0$ for elements $m_i, m_j \in
\mathsf m$, $$\rho(1\otimes x_1 \dashv_{\lambda_1} 1\otimes x_2)
= 1\otimes x_1 \dashv x_2+ \sum_{i=1}^k m_i\otimes \phi_i(x_1 \dashv x_2)
+ \sum_{i=1}^k m_i ( 1\otimes \psi_i^1([21]; x_1, x_2)).$$ Similarly, $$\rho(1\otimes x_1 \vdash_{\lambda_2} 1\otimes x_2)
= 1\otimes x_1 \vdash x_2+ \sum_{i=1}^k m_i\otimes \phi_i(x_1 \vdash x_2)
+ \sum_{i=1}^k m_i ( 1\otimes \psi_i^1([12]; x_1, x_2)).$$
Again, $$\begin{aligned}
\rho(1\otimes x_1)\dashv_{\lambda_2}&\ \rho(1\otimes x_2)\\
=& 1\otimes (x_1\dashv x_2) + \sum_{i=1}^k m_i
\otimes (\psi_i^2([21]; x_1, x_2)) + \sum_{i=1}^k m_i
\otimes \big(x_1\dashv \phi_i(x_2)\big)\\
& +\sum_{i=1}^k m_i \otimes (\phi_i(x_1)\dashv x_2),\end{aligned}$$ and $$\begin{aligned}
\rho(1\otimes x_1)\vdash_{\lambda_2}\ & \rho(1\otimes x_2)\\
= & 1\otimes (x_1\vdash x_2) +
\sum_{i=1}^k m_i \otimes (\psi_i^2([21]; x_1, x_2)) +
\sum_{i=1}^k m_i \otimes \big(x_1\vdash \phi_i(x_2)\big) \\
& +\sum_{i=1}^k m_i \otimes (\phi_i(x_1)\vdash x_2).\end{aligned}$$
Thus, the following are equivalent: $$\begin{aligned}
\text{a)}&\quad \rho( 1\otimes x_1 \dashv_{\lambda_1} 1\otimes x_2)
= \rho( 1\otimes x_1) \dashv_{\lambda_2} \rho( 1\otimes x_2)\\
\text{b)}&\quad \sum_{i=1}^k m_i \otimes (\psi_i^2([21]; x_1, x_2)
- \psi_i^1[21]; x_1, x_2)) +
\sum_{i=1}^k m_i \otimes \delta \phi_i([21]; x_1, x_2)=0\\
\text{c)}&\quad \psi_i^1([21]; x-1, x_2) - \psi_i^2([21]; x_1, x_2) = \delta \phi_i ([21]; x_1, x_2)\end{aligned}$$
and similarly these are equivalent, too: $$\begin{aligned}
\text{a$'$)}&\quad \rho( 1\otimes x_1 \vdash_{\lambda_1} 1\otimes x_2)
= \rho( 1\otimes x_1) \vdash_{\lambda_2} \rho( 1\otimes x_2),\\
\text{b$'$)}&\quad \sum_{i=1}^k m_i \otimes (\psi_i^2([12]; x_1, x_2)
- \psi_i^1[12]; x_1, x_2))
+ \sum_{i=1}^k m_i \otimes \delta \phi_i([12]; x_1, x_2)=0\\
\text{c$'$)}&\quad \psi_i^1([12]; x-1, x_2) - \psi_i^2([12]; x_1, x_2) = \delta \phi_i ([12]; x_1, x_2).\end{aligned}$$
Hence, $$\alpha_{\lambda_1, \xi_i}- \alpha_{\lambda_2, \xi_i}
= \delta \phi_i\quad \text{for $i\in\{1, 2,\ldots, k\}$}
\qquad \text{if and only if}\qquad
a_{\lambda_1}= a_{\lambda_2}.$$ This proves step 1.
[**Step 2**]{}. Let $$\phi= \mbox{id} \oplus a_\lambda': K\oplus
HY^2(D,D)'\longrightarrow K\oplus\mathsf m= A.$$ Claim: $\phi_*
\eta_D$ is equivalent to $\lambda$. It follows from definitions that $\alpha_{\phi_* \eta_D}= \mu\circ a_{\lambda}$. Thus, $a_{\phi_* \eta_D}= a_\lambda$. Hence by step 1, $\phi_* \eta_D$ and $\lambda$ are isomorphic. This completes the proof of Proposition 4.1.
Let $A$ be a local algebra with dim$(A/{\mathsf m}^2)<\infty.$ Then, $A/{\mathsf m}^2$ is also local with the maximal ideal ${\mathsf
m}/{\mathsf m}^2$, and $({\mathsf m}/{\mathsf m}^2)^2=0.$
The linear dual space $\text{Hom}({\mathsf m}/{\mathsf m}^2, K)$ is called the *tangent space* of $A$, and is denoted by $TA$.
\[diff\] Let $\lambda$ be a deformation of $D$ with base $A$. Then the mapping $$a_{\pi_*\lambda}: TA=({\mathsf m}/{\mathsf m}^2)'\longrightarrow HY^2(D,D),$$ where $\pi$ is the projection $A\longrightarrow A/{\mathsf m}^2,$ is called the *differential* of $\lambda$ and is denoted by $d\lambda.$
A formal deformation $\eta$ of a dialgebra $D$ with base $B$ is called *miniversal* if
1. for any formal deformation ${\lambda}$ of a dialgebra $D$ with any local base $A$ there exists a homomorphism $f: B \rightarrow A$ such that the deformation $\lambda$ is equivalent to $f_* \eta$;
2. with the above notations if $A$ satisfies the condition $\mathsf
m^2=0$, then $f$ is unique.
If $\eta$ satisfies only condition (1), then it is called *versal*.
The following proposition takes its shape from the general results of Schlessinger [@Sch]. It was first shown for the case of Lie algebras in [@Fi1], and stated for Leibniz algebras in [@FMM]. It is straightforward to see that it is true for the case of dialgebras, too.
If the dimension of $HY^2(D,D)$ is finite, then there exists a miniversal deformation of the dialgebra $D$.
Some Facts about Harrison Cohomology
====================================
Let $A$ denote a commutative algebra over $K$. In this section we shall state a few results, without proof [@H], about Harrison cohomology groups of $A$ with coefficients in a $A$-module $M$. Let $Ch(A)=\{Ch_q(A), \delta\}$ denote the Harrison complex of $A$.
For an $A$-module $M$, the Harrison homology and cohomology of $A$ with coefficients in $M$ are defined as follows: $$\begin{array}{ccc}
&H^{Harr}_q(A;M)&=H_q(Ch(A)\otimes M),\\
&H_{Harr}^q(A; M)&=H^q(\text{Hom}(Ch(A), M);
\end{array}$$
1. $H_{Harr}^1(A;M)$ is the space of derivations $A\rightarrow M$.
2. Elements of $H_{Harr}^2(A;M)$ correspond bijectively to isomorphism classes of extensions $0\rightarrow M\rightarrow B\rightarrow A\rightarrow0$ of the algebra $A$ by means of $M$.
\[tan\] If $A$ is a local algebra with the maximal ideal $\mathsf m$, then $$H_{Harr}^1(A;K)=(\mathsf m/\mathsf m^2)'=TA.$$
Suppose $0\rightarrow M_r
\stackrel{\stackrel{-}{i}}{\rightarrow}B_{r-1}
\stackrel{\stackrel{-}{p}}{\rightarrow} A\rightarrow 0$ is an $r$-dimensional extension of $A$. Then there is a $(r-1)$-dimensional extension $0\rightarrow M_{r-1}
\stackrel{i}{\rightarrow}B_r \stackrel{p}{\rightarrow}
A\rightarrow 0$ of $A$ and a $1$-dimensional extension $0\rightarrow K \stackrel{i'}{\rightarrow} B_r
\stackrel{p'}{\rightarrow}B_{r-1}\rightarrow 0$.
Let $0\rightarrow M\stackrel{i}{\rightarrow}B\stackrel{p}{\rightarrow}
A\rightarrow0$ be an extension of an algebra $A$ by $M$.
1. If $A$ has an identity then so does $B$.
2. If $A$ is local with the maximal ideal $\mathsf m$, then $B$ is local with the maximal ideal $p^{-1}(\mathsf m).$
Two extensions $B$ and $B'$ of the algebra $A$ by $M$ are said to be equivalent if there exists a $K$-algebra isomorphism $f: B\rightarrow B'$ such that the following diagram commutes.
$$\begin{array}{ccccccc}
0\longrightarrow & M & \stackrel{i_1}\longrightarrow & B &\stackrel{p_1}\longrightarrow & A & \longrightarrow 0\\
&&&&&&\\
&\downarrow \mbox{id} & & \downarrow f & & \downarrow \mbox{id}&\\
&&&&&&\\
0\longrightarrow & M & \stackrel{i_2}\longrightarrow & B' &\stackrel{p_2}\longrightarrow & A & \longrightarrow 0.
\end{array}$$
An equivalence from $B$ to $B$ is said to be an automorphism of $B$ over $A$.
\[aut\] $H_{Harr}^1(A;M)$ is isomorphic to the set of automorphisms of any given extension $0\rightarrow M\stackrel{i}{\rightarrow}B\stackrel{p}{\rightarrow}
A\rightarrow0$ of $A$ by $M$.
Obstructions to Extending Deformations
======================================
Let $A$ be a finite dimensional commutative, unital, local algebra with a fixed augmentation $\epsilon$, and let $\lambda$ be a deformation of a dialgebra $D$ with base $A$. Let $0\rightarrow K \stackrel{i}{\rightarrow}B \stackrel{p}{\rightarrow}A
\rightarrow 0$ be an extension of $A$, corresponding to a cohomology class $f\in H^2_{Harr}(A;K)$. Let $q: A\rightarrow B$ be a splitting. Let $\widehat{\epsilon}: B\rightarrow K$ be the augmentation of $B$. Let $I=i\otimes \mbox {id}~:
D=K\otimes D\rightarrow B\otimes D$ and $P=p\otimes \mbox {id}~:B\otimes
D\rightarrow A\otimes D$. Let $E= \widehat{\epsilon}\otimes \mbox {id}~:
B\otimes D\rightarrow K\otimes D=D$ and let $Q=q\otimes \mbox {id}~:
A\otimes D\rightarrow B\otimes D$. We define two $B$-bilinear operations $\{~,~\}_{\dashv}$, $\{~,~\}_{\vdash}$ on $B\otimes D$ as follows:
Let $l_1, l_2\in B\otimes D$. Define $$\{l_1, l_2\}_\dashv= Q\{P(l_1)\dashv_\lambda P(l_2)\}
+I[I^{-1}(l_1-Q\circ P(l_1))\dashv I^{-1}(l_2- Q\circ P(l_2))],$$ $$\{l_1, l_2\}_\vdash= Q\{P(l_1)\vdash_\lambda P(l_2)\} +I[I^{-1}(l_1-Q\circ
P(l_1))\vdash I^{-1}(l_2- Q\circ P(l_2))].$$ It is easy to verify that the two operations thus defined satisfy the following properties: $$\begin{aligned}
\label{op}
&(i)\ P\{l_1, l_2\}_* = P(l_1)* P(l_2),
&\text{where\ }& *\in \{\dashv, \vdash\}, l_1, l_2\in B\otimes D,\\
&(ii)~~~\{I(l), l_1\}_* = I[l*E(l_1)],
&\text{where\ }& *\in \{\dashv, \vdash\}, l \in D, l_1\in B\otimes D.\end{aligned}$$
Using the above two properties, one can show that $$\begin{array}{rcl}
E\{l_1, l_2\}_{\dashv}&=& E(l_1) \dashv E(l_2)\\
E\{l_1, l_2\}_{\vdash}&=& E(l_1) \vdash E(l_2)\ .
\end{array}$$
We define $$\begin{aligned}
\label{phi}
\phi([321];l_1, l_2, l_3)=& \{l_1, \{l_2, l_3\}_{\dashv}
\}_{\dashv}-\{\{l_1, l_2\}_{\dashv}, l_3\}_{\dashv},\\
\phi([312];l_1, l_2, l_3)=&\{\{l_1, l_2\}_{\dashv}, l_3\}_{\dashv}- \{l_1,
\{l_2, l_3\}_{\vdash} \}_{\dashv},\\
\phi([131];l_1, l_2, l_3)=& \{l_1, \{l_2, l_3\}_{\dashv}
\}_{\dashv}-\{\{l_1, l_2\}_{\vdash}, l_3\}_{\dashv},\\
\phi([213];l_1, l_2, l_3)=& \{\{l_1, l_2\}_{\dashv}, l_3\}_{\vdash}-\{l_1,
\{l_2, l_3\}_{\vdash} \}_{\vdash},\\
\phi([213];l_1, l_2, l_3)=&\{l_1, \{l_2, l_3\}_{\vdash}
\}_{\vdash}-\{\{l_1, l_2\}_{\vdash}, l_3\}_{\vdash}.\end{aligned}$$
It is easy to see that $\phi(y; l_1, l_2, l_3) \in \ker\,P$ for all $y \in Y_3.$ Also, note that if any $l_i \in \ker\, E, i\in
\{1,2,3\}$, then $\phi(l_1, l_2, l_3)=0.$ This defines the map $$\label{phibar}
\overline{\phi}: K[Y_3]\otimes D^{\otimes 3}=K[Y_3]\otimes ((B\otimes
D)/\ker\, E)^{\otimes 3}\rightarrow \ker\,P=D.$$ Thus $\overline{\phi}\in CY^3(D,D).$ One can check that $\delta
\overline{\phi}=0.$
Let $f'$ be cohomologous to $f$, and let $0\rightarrow K
\stackrel{i'}{\rightarrow}B' \stackrel{p'}{\rightarrow}A \rightarrow 0$ be the extension corresponding to $f'$, which is isomorphic to the extension corresponding to $f$. Since $B$ and $B'$ are isomorphic, without loss of generality, we shall work with $B$.
Let $\{~,~\}'_*, *\in \{\dashv, \vdash\}$ be another set of $B$-bilinear operations on $B\otimes D$, satisfying ($1$) and ($2$) above. Then $\{l_1,
l_2\}'_*- \{l_1, l_2\}_* \in \ker\,P, *\in \{\dashv, \vdash\}$ for all $l_1, l_2 \in B\otimes D.$ Also, $\{l_1, l_2\}'_*- \{l_1, l_2\}_* =0, *\in
\{\dashv, \vdash\}$ if $l_i \in \ker\,E, i\in \{1,2\}.$ This determines a map $\psi: K[Y_2]\otimes D^{\otimes 2}= K[Y_2]\otimes ((B\otimes
D)/\ker\, E)^{\otimes 2}\rightarrow \ker\,P=D.$ Thus, $\psi$ defines a $2$-cochain. Also, given an arbitrary $\psi\in CY^2(D,D)$, there exists an appropriate $\{~,~\}_*'$ such that $\psi$ can be obtained as $\{~,~\}_*'- \{~,~\}_*,$ where $*\in \{\dashv, \vdash\}.$
We remark here that if $\overline{\phi}$, $\overline{\phi}' \in CY^3(D,D)$ are the cochains corresponding to $\{~,~\}_*, \{~,~\}_*'$ in the sense of the construction above, then $$\overline{\phi}'-\overline{\phi}= \delta \psi.$$
Let ${\mathbb O}_\lambda (f) \in HY^3(D,D)$ be the cohomology class of the cochain $\overline{\phi}.$ We define the following linear map. $${\mathbb O}_\lambda : H^2_{Harr}(A,K)\longrightarrow HY^3(D,D),
~~~~~~~~~f\mapsto {\mathbb O}_\lambda(f).$$
We thus make the following proposition.
The deformation $\lambda$ with base $A$ can be extended to a deformation of the dialgebra $D$ with base $B$ if and only if ${\mathbb O}_\lambda (f)=0.$
The cohomology class ${\mathbb O}_\lambda (f)$ is called the *obstruction* to the extension of the deformation $\lambda$ from $A$ to $B$.
Extendible Deformations
=======================
Let $A$ be a finite dimensional commutative, unital, local algebra with a fixed augmentation $\epsilon$, and let $\lambda$ be a deformation of a dialgebra $D$ with base $A$. Let $0\rightarrow K \stackrel{i}{\rightarrow}B \stackrel{p}{\rightarrow}A
\rightarrow 0$ be an extension of $A$, corresponding to a cohomology class $f\in H^2_{Harr}(A;K)$. Following the same arguments as in [@FF], we can state the following proposition.
$HY^2(D,D)$ operates transitively on the set of equivalence classes of deformations $\mu$ of the dialgebra $D$ with base $B$ such that $p_*\mu=\lambda.$
We remark here that the group of automorphisms of the extension $0\rightarrow
K\stackrel{i}{\rightarrow}B\stackrel{p}{\rightarrow}A\rightarrow 0$ is $H_{Harr}^1(A;K),$ (\[aut\]) and $H_{Harr}^1(A;K)=(\mathsf m/\mathsf m^2)'=TA,$ (\[tan\]). Note that by \[diff\], there exists a map $d\lambda: TA\rightarrow
HY^2(D,D).$ The group of automorphisms of the extension $0\rightarrow
K\stackrel{i}{\rightarrow}B\stackrel{p}{\rightarrow}A\rightarrow 0$ operates on the set of equivalence classes of deformations $\mu$ such that $p_*\mu=
\lambda.$
We have the next proposition, the proof of which is straightforward.
The operation of $HY^2(D,D)$ on the set of equivalence classes of deformations $\mu$ such that $p_*\mu=\lambda$ and the operation of the group of automorphisms of the extension $0\rightarrow
K\stackrel{i}{\rightarrow}B\stackrel{p}{\rightarrow}A\rightarrow 0$ are related by the differential $d\lambda: TA\rightarrow HY^2(D,D).$ In other words, if $r: B\rightarrow B$ determines an automorphism of the extension $0\rightarrow
K\stackrel{i}{\rightarrow}B\stackrel{p}{\rightarrow}A\rightarrow 0$ which corresponds to an element $h\in H_{Harr}^1(A;K)=TA,$ then for any deformation $\mu$ of $D$ with base $B$ such that $p_*\mu=\lambda,$ the difference between the push-out $r_*\mu$ and $\mu$ is a cocycle of the cohomology class $d\lambda(h).$
Suppose that the differential map $d\lambda: TA \longrightarrow HY^2(D,D)$ is onto. Then the group of automorphisms of the extension $0\rightarrow
K\stackrel{i}{\rightarrow}B\stackrel{p}{\rightarrow}A\rightarrow 0$ operates transitively on the set of equivalence classes of deformations $\mu$ of $D$ with base $B$ such that $p_*\mu=\lambda.$
The proof of the following proposition is an imitation of the proof presented in [@FMM], for Leibniz algebras.
Let $A_1$ and $A_2$ be two finite dimensional local algebras with augmentations $\epsilon_1$ and $\epsilon_2$, respectively. Let $\phi:
A_2 \longrightarrow A_1$ be an algebra homomorphism with $\phi(1)=1$ and $\epsilon_1 \circ \phi= \epsilon_2$. Suppose $\lambda_2$ is a deformation of a dialgebra $D$ with base $A_2$ and $\lambda_1= \phi_*
\lambda_2$ is the push-out via $\phi$. Then the following diagram commutes: $$\begin{array}{rcl}
H^2_{Harr}(A_1;K) & \stackrel{\phi^*}\longrightarrow & H^2_{Harr}(A_2;K)\\
& & \\
\theta_{\lambda_1}\searrow & & \swarrow\theta_{\lambda_2}\\
& & \\
& HY^3(D;D) &
\end{array} \ .$$
[**Proof.**]{} Let $[f_{A_1}] \in H^2_{Harr}(A_1; K)$ correspond to the extension $$0\longrightarrow K \stackrel{i_1}{\longrightarrow} A'_1 \stackrel{p_1}{\longrightarrow}A_1 \longrightarrow 0.$$ Also, let $[f_{A_2}]= \phi^* ([f_{A_1}]) \in H^2_{Harr}(A_2; K)$ correspond to the extension $$0\longrightarrow K \stackrel{i_2}{\longrightarrow} A'_2 \stackrel{p_2}{\longrightarrow}A_2 \longrightarrow 0.$$ Let $q_k: A_k \longrightarrow A_k'$ be sections of $p_k$ for $k=1,2$. There exist $K$-module isomorphisms $A_k'\cong A_k \oplus K$. Let $(b,x)_{q_k}$ denote the inverse of $(b,x) \in A_k \oplus K$ under the isomorphisms. Define a linear map $\psi: A_2' \cong (A_2\oplus K) \longrightarrow A_1'\cong (A_1 \oplus K)$ by $\psi((a,x)_{q_2})= (\phi(a), x)_{q_1}$ for $(a,x)_{q_2}\in A_2'$. Thus we have a morphism of extensions
$$\begin{array}{ccccccc}
0\longrightarrow & K & \stackrel{i_2}\longrightarrow & A_2'&\stackrel{p_2}\longrightarrow & A_2 & \longrightarrow 0\\
&&&&&&\\
&\downarrow \mbox{id}& & \downarrow \psi && \downarrow \phi &\\
&&&&&&\\
0\longrightarrow & K & \stackrel{i_1}\longrightarrow & A_1'&\stackrel{p_1}\longrightarrow & A_1 & \longrightarrow 0.
\end{array}$$
Let $I_k= i_k \otimes id$, $P_k= p_k\otimes id$ and $E_k=
\widehat{\epsilon}_k \otimes id$, where $\widehat{\epsilon}_k =
\epsilon_k \circ p_k$ for $k= 1,2$. If $m_{A_k}$ denote the unique maximal ideal of $A_k$ then $m_{A_k'}= p_k^{-1}(m_{A_k})$ is the unique maximal ideal of $A_k'$. Let the basis of $m_{A_k}$ and $m_{A_k'}$ be $\{m_{k_i}\}_{1 \leq i\leq r_k}$ and $\{n_{k_i}\}_{1 \leq i\leq r_k+1}$ respectively, for $k=1,2$. Note that, $n_{k_j}= (m_{k_j}, 0)_{q_k}$ for $1\leq j\leq r_k$ and $n_{k_{r_k+1}}= (0,1)_{q_k}$. The dialgebra products on $A_2\otimes D$ is given by $$\begin{array}{rcl}
(1\otimes x_1)\dashv_{\lambda_2} (1\otimes x_2)&=& 1\otimes (x_1 \dashv x_2) + \sum_{i=1}^{r_2} m_{2_i}\otimes \psi^2_i([21]; x_1, x_2)\\
(1\otimes x_1)\vdash_{\lambda_2} (1\otimes x_2)&=& 1\otimes (x_1 \vdash x_2) + \sum_{i=1}^{r_2} m_{2_i}\otimes \psi^2_i([12]; x_1, x_2)
\end{array}$$ for $x_1, x_2 \in D$ and $\psi^2_i = \alpha_{{\lambda_2}, \xi_{2_i}}$, where $\{\xi_{2_i}\}$ is the dual basis of $\{m_{2_i}\}$.
Let $\phi(m_{2_i})= \sum_{j=1}^{r_1}c_{i,j}m_{1_j}$, $c_{i,j} \in K$ for $1\leq i\leq r_2$ and $1 \leq j \leq r_1$. Then the push-out $\lambda_1= \phi_* \lambda_2$ on $A_1\otimes D$ is defined by $$\begin{array}{rcl}
(1\otimes x_1)\dashv_{\lambda_1} (1\otimes x_2)&=& 1\otimes (x_1 \dashv x_2) + \sum_{i=1}^{r_2}(\sum_{j=1}^{r_1} c_{i,j} m_{1_j})\otimes \psi^2_i([21]; x_1, x_2)\\
&=& 1\otimes (x_1 \dashv x_2) + \sum_{i=1}^{r_1} m_{1_j}\otimes \psi^1_j([21]; x_1, x_2)\\
(1\otimes x_1)\vdash_{\lambda_1} (1\otimes x_2)&=& 1\otimes (x_1 \vdash x_2) + \sum_{i=1}^{r_2}(\sum_{j=1}^{r_1} c_{i,j} m_{1_j})\otimes \psi^2_i([12]; x_1, x_2)\\
&=& 1\otimes (x_1 \vdash x_2) + \sum_{i=1}^{r_1} m_{1_j}\otimes \psi^1_j([12]; x_1, x_2)
\end{array}$$ where $\psi^1_j\in CY^2(D,D)$ id defined by $$\begin{array}{rcl}
\psi^1_j([21]; x_1, x_2)&=& \sum_{i=1}^{r_2} c_{i,j} \psi^2_i([21]; x_1, x_2)\\
\psi^1_j([12]; x_1, x_2)&=& \sum_{i=1}^{r_2} c_{i,j} \psi^2_i([12]; x_1, x_2)
\end{array}$$ for $x_1, x_2 \in D$.
For any $2$-cochain $\chi \in CY^2(D,D)$, let us define $A_k'$ bilinear operations $\{, \}_{\dashv, k}, \{, \}_{\vdash, k}: (A_k'\otimes D)^{\otimes 2}\longrightarrow A_k' \otimes D$ by lifting $\lambda_k$, $$\begin{array}{rcl}
(1\otimes x_1)\dashv_{k} (1\otimes x_2)&=& 1\otimes (x_1 \dashv x_2) + \sum_{j=1}^{r_k} n_{k_j}\otimes \psi^k_j([21]; x_1, x_2)\\
&+& n_{k_{r_k+1}} \chi([21]; x_1, x_2)\\
(1\otimes x_1)\vdash_{k} (1\otimes x_2)&=& 1\otimes (x_1 \vdash x_2) + \sum_{j=1}^{r_k} n_{k_j}\otimes \psi^k_j([12]; x_1, x_2)\\
&+& n_{k_{r_k+1}} \chi([12]; x_1, x_2)
\end{array}$$ for $k=1,2$ and $x_1, x_2 \in D$. The operations $\{, \}_{\dashv, k}, \{, \}_{\vdash, k}$, for $k=1,2$ satisfy the conditions (i) and (ii) of \[op\].
We shall show that $\psi \otimes id: A_2' \otimes D \longrightarrow A_1' \otimes D$ preserves the liftings. It is enough to show that $$(\psi \otimes id)(1\otimes x_1 *_2 1\otimes x_2)
= \psi \otimes id(1\otimes x_1) *_1 \psi \otimes id(1\otimes x_2),$$ where $* \in \{\dashv, \vdash\}$ and $x_1, x_2 \in D$.
Now $$\begin{aligned}
(\psi \otimes id)(&1\otimes x_1 \dashv_2 1\otimes x_2)
=\psi(1)\otimes (x_1 \dashv x_2) \\
&+ \sum_{j=1}^{r_2}\psi(1) \psi(n_{2_j})\otimes \psi_j^2([21]; x_1, x_2)\
+ \psi(1) \psi(n_{2_{r_2+1}}) \otimes \chi([21]; x_1, x_2)\\
=&1\otimes (x_1 \dashv x_2)+ \sum_{j=1}^{r_2}
\Big(\sum_{i=1}^{r_1} c_{j,i} m_{1_i}\Big) \otimes \psi_j^2([21]; x_1,
x_2) + n_{1_{r_1+1}}\otimes \chi([21]; x_1, x_1),\end{aligned}$$ where we used that $\phi(m_{2_j})= \sum_{i=1}^{r_1} c_{j,i} m_{1_i}$ and $$\psi(n_{2_{r_2+1}})= \psi((0,1)_{q_2})= (\phi(0), 1)_{q_1}
= n_{1_{r_1+1}}.$$ Simplifying, we conclude that $$\begin{aligned}
(\psi \otimes& id)(1\otimes x_1 \dashv_2 1\otimes x_2)\\
&=\psi(1) \otimes (x_1 \dashv x_2)+
\sum_{i=1}^{r_1} \psi(1) m_{1_i} \otimes \psi_i^1([21]; x_1, x_2)
+ \psi(1) n_{1_{r_1+1}}\otimes \chi([21]; x_1, x_2)\\
&=(\psi(1)\otimes x_1) \dashv_1 (\psi(1)\otimes x_2) \\
&=\psi \otimes id(1\otimes x_1) \dashv_1 \psi \otimes id(1\otimes x_2).\end{aligned}$$
Similarly, we can show that $$(\psi \otimes id)(1\otimes x_1 \vdash_2 1\otimes x_2)= \psi
\otimes id(1\otimes x_1) \vdash_1 \psi \otimes id(1\otimes x_2).$$ Let $\phi_k$ be defined by the operations $\{, \}_{\dashv, k}, \{,
\}_{\vdash, k}$ as has been defined in \[phi\] and $\overline{\phi_k}$ the corresponding cocycle as in \[phibar\]. Since, $\psi(n_{2_{(r_2+1)}})= n_{1_{(r_1+1)}}$, we have $[\overline{\phi_2}]= [\overline{\phi_1}]$. Therefore, $$\theta_{\lambda_1}([f_{A_1}])=[\overline{\phi_1}] =
[\overline{\phi_2}]= \theta_{\lambda_2}([f_{A_2}])=
\theta_{\lambda_2}\circ \phi^*([f_{A_1}]).$$ Hence, $\theta_{\lambda_1}=\theta_{\lambda_2}\circ \phi^*$.
Construction of a Miniversal Deformation of a Dialgebra
=======================================================
An explicit description of the construction of a versal deformation of a Lie algebra is given in [@FF], and of a Leibniz algebra is given in [@FMM]. Here we sketch the construction, for the case of a dialgebra, following the same techniques developed in [@FF], [@FMM].
Start with a dialgebra $D$ with $\mbox{dim}(HY^2(D,D)) < \infty$. Consider the extension $$0\longrightarrow HY^2(D,D)' \stackrel{i}{\longrightarrow} C_1 \stackrel{p}
{\longrightarrow} C_0 \longrightarrow 0,$$ where $C_0=K$, $C_1= K\oplus HY^2(D,D)'$. Let $\eta_1$ denote the universal infinitesimal deformation with base $C_1$ as described in Section 4.
Suppose for some $k\geq 1$, we have constructed a finite dimensional local algebra $C_k$, and a deformation $\eta_k$ of $D$ with base $C_k$. Let $$\mu : H^2_{\text{\it Harr}}(C_k; K) \longrightarrow (Ch_2(C_k))'$$ be a homomorphism mapping a cohomology class into a cocycle representing the class. The dual map of $\mu$ $$f_{C_k}: Ch_2(C_k)\longrightarrow H^2_{\text{\it Harr}}(C_k; K)'$$ corresponds to the following extension of $C_k$: $$0\longrightarrow H^2_{\text{\it Harr}}(C_k; K)'
\stackrel{\overline{i}_{k+1}}{\longrightarrow} \overline{C}_{k+1}
\stackrel{\overline{p}_{k+1}}{\longrightarrow} C_k \longrightarrow 0.$$ The obstruction $\theta([f_{C_k}])\in H^2_{\text{\it Harr}}(C_k;
K)'\otimes HY^3(D,D)$ yields a map $\omega_k: H^2_{\text{\it Harr}}(C_k;
K)\longrightarrow HY^3(D,D)$, by adjunction property of tensor products, with the dual map $$\omega_k': HY^3(D,D)' \longrightarrow H^2_{\text{\it Harr}}(C_k; K)'.$$ This induces the following extension $$0\longrightarrow \text{coker}(\omega_k')\longrightarrow \overline{C}_{k+1}/\overline{i}_{k+1}\circ \omega_k'(HY^3(D,D)')\longrightarrow C_k \longrightarrow 0.$$ This yields the extension $$0\longrightarrow (\ker(\omega_k))' \stackrel{i_{k+1}}{\longrightarrow}C_{k+1}\stackrel{p_{k+1}}{\longrightarrow}C_k \longrightarrow 0$$ where $C_{k+1}= \overline{C}_{k+1}/ \overline{i}_{k+1}\circ \omega_k'(HY^3(D,D)')$ and $i_{k+1}, p_{k+1}$ are the mappings induced by $\overline{i}_{k+1}$ and $\overline{p}_{k+1}$ respectively. Along the same lines as in [@FF], [@FMM] we have the following proposition:
The deformation $\eta_k$ with base $C_k$ of a dialgebra $D$ admits an extension to a deformation with base $C_{k+1}$ which is unique up to an isomorphism and an automorphism of the extension $$0\longrightarrow (\ker(\omega_k))'\stackrel{i_{k+1}}{\longrightarrow}C_{k+1}\stackrel{p_{k+1}}{\longrightarrow}C_k \longrightarrow 0.$$
This process gives rise to a sequence of finite dimensional local algebras $C_k$ and deformations $\eta_k$ of the dialgebra $D$ with base $C_k$ $$K \stackrel{p_1}{\longleftarrow} C_1 \stackrel{p_2}{\longleftarrow} C_2 \stackrel{p_3}{\longleftarrow} \ldots \stackrel{p_k}{\longleftarrow} C_k \stackrel{p_{k+1}}{\longleftarrow} C_{k+1} \ldots$$ such that $p_{k+1}*\eta_{k+1}= \eta_k$. By taking the projective limit we obtain a formal deformation $\eta$ of $D$ with base $C=\overleftarrow{\displaystyle{\lim_{k\to\infty}}} C_k$.
Let $\dim\,(HY^2(D,D))=n$ and $K[[HY^2(D,D)']]$ denote the formal power series ring in $n$ variables. Also let $m$ denote the unique maximal ideal in $K[[HY^2(D,D)']]$, consisting of all elements with constant term zero. We have the following proposition, whose proof can be found in [@FMM].
The complete local algebra $C= \overleftarrow{\displaystyle{\lim_{k\to\infty}}} C_k$ can be described as $$C\cong K[[HY^2(D,D)']]/I,$$ where $I$ is an ideal contained in $m^2$.
Along the same lines as in [@FF], [@FMM], we state the following theorem, proof of which obeys the same techniques as developed in [@FF].
Let $D$ be a dialgebra with $\dim(HY^2(D,D))< \infty$. Then the formal deformation $\eta$ with base $C$ as described above is a miniversal deformation of $D$.
[Acknowledgements]{}
We would like to thank Professor Goutam Mukherjee for his interest in our work and the Indian Statistical Institute, Calcutta, for hospitality.
[99]{} Felipe, R. [*An analogue to Functional Analysis in Dialgebras*]{}, Int. Math. Forum 2 (22) (2007) 1069-1091.
Fialowski, A. [*An example of formal deformations of Lie algebras*]{}, NATO Conference on deformation theory of algebras and applications, 1986, Proceedings, Kluwer, Dordrecht (1988), 375-401.
Fialowski, A. and Fuchs, D [*Construction of Miniversal Deformations of Lie algebras*]{}, J. Funct. Anal. 161 (1999) 76-110.
Fialowski, A., Mandal, A., Mukherjee, G., [*Versal Deformations of Leibniz Algebras*]{} (to appear in J. K-Theory). ArXiv: math/QA0702476.
Frabetti, A. [*Dialgebra (co)homology with coefficients*]{}, in: J. -M. Morel, F. Takens, B. Teissier, eds., Dialgebras and Related Operads, Lecture Notes in Mathematics, Vol 1763, Springer, Berlin, 2001, pp. 67-103. A. Frabetti, [*Dialgebra homology of associative algebras*]{}, C. R. Acad. Sci. Paris 325 (1997) 135-140
Gerstenhaber, M. [*The Cohomology Structure of an Associative Ring*]{}, Ann. of Math. 78 (1963) 267-288. Ginzburg, V. and Kapranov, M. M. [*Koszul duality for operads*]{}, Duke Math Journal, 76(1) (1994) 203-272. Harrison, D.K., [*Commutative algebras and cohomology*]{}, Trans. Amer. Math. Soc. 104 (1962), 191-204. (1958), 450-459.
Loday, J.-L. [*Dialgebras*]{} in: J. -M. Morel, F. Takens, B. Teissier, eds., Dialgebras and Related Operads, Lecture Notes in Mathematics, Vol 1763, Springer, Berlin, 2001, pp. 7-66.
269-293. Majumdar, A. and Mukherjee, G. [*Deformation theory of dialgebras*]{}, K-theory 506 (2002), 1-28. Majumdar, A. and Mukherjee, G. [*Dialgebra cohomology as a G-algebra*]{}, Tran. Amer. Math. Soc. 356(6) (2004), 2443-2457. Schlessinger, M., [*Functors of Artin rings*]{}, Trans. Amer. Math. Soc. 130 (1968), 208-222. +-
[^1]: AMS Mathematics Subject Classification : 54H20, 57S25.
[^2]: The first author was supported by grants OTKA T043641 and T043034.
[^3]: The second author is supported by NBHM Post-doctoral fellowship.
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{
"pile_set_name": "ArXiv"
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---
author:
- 'F. Bournaud and P.-A. Duc'
date: 'Received 22 March 2006 / Accepted 10 May 2006'
title: From Tidal Dwarf Galaxies to Satellite Galaxies
---
Introduction
============
The properties of dwarf satellite galaxies that surround spirals and ellipticals are often used to constrain cosmological models. Their number has been actively debated since cosmological simulations have shown that the low-mass dark halos are much more numerous around their massive hosts than the dwarf galaxies identified around the Milky Way [e.g. @klypin99; @moore99]. Moreover, the spatial distribution of satellites can provide information on the mass and shape of dark haloes [@zaritsky; @brainerd05; @LK06]. Finally, they play, as building blocks, a major role on the physical evolution of massive galaxies with which they will eventually merge. It is generally assumed in such studies that dwarf galaxies have a cosmological primordial origin and were not formed recently. However, there is growing evidence that objects with masses typical of dwarf galaxies form in the tidal tails that surround colliding and merging spiral galaxies [e.g. @DM94; @DM98; @hibbard01; @mendes; @temporin03; @knierman]. Many of these so-called Tidal Dwarf Galaxies (TDGs) appear to be self-gravitating objects [@braine; @bournaud04]. If these tidal objects are long-lived, they could contribute to the total population of dwarf satellites in addition to primordial dwarfs. Their statistical properties would then be modified and the possible constraints on cosmological models would have to be updated.
Dwarf galaxies presently observed to form in tidal tails could however be short-lived. Indeed dynamical friction may cause a rapid orbital decay. @hibbard95 have shown that a large part of the once expelled tidal material falls back onto the parent spiral galaxies in a few hundreds of Myr, but the time-scale increases to Gyr in the outer regions; thus the tidal dwarfs formed there may have a significant lifetime. Also, dwarf galaxies may be disrupted by the tidal field of their progenitor even if the internal orbits of their stars are not resonant with their orbital period [e.g. @fleck]; this disruption process can take a few billion years depending on their mass, orbit, and concentration. Thus, the lifetime of tidal dwarfs is a priori rather uncertain and their cosmological importance far from being proven.
Actually, no real old Tidal Dwarf galaxies, still surviving after the merger of their progenitors and the vanishing of the umbilical cord linking them to their parents, has yet been unambiguously found although candidates were discussed in the literature [@hunter; @duc04iau and references therein]. Because of the difficulties to identify a tidal origin in evolved galaxies, numerical simulations appear to be unavoidable to predict their number. Sofar, numerical modeling has mainly been used to study the formation of objects in tidal tails. The gravitational collapse of pre-existing clouds or regions of tidal tails [@elmegreen; @BH92] can lead to the formation of clumps along tidal tails with typical masses of $10^{6-8}$ M$_{\sun}$. According to @wetz05 and @wetz06, a massive gaseous component is required to achieve this collapse. More massive accumulations of matter can form in the outer regions of tidal tails when dark halos are assumed to extend much further than the optical radius of galaxies [@bournaud03]. Their formation is initiated by a kinematical mechanism, and self-gravity can make these accumulations of matter collapse into dense star-forming objects [@duc04 hereafter D04]. Simulations assuming massive and extended gaseous disks in the colliding spirals produce objects that seem to become satellites galaxies [e.g. @cox D04] but the conditions for their survival have not yet been studied in detail. The fate of classical infalling satellites has been studied in numerous simulations [e.g. @mayer1; @mayer2; @mayer3]. In such models, the dwarfs were assumed to contain a somehow “shielding” layer of dark matter, while tidal dwarfs do not in a CDM scenario [@BH92]. @kroupa97 studied the survival of tidal-like dark-matter free dwarf satellites but started his simulations once the galaxies had already been formed and hence could not address the role of their initial distribution on their evolution. He argued that the objects produced in his model – with final masses of only $10^5$ M$_{\sun}$ – become long-lived after having lost up to 99% of their initial mass. These results cannot be easily extended to more massive objects since the latter undergo more dynamical friction and may have a different morphology. Hence, self-consistent simulations of the formation and subsequent evolution/survival of TDGs are still required to estimate the contribution of such objects to the population of dwarf satellite galaxies.
In this paper, we study the long-term survival of massive dwarf-like objects formed in tidal tails. A set of 96 numerical simulations of galactic encounters were carried out, varying the geometrical parameters of the collision and the relative masses of the colliding galaxies. They are used to predict the number and distribution of long-lived – with life expectancies greater than one billion year – tidal objects that will eventually look-like classical dwarf satellites. The numerical techniques and parameters are described in Sect 2. In Sect. 3, we detail our results on the formation criteria and survival time of the tidal objects; we then determine their characteristics once they have become long-lived satellite galaxies. In Sect. 4, we compare them with the properties of respectively the currently forming TDG candidates observed in young interacting systems and the satellites around massive hosts such as those identified in the Sloan Digital Sky Survey and in the Local Group. In particular we discuss rough calculations on the expected fraction of dwarfs of tidal origin among all classical ones. Our conclusions are summarized in Sect. 5.
Numerical simulations
=====================
Numerical techniques and parameters
-----------------------------------
The simulations presented in this paper use the same particle-mesh FFT code as in D04. In particuler, a sticky-particles scheme is used to model the ISM[^1] and star formation is described by a local Schmidt Law. Initial conditions for disk galaxies and dark haloes are largely similar to those described in D04. Galaxies are assumed to contain 15% of gas (gas-to-visible mass ratio, including gas outside the stellar disk). Each disk galaxy is embedded in an extended halo that maintains a flat rotation curve up to ten times the stellar disk radius, consistent with cosmological predictions.
We use a 512$^3$ cartesian grid to compute the gravitational potential, with a gravitational softening of 380 pc. The most massive galaxy is modeled by 10$^6$ particles for each component (stars, gas, dark matter). Its stellar mass is $2 \times 10^{11}$ M$_{\sun}$ and it contains 10% of gas. Its stellar disk radius is 15 kpc and its gaseous disk radius 45 kpc – HI is indeed observed much beyond the optical radius in spiral galaxies [e.g. @RH94]. The dark halo is modeled by a pseudo-isothermal sphere, with an asymptotic velocity of 205 km s$^{-1}$ and a core radius of 10 kpc. The circular velocity at the dark halo virial radius (280 kpc) is 220 km s$^{-1}$. All the timescales discussed in this paper are hence realistic for galaxies similar to the Milky-Way and must be rescaled accordingly for other galactic mass/virial velocities. The number of particles in the companion depends on its mass; its size is scaled by the square root of its mass to keep the stellar disk surface density constant.
Ninety-six galaxy mergers with various mass ratios and various orbital parameters have been simulated. The detailed parameters for each run are given in Appendix \[appen\]. The range of values explored for each parameter is summarized in Table \[params\]. In order to increase the statistical size of our sample of TDG candidates, we have privileged cases favorable to the formation of long-lived tidal dwarfs. For instance, because retrograde orbits are found not to form long-lived TDGs, we have mainly simulated prograde cases. Of course, we take into account this bias in our statistical study.
The simulations have been run until $2$ Gyr after the first pericenter of the relative orbit of the two galaxies. The ”merger” typically occurs 50–300 Myr after this pericenter. We were actually able to follow the evolution of the objects formed in tidal tails for more than 1.5 Gyr after their formation which is enough to know whether they have become long-lived dwarf satellites, i.e. objects orbiting around their massive host for a few dynamical times.
Parameter simulated range range most favorable to long-lived TDG formation
------------------- ----------------------- --------------------------------------------------
Relative velocity 50 to 320 km s$^{-1}$ 50 to 250 km s$^{-1}$
Impact parameter 15 to 200 kpc 30 to 200 kpc
Orbit inclination 0 to 60 degrees 0 to 40 degrees
Orbit orientation Prograde/Retrograde Prograde only
Mass ratio 10:1 to 1:10 4:1 to 1:8
Detection of massive substructures in tidal tails {#crit}
-------------------------------------------------
We analyzed the spatial distribution of matter at different epochs of the simulations. We identified the massive substructures along the tidal tails that may become long-lived “Tidal Dwarf Galaxies”. The tidal objects were sought in the gas component plus the young stars formed in situ after the periaster. Old stars from the parent’s disks were not considered as they seem to play a minor role in the formation of structures in tidal tails, at least in the most recent scenarii put forward: the kinematical pilling-up of gas at the tip of a tail (D04) or the dynamical formation of gravitational clumps in massive gaseous tidal tails [@wetz06].
The criterion for an ”object” to be detected is that the mass of gas and young stars within a diameter $d$ must be larger than a threshold $M_0$. Two combinations of mass threshold / maximal diameter have been used : $$M_0 = 3\times 10^8 \mathrm{\;within\;}d<6 \mathrm{\;kpc}$$ or $$M_0 = 1\times 10^8 \mathrm{\;within\;}d<3 \mathrm{\;kpc}$$ We do not use the combination $$M_0 = 1\times 10^8 \mathrm{\;within\;}d<6 \mathrm{\;kpc}$$ since it was sometimes found to include parts of tidal tails that visually do not correspond to any substructure. Objects of $10^8$ M$_{\sun}$ contain $2 \times 10^3$ to $10^4$ particles, depending of their gas fraction, so that objects above our mass threshold contain enough particles to be well resolved. The identification of objects was made around pixels with a density larger than $\mu_0=M_g/(\pi d^2/4)$, and the center of the spherical boundary (the 6 or 3 kpc diameter) was computed to maximize the included mass. At this stage, one object may in fact correspond to two different condensations of matter. We decided that one object must be separated in two if two components more massive than the mass threshold and separated by a density $\mu < \mu_0$ were found. Note that we have rarely observed objects that were first considered as double to merge into a single one, so that the somehow arbitrary threshold used to separate them was realistic.
The mass center of an object, used to determine its ”position”, was defined as the mass center of pixels with a density larger than $\mu_0$ within the object external radius. We only considered objects located outside 1.5 times the initial stellar disk radius of each parent spiral galaxy, in order not to include spiral arms or other features belonging to the outer disk. Objects located at smaller radii would anyway not survive for a few $10^8$ yr before merging and do not deserve being considered as potential TDGs. In all the paper, $t=0$ corresponds to the first pericenter between the two parent spiral galaxies, not to the beginning of the simulations. The criterion for the detection of massive substructures in tidal tails was first applied at $t=200$ Myr. Later on, every 100 Myr up to $t=2$ Gyr, each simulation was re-analyzed to know whether detected objects were still present (i.e., they were still obeying our mass/size and radial distance criterion) and identify new ones that were not formed at earlier times.
An example of the detection of sub-structures in a simulated merger, using our criteria, is shown in Fig. \[ex\_substructure\].
{width="14cm"}
Our detection algorithm based on a density threshold might be considered rather basic compared to more sophisticated algorithms, like the friend-to-friend one. Another approach could also have been to select only the gravitationally bound structures that were likely to become long-lived dynamically independent objects. However, the goal of our simulations is mainly to make relevant comparison with observations. Projection effects and a limited resolution make the identification of bound sub-structures in real systems difficult (see @bournaud04 and @HB04). Most of the so-called TDG candidates described in the literature were actually identified based on their shape rather than on kinematical data. Because we wished to treat our simulations as would be real images, we decided to use a criterion solely based on the apparent morphology. We thus first included some unbound objects, like observations do, but following their evolution, we could a posteriori reject them. Beside, this allowed us to study the process of tidal disruption and the falling back of tidal material.
Results: statistics on the simulated merging systems
====================================================
Number, lifetime and masses of tidal sub-structures
---------------------------------------------------
Over our sample of 96 simulations (184 parent spiral galaxies), we identified 593 massive substructures in tidal tails, i.e. an average of 3.2 tidal substructure per parent galaxy. 423 of these structures were first detected at $t=200$, and most the other ones at $t=300$ or $t=400$.
At $t=500$ Myr, 207 objects – about one third of all the detected structures – still exist. Conversely, the two other thirds have disappeared in less than 300 Myr. At $t=1$ Gyr, 143 objects are still present, and 119 at $t=2$ Gyr (20% of the total). 63% of the vanishing objects have fallen back onto their parent galaxies; the remaining objects ”disappeared” because their mass dropped below our $10^8$ M$_{\sun}$ threshold. This can be due to tidal disruption or to the fact that they were not gravitationally bound.
The distribution of their radius of formation (i.e. the radius at which they are first detected) along the tidal tails is shown on Fig. \[dist\_rad\]. It is roughly bimodal, with two thirds of the objects formed within the first half of the radial extent of the tail, while the other third is formed near the tip of the tails. A close analysis of the simulations (e.g., Fig. \[ex\_substructure\]) shows that the first kind of objects correspond to loose unbound structures and to gravitational clumps formed all along the tails (such as the ones described by @BH92 or @wetz06), while the second kind of objects correspond to massive accumulations of matter formed near the tip of tails according to the mechanism described by D04.
Each type of tidal objects appear to have different lifetimes:
- self-gravitating accumulations of material formed near the tip of tidal tails generally survive for several $10^8$ yr or a few Gyr
- other objects formed at smaller radii tend to disappear in less than 1 Gyr, and even often in only 500 Myr.
Both categories of objects also show differences in their respective masses, as shown in Fig. \[histo\_mass\]. Most structures detected at $t=200$ have a mass smaller than $10^9$ M$_{\sun}$; but those surviving at least $t=1$ Gyr have the highest masses, with 45% of them exceeding $10^9$ solar masses.
Therefore, according to our simulations, only the objects formed at the tip of tidal tails can become long-lived, massive, dwarf satellite galaxies. The less massive objects that condensed at smaller radii along the tails tend to fall back rapidly on their parent galaxies or rapidly lose most of their mass, probably contributing to the population of super stellar clusters or even globular clusters observed around merger remnants. Our simulations do not have the resolution to study them in details.
![Number of objects as a function of their [*formation*]{} radius, defined as the radius at which they are first detected, normalized by the total radial extent of the tidal tail – the tidal tail extent is defined to contain 98% of the mass that has been pulled out 1.5 times the initial gaseous disk radius. The results are shown for the objects existing at $t=200$ (green dotted), 500 (red dashed), and 2000 Myr (blue solid). The longest-lived objects are those formed near the tip of tidal tails, many of which still exist after 2 Gyr. On the other hand, gravitational clumps and other tidal debris formed at smaller radii rarely survive more than 500 Myr. []{data-label="dist_rad"}](fig2.eps){width="8cm"}
![Mass spectrum of the objects detected in tidal tails at $t=200$ Myr (green dashed) and $t=1$ Gyr (red solid). The most massive objects that have the longest life are formed near the tip of tidal tails.[]{data-label="histo_mass"}](fig3.eps){width="8cm"}
.
Constraints for the formation of long-lived tidal objects
---------------------------------------------------------
We now focus on the ”long-lived” objects able to survive at least 1 Gyr – those preferentially formed in the outermost regions of interacting systems. The table in Appendix \[appen\] indicates whether such objects are formed for each individual run and each parent galaxy. Table \[params\] summarizes the constraints obtained for the individual parameters of the collision (eventhough they are not fully independent). The three main restrictions can be summarized as follow: a given spiral galaxy can form long-lived tidal objects provided that:
- the orbit is [*prograde*]{}: the angle $\theta$ between the disturbed galactic disk spin and the companion orbital spin (see Figure 8 in @duc2000), is smaller than 90 degrees.
- its orbital plane is inclined by [*less than 40 degrees*]{}
- the mass of the companion is at least one fourth of that of the target galaxy (otherwise the tidal field cannot drive matter far enough from the parent galaxy) and less than 8 times the target galaxy mass (otherwise tidal objects may form but rapidly fall onto the very massive companion). In other words, for [*mass ratios in the*]{} 1:1–8:1 [*range*]{}, at least one of the two spiral galaxies can form long-lived objects. For mass ratios in the 1:1–4:1 range, long-lived objects can form in the material of both galaxies.
When these three conditions are achieved, most mergers will lead to the formation of long-lived tidal substructures, except those with small impact parameters (typically below 30 kpc), but the latter condition is a minor constraint since the small impact parameters are statistically far less frequent.
When a spiral galaxy merges with a companion inside the parameter range described above, according to the results given in Table \[appen\], on the average 1.9 objects are still visible at $t=500$, and 1.3 survive at least 1 Gyr.
Statistical properties of dwarf galaxy satellites of tidal origin
-----------------------------------------------------------------
### Spatial 3D distribution
![Distribution of the long-lived TDGs (surviving at least 1 Gyr) in the radius–inclination plane. The radius is measured from the progenitor galaxy center; the inclination is the latitude in the progenitor disk frame. Data are accumulated from 500 to 2000 Myr. The orbital tracks of two objects are shown. They have the same initial radius, but one (shown in red dashed) was formed on a more eccentric orbit attested by a larger radial excursion; it undergoes a larger radial decay.[]{data-label="points_r_i"}](fig4.eps){width="8cm"}
![Latitudinal distribution of long-lived TDGs around their progenitor galaxies (red), compared to an isotropic distribution (green dashed). TDGs are preferentially distributed towards the equatorial plane of their progenitor disk. This distribution has been computed with data accumulated from $t=500$ to $2000$ Myr, for TDGs that survive at least 1 Gyr. The anisotropy of the distribution increases with radius, as detailed on Fig.\[histo\_i\_4\].[]{data-label="histo_i"}](fig5.eps){width="8cm"}
As shown before, the long-lived tidal objects will behave as satellite galaxies, orbiting around their parent galaxies. We analyze in the following their spatial distribution, as derived from our simulations. To increase the size of the statistical sample, we have cumulated the data on the positions from $t=500$ to $2000$ Myr for objects that survive at least 1 Gyr. The number of ”detections” is then 1587, corresponding to 143 individual objects tracked at 6 to 16 epochs of their evolution depending of their lifetime. Cumulating data not only enables to increase the statistical size of the sample, but also to make the comparison with samples of real colliding galaxies that are observed at different times of their evolution. Of course, in the case of the simulations, not all the data points are statistically fully independent.
We measure at each time the radius and inclination with respect to the parent galaxy disk. This inclination $i$ is defined as the angle between (i) the parent galaxy disk, or, after the merger, the main flattening plane of the parent galaxy material[^2], and (ii) the line linking the parent galaxy mass center to the tidal dwarf mass center (Fig. \[schema\]). We show on Fig. \[points\_r\_i\] their distribution in the (radius, inclination) plane. The inclination distribution is shown on Fig. \[histo\_i\] for all the objects, and on Fig. \[histo\_i\_4\] for the objects separated into four quartiles depending on their initial radius (0–82, 82–121, 121–157 and $>$157 kpc).
The distribution of the long-lived tidal objects around their parent galaxies (or the associated merger remnants) is strongly anisotropic. The fraction of objects found at galactic latitudes larger than 60 degrees is 4 times smaller than what is expected for an isotropic distribution (Fig. \[histo\_i\]). Tidal dwarfs are mostly distributed in the equatorial plane of their progenitor disk. As mentioned earlier, they are preferentially formed when the orbital plane and the parent spiral galaxy plane are inclined by less than 40 degrees. In such cases, the tidal tails and the sub-structures formed within them are close to the disk plane. In a few rare cases, interactions with the progenitors and, potentially, with other tidal objects, drive some objects far from the parent disk plane.
The anisotropy of the distribution increases with radius, as seen on Fig. \[histo\_i\_4\]. Indeed, tidal tails are more extended in the close-to-coplanar cases and the objects forming at their tip are located at larger distances (Fig. \[n\_r\_i\]). Finally, the most prominent tidal substructures have a more anisotropic distribution. We measure an average value of the inclination $<i>=31$ degrees for objects less massive than $10^9$ M$_ {\sun}$, and $<i>=22$ for the more massive ones. This, again, is explained by the fact that the most massive objects are formed on low orbital inclinations (i.e. nearly-coplanar galactic encounters).
{width="16cm"}
{width="12cm"}
![Radial distribution of the long-lived objects, depending on the inclination of the companion orbital plane with respect to the progenitor disk plane: orbital inclinations smaller than 27 degrees (blue solid) and larger than 27 degrees (green dashed). Low-inclination orbits form the longest tidal tails and statistically produce TDGs at larger radii than more inclined orbits.[]{data-label="n_r_i"}](fig8.eps){width="8cm"}
### Orbit eccentricities
We determined the eccentricity of the orbits of the tidal satellites and followed its variation as a function of time. The ”eccentricity” of the orbit of a given object at a given epoch was defined as follow: the orbit that this object would have if the potential of the merging system did not evolve is computed. A maximal and a minimal radii, respectively $r_+$ and $r_-$ measured from the mass center of the merging/merged spirals, are found. The eccentricity $e$ is then derived assuming that the orbit is in first order elliptical: $e=(r_+ - r_-)/(r_+ + r_-)$. We show in Fig. \[eccent\] the distribution of $e$ for the long-lived objects when they are formed, i.e. at their first detection (generally at $t=200$ Myr). Nearly 90% of them were born on orbits with an axis ratio larger than 0.6; very eccentric orbits are rare. Objects with high eccentricities have a smaller pericenter radius, and are then exposed to a faster and more disruptive tidal field from the central galaxy; they also undergo a higher dynamical friction, causing them to quickly fall back onto their progenitor. At 2 Gyr, they have disappeared. The short-lived tidal structures, also presented in Fig. \[eccent\], had, without surprise, high eccentricities.
We illustrate this point in Fig. \[points\_r\_i\] where we show the orbital track of two objects, initially formed at similar radii, in similar colliding systems (2:1 mergers, run 5 and 6), but with different orbital eccentricities. The most eccentric tidal dwarf is found to undergo a stronger orbital decay, under the effects of dynamical friction. It also loses 18% of its mass from $t=0.5$ to 2 Gyr, while the less eccentric dwarf loses only 8% of its mass during the same period.
![Statistical distribution of the initial eccentricity (at $t=200$ Myr) of the objects able to survive at least 1 Gyr which became satellites (red dash-dotted histogram), and final eccentricity for those surviving at least 2 Gyr (green histogram). The dashed blue histogram shows the initial eccentricity of objects surviving less than 1 Gyr, renormalized to the same number of objects than for the red histogram.[]{data-label="eccent"}](fig9.eps){width="8cm"}
### Relative velocities
Since they move on low eccentricity orbits, the tidal satellites generally have a velocity close to the circular velocity of the parent gravitational system (dominated by the merged dark halo). The resulting distribution of the velocities at $t=1$ Gyr is shown on Fig. \[velo\]. It ranges between 50 and 400 km s$^{-1}$ (for a massive progenitor with a virial velocity of 220 km s$^{-1}$). Assuming that systems are observed under random orientation, we computed the distribution of the projected line-of-sight velocities (see Fig. \[velo\]). The average absolute value is $< |V_{\mathrm{LOS}}| > =115$ km s$^{-1}$ with respect to the mass center of the central merger remnant. In the case of young systems, a few $10^8$ yr after the merger, tidal tails are more easily seen edge-on [e.g. @HB04] so that the real line-of-sight distribution is not isotropic. This can statistically decrease the projection angle, which in turns increases the observed line-of-sight velocities. The observed distribution around young systems should then fall somewhere in between the two histograms shown on Fig. \[velo\]. Still, typical observed velocities will rarely exceed the order of 200-250 km s$^{-1}$.
![Velocity distribution of TDGs with respect to the mass center of the merging pair of massive galaxies: real (3-D) velocity (green dashed) and projected along the line-of-sight, in absolute value (blue solid). Data accumulated from $t=500$ to 2000 Myr, for TDGs surviving at least 1 Gyr.[]{data-label="velo"}](fig10.eps){width="8cm"}
### Dark matter content {#DMfrac}
We have measured the amount of dark matter inside the visible radius[^3] for tidal objects that survive more than 1 Gyr. We find that inside this radius, the dark-to-visible mass ratio is on the average 0.09. It is never more than 0.15, with the highest dark matter fractions generally found in the most massive TDGs. For comparison, massive spiral progenitors in our initial conditions have a dark-to-visible mass ratio of 0.65 inside their visible radius.
Hence, tidal dwarf galaxies are strongly deficient in dark matter, according to our simulations. @BH92 found a similar result; our tidal dwarfs contain slightly more dark matter because we use more extended dark halos around the massive progenitors, so that more dark matter is found in the vicinity of forming TDGs.
Discussion: comparison to observed TDGs and satellite galaxies {#obs}
==============================================================
Identifying young forming Tidal Dwarf Galaxies in colliding galaxies {#obs1}
--------------------------------------------------------------------
A large variety of star-forming entities have been reported in the tidal tails of real interacting systems: Super Star Cluster (SSCs), Giant HII Complexes (GHCs) [@gallagher; @knierman; @weilbacher1; @weilbacher2; @saviane; @LS04; @IP01; @sakai] and the more massive so-called “Tidal Dwarf Galaxies candidates” which, in fact, also correspond to a diversity of objects. However, several observed interacting systems are distinguished by the massive accumulations of matter, up to a few $10^9$ M$_{\sun}$, found at the tip of their tidal tails [e.g. @DM94; @DM98; @HB04]. In principle, they could be fake condensations caused by the projection of material along the line-of-sight in tidal tails seen edge-on. However their molecular gas content [@braine] and their large-scale kinematics provide evidences that they are real massive objects, while their internal kinematics, when resolved, indicates that these structures are gravitationally bound [@bournaud04]. Most of these tidal objects are observed at a time when they are probably still forming, a few hundred Myr after the beginning of the collision. Without any detailed observations and the help of evolutionary models, it is difficult to predict how they will evolve, and in particular whether they will become independent, stable, objects that would deserve to be called Tidal Dwarf Galaxies. Our large set of simulations show that among the objects formed at the base and the middle of the tidal tails, many are on rather high eccentric orbits which hampers considerably their ability to survive. Actually only a small fraction of them are still present after 500 Myr. About 75 percent of the objects more massive than $10^8$ M$_{\sun}$ – our mass threshold – survive less than 800 Myr. Such objects may be considered as TDG-candidates in on-going mergers, but will most likely be short-lived and not become dwarf galaxy satellites orbiting around the merger remnant for at least a few dynamical times. @kroupa97 studied the evolution of purely stellar low-mass satellite galaxies without dark matter (as expected for TDGs) in the potential well of the Galactic dark halo. He found that satellites on high eccentric orbits lose 10–20 % of their mass during each perigalactic passage. They nevertheless manage to survive reaching a stationary regime but their remnants contain then only 1 % of their initial mass, about $10^5$ M$_{\sun}$ in the simulations of @kroupa97. To have final masses typical of dwarf spheroidals – the less massive known galaxies –, they then should initially have had stellar masses as high as $10^9 - 10^{10} $ M$_{\sun}$, which can correspond only to the most massive objects, formed at the tip of tidal tails in our simulations. According to our simulations, the only objects that actually become genuine dwarf galaxies are those formed at large radii, and preferentially near the extremity of the tails (Fig. \[dist\_rad\]) via the mechanism studied by D04 which works for extended dark matter haloes. We then conclude that among the many structures observed in the tidal tails of young interacting systems, only the most massive ones that are formed near the tip of tails are likely to be the progenitors of Tidal Dwarf Galaxies, following the definition given e.g. by D04 or @HB04 that requires them to be long-lived.
One should however keep in mind the restrictions of these simulations. The produced TDGs were found to be long-lived essentially because they were orbiting on safe orbits (at large radii and low-eccentricities) and were not affected by a strong tidal field that could otherwise disrupt them. However we have not considered here the internal processes that could cause severe disturbances. Intensive feedback following an internal starburst may remove much gas from the TDGs [@dekel03]. Massive objects may also fragment into smaller ones in higher resolution models. We will present in a separate paper high resolution simulations zooming in onto TDGs that allowed us to conclude that, even taking into account their internal evolution, they are still long-lived objects that conserve most of their initial mass.
Identifying Tidal Dwarf Galaxies around merger remnants {#obs1b}
-------------------------------------------------------
Young forming Tidal Dwarf Galaxies are rather easily pinpointed as long as they are still linked to their parent galaxies by a tidal tail. As explained earlier, an HI accumulation or CO detection at its tip make a it a promising TDG candidate. However, once they become “isolated” and that the umbilical cord is broken – typically tidal tails dissolve in 300–500 Myr –, they become much more difficult to identify. Indeed, they resemble satellite galaxies orbiting the merger remnant.
Several criteria to establish a tidal origin were discussed in the literature: an unusually high metallicity received in in-heritage from their parent galaxies [@DM98], a deficiency of non–baryonic dark matter [@BH92] and thus a particular location in the Tully-Fischer diagram [@hunter]. Several quests for identifying old TDGs among the dwarf galaxy population were carried out (see @duc04iau and references therein), but sofar no dwarf satellite formed more than 1 Gyr ago during a collision has ever been unambiguously recognized, partly because the above mentioned criteria are often difficult to check.
The simulations presented here provide further and somehow simpler constraints to probe a tidal origin:
#### Mass criterion
The mass of TDGs is typically 0.1 to 1% the baryonic (gas+star) mass of their progenitor when the latter had a gas content typical of bright spiral galaxies before the merger. It can be at the very most a few percent in the case of gas-rich progenitors. Even more massive tidal galaxies were only able to form at high redshift when the parent spiral galaxies contained more gas. It is likely that such objects have already disappeared at redshift 0. They would be anyway difficult to identify from their metallicity which is not expected to be very high. We can apply the mass criterion to some nearby systems (though still young) where the presence of massive TDGs had been suspected but later debated. For instance, the northern TDG-candidate of Arp 105 [@duc97] has a mass of nearly 10 percent of the parent galaxy mass ($7\times10^9$ solar masses in HI, the dominant component). An object with such a mass is then unlikely to be a real tidal dwarf, unless its mass had been over-estimated due, for instance, to the contribution of projected foreground or background material along the line-of-sight. The presence of such a projection effect in the northern tail of Arp 105 is indeed suggested by the large-sale kinematics of the tail [@bournaud04]. The Southern TDG-candidate identified by @duc97, with a mass of 0.5% of its parent spiral, is much more likely to be a genuine tidal dwarf.
This mass criterion also suggests that the potential TDG-candidate NGC 3077 in the group of M 81, linked to M 81 by a gaseous bridge [@yun94] is unlikely to be of tidal origin: according to its infrared luminosity and estimated HI mass, the mass of this galaxy is indeed as high as 15–20% of the mass of M 81.
#### Spatial distribution
Statistically, the simulated TDGs tend to be concentrated towards the equatorial plane of their massive host galaxy. However, this property is not a strong one for individual cases, since one given TDG can be found in a polar plane (see Fig. \[histo\_i\]). Moreover, in the case of equal-mass mergers, the equatorial plane of the spiral progenitor may hardly be observationally determined. Additionally, TDGs are rarely found at more than 15 times the optical radius of their progenitor, i.e. typically 200-250 kpc around a bright early-type host galaxy. In compact galaxy groups, the interaction with other galaxies may kick them from the host potential, and TDGs may be found further out from their progenitor.
#### Velocity
The relative velocities of TDGs, with respect to their hosts, are of the order of the circular velocities in dark haloes. Projected along the line-of-sight, this rarely results in velocities larger than 200–250 km s$^{-1}$. Higher-velocity dwarf satellites are unlikely to be of tidal origin.
#### Dark matter content {#dark-matter-content}
Dwarf galaxies of tidal origin are expected to be deficient in dark matter (see Sect. \[DMfrac\] and @BH92). Their dark matter content is negligible compared to their progenitor and to the many dwarfs with measured high M/L ratios. Note however that this prediction is made within a pure CDM frame. The presence of a baryonic and dissipative dark matter component [e.g. @pfenniger] could radically change the result, for this component could participate to the formation of TDGs.
#### Properties of the host galaxy
Long-lived TDGs are formed during galaxy collisions in the range of mass ratio 1:1 to 8:1, and more efficiently between 1:1 and 4:1. According to recent simulations [@bournaud05; @naab06], the remnant of such barely-equal mass merger should have the morphology of an Sa or more likely an S0 or even that of an elliptical, unless the colliding galaxies were particularly gas rich, as in the Early Universe. Tidal satellites should then be found more frequently around early-type galaxies.
We did not explore parameters for which the parent galaxies do not merge and hence may remain spirals. In such fly-bys, which generally correspond to large impact parameters, the corotation radius is large; less material gets into tidal tails and the latter are shorter. If TDGs are produced at all, they will have lower masses. Indeed, the high-velocity or large-distance cases that we have simulated (that are the closest to fly-bys without mergers), correspond to less numerous and less massive tidal dwarfs than the average (see Appendix A).
Satellite galaxies around their hosts {#anis2D}
-------------------------------------
\[obs2\]
We have shown in the Section 3 that the distribution of TDGs around their progenitor galaxies is anisotropic. However, the projection on the sky may hide this property. Indeed, a TDG that in the 3D space lies in the equatorial plane of its massive progenitor, can be observed in the 2D projection aligned with the minor axis of its massive host, causing possible confusion with a dwarf satellite on a polar orbit.
We have statistically estimated the apparent angle $\theta$ between the host galaxy major axis and the apparent position of the TDG, defined on Fig. \[schema\]. This is the usual observational method used to study the distribution of dwarf satellites around bright galaxies [e.g. @yang06 and references therein]. To infer the distribution of $\theta$ from that of the latitude $i$ in three dimensions, we assumed that the host+TDG systems are observed under isotropically distributed line-of-sights. The calculation was made analytically rather than assuming a limited number of possible line-of-sights in simulations. The result is shown of Fig. \[aniso2D\] : after projection on the sky plane, the anisotropy in the distribution of TDGs is still visible.
Most studies of the distribution of dwarf satellites around massive host galaxies focus on the most massive dwarfs. For instance, the work of @brainerd05 based on SDSS data includes only satellites more massive than 1/8 of their host mass. Such objects are unlikely to be of tidal origin, since they are 1–2 orders of magnitude above the typical masses of long-lived TDGs. A study including lower mass objects has been performed by @yang06. These authors found an anisotropic distribution of dwarf satellites, similar to what is found the simulations tidal dwarfs. These results cannot be directly compared, since certainly not all the dwarfs are of tidal origin (see next section). However, TDGs may significantly contribute to forster the observed anisotropy in the distribution of dwarf satellites. Hence, the anisotropy of dwarf satellites may not only relate to the shape of dark haloes, but also to the tidal origin of some satellites. Therefore an anisotropic distribution could in principle be present even within spherical dark halos. Also note that @yang06 find that the distribution of satellites is more anisotropic around red, early-type host galaxies. This observational fact could be accounted for by the anisotropy of TDGs since tidal dwarfs are more likely to be found around early-type hosts.
![Distribution of the projected inclination of dwarf satellites of tidal origin w.r.t. their host (progenitors) galaxy plane: the angle $\theta$ between the dwarf galaxy and the host galaxy apparent major axis is measured (see Fig. \[schema\]). The distribution has been derived assuming that the systems are randomly projected on the sky. The average value of $\theta$ is 42.5 degrees, while an isotropic distribution corresponds to $\theta=45$ degrees. The slightly more anisotropic distribution observed in the Sloan Digitized Sky Survey by Yang et al. (2006) is also shown.[]{data-label="aniso2D"}](fig11.eps){width="8cm"}
Contributions of TDGs to the dwarf population {#fractdg}
---------------------------------------------
The rate of TDG formation in galaxy mergers cannot be directly deduced from our set of simulations, for we deliberately privileged favorable cases: the distribution of mass ratios and orbital parameters in our simulations is not aimed at being cosmologically realistic. However, we have derived the constraints on parameters for the formation of long-lived tidal dwarfs (Section 3.2) and this can be used to estimate the actual frequency at which TDGs are formed.
Our simulations indicate that on average a bright spiral galaxy produces 1.3 tidal dwarf (considering only those that survive at least 1 Gyr), whenever it undergoes a merger with a galaxy between 1/4 and 8 times its mass, on a prograde orbit, with an orbital plane inclined up to 40 degrees with respect to its disk plane. Assuming that real orbits for galactic encounters have an isotropic distribution, this corresponds to a production rate of 0.4 TDG per merging spiral. For mergers in the 1:1–4:1 range of mass ratios, both spirals can form long-lived TDGs, and the production rate increases to 0.8 TDG per merging pair.
This estimate [*a priori*]{} assumes that the distribution of impact parameters and initial velocities in our sample of simulation is statistically representative of real colliding systems, which is probably not the case. However, our results summarized in Table 1 show these two parameters have a minor influence on the formation of TDGs. Then, even if their distribution in our sample is not realistic, this does not result in an important bias on the estimated rate of TDG production.
@okazaki claimed that all dwarfs in the Universe could have a tidal origin. Their estimate relied however on a production rate of 1 to 2 tidal dwarfs having a live time of at least 10 Gyr, which is actually much higher than that predicted by our simulations.
The number of TDGs versus time in our simulations (207 at $t=500$ Myr, 143 at $t=1$ Gyr, and 119 at $t=2$ Gyr) fits well with an exponential decay and a lifetime of 2.5 Gyr, so that only 20 percent of the TDGs actually survive up to 10 Gyr. Therefore, our best production rate – 0.8 TDG with life of 1 Gyr per merging pair – corresponds to just 0.1–0.2 TDG with a life of 10 Gyr, ten times lower than the value assumed by @okazaki. Using their other hypotheses, in particular the evolution of the number of mergers with redshift, we conclude that at most 10% of the dwarf galaxy population could be of tidal origin.
In fact, the study by @okazaki implicitly covers a larger range of mass ratios. Only about one third of the mergers are likely to fall within the 1:1–8:1 range favorable to long-lived TDGs. Since the latter estimate is very tentative, we will simply conclude that a few percent of the dwarf galaxy population can be of tidal origin. On the other hand, we did not take into account the fact that more distant galaxies were more gas rich and had probably an HI disk much more extended than in local spirals, two properties which may enhance the TDG production rate.
Only few observational studies have sofar tried to quantify the contribution of old, “detached” (thus long–lived) TDGs to the dwarf population. @DD03 did not found any excess of dwarfs around a sample of strongly interacting systems and concluded that the contribution of long-lived TDGs should be rather minor. On the other hand, in denser environments, multiple collisions could kick tidal dwarfs from the progenitor potential and thus contribute to increase their life expectancy. @Hunsberger96 probed the dwarf population of a sample of compact groups of galaxies and concluded that 50% of them were probably tidal objects. The production of tidal dwarfs in the cluster environment could be hampered by the fact that tidal tails develop in a less efficient way and tend to be more diffuse in massive potential wells [@Mihos-IAUS217]. However such objects may have formed in sub-groups falling in the cluster. Arp 105 [@DM94], NGC 5291 [@DM98], IC 1182 [@Iglesias-Paramo03] are instances of systems belonging to clusters or sub-structures within clusters where young TDG candidates have actually been identified. The presence of a large population of cluster dwarf ellipticals in Coma and Fornax [@Poggianti01; @Rakos00], of dwarf irregulars in Hydra-I and Hercules [@Duc01; @Iglesias-Paramo03] having deviant – high – metallicities for their luminosity could be interpreted as hints for a tidal origin for some of them although other mechanisms could explain these anomalies [@Conselice03].
Looking for TDGs in the Local Group {#obs3}
-----------------------------------
The distribution of the dwarf satellites of the Milky-Way is observed to be anisotropic within so-called Great Plane(s) [@LB76; @K79; @Maj94]. This fact was exploited by @kroupa05 and @sawa to claim that the dSphs around the MW could be of tidal origin. However, the satellites are actually not concentrated towards the disk plane, but along one or two nearly perpendicular directions. Our simulations indicate that such a distribution is a priori in conflict with the hypothesis that the M-W dSphs are old TDGs since the latter are statistically concentrated towards the equatorial plane of their progenitor[^4] ... unless the parent galaxy is not the M-W. Let us imagine that long ago a spiral with a disk perpendicular to the M-W disk, orbited in a polar plane w.r.t the M-W and produced in that plane long-lived TDGs. Nowadays, we do not see the disk of this progenitor, either because it has flown far away or it has merged with the M-W disk. According to our simulations, the large number of putative TDGs – more than 10 dSphs are orbiting around the M-W – indicate that either the collision at their origin occurred less than 1 Gyr ago, before most of them are destroyed – however, no sign of such recent encounter is clearly visible, except the LMC/SMC/interaction –, or that several mergers, all with specific orbits included in the Great Plane(s), occurred long ago and produced each 1–2 long–lived TDGs. However, the present morphology of the Galaxy is inconsistent with the idea that it has merged in the past with several companions with mass ratios of at least 8:1, the limits we found to produce a TDG: if this had occurred, the M-W would have become an S0 or E galaxy [e.g. @bournaud05].
In that respect, M 31 is more likely to host tidal satellites; its morphology has an earlier type and numerous signs of possible past interactions, such as long stellar streams, were found in its outer most regions [@ibata01]. @koch06 discuss the three-dimensional location of the satellite galaxies around M 31. The global distribution of satellites in this system do not show any clear concentration towards a particular plane, but the early-type dwarfs and the dSphs in particular seem to be concentrated towards a polar plane. As for the case of the Milky-Way, this distribution would require that several galaxy mergers have occurred on this configuration, which makes the tidal scenario for the origin of M 31’s dwarfs unlikely. Therefore, just from the distribution and number of the dSphs located in the vicinity of the M-W, and the statistical results obtained with our simulations, one should conclude that it is unlikely that the majority of them are of tidal origin. This is also the case for M 31. Other scenarios to explain their distribution in great planes [e.g. @koch06; @LK06] are then more realistic.
However, other, somehow more direct, criteria to probe a tidal origin should be investigated. For instance, a deviation from the luminosity–metallicity relation [@duc2000] due to pre-enrichment processes in the parent’s gaseous disk or the absence of a non baryonic dark matter component (see Sect. \[DMfrac\]) are clear characteristics of TDGs.
In that respect, the Local Group dwarfs were subject of active debates. They are usually considered as being the objects showing the highest M/L ratios (up to 100) and thus containing the highest quantities of dark matter, which would argue against a tidal origin, at least in a CDM context. Nevertheless the accurateness of the estimate of their total mass is controversial, and claims were made that it had been largely over-estimated due to the lack of statistics – the dynamical mass is derived from the velocity dispersion of a limited number of individual stars – or due to the tidal shaking of the stars [@kroupa05]. Beside, the Local Group dwarfs are remarkable for their variety of M/L ratios, chemical enrichment and more globally star formation history despite being all in the same environment [@grebel01]. This leaves room for a tidal origin in some of them.
A note on merger induced globular clusters
------------------------------------------
In our analysis, we only considered sub-structures with masses exceeding $10^8$ M$_{\sun}$ to make sure they contained enough particles and had a fair resolution. Our numerical study did not aim at making predictions on lower mass objects, in particular the potentially forming globular clusters[^5]. Objects of masses typically $10^5-10^7$ M$_{\sun}$ could be numerous in tidal tails, and their survival less problematic than the objects with typical masses of $10^8$ M$_{\sun}$. Indeed, if concentrated enough, they will be less affected by tidal disruption, undergo less dynamical friction and face a slower orbital decay [@miocchi06]. @kroupa97 studied the evolution of $10^5$ M$_{\sun}$ dark matter free satellites and concluded that they could be long-lived, as well as globular clusters possibly formed in mergers. Higher resolution simulations including a description of the formation process are needed to study this scenario of globular cluster formation in mergers.
Conclusion
==========
In this paper, we have studied the formation and the evolution of Tidal Dwarf Galaxies and related candidates, in about 100 numerical simulations of collisions of spiral galaxies, with gas content typical of gas-rich low to moderate redshift spirals. The identification of massive ($M > 10^8$ M$_{\sun}$) substructures formed in tidal tails, and their follow-up during 2 Gyr, have shown that:
- Among the numerous tidal objects, those susceptible to survive more than 1 Gyr were formed in the outer parts of tidal tails; they also turn out to be the most massive ones. Gravitational clumps which grow all along the tails rarely survive more than 500 Myr, mainly because they are formed on more eccentric orbits, and generally fall back onto their progenitors or are disrupted by the tidal field losing much of their initial mass. These objects should not be considered as genuine Tidal Dwarf Galaxies, according to the definition of @duc04iau or @HB04, even if they have, at the moment they are observed, the apparent characteristics of young forming dwarf galaxies.
- Long-lived tidal dwarf galaxies are preferentially formed on prograde, close-to-coplanar orbits (inclination smaller than $\sim$ 40 degrees). In such cases, more than 2 long-lived TDGs are rarely formed from one progenitor galaxy. These results have led us to the conclusion that a small but significant fraction of a few percent of dwarf satellites could be of tidal origin. The fraction of tidal dwarfs can be larger around early-type hosts that have undergone numerous mergers or in rich environments where tidal dwarfs can be longer-lived.
- The statistical properties of the long-lived TDGs, those that become dwarf satellites, have been studied. They are typically found within 10–15 optical radii from their progenitor, with moderate relative velocities, and masses at the very most a few percent of their progenitor mass (or about 10% of their progenitor initial gaseous mass). Their spatial distribution is anisotropic, flattened towards the equatorial plane of the parent spiral galaxy. Would the contribution of TDGs be important in some environments, they would then contribute to foster the anisotropy already observed for the distribution of the general dwarf satellite population.
- Given their number and spatial distribution in polar Great Planes, the majority of the satellites of the Milky-Way and M 31 are unlikely to all be of tidal origin.
These results can help identifying, in real young interacting systems, the tidal objects that are the most likely to be long-lived and provide hints on the location of the long searched ”old” TDGs which, once the tidal tails linking them to their parents have disappeared, are difficult to identify. If searched randomly without any selection criteria, tens of dwarf satellites would have to be analyzed for a few TDGs to be found.
Galaxy mergers can also produce a significant number of smaller mass objects like globular clusters, but the resolution of the present simulations did not enable us to study them in detail. Also, the internal properties of TDGs have not been studied in this paper. They will be the subject of a forthcoming study using higher resolution simulations, dedicated to resolve the internal dynamics of tidal substructures, in particular those that are now known to be long-lived.
We are grateful to an anonymous referee for a careful reading of the manuscript, and to Françoise Combes for her comments on a previous version. The numerical simulations were carried out on the NEC-SX6 vectorial computer of the CEA/CCRT, and we are most grateful to Frederic Masset for helping with access to this computing center. This research was led within the Horizon project (`http://www.projet-horizon.fr`).
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Parameters and results of numerical runs {#appen}
========================================
We give in Table \[table\_run\] the parameters used for each simulation. The mass ratio is $M$. The orientation of the orbit, prograde (P) or retrograde (R) is given for the most massive galaxy and the less massive one respectively. The impact parameter $R$ and relative velocity at an infinite distance $V$ are computed assuming that dynamical friction is negligible before the beginning of the simulation. The initial inclination of the orbital plane to the disk plane is given by $i_1$ and $i_2$ for each colliding galaxy. $N_1$ and $N_2$ are the number of long-lived tidal dwarf galaxies (surviving at least 1 Gyr) more massive than $10^8$ M$_{\sun}$ formed in the material of each parent galaxy. Galaxy number 1 is the most massive one.
Run M Orient. $R$ $V$ $i_1$ $i_2$ $N_1$ $N_2$
----- --- --------- ----- ----- ------- ------- -- ------- -------
0 1 P/P 4 150 0 0 2 2
1 1 R/R 4 150 0 0 - -
2 1 P/R 4 150 0 0 2 -
3 2 P/P 4 150 0 0 1 1
4 2 R/R 4 150 0 0 - -
5 2 P/R 4 150 0 0 1 -
6 5 P/P 4 150 0 0 1 1
7 5 R/P 4 150 0 0 - 2
8 2 P/P 4 150 35 35 1 1
9 2 P/R 4 150 35 35 2 -
10 2 P/P 4 150 15 15 1 2
11 2 P/P 4 150 45 45 1 -
12 2 P/P 4 150 60 60 - -
13 5 P/P 4 150 35 35 - 2
14 5 P/R 4 150 35 35 - -
15 5 P/P 4 150 15 15 1 2
16 5 P/P 4 150 45 45 - -
17 5 P/P 4 150 60 60 - -
18 2 P/P 1 150 30 30 - 1
19 2 P/P 2 150 30 30 - 3
20 2 P/P 6 150 30 30 1 1
21 2 P/P 9 150 30 30 1 2
22 2 P/P 1 80 30 30 - 1
23 2 P/P 2 80 30 30 1 2
24 2 P/P 6 80 30 30 1 2
25 2 P/P 9 80 30 30 1 2
26 2 P/P 12 80 30 30 2 1
27 2 P/P 7 30 30 30 - -
28 2 P/P 7 50 30 30 - 2
29 2 P/P 7 80 30 30 1 1
30 2 P/P 7 110 30 30 1 2
31 2 P/P 7 150 30 30 1 1
32 2 P/P 7 190 30 30 1 2
33 2 P/P 7 250 30 30 1 1
34 2 P/P 7 320 30 30 - -
35 5 P/P 7 30 30 30 - 1
36 5 P/P 7 50 30 30 - 2
37 5 P/P 7 80 30 30 - 1
38 5 P/P 7 110 30 30 - 1
39 5 P/P 7 150 30 30 - 2
40 5 P/P 7 190 30 30 - 1
41 5 P/P 7 250 30 30 - 1
42 5 P/P 7 320 30 30 - 1
43 5 P/P 1 150 30 30 - 1
44 5 P/P 2 150 30 30 - 1
45 5 P/P 6 150 30 30 - 1
46 5 P/P 9 150 30 30 1 2
47 5 P/P 1 80 30 30 - -
48 5 P/P 2 80 30 30 - 1
49 5 P/P 6 80 30 30 1 1
50 5 P/P 9 80 30 30 - 2
51 5 P/P 12 80 30 30 - 1
52 8 R/R 5 70 0 0 - -
53 8 R/R 8 70 0 0 - 1
53 8 R/R 12 70 0 0 - 1
54 8 R/R 5 220 0 0 - -
55 8 R/R 8 220 0 0 - -
Run M Orient. $R$ $V$ $i_1$ $i_2$ $N_1$ $N_2$
----- ---- --------- ----- ----- ------- ------- -- ------- -------
56 2 P/P 10 60 0 0 1 2
57 2 P/P 10 220 0 0 1 1
58 2 P/P 10 60 45 45 - -
59 2 P/P 10 220 45 45 - -
60 4 P/P 10 60 0 0 1 1
61 4 P/P 10 220 0 0 1 2
62 4 P/P 10 60 45 45 - 1
63 4 P/P 10 220 45 45 - -
64 2 P/P 2 60 0 0 1 1
65 2 P/P 2 250 0 0 - -
66 2 P/P 2 60 45 45 - 1
67 2 P/P 2 250 45 45 - -
68 1 P/P 5 70 0 0 1 2
69 1 P/P 5 250 0 0 1 1
70 1 P/P 5 70 45 45 - 1
71 1 P/P 5 250 45 45 - -
72 1 P/P 5 250 40 40 - -
73 2 P/P 5 250 40 40 - -
74 3 P/P 5 250 40 40 - 1
75 3 P/P 1 170 30 30 - -
76 3 P/P 2 170 30 30 - 1
77 3 P/P 6 170 30 30 2 1
78 3 P/P 9 170 30 30 1 1
79 3 P/P 1 90 30 30 - 1
80 3 P/P 2 90 30 30 - 1
81 3 P/P 6 90 30 30 1 2
82 3 P/P 9 90 30 30 1 1
83 3 P/P 12 90 30 30 1 1
84 1 P/P 1 170 10 20 - -
85 1 P/P 2 170 10 20 1 1
86 1 P/P 6 170 10 20 1 1
87 1 P/P 9 170 10 20 1 2
88 1 P/P 1 90 10 20 2 1
89 1 P/P 2 90 10 20 1 1
90 1 P/P 6 90 10 20 2 2
91 1 P/P 9 90 10 20 1 2
92 1 P/P 12 90 10 20 1 1
93 10 P/P 4 120 10 10 - -
94 10 P/P 8 120 10 10 - -
95 10 P/P 6 180 10 10 - -
[^1]: D04 explained that the modeling of the ISM is not critical for the formation of massive tidal objects, and indeed @cox have obtained the formation of comparable objects with an SPH code
[^2]: In major mergers forming elliptical galaxies, the main flattening plane of the remnant is often found to correspond to the orbital plane, more than to the parent galaxy plane. However, since TDG-forming cases correspond to low orbital inclinations, the parent galaxy plane and the orbital plane are never very different.
[^3]: defined here as the radius containing 90% of the baryonic mass (stars+gas) of the object
[^4]: though high latitudes TDGs can also be present; see Fig.\[histo\_i\]
[^5]: Note though on Fig. 1 the presence of low mass objects in the vicinity of the merger remnant.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
For some years it was believed that for “connectivity” problems such as <span style="font-variant:small-caps;">Hamiltonian Cycle</span>, algorithms running in time $2^{O({{\mathbf{tw}}})}\cdot n^{O(1)}$ –called *single-exponential*– existed only on planar and other sparse graph classes, where ${{\mathbf{tw}}}$ stands for the treewidth of the $n$-vertex input graph. This was recently disproved by Cygan *et al*. \[FOCS 2011\], Bodlaender *et al*. \[ICALP 2013\], and Fomin *et al*. \[SODA 2014\], who provided single-exponential algorithms on general graphs for essentially all connectivity problems that were known to be solvable in single-exponential time on sparse graphs. In this article we further investigate the role of planarity in connectivity problems parameterized by treewidth, and convey that several problems can indeed be distinguished according to their behavior on planar graphs. Known results from the literature imply that there exist problems, like <span style="font-variant:small-caps;">Cycle Packing</span>, that [*cannot*]{} be solved in time $2^{o({{\mathbf{tw}}}\log {{\mathbf{tw}}})} \cdot n^{O(1)}$ on general graphs but that can be solved in time $2^{O({{\mathbf{tw}}})} \cdot n^{O(1)}$ when restricted to planar graphs. Our main contribution is to show that there exist problems that can be solved in time $2^{O({{\mathbf{tw}}}\log {{\mathbf{tw}}})} \cdot n^{O(1)}$ on general graphs but that [*cannot*]{} be solved in time $2^{o({{\mathbf{tw}}}\log {{\mathbf{tw}}})} \cdot n^{O(1)}$ even when restricted to planar graphs. Furthermore, we prove that <span style="font-variant:small-caps;">Planar Cycle Packing</span> and <span style="font-variant:small-caps;">Planar Disjoint Paths</span> cannot be solved in time $2^{o({{\mathbf{tw}}})} \cdot n^{O(1)}$. The mentioned negative results hold unless the ETH fails. We feel that our results constitute a first step in a subject that can be further exploited.
**Keywords:** parameterized complexity, treewidth, connectivity problems, single-exponential algorithms, planar graphs, dynamic programming.
author:
- Julien Baste and Ignasi Sau
bibliography:
- 'Bib\_SparseDP.bib'
title: 'The role of planarity in connectivity problems parameterized by treewidth[^1] '
---
Introduction {#sec:intro}
============
Notation and preliminaries {#sec:prelim}
==========================
Problems of Type 2 {#sec:type2}
==================
Problems of Type 3 {#sec:type3}
==================
Lower bound for Planar Disjoint Paths {#sec:pdp}
======================================
[^1]: This work was supported by the ANR French project AGAPE (ANR-09-BLAN-0159) and the Languedoc-Roussillon Project “Chercheur d’avenir” KERNEL.
|
{
"pile_set_name": "ArXiv"
}
|
---
address: 'Max-Planck-Institut für Physik, Föhringer Ring 6, D-80805 Munich, Germany'
author:
- Sandra Kortner on behalf of the ATLAS Collaboration
title: Search for the Standard Model Scalar Boson with the ATLAS detector
---
Introduction
============
One of the main missing pieces of puzzle in the Standard Model (SM) of particle physics relates to the electroweak symmetry breaking mechanism [@EnglertBrout; @Higgs1; @Higgs2; @GHK] which predicts the existence of a new, yet undiscovered scalar boson, the Higgs boson. Electroweak precision measurements at the LEP, SLD and Tevatron experiments [@lepindirect] set an indirect upper limit on the Higgs boson mass of $m_H<$ 152 GeV at the 95% confidence level (CL). The existence of a Higgs boson in the mass regions with $m_H<$ 114.4 GeV and 156 GeV $<m_H<$ 177 GeV is excluded at the 95% CL by the LEP [@LEP] and Tevatron [@Tevatron] experiments, respectively. In the following, the latest results from direct searches for the SM Higgs boson with the ATLAS detector at the Large Hadron Collider (LHC) are presented. The results superseed the ones published in Ref.[@previous_comb] and are based on the full dataset of $pp$ collision data recorded in 2011 with an integrated luminosity of up to 4.9 fb$^{-1}$ at a center-of-mass energy of $\sqrt{s}$=7 TeV.
Search overview
===============
The search for the SM Higgs boson is performed in the mass range from 110 GeV to 600 GeV combining several search channels summarized in Table \[channel\_summary\]. The Higgs boson decays into vector boson pairs allow for the search in the entire mass range, while the $H\to~ \gamma\gamma$, $H\to~\tau^+\tau^-$ and $H\to b\bar{b}$ decay modes provide an additional sensitivity for a low-mass Higgs boson ($m_H<$ 150 GeV). As opposed to other channels, the $H\to \gamma\gamma$ and $H\to ZZ\to 4\ell$ decay modes are distinguished through their high signal mass resolution. Each search channel is divided into several exclusive sub-channels with different background composition or signal-to-background ratios.
Search channel Nr. of subchannels $m_H$ range (GeV) L (fb$^{-1}$) Reference
----------------------------------- -------------------- ------------------- --------------- -----------
$H\to \gamma\gamma$ 9 110-150 4.9 [@gg]
$H\to \tau\tau$ 12 110-150 4.7 [@tautau]
$VH, H\to bb$ 11 110-130 4.7 [@bb]
$H\to WW^{(*)}\to \ell\nu\ell\nu$ 9 110-600 4.7 [@lvlv]
$H\to ZZ^{(*)}\to 4\ell$ 4 110-600 4.8 [@4l]
$H\to ZZ\to\ell\ell\nu\nu$ 4 200-600 4.7 [@llvv]
$H\to ZZ\to\ell\ell jj$ 2 200-600 4.7 [@llqq]
$H\to WW\to\ell\nu jj$ 4 300-600 4.7 [@lvqq]
: Individual channels in search for the SM Higgs boson along with the corresponding number of sub-channels, explored range of Higgs boson masses, integrated luminosity and references to public documentation.\[tab:exp\][]{data-label="channel_summary"}
The exclusion limits on the Higgs boson production cross section are set based on the mass spectra of the final decay products, using the modified frequentist approach ($CL_S$) [@cls]. The $p_0$-value, i.e. the probability that the expected background fluctuates as high as the observed number of events or higher, has been evaluated for each hypothesized mass $m_H$ using a frequentist approach [@p0]. The Higgs boson production cross sections and decay branching ratios, as well as the corresponding theory uncertainties are taken into account and are summarized in Ref. [@YellowReport1]. The QCD scale uncertainties depend on the mass $m_H$ and typically amount to $^{+12}_{-8}$% for the dominant gluon fusion ($ggH$) production mode, $\pm$1% for the production via vector boson fusion (VBF) and in association with vector bosons (VH), and $^{+3}_{-9}$% for the associated production with top-quark pairs ($t\bar{t}H$). The uncertainties related to the parton distribution functions (PDF) amount to $\pm$8% for the predominantly gluon-initiated $ggH$ and $t\bar{t}H$ processes, and $\pm$4% for the predominantly quark-initiated VBF and VH processes.
Searches in the low-$m_H$ region (110 GeV - 150 GeV)
====================================================
$H\to \gamma\gamma$
-------------------
Despite of a very small branching ratio of about 0.2%, the $H\to \gamma\gamma$ decay channel provides the highest sensitivity for the Higgs boson search in the mass region $m_H\lesssim$ 120 GeV. Selection of two isolated photons with high transverse energy has been optimized for the supression of the reducible photon-jet and jet-jet backgrounds with one or two misidentified jets. The irreducible background with true diphoton events comprises about 70% of all selected events in the explored mass range from 100 GeV to 150 GeV. The analysis is divided into 9 sub-categories based on the photon pseudorapidity, conversion status and $p_{Tt}^{\gamma\gamma}$, i.e. the component of the diphoton $p_T$ orthogonal to the di-photon thrust axis. Excellent diphoton invariant mass resolution of 1% to 2% allows for the search of a narrow di-photon invariant mass peak from the Higgs boson decays on the continuum background. The inclusive invariant diphoton mass spectrum is shown in Figure \[fig:gg\] (left) together with the total expected background contribution. The background contributions are estimated separately in each sub-category from the fit of an exponential function to the diphoton invariant mass spectrum, while the signal shape is modelled by the sum of a Crystal Ball and a Gaussian function.
The observed and expected exclusion limits at the 95% CL on the Higgs boson production in units of the SM cross section are shown in Figure \[fig:gg\] (right). A SM Higgs boson is excluded at the 95% CL in the mass ranges of 113 GeV-115 GeV and 134.5 GeV-136 GeV. The largest excess with respect to the background-only hypothesis is observed at 126.5 GeV with a local significance of 2.8$\sigma$. The corresponding global significance is 1.5$\sigma$ after accounting for the look-elsewhere-effect ($LEE$) [@LEE] in the entire mass range explored with this decay channel.
$H\to\tau^+\tau^-$
------------------
The mass range from 100 GeV to 150 GeV can also be probed by the less sensitive $H~\to~\tau^+\tau^-$ decay channel with leptonic and hadronic decay modes of the two tau-leptons resulting in the fully-leptonic $\ell\ell 4\nu$, semi-leptonic $\ell\tau_{had} 3\nu$ and fully-hadronic $\tau_{had}\tau_{had} 2\nu$ final states. For an optimal analysis sensitivity, events are separated into 12 mutually exclusive sub-channels based on the lepton flavour and jet multiplicity, as well on the event topologies characteristic for the VBF and VH signal production mechanisms. The reconstructed mass shape of the dominant $Z~\to~\tau^+\tau^-$ background contribution is estimated by means of an embedding technique in which muons from $Z$ decay events are substituted by simulated tau-lepton decays. The normalization of this background is obtained from the simulation. Background processes with jets misidentified as leptons or as tau-jets are estimated from the signal-free control data with reversed lepton isolation and tau-jet identification criteria, or samples with a same-sign charge of the two tau-lepton decay products. All other background contributions are estimated from simulation. Depending on the sub-channel the $\tau\tau$ invariant mass distribution used for the limit setting is reconstructed using the effective mass, collinear approximation or a Missing Mass Calculator [@MMC] technique. The 95% CL exclusion limits on the SM Higgs boson production are shown in Figure \[fig:tt\_bb\]. The observed limits vary from 2.5 to 11.9 times the predicted SM cross section over the entire explored mass range.
$VH, H\to b\bar{b}$
-------------------
In order to supress the extremely large multijet background contribution, the VH production mode is used to explore the $H\to b\bar{b}$ decay channel in the mass range from 100 GeV to 130 GeV. The search is divided into 11 sub-channels based on the decay mode of the associated vector boson and its transverse momentum: $\ell\nu b\bar{b}$ with four $p_T^W$ bins, $\ell\ell b\bar{b}$ with four $p_T^Z$ bins and $\nu\bar{\nu} b\bar{b}$ with three bins of the mising transverse energy. Since the Higgs and the vector boson tend to recoil away from each other, the highest sensitivity is reached in bins with highest transverse momenta of the vector bosons. The dominant backround contributions originate from the $Z$+jets, $W$+jets, $t\bar{t}$, diboson and multijet processes. The shape of all but the last background process is obtained from the simulation, while the dedicated control data samples are used to estimate the shape of the multijet contribution and the normalization of all mentioned processes. The exclusion limits shown in Figure \[fig:tt\_bb\] are set based on the di-$b$-jet invariant mass distribution. The observed upper limits at the 95% CL on Higgs boson production vary from of 2.7 to 5.3 times the SM cross section in the entire mass range explored by this channel.
Searches in the high-$m_H$ region (200 GeV - 600 GeV)
=====================================================
A high-mass Higgs boson will predominantly decay into a pair of vector bosons. Final states with a non-leptonic decay of one of the two vector boson have an advantage of a higher decay rate with respect to fully leptonic channels. At the same time, the backround level can still be controlled using the invariant mass constraint of both vector bosons.
The $H\to ZZ\to \ell\ell\nu\nu$ decay channel provides the highest sensitivity in the high-$m_H$ region due to the high transverse momenta of leptons and neutrinos from the $Z$ boson decays. The search is performed separately in $ee\nu\nu$ and $\mu\mu\nu\nu$ sub-channels, both separated for low and high pile-up environment to account for the different resolution of the missing transverse energy. The invariant mass of the pair of the charged leptons is required to agree with the $Z$ boson mass within 15 GeV. Different selection cuts are applied for searches below and above $m_H$=280 GeV, accounting for the different level of boost of the $Z$ boson from the Higgs boson decay. The main background processes are suppressed mostly by the cuts on the opening angle of the two leptons from the $Z$ boson decay and on the missing transverse energy. The contribution of the dominant $ZZ$ background is estimated from simulation, while dedicated control data are used for other processes ($WZ$, $t\bar{t}$ and $W/Z$+jets). The transverse mass distribution is used for the limit setting. The observed (expected) exclusion limit at the 95% CL covers the mass range from 320 GeV to 560 GeV (260 GeV to 490 GeV), as shown in Figure \[fig:llvv\_llqq\] (left).
Somewhat lower sensitivity is reached in the $H\to ZZ\to \ell\ell jj$ channel which is divided into two sub-channels, one with maximum one jet tagged as a $b$-jet and one with two $b$-jets. Candidate events must have a lepton pair and a dijet pair with invariant masses compatible with the $Z$ boson mass. The dominant $t\bar{t}$ and $Z+jets$ backgrounds are suppressed by the cuts on the missing transverse energy and the opening angle between the two jets. The exclusion limit based on the $m_{\ell\ell jj}$ invariant mass distribution is shown in Figure \[fig:llvv\_llqq\] (right). Expected exclusion limit at the 95% CL covers the mass range from 360 GeV to 400 GeV, while the observed upper limit excludes the mass range from 300 GeV to 310 GeV and 360 GeV to 400 GeV.
The $H\to WW\to \ell\nu jj$ channel is separated into six sub-channels based on the lepton flavour and the jet multiplicity (0, 1 and 2 jets with a VBF-like topology). The invariant mass of the dijet pair must be compatible with the $W$ boson mass and a mass constraint $m_{\ell\nu}$=$m_W$ is applied to determine the $z$-component of the neutrino momentum. The limit setting is based on the resulting $m_{\ell\nu jj}$ invariant mass distribution. At present the channel is not sensitive enough to exclude the SM Higgs boson in the explored mass range from 300 GeV to 600 GeV at the 95% CL.
Searches in the full $m_H$ region (110 GeV - 600 GeV)
=====================================================
The fully leptonic Higgs boson decays into two vector bosons provide a good handle for the background processes even in the case that one of the vector bosons is off-shell. These final states can therefore be used for the search in the entire mass range from 110 GeV to 600 GeV.
$H\to WW^{(*)}\to \ell\nu\ell\nu$
---------------------------------
The Higgs boson search in the $H\to WW^{(*)}\to \ell\nu\ell\nu$ channel provides the highest sensitivity in a wide range of hypothesized Higgs boson masses. The analysis is divided in 9 sub-channels: $ee\nu\nu$, $\mu\mu\nu\nu$ and $e\mu\nu\nu$, each separated by the jet multiplicity into final states with no jets, 1 jet or two jets with a VBF-like topology. The channel is characterized by a broad transverse mass distribution $m_T$ for the Higgs signal due to the presence of two neutrinos in the final state. The main reducible backgrounds ($W/Z$+jets, multijets, $t\bar{t}$) are suppressed by the lepton isolation and $b$-jet veto requirement, as well as the cuts on the missing transverse energy. The dominant $WW$ background contribution can be discriminated by means of topological cuts on the invariant dilepton mass and the opening angle between the two leptons. All background contributions remaining after the full selection are normalized using dedicated control regions and extrapolated to the signal region relying on predictions from the simulation. The transverse mass distribution is used in each sub-channel for the limit setting. As an example, the $m_T$ distribution in the 0-jet sub-channel is shown in Figure \[fig:ww\] (left). The 95% CL exclusion limit with all sub-channels combined is shown in Figure \[fig:ww\] (right). No significant excess of events is observed. A SM Higgs boson with a mass in the range between 130 GeV and 260 GeV is excluded at the 95% CL, while the expected exclusion interval ranges from 127 GeV to 234 GeV.
$H\to ZZ^{(*)}\to 4\ell$
------------------------
The Higgs boson decays into four-leptons provide a clean signature with a very low level of background. After the lepton isolation and impact parameter cuts, the only remaining background for $m_H\gtrsim$200 GeV is the irreducible $ZZ$ process. For $m_H\lesssim$200 GeV, there is a small additional contribution of the reducible $Zb\bar{b}$, $Z$+jet and $t\bar{t}$ backgrounds estimated from the control data samples. The channel is characterized by high experimental invariant mass resolution of 1.5% to 2% up to $m_H$=350 GeV, after which the natural Higgs boson decay width starts to dominate. A high signal efficiency is ensured by a high lepton reconstruction and identification efficiency, as well as the low transverse momentum cut of 7 GeV on the two subleading leptons. The mass of one dilepton pair is required to be compatible with the $Z$ boson mass, while the $m_H$-dependent invariant mass cut is applied on the second lepton pair. The analysis is divided into four sub-channels based on lepton flavours: $4\mu$, $2e2\mu$, $2\mu 2e$ and $4e$, where the first two leptons are assigned to an on-shell $Z$ boson. Figure \[fig:4l\] (left) shows the inclusive four-lepton invariant mass distribution after all selection cuts.
A total of 62$\pm$9 candidates are expected from background processes which is in good agreement with 71 observed candidate events. The corresponding 95% CL exclusion limit is shown in Figure \[fig:4l\] (right). The observed exclusion covers the mass regions from 134 GeV to 156 GeV, 182 GeV to 233 GeV, 256 GeV to 265 GeV and 268 GeV to 415 GeV. The expected exclusion ranges are 136 GeV to 157 GeV and 184 GeV to 400 GeV. The most significant upward deviations from the background-only hypothesis are observed for $m_H$ = 125 GeV with a local significance of 2.1$\sigma$, $m_H$ = 244 GeV with a local significance of 2.2$\sigma$ and $m_H$ = 500 GeV with a local significance of 2.1$\sigma$. Once the look-elsewhere-effect is considered, none of the observed local excesses is significant.
Combined results
================
The results of the statistical combination of all individual search channels are summarized in Ref. [@combination]. The expected and observed limits obtained are shown in Figure \[fig:comb\].
SM Higgs boson masses between 120 GeV and 555 GeV are expected to be excluded at the 95% CL or higher. The observed 95% CL exclusion regions range from 110.0 GeV to 117.5 GeV, 118.5 GeV to 122.5 GeV, and 129 GeV to 539 GeV. The mass regions between 130 GeV and 486 GeV are excluded at the 99% CL.
The mass region around $m_H$=126 GeV cannot be excluded due to an observed excess of events compared to the expected background contribution. The $p_0$-values in the given mass region are shown in Figure \[fig:comb2\] for individual channels and their combination. The most significant excesses are observed in the two channels with high mass resolution, $H\to \gamma\gamma$ and $H\to ZZ^{(*)}\to 4\ell$. The largest combined local significance of the observed excess is 2.5$\sigma$, while the expected significance in the presence of a SM Higgs boson with $m_H$=126 GeV is 2.9$\sigma$. The global probability for such an excess to occur anywhere in the entire explored Higgs boson mass region from 110 GeV to 600 GeV is approximately 30%. The excess corresponds to the best-fit signal strength $\mu$ of approximately 0.9$^{+0.4}_{-0.3}$, which is compatible with the signal strength expected from a SM Higgs boson at that mass, as shown in Figure \[fig:comb2\] (right).
Summary
=======
The full dataset of $pp$ collision data recorded with the ATLAS detector at the LHC in 2011 has been studied in search for the SM Higgs boson, combining several Higgs boson decay channels in the $m_H$ range from 110 GeV to 600 GeV. A SM Higgs boson is excluded at the 95% CL in a wide mass range. The exclusion was not possible for $m_H$ around 126 GeV, due to an observed excess of events at the local significance level of 2.5$\sigma$. More data collected in 2012 will be needed to better understand the origin of the described excess.
References {#references .unnumbered}
==========
[99]{}
F. Englert, R. Brout, .
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G. S. Guralnik, C. R. Hagen, T. W. B. Kibble, .
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R. Barate et al, .
CDF and D0 Collaborations, arXiv:1203.3774.
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ATLAS Collaboration, .
ATLAS Collaboration, ATLAS-CONF-2012-014, http://cdsweb.cern.ch/record/1429662.
ATLAS Collaboration, ATLAS-CONF-2012-015, http://cdsweb.cern.ch/record/1429664.
ATLAS Collaboration, ATLAS-CONF-2012-012, http://cdsweb.cern.ch/record/1429660.
ATLAS Collaboration, .
ATLAS Collaboration, ATLAS-CONF-2012-016, http://cdsweb.cern.ch/record/1429665.
ATLAS Collaboration, ATLAS-CONF-2012-017, http://cdsweb.cern.ch/record/1429666.
ATLAS Collaboration, ATLAS-CONF-2012-018, http://cdsweb.cern.ch/record/1429667.
A. L. Read, .
ATLAS and CMS Collaborations, LHC Higgs Combination Group, ATL-PHYS-PUB-2011-011, https://cdsweb.cern.ch/record/1375842.
LHC Higgs Cross Section Working Group, S. Dittmaier, C. Mariotti, G. Passarino, and R. Tanaka (Eds.), arXiv:1101.0593 \[hep-ph\].
E. Gross, O. Vitells, .
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ATLAS Collaboration, ATLAS-CONF-2012-019, http://cdsweb.cern.ch/record/1430033.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The longitudinal transport problem (current is applied parallel to some bias magnetic field) in type-II superconductors is analyzed theoretically. Based on analytical results for simplified configurations and relying on numerical studies for general scenarios, it is shown that a remarkable inversion of the current flow in a surface layer may be predicted under a wide set of experimental conditions. Strongly inhomogeneous current density profiles, characterized by [*enhanced*]{} transport towards the center and reduced or even negative values at the periphery of the conductor are expected when the physical mechanisms of flux depinning and consumption (via line cutting) are recalled. A number of striking collateral effects such as local and global paramagnetic behavior are predicted. Our geometrical description of the macroscopic material laws allows a pictorial interpretation of the physical phenomena underlying the transport backflow.'
author:
- 'H. S. Ruiz'
- 'C. López'
- 'A. Badía–Majós'
title: 'Inversion mechanism for the transport current in type-II superconductors'
---
Introduction
============
Type-II superconductors under the action of a transport current and a longitudinal magnetic field may exhibit the counter intuitive phenomenon of negative resistance within a certain set of experimental conditions. This property, together with other intriguing phenomena, such as the observation of paramagnetic moments, and the [*compression*]{} of the transport current by the action of a parallel magnetic field, have been reported in the course of intense experimental and theoretical activities.[@parallelE; @walmsley; @cave; @matsushita; @parallelT; @voloshin; @fisher] Most of these works were primarily concerned with the arrangement of the macroscopic current density ${\bf J}$ along the so-called nearly [*force free*]{} trajectories. Recall that if ${\bf J}$ is [*nearly parallel*]{} to the magnetic induction ${\bf B}$, moderate or weak pinning forces are needed for avoiding the detrimental flux-flow losses related to the drift of flux tubes driven by the magnetostatic force (${\bf J}\times{\bf B}$ per unit volume). More specifically, negative voltages have been observed by different groups[@walmsley; @cave; @matsushita] when recording the current-voltage characteristics at specific locations on the surface of the sample (central region). In addition, the striking effect takes place within a definite interval of applied magnetic fields.
Within such a complex scenario, it was recognized early on that the observations could only be understood if new dissipation mechanisms, additional to the flux flow phenomena were considered. In particular, a prominent role happens to be played by the flux-line cutting (crossing and recombination) between adjacent tilted vortices.[@parallelT; @leblanc] Nevertheless, certain facts still remain to be fully understood. Thus, the challenging problem of inhomogeneous electric fields, even changing sign along the specimen surface is hitherto open. On the other hand, important issues as the consideration of irreversible effects related to the thresholds for flux depinning and cutting phenomena have not been reported yet. In this contribution, we investigate the influence of such mechanisms on the establishment of critical negative current structures within the superconducting state. This is to be considered as a step forward for gaining knowledge on the processes that operate just previous to the dissipation regime. Within such a physical scenario, the concentration of transport current towards the center of the sample and the appearance of negative flow at the surface will be predicted for a certain range of experimental conditions. To be specific, the application of different components of magnetic field, their sequence and characteristic values will be identified as relevant issues for the observation of negative currents.
The article is organized as follows. In Sec.\[secCS\] we put forward the basic ideas about the theoretical approach used, that is a general critical state theory for type-II superconductors. An idealized slab geometry arrangement is proposed, aimed to introduce the lowest level of complexity for our purposes. Then, in Sec.\[secAnalytic\] we perform a simplified analytical evaluation that allows to capture the main underlying physical matters. In Sec.\[secNumeric\] a quasi-3D statement of the problem is solved by numerical means. This is needed for the consideration of inhomogeneities over the sample. A discussion about the scope of our investigation within the problem of negative currents is given in Sec.\[secDisc\]
Critical state approximation {#secCS}
============================
Classical Maxwell equation approach
-----------------------------------
The fundamental concept on which the critical state theory relies is that, in many cases, the experimental conditions allow to analyze the evolution of the system in the quasistationary regime. Thus, Ampere’s law becomes $\nabla \times {\bf H} = {\bf J}$, and determines the distribution of supercurrents within the sample. When some external excitation (magnetic field and/or transport current) applies, the quasistationary evolution between successive equilibrium states is ruled by Faraday’s law $\nabla \times {\bf E} = - \partial _t {\bf B}$. Here the induced transient electric field is determined through an appropriate material relation ${\bf J}({\bf E})$, and is used to update the profile of $\bf J$.
The material law encodes the mechanisms related to the breakdown of magnetostatic equilibrium, as well as the dissipation modes operating in the transient from one state to the other. In this sense, there have been a number of theoretical proposals, and among them (see Refs. and the citations therein), here we choose the so-called [*double critical state model*]{} (DCSM).[@dcsm] This approach allows a straightforward connection between the mesoscopic flux depinning and cutting phenomena, and the field equations for the coarse grained quantity $\bf J$. On the one hand, the model establishes the critical conditions $|{\bf J}_{\perp}|\leq J_{c\perp}$ and $|{\bf J}_{\parallel}|\leq J_{c\parallel}$ that relate to (i) the maximum pinning force on the vortex lines ($|{\bf J}\times{\bf B}|= J_{\perp}B\leq F_{\rm p,max}$) and (ii) to the maximum variation of the tilt angle between vortices (notice that from Eq.(\[eq:ampere\]) one has $d\alpha\propto J_{\parallel}$).[@bcw] On the other hand, as it was thoroughly discussed in Ref., the model also provides a rule that fixes the [*trajectory*]{} of the system through the dissipation excursion towards the new equilibrium state. Thus, corresponding to a very sharp transition from the superconducting state to some regime of high losses, one can argue that the final state current density verifies a maximum projection law relative to the transient electric field, i.e.: ${\rm max}\,\,({\bf J}\cdot\hat{\bf E})$ that ensures the fastest return to equilibrium. Notice that, in 1D situations (infinite slab with a single component applied magnetic field) this is trivially verified because by symmetry one has ${\bf J}\parallel{\bf E}$ and both perpendicular to ${\bf B}$. In other words, one has $J_{\perp}={\rm sgn}(E_{\perp})J_{c\perp}$ with $E_{\perp}$ standing for the component of ${\bf E}$ along the direction ${\bf B}\times ({\bf J}\times{\bf B})$. In more general configurations, the maximum projection condition is not so simple due to the vectorial character of the problem. Thus, within the DCSM framework, the current density must transit from one state to another that fulfill the conditions $|{\bf J}_{\perp}|\leq J_{c\perp}$ and $|{\bf J}_{\parallel}|\leq J_{c\parallel}$. This may be expressed by a relation of the kind ${\bf J}\in\Delta$ with $\Delta$ having a rectangular section of size $2J_{c\parallel}\times 2J_{c\perp} $ in this case. In summary, the critical state model in general 3D systems is posed by the system of equations $$\begin{aligned}
\label{eq:maxwell}
\nabla \times {\bf E} &=& - \partial _t{\bf B} \qquad ;\,
\nabla \times {\bf H} = {\bf J}\;\; ({\bf B}=\mu_{0}{\bf H} )
\nonumber\\
\nabla\cdot{\bf B} &=& 0 \qquad \qquad;\,{\rm max}\,\,{\bf J}\cdot\hat{\bf E} \,\,\,{\rm with}\,\,\, {\bf J}\in \Delta\, .\end{aligned}$$ Notice that, as equilibrium magnetization is usually neglected in the critical state regime, ${\bf B}=\mu_{0}{\bf H}$ is used.
The integration of the above system of equations supplemented by appropriate boundary conditions may be cumbersome, so that an alternative formulation has been frequently used, that is fully equivalent, and states the problem in a variational form. It is briefly explained in the forthcoming paragraphs.
Variational statement of the critical state problem
---------------------------------------------------
From the mathematical point of view, the above problem is equivalent to the incremental minimization of the functional (field Lagrangian) $$\label{eq:minprin}
{L}[{\bf H}] \equiv
\int_{\IR ^3}\left[\frac{\mu_0}{2}(\Delta{\bf H})^2 + {\bf p}\cdot(\nabla \times
{\bf H} -{\bf J})\right]d^{3}{\bf r} \, .$$
Here, one introduces the variable ${\bf p}$ as a [*Lagrange multiplier*]{} for enforcing Ampère’s law. $\Delta{\bf H}$, on the other hand, represents the magnetic field increment for the time step under consideration. Additionally, the algebraic condition ${\bf J}\in \Delta$ should be fulfilled in the minimization.
Notice that the Euler–Lagrange equations for this variational problem are
$$\begin{aligned}
\partial _{\bf p} {\cal L} = &\, 0\, & \Rightarrow \nabla \times {\bf H} - {\bf J}
\nonumber\\ \nonumber\\
\partial _{\bf H}{\cal L} - \partial ^i \left( \frac {\partial {\cal L}}{\partial _i {\bf H}} \right) = &\, 0\, & \Rightarrow \mu _0
\Delta {\bf H} = -
\nabla \times {\bf p}\end{aligned}$$
that identify ${\bf p} \simeq {\bf E} \Delta t$. From the mathematical point of view, here $\bf J$ is no longer a variable, but plays the role of a parameter to be adjusted in a direct algebraic minimization, i.e.: $${\rm max}\,\,(\hat{\bf E}\cdot{\bf J}) \Leftrightarrow {\rm max} \,\,({\bf p}\cdot{\bf J})$$
As one can see, both Ampere and Faraday’s laws are included in the variational formulation, as well as any domain $\Delta$ for the critical current material law. Technically, we emphasize that the shorter is the path step, the better agreement with the standard Maxwell equation formulation. On the other hand, numerical methods for discrete constrained minimization can be used as an alternative to the integration methods for Maxwell’s equations, that happends to be very convenient for dealing with 3D problems.
Application: 2D and 3D slab geometry.
-------------------------------------
A further advantage of the methodology introduced above is that the constraint relation ${\bf J}\in\Delta$ allows a pictorial representation that provides a useful tool for the understanding of the current flow structures that arise in the longitudinal configurations. Notice, that in our case (DCSM conditions) $\Delta$ may be depicted by the cylindrical region in Fig.\[fig\_1\].
Regarding the specific details about the mathematical technique for obtaining the numerical solution of Eq.(\[eq:minprin\]), the interested reader is addressed to our Ref.. There, we analyzed a number of situations that are easily translated to the study within this work. Here, we will just mention that our proposal consists of transforming the volume integral over the whole space in (\[eq:minprin\]), into a double integration over the sample’s volume. Then, upon discretization, one solves for the distribution of current in a proper set of circuits, under the corresponding constraints for the components of ${\bf J}$ parallel and perpendicular to the local magnetic field. Thus, when one chooses the infinite slab geometry depicted in Fig.\[fig\_1\], such circuits are naturally defined by a collection of current layers within the $xy$-plane, and each carrying a current density given by $[J_{x}(z_i),J_{y}(z_i)]$, with $z_i$ the position of the layer. Notice that, owing to the planar translational symmetry, a $z$-component of ${\bf J}$ may be ruled out.
[ Better resolution in original paper. Panel (a): magnetic process considered in this work. A magnetic field $H_{z0}$ is applied perpendicular to the surface of a superconducting slab, that is later subjected to a transport current along the $y$-axis and to an increasing field $H_{y0}$. Panel (b): the critical current restriction is represented by a cylindrical region $\Delta$ around the local magnetic field axis (length $2J_{c\parallel}$ and radius $J_{c\perp}$). $\Delta_p$ is the projection onto the slab ($xy$)-plane. $\gamma$ is the angle between the field and the $z$-axis. The lower inset depicts an element of the slab and the current backflow.](fig1_rescaled.eps "fig:"){width="37.50000%"}]{}
We emphasize that the slab geometry enables to study the appearance of the focused physical phenomena with the least mathematical complication, i.e:. this configuration allows to clearly determine the mechanisms related to the negative currents. As a main fact it will be established that, when building the parallel configuration, the response of the superconductor depends on the limitations for the current density established by the depinning threshold $J_{c\perp}$, on the orientation of the local magnetic field, and eventually on the flux cutting restriction $J_{c\parallel}$. This is easily understood at a qualitative level just by glancing at Fig.\[fig\_1\]. The critical current restriction is given by the region $\Delta_{\rm p}$ that is the intersection between the cylinder $\Delta$ and the $xy$-plane, where the current flows. For moderate values of the angle $\gamma$ between the local magnetic field and the $z$-axis, $\Delta_{\rm p}$ is an ellipse of semi-axes $J_{c\perp}$ and $J_{c\parallel}^{\rm p}$ with
$$J_{c\parallel}^{\rm p}={J_{c\perp}}/{\cos{\gamma}}=J_{c\perp}{\sqrt{H_{x}^{2}+H_{y}^{2}+H_{z}^{2}}}/{H_{z}} \, ,$$
An increase of the in-plane magnetic field component will result in a tilt of the cylinder, by an increase of the angle $\gamma$. Note in particular that, initially, the maximum value of the in-plane parallel current density, $J_{c\parallel}^{\rm p}$, grows with the angle $\gamma$, independent of $J_{c\parallel}$ (which is, thus, absent from the theory) until the maximum value $\sqrt{J_{c\perp}^{2}+J_{c\parallel}^{2}}$ is reached. Then, the ellipse is truncated and eventually would be practically a rectangle of size $2J_{c\parallel}\times 2J_{c\perp}$ when $\gamma \to\pi/2$. Outstandingly, for large values of $\chi\equiv J_{c\parallel}/J_{c\perp}$ (long cylinders), the critical current along the parallel axis, $J_{c\parallel}^{\rm p}$ increases more and more as the weight of $H_{z0}$ decreases and, furthermore, this quantity is always beyond the individual values $J_{c\perp}$ and $J_{c\parallel}$
Simplified analytical model {#secAnalytic}
===========================
Here, we show that some of the experimental features that will be obtained later on, from numerical calculations, may be already predicted by a simplified analytical model. Let us consider the excitation process depicted in Fig.\[fig\_1\] for the particular case $H_{z0}=0 \Rightarrow \gamma=\pi /2$ (the region $\Delta_{\rm p}$ is a rectangle with axes defined by the directions parallel and perpendicular to ${\bf H}$).
Governing equations
-------------------
Ampère’s law takes the following form for the infinite slab geometry considered in this work $$%
\label{eq:amperexy}
-\frac{dH_{y}}{dz}=J_{x}
\quad ; \quad
\frac{dH_{x}}{dz}=J_{y} \, .
%$$ On the other hand, following the theory issued in Ref. , one can show that such expressions may be transformed into the polar form $$%
\label{eq:ampere}
-H\frac{d\alpha}{dz}=J_{\parallel}^{\rm p}
\quad ; \quad
\frac{dH}{dz}=J_{\perp}^{\rm p}
%$$ with $H=\sqrt{H_{x}^{2}+H_{y}^{2}}$ the modulus of the magnetic field vector, and $\alpha = {\rm atan}(H_y/H_x)$ the angle between such vector and the $x$-axis.
Now, the thresholds of flux depinning and cutting imply the in-plane conditions $$%
\label{eq:critical}
|J_{\parallel}^{\rm p}|\leq J_{c\parallel}^{\rm p}(\gamma =\pi /2)=J_{c\parallel}
\quad ; \quad
|J_{\perp}^{\rm p}|\leq J_{c\perp} \, .
%$$ It is apparent that, in general, Eq.(\[eq:ampere\]) and the conditions in Eq.(\[eq:critical\]) would not straightforwardly lead to the solution of the problem. Typically, one should also use Faraday’s law, either by explicit introduction of the related electric fields (as in Ref. ), or by our variational statement. Nevertheless, in this case ($\gamma =\pi/2\Rightarrow J_{c\parallel}^{\rm p}=J_{c\parallel}$), the resolution noticeably simplifies. In fact, for the situation considered, we will have a combination of the cases $J_{\parallel}^{\rm p}=0,\pm J_{c\parallel}$ and $J_{\perp}^{\rm p}=0,J_{c\perp}$ and integration of Eq.(\[eq:ampere\]) is straightforward. For further mathematical ease, we will also consider $J_{c\parallel}$ and $J_{c\perp}$ to be field independent constants in this work.
The following normalization, based on the physical parameters that define the problem, will be used: $\vec{\jmath}\equiv {\bf J}/J_{c\perp}$, ${\bf h}\equiv {\bf H}/J_{c\perp}a$ and ${\tt z}\equiv z/a$ ($a$ is the thickness of the slab). The origin of coordinates will be taken at the center of the sample.
Following the notation introduced in Ref. we will refer to different zones within the sample that are, in brief, macroscopic regions where well defined dissipation mechanisms occur. Inserting our normalized units, there can be T zones, where only flux depinning (transport) occurs (${j}_{\parallel}=0\, ,{j}_{\perp}=\pm 1$), C zones, where only flux cutting occurs (${j}_{\parallel}=\pm\chi\, ,{j}_{\perp}=0$), and CT zones where both transport and cutting occur (${j}_{\parallel}=\pm\chi\, , {j}_{\perp}=\pm 1$). Finally, one will have O zones where neither flux transport nor cutting take place (${j}_{\parallel}=0\, , {j}_{\perp}=0$). Introducing these possibilities in Eqs.(\[eq:ampere\]) and (\[eq:critical\]) one gets the following cases for the incremental behavior of the magnetic field in polar components $$\begin{aligned}
%
dh=\left\{
%
\begin{array}{rr}
%
0\qquad \rm{(O,C)}&
\\
\pm\, dz\;\; \rm{(T,CT)}&
\end{array}
\right.
%
\!; \;
d\alpha=\left\{
%
\begin{array}{rr}
%
0\qquad\qquad \rm{(O,T)}&
\\
\pm\,({\chi}/{h})\,dz\;\; \rm{(C,CT)}&
\end{array}
\right. \, ,
%\end{aligned}$$ and all that remains for obtaining the penetration profiles is to solve successively (integrate) for $h$ and $\alpha$ with the corresponding boundary conditions (evolutionary surface values $h_{0},\alpha_{0}$). The case selection has to be made according to Lenz’s law. We note in passing that further specification related to the sign is usually included in the notation. Thus, a T$_{+}$ zone will exactly mean $dh = +dz$.
Magnetic process
----------------
[ Penetration of the magnetic field components and rotation angle in the longitudinal transport experiment ($H_{z0}=0$) for a superconducting slab of thickness $2a$, as calculated from Eq.(\[eq:ampere\]). The zone structure induced by increasing the field $H_{y0}$ is marked upon some of the curves. The dashed line corresponds to the unstable regime (see text). Dimensionless units for $h$ and $z$ are defined in the text.](prl_ruiz_fig2_resub.eps "fig:"){width="45.00000%"}]{}
### Application of current
To start with, the application of the transport current along the $y$-axis produces a T$_{+}$ zone $$\begin{aligned}
dh= dz \quad ; \quad d\alpha = 0
\nonumber\\
{\Downarrow}\qquad\qquad
\\
h=h_{x0}+{\tt z}-1\quad ; \quad \alpha=0 \, ,
\nonumber\end{aligned}$$ that penetrates from the surface until the point where $h$ equals $0$, i.e.: ${\tt z}_{p0}=1-I_{tr}$. In our units, ${\tt z}_{p0}=0.5$ for $I_{tr}=h_{x0}=0.5$. An O zone appears in the inner region $0<{\tt z}<{\tt z}_{p0}$ as far as $I_{tr}<1$.
### Application of parallel field: initial steps
The above situation remains valid until $h_{y0}$ is applied. Then, upon increasing $h_{y0}$, flux line rotation starts on the surface and the perturbation propagates towards the center in the form of a C$_{-}$T$_{+}$ zone defined by $$\begin{aligned}
dh= dz \quad ; \quad d\alpha = -\chi/h\, dz\qquad\qquad
\nonumber\\
{\Downarrow}\qquad\qquad\qquad\qquad\qquad
\\
h=h_{x0}+{\tt z}-1\; ; \; \alpha=\alpha_{0}+\chi{\rm ln}[1+({\tt z}-1)/h_{0}] \, ,
\nonumber\end{aligned}$$ that covers the range ${\tt z}_{c}^{-}<{\tt z}<1$, defined by $\alpha=0\Rightarrow{\tt z}_{c}^{-}=1+h_{0}[{\rm exp}(-\alpha_{0}/\chi)-1]$. The former T$_{+}$ zone is pushed towards the center and occupies the interval ${\tt z}_{p}^{-}<{\tt z}<{\tt z}_{c}^{-}$ with ${\tt z}_{p}^{-}=1-h_{0}$. Finally, an O zone fills the core $0<{\tt z}<{\tt z}_{p}^{-}$.
The upper panes of Fig.\[fig\_2\] sketch the above described structure marked on the cartesian components of the magnetic field vector. The transition points between the different regimes are highlighted.
### Application of parallel field: instability at the center
The O/T$_{+}$/C$_{-}$T$_{+}$ structure remains valid until the condition ${\tt z}_{p}^{-}=0\Leftrightarrow h_{0}=1$ is reached, i.e.: the modulus of ${\bf h}$ penetrates completely within the interval $0<{\tt z}<1$. Then, the O zone disappears, and a T$_{+}$/C$_{-}$T$_{+}$ structure fills the sample. We emphasize that this configuration becomes unstable owing to the boundary condition $h_{x}({\tt z}=0)=0$ that is dictated by the symmetry of $j_y$ around the center. Thus, corresponding to the even behavior of $j_y({\tt z})$, $h_{x}({\tt z})$ must be an odd function. In physical terms, flux vortices penetrate from the surface with some orientation given by the components of the vector $(h_{x},h_{y})$. Owing to the critical condition for the penetration of the field $dh/dz = 1$, as soon as the modulus reaches the centre, flux rotation must take place there. This is needed for accommodating the vector to the condition ${\bf h}({\tt z}=0)=(0,h_{y}({\tt z}=0))=(0,h({\tt z}=0))$. On the other hand, as the angle variation is determined by the value of $J_{c\parallel}$, a jump is induced at the centre, i.e.: $\alpha({\tt z}=0)\to\pi /2$, and the related instability may be visualized by a critical C$_{+}$T$_{+}$/T$_{+}$/C$_{-}$T$_{+}$ profile (dashed line in Fig. \[fig\_2\]) in which the field angle decreases from its surface value $\alpha_{0}$ to $0$ in the C$_{-}$T$_{+}$ region, then keeps null within the T$_{+}$ zone, and suddenly increases to the value $\pi/2$ in the inner C$_{+}$T$_{+}$ band defined by $$\begin{aligned}
dh= dz \quad ; \quad d\alpha = \chi/h\, dz\qquad\qquad\quad
\nonumber\\
{\Downarrow}\qquad\qquad\qquad\qquad\qquad
\\
h=h_{x0}+{\tt z}-1\; ; \; \alpha=\pi/2-\chi{\rm ln}[1+{\tt z}/(h_{0}-1)]\, .
\nonumber\end{aligned}$$ In fact, a C$_{+}$T$_{+}$/C$_{-}$T$_{+}$ structure is stabilized with the intersection between regions at the point \[$\alpha^{{+},{+}}({\tt z}_{\rm v})=\alpha^{{-},{+}}({\tt z}_{\rm v})$\] given by ${\tt z}_{\rm v}= 1-h_{0}+\surd{h_{0}(h_{0}-1){\rm exp}[(\pi/2-\alpha_{0})/\chi]}$. Note that, upon further increasing $h_{y0}$ the point ${\tt z}_{\rm v}$ follows the rule ${\tt z}_{\rm v}(h_{y0}\to\infty)\to (1+h_{x0}/\chi)/2$. All these features have been marked in the lower pane of Fig. \[fig\_2\] .
### Physical phenomena
The previous results allow to identify the following properties as $h_{y0}$ is increased: (i) the appearance of a surface layer with negative transport current density (mind the slope of $h_{x}$ in Fig.(\[fig\_2\]) in view of Eq.(\[eq:amperexy\])), and (ii) the applied magnetic field [*re-entry*]{} as related to the inner C$_{+}$T$_{+}$ zone. These features will be confirmed along the forthcoming paragraphs, where the numerical solution of the problem is presented. Additionally, we will show that the inclusion of a third component of the magnetic field ($h_{z0}\neq 0$ in what follows) allows to unveil further details reported in the literature. In particular, the occurrence of the negative current phenomenon at specific locations on the surface of the sample and for a given range of applied magnetic field will be understood within the 3 dimensional scenario.
Numerical results {#secNumeric}
=================
Next, we detail the results obtained numerically for different material laws, as related to the selection of the critical current region $\Delta$. We restrict the plots to the limiting cases $j_{c\parallel}\to\infty$ and $j_{c\parallel} = 1$ (infinite and unit aspect ratio, or also named after T and CT states for obvious reasons). The information of interest for intermediate values is straightforwardly interpolated.
From the technical side, we must point out that when minimizing $L$ \[see Eq.(\[eq:minprin\])\] a slightly smoothed version of the cylindrical region $\Delta$ has been considered by means of a [*superelliptic*]{} relation[@superellipse] given by $$\label{eq:superellipse}
j_{\perp}^{2n}+(j_{\parallel}/\chi)^{2n}\leq 1$$ with $n=4$. This statement performs with a high stability from the numerical point of view. Figs. \[fig\_3\] and \[fig\_4\] display the main features obtained for the longitudinal transport experiment when the third component of the magnetic field ($h_{z0}$) is incorporated. First, we will analyze the properties of the field (${\bf h}(z)$) and current density (${\vec{\jmath}\,(z)}$) profiles, for a longitudinal configuration built in the fashion described in Fig.(\[fig\_1\]). Fig.(\[fig\_3\]) contains the behavior of these quantities as $h_{y0}$ (applied parallel field) is increased, subsequent to the application of the transport current. This is done for a low and a high value of the perpendicular magnetic field $h_{z0}$. For the meaning of low and high recall that, along this work, the units are relative to the characteristic penetration field value $H^{*}=J_{c\perp}a$
Field and current density penetration profiles
----------------------------------------------
[{width="100.00000%"}]{}
The curves in Fig.(\[fig\_3\]) provide a basic mapping of the physical conditions in which negative currents occur. For completeness and for quantitative purposes, we have included both the field and current density profiles. Nevertheless, they are related by Ampère’s law (Eq.(\[eq:amperexy\])) as one can easily check at qualitative level, i.e.: in the slab geometry $J_{x,y}$ is the slope of $H_{y,x}$ respectively.
Recall that negative values for the transport current density $j_{y}$ are neither obtained for the T or CT states when $h_{z0}$ is high ($h_{z0}\gtrsim 50$) until extreme values of the longitudinal field ($h_{y0}\gtrsim 1000$) are reached. On the contrary, one can early find negative current flow for both cases when $h_{z0}=1$. If $j_{\parallel}$ is unbounded (T states) the ${j}_{y}(z)$ structure becomes rather inhomogeneous as $h_{y0}$ increases and takes the form of a highly positive layer in the center [*shielded*]{} by a prominent negative region. When $j_{\parallel}$ is bounded (CT states) one observes a negative layer at the surface that eventually disappears when $h_{y0}$ increases more and more ($h_{y0}>50$).
Some fine structure details are also to be noticed: (i) for the magnetic process under consideration, the [*partial penetration regime*]{} in which the flux free core progressively shrinks to zero (curves labelled $h_{y0}=0.005\, \cdots\, 0.845$) is practically independent of the critical current model (region) used, (ii) the peaked structure of $j_{y}(z)$ for the T-states at $h_{z0}=1$ (curves labelled $h_{y0}=10 \,\cdots\, 300$) is accompanied by a similar behavior in $j_{x}(z)$ that relates to a subtle magnetic field reentry phenomenon in $h_{y}(z)$ as outlined in the plot, (iii) the negative values of ${j}_{y}(z)$ are obtained for smaller and smaller $h_{y0}$ as $h_{z0}$ also decreases. In fact, negative values can happen even for the partial penetration regime ($h_{y0}\lesssim 0.845$) when $h_{z0}$ tends to $0$, in accordance with the analytical model presented before (Sec.(\[secAnalytic\])).
Experimental quantities
-----------------------
[{width="100.00000%"}]{}
For a closer connection with real experiments, we have also calculated the sample’s magnetic moment ${\bf M}$ as a function of the longitudinal field. The fingerprints of negative current flow will be identified. Fig. \[fig\_4\] displays the magnetization process of the slab as a function of the applied longitudinal field amplitude $h_{y0}$. ${\bf M}$ (in units of $J_{c\perp}a^{2}$), as well as the transport current density at the center $j_{y}(0)$ and at the surface $j_{y}(a)$ are displayed.
Several features are to be identified:
\(i) Unlimited growth of $M_{x}(h_{y0})$ and $j_{y}(0)$ occurs for the T-states, in which $j_{\parallel}$ is unbounded. On the other hand, the appearance of a peak structure in $M_{x}(h_{y0})$ correlates with a maximum value of the transport current density at the center of the slab for the CT states. The obtained maximum value $j_{y}^{max}(0)= 1.2968$ corresponds to the optimal orientation of the region $\Delta$ in which the biggest distance within the superelliptic hypothesis is reached. Such situation is sketched in Fig.\[fig\_5\] and one may check the numeric result from the expression $${\rm max}\; j_{c\parallel}^{p}=(1+\chi^{2n/(n-1)})^{(n-1)/2n} \, ,$$ when the choices $\chi=1\, , n=4$ are used. The above formula is obtained from Eq.(\[eq:superellipse\]) after straightforward calculations. Notice that, as a limiting case, it produces the expected value $2^{1/2}$ for the diagonal of a perfect square (i.e.: $n\to\infty$ in Eq.(\[eq:superellipse\])).
\(ii) For the unbounded case, in the low $h_{z0}$ regime, the negative current density at the surfaces stabilizes towards the value $j_{y}(a)=-0.5$.
\(iii) For the bounded case, and moderate or low $h_{z0}$, the transport current at the surface stabilizes towards the value $j_{y}(a)=0.422$ whether or not it has been negative along the ramp of applied longitudinal field.
\(iv) As a general rule, the smaller the value of $h_{z0}$, the sooner the negative transport current is found. In the CT case, this also increases the range of longitudinal field for which negative values are observed.
[ Side view of the critical current region $\Delta$ rotated by an angle $\gamma$ that produces a maximal parallel current at the $xy$-plane (this plot is a specific longitudinal section of Fig.\[fig\_1\]). The precise orientation takes place for a definite value of the applied magnetic field $h_{y0}$. The superelliptical region considered in this work has been plotted together with the strictly CT model.](prl_ruiz_fig5_resub3.eps "fig:"){width=".50\textwidth"}]{}
Discussion {#secDisc}
==========
Within the previous sections we have displayed a number of cases in which negative transport layers are predicted for type-II superconductors if the longitudinal field configuration (${\bf J}\parallel{\bf B}$) is induced by some external process. Our theoretical investigations allow to identify the following relevant aspects for the appearance of such phenomenon:
\(i) the physical mechanisms of [*flux cutting and depinning*]{}, that may be modelled by the thresholds for the components of ${\bf J}$ parallel and perpendicular to the local magnetic field. In this sense, we have shown that negative currents are much easier observed for materials in which $J_{c\parallel}$ and $J_{c\perp}$ are independent and $J_{c\parallel}\gg J_{c\perp}$. On the other hand, additional calculations (not displayed) imply that alternative ansatzs as the isotropic model ($J_{\parallel}^2+J_{\perp}^2 \leq J_{c}^2$, i.e.: the difference between the mechanisms responsible for the thresholds $J_{c\parallel}$ and $J_{c\perp}$ are not relevant and the region $\Delta$ is a circle) cannot predict such behavior. However, if some interaction is allowed between the cutting and depinning thresholds \[moderately smoothed $J_{\perp}(J_{\parallel})$ relation, i.e.: the region $\Delta$ is a superellipsoid\], the negative current flow will occur for some range of conditions.
[ Penetration of a magnetic field parallel to the axis of a finite superconducting cylinder (half longitudinal section is shown for symmetry reasons). The component of the magnetic field perpendicular to the lateral surface is visualized by a set of arrows with normalized lengths.The dashed line represents the symmetry axis.](prl_ruiz_fig6_resub3.eps "fig:"){width=".35\textwidth"}]{}
\(ii) The [*sequence in which the external excitation components are applied*]{} to the superconductor. Thus, as one could expect from the idea that in the critical state all changes proceed from the surface toward the centre of the sample, negative current structures are enhanced when the applied magnetic field is applied after originally building a subcritical transport profile. Recall that in the situations depicted in Fig.\[fig\_3\], the initial current flow is compressed more and more, even until compensating negative values are needed at the surface, for maintaining the biased transport current. Along these lines, we should comment that when simulating experiments in which the transport current is applied subsequent to the field, our theory does not predict negative flow values at all. On the contrary, in such cases, what one gets is a [*compression*]{} of the original field penetration profile, until the increasing transport current leads to dissipation.
Additional physical considerations can be done so as to cover the full experimental scenario. In particular, though our analysis has been done within the infinite slab geometry, one can straightforwardly argue about the extrapolation to real experiments. Thus, the inclusion of the third component of the magnetic field $H_{z0}$ relates to the last relevant aspect:
\(iii) The importance of the [*finite size effects*]{}. Notice that from our numerical calculations, one can predict that negative currents should be more prominent in those regions of the sample where the component of ${\bf H}$ perpendicular to the current layers is less important. This will occur at the central region of the sample’s lateral surface, where end effects are minimal. Thus, considering that a real sample in a longitudinal configuration will be typically a rod with field and transport along the axis, the above idea is straightforwardly shown by plotting the penetration of an axial field in a finite cylinder. This has been done in Fig.\[fig\_6\]. The plot shows the distortion of the magnetic field, shielded by the induced supercurrents. Just for visual purposes, we have superimposed the horizontal component of the magnetic field, along the lateral side of the cylinder. It is apparent that the normal component of ${\bf H}$ will be enhanced close to the bases and tend to zero at the central region. Thus, inhomogeneous surface current densities, with negative flow at the mid part should be expected.
Concluding remarks {#secConc}
==================
In this article, we have shown that the counterintuitive effect of negative current flow in type-II superconductors may be predicted and quantified by means of the critical state theory. For restricted situations (infinite slab geometry and fields parallel to the surface), the prediction may even be done within a simplified analytical model. Three dimensional effects may be incorporated by numerical methods when a third component of the magnetic field, perpendicular to the surface of the slab is considered. The analysis of this situation has allowed to conclude that negative transport is enhanced for superconductors in which the flux cutting barrier is much above the depinning value, and at those regions of the sample where the magnetic field is basically parallel to the surface (central part in cylinder geometry).
Extensions of this work are planned along two lines: (i) the actual evaluation of the longitudinal problem in finite length samples (field and transport along the axis of a rod), and (ii) the extrapolation of the current density profiles beyond the critical state threshold, so as to include the dissipation effects.
Acknowledgment {#acknowledgment .unnumbered}
==============
This work was supported by the Spanish CICyT project MAT2008-05983-C03-01 and the DGA grant PI049/08. H. S. Ruiz acknowledges a grant from the Spanish CSIC (JAE program).
[99]{}
A. M. Campbell and J. E. Evetts, Adv. Phys. [**21**]{}, 199 (1972) and references therein; [T. Ezaki, K. Yamafuji, and F. Irie]{}, J. Phys. Soc. Japan [**40**]{}, 1271 (1976); , J. Phys. F [**2**]{}, 510 (1971); [D. G. Walmsley and W. E. Timms]{}, J. Phys. F[**7**]{}, 2373 (1977); , Phil. Mag. B [**37**]{}, 111 (1978). , Physica C [**298**]{}, 115 (1998). , J. Low Temp. Phys. [**38**]{}, 353 (1980); [E. H. Brandt]{}, J. Low Temp. Phys. [**39**]{}, 41 (1980) , JEPT Lett. [**53**]{}, 115 (1991). , Solid State Comm. [**103**]{}, 313 (1997). , Phys. Rev. Lett. [**71**]{}, 3367 (1993). , Phys. Rev. Lett. [**87**]{}, 127004 (2001). A. Badía-Majós, C. López, and H. S. Ruiz, Phys. Rev. B [**80**]{}, 144509 (2009). , Phys. Rev. B [**30**]{}, 5041 (1984). , J. Low Temp. Phys. [**37**]{}, 43 (1979); [J. R. Clem]{}, Phys. Rev. B [**26**]{}, 2463 (1982).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In nice environments, such as Lipschitz or chord-arc domains, it is well-known that the solvability of the Dirichlet problem for an elliptic operator in $L^p$, for some finite $p$, is equivalent to the fact that the associated elliptic measure belongs to the Muckenhoupt class $A_\infty$. In turn, any of these conditions occurs if and only if the gradient of every bounded null solution satisfies a Carleson measure estimate. This has been recently extended to much rougher settings such as those of 1-sided chord-arc domains, that is, sets which are quantitatively open and connected with a boundary which is Ahlfors-David regular. In this paper, we work in the same environment and consider a qualitative analog of the latter equivalence showing that one can characterize the absolute continuity of the surface measure with respect to the elliptic measure in terms of the finiteness almost everywhere of the truncated conical square function for any bounded null solution. As a consequence of our main result particularized to the Laplace operator and some previous results, we show that the boundary of the domain is rectifiable if and only if the truncated conical square function is finite almost everywhere for any bounded harmonic function. Also, we obtain that for two given elliptic operators $L_1$ and $L_2$, the absolute continuity of the surface measure with respect to the elliptic measure of $L_1$ is equivalent to the same property for $L_2$ provided the disagreement of the coefficients satisfy some quadratic estimate in truncated cones for almost everywhere vertex. Finally for the case on which $L_2$ is either the transpose of $L_1$ or its symmetric part we show the equivalence of the corresponding absolute continuity upon assuming that the antisymmetric part of the coefficients has some controlled oscillation in truncated cones for almost every vertex.'
address:
- |
Mingming Cao\
Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM\
Consejo Superior de Investigaciones Cient[í]{}ficas\
C/ Nicolás Cabrera, 13-15\
E-28049 Madrid, Spain
- |
José Mar[í]{}a Martell\
Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM\
Consejo Superior de Investigaciones Cient[í]{}ficas\
C/ Nicolás Cabrera, 13-15\
E-28049 Madrid, Spain
- |
Andrea Olivo\
Departamento de Matemática\
Facultad de Ciencias Exactas y Naturales\
Universidad de Buenos Aires and IMAS-CONICET\
Pabellón I (C1428EGA), Ciudad de Buenos Aires, Argentina
author:
- Mingming Cao
- 'José Mar[í]{}a Martell'
- Andrea Olivo
title: 'Elliptic measures and square function estimates on 1-sided chord-arc domains'
---
[^1]
Introduction {#sec:intro}
============
A classical theorem of F. and M. Riesz [@RR] states that $$\label{eq:Riesz}
\begin{array}{c}
\w \ll \mathcal{H}^1|_{\pom}\ll \w \text{ on } \partial \Omega \text{ for any simply connected } \\[0.2cm]
\text{domain $\Omega \subset \R^2$ with a rectifiable boundary},
\end{array}$$ where $\w$ denotes the harmonic measure relative to the domain $\Omega$. A quantitative version of this result was obtained later by Lavrentiev [@Lav] who showed that in a chord-arc domain in the plane, harmonic measure is quantitatively absolutely continuous with respect to the arc-length measure, that is, harmonic measure is an $A_\infty$ weight with respect to surface measure. After these two fundamental results there has been many authors seeking to find necessary and sufficient geometric criteria for the absolute continuity, or its quantitative version, of harmonic measure with respect to surface measure on the boundary of a domain in higher dimensions. In general, those can be divided into two categories: quantitative and qualitative.
In the quantitative category it has been recently established that if $\Omega\subset\mathbb{R}^{n+1}$, $n\ge 2$, is a 1-sided CAD (chord-arc domain, cf. Definition \[defi:NTA\]), then the following are equivalent: $$\label{eq:UR}
\begin{split}
&{\rm (a)}\quad \partial \Omega \text{ is uniformly rectifiable}, \\
&{\rm (b)}\quad \Omega \text{ satisfies the exterior corkscrew condition, hence it is a CAD}, \\
&{\rm (c)}\quad \w \in A_{\infty}(\sigma).
\end{split}$$ Here, $\sigma=\mathcal{H}^{n}|_{\pom}$ denotes the surface measure and $A_{\infty}(\sigma)$ is as mentioned above the scale-invariant version of absolute continuity. The direction (a) implies (b) was shown by Azzam, Hofmann, Nyström, Toro, and the second named author of the present paper in [@AHMNT]. That (b) implies (c) was proved by David and Jerison in [@DJ], and independently by Semmes in [@Se]. Also, (a) implies (c) was proved by Hofmann and the second author of this paper in [@HM]. Both, jointly with Uriarte-Tuero [@HMU] also established that (c) implies (a). The equivalent statements in reveal the close connection between the regularity of the boundary of a domain and the good behavior of harmonic measure with respect to surface measure. In addition, connects several known results, including the extension of [@RR] on Lipschitz domain [@D2], $L^p_1$ domain [@JK1] and $\BMO_1$ domain [@JK2].
For divergence form elliptic operators $Lu=-\div(A \nabla u)$ with real variable coefficients, that $(b)$ implies $(c)$ (with the elliptic measure $\omega_L$ in place of $\omega$) was proved by Kenig and Pipher in [@KP] under some Carleson measure estimate assumption for the matrix of coefficients $A$. The converse, that is, the fact that $(c)$ implies $(b)$ on a 1-sided CAD for the Kenig-Pipher class has been recently obtained by Hofmann, the second author of the present paper, Mayboroda, Toro, and Zhao in [@HMMTZ1; @HMMTZ2] (see also [@HMT1] for a previous result in a smaller class of operators). In another direction, it was shown in [@CHMT] that for any real (not necessarily non-symmetric) elliptic operator $L$, $\w_L \in A_{\infty}(\sigma)$ is equivalent to the so-called Carleson measure estimates, that is, every bounded weak null solution of $L$ satisfies Carleson measure estimates.
On the other hand, the qualitative version of has been also studied extensively. In contrast with , some counterexamples have been presented to show how the absolute continuity of harmonic measure is indeed affected by the topology/geometry of the domain and its boundary.
- Example 1. Lavrentiev constructed in [@Lav] a simply connected domain $\Omega \subset \R^2$ and a set $E \subset \partial \Omega$ such that $E$ has zero arclength, but $\w(E)>0$.
- Example 2. Bishop and Jones in [@BJ] found a uniformly rectifiable set $E$ on the plane and some subset of $E$ with zero arc-length which carries positive harmonic measure relative to the domain $\re^2\setminus E$.
- Example 3. Wu proved in [@W] that there exists a topological ball $\Omega \subset \R^3$ and a set $E \subset \partial \Omega$ lying on a 2-dimensional hyperplane so that Hausdorff dimension of $E$ is 1 (which implies $\sigma(E)=0$) but $\w(E)>0$.
- Example 4. In [@AMT], Azzam, Mourgoglou and Tolsa obtained that for all $n \geq 2$, there are a Reifenberg flat domain $\Omega \subset \R^{n+1}$ and a set $E \subset \partial \Omega$ such that $\w(E)>0=\sigma(E)$.
Compared with , Examples 1 and 2 indicate that both the regularity of the boundary and the connectivity of the domain seem to be necessary for absolute continuity to occur. However, Examples 3 and 4 say that $\w \ll \sigma$ fails in the presence of some connectivity assumption. Indeed, a quantitative form of path connectedness is contained in Example 4 since Reifenberg flat domains which are sufficiently flat are in fact NTA domains (cf. Definition \[defi:NTA\]), see [@KT Theorem 3.1]. Taking into consideration these, it is natural to investigate what extra mild assumptions are necessary to obtain the absolute continuity of harmonic measure.
It was shown by McMillan [@M Theorem 2] that for bounded simply connected domains $\Omega \subset \C$, $\w \ll \sigma \ll \w$ on the set of cone points. Later, Bishop and Jones [@BJ] obtained that for any simply connected domain $\Omega \subset \R^2$ and curve $\Gamma$ of finite length, $\w \ll \sigma$ on $\partial \Omega \cap \Gamma$. That result refined the conclusions in [@O p. 471] and [@KW Theorem 3] where $\Gamma$ was a line and a quasi-smooth curve respectively. Beyond that, in a Wiener regular domain with large complement (cf. [@AAM Definition 1.5]), Akman, Azzam and Mourgoglou [@AAM] gave a characterization of sets of absolute continuity in terms of the cone point condition and the rectifiable structure of elliptic measure. Let us point out that in all of the just mentioned results, the absolute continuity happens locally. In the case of the whole boundary, for every Lipschitz domain Dahlberg [@D1] proved that harmonic measure belongs to the reverse Hölder class with exponent $2$ with respect to surface measure, this, in turn, yields $\w \ll \sigma \ll \w$ holds. This was extended to the setting of CAD domains in [@DJ; @Se]. For general NTA domains $\Omega \subset \mathbb{R}^{n+1}$, $n \geq 1$, Badger [@B] proved that $\sigma \ll \w$ if the boundary $\partial \Omega$ has finite surface measure. When $\Omega$ is a 1-sided CAD, Akman, Badger, Hofmann and the second author established in [@ABHM] that $\partial \Omega$ is rectifiable if and only if $\sigma \ll \w$ on $\partial \Omega$, which is also equivalent to the fact that $\partial \Omega$ possesses exterior corkscrew points in a qualitative way and that $\partial \Omega$ can be covered $\sigma$-a.e. by a countable union of portions of boundaries of bounded chord-arc subdomains of $\Omega$. Based on a qualitative Carleson measure condition, they also got that the same conclusions hold for some class of elliptic operators with regular coefficients. The remarkable result in [@AHMMMTV] proved that, in any dimension and in the absence of any connectivity condition, any piece of the boundary with finite surface measure is rectifiable, provided surface measure is absolutely continuous with respect to harmonic measure on that piece. The converse was treated in [@ABoHM] by Akman, Bortz, Hofmann assuming that the boundary has locally finite surface measure and satisfies some weak lower Ahlfors regular condition.
Motivated by the previous work, the purpose of this article is to find characterizations of the absolute continuity of surface measure with respect to elliptic measure for real second order divergence form uniformly elliptic operators. Our main goal is to establish the equivalence between the absolute continuity and the finiteness almost everywhere of the conical square function applied to any bounded weak solution. To set the stage let us give few definitions (see Section \[sec:pre\] for more definitions and notation). The conical square function is defined as $$\begin{aligned}
S_{\alpha}u(x) := \left(\iint_{\Gamma_{\alpha}(x)}
|\nabla u(Y)|^2 \delta(Y)^{1-n} dY \right)^{\frac12}, \qquad x\in\pom,\end{aligned}$$ where $\delta(\cdot)=\dist(\cdot\,,\pom)$ and the cone $\Gamma_{\alpha}(x)$ with vertex at $x\in\pom$ and aperture $\alpha>0$ is given by $$\begin{aligned}
\Gamma_{\alpha}(x) = \{Y \in \Omega: |Y-x|<(1+\alpha) \delta(Y) \}.\end{aligned}$$ Similarly, we define the truncated square function $S_{\alpha}^r$ by integrating over the truncated cone $\Gamma_{\alpha}^r(x) := \Gamma_{\alpha}(x) \cap B(x, r)$ for any $r>0$.
Our main result is a qualitative analog of [@KKiPT] and [@CHMT Theorem 1.1]. More precisely, condition is a qualitative analog of $\omega_L\in A_\infty(\sigma)$ —or equivalently $\sigma\in A_\infty(\omega_L)$— while condition , or , or is a qualitative version of the so-called Carleson measure condition, which is in turn equivalent to some local scale-invariant $L^2$ estimate for the truncated conical square function.
\[thm:abs-cont\] Let $\Omega \subset \R^{n+1}$, $n\ge 2$, be a $1$-sided CAD (cf. Definition \[defi:NTA\]) and write $\sigma:=\H^n |_{\partial \Omega}$. There exists $\alpha_0>0$ (depending only on the $1$-sided CAD constants) such that for each fixed $\alpha \geq \alpha_0$ and for every real (not necessarily symmetric) elliptic operator $Lu=-\div(A \nabla u)$ the following statements are equivalent:
=0.2cm =.2cm
\[list:wL\] $\sigma \ll \w_L$ on $\partial \Omega$.
\[list:si-wL-si\] $\partial \Omega=\bigcup_{N\geq 0} F_N$, where $\sigma(F_0)=0$ and for each $N \geq 1$ there exists $C_N>1$ such that $$\begin{aligned}
C_N^{-1} \sigma(F) \leq \w_L(F) \leq C_N \sigma(F), \quad \forall\,F \subset F_N.
\end{aligned}$$
\[list:Sr-L2\] $\partial \Omega=\bigcup_{N \geq 0} F_N$, where $\sigma(F_0)=0$, for each $N \geq 1$, $F_N=\partial \Omega \cap \partial \Omega_N$ for some bounded $1$-sided CAD $\Omega_N \subset \Omega$, and $S_{\alpha}^r u \in L^2(F_N, \sigma)$ for every weak solution $u \in W_{\loc}^{1,2}(\Omega) \cap L^{\infty}(\Omega)$ of $Lu=0$ in $\Omega$ and for all (or for some) $r>0$.
\[list:Sr\] $S_{\alpha}^r u(x)<\infty$ for $\sigma$-a.e. $x \in \partial \Omega$ for every weak solution $u \in W_{\loc}^{1,2}(\Omega)\cap L^{\infty}(\Omega)$ of $Lu=0$ in $\Omega$ and for all (or for some) $r>0$.
\[list:Srx\] For every weak solution $u \in W_{\loc}^{1,2}(\Omega)\cap L^{\infty}(\Omega)$ of $Lu=0$ in $\Omega$ and for $\sigma$-a.e. $x \in \partial \Omega$ there exists $r_x>0$ such that $S_{\alpha}^{r_x} u(x)<\infty$.
We would like to make the following observation regarding the parameter $\alpha$ in the previous statement. Note first that if one of the conditions , , or holds for some $\alpha>0$, then the same condition is automatically true for all $\alpha'\le \alpha$. Thus, or implies , , or holds for all $\alpha>0$. On the other hand, for the converse implications we need to make sure that $\alpha$ does not get too small to prevent having empty cones, in which case the corresponding assumption trivially holds.
When turning to the harmonic measure, we obtain the following connection between the rectifiability of the boundary, the absolute continuity of surface measure with respect to harmonic measure, and the square functions estimates for harmonic functions.
\[thm:harmonic\] Let $\Omega \subset \R^{n+1}$, $n\ge 2$, be a $1$-sided CAD and write $\sigma:=\H^n |_{\partial \Omega}$. There exists $\alpha_0>0$ (depending only on the $1$-sided CAD constants) such that for each fixed $\alpha \geq \alpha_0$ if we write $\w$ to denote the harmonic measure for $\Omega$ then the following statements are equivalent:
=0.2cm =.2cm
\[list:rect\] $\partial \Omega$ is rectifiable, that is, $\sigma$-almost all of $\partial\Omega$ can be covered by a countable union of $n$-dimensional (possibly rotated) Lipschitz graphs.
\[list:w\] $\sigma \ll \w$ on $\partial \Omega$.
\[list:harmonic\] $S_{\alpha}^r u(x)<\infty$ for $\sigma$-a.e. $x \in \partial \Omega$ for every bounded harmonic function $u \in W_{\loc}^{1,2}(\Omega)$ and for all (or for some) $r>0$.
The equivalence of and was established in [@ABHM], while Theorem \[thm:abs-cont\] readily gives that is equivalent to .
As an application of Theorem \[thm:abs-cont\], we can obtain some additional results. The first deals with perturbations (see [@AHMT; @CHM; @CHMT; @D3; @FJK; @F; @FKP; @MPT1; @MPT2]) and should be compared with its quantitative version in the 1-sided CAD setting [@CHMT Theorems 1.3]. We note that our next result provides also a quantitative version of the work by Fefferman in [@F] who showed that in the unit ball if the right hand side of is an essentially bounded function (rather than knowing that is finite almost everywhere) then one has $\omega_{L_0} \in A_\infty(\sigma)$ if and only if $\omega_{L_1} \in A_\infty(\sigma)$.
\[thm:perturbation\] Let $\Omega \subset \R^{n+1}$, $n\ge 2$, be a $1$-sided CAD and write $\sigma:=\H^n |_{\partial \Omega}$. There exists $\alpha_0>0$ (depending only on the $1$-sided CAD constants) such that if the real (not necessarily symmetric) elliptic operators $L_0 u = -\div(A_0 \nabla u)$ and $L_1 u = -\div(A_1 \nabla u)$ satisfy for some $\alpha \ge\alpha_0$ and for some $r>0$ $$\begin{aligned}
\label{eq:rhoAA}
\iint_{\Gamma_{\alpha}^{r}(x)}
\frac{\varrho(A_0, A_1)(X)^2}{\delta(X)^{n+1}} dX < \infty,
\quad \sigma \text{-a.e. } x \in \partial \Omega,
\end{aligned}$$ where $$\varrho(A_0, A_1)(X) :=\sup_{Y \in B(X, \delta(X)/2)} |A_0(Y) - A_1(Y)|, \quad X \in \Omega,$$ then $\sigma \ll \w_{L_0}$ if and only if $\sigma \ll \w_{L_1}$.
Our second application of Theorem \[thm:abs-cont\] allows us to establish a connection between the absolute continuity properties of the elliptic measures of an operator, its adjoint and/or its symmetric part. Given $Lu=-\div(A \nabla u)$ a real (not necessarily symmetric) elliptic operator, we let $L^{\top}$ denote the transpose of $L$, and let $L^{\rm sym}=\frac{L+L^{\top}}{2}$ be the symmetric part of $L$. These are respectively the divergence form elliptic operators with associated matrices $A^\top$ (the transpose of $A$) and $A^{\rm sym}=\frac{A+A^{\top}}{2}$. In this case, the following result is a qualitative version of [@CHMT Theorem 1.6].
\[thm:wL-wLT\] Let $\Omega \subset \R^{n+1}$, $n\ge 2$, be a $1$-sided CAD and write $\sigma:=\H^n |_{\partial \Omega}$. There exists $\alpha_0>0$ (depending only on the $1$-sided CAD constants) such that if $Lu=-\div(A \nabla u)$ is a real (not necessarily symmetric) elliptic operator, and we assume that $(A-A^{\top}) \in \Lip_{\loc}(\Omega)$ and that for some $\alpha\ge \alpha_0$ and for some $r>0$ one has $$\label{eq:divCAA}
\iint_{\Gamma_{\alpha}^{r}(x)}
\left|\div_C(A-A^{\top})(X)\right|^2 \delta(X)^{1-n} dX<\infty,
\quad \sigma \text{-a.e. } x \in \partial \Omega,$$ where $$\begin{aligned}
\div_C(A-A^{\top})(X)=\bigg(\sum_{i=1}^{n+1} \partial_i(a_{i,j}-a_{j,i})(X) \bigg)_{1 \leq j \leq n+1},
\quad X \in \Omega,
\end{aligned}$$ then $\sigma \ll \w_L$ if and only if $\sigma \ll \w_{L^{\top}}$ if and only if $\sigma \ll \w_{L^{\rm sym}}$.
The paper is organized as follows. In Section \[sec:pre\], we present some preliminaries, definitions, and some background results that will be used throughout the paper. Section \[sec:proof-abs\] is devoted to showing Theorem \[thm:abs-cont\]. Finally, in Section \[sec:perturbation\], applying Theorem \[thm:abs-cont\] $\eqref{list:wL} \Leftrightarrow \eqref{list:Sr}$, we obtain a more general perturbation result about the absolute continuity of surface measure with respect to elliptic measure and then prove Theorems \[thm:perturbation\] and \[thm:wL-wLT\].
Preliminaries {#sec:pre}
=============
Notation and conventions
------------------------
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Our ambient space is $\re^{n+1}$, $n\ge 2$.
We use the letters $c$, $C$ to denote harmless positive constants, not necessarily the same at each occurrence, which depend only on dimension and the constants appearing in the hypotheses of the theorems (which we refer to as the “allowable parameters”). We shall also sometimes write $a\lesssim b$ and $a\approx b$ to mean, respectively, that $a\leq C b$ and $0<c\leq a/b\leq C$, where the constants $c$ and $C$ are as above, unless explicitly noted to the contrary. Moreover, if $c$ and $C$ depend on some given parameter $\eta$, which is somehow relevant, we write $a\lesssim_\eta b$ and $a\approx_\eta b$. At times, we shall designate by $M$ a particular constant whose value will remain unchanged throughout the proof of a given lemma or proposition, but which may have a different value during the proof of a different lemma or proposition.
Given $E \subset \R^{n+1}$ we write $\diam(E)=\sup_{x, y \in E}|x-y|$ to denote its diameter.
Given a domain (i.e., open and connected) $\Omega\subset\re^{n+1}$, we shall use lower case letters $x,y,z$, etc., to denote points on $\partial\Omega$, and capital letters $X,Y,Z$, etc., to denote generic points in $\re^{n+1}$ (especially those in $\Omega$).
The open $(n+1)$-dimensional Euclidean ball of radius $r$ will be denoted $B(x,r)$ when the center $x$ lies on $\partial\Omega$, or $B(X,r)$ when the center $X\in\re^{n+1}\setminus \partial\Omega$. A “surface ball” is denoted $\Delta(x,r):=B(x,r)\cap \partial\Omega$, and unless otherwise specified it is implicitly assumed that $x\in\partial\Omega$. Also if $\partial\Omega$ is bounded, we typically assume that $0<r\lesssim\diam(\partial\Omega)$, so that $\Delta=\partial\Omega$ if $\diam(\partial\Omega)<r\lesssim\diam(\partial\Omega)$.
Given a Euclidean ball $B$ or surface ball $\Delta$, its radius will be denoted $r_B$ or $r_\Delta$ respectively.
Given a Euclidean ball $B=B(X,r)$ or a surface ball $\Delta=\Delta(x,r)$, its concentric dilate by a factor of $\kappa>0$ will be denoted by $\kappa B=B(X,\kappa r)$ or $\kappa\Delta=\Delta(x,\kappa r)$.
For $X\in\re^{n+1}$, we set $\delta(X):=\dist(X,\partial\Omega)$.
We let $\H^n$ denote the $n$-dimensional Hausdorff measure, and let $\sigma:=\H^n |_{\partial \Omega}$ denote the surface measure on $\partial \Omega$.
For a Borel set $A\subset\re^{n+1}$, we let $\inter(A)$ denote the interior of $A$, and $\overline{A}$ denote the closure of $A$. If $A\subset \partial\Omega$, $\inter(A)$ will denote the relative interior, i.e., the largest relatively open set in $\partial\Omega$ contained in $A$. Thus, for $A\subset \partial\Omega$, the boundary is then well defined by $\partial A:=\overline{A}\setminus\inter(A)$.
For a Borel set $A\subset \partial\Omega$ with $0<\sigma(A)<\infty$, we write $\fint_{A}f\,d\sigma:=\sigma(A)^{-1}\int_A f\,d\sigma$.
We shall use the letter $I$ (and sometimes $J$) to denote a closed $(n+1)$-dimensional Euclidean cube with sides parallel to the coordinate axes, and we let $\ell(I)$ denote the side length of $I$. We use $Q$ to denote a dyadic “cube” on $\partial \Omega$. The latter exist, given that $\partial \Omega$ is ADR (see [@DS1], [@Ch], and enjoy certain properties which we enumerate in Lemma \[lem:dyadic\] below).
Some definitions
----------------
We say that a closed set $E \subset \R^{n+1}$ is $n$-dimensional Ahlfors-David regular (or simply ADR) if there is some uniform constant $C\ge 1$ such that $$C^{-1} r^n \leq \H^n(E \cap B(x, r)) \leq C r^n, \qquad \forall\,x \in E, \ r \in (0, 2\,\diam(E)).$$
\[def:CKS\] We say that an open set $\Omega \subset \R^{n+1}$ satisfies the Corkscrew condition if for some uniform constant $c \in (0, 1)$, and for every surface ball $\Delta:=\Delta(x, r)$ with $x \in \partial \Omega$ and $0< r < \diam(\partial \Omega)$, there is a ball $B(X_{\Delta}, cr) \subset B(x, r) \cap \Omega$. The point $X_{\Delta} \in \Omega$ is called a “Corkscrew point” relative to $\Delta$. We note that we may allow $r < C \diam(\partial \Omega)$ for any fixed $C$, simply by adjusting the constant $c$.
We say that an open set $\Omega$ satisfies the Harnack Chain condition if there is a uniform constant $C$ such that for every $\rho>0$, $\Lambda \geq 1$, and every pair of points $X, X' \in \Omega$ with $\min\{\delta(X), \delta(X')\} \geq \rho$ and $|X-X'| < \Lambda \rho$, there is a chain of open balls $B_1, \ldots, B_N \subset \Omega$, $N \leq C(\Lambda)$, with $X \in B_1$, $X' \in B_N$, $B_k \cap B_{k+1} \neq \emptyset$, $C^{-1} \diam(B_k) \leq \dist(B_k, \partial \Omega) \leq C \diam(B_k)$. Such a sequence is called a “Harnack Chain”.
We remark that the Corkscrew condition is a quantitative, scale invariant version of openness, and the Harnack Chain condition is a scale invariant version of path connectedness.
\[defi:NTA\] We say that $\Omega$ is a $1$-sided NTA (non-tangentially accessible) domain if $\Omega$ satisfies both the Corkscrew and Harnack Chain conditions. Furthermore, we say that $\Omega$ is an NTA domain if it is a $1$-sided NTA domain and if, in addition, $\R^{n+1} \setminus \overline{\Omega}$ satisfies the Corkscrew condition. If a $1$-sided NTA domain, or an NTA domain, has an ADR boundary, then it is called a 1-sided CAD (chord-arc domain) or a CAD, respectively.
Dyadic grids and sawtooths {#section:dyadic}
--------------------------
We give a lemma concerning the existence of a “dyadic grid”, which was proved in [@DS1; @DS2; @Ch].
\[lem:dyadic\] Suppose that $E \subset \R^{n+1}$ is an $n$-dimensional $ADR$ set. Then there exist constants $a_0>0$, $\gamma>0$, and $C_1 < 1$ depending only on $n$ and the $ADR$ constant such that, for each $k \in \Z$, there is a collection of Borel sets (cubes) $$\begin{aligned}
\D_k =\{Q_j^k \subset E: j \in \mathfrak{J}_k\}\end{aligned}$$ where $\mathfrak{J}_k$ denotes some (possibly finite) index set depending on $k$, satisfying:
[$({\alph{enumi}})$]{}
=0.1cm =.2cm
$E=\bigcup_j Q_j^k$, for each $k \in \Z$.
If $m \geq k$, then either $Q_i^m \subset Q_j^k$ or $Q_i^m \cap Q_j^k = \emptyset$.
For each $(j, k)$ and each $m<k$, there is a unique $i$ such that $Q_j^k \subset Q_i^m$.
$\diam(Q_j^k) \leq C_1 2^k$.
Each $Q_j^k$ contains some surface ball $\Delta(x_j^k, a_0 2^{-k}):=B(x_j^k, a_0 2^{-k}) \cap E$.
$\H^n(\{x \in Q_j^k: \dist(x, E \backslash Q_j^k) \leq 2^{-k} a\}) \leq C_1 a^{\gamma} \H^n(Q_j^k)$ for all $k, j$ and $a \in (0, a_0)$.
A few remarks are in order concerning this lemma.
1. In the setting of a general space of homogeneous type, this lemma has been proved by Christ [@Ch], with the dyadic parameter $1/2$ replaced by some constant $\delta \in (0, 1)$. In fact, one may always take $\delta=1/2$ (cf. [@HMMM Proof of Proposition 2.12]). In the presence of the Ahlfors-David property, the result already appears in [@DS1; @DS2].
2. For our purposes, we may ignore those $k \in \Z$ such that $2^{-k} \gtrsim \diam(E)$, in the case that the latter is finite.
3. We shall denote by $\D(E)$ the collection of all relevant $Q^k_j$, i.e., $$\begin{aligned}
\D(E) := \bigcup_{k \in \Z} \D_k,\end{aligned}$$ where, if $\diam(E)$ is finite, the union runs over those $k$ such that $2^{-k} \lesssim \diam(E)$.
4. For a dyadic cube $Q \in \D_k$, we shall set $\ell(Q)=2^{-k}$, and we shall refer to this quantity as the “length” of $Q$. Evidently, $\ell(Q) \simeq \diam(Q)$. We set $k(Q)=k$ to be the dyadic generation to which $Q$ belongs if $Q \in \D_k$; thus, $\ell(Q)=2^{-k(Q)}$.
5. Properties $(d)$ and $(e)$ imply that for each cube $Q \in \D_k$, there is a point $x_Q \in E$, a Euclidean ball $B(x_Q, r_Q)$ and a surface ball $\Delta(x_Q, r_Q) := B(x_Q, r_Q) \cap E$ such that $c\ell(Q) \leq r_Q \leq \ell(Q)$, for some uniform constant $c > 0$, and $$\begin{aligned}
\label{eq:Q-DQ}
\Delta(x_Q, 2r_Q) \subset Q \subset \Delta(x_Q, Cr_Q), \end{aligned}$$ for some uniform constant $C>1$. We shall write $$\begin{aligned}
\label{eq:BQ}
B_Q := B(x_Q, r_Q),\quad \Delta_Q := \Delta(x_Q, r_Q),
\quad \widetilde{\Delta}_Q := \Delta(x_Q, Cr_Q), \end{aligned}$$ and we shall refer to the point $x_Q$ as the “center” of $Q$.
6. Let $\Omega \subset \R^{n+1}$ be an open set satisfying the corkscrew condition and such that $\partial \Omega$ is ADR. Given $Q \in \D(\partial \Omega)$, we define the “corkscrew point relative to $Q$” as $X_Q:=X_{\Delta_Q}$. We note that $$\delta(X_Q) \simeq \dist(X_Q, Q) \simeq \diam(Q).$$
We next introduce the notation of “Carleson region” and “discretized sawtooth” from [@HM Section 3]. Given a dyadic cube $Q \in \D(E)$, the “discretized Carleson region” $\D_Q$ relative to $Q$ is defined by $$\D_Q := \{Q' \in \D(E) : Q' \subset Q\}.$$ Let $\F=\{Q_j\} \subset \D(E)$ be a pairwise family of disjoint cubes. The “global discretized sawtooth” relative to $\F$ is the collection of cubes $Q \in \D(E)$ that are not contained in any $Q_j \in \F$, that is, $$\D_{\F} :=\D(E) \setminus \bigcup_{Q_j \in \F} \D_{Q_j}.$$ For a given cube $Q \in \D(E)$, we define the “local discretized sawtooth” relative to $\F$ is the collection of cubes in $\D_Q$ that are not contained in any $Q_j \in \F$ of, equivalently, $$\D_{\F, Q} :=\D_Q \setminus \bigcup_{Q_j \in \F} \D_{Q_j}=\D_{\F} \cap \D_Q.$$
We also introduce the “geometric” Carleson regions and sawtooths. In the sequel, $\Omega \subset \R^{n+1}$, $n \geq 2$, is a $1$-sided CAD. Given $Q \in \D:=\D(\partial \Omega)$ we want to define some associated regions which inherit the good properties of $\Omega$. Let $\W=\W(\Omega)$ denote a collection of (closed) dyadic Whitney cubes of $\Omega$, so that the cubes in $\W$ form a covering of $\Omega$ with non-overlapping interiors, which satisfy $$4 \diam(I) \leq \dist(4I, \partial \Omega) \leq \dist(I, \partial \Omega) \leq 40 \diam(I), \quad \forall\,I \in \W,$$ and also $$(1/4) \diam(I_1) \leq \diam(I_2) \leq 4 \diam(I_1), \quad\text{whenever $I_1$ and $I_2$ touch}.$$ Let $X(I)$ be the center of $I$ and $\ell(I)$ denote the sidelength of $I$.
Given $0<\lambda<1$ and $I \in \W$, we write $I^*=(1+\lambda)I$ for the “fattening” of $I$. By taking $\lambda$ small enough, we can arrange matters, so that for any $I, J \in \W$, $$\begin{aligned}
\dist(I^*, J^*) & \simeq \dist(I, J), \\
\operatorname{int}(I^*) \cap \operatorname{int}(J^*) \neq \emptyset &
\Longleftrightarrow \partial I \cap \partial J \neq \emptyset.
\end{aligned}$$ (The fattening thus ensures overlap of $I^*$ and $J^*$ for any pair $I, J \in \W$ whose boundaries touch, so that the Harnack chain property then holds locally, with constants depending upon $\lambda$, in $I^* \cap J^*$.) By choosing $\lambda$ sufficiently small, say $0<\lambda<\lambda_0$, we may also suppose that there is a $\tau \in (1/2, 1)$ such that for distinct $I, J \in \W$, we have that $\tau J \cap I^{*}=\emptyset$. In what follows we will need to work with the dilations $I^{**}=(1+2\lambda)I$ or $I^{***}=(1+4\lambda)I$, and in order to ensure that the same properties hold we further assume that $0<\lambda<\lambda_0/4$.
Given $\vartheta\in\mathbb{N}$, for every cube $Q \in \D$ we set $$\label{eq:WQ}
\W_Q^\vartheta :=\left\{I \in \W: 2^{-\vartheta}\ell(Q) \leq \ell(I) \leq 2^\vartheta\ell(Q), \text { and } \dist(I, Q) \leq 2^\vartheta \ell(Q) \right\}.$$ We will choose $\vartheta\ge \vartheta_0$, with $\vartheta_0$ large enough depending on the constants of the corkscrew condition (cf. Definition \[def:CKS\]) and in the dyadic cube construction (cf. Lemma \[lem:dyadic\]), so that $X_Q \in I$ for some $I \in \W_Q^\vartheta$, and for each dyadic child $Q^j$ of $Q$, the respective corkscrew points $X_{Q^j} \in I^j$ for some $I^j \in \W_Q^\vartheta$. Moreover, we may always find an $I \in \W_Q^\vartheta$ with the slightly more precise property that $\ell(Q)/2 \leq \ell(I) \leq \ell(Q)$ and $$\W_{Q_1}^\vartheta \cap \W_{Q_2}^\vartheta \neq \emptyset, \quad \text { whenever }
1 \leq \frac{\ell(Q_2)}{\ell(Q_1)} \leq 2, \text { and } \dist(Q_1, Q_2) \leq 1000 \ell(Q_2).$$
For each $I \in \W_Q^\vartheta$, we form a Harnack chain from the center $X(I)$ to the corkscrew point $X_Q$ and call it $H(I)$. We now let $\W_{Q}^{\vartheta, *}$ denote the collection of all Whitney cubes which meet at least one ball in the Harnack chain $H(I)$ with $I \in \W_Q$, that is, $$\W_{Q}^{\vartheta, *}:=\{J \in \W: \text{ there exists } I \in \W_Q \text{ such that } H(I) \cap J \neq \emptyset\}.$$ We also define $$U_{Q}^\vartheta :=\bigcup_{I \in \W_{Q}^{\vartheta, *}}(1+\lambda) I=: \bigcup_{I \in \W_{Q}^{\vartheta, *}} I^{*}.$$ By construction, we then have that $$\W_{Q}^\vartheta \subset \W_{Q}^{\vartheta, *} \subset \W \quad \text{and}\quad X_Q \in U_Q^{\vartheta}, \quad X_{Q^{j}} \in U_{Q}^{\vartheta},$$ for each child $Q^j$ of $Q$. It is also clear that there is a uniform constant $k^*$ (depending only on the $1$-sided CAD constants and $\vartheta$) such that $$\begin{aligned}
2^{-k^*} \ell(Q) \leq \ell(I) \leq 2^{k^*}\ell(Q), &\quad \forall\,I \in \W_{Q}^{\vartheta, *},
\\X(I) \rightarrow_{U_Q^\vartheta} X_Q, &\quad \forall\,I \in \W_{Q}^{\vartheta, *},
\\\dist(I, Q) \leq 2^{k^*} \ell(Q), &\quad \forall\,I \in \W_{Q}^{\vartheta, *}. \end{aligned}$$ Here, $X(I) \to_{U_Q^\vartheta} X_Q$ means that the interior of $U_Q^\vartheta$ contains all balls in Harnack Chain (in $\Omega$) connecting $X(I)$ to $X_Q$, and moreover, for any point $Z$ contained in any ball in the Harnack Chain, we have $\dist(Z, \partial \Omega) \simeq \dist(Z, \Omega \setminus U_Q^\vartheta)$ with uniform control of implicit constants. The constant $k^*$ and the implicit constants in the condition $X(I) \to_{U_Q^\vartheta} X_Q$, depend on at most allowable parameter, on $\lambda$, and on $\vartheta$. Moreover, given $I \in \W$ we have that $I \in \W^{\vartheta,*}_{Q_I}$, where $Q_I \in \D(\partial \Omega)$ satisfies $\ell(Q_I)=\ell(I)$, and contains any fixed $\widehat{y} \in \partial \Omega$ such that $\dist(I, \partial \Omega)=\dist(I, \widehat{y})$. The reader is referred to [@HM] for full details. We note however that in that reference the parameter $\vartheta$ is fixed. Here we need to allow $\vartheta$ to depend on the aperture of the cones and hence it is convenient to include the superindex $\vartheta$.
For a given $Q \in \D$, the “Carleson box” relative to $Q$ is defined by $$T_{Q}^\vartheta :=\operatorname{int}\bigg(\bigcup_{Q' \in \D_Q} U_{Q'}^\vartheta\bigg).$$ For a given family $\F=\{Q_j\}$ of pairwise disjoint cubes and a given $Q \in \D(\partial \Omega)$, we define the “local sawtooth region” relative to $\F$ by $$\Omega_{\F, Q}^\vartheta :=\inter\bigg(\bigcup_{Q' \in \D_{\F, Q}} U_{Q'}^\vartheta\bigg)
=\inter\bigg(\bigcup_{I \in \W_{\F, Q}^\vartheta} I^* \bigg),$$ where $\W_{\F, Q}^\vartheta:=\bigcup_{Q' \in \D_{\F, Q}} \W^{\vartheta,*}_Q$. Analogously, we can slightly fatten the Whitney boxes and use $I^{**}$ to define new fattened Whitney regions and sawtooth domains. More precisely, for every $Q \in \D(\partial \Omega)$, $$T^{\vartheta,*}_Q :=\inter\bigg(\bigcup_{Q' \in \D_Q} U_{Q'}^\vartheta\bigg), \quad
\Omega^{\vartheta,*}_{\F, Q} :=\inter\bigg(\bigcup_{Q' \in \D_{\F, Q}} U^{\vartheta,*}_{Q'}\bigg), \quad
U^{\vartheta,*}_Q :=\bigcup_{I \in \W^{\vartheta,*}_Q} I^{**}.$$ Similarly, we can define $T^{\vartheta,**}_Q$, $\Omega^{\vartheta,**}_{\F, Q}$ and $U^{\vartheta,**}_Q$ by using $I^{***}$ in place of $I^{**}$. For later use, we recall that [@HM Proposition 6.1]: $$\label{eq:boundary}
Q \backslash \bigg(\bigcup_{Q_j \in \F} Q_j\bigg)
\subset \partial \Omega \cap \partial \Omega_{\F, Q}^\vartheta
\subset \overline{Q} \backslash \bigg(\bigcup_{Q_j \in \F} \inter(Q_j)\bigg).$$
Following [@HM], one can easily see that there exist constants $0<\kappa_1<1$ and $\kappa_0 \geq \max\{2C, 4/c\}$ (with $C$ the constant in , and $c$ such that $c \ell(Q) \leq r_Q$), depending only on the allowable parameters and on $\vartheta$, so that $$\begin{aligned}
\label{eq:kappa}
\kappa_1B_Q \cap \Omega \subset T_Q^\vartheta \subset T^{\vartheta,*}_Q \subset T^{\vartheta,**}_Q \subset \overline{T^{\vartheta,**}_Q}
\subset \kappa_0 B_Q \cap \overline{\Omega} =: \frac12 B^*_Q \cap \overline{\Omega}, \end{aligned}$$ where $B_Q$ is defined as in .
PDE estimates
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Now we recall several facts concerning the elliptic measures and the Green functions. For our first results we will only assume that $\Omega \subset \R^{n+1}$, $n \geq 2$, is an open set, not necessarily connected, with $\partial \Omega$ being ADR. Later we will focus on the case where $\Omega$ is a $1$-sided CAD.
Let $Lu = - \div(A \nabla u)$ be a variable coefficient second order divergence form operator with $A(X)=(a_{i, j}(X))_{i,j=1}^{n+1}$ being a real (not necessarily symmetric) matrix with $a_{i, j} \in L^{\infty}(\Omega)$ for $1 \leq i, j \leq n+1$, and $A$ uniformly elliptic, that is, there exists $\Lambda \geq 1$ such that $$\begin{aligned}
\Lambda^{-1} |\xi|^2 \leq A(X) \xi \cdot \xi,\quad |A(X) \xi \cdot \eta| \leq \Lambda |\xi| |\eta|,
\quad \forall\,\xi, \eta \in \R^{n+1} \text{ and a.e.~}X \in \Omega.\end{aligned}$$
In what follows we will only be working with this kind of operators, we will refer to them as “elliptic operators” for the sake of simplicity. We write $L^{\top}$ to denote the transpose of $L$, or, in other words, $L^{\top}u = -\div(A^{\top}\nabla u)$ with $A^{\top}$ being the transpose matrix of $A$.
We say that a function $u \in W_{\loc}^{1,2}(\Omega)$ is a weak solution of $Lu=0$ in $\Omega$, or that $Lu=0$ in the weak sense, if $$\begin{aligned}
\iint_{\Omega} A(X) \nabla u(X) \cdot \nabla \phi(X)=0, \quad \forall\,\phi \in C_c^{\infty}(\Omega). \end{aligned}$$ Here and elsewhere $C_c^{\infty}(\Omega)$ stands for the set of compactly supported smooth functions with all derivatives of all orders being continuous.
Associated with the operators $L$ and $L^{\top}$, one can respectively construct the elliptic measures $\{\w_L^X\}_{X \in \Omega}$ and $\{\w_{L^\top}^X\}_{X \in \Omega}$, and the Green functions $G_L$ and $G_{L^{\top}}$ (see [@HMT2] for full details). We next present some definitions and properties that will be used throughout this paper.
The following lemmas can be found in [@HMT2].
Suppose that $\Omega \subset \R^{n+1}$, $n\ge 2$, is an open set such that $\partial \Omega$ is ADR. Given an elliptic operator $L$, there exist $C>1$ (depending only on dimension and on the ellipticity of $L$) and $c_{\theta}>0$ (depending on the above parameters and on $\theta \in (0, 1)$) such that $G_L$, the Green function associated with $L$, satisfies $$\begin{aligned}
G_L(X, Y) &\leq C|X-Y|^{1-n}; \\
c_{\theta} |X-Y|^{1-n} \leq G_L(X, Y), \quad &\text{if } |X-Y| \leq \theta \delta(X), \theta \in(0,1); \\
G_L(\cdot, Y) \in C(\overline{\Omega} \setminus \{Y\}) &\text{ and }
G_L(\cdot, Y)|_{\partial \Omega} \equiv 0, \forall\,Y \in \Omega; \\
G_L(X, Y) \geq 0, &\quad \forall\,X, Y \in \Omega, X \neq Y; \\
G_L(X, Y)=G_{L^{\top}}(Y, X), &\quad \forall\,X, Y \in \Omega, X \neq Y; \end{aligned}$$ Moreover, $G_L(\cdot, Y) \in W^{1,2}_{\loc}(\Omega \setminus \{Y\})$ for every $Y \in \Omega$, and satisfies $LG_L(\cdot, Y)=\delta_Y$ in the weak sense in $\Omega$, that is, $$\iint_{\Omega} A(X) \nabla_{X} G_L(X, Y) \cdot \nabla \Phi(X) dX = \Phi(Y), \quad \forall\,\Phi \in C_c^{\infty}(\Omega).$$
Finally, the following Riesz formula holds $$\iint_{\Omega} A^{\top}(Y) \nabla_{Y} G_{L^{\top}}(Y, X) \cdot \nabla \Phi(Y) dY
= \Phi(X) - \int_{\partial \Omega} \Phi d\w_L^X,$$ for a.e. $X \in \Omega$ and for every $\Phi \in C_c^{\infty} (\R^{n+1})$.
Suppose that $\Omega \subset \R^{n+1}$, $n\ge 2$, is a $1$-sided CAD. Let $L$ be an elliptic operator. There exists a constant $C$ (depending only on the dimension, the $1$-sided CAD constants and the ellipticity of $L$) such that for every ball $B_0:=B(x_0, r_0)$ with $x_0 \in \partial \Omega$ and $0<r_0<\diam(\partial \Omega)$, and $\Delta_0=B_0\cap\pom$ we have the following properties:
=0.2cm =.2cm
There holds $$\begin{aligned}
\label{Bourgain}
\w_L^Y(\Delta_0) \geq 1/C, \quad \forall\,Y \in \Omega \cap B(x_0, C^{-1}r_0). \end{aligned}$$
If $B=B(x, r)$ with $x \in \partial \Omega$ is such that $2B \subset B_0$, then for any $X \in \Omega \setminus B_0$, $$\begin{aligned}
\label{eq:comparison}
C^{-1} \w_L^{X}(\Delta) \leq r^{n-1} G_L(X, X_{\Delta}) \leq C \w_L^{X}(\Delta). \end{aligned}$$
If $X \in \Omega \backslash 4B_0$, then we have $$\begin{aligned}
\label{eq:doubling}
\w_L^X(2\Delta_0) \leq C \w_L^X(\Delta_0). \end{aligned}$$
Proof of Theorem \[thm:abs-cont\] {#sec:proof-abs}
=================================
The goal of this section is to prove Theorem \[thm:abs-cont\]. We start with the following observation which will be used throughout the paper:
\[remark:truncations\] For every $\alpha>0$, $0<r<r'$, and $\varpi\in \R$, if $F \subset\pom$ is a bounded set and $v\in L^2_{\loc}(\Omega)$, then $$\label{est:two-trunc}
\sup_{x\in F}\iint_{\Gamma_{\alpha}^{r'}(x) \setminus \Gamma_{\alpha}^{r}(x)}
|v(Y)|^2 \delta(Y)^{\varpi} dY
<\infty.$$ To see this we first note that since $F$ is bounded we can find $R$ large enough so that $F\subset B(0,R)$. Then, if $x\in F$ one readily sees that $$\begin{aligned}
\Gamma_{\alpha}^{r'}(x) \backslash \Gamma_{\alpha}^{r}(x)
\subset \overline{B(0, r'+R)} \cap
\Big\{Y \in \Omega: \frac{r}{1+\alpha} \leq \delta(Y) \leq r' \Big\} =:K. \end{aligned}$$ Note that $K \subset \Omega$ is a compact set. Then, since $v \in L^2_{\loc}(\Omega)$, we conclude that $$\begin{aligned}
\label{eq:Gr-Gr0}
\sup_{x\in F}\iint_{\Gamma_{\alpha}^{r'}(x) \setminus \Gamma_{\alpha}^{r}(x)}
|v(Y)|^2 \delta(Y)^{\varpi} dY
\leq \max\left\{r',\frac{1+\alpha}{r}\right\}^{|\varpi|} \iint_{K} |v(Y)|^2 dY<\infty. \end{aligned}$$
We can now proceed to prove Theorem \[thm:abs-cont\]. We first note that it is immediate to see that $\eqref{list:si-wL-si}\Longrightarrow\eqref{list:wL}$, $\eqref{list:Sr-L2}\Longrightarrow\eqref{list:Sr}$, and $\eqref{list:Sr}\Longrightarrow\eqref{list:Srx}$. Moreover, yields easily $\eqref{list:Srx}\Longrightarrow\eqref{list:Sr}$. Thus, it suffices to prove the following implications: $$\begin{aligned}
\eqref{list:wL} \Longrightarrow \eqref{list:Sr-L2}, \qquad
\eqref{list:wL} \Longrightarrow \eqref{list:si-wL-si}, \qquad\text{and}\qquad
\eqref{list:Sr} \Longrightarrow \eqref{list:wL}. \end{aligned}$$
Proof of $\eqref{list:wL} \Longrightarrow \eqref{list:Sr-L2}$ {#sec:a-c}
-------------------------------------------------------------
Assume that $\sigma \ll \w_L$. Fix and arbitrary $Q_0 \in \D_{k_0}$ where $k_0 \in \Z$ is taken so that $2^{-k_0}=\ell(Q_0) < \diam(\partial \Omega)/M_0$, where $M_0$ is large enough and will be chosen later. From the construction of $T_{Q_0}^{\vartheta}$ one can easily see that $T_{Q_0}^{\vartheta} \subset \frac12B_{Q_0}^*:=\kappa_0 B_{Q_0}$, see . Let $X_0:=X_{M_0 \Delta_{Q_0}}$ be an interior corkscrew point relative to $M_0 \Delta_{Q_0}$ so that $X_0 \notin 4B_{Q_0}^*$ provided that $M_0$ is taken large enough depending on the allowable parameters. Since $\partial \Omega$ is ADR, and Harnack’s inequality give that $\w_L^{X_0}(Q_0) \geq C_0^{-1}$, where $C_0>1$ depends on 1-sided CAD constants and $M_0$. We now normalize the elliptic measure and the Green function as follows $$\begin{aligned}
\label{eq:normalize}
\w:= C_0 \sigma(Q_0) \w_L^{X_0} \quad\text{ and }\quad
\G(\cdot) := C_0 \sigma(Q_0) G_L(X_0, \cdot). \end{aligned}$$ The hypothesis $\sigma \ll \w_L$ implies that $\sigma \ll \w$. Note that $1 \leq \frac{\w(Q_0)}{\sigma(Q_0)} \leq C_0$. Let $N > C_0$ and let $\F_N^+ :=\{Q_j\} \subset \D_{Q_0} \backslash \{Q_0\}$, respectively, $\F_N^- :=\{Q_j\} \subset \D_{Q_0} \backslash \{Q_0\}$, be the collection of descendants of $Q_0$ which are maximal (and therefore pairwise disjoint) with respect to the property that $$\begin{aligned}
\label{eq:stopping}
\frac{\w(Q_j)}{\sigma(Q_j)} < \frac{1}{N}, \quad\text{ respectively }\quad
\frac{\w(Q_j)}{\sigma(Q_j)} >N. \end{aligned}$$ Write $\F_N=\F_N^+\cup\F_N^-$ and note that $\F_N^+\cap\F_N^-=\emptyset$. By maximality, one has $$\begin{aligned}
\label{eq:NN}
\frac{1}{N}\leq \frac{\w(Q)}{\sigma(Q)} \leq N, \quad \forall\,Q \in \D_{\F_N, Q_0}. \end{aligned}$$ Write for every $N>C_0$, $$\begin{aligned}
\label{eq:E0N-EN}
E_N^\pm := \bigcup_{Q \in \F_N^\pm} Q,
\qquad
E_N^0=E_N^+\cup E_N^-, \qquad
E_N := Q_0\setminus E_N^0,\end{aligned}$$ and $$\begin{aligned}
\label{eq:Q-decom}
Q_0=
\bigg(\bigcap_{N>C_0} E_N^0\bigg)\cup \bigg(\bigcup_{N>C_0} E_N \bigg)
=: E_0\cup \bigg(\bigcup_{N>C_0} E_N \bigg)
. \end{aligned}$$ By and [@HM Proposition 6.3], we have $$\begin{aligned}
\label{eq:EN-FN}
E_N \subset F_N:=\partial \Omega \cap \partial \Omega_{\F_N, Q_0}^{\vartheta}
\qquad\text{and}\qquad \sigma(F_N \backslash E_N)=0.\end{aligned}$$ Note that [@HM Lemma 3.61] yields that $\Omega_{\F_N, Q_0}^\vartheta$ is a bounded $1$-sided CAD for any $\vartheta\ge \vartheta_0$.
Let $x \in E_{N+1}^{ \pm}$. Then there exists $Q_x \in \F_{N+1}^\pm$ such that $x \in Q_x$. By , we have $$\begin{aligned}
\frac{\w(Q_x)}{\sigma(Q_x)} < \frac{1}{N+1} <\frac{1}{N} \quad\text{if $Q_x \in \F_{N+1}^+$, or }\quad
\frac{\w(Q_x)}{\sigma(Q_x)} >N+1>N \quad\text{if $Q_x \in \F_{N+1}^-$}. \end{aligned}$$ Therefore, the maximality of the cubes in $\F_N^\pm$ gives that $Q_x \subset Q'_x$ for some $Q'_x \in \F_N^\pm$ with $x \in Q'_x \subset E_N^\pm$. This shows that $\{E_N^+\}_N$, $\{E_N^-\}_N$, and $\{E_N^0\}_N$ are decreasing sequence of sets. This, together with the fact that $\omega(E_N^\pm)\le \omega(Q_0)<\infty$ and $\sigma(E_N^\pm)\le \sigma(Q_0)<\infty$, implies that $$\label{wrqfawfvrw}
\omega\bigg(\bigcap_{N>C_0} E_N^\pm\bigg)=\lim_{N\to\infty} \omega(E_N^\pm),
\qquad
\sigma\bigg(\bigcap_{N>C_0} E_N^\pm\bigg)=\lim_{N\to\infty} \sigma(E_N^\pm).$$
Our next goal is to show that $\sigma(E_0)=0$. To see this we note that by $$\omega(E_N^+)
=
\sum_{Q\in \F_N^+} \omega(Q)
<\frac1N\sum_{Q\in \F_N^+} \sigma(Q)
=\frac1N\sigma(E_N^+)
\le
\frac1N\sigma(Q_0)$$ and, by $$\omega\bigg(\bigcap_{N>C_0} E_N^+\bigg)=\lim_{N\to\infty} \omega(E_N^+)=0.$$ Use this, the fact that $\sigma \ll \w$, and to derive $$0=\sigma\bigg(\bigcap_{N>C_0} E_N^+\bigg)=\lim_{N\to\infty} \sigma(E_N^+).$$ On the other hand, yields $$\sigma(E_N^-)
=
\sum_{Q\in \F_N^-} \sigma(Q)
<\frac1N\sum_{Q\in \F_N^-} \omega(Q)
=\frac1N\omega(E_N^-)
\le
\frac1N\omega(Q_0).$$ All these, together with and the fact that $\{E_N^0\}_N$ is decreasing sequence of sets with $\sigma(E_N^0)\le\sigma(Q_0)<\infty$, give $$\label{34fravcrv}
\sigma(E_0)
=
\lim_{N\to\infty}
\sigma(E_N^0)
\le
\lim_{N\to\infty}
\sigma(E_N^+)+\lim_{N\to\infty}\sigma(E_N^-)
=0,$$ hence $\sigma(E_0)=0$.
Now we are going to bound the square function in $L^2(F_N, \sigma)$. Let $u \in W_{\loc}^{1,2}(\Omega) \cap L^{\infty}(\Omega)$ be a weak solution of $Lu=0$ in $\Omega$. Let $\vartheta\ge \vartheta_0$ and note that by , we see that $2B_Q \subset B_{Q_0}^*$. Recalling and the fact $X_0 \not\in 4B_{Q_0}^*$, we use , , , Harnack’s inequality, and the fact that $\pom$ is ADR to conclude that $$\begin{aligned}
\label{eq:w-sig-com}
\frac{\G(X)}{\delta(X)} \simeq \frac{\w(Q)}{\sigma(Q)} \simeq_N 1, \end{aligned}$$ for all $X \in I^*$ with $I \in \W_Q^{\vartheta,*}$ and $Q \in \D_{\F_N, Q_0}$. This and the definition of $\Omega_{\F_N, Q_0}^\vartheta$ yield $$\begin{aligned}
\label{eq:u-G}
\iint_{\Omega_{\F_N, Q_0}^\vartheta} & |\nabla u(Y)|^2 \delta(Y) dY
\lesssim_N \iint_{\Omega_{\F_N, Q_0}^\vartheta} |\nabla u(Y)|^2 \G(Y) dY. \end{aligned}$$ For every $M \geq 1$, we set $\F_{N, M}$ to be the family of maximal cubes of the collection $\F_N$ augmented by adding all the cubes $Q \in \D_{Q_0}$ such that $\ell(Q) \leq 2^{-M} \ell(Q_0)$. This means that $Q \in \D_{\F_{N,M}, Q_0}$ if and only if $Q \in \D_{\F_N, Q_0}$ and $\ell(Q)>2^{-M}\ell(Q_0)$. Observe that $\D_{\F_{N,M}, Q_0} \subset \D_{\F_{N, M'}, Q_0}$ for all $M \leq M'$, and hence $\Omega_{\F_{N,M}, Q_0}^\vartheta \subset \Omega_{\F_{N,M'}, Q_0}^\vartheta \subset \Omega_{\F_N, Q_0}^\vartheta$. This, together with the monotone convergence theorem, gives $$\begin{aligned}
\label{eq:lim-FM}
\iint_{\Omega_{\F_N, Q_0}^\vartheta} |\nabla u(Y)|^2 \G(Y) dY
=\lim_{M \to \infty} \iint_{\Omega_{\F_{N,M}, Q_0}^\vartheta} |\nabla u(Y)|^2 \G(Y) dY. \end{aligned}$$ Invoking [@CHMT Proposition 3.58], one has $$\begin{aligned}
\label{eq:int-parts}
\iint_{\Omega_{\F_{N,M}, Q_0}^\vartheta} |\nabla u(Y)|^2 \G(Y) dY
\lesssim_{N} \sigma(Q_0) \simeq 2^{-k_0 n}, \end{aligned}$$ where the implicit constants are independent of $M$. Consequently, combining , and , we deduce that $$\begin{aligned}
\label{eq:saw-square}
\iint_{\Omega_{\F_N, Q_0}^\vartheta} & |\nabla u(Y)|^2 \delta(Y) dY \leq C_N. \end{aligned}$$
To continue, we recall the dyadic square function defined in [@HMU Section 2.3]: $$\begin{aligned}
S_{Q_0}^\vartheta u(x) := \left(\iint_{\Gamma_{Q_0}^\vartheta(x)} |\nabla u(Y)|^2 \delta(Y)^{1-n} dY \right)^{1/2},
\text{ where } \Gamma_{Q_0}^\vartheta(x) := \bigcup_{x \in Q \in \D_{Q_0}} U_{Q}^\vartheta. \end{aligned}$$ Note that if $Q \in \D_{Q_0}$ is so that $Q \cap E_N \neq \emptyset$, then necessarily $Q \in \D_{\F_N, Q_0}$, otherwise, $Q \subset Q' \in \F_N$, hence $Q \subset Q_0 \backslash E_N$. In view of , we have $$\begin{aligned}
\label{eq:SQk-ENk}
\int_{E_N} S_{Q_0}^\vartheta u(x)^2 d\sigma(x)
&=\int_{E_N} \iint_{\bigcup\limits_{x \in Q \in \D_{Q_0}} U_Q^\vartheta}
|\nabla u(Y)|^2 \delta(Y)^{1-n} dY d\sigma(x)
\nonumber \\&\lesssim \sum_{Q \in \D_{Q_0}} \ell(Q)^{-n}
\sigma(Q \cap E_N) \iint_{U_Q^\vartheta} |\nabla u(Y)|^2 \delta(Y) dY
\nonumber \\&\lesssim \sum_{Q \in \D_{\F_N, Q_0}}
\iint_{U_Q^\vartheta} |\nabla u(Y)|^2 \delta(Y) dY
\nonumber \\&\lesssim \iint_{\Omega_{\F_N, Q_0}^\vartheta} |\nabla u(Y)|^2 \delta(Y) dY
\leq C_{N}, \end{aligned}$$ where we have used that the family $\{U_Q^\vartheta\}_{Q\in\D}$ has bounded overlap. This along with the last condition in yields $$\begin{aligned}
\label{eq:square-FNj}
S_{Q_0}^\vartheta u \in L^2(F_N, \sigma), \qquad\forall\, \vartheta\ge \vartheta_0.\end{aligned}$$
We next claim that fixed $\alpha>0$, we can find $\vartheta$ sufficiently large depending on $\alpha$ such that for any $r_0 \ll 2^{-k_0}$, $$\begin{aligned}
\label{eq:Sa-SQ}
S_{\alpha}^{r_0}u(x) \leq S_{Q_0}^\vartheta u(x), \quad x \in Q_0. \end{aligned}$$ It suffices to show $\Gamma_{\alpha}^{r_0}(x) \subset \Gamma_{Q_0}^\vartheta(x)$ for any $x \in Q_0$. Indeed, let $Y \in \Gamma_{\alpha}^{r_0}(x)$. Pick $I \in \W$ so that $Y \in I$, and hence, $\ell(I) \simeq \delta(Y) \leq |Y-x|<r_0 \ll 2^{-k_0} = \ell(Q_0)$. Pick $Q_I \in \D_{Q_0}$ such that $x \in Q_I$ and $\ell(Q_I)=\ell(I) \ll \ell(Q_0)$. Thus, one has $$\begin{aligned}
\dist(I, Q_I) \leq |Y-x| < (1+\alpha) \delta(Y) \leq C(1+\alpha) \ell(I) = C(1+\alpha) \ell(Q_I). \end{aligned}$$ Recalling , if we take $\vartheta\ge \vartheta_0$ large enough so that $$\label{eq:c0-alpha}
2^\vartheta \geq C(1+\alpha),$$ then $Y \in I \in \W_{Q_I}^\vartheta \subset \W^{\vartheta,*}_{Q_I}$. The latter gives that $Y \in U_{Q_I}^\vartheta \subset \Gamma_{Q_0}^\vartheta(x)$ and consequently holds. We should mention that the dependence of $\vartheta$ on $\alpha$ implies that all the sawtooth regions $\Omega_{\F_N, Q_0}^\vartheta$ above as well as all the implicit constants depend on $\alpha$.
To complete the proof we note that, it follows from and that $S_{\alpha}^{r_0} u \in L^2(F_N, \sigma)$. This together with Remark \[remark:truncations\] easily yields $$\label{eq:Sr-u-L2-FN}
S_{\alpha}^r u \in L^2(F_N, \sigma),\quad \text{for any } r>0.$$
We note that the previous argument has been carried out for an arbitrary $Q_0\in \D_{k_0}$. Hence, using , , and with $Q_k\in \D_{k_0}$, we conclude, with the induced notation, that $$\begin{aligned}
\label{eq:EE-FF}
\partial \Omega = \bigcup_{Q_k \in \D_{k_0}} Q_k
&=\bigg(\bigcup_{Q_k \in \D_{k_0}} E^k_0\bigg) \bigcup
\bigg(\bigcup_{Q_k \in \D_{k_0}} \bigcup_{N>C_0} E^k_N \bigg)
\nonumber \\&=\bigg(\bigcup_{Q_k \in \D_{k_0}} E^k_0\bigg) \bigcup
\bigg(\bigcup_{Q_k \in \D_{k_0}} \bigcup_{N>C_0} F^k_N \bigg)
=: F_0 \cup \bigg(\bigcup_{k, N} F^k_N \bigg), \end{aligned}$$ where $\sigma(F_0)=0$ and $F^k_N=\partial \Omega \cap \partial \Omega_{\F^k_N, Q_k}$ where each $\Omega_{\F^k_N, Q_k} \subset \Omega$ is a bounded 1-sided CAD . Combining and with $F_N^k$ in place of $F_N$, the proof of $\eqref{list:wL} \Rightarrow \eqref{list:Sr-L2}$ is complete.
Proof of $\eqref{list:wL} \Longrightarrow\eqref{list:si-wL-si}$
---------------------------------------------------------------
We borrow some idea from [@TZ] and address some small inaccuracies that do not affect their conclusion. We follow and use the notation from the proof of $\eqref{list:wL}\Longrightarrow \eqref{list:Sr-L2}$. As before we fix an arbitrary cube $Q_0 \in \D_{k_0}$ and an integer $N>C_0$. Recall that the family $\F_N$ of stopping cubes is constructed in and $E_N$ is defined in . We claim that there exists $C_N>1$ such that $$\begin{aligned}
\label{eq:EEE}
C_N^{-1} \sigma(F) \leq \w_L(F) \leq C_N \sigma(F), \quad \forall\,F \subset E_N. \end{aligned}$$ Assuming this momentarily, and applying it to every $Q_k\in \D_{k_0}$ we readily get with the help of and the fact that $\sigma(F_0)=0$.
Let us then focus on justifying . To ease the notation, we write $E:=E_N= E_{\text{good}}\cup E_{\text{bad}}$, where $$\begin{aligned}
E_{\text{good}} &:= \{x \in E: \exists\, Q_x \in \D_{Q_0} \text{ with } x \in Q_x \text{ and }Q_x \subset E\},
\quad
E_{\text{bad}} := E \backslash E_{\text{good}}.\end{aligned}$$ Let $x \in E_{\text{good}}$. Then there exists $Q_x \in \D_{Q_0}$ with $x\in Q_x \subset E$. Let $Q_x^* \in \D_{Q_0}$ be the maximal cube with $Q_x^* \supset Q_x$ such that $Q_x^* \subset E$. Let $\F'\subset\D_{Q_0}$ be the collection of the maximal cubes $Q_x^*$ for $x \in E_{\text{good}}$. Then $\F'$ is pairwise disjoint and $$\begin{aligned}
\label{eq:Fgood}
E_{\text{good}} = \bigcup_{Q \in \F'} Q. \end{aligned}$$ Observe that $$\begin{aligned}
\label{eq:NN-max}
\frac{1}{N} \leq \frac{\w(Q')}{\sigma(Q')} \leq N,
\qquad\forall\, Q'\subset Q\in\F', \end{aligned}$$ since $Q' \subset Q \subset E_{\text{good}} \subset E=Q_0\setminus \bigcup_{Q_j\in\F_N} Q_j$ implies that $Q' \in \D_{\F_N, Q_0}$ and follows at once from .
Set $h=d\sigma/d\w$ and $$\begin{aligned}
\label{eq:L0-def}
L_0 := \left\{x \in Q_0: \fint_{Q_x} |h(y)-h(x)| \ d\w(y) \to 0, \D_{Q_0} \ni Q_x \searrow \{x\} \right\}\end{aligned}$$ and let us claim that $$\begin{aligned}
\label{eq:L0-points}
\w(Q_0 \backslash L_0)=0.\end{aligned}$$ Assuming this momentarily and using again that $\sigma \ll \w$, we obtain $$\label{eq:sig-Q0-L0}
\sigma(Q_0 \backslash L_0)=0.$$ Let us then show . Since $h \in L^1(Q_0, \w)$, there exists $h_j \in C_c(\partial \Omega)$ such that $\|h_j-h\|_{L^1(Q_0, \w)} \to 0$, as $j \to \infty$. Thus, for any $j$, we have $$\begin{aligned}
0&\leq\limsup_{Q_x \searrow \{x\}} \fint_{Q_x} |h-h(x)| d\w
\\&\leq \limsup_{Q_x \searrow \{x\}} \fint_{Q_x} |h-h_j| d\w +
\limsup_{Q_x \searrow \{x\}} \fint_{Q_x} |h_j-h_j(x)| d\w + |h_j(x)-h(x)|
\\&\leq M_{Q_0, \w}^{\text{dya}}(h-h_j)(x) + |h_j(x)-h(x)|, \end{aligned}$$ where the dyadic maximal operator $M_{Q_0, \w}^{\text{dya}}$ is defined by $$\begin{aligned}
M_{Q_0, \w}^{\text{dya}}f(x) := \sup_{x \in Q \in \D_{Q_0}}
\frac{1}{\w(Q)} \int_{Q} |f| d\w. \end{aligned}$$ As a consequence, for any fixed $\lambda>0$, $$\begin{aligned}
&\w\bigg(\bigg\{x \in Q_0: \limsup_{Q_x \searrow \{x\}} \fint_{Q_x} |h-h(x)| d\w>\lambda \bigg\}\bigg)
\\&\qquad\leq \w(\{x \in Q_0: M_{Q_0, \w}^{\text{dya}}(h-h_j)(x)>\lambda/2 \})
+ \w(\{x \in Q_0: |h_j(x)-h(x)|>\lambda/2 \})
\\&\qquad\lesssim \frac{1}{\lambda} \int_{Q_0} |h-h_j| d\w \to 0, \end{aligned}$$ as $j \to \infty$, where we used the weak-$(1, 1)$ type boundedness of $M_{Q_0, \w}^{\text{dya}}$ with respect to $\w$. Accordingly, we obtain as desired : $$\begin{gathered}
\label{eq:L-point}
\w(Q_0 \backslash L_0) = \w\bigg(\bigg\{x \in Q_0:
\limsup_{Q_x \searrow \{x\}} \fint_{Q_x} |h-h(x)| d\w>0 \bigg\}\bigg)
\\\leq \sum_{k \geq 1} \w\bigg(\bigg\{x \in Q_0:
\limsup_{Q_x \searrow \{x\}} \fint_{Q_x} |h-h(x)| d\w>\frac{1}{k} \bigg\}\bigg) = 0. \end{gathered}$$
For any $x \in E_{\text{bad}}$ and for every $Q_x \in \D_{Q_0}$ with $Q_x \ni x$, one has $Q_x \not\subset E=Q_0 \backslash \bigcup_{Q_j \in \F_N}Q_j$. This gives that $Q_x \cap Q_j \neq \emptyset$ for some $Q_j \in \F_N$. If $Q_x \subset Q_j$, then $x \in Q_j \in \F_N$, which contradicts that $x \in E=Q_0 \backslash \bigcup_{Q_j \in \F_N}Q_j$. Hence, $Q_j \subsetneq Q_x \subset Q_0$ and by the maximality of $Q_j$, we have $$\begin{aligned}
\frac{1}{N} \leq \frac{\w(Q_x)}{\sigma(Q_x)} \leq N. \end{aligned}$$ Together with the definition of $L_0$ in , this yields that $$\begin{aligned}
\label{eq:NN-h(x)}
\frac{1}{N} \leq h(x) \leq N, \qquad \forall\, x \in E_{\text{bad}} \cap L_0. \end{aligned}$$
To continue, let $F \subset E \subset Q_0$ be a Borel set. We write $$\begin{aligned}
E &= E_{\text{good}} \cup (E_{\text{bad}} \cap L_0) \cup (E_{\text{bad}} \backslash L_0)
=:E_{\text{good}} \cup E^1_{\text{bad}} \cup E^0_{\text{bad}},
\\F &= (F \cap E_{\text{good}}) \cup (F \cap E^1_{\text{bad}}) \cup (F \cap E^0_{\text{bad}})
=:F_{\text{good}} \cup F^1_{\text{bad}} \cup F^0_{\text{bad}}. \end{aligned}$$ Recalling and , we have $\w(Q_0 \backslash L_0)=\sigma(Q_0 \backslash L_0)=0$. Thus, $$\begin{aligned}
\label{eq:bad-bad-0}
\w(F^0_{\text{bad}})=\sigma(F^0_{\text{bad}})=0.\end{aligned}$$ Using the fact $F^1_{\text{bad}} \subset E^1_{\text{bad}}$ and , we deduce that $$\begin{aligned}
\label{eq:bad-bad-1}
\frac{1}{N} \w(F^1_{\text{bad}}) \leq \sigma(F^1_{\text{bad}})
=\int_{F^1_{\text{bad}}} h \ d\w \leq N \w(F^1_{\text{bad}}). \end{aligned}$$
In order to handle $F_{\text{good}}$, we fix $Q \in \F'$ with $\w(F \cap Q)>0$ and set $F_Q:=F \cap Q$. Since both $\w$ and $\sigma$ are regular, there exist a closed set $F'_Q \subset F_Q$ and an open set $F''_Q \supset F_Q$ such that $$\begin{aligned}
\label{eq:regular}
\w(F''_Q \backslash F'_Q) + \sigma(F''_Q \backslash F'_Q) \leq \frac{1}{2N} \w(F_Q). \end{aligned}$$ Let $x \in F''_Q \cap Q$. Since $F''_Q$ is open, there exists $r_x>0$ such that $\Delta(x, r_x) \subset F''_Q$. Take $Q_x \in \D_Q$ with $x \in Q_x$ and $\ell(Q_x) \ll r_x$ such that $Q_x \subset \Delta(x, r_x)$. Then, $Q_x \subset F''_Q \cap Q$. Let $Q^*_x \in \D_{Q}$ be the maximal cube such that $Q_x \subset Q^*_x\subset F''_Q \cap Q$. Let $\F_Q$ be the collection of maximal cubes $Q^*_x$ for $x \in F''_Q \cap Q$. One can then easily see that $\F_Q$ is a pairwise disjoint family with $$\begin{aligned}
\label{eq:E''Q-Q'}
F''_Q \cap Q = \bigcup_{Q' \in \F_Q} Q'. \end{aligned}$$ Note that for any cube $Q' \in \F_Q$, one has $Q' \subset Q\in \F'$. Hence, yields $$\begin{aligned}
\frac{1}{N} \leq \frac{\w(Q')}{\sigma(Q')} \leq N, \qquad \forall\,Q' \in \F_Q. \end{aligned}$$ This, combined with and , immediately implies that $$\begin{gathered}
\w(F_Q) \leq \w(F''_Q \cap Q) = \sum_{Q' \in \F_Q} \w(Q')
\leq N \sum_{Q' \in \F_Q} \sigma(Q') = N \sigma(F''_Q \cap Q)
\\\leq N (\sigma(F''_Q \backslash F'_Q) + \sigma(F'_Q) )
\leq \frac12 \w(F_Q) + N \sigma(F_Q), \end{gathered}$$ and hence, $$\begin{aligned}
\label{eq:EQ-EQ-1}
\w(F_Q) \leq 2N \sigma(F_Q). \end{aligned}$$ Analogously, $$\begin{gathered}
\label{eq:EQ-EQ-2}
\sigma(F_Q) \leq \sigma(F''_Q \cap Q) = \sum_{Q' \in \F_Q} \sigma(Q')
\leq N \sum_{Q' \in \F_Q} \w(Q') = N \w(F''_Q \cap Q)
\\\leq N(\w(F''_Q \backslash F'_Q) + \w(F'_Q) )
\leq \frac{1}{2} \w(F_Q) + N \w(F_Q) \leq 2N \w(F_Q), \end{gathered}$$ Therefore, collecting and , we have proved that $$\begin{aligned}
\label{eq:EQ-EQ}
\frac{1}{2N} \sigma(F \cap Q) \leq \w(F \cap Q) \leq 2N \sigma(F \cap Q),
\quad \forall\,Q \in \F',\ \w(F \cap Q)>0. \end{aligned}$$
Let us turn our attention to $F_{\text{good}}$. The second inequality in and give that $$\begin{aligned}
\w(F_{\text{good}})
=\sum_{Q \in \F': \w(F \cap Q)>0} \w(F \cap Q)
\leq 2N \sum_{Q \in \F'} \sigma(F \cap Q)
=2N \sigma(F_{\text{good}}). \end{aligned}$$ This, , and readily yield $$\begin{aligned}
\label{eq:EE-1}
\w(F) &= \w(F_{\text{good}}) + \w(F^1_{\text{bad}}) + \w(F^0_{\text{bad}})
\leq 2N \sigma(F_{\text{good}}) + N \sigma(F^1_{\text{bad}})
\leq 2N \sigma(F). \end{aligned}$$ Similarly, using again , , , the fact $\sigma \ll \w$, and the first inequality in , we conclude that $$\begin{aligned}
\label{eq:EE-2}
\sigma(F) &= \sigma(F_{\text{good}}) + \sigma(F^1_{\text{bad}}) + \sigma(F^0_{\text{bad}})
\nonumber \\&\leq \sum_{Q \in \F': \sigma(F \cap Q)>0} \sigma(F \cap Q) + N \w(F^1_{\text{bad}})
\nonumber \\&\leq \sum_{Q \in \F': \w(F \cap Q)>0} \sigma(F \cap Q) + N \w(F^1_{\text{bad}})
\nonumber \\&\leq 2N \sum_{Q \in \F'} \w(F \cap Q) + N \w(F^1_{\text{bad}})
\leq 2N \w(F). \end{aligned}$$ Consequently, the desired estimate follows from , and . Thus, holds.
Proof of $\eqref{list:Sr}\Longrightarrow\eqref{list:wL}$
--------------------------------------------------------
Given $Q_0 \in \D$ and for any $\eta \in (0, 1)$, we define the modified dyadic square function $$\begin{aligned}
S_{Q_0}^{\vartheta_0,\eta} u(x) := \left(\iint_{\Gamma_{Q_0}^{\vartheta_0,\eta}(x)}
|\nabla u(Y)|^2 \delta(Y)^{1-n} dY \right)^{1/2}, \end{aligned}$$ where the modified non-tangential cone $\Gamma_{Q_0}^{\vartheta_0,\eta}(x)$ is given by $$\begin{aligned}
\Gamma_{Q_0}^{\vartheta_0,\eta}(x) := \bigcup_{x \in Q \in \D_{Q_0}} U_{Q, \eta^3}^{\vartheta_0}, \qquad
U_{Q, \eta^3}^{\vartheta_0} = \bigcup_{\substack{Q' \in \D_Q \\ \ell(Q')>\eta^3 \ell(Q)}} U_{Q'}^{\vartheta_0}. \end{aligned}$$ Here we recall that $\vartheta_0$ depends on the 1-sided CAD constants (see Section \[section:dyadic\]).
The following lemma from [@CHMT] is crucial to our proof.
\[lem:square\] There exist $0<\eta \ll 1$ (depending only on dimension, the $1$-sided CAD constants and the ellipticity of $L$) and $\beta_0 \in (0, 1)$, $C_{\eta} \geq 1$ both depending on the same parameters and additionally on $\eta$, such that for every $Q_0 \in \D$, for every $0<\beta<\beta_0$, and for every Borel set $F \subset Q_0$ satisfying $\w_L^{X_{Q_0}}(F ) \leq \beta \w_L^{X_{Q_0}}(Q_0)$, there exists a Borel set $S \subset Q_0$ such that the bounded weak solution $u(X)=\w_L^{X}(S)$ satisfies $$\begin{aligned}
S_{Q_0}^{\vartheta_0,\eta} u(x) \geq C_{\eta}^{-1} (\log \beta^{-1})^{\frac12},\qquad \forall\,x \in F. \end{aligned}$$
Assume that holds. In order to prove that $\sigma \ll \w_L$ on $\partial \Omega$, from Lemma \[lem:dyadic\] (a), it suffices to show that for any given $Q_0 \in \D$, $$\begin{aligned}
\label{eq:abs-cont-Q0}
F \subset Q_0,\quad \w_L(F)=0 \quad \Longrightarrow \quad \sigma(F)=0. \end{aligned}$$
Consider then $F \subset Q_0$ with $\w_L(F)=0$. By the mutually absolute continuity between elliptic measures, one has $\w_L^{X_{Q_0}}(F )=0 \leq \beta \w_L^{X_{Q_0}}(Q_0)$ for every $\beta>0$. Lemma \[lem:square\] applied to $F$ yields that for every $0<\beta<\beta_0$ there exists a Borel set $S_{\beta} \subset Q_0$ such that $u_{\beta}(X)=\w_L^{X}(S_{\beta})$ satisfies $$\begin{aligned}
\label{eq:S-lower}
\inf_{x \in F} S_{Q_0}^{\vartheta_0,\eta} u_{\beta}(x) \geq C_{\eta}^{-1} (\log \beta^{-1})^{\frac12}. \end{aligned}$$ To continue, we claim that there exist $\alpha_0>0$ and $r>0$ such that $$\begin{aligned}
\label{eq:cone-cone}
\Gamma_{Q_0}^{\vartheta_0,\eta}(x) \subset \Gamma_{\alpha}^{r}(x),
\qquad \forall\,x \in Q_0 \text{ and }\forall\,\alpha \geq \alpha_0. \end{aligned}$$ Indeed, let $Y \in \Gamma_{Q_0}^{\vartheta_0,\eta}(x)$. By definition, there exist $Q \in \D_{Q_0}$ and $Q' \in \D_Q$ with $\ell(Q')>\eta^3 \ell(Q)$ such that $Y \in U_{Q'}^{\vartheta_0}$ and $x \in Q$. Then $Y \in I^*$ for some $I \in \W_{Q'}^{\vartheta_0,*}$, and hence, $$\begin{aligned}
\label{eq:sidelength}
\delta(Y) \simeq \ell(I) \simeq \ell(Q')\le \ell(Q)<\eta^{-3}\ell(Q').\end{aligned}$$ This further implies that $$\begin{gathered}
\label{eq:Yx}
|Y-x| \leq \diam(I^*) + \dist(I, x) \leq \diam(I^*) + \dist(I, Q') + \diam(Q)
\\
\lesssim 2^{k^*}\ell(Q')+\ell(Q)
\lesssim \ell(Q), \end{gathered}$$ where $k^*$ depends on the $1$-sided CAD constants (see Section \[section:dyadic\]). Combining with , we get $$\begin{aligned}
|Y-x| \leq C_1 \ell(Q_0) =: r/2 \quad \text{ and }\quad
|Y-x| \leq (1+C_{1, \eta}) \delta(Y) =:(1+\alpha_0) \delta(Y), \end{aligned}$$ where $C_1$ depends only on the allowable parameters, and $C_{1,\eta}$ depends only on the allowable parameters and also on $\eta$. Eventually, this justifies .
By , we have $$\begin{aligned}
\label{eq:boundary-par}
Q_0 = \bigcup_{N=0}^{\infty} E_N, \qquad \sigma(E_0)=0, \end{aligned}$$ where $$\begin{aligned}
E_0 :=\{x \in Q_0: S_{\alpha}^r u_{\beta}(x)=\infty\} \quad\text{and}\quad
E_N :=\{x \in Q_0: S_{\alpha}^r u_{\beta}(x)<N\}. \end{aligned}$$ Invoking then and we therefore obtain $$\begin{gathered}
\sigma(F \cap E_N) \leq C_{\eta}^2 (\log \beta^{-1})^{-1}
\int_{F \cap E_N} S_{Q_0}^{\vartheta_0,\eta} u_{\beta}(x)^2 d\sigma(x)
\\\leq C_{\eta}^2 (\log \beta^{-1})^{-1}
\int_{E_N} S_{\alpha}^r u_{\beta}(x)^2 d\sigma(x)
\leq C_{\eta}^2 N^2 \sigma(Q_0) (\log \beta^{-1})^{-1} \longrightarrow 0, \end{gathered}$$ as $\beta \to 0^{+}$. This shows that $\sigma(F \cap E_N)=0$ for every $N \geq 1$ and follows at once from . This completes the proof of $\eqref{list:Sr}\Longrightarrow\eqref{list:wL}$ and hence that of Theorem \[thm:abs-cont\].
Proof of Theorems \[thm:perturbation\] and \[thm:wL-wLT\] {#sec:perturbation}
=========================================================
In order to prove Theorems \[thm:perturbation\] and \[thm:wL-wLT\], we will make use of Theorem \[thm:abs-cont\] and show that the truncated square function is finite $\sigma$-a.e. for every bounded weak solution. Indeed, we are going to show the following more general result, which is a qualitative version of [@CHMT Theorem 4.13].
\[thm:AAAD\] Let $\Omega \subset \R^{n+1}$, $n \ge 2$, be a $1$-sided CAD. There exists $\widetilde{\alpha}_0>0$ (depending only on the $1$-sided CAD constants) such that if $L_0 u = -\div(A_0 \nabla u)$ and $L_1 u = -\div(A_1 \nabla u)$ are real (not necessarily symmetric) elliptic operators such that $A_0-A_1=A+D$, where $A, D \in L^{\infty}(\Omega)$ are real matrices satisfying the following conditions:
1. there exist $\alpha_1 \geq \widetilde{\alpha}_0$ and $r_1>0$ such that $$\begin{aligned}
\label{eq:a(X)-delta}
\iint_{\Gamma_{\alpha_1}^{r_1}(x)} a(X)^2 \delta(X)^{-n-1} dX < \infty,
\qquad \sigma \text{-a.e. } x \in \partial \Omega, \end{aligned}$$ where $$\begin{aligned}
a(X):=\sup_{Y \in B(X, \delta(X)/2)}|A(Y)|, \qquad X \in \Omega;\end{aligned}$$
2. $D \in \Lip_{\loc}(\Omega)$ is antisymmetric and there exist $\alpha_2 \geq \widetilde{\alpha}_0$ and $r_2>0$ such that $$\begin{aligned}
\label{eq:divCD}
\iint_{\Gamma_{\alpha_2}^{r_2}(x)} |\div_{C} D(X)|^2 \delta(X)^{1-n} dX < \infty,
\quad \sigma \text{-a.e. } x \in \partial \Omega; \end{aligned}$$
then $\sigma \ll \w_{L_0}$ if and only if $\sigma \ll \w_{L_1}$.
By symmetry, it suffices to assume that $\sigma \ll \w_{L_0}$ and prove $\sigma \ll \w_{L_1}$. Let $u \in W_{\loc}^{1,2}(\Omega) \cap L^{\infty}(\Omega)$ be a weak solution of $L_1u=0$ in $\Omega$ and $\|u\|_{L^{\infty}(\Omega)}=1$. Applying Theorem \[thm:abs-cont\] $\eqref{list:Sr} \Rightarrow \eqref{list:wL}$ to $u$, we are reduced to showing that for some $r>0$, $$\begin{aligned}
S_{\alpha_0}^ru(x) < \infty, \qquad \text{for $\sigma $-a.e. } x \in \partial \Omega, \end{aligned}$$ where $\alpha_0$ is given in Theorem \[thm:abs-cont\]. Proceeding as in Section \[sec:a-c\] and invoking , it suffices to see that for every fixed $Q_0 \in \D_{k_0}$ and for some fixed large $\vartheta$ (which depends on $\alpha_0$ and hence solely on the 1-sided CAD constants) one has $$\begin{aligned}
\label{eq:Q0-SQ0u-EN}
Q_0 = \bigcup_{N \geq 0} \widehat{E}_N,\quad \sigma(\widehat{E}_0)=0
\quad\text{and}\quad S_{Q_0}^\vartheta u \in L^2(\widehat{E}_N,\sigma), \ \forall\,N \geq 1. \end{aligned}$$
Fix then $Q_0 \in \D_{k_0}$, where $k_0$ is given in the beginning of Section \[sec:a-c\]. We use the normalization in with $L=L_0$ and the family $\F_N$ of stopping cubes constructed in . Set $$\begin{aligned}
S_{Q_0}\gamma^\vartheta(x) := \bigg(\sum_{x \in Q \in \D_{Q_0}} \gamma_{Q} ^\vartheta\bigg)^{1/2},\end{aligned}$$ where for every $Q\in\D_{Q_0}$ we write $$\begin{aligned}
\gamma_Q^\vartheta:=
\iint_{U^{\vartheta, *}_Q} a(X)^2 \delta(X)^{-n-1} dX
+ \iint_{U^{\vartheta, *}_Q} |\div_{C} D(X)|^2 \delta(X)^{1-n} dX. \end{aligned}$$ We claim that there exist $\widetilde{\alpha}_0>0$ and $\widetilde{r}>0$ such that $$\label{eq:TQ-Tar}
\Gamma^{\vartheta,*}_{Q_0}(x)
:=\bigcup_{x \in Q \in \D_{Q_0}} U^{\vartheta, *}_{Q}
\subset \Gamma_{\widetilde{\alpha}_0}^{\widetilde{r}}(x),\quad x \in \partial \Omega.$$ Indeed, let $Y \in \Gamma^{\vartheta,*}_{Q_0}(x)$. Then, there exists $Q \in \D_{Q_0}$ with $Q \ni x$ and $I \in \W^{\vartheta,*}_Q$ such that $Y \in I^{**}$. Using these, one has $$\begin{aligned}
|Y-x| \leq \diam(I^{**}) + \dist(I, Q) + \diam(Q) \lesssim_\vartheta \ell(Q) \simeq_\vartheta \ell(I) \simeq \delta(Y), \end{aligned}$$ which implies $$|Y-x| < C_1 \delta(Y)=:(1+\widetilde{\alpha}_0) \delta(Y) \quad\text{and}\quad
|Y-x| < C_2 \ell(Q_0)=C_2 2^{-k_0} =: \widetilde{r},$$ where both $C_1$ and $C_2$ depend only on the allowable parameters —note that they depend on $\vartheta$, hence on the 1-sided CAD constants. Thus, holds for the choice of $\widetilde{\alpha}_0$ and $\widetilde{r}$, and as a result $$\begin{aligned}
\label{eq:SQ0-a.e.}
&S_{Q_0}\gamma^\vartheta(x)^2 \quad\lesssim
\iint_{\Gamma^{\vartheta,*}_{Q_0}(x) } a(X)^2 \delta(X)^{-n-1} dX
+ \iint_{\Gamma^{\vartheta,*}_{Q_0}(x) } |\div_{C} D(X)|^2 \delta(X)^{1-n} dX
\nonumber\\&\quad\le \iint_{\Gamma_{{\alpha}_1}^{\max\{\widetilde{r}, r_1\}}(x)} a(X)^2 \delta(X)^{-n-1} dX
+ \iint_{\Gamma_{{\alpha}_2}^{\max\{\widetilde{r},r_2\}}(x)} |\div_{C} D(X)|^2 \delta(X)^{1-n} dX
\nonumber\\&\quad< \infty, \quad \text{for $\sigma $-a.e.~} x \in Q_0, \end{aligned}$$ where we have used that the fact that the family $\{U^{\vartheta,*}_Q\}_{Q \in \D}$ has bounded overlap, that $\alpha_1, \alpha_2\ge \widetilde{\alpha}_0$ and the last estimate follows from , together with Remark \[remark:truncations\].
Given $N>C_0$ ($C_0$ is the constant that appeared in Section \[sec:a-c\]), let $\widetilde{\F}_N\subset \D_{Q_0}$ be the collection of maximal cubes (with respect to the inclusion) $Q_j \in \D_{Q_0}$ such that $$\begin{aligned}
\label{eq:FN-stopping}
\sum_{Q_j \subset Q \in \D_{Q_0}} \gamma_{Q}^\vartheta > N^2.\end{aligned}$$ Observe that $$\label{eq:Sa<N}
S_{Q_0}\gamma^\vartheta(x) \leq N, \qquad \forall\,x \in Q_0 \backslash \bigcup_{Q_j \in \widetilde{\F}_N}Q_j.$$ Otherwise, there exists a cube $Q_x\ni x$ such that $\sum_{Q_x \subset Q \in \D_{Q_0}} \gamma_Q^\vartheta > N^2$, hence $x \in Q_x \subset Q_j$ for some $Q_j \in \widetilde{\F}_N$, which is a contradiction.
We next set $$\begin{aligned}
\label{eq:E0-tilde-measure:a}
\widetilde{E}_0 :=
\bigcap_{N>C_0} \widetilde{E}_N^0
:=
\bigcap_{N>C_0} \bigg(\bigcup_{Q_j \in \widetilde{\F}_N} Q_j\bigg). \end{aligned}$$ Let $x \in \widetilde{E}^0_{N+1}$. Then there exists $Q_x \in \widetilde{\F}_{N+1}$ such that $x \in Q_x$. By , one has $$\begin{aligned}
\sum_{Q_x \subset Q \in \D_{Q_0}} \gamma_{Q}^\vartheta > (N+1)^2>N^2.\end{aligned}$$ Therefore, the maximality of the cubes in $\widetilde{\F}_N$ gives that $Q_x \subset Q'_x$ for some $Q'_x \in \widetilde{\F}_N$ with $x \in Q'_x \subset \widetilde{E}_0^N$. This shows that $\{\widetilde{E}_N^0\}_N$ is a decreasing sequence of sets, and since $\widetilde{E}_N^0\subset Q_0$ for every $N$ we conclude that $$\begin{aligned}
\omega(\widetilde{E}_0 )=\lim_{N\to\infty} \omega(\widetilde{E}_N^0),
\qquad
\sigma(\widetilde{E}_0 )=\lim_{N\to\infty} \sigma(\widetilde{E}_N^0).\end{aligned}$$ Note that for every $N>C_0$, if $x \in \widetilde{E}_0$ there exists $Q_x^{N} \in \widetilde{\F}_N$ such that $Q_x^N \ni x$. By the definition of $\widetilde{\F}_N$, we have $$\begin{aligned}
S_{Q_0}\gamma^\vartheta(x)^2 = \sum_{x \in Q \in \D_{Q_0}} \gamma_Q^\vartheta
\geq \sum_{Q_x^N \subset Q \in \D_{Q_0}} \gamma_Q^\vartheta > N^2, \end{aligned}$$ and, therefore,
$$\begin{gathered}
\label{eaf4a3f4}
\sigma(\widetilde{E}_0 )
=
\lim_{N\to\infty} \sigma(\widetilde{E}_N^0)
\le \lim_{N\to\infty} \sigma(\{x \in Q_0: S_{Q_0}\gamma^\vartheta(x)>N\})
\\=\sigma(\{x \in Q_0: S_{Q_0}\gamma^\vartheta(x)=\infty\})=0,\end{gathered}$$
by .
To proceed, let $\widehat{\F}_N$ be the collection of maximal, hence pairwise disjoint, cubes in $\F_N \cup \widetilde{\F}_N$. Note that $\D_{\widehat{\F}_N, Q_0} \subset \D_{\F_N, Q_0} \cap \D_{\widetilde{\F}_N, Q_0}$. This along with yields $$\begin{aligned}
\label{eq:NN-Fhat}
\frac{1}{N}\leq \frac{\w(Q)}{\sigma(Q)} \leq N, \quad \forall\,Q \in \D_{\widehat{\F}_N, Q_0}. \end{aligned}$$ We next set $$\begin{aligned}
\label{eq:E0-tilde-measure:aaa}
\widehat{E}_0 :=
\bigcap_{N>C_0} \widehat{E}_N^0
:=
\bigcap_{N>C_0 } \bigg(\bigcup_{Q_j \in \widehat{\F}_N} Q_j\bigg). \end{aligned}$$ Note that $\widehat{\F}_N\subset\F_N \cup \widetilde{\F}_N$ and also that if $Q\in \F_N \cup \widetilde{\F}_N$ then there exists $Q'\in \F_N \cup \widetilde{\F}_N$ so that $Q\subset Q'$. This shows that $\widehat{E}_N^0 = {E}_N^0 \cup \widetilde{E}_N^0$, where ${E}_N^0$ and $\widetilde{E}_N^0$ are defined in and respectively. As we showed that $\{E_N^0\}_N$ and $\{\widetilde{E}_N^0\}_N$ are decreasing sequence of sets, then so is $\{\widehat{E}_N^0\}_N$. This together with the fact that $\widehat{E}_N^0\subset Q_0$ lead to $$\begin{aligned}
\sigma(\widehat{E}_0)
=\lim_{N \to\infty} \sigma(\widehat{E}_N^0)
\le\lim_{N \to\infty} \sigma({E}_N^0)
+\lim_{N \to\infty} \sigma(\widetilde{E}_N^0)
=0, \end{aligned}$$ as shown in and , hence $\sigma(\widehat{E}_0)=0$.
Next we write $$\begin{aligned}
\label{eq:Q0-Ehat}
Q_0=
\widehat{E}_0\cup \bigg(\bigcup_{N>C_0} \widehat{E}_N \bigg)
:=
\widehat{E}_0\cup \bigg(\bigcup_{N>C_0} (Q_0\setminus \widehat{E}_N^0) \bigg).\end{aligned}$$ Therefore, to get , we are left with proving $$\label{eq:SQ0u-EN-tilde}
S_{Q_0}^\vartheta u \in L^2(\widehat{E}_N, \sigma), \quad \forall\,N>C_0.$$ With this goal in mind, we apply , and proceed as in the proof of and , to conclude that $$\begin{aligned}
\label{eq:S-delta-G}
\int_{\widehat{E}_N} S_{Q_0}^\vartheta u(x)^2 d\sigma(x)
\lesssim \iint_{\Omega_{\widehat{\F}_N, Q_0}^\vartheta} |\nabla u|^2 \delta \ dY
\lesssim_N \iint_{\Omega_{\widehat{\F}_{N}, Q_0}^\vartheta} |\nabla u|^2 \G \ dY. \end{aligned}$$ As in Section \[sec:a-c\], for every $M \geq 1$, we consider the pairwise disjoint collection $\widehat{\F}_{N,M}$ that is the family of maximal cubes of the collection $\widehat{\F}_N$ augmented by adding all the cubes $Q \in \D_{Q_0}$ such that $\ell(Q) \leq 2^{-M} \ell(Q_0)$. In particular, $Q \in \D_{\widehat{\F}_{N,M}, Q_0}$ if and only if $Q \in \D_{\widehat{\F}_N, Q_0}$ and $\ell(Q)>2^{-M}\ell(Q_0)$. Moreover, $\D_{\widehat{\F}_{N,M}, Q_0} \subset \D_{\widehat{\F}_{N,M'}, Q_0}$ for all $M \leq M'$, and hence $\Omega_{\widehat{\F}_{N,M}, Q_0}^\vartheta \subset \Omega_{\widehat{\F}_{N,M'}, Q_0}^\vartheta \subset \Omega_{\widehat{\F}_N, Q_0}^\vartheta$. Then the monotone convergence theorem implies $$\begin{aligned}
\label{eq:JM-lim}
\iint_{\Omega_{\widehat{\F}_{N}, Q_0}^\vartheta} |\nabla u|^2 \G \ dY
=\lim_{M \to \infty}\iint_{\Omega_{\widehat{\F}_{N,M}, Q_0}^\vartheta} |\nabla u|^2 \G \ dY
=:
\lim_{M \to \infty} \mathcal{J}_M.\end{aligned}$$ To continue with the proof, we are going to follow [@CHMT Proof of Proposition 4.18]. Let $\Psi\in C^\infty_c(\R^{n+1})$ be the smooth cut-off function associated with the sawtooth domain $\Omega_{\widehat{\F}_{N, M}, Q_0}^\vartheta$ (see [@CHMT Lemma 3.61] or [@HMT1 Lemma 4.44]) and note that since $\Psi\gtrsim 1$ in $\Omega_{\widehat{\F}_{N,M}}^{\vartheta}$ we have $$\begin{aligned}
\label{eq:JM-lim-tilde}
\mathcal{J}_M \lesssim \widetilde{\mathcal{J}}_M
:= \iint_{\Omega} |\nabla u|^2 \G \Psi^2\ dY. \end{aligned}$$ Note that $\widetilde{\mathcal{J}}_M<\infty$ because $\supp \Psi\subset \overline{\Omega_{\widehat{\F}_{N, M}, Q_0}^{\vartheta,*}} \subset \Omega$ and $u \in W^{1,2}_{\loc}(\Omega)$. A careful examination of [@CHMT Proof of Proposition 4.18] gives $$\begin{gathered}
\widetilde{\mathcal{J}}_M \lesssim_N \sigma(Q_0) + \widetilde{\mathcal{J}}_M^{\frac12}
\left(\iint_{\Omega^{\vartheta,*}_{\widehat{\F}_{N,M}, Q_0}} \frac{a(X)^2}{\delta(X)} dX \right)^{\frac12}
\\
+ \widetilde{\mathcal{J}}_M^{\frac12} \sigma(Q_0)^{\frac12}
+ \sigma(Q_0)^{\frac12}
\left(\iint_{\Omega^{\vartheta,*}_{\widehat{\F}_{N,M}, Q_0}} |\div_C D(X)|^2 \delta(X) dX \right)^{\frac12}. \end{gathered}$$ In turn, applying Young’s inequality and hiding, we readily get $$\begin{aligned}
\label{eq:JM}
\widetilde{\mathcal{J}}_M \lesssim_N \sigma(Q_0) +
\iint_{\Omega^{\vartheta,*}_{\widehat{\F}_N, Q_0}} \frac{a(X)^2}{\delta(X)} dX
+ \iint_{\Omega^{\vartheta,*}_{\widehat{\F}_N, Q_0}} |\div_C D(X)|^2 \delta(X) dX, \end{aligned}$$ where the implicit constant is independent of $M$. Collecting , , , and , we obtain $$\begin{aligned}
\label{eq:S-dis}
&\int_{\widehat{E}_N} S_{Q_0}^\vartheta u(x)^2 d\sigma(x)
\lesssim \sigma(Q_0) + \iint_{\Omega^{\vartheta,*}_{\widehat{\F}_N, Q_0}} \frac{a(X)^2}{\delta(X)} dX
+ \iint_{\Omega^{\vartheta,*}_{\widehat{\F}_N, Q_0}} |\div_C D(X)|^2 \delta(X) dX
\nonumber \\&\qquad\leq\sigma(Q_0) + \sum_{Q \in \D_{\widehat{\F}_N, Q_0}}
\bigg(\iint_{U^{\vartheta,*}_Q} \frac{a(X)^2}{\delta(X)} dX
+
\iint_{U^{\vartheta,*}_Q} |\div_C D(X)|^2 \delta(X)dX\bigg)
\nonumber \\&\qquad\lesssim \sigma(Q_0) + \sum_{Q \in \D_{\widehat{\F}_N, Q_0}} \gamma_Q^\vartheta \sigma(Q), \end{aligned}$$ where we used that $\sigma(Q) \simeq \ell(Q)^n \simeq \delta(X)^n$ for every $X \in U^{\vartheta,*}_Q$. On the other hand, $$\begin{gathered}
\label{eq:dis}
\sum_{Q \in \D_{\widehat{\F}_N, Q_0}} \gamma_Q^\vartheta \sigma(Q)
=\int_{Q_0} \sum_{x \in Q \in \D_{\widehat{\F}_N, Q_0}} \gamma_Q^\vartheta d\sigma(x)
\\\leq \int_{\widehat{E}_N} S_{Q_0} \gamma^\vartheta(x)^2 d\sigma(x)
+ \sum_{Q_j \in \widehat{\F}_N} \sum_{Q \in \D_{\widehat{\F}_N, Q_0}} \gamma_Q^\vartheta\,\sigma(Q\cap Q_j). \end{gathered}$$ As observed above $\widetilde{E}_N^0\subset \widehat{E}_N^0$, hence, leads to $$\begin{aligned}
\label{eq:dis-1}
\int_{\widehat{E}_N} S_{Q_0} \gamma^\vartheta(x)^2 d\sigma(x)
\leq N^2 \sigma(Q_0). \end{aligned}$$ In order to control the second term in , we fix $Q_j \in \widehat{\F}_N$. Note that if $Q \in \D_{\widehat{\F}_N, Q_0}$ is so that $Q \cap Q_j \neq \emptyset$ then necessarily $Q_j \subsetneq Q$. Write $\widehat{Q}_j$ for the dyadic father of $Q_j$, that is, $\widehat{Q}_j$ is the unique dyadic cube containing $Q_j$ with $\ell(\widehat{Q}_j)=2\ell(Q_j)$. We claim that $$\label{eq:QjQ-N2}
\sum_{\widehat{Q}_j \subset Q \in \D_{Q_0}} \gamma_Q^\vartheta =\sum_{Q_j \subsetneq Q \in \D_{Q_0}} \gamma_Q^\vartheta \leq N^2.$$ Otherwise, recalling the construction of $\widetilde{\F}_N$ in , it follows that $\widehat{Q}_j \subset Q'$ for some $Q' \in \widetilde{\F}_N$. From the definition of $\widehat{\F}_N$, we then have that $Q' \subset Q''$ for some $Q'' \in \widehat{\F}_N$. Consequently, $Q_j \subsetneq Q''$ with $Q_j, Q'' \in \widehat{\F}_N$ contradicting the maximality of the family $\widehat{\F}_N$. Then it follows from that $$\begin{gathered}
\label{43r43r4}
\sum_{Q_j \in \widehat{\F}_N} \sum_{Q \in \D_{\widehat{\F}_N, Q_0}} \gamma_Q^\vartheta \sigma(Q \cap Q_j)
=\sum_{Q_j \in \widehat{\F}_N} \sigma(Q_j) \sum_{Q_j \subsetneq Q \in \D_{Q_0}} \gamma_Q^\vartheta
\\
\leq N^2 \sum_{Q_j \in \widehat{\F}_N} \sigma(Q_j)
\leq N^2 \sigma \bigg(\bigcup_{Q_j \in \widehat{\F}_N} Q_j \bigg)
\leq N^2 \sigma(Q_0). \end{gathered}$$ Collecting , , , and , we deduce that $$\begin{aligned}
\int_{\widehat{E}_N} S_{Q_0}u(x)^2 d\sigma(x)
\leq C_N \sigma(Q_0) \simeq C_N 2^{-k_0 n}.\end{aligned}$$ This shows and completes the proof of Theorem \[thm:AAAD\].
Now let us see how we deduce Theorems \[thm:perturbation\] and \[thm:wL-wLT\] from Theorem \[thm:AAAD\].
Let $L_0$ and $L_1$ be the elliptic operators given in Theorem \[thm:perturbation\]. If we take $A=A_0-A_1$ and $D=0$ in Theorem \[thm:AAAD\], then coincides with the assumption and holds automatically. Therefore, Theorem \[thm:perturbation\] immediately follows from Theorem \[thm:AAAD\].
Let $A$ be the matrix as stated in Theorem \[thm:wL-wLT\]. If we take $A_0=A$, $A_1=A^{\top}$, $\widetilde{A}=0$ and $D=A-A^{\top}$ in Theorem \[thm:AAAD\], then one has $A_0-A_1=\widetilde{A}+D$ with $D \in \Lip_{\loc}(\Omega)$ antisymmetric, holds trivially and agrees with . Thus, Theorem \[thm:AAAD\] implies that $\sigma \ll \w_L$ if and only if $\sigma \ll \w_{L^{\top}}$.
Similarly, the conclusion that $\sigma \ll \w_L$ if and only if $\sigma \ll \w_{L^{\rm sym}}$ follows if we set $A_0=A$, $A_1=(A+A^{\top})/2$, $\widetilde{A}=0$ and $D=(A-A^{\top})/2$.
[00]{}
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[^1]: The first and second authors financial support from the Spanish Ministry of Science and Innovation, through the “Severo Ochoa Programme for Centres of Excellence in R&D” (SEV-2015-0554) and from the Spanish National Research Council, through the “Ayuda extraordinaria a Centros de Excelencia Severo Ochoa” (20205CEX001). They also acknowledge that the research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ ERC agreement no. 615112 HAPDEGMT. The third author was supported by PIP 112201501003553 (CONICET) and has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 777822. The last author would like to express her gratitude to the first two authors and the Instituto de Ciencias Matemáticas (ICMAT), for their support and hospitality.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Pattern-equivariant (PE) cohomology is a well-established tool with which to interpret the Čech cohomology groups of a tiling space in a highly geometric way. In this paper we consider homology groups of PE infinite chains. A generalised setting in which to consider PE homology and cohomology is established. We prove Poincaré duality between the two under certain conditions, which applies to examples such as the translational hull of an FLC tiling. So PE chains may be used to visualise topological invariants of tilings. The PE homology groups for a Euclidean tiling based upon chains which are PE with respect to the group of orientation preserving rigid motions exhibit a singular behaviour at points of rotational symmetry, which often adds extra torsion to the calculated invariants. We present an efficient method of computation of the PE (co)homology groups for hierarchical tilings.'
address: |
Department of Mathematics\
University of York\
Heslington, York, YO10 5DD\
UK
author:
- 'James J. Walton'
title: 'Pattern-equivariant homology'
---
Introduction {#introduction .unnumbered}
============
Periodic patterns of Euclidean space have been of great important to mathematics, and aesthetics, for centuries. However, the geometries of periodic patterns are well understood, and there is a rich class of patterns, such as the Penrose tilings, which exhibit intricate internal structure without possessing translational symmetry. Aperiodic tilings enjoy connections with areas of mathematics such as mathematical logic [@LafWei08]—as established by Berger’s proof of the undecidability of the domino problem [@Berg66], Diophantine approximation [@ABEI01; @BerSie05; @HKW14; @HKW15], the structure of attractors [@ClaHun12] and symbolic dynamics [@Sch01], but also to physical applications, most notably to solid state physics in the wake of the discovery of quasicrystals by Shechtman et al. [@SBGC84].
A full understanding of a periodic tiling, modulo locally defined reversible redecorations, amounts to an understanding of its symmetry group. In the aperiodic setting, the complexity and incredible diversity of examples demands a multifaceted approach. Techniques from the theory of groupoids [@BJS10], semigroups [@KelLaw00], $C^*$–algebras [@AndPut98], dynamical systems [@ClaSad06; @Kel08], ergodic theory [@Rad97] and shape theory [@ClaHun12] find natural rôles in the field, and of course these tools have tightly knit connections to each other [@KelPut00]. One approach to studying a given aperiodic tiling $T$ is to associate to it a moduli space $\Omega_T$, sometimes called the tiling space, of locally indistinguishable tilings; see Sadun’s book [@Sad08] for an accessible introduction to the theory. A central goal is then to formulate methods of computing topological invariants of $\Omega_T$, and to describe what these invariants actually tell us about the original tiling $T$. An important perspective, particularly for the latter half of this objective, is provided by Kellendonk and Putnam’s theory of pattern-equivariant (PE) cohomology [@Kel03; @KelPut06]. The PE cohomology groups allow for an intuitive geometric description of the Čech cohomology $\check{H}^\bullet(\Omega_T)$ of the tiling space. Over $\mathbb{R}$ coefficients the PE cochain groups may be defined using PE differential forms [@Kel03], and over general Abelian coefficients, when the tiling has a cellular structure, with PE cellular cochains [@Sad07].
In this paper we introduce the *pattern-equivariant homology groups* of a tiling. The PE chain complex is essentially defined by replacing the cellular coboundary maps of the PE cellular cochain complex with the cellular boundary maps. So the elements of the PE chain groups are based on infinite cellular chains (sometimes called Borel–Moore chains) which respect the internal symmetries of the pattern.
Although the PE homology groups may be defined using singular chains without reference to a specific cellular structure for the underlying pattern [@Wal14], in this exposition we shall avoid the singular theory and work exclusively with cellular chains. We review the combinatorial theory of regular CW complexes in §\[sec: Preliminaries\], which one may think of as a generalisation of the theory of abstract simplicial complexes. This allows us, in §\[sec: SIS\], to introduce the notion of a *system of internal symmetries* (*SIS*, for short) over a regular CW complex. Constructions such as the PE cochain complex or the inverse limit of approximants associated to a cellular tiling extend to these objects. The setting applies naturally to Euclidean as well as non-Euclidean examples, such as hyperbolic tilings or Bowers and Stephenson’s conformal tilings [@BowSte97], and even to more abstract examples not canonically described by the internal symmetries of a geometric tiling of space, such as those associated to solenoids (see Example \[ex: dyadic\] and §\[solenoid\]). Whilst SIS’s are introduced primarily as a convenient notational tool in which to frame our arguments, we believe that the notion should be of particular use in defining invariants for tilings in the presence of non-trivial isotropy.
In §\[sec: PE (Co)homology\] we define the PE homology and cohomology groups of an SIS, and show their invariance under barycentric subdivision. We then relate the two through the following Poincaré duality result:
[**Theorem \[thm: PEPD\]**]{} [*Let ${\mathfrak}{T}$ be an SIS defined over an oriented homology $d$–manifold. If the internal symmetries of ${\mathfrak}{T}$ are orientation preserving then we have PE Poincaré duality $H^\bullet({\mathfrak}{T};\mathbb{R}) \cong H_{d-\bullet}({\mathfrak}{T};\mathbb{R})$ over divisible coefficients. Moreoever, if ${\mathfrak}{T}$ has trivial local isotropy, then we have PE Poincaré duality $H^\bullet({\mathfrak}{T}) \cong H_{d-\bullet}({\mathfrak}{T})$ over general Abelian coefficients.*]{}
For example, we have Poincaré duality between the Čech cohomology of the translational hull of an FLC tiling $T$ of $\mathbb{R}^d$ and the PE homology of $T$ with respect to translations.
So the PE homology groups provide highly geometric descriptions of pre-established topological invariants associated to aperiodic tilings; see, for example, Figure \[fig: Penrose\]. When one wishes to incorporate rotational symmetries, so that ${\mathfrak}{T}$ may possess non-trivial local isotropy, the PE Poincaré duality of Theorem \[thm: PEPD\] generally fails over non-divisible coefficients and the PE homology provides a new invariant. The PE homology often picks up extra torsion elements to the PE cohomology. Although we demonstrate in §\[subsec: mod\] how one may modify the PE homology groups so as to regain duality, we consider this extra torsion as a phenomenon of potential interest. Indeed, in forthcoming work [@Wal15] (see also [@Wal14]) we will show how this extra torsion may be incorporated into a spectral sequence converging to the Čech cohomology $\check{H}^\bullet(\Omega_T^{\text{rot}})$ of the Euclidean hull of a $2$–dimensional tiling.
In §\[sec: PE Homology of Hierarchical Tilings\] we present a method of computation for the PE homology of a hierarchical tiling, along with some worked through examples. The method is applicable to a broad range of tilings, including Euclidean ‘mixed substitution tilings’ (see [@GahMal13]) but also non-Euclidean examples, such as Bowers and Stephenson’s combinatorial pentagonal tilings (see [@BowSte97] and §\[Pentagonal\]). The ‘approximant homology groups’ of the computation and the ‘connecting maps’ between them have a direct description in terms of the combinatorics of the star-patches. In [@Gon11], Gonçalves used the duals of these approximant chain complexes for a computation of the $K$–theory of the $C^*$–algebra of the stable equivalence relation of a substitution tiling. Our method of computation of the PE homology groups seems to confirm the observation there of a certain duality between these $K$–groups and the $K$–theory of the tiling space.
Acknowledgements {#acknowledgements .unnumbered}
----------------
The author wishes to thank John Hunton, Alex Clark, Lorenzo Sadun and Dan Rust for numerous helpful discussions, and the anonymous referee for their valuable suggestions. We also acknowledge the financial support of the EPSRC.
Preliminaries {#sec: Preliminaries}
=============
Regular CW Complexes
--------------------
We shall denote by $E^n$ the open unit disc, by $B^n$ the closed unit disc and by $S^{n-1}$ the unit $(n-1)$–sphere of $\mathbb{R}^n$. An *open $n$–cell* of a topological space $X$ is a subspace $e \subseteq X$ which is homeomorphic to $E^n$.
Let $X$ be a Hausdorff topological space and ${\mathcal}{K} = \{e_i \mid i \in {\mathcal}{I}\}$ be a partition of $X$ as a disjoint union of open cells $e_i$. Denote the pair of $X$ and ${\mathcal}{K}$ by $X_{\mathcal}{K}$, and the *$n$–skeleton* $X_{\mathcal}{K}^n \subseteq X$ by $$X_{\mathcal}{K}^n \coloneqq \coprod_{i \in {\mathcal}{I} : \text{dim}(e_i) \leq n} e_i.$$ Then $X_{\mathcal}{K}$ is called a *CW complex* if:
1. For $e_i \in {\mathcal}{K}$ there exists a continuous map (called a *characteristic map*) $f \co B^n \to X$ which restricts to a homeomorphism $f \restriction_{E^n} \co E^n \to e_i$ and is such that $\operatorname{\text{im}}(f \restriction_{S^{n-1}})$ $\subseteq$ $X_{\mathcal}{K}^{n-1}$.
2. For $e_i \in {\mathcal}{K}$ the closure $\overline{e_i}$ intersects only finitely many elements of ${\mathcal}{K}$.
3. A subset $A \subseteq X$ is closed if and only if $A \cap \overline{e_i}$ is closed in $X$ for all $e_i \in {\mathcal}{K}$.
Whilst CW complexes are ‘nice’ topological spaces which may be constructed by inductively attaching cells, there are not many constraints imposed on the characteristic maps governing how cells may be attached. In contrast, for a space with a *triangulation*, its topology is totally described up to homeomorphism by the combinatorial data of the simplexes and their incidences, i.e., the data of which simplexes are faces of others. Simplicial complexes are too restrictive to be practical as a starting point for us here; most well-known tilings are naturally cellular but not simplicial. A good middle-ground between the versatility of CW complexes and the combinatorial nature of simplicial complexes seems to be the setting of *regular* CW complexes:
A CW complex $X_{\mathcal}{K}$ is called *regular* if its characteristic maps may be chosen to be homeomorphisms.
For a CW complex $X_{\mathcal}{K}$ and open $n$–cell $e_i \in {\mathcal}{K}$, the closure $\overline{e_i}$ of $e_i$ in $X$ will be called a *closed $n$–cell* of $X_{\mathcal}{K}$. In the case that $X_{\mathcal}{K}$ is regular, each closed $n$–cell is homeomorphic to the closed $n$–disc $B^n$. Note, however, that the closed cells being homeomorphic to $n$–discs does not guarantee that a CW complex is regular.
CW Posets {#subsec: CW Posets}
---------
We shall now justify the assertion that regular CW complexes are combinatorial in character, and in so doing set up some important notation for the remainder of the paper. The main idea (see for example Björner [@Bjo84]) is that one may associate to a regular CW complex $X_{\mathcal}{K}$ a *face poset* which uniquely identifies $X_{\mathcal}{K}$ up to cellular homeomorphism.
Let $X_{\mathcal}{K}$ be a CW complex. We define the *face poset* ${\mathscr}{F}({\mathcal}{K})$ by setting the underlying set of ${\mathscr}{F}({\mathcal}{K})$ to be the set of closed cells of $X_{\mathcal}{K}$ and let $a \preceq b$ in ${\mathscr}{F}({\mathcal}{K})$ if and only if $a \subseteq b$.
The above allows one to associate to any CW complex a poset. Conversely, given any poset, there is a natural way to associate to it an abstract simplicial complex, and hence a topological space:
\[def: order complex\] For a poset ${\mathscr}{P}$, we define its *order complex* ${\mathscr}{P}_\Delta$ as follows. The underlying set of ${\mathscr}{P}_\Delta$ is given by the set of *finite chains* of finite, non-empty, linearly ordered subsets of ${\mathscr}{P}$ (i.e., subsets $\{a_1,\ldots,a_n\}$ where $a_i \prec a_{i+1}$). For two such chains $A,B$, we set $A \preceq B$ in ${\mathscr}{P}_\Delta$ if $A \subseteq B$.
An *abstract simplicial complex* is a non-empty set $V$ (called the *vertex set*) together with a collection ${\mathscr}{S}$ of non-empty subsets of $V$ for which:
1. $\{v\} \in {\mathscr}{S}$ for all $v \in V$.
2. If $B \in {\mathscr}{S}$ and $\emptyset \neq A \subseteq B$, then $A \in {\mathscr}{S}$ also.
Clearly the order complex ${\mathscr}{P}_\Delta$ of any poset ${\mathscr}{P}$ is an abstract simplicial complex with vertex set the underlying set of ${\mathscr}{P}$. To an abstract simplicial complex ${\mathscr}{S}$ one may associate a topological space $|{\mathscr}{S}|$, called the *geometrical realisation of ${\mathscr}{S}$*. The following shows that a regular CW complex is determined, up to cellular homeomorphism, by the combinatorics of its closed cells and their incidences (c.f., Björner [@Bjo84 §3], Lundell and Weingram [@LunWei69 Theorem 1.7] or Massey [@Mas91 Chapter IX §6]):
Let $X_{\mathcal}{K}$ be a regular CW complex. Then there exists a homeomorphism $h \co |{\mathscr}{F}({\mathcal}{K})_\Delta| \to X$. Moreover, $h$ may be chosen so that:
1. For a closed cell $a$ of ${\mathcal}{K}$, the subspace of $|{\mathscr}{F}({\mathcal}{K})_\Delta|$ corresponding to the set of simplexes $\{\{a_0,\ldots,a_n\} \in {\mathscr}{F}({\mathcal}{K})_\Delta \mid a_n \preceq a\}$ is mapped by $h$ onto $a$.
2. For an open cell $e \in {\mathcal}{K}$, the subspace of $|{\mathscr}{F}({\mathcal}{K})_\Delta|$ corresponding to the set of simplexes $\{\{a_0,\ldots,a_n\} \in {\mathscr}{F}({\mathcal}{K})_\Delta \mid a_n=\overline{e}\}$ is mapped by $h$ onto $e$.
So a regular CW complex may be subdivided into a simplicial complex in a way which respects the structure of the original cell decomposition. It follows from these considerations that there is effectively no loss of information in passing from a regular CW complex $X_{\mathcal}{K}$ to its face poset ${\mathscr}{F}({\mathcal}{K})$. We shall embrace this philosophy in what follows by working chiefly with face posets of regular CW complexes, which shall frequently simplify our notation and arguments. A poset which is isomorphic to the face poset of a regular CW complex is called a *CW poset*. We shall refer to the order complex ${\mathscr}{F}_\Delta$ of a CW poset ${\mathscr}{F}$ as *the barycentric subdivision of ${\mathscr}{F}$*. We denote by $|{\mathscr}{F}|$ its geometric realisation, which by the above one may unambiguously define, up to homeomorphism, to be $|{\mathscr}{F}_\Delta|$ or as any space $X$ possessing a regular CW decomposition ${\mathcal}{K}$ with ${\mathscr}{F}({\mathcal}{K}) \cong {\mathscr}{F}$. Elements $a \in {\mathscr}{F}$ of a CW poset shall be called *cells*. If $a \preceq b$ then we shall say that $a$ is a *face* of $b$, and that $b$ is a *coface* of $a$. For $a \in {\mathscr}{F}$, its dimension may be defined by the cardinality (minus one) of any maximal chain $\{a_0,\ldots,a_{\text{dim}(c)}=a\}$ terminating in $a$ (note that we do not include the ‘empty face’ here). We shall allow ourselves to denote $a$ by $a^n$ when it has dimension $n$. CW posets exhibit the $\diamond$–property, taking its name from the corresponding Hasse diagram: for any $a^n \prec c^{n+2}$ there are precisely two $(n+1)$–cells $b_i^{n+1}$ satisfying $a^n \prec b_i^{n+1} \prec c^{n+2}$.
Let ${\mathscr}{F}$ be a CW poset and $S \subseteq {\mathscr}{F}$. Then we define the *closure* of $S$ to be the sub-poset $$\overline{S} \coloneqq \{a \in {\mathscr}{F} \mid \exists s \in S : a \preceq s\}.$$ Note that $\overline{S}$ is itself a CW poset. For $a \in {\mathscr}{F}$ we write $\overline{a}$ for the *cell poset* $\overline{\{a\}}$ of faces of $a$. We call $S$ a *subcomplex* if $S = \overline{S}$ (that is, if $S$ is downwards closed).
For $a \in {\mathscr}{F}$ we define the (*open*) *star of $a$* to be the sub-poset of cells $$\text{St}(a) \coloneqq \{a' \in {\mathscr}{F} \mid a' \succeq a\}.$$ We shall refer to $\overline{\text{St}(a)}$ as the *star complex of $a$*. We call ${\mathscr}{F}$ *pure* (*of dimension $d$*) if $\text{St}(a)$ contains a $d$–cell, and no cells of strictly greater dimension, for all $a \in {\mathscr}{F}$.
We define the following collections of simplexes of ${\mathscr}{F}_\Delta$:
1. The *closed cell* $a_\Delta \coloneqq \{\{a_0,\ldots,a_n\} \mid a_n \preceq a\}$.
2. The *open cell* $a_\Delta^\circ \coloneqq \{\{a_0, \ldots, a_n\} \mid a_n=a\}$.
3. The *boundary* $\partial a_\Delta \coloneqq \Delta(a) - \Delta(a)^\circ = $ $\{\{a_0,\ldots,a_n\} \mid a_n \precneq a\}$.
4. The *closed dual cell* $\widehat{a}_\Delta \coloneqq \{\{a_0,\ldots,a_n\} \mid a \preceq a_0\}$.
5. The *open dual cell* $\widehat{a}_\Delta^\circ \coloneqq \{\{a_0,\ldots,a_n\} \mid a = a_0\}$.
6. The *dual boundary* $\partial \widehat{a}_\Delta \coloneqq \widehat{a}_\Delta - \widehat{a}_\Delta^\circ =$ $\{\{a_0,\ldots,a_n\} \mid a \precneq a_0\}$.
7. The *$k$–skeleton* ${\mathscr}{F}^k_\Delta \coloneqq \{s \in {\mathscr}{F}_\Delta \mid s \in a_\Delta : \text{dim}(a) = k\} = $ $\{s \in {\mathscr}{F}_\Delta \mid s \in a_\Delta^\circ : \text{dim}(a) \leq k\}$.
8. If ${\mathscr}{F}$ is pure of dimension $d$, then we define the *dual $k$–skeleton* $\widehat{{\mathscr}{F}}_\Delta^k \coloneqq \{s \in {\mathscr}{F}_\Delta \mid s \in \widehat{a}_\Delta : \text{dim}(a) = (d-k)\} =$ $\{s \in {\mathscr}{F}_\Delta \mid s \in \hat{a}_\Delta^\circ : \text{dim}(a) \geq (d-k)\}$.
Let ${\mathscr}{F}$ be a CW poset. We shall say that ${\mathscr}{F}$ is *finite-dimensional* if there exists some $d$ for which $\text{dim}(a) \leq d$ for all $a \in {\mathscr}{F}$. We call ${\mathscr}{F}$ *locally finite* if $\text{St}(a)$ is finite for all $a \in {\mathscr}{F}$.
*Henceforth, all CW posets considered shall be taken to be finite-dimensional and locally finite.*
Homology and Cohomology of Regular CW Complexes
-----------------------------------------------
Given a chain complex $C_\bullet$ we shall denote its homology, as a $\mathbb{Z}$ graded collection of groups, by $H(C_\bullet)$. The degree $n$ homology group shall be denoted by $H_n(C_\bullet)$. Given a CW complex $X_{\mathcal}{K}$ one may compute the singular homology of $X$ using instead the cellular chain complex $C_\bullet(X_{\mathcal}{K})$. As a group $C_n(X_{\mathcal}{K})$ is isomorphic to the free Abelian group generated by the set of $n$–cells of ${\mathcal}{K}$. To determine the boundary maps $\partial_n \co C_n(X_{\mathcal}{K}) \to C_{n-1}(X_{\mathcal}{K})$, one uses the formula $$\partial_n(b^n) = \sum_{a^{n-1} : a^{n-1} \preceq b^n} [a^{n-1},b^n] a^{n-1}$$ where the elements $a^{n-1}$ and $b^n$ in the above correspond to cells of ${\mathcal}{K}$, considered as elements of $C_n(X_{\mathcal}{K})$ with their preferred orientations. One then defines $\partial_n$ by extending from the above formula linearly. The *incidence numbers* $[a,b]$ of the above are integers determined by a choice of orientation for each cell and the characteristic maps of the CW complex. A practical motivation for using *regular* CW complexes $X_{\mathcal}{K}$ is that a choice of orientations for the cells of ${\mathcal}{K}$ determines and is determined by incidence numbers $[a,b]$ satisfying a few simple and intuitive axioms in terms of the CW poset ${\mathscr}{F}({\mathcal}{K})$ (see Cooke and Finney [@CooFin67 Chapter II, §1] or Massey [@Mas91 Chapter IX, §7]).
The above approach allows one to compute $H_\bullet(X_{\mathcal}{K})$ directly from ${\mathscr}{F}({\mathcal}{K})$. However, since we shall need to be able to compare orientations of cells under cellular isomorphism, we still require a notion of orientation for cells. One way is to define an orientation on a cell $a^n \in {\mathscr}{F}$ to be a simplicial $n$–chain of $a_\Delta$ with coefficients $+1$ or $-1$ and boundary supported on the subcomplex $\partial a_\Delta$. Such a chain is determined by choosing any chain $\{a_0,\ldots,a_n=a\}$ of length $(n+1)$ and assigning it coefficient $+1$ or $-1$. There are precisely two orientations of any given cell, we call these two orientations *opposite*. An orientation $\omega_b$ of a cell $b$ defines an orientation $\omega \hspace{-4pt} \downarrow_a^b$ of any codimension one face $a$ of $b$ in a canonical way. In the case that ${\mathscr}{F}$ is simplicial, one may associate to each orientation $\omega_{a^n}$ a simplicial orientation in a way which is natural with respect to restrictions to faces (recall that a simplicial orientation $\pm 1 [v_0,\ldots,v_n]$ is a signed and ordered list of the vertices of $a^n$, taken up to the relation of permutation $\pm 1 [v_0,\ldots,v_n] \simeq (\pm 1)^{\text{sgn}(\sigma)} \sigma[v_0,\ldots,v_n]$. Such an orientation restricts to any face by omission of the corresponding vertex).
The usual cellular chain groups $C_i(X_{\mathcal}{K})$ are based on finite chains, each supported on a compact subcomplex of $X_{\mathcal}{K}$. Finite chains will not be of much use to us in representing invariants on infinite tilings. However, for a locally finite CW complex one may define homology groups based on infinite cellular chains, sometimes known as Borel–Moore chains[^1].
Let ${\mathscr}{F}$ be a CW poset. We define the *cellular Borel–Moore chain complex* $C^{\text{BM}}_\bullet({\mathscr}{F})$ as follows. Set $C^{\text{BM}}_n({\mathscr}{F})$ to be the Abelian group of functions $$\sigma \co \{\text{oriented } n\text{--cells of } {\mathscr}{F}\} \to \mathbb{Z}$$ for which $\sigma(\omega_a^+) = -\sigma(\omega_a^-)$ for opposite orientations $\omega_a^+$ and $\omega_a^-$ of a cell $a$. Of course, $\sigma_1 + \sigma_2$ is defined by setting $(\sigma_1 + \sigma_2)(\omega) \coloneqq \sigma_1(\omega) + \sigma_2(\omega)$. We define the boundary maps $\partial_n \co C_n^{\text{BM}}({\mathscr}{F}) \to C_{n-1}^{\text{BM}}({\mathscr}{F})$ by setting $$\partial_n(\sigma)(\omega_a) \coloneqq \sum_{\omega_b : \omega_a = \omega \hspace{-1pt} \downarrow_a^b} \sigma(\omega_b)$$ and extending linearly. The homology $H^{\text{BM}}_\bullet({\mathscr}{F}) \coloneqq H(C_\bullet^{\text{BM}}({\mathscr}{F}))$ of this chain complex will be called the *Borel–Moore homology of ${\mathscr}{F}$*.
The boundary maps are well-defined due to our running assumption that the CW posets under consideration are locally finite. By setting an orientation for each cell $a \in {\mathscr}{F}$, one has a canonical isomorphism $$C_n^{\text{BM}}({\mathscr}{F}) \cong \prod_{a \in {\mathscr}{F} : \text{dim}(a)=n} \mathbb{Z}.$$ With this identification, the boundary maps agree on the ‘generators’ with the usual cellular boundary maps for finite cellular chains.
Given a cell $a$ and a codimension one coface $b$ of $a$, an orientation $\omega_a$ of $a$ induces an orientation $\omega \hspace{-4pt} \uparrow_a^b$ of $b$.
Let ${\mathscr}{F}$ be a CW poset. Then we define the *cellular cochain complex* $C^\bullet({\mathscr}{F})$ as follows. Set $C^n({\mathscr}{F})$ to be the Abelian group of functions $$\psi \co \{\text{oriented } n \text{-cells of } {\mathscr}{F} \} \to \mathbb{Z}$$ for which $\psi(\omega_a^+)=-\psi(\omega_a^-)$ for opposite orientations $\omega_a^+$ and $\omega_a^-$ (so $C_n^{\text{BM}}({\mathscr}{F}) = C^n({\mathscr}{F})$). We define the coboundary maps $\delta^n \co C^n({\mathscr}{F}) \to C^{n+1}({\mathscr}{F})$ by setting $$\delta^n(\psi)(\omega_b) \coloneqq \sum_{\omega_a : \omega_b = \omega \hspace{-1pt} \uparrow_a^b} \psi(\omega_a)$$ and extending linearly. The cohomology $H^\bullet({\mathscr}{F}) \coloneqq H(C^\bullet({\mathscr}{F}))$ of this cochain complex will be called the *cohomology of ${\mathscr}{F}$*.
Let $\sigma$ be a Borel–Moore chain of the CW poset ${\mathscr}{F}$. Given a set of cells $S \subseteq {\mathscr}{F}$, we shall write $\sigma \restriction S$ to denote the restriction of $\sigma$ to $S$, which we consider to be a chain of the subcomplex $\overline{S}$.
A *cellular isomorphism* of CW posets ${\mathscr}{F}$ and ${\mathscr}{G}$ is a bijection $\Phi$ between the underlying sets of ${\mathscr}{F}$ and ${\mathscr}{G}$ for which $\Phi$ and $\Phi^{-1}$ are order-preserving. Such a cellular isomorphism induces another $\Phi_\Delta \co {\mathscr}{F}_\Delta \to {\mathscr}{G}_\Delta$ between barycentric subdivisions by setting $$\Phi_\Delta(\{a_0,\ldots,a_k\}) \coloneqq \{\Phi(a_0),\ldots,\Phi(a_k)\}.$$ Pushforwards of chains are denoted by $\Phi_*(\sigma)$. We shall frequently abuse notation by writing $\Phi_*$ in place of $(\Phi_\Delta)_*$. For example, given a cellular isomorphism between cell posets $\Phi \co \overline{a} \to \overline{b}$ and an orientation $\omega_a$ of $a$, we may push forward the orientation on $a$ to the orientation $\Phi_*(\omega_a) \coloneqq (\Phi_\Delta)_*(\omega_a)$ on $b$; here, we are considering the orientation $\omega_a$ as a simplicial $n$–cycle of the relative pair $(a_\Delta,\partial a_\Delta)$.
Recall that a *quasi-isomorphism* is a (co)chain map which induces isomorphisms between (co)homology groups.
\[prop: BCS\] Let ${\mathscr}{F}$ be a CW poset. Then there exist quasi-isomorphisms: $$\begin{gathered}
\iota_\bullet \co C_\bullet^{\text{BM}}({\mathscr}{F};G) \to C_\bullet^{\text{BM}}({\mathscr}{F}_\Delta)\\
\iota^\bullet \co C^\bullet({\mathscr}{F}_\Delta;G) \to C^\bullet({\mathscr}{F})\end{gathered}$$
Whilst this proposition is standard, we shall provide the essential details of the proof which shall be useful later when restricting to pattern-equivariant subcomplexes. A chain $\sigma \in C_n^{\text{BM}}({\mathscr}{F})$ canonically determines a simplicial chain $\sigma_\Delta \in C_n^{\text{BM}}({\mathscr}{F}_\Delta)$ by setting $(\sigma_\Delta \restriction a_\Delta) = \sigma(\omega_a) \cdot \omega_a$ for any orientation $\omega_a$ (considered as a simplicial $n$–chain) on an $n$–cell $a$. This defines a simplicial chain supported on the $n$–skeleton ${\mathscr}{F}_\Delta^n$ whose boundary is supported on the $(n-1)$–skeleton ${\mathscr}{F}_\Delta^{n-1}$. Such chains form a sub-chain complex $$\cdots\xleftarrow{\partial_n} H_n^{\text{BM}}({\mathscr}{F}^n_\Delta,{\mathscr}{F}^{n-1}_\Delta) \xleftarrow{\partial_{n+1}} H_{n+1}^{\text{BM}}({\mathscr}{F}^{n+1}_\Delta,{\mathscr}{F}^n_\Delta) \xleftarrow{\partial_{n+1}} \cdots$$ of $C_\bullet^{\text{BM}}({\mathscr}{F}_\Delta)$. It essentially follows from the definitions that the map $(-)_\Delta$ induces a chain isomorphism from $C_\bullet^{\text{BM}}({\mathscr}{F})$ to the above relative complex.
The map $\iota_\bullet$ is then induced by the canonical inclusion of the above relative complex into the chain complex $C_\bullet({\mathscr}{F}_\Delta)$. Using the usual diagram chases, the result follows from the fact that the homologies of the relative pairs $H_n^{\text{BM}}({\mathscr}{F}^k_\Delta,{\mathscr}{F}^{k-1}_\Delta)$ vanish for $n \neq k$. The proof for cohomology is analogous.
The cellular Borel–Moore chain complexes of ${\mathscr}{F}$ defined above are based on $\mathbb{Z}$ coefficients. One may also define the chain complexes over $G$ coefficients, where $G$ is an arbitrary (discrete) Abelian group. We write $C^{\text{BM}}_\bullet({\mathscr}{F};G)$ for the cellular chain complex of ${\mathscr}{F}$ over $G$ coefficients, where the degree $n$ chain group is based instead on functions $$\sigma \co \{\text{oriented } n\text{--cells of } {\mathscr}{F}\} \to G.$$ One may similarly define the cellular cochain complexes over $G$ coefficients, denoted $C^\bullet({\mathscr}{F};G)$. We shall frequently drop the notation of the coefficient group where it is clear from context which coefficient group is in question.
Poincaré Duality
----------------
The classical ‘cell, dual cell proof’ of Poincaré duality, whilst usually stated in the simplicial setting (see for example Munkres [@Mun84]) follows over almost word-for-word to the regular CW setting, which we shall briefly recall here.
Let ${\mathscr}{S}$ be a simplicial complex which possesses a partial ordering on its vertex set which linearly orders the vertices of each simplex. Then we define a homomorphism $$\cap \co C^p({\mathscr}{S}) \otimes C_{p+q}^{\text{BM}}({\mathscr}{S}) \to C_p^{\text{BM}}({\mathscr}{S})$$ by setting $$\psi^p \cap ([v_0,\ldots,v_{p+q}]) \coloneqq [v_0,\ldots,v_q] \cdot \psi^p([v_q,\ldots,v_{p+q}])$$ and extending linearly. Here, simplicial chains are written with the elements in reverse order, so that $v_0 > v_1 > \ldots > v_{p+q}$.
One may easily check that the cap product satisfies $$\partial(\psi^p \cap \sigma_{p+q}) = (-1)^q(\delta(\psi^p) \cap \sigma_{p+q}) + \psi^p \cap \partial(\sigma_{p+q}).$$ It follows that the cap product induces a homomorphism $$\cap \co H^p({\mathscr}{S}) \otimes H_{p+q}^{\text{BM}}({\mathscr}{S}) \to H_q^{\text{BM}}({\mathscr}{S}).$$
Let ${\mathscr}{F}$ be a pure $d$–dimensional CW poset and $G$ be some coefficient ring. Given a choice of orientation of each $d$–cell of ${\mathscr}{F}$, there is an associated Borel–Moore $d$–chain given by assigning coefficient $+1$ to each preferred orientation. We shall say that ${\mathscr}{F}$ is *oriented* if it is equipped with a Borel–Moore cycle $\Gamma$, called a *fundamental class*, associated to some choice of orientation of each $d$–cell.
One may canonically identify the chain complex $C_\bullet^{\text{BM}}({\mathscr}{F})$ with the sub-chain complex of $C_\bullet^{\text{BM}}({\mathscr}{F}_\Delta)$ which in degree $n$ consists of simplicial $n$–chains supported on ${\mathscr}{F}_\Delta^n$ and with boundaries supported on ${\mathscr}{F}_\Delta^{n-1}$. In a similar way we may define the *dual chain complex* $C_\bullet^{\text{BM}}(\widehat{{\mathscr}{F}})$:
Let ${\mathscr}{F}$ be a pure CW poset. Then we define the *dual chain complex* $C_\bullet^{\text{BM}}(\widehat{{\mathscr}{F}};G)$ to be the sub-chain complex of $C_\bullet^{\text{BM}}({\mathscr}{F}_\Delta;G)$ which in degree $n$ consists of simplicial $n$–chains supported on the dual $n$–skeleton $\widehat{{\mathscr}{F}}^n_\Delta$ and with boundaries supported on the dual $(n-1)$–skeleton $\widehat{{\mathscr}{F}}_\Delta^{n-1}$.
A *homology* $d$*–manifold* is a locally compact topological space $X$ for which its local singular homology groups $H_i(X,X-\{x\})$, for any $x \in X$, are trivial for $i \neq d$ and are isomorphic to $\mathbb{Z}$ for $i=d$.
Of course, it is instructive to think of the dual chain complex as a cellular chain complex associated to the ‘dual cell decomposition’. It is worth reiterating, however, that even for a triangulated $d$–manifold, the dual cell decomposition need not be a CW complex.
Let ${\mathscr}{F}$ be a CW poset for a homology $d$–manifold $|{\mathscr}{F}|$. Fix a coefficient ring $G$ and suppose that $C_d^{\text{BM}}({\mathscr}{F};G)$ possesses a fundamental class $\Gamma$. Then:
1. The cap product with the fundamental class induces a cochain isomorphism $- \cap \Gamma \co C^\bullet({\mathscr}{F};G) \to C_{d-\bullet}^{\text{BM}}(\widehat{{\mathscr}{F}};G)$.
2. The canonical inclusion $\iota \co C_\bullet^{\text{BM}}(\widehat{{\mathscr}{F}};G) \to C_\bullet^{\text{BM}}({\mathscr}{F}_\Delta;G)$ is a quasi-isomorphism.
Hence, there exists an isomorphism $H^\bullet({\mathscr}{F};G) \cong H_{d-\bullet}^{\text{BM}}({\mathscr}{F};G)$.
For concreteness and later reference we shall make precise how the map $- \cap \Gamma$ in the above is explicitly defined. For a cochain $\psi \in C^n({\mathscr}{F};G)$ one may define a simplicial cochain $\psi_\Delta \in C^n({\mathscr}{F}_\Delta^n,{\mathscr}{F}_\Delta^{n-1};G)$ which, for an orientation $\omega_c$ of $n$–cell $c$, evaluates on the corresponding sum of oriented $n$–simplexes the value $\psi(\omega_c)$. Similarly (as in the proof of Proposition \[prop: BCS\]) to a chain $\sigma \in C_n^{\text{BM}}({\mathscr}{F};G)$ one may associate a simplicial chain $\sigma_\Delta \in C_n^{\text{BM}}({\mathscr}{F}_\Delta;G)$. Note that the barycentric subdivision ${\mathscr}{F}_\Delta$ inherits a natural partial order on its vertices from ${\mathscr}{F}$. Then $\psi \cap \Gamma$ is defined to be $\psi_\Delta \cap \Gamma_\Delta \in C_{d-n}^{\text{BM}}(\widehat{{\mathscr}{F}})$. The work in showing that this induces the required isomorphism is in showing that the dual $n$–cells modulo their boundaries have simplicial homology in degree $n$ freely generated by a relative fundamental class. For $2$ one needs to show that the relative homology groups of the neighbouring levels of the skeleta of the dual cell decomposition vanish in the expected degrees, in analogy to a CW decomposition.
Systems of Internal Symmetries {#sec: SIS}
==============================
A tiling possessing few global symmetries may be far from totally random. There exist interesting examples of tilings—the most famous being the Penrose tilings—which, despite having no translational symmetries, have the property that any given finite motif of the pattern, no matter how large, may be found with only bounded gaps across the entire pattern. This is a property known as recurrence or repetitivity. So, without possessing any *global* symmetries, a tiling may possess a rich structure of *internal* symmetries between finite portions of it. Forgetting the precise decoration in question, the essential structure of a given tiling is recorded by a system of pseudogroups or groupoids keeping track of which points of the pattern are equivalent to a certain radius and by which local morphisms.
These ideas motivate the central definition of this section, given below. Recall that a poset $(\Lambda, \leq)$ is called a *directed set* if for all $\lambda_1$, $\lambda_2 \in \Lambda$ there exists some $\lambda \in \Lambda$ with both $\lambda_1$, $\lambda_2 \leq \lambda$. We shall often say that a property holds *for sufficiently large $\lambda$*, which shall always mean that there exists some $\lambda' \in \Lambda$ for which the property holds for any $\lambda \geq \lambda'$.
\[def: SIS\] A *system of internal symmetries* (or *SIS*, for short) ${\mathfrak}{T}$ consists of the following data:
- A CW poset ${\mathscr}{T}$.
- A directed set $(\Lambda,\leq)$ called the *radius poset*.
- For each $\lambda \in \Lambda$ and each pair of cells $a,b \in {\mathscr}{T}$ a set ${\mathfrak}{T}_{a,b}^\lambda$ of cellular isomorphisms $\Phi \co \overline{\text{St}(a)} \to \overline{\text{St}(b)}$ sending $a$ to $b$. We denote the collection of all such morphisms by ${\mathfrak}{T}^\lambda$.
This data is required to satisfy the following:
1. For all $\lambda \in \Lambda$ and $a \in {\mathscr}{T}$ we have that $\text{Id}_{\overline{\text{St}(a)}} \in {\mathfrak}{T}_{a,a}^\lambda$.
2. For all $\lambda \in \Lambda$ and $\Phi \in {\mathfrak}{T}_{a,b}^\lambda$ we have that $\Phi^{-1} \in {\mathfrak}{T}_{b,a}^\lambda$.
3. For all $\lambda \in \Lambda$, $\Phi_1 \in {\mathfrak}{T}_{a,b}^\lambda$ and $\Phi_2 \in {\mathfrak}{T}_{b,c}^\lambda$, we have that $\Phi_2 \circ \Phi_1 \in {\mathfrak}{T}_{a,c}^\lambda$.
4. For all $\lambda_1 \leq \lambda_2$ we have that ${\mathfrak}{T}^{\lambda_1} \supseteq {\mathfrak}{T}^{\lambda_2}$.
5. For all $\lambda \in \Lambda$ there exists some $\lambda_{\text{res}} \geq \lambda$ satisfying the following. Given any $b \in {\mathscr}{T}$ and face $a$ of $b$, every morphism of ${\mathfrak}{T}^{\lambda_{\text{res}}}_{a,-}$ restricts to a morphism of ${\mathfrak}{T}^\lambda_{b,-}$.
6. Dually, for all $\lambda \in \Lambda$ there exists some $\lambda_{\widehat{\text{res}}}$ satisfying the following. Given any $a \in {\mathscr}{T}$ and coface $b$ of $a$, every morphism of ${\mathfrak}{T}_{b,-}^{\lambda_{\widehat{\text{res}}}}$ is a restriction of some morphism of ${\mathfrak}{T}_{a,-}^\lambda$.
The CW poset ${\mathscr}{T}$ is to be thought of as the underlying complex on which the tiling is supported. The radius poset $(\Lambda, \leq)$ will usually correspond to $\Lambda = \mathbb{R}_{>0}$ (as a ‘radial distance’) or $\mathbb{N}_0$ (as a ‘combinatorial distance’), although it will sometimes be helpful to allow some more flexibility. Then a morphism $\Phi \in {\mathfrak}{T}_{a,b}^\lambda$ is to be thought of as recording the fact that cells $a$ and $b$ are equivalent to radius $\lambda$ in the tiling, via a morphism which is described locally by $\Phi$. Such morphisms should include the identity morphism and be invertible and composable, as dictated by the groupoid axioms (G1), (G2) and (G3), respectively. The inclusion axiom (Inc) simply states that if two cells are equivalent to radius $\lambda_2$ in the tiling, via morphism $\Phi$, then they are still equivalent via $\Phi$ to any smaller radius $\lambda_1 \leq \lambda_2$. In short, for $\lambda_1 \leq \lambda_2$ we have an inclusion of groupoids $\iota \co {\mathfrak}{T}^{\lambda_2} \hookrightarrow {\mathfrak}{T}^{\lambda_1}$. The final two axioms (Res) and (CoRes) of restriction and corestriction establish a coherence between the cellular structure of ${\mathscr}{T}$ and the restrictions between the various morphisms of ${\mathfrak}{T}$.
Motivating Examples
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### Euclidean Tilings {#subsubsec: Euclidean Tilings}
A *tiling* $T$ of $\mathbb{R}^d$ is a collection of subsets $t \subset \mathbb{R}^d$, called *tiles*, satisfying:
1. The tiles of $T$ are compact and equal to the closures of their interiors.
2. $\mathbb{R}^d = \bigcup_{t \in T} t$.
3. Distinct $t_1,t_2 \in T$ intersect on at most their boundaries.
Sometimes one may wish to endow the tiles with ‘labels’ (or ‘colours’), so as to distinguish patches of tiles which are geometrically equivalent but are not so with their labels c.f., Wang tilings of decorated unit squares of $\mathbb{R}^2$.
The collection of closed $d$–cells of a regular CW decomposition ${\mathcal}{T}$ of $\mathbb{R}^d$ is a tiling, which we shall call a *cellular tiling*. There are some natural systems of internal symmetries defined over ${\mathscr}{F}({\mathcal}{T}) \coloneqq {\mathscr}{T}$.
Firstly, set the radius poset as $(\mathbb{R}_{>0},\leq)$, with the usual linear ordering. If we wish to compare patches of tiles only up to translation, then we define ${\mathfrak}{T}^1$ as follows. For $a \in {\mathscr}{T}$ and $r \in \mathbb{R}_{>0}$, denote by $P(a,r)$ the ‘patch’ of cells within radius $r$ of $a$. This may be defined in a variety of essentially equivalent ways (up to tail-equivalence, see Definition \[def: tail-equivalence\]). For example, one may set $$P(a,r) \coloneqq \overline{ \{a' \in {\mathscr}{T} \mid d(a,a') \leq r\} }$$ where the distance $d(a,a')$ is the infimum of (Euclidean) distances between points of $a$ and $a'$. Then set $\Phi \in ({\mathfrak}{T}^1)_{a,b}^r$ if and only if $\Phi$ is a cellular isomorphism between the star complexes of $a$ and $b$ induced by a translation taking $P(a,r)$ to $P(b,r)$ and preserving labels of cells, in case we have labelled the cells of our tiling.
One may instead wish to compare patches using orientation preserving isometries (which we shall call *rigid motions*). In this case, we define the SIS ${\mathfrak}{T}^0$ by setting $\Phi \in ({\mathfrak}{T}^0)_{a,b}^r$ if and only if $\Phi$ is a cellular isomorphism between the star complexes of $a$ and $b$ induced by a rigid motion taking $P(a,r)$ to $P(b,r)$.
For the purpose of studying topological invariants of these tilings, the restriction to cellular tilings is not a severe one. For any Euclidean tiling $T$ (whose tiles have bounded in and out-radii), by making a selection of puncture for each cell one may define a Delone set $P_T$, and to that an associated Voronoi tiling ${\mathcal}{V}(P_T)$. The tiling ${\mathcal}{V}(P_T)$ defines a regular CW decomposition of $\mathbb{R}^d$ of convex polytopal cells and, if one chooses the punctures carefully (or perhaps endows them with a colouring), the tilings ${\mathcal}{V}(P_T)$ and $T$ are equivalent in a very rigid sense (they are ‘MLD’, see [@Sad08 Chapter 1, §3]).
### Combinatorial Tilings {#subsubsec: Combinatorial Tilings}
One may mimic the constructions above for the case that ${\mathscr}{T}$ is a general CW poset, not necessarily coming from a cellular tiling of Euclidean space. One would usually think of a combinatorial tiling as defining a *pure* $d$–dimensional complex, in which case it makes sense to think of the $d$–cells as the tiles, although this is not actually necessary here. We see that we need the notions of:
1. Patches of size $\lambda$.
2. Labelling of cells.
3. ‘Allowed symmetries’ of patches (c.f., Euclidean translations/rigid motions).
For item one, we may define a combinatorial distance between cells. Inductively define $$\begin{gathered}
\text{St}^0(a) \coloneqq \overline{\text{St}(a)},\\
\text{St}^n(a) \coloneqq \overline{\{a'' \in {\mathscr}{T} \mid a'' \in \text{St}(a') \text{ for some } a' \in \text{St}^{n-1}(a)}.\end{gathered}$$ So for $n \in \mathbb{N}_0$ and $a \in {\mathscr}{T}$ one may define the *$n$–patch at $a$* to be $P(a,n) \coloneqq \text{St}^n(a)$. For example, for a cellular tiling of $\mathbb{R}^d$ and $d$–cell $a$ of it, we have that $P(a,n)$ corresponds to what is often named the “$n$–collared patch of $a$” (c.f., the construction of the Gähler approximants; see [@Sad08 Chapter 2, §4]).
One may deal with items two and three together as follows. We let $L$ be a groupoid, with objects the cells of ${\mathscr}{T}$ and with morphisms $L_{a,b}$ between $a$ and $b$ cellular isomorphisms between cell posets $\overline{a}$ and $\overline{b}$. So we demand that $L$ contains identity morphisms, and is closed under composition and inversion. This allows one to label cells and to rule out certain cellular isomorphisms as non-allowed symmetries. For example, if ${\mathscr}{T}$ is pure of dimension $d$ and is oriented then one could allow for only orientation preserving morphisms between $d$–cells. We then define ${\mathfrak}{T}^L$ over ${\mathscr}{T}$ and radius poset $(\mathbb{N}_0,\leq)$ by setting $\Phi \in ({\mathfrak}{T}^L)_{a,b}^n$ if and only if $\Phi$ is a cellular isomorphism between the star complexes of $a$ and $b$ induced by a cellular isomorphism between $P(a,n)$ and $P(b,n)$, sending $a$ to $b$, whose restrictions to each cell of $P(a,n)$ are elements of the allowed morphisms of $L$.
### Barycentric Subdivisions
Recall from Definition \[def: order complex\] that one may associate to a CW poset ${\mathscr}{F}$ its barycentric subdivision ${\mathscr}{F}_\Delta$. Given a system of internal symmetries ${\mathfrak}{T}$ over a CW poset ${\mathscr}{T}$ and radius poset $\Lambda$, we may define another, its *barycentric subdivision* ${\mathfrak}{T}_\Delta$, defined over the CW poset ${\mathscr}{T}_\Delta$ and with radius poset $\Lambda$. To do so, firstly recall that given $\Phi \in {\mathfrak}{T}_{a,b}^\lambda$ we may define a cellular isomorphism $\Phi_\Delta$ between the barycentric subdivisions of the star complexes of $a$ and $b$ (see §\[subsec: CW Posets\]). Given a simplex $s = \{a_0,\ldots,a_n\}$ of ${\mathscr}{F}_\Delta$ we have that $$\overline{\text{St}(s)} = \{ s'' \in {\mathscr}{T}_\Delta \mid \exists s' : s \subseteq s' \supseteq s''\} \subseteq (\overline{\text{St}(a_n)})_\Delta.$$ So we may define ${\mathfrak}{T}_\Delta$ by setting $\Phi \in ({\mathfrak}{T}_\Delta)_{s,t}^\lambda$ if and only if there exists some $\widetilde{\Phi} \in {\mathfrak}{T}^\lambda$ between the star complexes of the top entries of $s$ and $t$ for which $\widetilde{\Phi}_\Delta$ restricts to $\Phi$.
Let ${\mathfrak}{T}$ be an SIS. Then ${\mathfrak}{T}_\Delta$ is also an SIS.
The first three groupoid axioms for an SIS follow easily from the fact that $\text{Id}_{\overline{\text{St}(c)}_\Delta}= (\text{Id}_{\overline{\text{St}(c)}})_\Delta$, $(\Phi_2)_\Delta \circ (\Phi_1)_\Delta = (\Phi_2 \circ \Phi_1)_\Delta$ and $(\Phi_\Delta)^{-1} = (\Phi^{-1})_\Delta$. Similarly, the fourth axiom (Inc) follows trivially.
For the fifth axiom (Res) of restriction, let $\lambda \in \Lambda$ and $s \preceq t$ be simplexes of ${\mathscr}{T}_\Delta$ whose vertices of maximal dimension are $a,b \in {\mathscr}{T}$, respectively. Set $\lambda_{\text{res}}$ as in axiom (Res) for ${\mathfrak}{T}$. Suppose that $\Phi \in ({\mathfrak}{T}_\Delta)_{s,-}^{\lambda_{\text{res}}}$. Then $\Phi$ is a restriction to the star complex of $s$ of $\widetilde{\Phi}_\Delta$ for some $\widetilde{\Phi} \in {\mathfrak}{T}_{a,-}^{\lambda_{\text{res}}}$. So, by (Res), the restriction $\Phi'$ of $\widetilde{\Phi}$ to the star complex of $b$ is an element of ${\mathfrak}{T}_{b,-}^\lambda$. The restriction of $\Phi'_\Delta$ to the star complex of $t$ is an element of $({\mathfrak}{T}_\Delta)_{t,-}^\lambda$ which is a restriction of $\Phi$, establishing axiom (Res) for ${\mathfrak}{T}_\Delta$. Axiom (CoRes) is proved analogously.
PE (Co)homology and PE Poincaré Duality {#sec: PE (Co)homology}
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PE (Co)homology
---------------
Given a system of internal symmetries, one may consider the Borel–Moore chains, or cochains, which assign oriented coefficients to cells in a way which depends only on the equivalence classes of the cells to some sufficiently large radius:
Let ${\mathfrak}{T}$ be an SIS over the CW poset ${\mathscr}{T}$ and radius poset $\Lambda$. We shall say that a Borel–Moore chain $\sigma \in C_n({\mathscr}{T})$ is *pattern-equivariant* (*PE*) *to radius $\lambda \in \Lambda$* if for any $\Phi \in {\mathfrak}{T}^\lambda_{a,b}$ between $n$–cells $a$ and $b$ we have that $\sigma(\omega_a) = \sigma(\Phi_*(\omega_a))$, where $\omega_a$ is some orientation on $a$. Similarly, we say that a cochain $\psi \in C^n({\mathscr}{T})$ is *pattern-equivariant to radius $\lambda$* if, for any $\Phi \in {\mathfrak}{T}_{a,b}^\lambda$ we have that $\psi(\omega_a)=\psi(\Phi_*(\omega_a))$.
The (co)boundary of a PE (co)chain is PE.
Let $\sigma$ be a Borel–Moore $n$–chain which is PE to radius $\lambda$. We must check that there exists some $\lambda'$ for which, for any two $(n-1)$–cells $a$, $b$ and $\Phi \in {\mathfrak}{T}_{a,b}^{\lambda'}$, we have that $\partial(\sigma)(\omega_a) = \partial(\sigma)(\Phi_*(\omega_a))$. In fact, we claim that $\lambda' = \lambda_{\text{res}}$ as in axiom (Res) of Definition \[def: SIS\] for ${\mathfrak}{T}$ will do.
Indeed, given $\Phi \in {\mathfrak}{T}_{a,b}^{\lambda_{\text{res}}}$, the restriction of $\Phi$ to each star complex of $n$–cell containing $a$ is an element of ${\mathfrak}{T}^\lambda$. It follows, by the fact that $\sigma$ is PE to radius $\lambda$, that the cellular isomorphism $\Phi$ not only maps the star complex at $a$ to the star complex at $b$, but it also maps $\sigma \restriction \overline{\text{St}(a)}$ to $\sigma \restriction \overline{\text{St}(b)}$. Since $\partial(\sigma)(\omega_a)$ only depends on the restriction of $\sigma$ to $\text{St}(a)$ (and similarly for $b$), we see that $\partial(\sigma)(\omega_a) = \partial(\sigma)(\Phi_*(\omega_a))$, so $\partial(\sigma)$ is PE to radius $\lambda_{\text{res}}$. The proof that the coboundary of a PE cochain is PE is analogous; one implements axiom (CoRes) instead.
It follows from the above that the chain complex $C_\bullet^{\text{BM}}({\mathscr}{T})$ restricts to a sub-chain complex $C_\bullet({\mathfrak}{T})$ of PE chains. So we may define the *pattern-equivariant homology* of an SIS ${\mathfrak}{T}$ to be $H_\bullet({\mathfrak}{T}) \coloneqq H(C_\bullet({\mathfrak}{T}))$. We similarly have the cochain complex $C^\bullet({\mathfrak}{T})$ of PE cochains and associated *pattern-equivariant cohomology* $H^\bullet({\mathfrak}{T}) \coloneqq H(C^\bullet({\mathfrak}{T}))$.
Consider the simple example of the periodic tiling $T$ of $\mathbb{R}^2$ by unit squares, each with corners lying on the integer lattice $\mathbb{Z}^2$. This defines a CW poset ${\mathscr}{T}$ and the SIS ${\mathfrak}{T}^1$ based upon comparison of patches via translation (see §\[subsubsec: Euclidean Tilings\]). There are naturally four types of cells of ${\mathscr}{T}$: vertices, horizontal edges, vertical edges and faces. The groupoid $({\mathfrak}{T}^1)^r$ has the same description for each radius $r \in \mathbb{R}_{>0}$: we have precisely one morphism (induced by translation) in each $({\mathfrak}{T}^1)_{a,b}^\lambda$ whenever $a,b$ are of the same type, and $({\mathfrak}{T}^1)_{a,b}^\lambda = \emptyset$ otherwise. An $n$–(co)chain being PE simply means that all $n$–cells of the same type have the same (oriented) coefficient. So $C_i({\mathfrak}{T}) \cong \mathbb{Z}$, $\mathbb{Z}^2$ and $\mathbb{Z}$ for $i=0,1,2$, respectively (and similarly for PE cohomology). The (co)boundary maps are easily calculated to be trivial, and so the PE (co)homology groups have the same description. The PE cohomology is isomorphic to the cohomology of the tiling space of $T$ (see §\[subsec: Tiling Spaces\]), which is the $2$–torus $\mathbb{T}^2$, and the PE homology is Poincaré dual to the PE cohomology; see Theorem \[thm: PEPD\].
\[def: tail-equivalence\] Let ${\mathfrak}{T}_1$ and ${\mathfrak}{T}_2$ be two SIS’s, defined over the same CW poset and radius posets $\Lambda_1$ and $\Lambda_2$, respectively. If for all $\lambda_2 \in \Lambda_2$ there exists some $\lambda_1 \in \Lambda_1$ for which ${\mathfrak}{T}_1^{\lambda_1} \subseteq {\mathfrak}{T}_2^{\lambda_2}$, and vice versa, we say that ${\mathfrak}{T}_1$ and ${\mathfrak}{T}_2$ are *tail-equivalent*.
Clearly tail-equivalence defines an equivalence relation on SIS’s defined over a fixed CW poset ${\mathscr}{T}$ and, for ${\mathfrak}{T}_1$ and ${\mathfrak}{T}_2$ tail-equivalent, a (co)-chain is PE in ${\mathfrak}{T}_1$ if and only if it is PE in ${\mathfrak}{T}_2$.
Let $T$ be a cellular tiling of $\mathbb{R}^d$ whose tiles have bounded in and out-radii. Then whether one defines ${\mathfrak}{T}^1$ or ${\mathfrak}{T}^0$ in terms of patches of radius $r \in \mathbb{R}_{>0}$ or in terms of $n$–collared patches for $n \in \mathbb{N}_{>0}$ does not matter up to tail-equivalence.
The groups ${\mathfrak}{T}^\lambda_{a,a}$ of ${\mathfrak}{T}^\lambda$ shall be called the *local isotropy groups*. If ${\mathfrak}{T}^\lambda_{a,a} = \{\text{Id}_{\overline{\text{St}(a)}} \}$ for all $a \in {\mathscr}{T}$ and sufficiently large $\lambda$, then we shall say that ${\mathfrak}{T}$ has *trivial local isotropy*.
Denote by $\widetilde{{\mathfrak}{T}}^\lambda$ the collection of restrictions of ${\mathfrak}{T}^\lambda$ to source cells, so that $\widetilde{\Phi} \in \widetilde{{\mathfrak}{T}}^\lambda$ if and only if $\widetilde{\Phi} = \Phi \restriction \overline{a}$ for some $\Phi \in {\mathfrak}{T}^\lambda_{a,-}$ (we shall always implicitly ‘corestrict’ cellular isomorphisms to the appropriate range). We write $\widetilde{\Phi} \in \widetilde{{\mathfrak}{T}}^\lambda_{a,b}$ to specify that $\widetilde{\Phi}$ is the restriction of some $\Phi \in {\mathfrak}{T}^\lambda_{a,b}$ between $\overline{a}$ and $\overline{b}$. Then each $\widetilde{{\mathfrak}{T}}^\lambda$ is a groupoid. We say that ${\mathfrak}{T}$ has *trivial cell isotropy* if $\widetilde{{\mathfrak}{T}}^\lambda_{a,a} = \{\text{Id}_{\overline{a}}\}$ for all $a \in {\mathscr}{T}$ and sufficiently large $\lambda$. Note that the isotropy groups of cells $\widetilde{{\mathfrak}{T}}^\lambda_{a,a}$ are quotient groups of the local isotropy groups ${\mathfrak}{T}^\lambda_{a,a}$.
A commutative ring $G$ has *division by $n$* for $n \in \mathbb{N}$ if $n \cdot 1$ is invertible in $G$.
\[prop: PE BCS\] Let ${\mathfrak}{T}$ be an SIS, and fix a coefficient ring $G$. Suppose that, for sufficiently large $\lambda \in \Lambda$, we have that $G$ has division by the order of the cell isotropy group $\#(\widetilde{{\mathfrak}{T}}^\lambda_{a,a})$ for all cells $a \in {\mathscr}{T}$. Then we have quasi-isomorphisms: $$\begin{gathered}
\iota_\bullet \co C_\bullet({\mathfrak}{T};G) \to C_\bullet({\mathfrak}{T}_\Delta;G)\\
\iota^\bullet \co C^\bullet({\mathfrak}{T}_\Delta;G) \to C^\bullet({\mathfrak}{T};G)\end{gathered}$$
Mimicking the proof Proposition \[prop: BCS\], upon restricting to PE chains we need to show that
1. $H_n({\mathfrak}{T}^k_\Delta,{\mathfrak}{T}^{k-1}_\Delta) = 0$ for $n \neq k$
2. We have a canonical isomorphism of chain complexes $C_\bullet({\mathfrak}{T}) \cong H_\bullet({\mathfrak}{T}^\bullet_\Delta,{\mathfrak}{T}^{\bullet-1}_\Delta)$
(the notation of which shall be explained below). For the proof of the quasi-isomorphism in homology, we shall use a simple ‘averaging trick’, implementing the divisibility of $G$, to show $1$. One may obtain part $2$ in homology ‘for free’. In the cohomology case, the situation is reversed, and one essentially needs the divisibility of $G$ for part $2$.
Let $C_n({\mathfrak}{T}^k_\Delta,{\mathfrak}{T}^{k-1}_\Delta)$ be the group of equivalence classes of PE chains of ${\mathfrak}{T}_\Delta$ supported on ${\mathscr}{T}^k_\Delta$, where we identify two such chains if they agree away from${\mathscr}{T}_\Delta^{k-1}$. The usual boundary maps induce boundary maps between the relative groups (for fixed $k$), so one may define the homology groups $H_n({\mathfrak}{T}^k_\Delta,{\mathfrak}{T}^{k-1}_\Delta)$; we wish to show that these groups are trivial for $n \neq k$.
So let $n < k$ and $\sigma \in C_n({\mathfrak}{T}^k_\Delta)$ be a PE simplicial $n$–chain supported on ${\mathscr}{T}_\Delta^k$ for which $\partial(\sigma)$ is supported on the ${\mathscr}{T}_\Delta^{k-1}$. The chain $\sigma$ is PE to some radius $\lambda$, which we may assume to be sufficiently large relative to the condition of the proposition. We know that there exists some Borel–Moore $(n+1)$–chain $\tau$ for which $\sigma + \partial(\tau)$ is supported on the $(k-1)$–skeleton, we just need to show that $\tau$ may be chosen to be PE.
Say that $k$–cells $a,b \in {\mathscr}{T}$ are equivalent to radius $\lambda$ if ${\mathfrak}{T}^\lambda_{a,b} \neq \emptyset$, which defines equivalence relations on the $k$–cells by the groupoid axioms for ${\mathfrak}{T}$. Choose an equivalence class $[a]$ of cells and representative $a \in [a]$ for radius $\lambda$. The restriction of $\sigma$ to $a^\circ_\Delta$ is homologous to a chain supported on $\partial a_\Delta$ via some $(n+1)$–chain $\tau_a$ supported on $a_\Delta^\circ$. That is, we have a chain $\tau_a$ supported on $a^\circ_\Delta$ for which $$(\sigma \restriction a_\Delta^\circ) + \partial(\tau_a) = \sigma_{\partial(a)},$$ for some chain $\sigma_{\partial(a)}$ supported on $\partial a_\Delta \subseteq {\mathscr}{T}_\Delta^{k-1}$. Define the PE $(n+1)$–chain $\tau_{[a]}$ by setting $$\tau_{[a]} \coloneqq \sum_{\Phi \in \widetilde{{\mathfrak}{T}}_{a,-}^\lambda} \#(\widetilde{{\mathfrak}{T}}^\lambda_{a,a})^{-1} \cdot \Phi_*(\tau_a),$$ where the sum is taken over all morphisms of $\widetilde{{\mathfrak}{T}}^\lambda$ with source $a$. To show that the above is PE, note firstly that $\tau_{[a]}$ is supported on the cells which are equivalent to $a$ to radius $\lambda$. So it suffices to show that the morphisms $\Phi \in {\mathfrak}{T}_{b,c}^\lambda$, where $[a]=[b]=[c]$, sends $\tau_{[a]}$ restricted to $b$ to $\tau_{[a]}$ restricted to $c$. Indeed, we have that: $$\begin{gathered}
\Phi_*(\tau_{[a]}\restriction b^\circ_\Delta) = \Phi_*(\bigg(\sum_{\Phi' \in \widetilde{{\mathfrak}{T}}_{a,-}^\lambda} \#(\widetilde{{\mathfrak}{T}}^\lambda_{a,a})^{-1} \cdot \Phi'_*(\tau_a)\bigg)\restriction b^\circ_\Delta) =\\
\sum_{\Phi' \in \widetilde{{\mathfrak}{T}}_{a,b}^\lambda} \#(\widetilde{{\mathfrak}{T}}^\lambda_{a,a})^{-1} \Phi_* \circ \Phi'_*(\tau_a) = \sum_{\Phi' \in \widetilde{{\mathfrak}{T}}_{a,c}^\lambda} \#(\widetilde{{\mathfrak}{T}}^\lambda_{a,a})^{-1} \Phi'_*(\tau_a) = \tau_{[a]} \restriction c^\circ_\Delta\end{gathered}$$ It follows that $\tau_{[a]}$ is PE in ${\mathfrak}{T}_\Delta$. Since $\sigma$ is PE to radius $\lambda$, we have that $\Phi_*(\sigma \restriction a^\circ_\Delta) = \sigma \restriction b^\circ_\Delta$ for all $k$–cells $b$ and $\Phi \in {\mathfrak}{T}_{a,b}^\lambda$. Hence, for any cell $b$ with $[a]=[b]$, we have that: $$\begin{gathered}
(\sigma \restriction b^\circ_\Delta) + \partial(\tau_{[a]} \restriction b^\circ_\Delta) =
(\sigma \restriction b^\circ_\Delta) +
\sum_{\Phi \in \widetilde{{\mathfrak}{T}}_{a,b}^\lambda} \#(\widetilde{{\mathfrak}{T}}^\lambda_{a,a})^{-1} \Phi_*(\partial(\tau_a)) = \\
(\sigma \restriction b^\circ_\Delta) + \sum_{\Phi \in \widetilde{{\mathfrak}{T}}_{a,b}^\lambda} \#(\widetilde{{\mathfrak}{T}}^\lambda_{a,a})^{-1}((-\sigma \restriction b^\circ_\Delta) + \Phi_*(\sigma_{\partial(a)})) =
\sum_{\Phi \in \widetilde{{\mathfrak}{T}}_{a,b}^\lambda} \#(\widetilde{{\mathfrak}{T}}^\lambda_{a,a})^{-1} \cdot \Phi_*(\sigma_{\partial(a)})\end{gathered}$$ This final term is a chain supported on the boundary of $b$.
Define the chain $\tau$ as the sum of chains $\tau_{[a]}$ taken over all equivalence classes of cells. Then it follows from the above that $\tau$ is PE and that $\sigma + \partial(\tau)$ is a chain supported on ${\mathscr}{T}_\Delta^{k-1}$, as desired.
The chain isomorphism $C_\bullet({\mathfrak}{T}) \cong H_\bullet({\mathfrak}{T}_\Delta^\bullet,{\mathfrak}{T}_\Delta^{\bullet-1})$ of part $2$ above is defined by the chain map $(-)_\Delta$ which canonically associates to a cellular chain its corresponding relative simplicial cycle (see the proof of Proposition \[prop: BCS\]). It is easily checked that $(-)_\Delta$ restricts to an isomorphism between the PE chain complexes.
We shall omit the proof in the cohomology case, which is similar. We note, however, that for part $2$ one needs to implement the divisibility of $G$. Indeed, in this case, an $n$–cochain $\psi_\Delta$ is defined for an $n$–cochain $\psi$ with $\psi(\omega_c)=g$ by making a choice of simplicial $n$–cochain for each closed $n$–cell $c_\Delta$ for which the oriented coefficients *sum* to $g$. For non-divisible coefficient ring $G$, it may not be possible to define such a cochain to be PE, even if $\psi$ is, if the cells have non-trivial isotropy in ${\mathfrak}{T}$.
For an SIS ${\mathfrak}{T}$ we have that ${\mathfrak}{T}_\Delta$ has trivial cell isotropy (indeed, any element of $(\widetilde{{\mathfrak}{T}}_\Delta)_{s,s}^\lambda$ must preserve the canonical total ordering of the vertices of the simplex $s$). So by the above proposition, the PE (co)homology is stable under barycentric subdivision after passing to the first barycentric subdivision. We view the PE (co)homology of ${\mathfrak}{T}$ in the case of having non-trivial cell isotropy as being potentially ‘incorrect’. When ${\mathfrak}{T}$ has trivial cell isotropy then the PE cohomology of ${\mathfrak}{T}$ will correspond to the Čech cohomology of an associated tiling space (see the following subsection). The PE homology will only necessarily correspond to a singular version in general when ${\mathfrak}{T}$ has trivial cell isotropy [@Wal14].
Tiling Spaces {#subsec: Tiling Spaces}
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For an FLC tiling $T$ of $\mathbb{R}^d$, there is an associated topological space $\Omega^1$, called the *translational hull* or *tiling space* of $T$ [@Sad08]. It may be defined by taking the collection $T+\mathbb{R}^d$ of translates of the tiling and then taking the completion of this collection with respect to the tiling metric, which deems two tilings to be ‘close’ if they are identical to a ‘large’ radius about the origin, up to a ‘small’ perturbation. Taking the completion is the same as considering all tilings ‘locally indistinguishable’ from $T$, tilings for which any finite sub-patch appears as a translate of a sub-patch of $T$.
As briefly justified in §\[subsubsec: Euclidean Tilings\], for the purpose of studying a tiling $T$ through its associated tiling space $\Omega^1$, it does no harm to restrict to the case that $T$ is cellular. Then the tiling space $\Omega^1$ is homeomorphic to an inverse limit of CW complexes, called *approximants*, which may be constructed by glueing together “collared tiles” (in the Gähler construction, see [@Sad08 Chapter 2, §4]) or by “collaring points” (in the Barge–Diamond–Hunton–Sadun construction, see [@BDHS10 §3]). In direct analogy to the former approach, for any SIS there is an associated inverse limit of CW complexes. Suppose that ${\mathfrak}{T}$ is an SIS, and that $\lambda_{\widehat{\text{res}}}$ may be set to be $\lambda$ in axiom (Res) of Definition \[def: SIS\]. Note that we may always choose a tail-equivalent SIS for which this is the case. In addition, suppose that the isotropy of cells is trivial, which will at least always be the case after passing to the barycentric subdivision. Recall from the previous subsection that $\widetilde{{\mathfrak}{T}}^\lambda$ denotes the collection of morphisms of ${\mathfrak}{T}^\lambda$ restricted to source cells. Then each $\widetilde{{\mathfrak}{T}}^\lambda$ satisfies:
1. For each $a \in {\mathscr}{T}$, the identity morphism $\text{Id}_{\overline{a}} \in \widetilde{{\mathfrak}{T}}^\lambda$.
2. For each $\Phi \in \widetilde{{\mathfrak}{T}}^\lambda$, we have that $\Phi^{-1} \in \widetilde{{\mathfrak}{T}}^\lambda$.
3. For $\Phi_1 \in \widetilde{{\mathfrak}{T}}_{a,b}^\lambda$ and $\Phi_2 \in \widetilde{{\mathfrak}{T}}_{b,c}^\lambda$, we have that $\Phi_2 \circ \Phi_1 \in \widetilde{{\mathfrak}{T}}^\lambda_{a,c}$.
4. If $\Phi \in \widetilde{{\mathfrak}{T}}_{c,c}^\lambda$, then $\Phi$ is the identity morphism.
5. If $\Phi \in \widetilde{{\mathfrak}{T}}_{b,-}^\lambda$ and $b$ is a coface of $a$, then the restriction of $\Phi$ to $a$ is an element of $\widetilde{{\mathfrak}{T}}^\lambda$.
By passing to a geometric realisation of the above system of morphisms, the above axioms precisely specify a *family of identifications* of the cells ${\mathscr}{T}$, see [@CooFin67 Chapter III]. One may consider a quotient of $|{\mathscr}{T}|$ by identifying the cells via the morphisms of ${\mathfrak}{T}^\lambda$. The quotient inherits a CW decomposition, although not necessarily a regular one (without further barycentric subdivision).
The quotient spaces $K_\lambda = |{\mathscr}{T}| / {\mathfrak}{T}^\lambda$ shall be called *approximants*. Denote the quotient map from $|{\mathscr}{T}|$ to such an approximant by $\pi_\lambda$. For $\lambda \leq \mu$, since ${\mathfrak}{T}^\lambda \supseteq {\mathfrak}{T}^\mu$, we have cellular quotient maps $\pi_{\lambda,\mu} \co K_\mu \to K_\lambda$ given by making any extra identifications of $\widetilde{{\mathfrak}{T}}^\lambda$ that may not have been identifications for $\widetilde{{\mathfrak}{T}}^\mu$. Then ${\mathfrak}{T}$ defines an inverse system $(K_\lambda, \pi_{\lambda,\mu})$ indexed over the directed set $\Lambda$, whose inverse limit will be denoted by $\Omega({\mathfrak}{T})$ and is called the *tiling space of ${\mathfrak}{T}$*.
Let ${\mathfrak}{T}$ be an SIS satisfying conditions $4$ and $5$ above. If, in addition, there are only a finite number of equivalence classes of cells of $\tilde{{\mathfrak}{T}}^\lambda$ for each $\lambda \in \Lambda$, then there exists an isomorphism $\check{H}^\bullet(\Omega({\mathfrak}{T})) \cong H^\bullet({\mathfrak}{T})$.
The proof is essentially the same as that of [@Sad07 Theorem 4]. Under the above hypotheses, $\Omega({\mathfrak}{T})$ is an inverse limit over $\Lambda$ of compact Hausdorff spaces, and so is itself compact (and Hausdorff). From the compactness of $\Omega({\mathfrak}{T})$, it is easy to show that $\check{H}^\bullet(\Omega({\mathfrak}{T})) \cong \varinjlim(\check{H}^\bullet(K_\lambda,\pi_{\lambda,\mu}^*)$; we note that compactness cannot be dropped here, there are inverse limits of non-compact CW complexes for which this does not hold. Čech cohomology is naturally isomorphic to singular cohomology on the subcategory of topological spaces which are homotopy equivalent to CW complexes. So we have that $$\check{H}^\bullet(\Omega({\mathfrak}{T})) = \check{H}^\bullet(\varprojlim(K_\lambda,\pi_{\lambda,\mu})) \cong \varinjlim(\check{H}^\bullet(K_\lambda),\pi^*_{\lambda,\mu}) \cong \varinjlim(H^\bullet(K_\lambda),\pi^*_{\lambda,\mu}),$$ where $H^\bullet(K_\lambda)$ is the cellular cohomology of $K_\lambda$. Taking cohomology commutes with taking the direct limit, and so the above is isomorphic to the cohomology of the direct limit cochain complex $\varinjlim(C^\bullet(K_\lambda),\pi^*_{\lambda,\mu})$. A cellular cochain $\psi \in C^n({\mathscr}{T})$ is PE if and only if it is the pullback $\pi^*_\lambda(\widetilde{\psi})$ of a cellular cochain $\widetilde{\psi} \in C^\bullet(K_\lambda)$ for some $\lambda \in \Lambda$. We see that the maps $\pi_\lambda^*$ induce a cochain isomorphism from the cochain complex $\varinjlim(C^\bullet(K_\lambda),\pi_{\lambda,\mu}^*)$ to $C^\bullet({\mathfrak}{T})$ and so $\check{H}^\bullet(\Omega({\mathfrak}{T})) \cong H^\bullet({\mathfrak}{T})$.
Let $T$ be an FLC cellular tiling of $\mathbb{R}^d$ and consider the SIS ${\mathfrak}{T}^1$ based upon comparison of patches via translations (see §\[subsubsec: Euclidean Tilings\]). Finite local complexity (FLC) here means that there are only finitely many patches of any given radius $r$ up to translational equivalence. The tiling space $\Omega({\mathfrak}{T}^1)$ is an inverse limit of approximants constructed by identifying cells when they agree to a certain radius in the tiling up to translation. This is precisely the translational hull of $T$, which in [@BDHS10] is denoted by $\Omega^1$. So we have that $\check{H}^\bullet(\Omega^1) \cong H^\bullet({\mathfrak}{T}^1)$, as shown in [@Sad07].
Similarly, one may define the SIS ${\mathfrak}{T}^0$ based upon comparison of patches via rigid motions. Suppose that $T$ is *rotationally FLC*, which means that for each $r>0$ there are only a finite number of patches of size $r$ up to rigid motion (Radin’s pinwheel tilings are the standard examples of tilings which are rotationally FLC, but are not FLC with respect to translations). The space $\Omega({\mathfrak}{T}^0)$ is an inverse limit of approximants built by identifying cells whenever they agree to a certain radius in the tiling up to rigid motion. We require here that the cells have trivial isotropy in ${\mathfrak}{T}^0$, which is always the case after passing to the barycentric subdivision. The space $\Omega({\mathfrak}{T}^0)$ corresponds to the space $\Omega^0 \cong \Omega^{\text{rot}}/\text{SO}(d)$ of [@BDHS10], where $\Omega^{\text{rot}}$ is the full Euclidean hull of $T$, which is acted upon by $\text{SO}(d)$ by rotations. So again, the Čech cohomology of this space may be visualised using PE cochains: $\check{H}^\bullet(\Omega^0) \cong H^\bullet({\mathfrak}{T}^0)$, c.f., [@Ran06; @Sad08].
\[ex: dyadic\] Let ${\mathscr}{S}^d$ be the CW poset corresponding to the periodic tiling of $\mathbb{R}^d$ by unit hypercubes with corners lying on the integer lattice $\mathbb{Z}^d$. We define an SIS ${\mathfrak}{D}_2^d$ over ${\mathscr}{S}^d$ and radius poset $(\mathbb{N}_0,\leq)$ as follows. Let $\Phi \in ({\mathfrak}{D}_2^d)_{a,b}^n$ if and only if $\Phi$ is a cellular isomorphism between the star complexes of $a$ and $b$ induced by translation by a vector of $2^n\mathbb{Z}^d$.
It is easy to see that each approximant $K_n$ is homeomorphic to the $d$–torus $\mathbb{T}^d$ and that the map $\pi_{i,j}$ for $i \leq j$ corresponds to the self-map of $\mathbb{T}^d$ induced by the $\times 2^{j-i}$ map on $\mathbb{R}^d$. So the tiling space $\Omega({\mathfrak}{D}_2^d)$ is the $d$–dimensional dyadic solenoid $\mathbb{D}^d_2 \coloneqq \varprojlim(\mathbb{T}^d,\times 2)$. A cochain $\psi \in C^n({\mathscr}{S}^d)$ is PE if and only if there exists some $n \in \mathbb{N}_0$ for which $\psi$ is invariant under the action of translation of $2^n\mathbb{Z}^d$. The PE cohomology is isomorphic to the Čech cohomology of the tiling space $\check{H}^\bullet(\Omega({\mathfrak}{D}_2^d)) \cong H^\bullet(\mathbb{D}_2^d)$. This example may be thought of as a hierarchical tiling, as described in §\[solenoid\], where the cells ‘know’ of their location in the hierarchy despite this information not being determined geometrically by the underlying tiling. The ‘supertilings’ $T_n$ in the hierarchy are periodic tilings of hypercubes of side-length $2^n$, which are the ‘supertiles’ of the non-recognisable substitution of a unit hypercube into $2^d$ copies of half the side-length. Alternatively, it may be realised as a tiling whose tiles are labelled by elements of a compact metric space, see [@PrFSad14ILC Example 3].
PE Poincaré Duality
-------------------
As we have seen, the PE cohomology of an SIS ${\mathfrak}{T}$ corresponds to the Čech cohomology of an associated inverse limit space $\Omega({\mathfrak}{T})$, which in the main cases of interest (for example, for the translational hull of an FLC tiling) corresponds to a naturally defined moduli space of tilings. We now wish to show that the PE homology, in certain cases, is related to the PE cohomology through Poincaré duality.
Let ${\mathfrak}{T}$ be an SIS over a pure CW poset ${\mathscr}{T}$. We define the chain complex $C_\bullet(\widehat{{\mathfrak}{T}};G)$ to be the sub-chain complex of $C_\bullet({\mathfrak}{T}_\Delta;G)$ which in degree $n$ consists of PE simplicial $n$–chains supported on the dual $n$–skeleton $\widehat{{\mathscr}{T}}_\Delta^n$ and with boundaries supported on the dual $(n-1)$–skeleton $\widehat{{\mathscr}{T}}_\Delta^{n-1}$.
\[thm: PEPD\] Let ${\mathfrak}{T}$ be an SIS over a CW poset ${\mathscr}{T}$ for a homology $d$–manifold $|{\mathscr}{T}|$. Set a coefficient ring $G$ and assume that there exists a PE fundamental class $\Gamma \in C_d({\mathfrak}{T};G)$ (that is, $\Gamma$ is a Borel–Moore fundamental class of $C_d^{\text{BM}}({\mathscr}{T};G)$ and the morphisms of ${\mathfrak}{T}^\lambda$ between $d$–cells are orientation preserving for sufficiently large $\lambda$). Then the cap product with the fundamental class induces a cochain isomorphism $$- \cap \Gamma \co C^\bullet({\mathfrak}{T};G) \to C_{d-\bullet}(\widehat{{\mathfrak}{T}};G).$$ Furthermore, if for sufficiently large $\lambda$ we have that $G$ has division by the order of local isotropy $\#({\mathfrak}{T}_{a,a}^\lambda)$ for all cells $a \in {\mathscr}{T}$, then there exists a quasi-isomorphism $$\iota_\bullet \co C_\bullet(\widehat{{\mathfrak}{T}};G) \to C_\bullet({\mathfrak}{T}_\Delta;G).$$ Hence, in this case, we have PE Poincaré duality $H^\bullet({\mathfrak}{T};G) \cong H_{d-\bullet}({\mathfrak}{T};G)$.
Each $\Phi \in {\mathfrak}{T}_{a,b}^\lambda$ induces a cellular isomorphism $\widehat{\Phi}$ between the dual cells $\widehat{a}_\Delta$ and $\widehat{b}_\Delta$ by setting $$\widehat{\Phi}(\{a_0,\ldots,a_n\}) \coloneqq \{\Phi(a_0),\ldots,\Phi(a_n)\}$$ for a simplex $\{a_0,\ldots,a_n\} \in \widehat{a}_\Delta$. It is simple to check that the above is well-defined and that:
1. $\text{Id}_{\widehat{a}_\Delta} = \widehat{\text{Id}_{\overline{\text{St}(a)}}}$
2. $\widehat{\Phi_2} \circ \widehat{\Phi_1} = \widehat{\Phi_2 \circ \Phi_1}$
3. $(\widehat{\Phi})^{-1} = \widehat{\Phi^{-1}}$
It follows from the above and the groupoid axioms of ${\mathfrak}{T}$ that we may define the groupoids $\widehat{{\mathfrak}{T}}^\lambda$ which have as objects the dual cells of ${\mathscr}{T}$ and as morphisms the duals $\widehat{\Phi}$ of the morphisms of ${\mathfrak}{T}^\lambda$, analogously to the groupoids $\widetilde{{\mathfrak}{T}}^\lambda$.
The relative pair $(\widehat{a}_\Delta,\partial \widehat{a}_\Delta)$ of the dual of an $n$–cell modulo the dual boundary has an orientation, a simplicial $(d-n)$–chain with coefficients $\pm 1$ supported on $\widehat{a}_\Delta$ and boundary supported on $\partial \widehat{a}_\Delta$. A chain $\sigma \in C_n^{\text{BM}}(\widehat{{\mathscr}{T}})$ is an element of $C_n(\widehat{{\mathfrak}{T}})$ if and only if there exists some $\lambda$ for which, for any $\widehat{\Phi} \in \widehat{{\mathfrak}{T}}^\lambda_{\widehat{a}_\Delta,\widehat{b}_\Delta}$, we have that $\widehat{\Phi}_*(\sigma \restriction \widehat{a}_\Delta) = \sigma \restriction \widehat{b}_\Delta$.
Recall that to explicitly define $\psi \cap \Gamma$ for $\psi \in C^n({\mathscr}{T})$, one firstly defines the simplicial cochain $\psi_\Delta \in C^n({\mathscr}{T}_\Delta^n,{\mathscr}{T}_\Delta^{n-1})$ and then take its cap product (with respect to the canonical partial order of the vertices of ${\mathscr}{T}_\Delta$) with the simplicial cycle $\Gamma_\Delta \in C_d({\mathscr}{T}_\Delta)$. Then we have that $\psi \in C^n({\mathscr}{T})$ is PE if and only if $\psi \cap \Gamma$ is PE. Indeed, suppose that all of the elements of ${\mathfrak}{T}^{\lambda'}$ preserve the orientations of $d$–cells, which is true for sufficiently large $\lambda'$ by the assumption of the theorem and axiom (Res) of Definition \[def: SIS\]. Then for any $\Phi \in {\mathfrak}{T}_{a,b}^\lambda$ for $n$–cells $a$ and $b$ with $\lambda \geq \lambda'$, we have that $$\Phi_*(\Gamma \restriction \overline{\text{St}(a)}) = \Gamma \restriction \overline{\text{St}(b)}.$$ Since $\Phi$ transports the restriction of the chain $\Gamma$ at the star of $a$ to its restriction at the star of $b$, as well as the canonical partial ordering of the vertices, the result quickly follows from the classical pairing of the cell orientation with the corresponding dual cell orientation induced by the cap product.
The proof of the second quasi-isomorphism is entirely analogous to the proof of Proposition \[prop: PE BCS\], where we use $\widehat{{\mathfrak}{T}}$ in replacement of $\widetilde{{\mathfrak}{T}}$. To use the ‘averaging trick’ one needs that, for sufficiently large $\lambda$, the coefficient ring $G$ has division by the order of the isotropy of dual cells $\#(\widehat{{\mathfrak}{T}}_{\widehat{a}_\Delta,\widehat{a}_\Delta}^\lambda)$. Note that each $\widehat{{\mathfrak}{T}}_{\widehat{a}_\Delta,\widehat{a}_\Delta}^\lambda$ is a quotient group of the local isotropy group ${\mathfrak}{T}_{a,a}^\lambda$. Hence, $G$ eventually has division by the order of the isotropy of dual cells whenever it eventually has division by the order of the local isotropy of ${\mathfrak}{T}$. In this case, similarly, $G$ eventually has division by the order of the isotropy of cells $\#(\widetilde{{\mathfrak}{T}}_{a,a}^\lambda)$ and so by Proposition \[prop: PE BCS\] the PE homology is invariant under barycentric subdivision. In summation, we have the following quasi-isomorphisms $$C^\bullet({\mathfrak}{T}) \to C_{d-\bullet}(\widehat{{\mathfrak}{T}}) \to C_{d-\bullet}({\mathfrak}{T}_\Delta) \leftarrow C_{d-\bullet}({\mathfrak}{T})$$ and so we have PE Poincaré duality $H^\bullet({\mathfrak}{T}) \cong H_{d-\bullet}({\mathfrak}{T})$.
PE Poincaré duality allows one to visualise topological invariants of tiling spaces using geometric chains supported on the original tiling. Since PE Poincaré duality is simply a restriction of usual Poincaré duality, it is also often possible to express the product structure on the cohomology groups as an intersection product in PE homology.
As we shall see in examples to follow, PE Poincaré duality often fails in the case that the SIS has local isotropy. We shall show in §\[subsec: mod\] that one may modify the definition of PE homology so as to regain duality, although we consider the extra torsion elements seen in the PE homology as invariants of potential interest, which we shall utilise in forthcoming work [@Wal15].
Let $T$ be an FLC regular cellular tiling of $\mathbb{R}^d$ and associate to it the SIS ${\mathfrak}{T}^1$ (see §\[subsubsec: Euclidean Tilings\]). Clearly ${\mathfrak}{T}^1$ has trivial local isotropy, and so we have Poincaré duality $H^\bullet({\mathfrak}{T}^1) \cong H_{d-\bullet}({\mathfrak}{T}^1)$. The PE cohomology is isomorphic to the Čech cohomology $\check{H}^\bullet(\Omega({\mathfrak}{T}^1))$ of the translational hull of $T$.
Now consider ${\mathfrak}{T}^0$, which is defined by allowing comparison of patches using rigid motions instead of just translations. Assume that the cells have trivial isotropy in ${\mathfrak}{T}^0$, which will at least always be the case by passing to a barycentric subdivision. In this case we still have that $H^\bullet({\mathfrak}{T}^0)$ is isomorphic to the Čech cohomology of the tiling space $\check{H}^\bullet(\Omega({\mathfrak}{T}^0))$. However, it is not necessarily true that we have Poincaré duality $H^\bullet({\mathfrak}{T}^0) \cong H_{d-\bullet}({\mathfrak}{T}^0)$. If there exists a rotationally invariant tiling of the tiling space, then for all $r > 0$ there will exist some cell $a \in {\mathscr}{T}$ with $\#({\mathfrak}{T}_{a,a}^r) > 1$ so Theorem \[thm: PEPD\] does not apply. One still has Poincaré duality over divisible coefficients, however. Alternatively, one may restore duality by modifying the PE chain complex, see §\[subsec: mod\].
\[ex: triangle\] Let $T$ be a periodic tiling of $\mathbb{R}^2$ of equilateral triangles. For ${\mathfrak}{T}^1$ we have Poincaré duality $\check{H}^\bullet(\Omega({\mathfrak}{T}^1))$ $\cong$ $H^\bullet({\mathfrak}{T}^1)$ $\cong$ $H_{d-\bullet}({\mathfrak}{T}^1)$, since $\#({\mathfrak}{T}_{a,a}^\lambda)=1$ for all $a \in {\mathscr}{T}$. The space $\Omega({\mathfrak}{T}^1)$ is the translational hull of $T$, which is the $2$–torus $\mathbb{T}^2$.
To calculate invariants for ${\mathfrak}{T}^0$, because of cell isotropy we pass to the barycentric subdivision ${\mathfrak}{T}_\Delta^0$. In this case, we do not have Poincaré duality: $H_i({\mathfrak}{T}^0_\Delta) \cong H^{2-i}({\mathfrak}{T}^0_\Delta) \cong 0,\mathbb{Z}$ for $i=1,2$, respectively, but for $i=0$ we have that $H_0({\mathfrak}{T}_\Delta^0) \cong \mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/3\mathbb{Z}$ whereas $H^2({\mathfrak}{T}_\Delta^0) \cong \mathbb{Z}$. We have that $H^\bullet({\mathfrak}{T}_\Delta^0) \cong \check{H}^\bullet(\Omega({\mathfrak}{T}_\Delta^0))$, where $\Omega({\mathfrak}{T}_\Delta^0) \cong S^2$ is the $2$–sphere.
We do not have Poincaré duality between the PE cohomology and PE homology here, since cells have *local* isotropy, rotational symmetries at the $0$–cells. Carrying out similar computations for a periodic square tiling $S$ and its associated SIS ${\mathfrak}{S}_\Delta^0$, one computes that $H_0({\mathfrak}{S}_\Delta^0) \cong \mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/4\mathbb{Z}$. In particular, we see that the PE homology is not a topological invariant of the tiling space $\Omega({\mathfrak}{S}_\Delta^0) \cong \Omega({\mathfrak}{T}_\Delta^0) \cong S^2$. It is not hard to show that, over divisible coefficients $G$, the PE homology $H_\bullet({\mathfrak}{T}^0;G)$ for a tiling $T$ of $\mathbb{R}^d$ corresponds to the subgroup of $H_\bullet({\mathfrak}{T}^1;G)$ of elements represented by ‘rotationally invariant chains’ (c.f., [@BDHS10 Theorem 7]). We see from this example that this is no longer necessarily true over non-divisible coefficients.
The name *pattern-equivariant* homology was chosen so as to be consistent with the common usage of the term in the field of aperiodic tilings. However, in the context of the above example it seems more logical, linguistically, for the pattern-equivariant cohomology groups to refer to the $G$–equivariant cohomology groups, where $G$ is the symmetry group of the tiling acting on $\mathbb{R}^2$ by rigid motions. These groups will correspond to the group cohomology of $G$ (since $\mathbb{R}^2$ is contractible), or the cohomology of $\mathbb{R}^2/G$ as an orbifold. Thus, the term pattern-*invariant* may be neater in this context. In fact, it should not be too difficult to define a cohomology theory associated to our SIS’s ${\mathfrak}{T}$ in general, which is to the PE cohomology (as defined here) as orbifold cohomology is to the coarse cohomology of its underlying space. One would expect the PE homology groups here to be relaying some extra information about such groups through the extra torsion elements that they possess.
We further remark that the term *homology* in PE homology is intended to reflect the geometric construction of these groups, rather than to refer to any functorial properties.
Modified PE Homology Groups {#subsec: mod}
---------------------------
As we have seen, pattern-equivariant Poincaré duality can fail for SIS’s in the presence of non-trivial local isotropy. We shall now describe how one may modify the definition of PE homology so as to regain duality with the PE cohomology.
To simplify discussion, we shall restrict to the case where ${\mathfrak}{T}$ is defined over a $2$–dimensional CW poset ${\mathscr}{T}$. We assume that $|{\mathscr}{T}|$ is an orientable $2$–manifold and that the elements of ${\mathfrak}{T}^\lambda$ preserve orientations of $2$–cells. We shall set our coefficient group to be $\mathbb{Z}$ throughout. Furthermore, we assume that the isotropy of cells is trivial which, with our previous assumptions, is equivalent to ${\mathfrak}{T}_{a,a}^\lambda = \{ \text{Id}_{\overline{\text{St}(a)}} \}$ for sufficiently large $\lambda$ and all $a \in {\mathscr}{T}$ with $\text{dim}(a) > 0$ (so the non-trivial local isotropy is concentrated at the vertices).
We shall write $\sigma \in C_n^\dagger({\mathfrak}{T})$ to mean that $\sigma$ is PE with respect to ${\mathfrak}{T}$ and, if $n = 0$, then in addition we require that there exists some $\lambda$ for which, for each vertex $v \in {\mathscr}{T}$ with orientation $\omega_v$, we have that $\sigma(\omega_v) = \#({\mathfrak}{T}_{v,v}^\lambda) \cdot k$ for some $k \in \mathbb{Z}$. We claim that for $\sigma \in C_n^\dagger({\mathfrak}{T})$ we have that $\partial(\sigma) \in C_{n-1}^\dagger({\mathfrak}{T})$. We know that the boundary of a PE chain is PE, so we need only check that for $\sigma \in C_1({\mathfrak}{T})$ there exists some $\lambda$ for which $\sigma(\omega_v)$ is divisible by $\#({\mathfrak}{T}_{v,v}^\lambda)$ for any vertex $v$. Indeed, suppose that $\sigma$ has PE radius $\lambda$. Then $\sigma \restriction \text{St}(v)$ is invariant under $\#({\mathfrak}{T}_{v,v}^{\lambda_{\text{res}}})$–fold rotation and so $\partial(\sigma)(\omega_v)$ is some multiple of $\#({\mathfrak}{T}_{v,v}^{\lambda_{\text{res}}})$. It follows that $C_\bullet^\dagger({\mathfrak}{T})$ is a subcomplex of $C_\bullet({\mathfrak}{T})$. Define the *modified PE homology groups* $H_\bullet^\dagger({\mathfrak}{T}) \coloneqq H(C_\bullet^\dagger({\mathfrak}{T}))$. Then, with this modification, one has PE Poincaré duality:
Let ${\mathfrak}{T}$ be an SIS over the CW poset ${\mathscr}{T}$ of an orientable homology $d$–manifold $|{\mathscr}{T}|$ which possesses a fundamental class $\Gamma \in C_d^{\text{BM}}({\mathscr}{T})$. Suppose that the elements of ${\mathfrak}{T}_{a,a}^\lambda$ are orientation preserving on the $2$–cells of ${\mathscr}{T}$ and that ${\mathfrak}{T}$ has trivial cell isotropy (but not necessarily trivial local isotropy). Then we have Poincaré duality $H^\bullet({\mathfrak}{T}) \cong H_{d-\bullet}^\dagger({\mathfrak}{T})$.
The proof is a simple modification of the proof for Theorem \[thm: PEPD\]. One needs to show that there exist the following quasi-isomorphisms: $$\begin{gathered}
\iota_\bullet \co C_\bullet^\dagger({\mathfrak}{T}) \to C_\bullet^{\dagger}({\mathfrak}{T}_\Delta)\\
- \cap \Gamma \co C^\bullet({\mathfrak}{T}) \to C_{d-\bullet}^\dagger(\widehat{{\mathfrak}{T}})\\
\widehat{\iota}_\bullet \co C^\dagger_\bullet(\widehat{{\mathfrak}{T}}) \to C_\bullet^\dagger({\mathfrak}{T}_\Delta)\end{gathered}$$ The first two follow with almost no alterations to the proofs of Proposition \[prop: PE BCS\] and Theorem \[thm: PEPD\]. Poincaré duality fails before the modification of the PE chain complex because of the final chain map not necessarily being a quasi-isomorphism. It may not be true that $H_0(\widehat{{\mathfrak}{T}}^2,\widehat{{\mathfrak}{T}}^1) = 0$, since a $2$–dimensional dual cell may have non-trivial isotropy to radius $\lambda$ when the corresponding $0$–cell of ${\mathscr}{T}$ has non-trivial $n$–fold local isotropy. In this case $0$–chains can be ‘trapped’ modulo $n$ at the barycentre of the dual $2$–cell. However, by passing to $C_\bullet^\dagger$ this problem is resolved, since then $0$–chains at such a vertex may only be assigned coefficient some multiple of the order of that isotropy.
The only non-trivial chain group of the relative chain complex of the pair of $C_\bullet({\mathfrak}{T})$ and $C_\bullet^\dagger({\mathfrak}{T})$ is a torsion group in degree zero. For example, for the SIS ${\mathfrak}{T}^0$ associated to a rotationally FLC $2$–dimensional tiling $T$ of $\mathbb{R}^d$, this degree zero group is isomorphic to $\prod_{T_i}\mathbb{Z}/n_i \mathbb{Z}$, where the sum is taken over all rotation classes of tilings $T_i$ of $\Omega({\mathfrak}{T}^0)$ with $n_i$–fold rotational symmetry at the origin (in the highly non-standard case that there exist uncountably many such tilings, one should replace each uncountable product of torsion factors with a countable one). It follows that $$H_i({\mathfrak}{T}) \cong H_i^\dagger({\mathfrak}{T}) \cong H^{d-i}({\mathfrak}{T}) \cong \check{H}^{d-i}(\Omega({\mathfrak}{T}))$$ for $i \neq 0$ and that $H_0({\mathfrak}{T})$ is an extension of $\check{H}^2(\Omega({\mathfrak}{T}))$ over a torsion group determined by the rotational symmetries of vertices of $T$. Similar techniques will work in higher dimensions, but in this case the coefficients assigned to higher dimensional cells will need to be restricted. For example, for $d=3$ one would need to take into account rotational symmetries of patches about $1$–cells. One would expect in the general case for there to exist a Zeeman type spectral sequence [@Zee63] relating the PE homology to the PE cohomology, and vice versa, in terms of local coefficients determined by the rotational symmetries of the tiling.
Consider again the SIS ${\mathfrak}{T}_\Delta^0$ associated to the periodic triangle tiling $T$ of Example \[ex: triangle\]. The chain group $$C_0({\mathfrak}{T}_\Delta^0) \cong \mathbb{Z}v \oplus \mathbb{Z}e \oplus \mathbb{Z}f$$ is generated over $\mathbb{Z}$ by the $3$ distinct equivalence classes of $0$–cells of ${\mathscr}{T}_\Delta$, the barycentres $v$ of vertices, $e$ of edges and $f$ faces of $T$. Since these vertices have rotational symmetry of order $6$, $2$ and $3$, respectively, we have that $$C_0^\dagger({\mathfrak}{T}_\Delta^0) \cong 6\mathbb{Z}v \oplus 2\mathbb{Z}e \oplus 3\mathbb{Z}f.$$ One easily computes the corresponding homology group in degree zero to be $H_0^\dagger({\mathfrak}{T}_\Delta^0) \cong \mathbb{Z}$, which is in agreement with the fact that $$H^2(S^2) \cong \check{H}^2(\Omega({\mathfrak}{T}_\Delta^0)) \cong H^2({\mathfrak}{T}_\Delta^0) \cong H_0^\dagger({\mathfrak}{T}_\Delta^0) \cong \mathbb{Z}.$$
PE Homology of Hierarchical Tilings {#sec: PE Homology of Hierarchical Tilings}
===================================
In this final section we shall define a method of computation for the PE invariants of a tiling equipped with a certain type of hierarchical structure. It shall be applicable to recognisable substitution tilings of Euclidean space, such as those considered in [@AndPut98; @BDHS10] (see §\[Fibonacci\], §\[Thue–Morse\] and §\[Penrose\]) but also to non-Euclidean ‘combinatorial substitutions’, such as Bowers and Stephenson’s ‘regular pentagonal tilings’ [@BowSte97] (see §\[Pentagonal\]). It shall also be applicable to mixed-substitution systems [@GahMal13] (see §\[Arnoux–Rauzy\]). The method is best illustrated through examples, so we recommend the reader to skip the preliminary details of the general setting on a first reading and to head to §\[subsec: Example Computations\], where the method is applied to a broad range of examples.
Hierarchical Systems of Tilings
-------------------------------
The two main approaches to producing interesting examples of aperiodic tilings, such as the Penrose tilings, are through the cut-and-project method (see [@FHK02]) and through tiling substitutions (see [@AndPut98]); we shall focus on the latter. A substitution rule consists of a finite collection of *prototiles* of $\mathbb{R}^d$, a rule for subdividing these prototiles and an expanding dilation which, when applied to the subdivided prototiles, defines patches of translates of the original prototiles. By iterating this procedure and inflating, one produces larger and larger patches of tiles, each of which is a translate of one of the originals. A tiling is said to be *admitted by the substitution rule* if every bounded patch is a sub-patch of a translate of some iteratively substituted prototile.
Under certain conditions on the substitution rule, tilings admitted by it exist and, in addition, for each such tiling $T_0$ there is a *supertiling* $T_1$, based on inflated versions of the prototiles, which subdivides to $T_0$ and is itself a (dilation of an) admitted tiling. So $T_0$ has a hierarchical structure: there is an infinite list of substitution tilings $T_0$, $T_1$, $T_2$, …for which the tiles of $T_0$ may be grouped to form the supertiles of $T_1$, the supertiles of $T_1$ may be grouped into the super$^2$tiles of $T_2$, …and so on.
There are several ways in which this process may be generalised. Firstly, one may allow for general rigid motions instead of just translations in the construction, such as in Radin’s pinwheel tilings where tiles point in infinitely many directions. Secondly, one may consider hierarchical tilings of more general spaces than of just Euclidean space. An interesting set of examples are given by Bowers and Stephenson’s pentagonal tilings [@BowSte97]. Thirdly, the requirement that the substitute of a prototile has precisely the same support is not entirely necessary, see for example the Penrose kite and dart substitution. Finally, instead of using a single substitution rule to define the supertiles, one may choose a finite list of distinct substitutions and an infinite sequence in which to apply them. In the one dimensional case, these symbolic sequences are often known as $S$–adic systems. In general dimensions, the tilings that arise are known as mixed or multi-substitution tilings [@GahMal13]. Passing to the setting of mixed substitutions adds far more generality; for example, every Sturmian word may be defined using a mixed substitution system, whereas the only Sturmian words which may be expressed as purely substitutive tilings are those associated to quadratic irrationals. In contrast to the purely substitutive case, the family of one dimensional mixed substitution tilings exhibit an uncountable number of distinct isomorphism classes of degree one Čech cohomology groups [@Rus15]. In a mixed substitution tiling the tiles group together to the supertiles, those into super$^2$tiles, and so on, just as in the purely substitutive case, but now the rules connecting the various levels of the hierarchy are not constant. A general framework which captures this idea is laid out in [@PrFSad14].
To attempt to systematically deal with such a range of examples, we shall take the list of supertilings $T_0$, $T_1$, $T_2$, …as a description of the hierarchy of $T_0$, instead of some underlying set of substitution rule(s). To begin setting up notation, let ${\mathscr}{T}$ be a CW poset; the cells of the supertilings will be built out of the cells of ${\mathscr}{T}$. Then fix directed sets $\Lambda$ (as a directed set of radii) and ${\mathcal}{I}$ (which shall index the tilings). Suppose that for each $\alpha \in {\mathcal}{I}$ one has an SIS ${\mathfrak}{T}_\alpha$ defined over ${\mathscr}{T}$ and radius poset $\Lambda$ for which, given $\lambda \in \Lambda$ and $\alpha \leq \beta$ in ${\mathcal}{I}$, we have that ${\mathfrak}{T}_\alpha^\lambda \supseteq {\mathfrak}{T}_\beta^\lambda$ (so any radius $\lambda$ internal symmetry of ${\mathfrak}{T}_\beta$ is also a radius $\lambda$ internal symmetry of any ${\mathfrak}{T}_\alpha$ below it in the hierarchy). Then we may define an SIS ${\mathfrak}{T}_\infty$ over ${\mathscr}{T}$ with radius poset $\Lambda \times {\mathcal}{I}$, where $(\lambda,\alpha) \leq (\mu,\beta)$ in $\Lambda \times {\mathcal}{I}$ if $\lambda \leq \mu$ and $\alpha \leq \beta$. To do this, we set $\Phi \in ({\mathfrak}{T}_\infty)^{(\lambda,\alpha)}$ whenever $\Phi \in {\mathfrak}{T}_\alpha^\lambda$. So a Borel–Moore chain is PE in ${\mathfrak}{T}_\infty$ if and only if it is PE in some ${\mathfrak}{T}_\alpha$.
We wish to specialise this construction of ${\mathfrak}{T}_\infty$ to the case where each ${\mathfrak}{T}_\alpha$ corresponds to the internal symmetries of some combinatorial tiling. So suppose that, in addition to the ‘base’ CW poset ${\mathscr}{T}$, we have CW posets ${\mathscr}{T}_\alpha$ for each $\alpha \in {\mathcal}{I}$, corresponding to the CW posets of the supertilings. We assume that to each cell $a \in {\mathscr}{T}_\alpha$ there is an associated subcomplex $a_{\mathscr}{T}$ of ${\mathscr}{T}$. We demand that $|a_{\mathscr}{T}| \cong B^n$ for an $n$–cell $a \in {\mathscr}{T}_\alpha$, and that $a \preceq b$ in ${\mathscr}{T}_\alpha$ if and only if $a_{\mathscr}{T} \subseteq b_{\mathscr}{T}$. In short, each ${\mathscr}{T}_\alpha$ corresponds to a regular CW decomposition of the space $|{\mathscr}{T}|$, all of them having ${\mathscr}{T}$ as a common refinement.
To each of the CW posets ${\mathscr}{T}_\alpha$ we wish to associate an SIS ${\mathfrak}{T}_\alpha$, analogously to the construction of §\[subsubsec: Combinatorial Tilings\], but over the base CW poset ${\mathscr}{T}$. We may want to label cells or to rule out certain morphisms between cells as being non-allowed symmetries, such as, for example, when we wish to consider only translations between patches of a Euclidean tiling, or orientation preserving rigid motions between patches. So we suppose that each ${\mathscr}{T}_\alpha$ is equipped with a groupoid $L_\alpha$ of allowed morphisms, whose elements are the cells of ${\mathscr}{T}_\alpha$ and morphisms between them are given by cellular isomorphisms. By a cellular isomorphism between $a,b \in {\mathscr}{T}_\alpha$, we shall in fact always mean a cellular isomorphism $\Phi \co a_{\mathscr}{T} \to b_{\mathscr}{T}$ in ${\mathscr}{T}$ which induces a cellular isomorphism $\Phi' \co \overline{a} \to \overline{b}$ in ${\mathscr}{T}_\alpha$ (and shall usually drop the relevant notation indicating this). When we later introduce our homology calculations, we shall always assume that our coefficient group $G$ has division by the order of the isotropy groups of each $L_\alpha$.
Given a cell $a \in {\mathscr}{T}$ and $n \in \mathbb{N}_0$, we let $P_\alpha(a,n)$ be the closure of the set of cells in ${\mathscr}{T}_\alpha$ which contain a cell of ${\mathscr}{T}$ in their interiors within combinatorial distance $n$ of $a$ (see §\[subsubsec: Combinatorial Tilings\] for a notion of combinatorial distance between cells). So we are measuring sizes of patches using the cells of ${\mathscr}{T}$, not of ${\mathscr}{T}_\alpha$. Then given $a$, $b \in {\mathscr}{T}$, let $\Phi \in ({\mathfrak}{T}_\alpha)_{a,b}^n$ if and only if $\Phi$ is the restriction to $\overline{\text{St}(a)}$ of some cellular isomorphism $\widetilde{\Phi}$ between $P_\alpha(a,n)$ and $P_\alpha(b,n)$ which is permitted by the allowed morphisms of $L_\alpha$.
We shall call the collection of the above data a *hierarchical system* if for $\alpha \leq \beta$ we have that ${\mathfrak}{T}_\alpha \supseteq {\mathfrak}{T}_\beta$, and we collect this data into the SIS ${\mathfrak}{T}_\infty$ as constructed in greater generality above. This condition may be thought of as saying that, for $\alpha \leq \beta$, the radius $n$ patch of tiles of ${\mathscr}{T}_\alpha$ at a cell $a \in {\mathscr}{T}$ is determined by the $n$–patch of $a$ in ${\mathscr}{T}_\beta$; compare to the case of an ordered list $T_0$, $T_1$, $T_2$, …of supertilings arising from a substitution as discussed above.
In many cases of interest, there is a local rule for ‘undoing’ the subdivision, so that a patch $P_\beta(c,n)$ may always be determined up to equivalence using only the information of the patch $P_\alpha(c,m)$ below it in the hierarchy, for possibly larger radius $m$. This case would correspond to a *recognisable* substitution, and in the language here this simply means that each ${\mathfrak}{T}_\alpha$ is tail-equivalent to ${\mathfrak}{T}_\infty$ (and in the language of tiling theory that the tilings corresponding to ${\mathfrak}{T}_\alpha$ are MLD [@Sad08 Chapter 1, §3]). So in this case a chain is PE in *some* ${\mathfrak}{T}_\alpha$ if and only if it is PE in *any given* ${\mathfrak}{T}_\alpha$, and calculation of invariants of ${\mathscr}{T}_\infty$ are tantamount to calculating the invariants of any ${\mathfrak}{T}_\alpha$ in the hierarchy. In the non-recognisable case, moving up the hierarchy adds more information, so that a tiling ${\mathscr}{T}_\beta$ may ‘know’ something that a tiling ${\mathscr}{T}_\alpha$ does not, see the solenoid example §\[solenoid\].
Growth of Cells
---------------
To apply the main theorem of this section, which determines a method of computation of the PE homology of ${\mathfrak}{T}_\infty$, we shall need the cells of each ${\mathscr}{T}_\alpha$ to become large relative to the cells of ${\mathscr}{T}$ as $\alpha$ increases. The precise condition that we require is defined here; we note that this condition will hold for any of the examples that we are actually interested in here, such as polytopal (mixed) substitution tilings of Euclidean space.
For each $\alpha \in {\mathcal}{I}$, $k \in \{0,\ldots,d\}$ and $x \in {\mathscr}{T}$ contained in the $k$–skeleton of ${\mathscr}{T}_\alpha$, let $N^{\alpha,k}(x) \subseteq {\mathscr}{T}_\alpha$ be a subset of cells. We think of the cells of $N^{\alpha,k}(x)$ as being ‘near to $x$’, so we shall always assume that if $x \in a_{\mathscr}{T}$ for $a \in {\mathscr}{T}_\alpha$ then $a \in N^{\alpha,k}(x)$. Fix a coefficient ring $G$ which has division by the order of isotropy for each $L_\alpha$. For $\sigma \in C_n^{\text{BM}}({\mathscr}{T};G)$, we write $\sigma \in N_n^{\alpha,k}$ to mean that $\sigma$ is supported on the subcomplex corresponding to the $k$–skeleton of ${\mathscr}{T}_\alpha$ and is such that, for any $n$–cell $x \in {\mathscr}{T}$ of this subcomplex and any combinatorial isomorphism $\Phi \co X \to Y$ between subcomplexes of ${\mathscr}{T}_\alpha$ (permitted by the allowed morphisms of $L_\alpha$) with $N^{\alpha,k}(x) \subseteq X$, we have that $\sigma(\omega_x) = \sigma(\Phi_*(\omega_x))$ for an orientation $\omega_x$ of $x$. We think of the chains of $N_n^{\alpha,k}$ as the chains which assign coefficients to cells of ${\mathscr}{T}$ in a way which only depends locally on the cells of ${\mathscr}{T}_\alpha$.
\[def: expansive\] We shall say that the hierarchical system ${\mathfrak}{T}_\infty$ is *expansive* if, for any $\alpha \in {\mathcal}{I}$ and $m \in \mathbb{N}_0$, we can choose $\beta \geq \alpha$ and $N^{\beta,k}(x)$, as above, so that:
1. Each radius $m$ patch $P_\beta(x,m) \subseteq N^{\beta,d}(x)$.
2. For each $k$–cell $a \in {\mathscr}{T}_\beta$, there exists some $k$–cell $x \in a_{\mathscr}{T}$ for which $N^{\beta,k}(x) \subseteq \overline{\text{St}(a)}$.
3. For any $\sigma \in N_n^{\beta,k}$ with $n<k$ and $\partial(\sigma)$ supported on the $(k-1)$-skeleton of ${\mathscr}{T}_\beta$, there exists some PE chain $\tau \in C_{n+1}({\mathfrak}{T}_\infty)$ for which $\sigma + \partial(\tau) \in N_n^{\beta,k-1}$.
\[ex: expansive Euc\] Let ${\mathcal}{K}$ be a CW decomposition of $\mathbb{R}^d$. We shall call ${\mathcal}{K}$ *polytopal* if each cell $a \in {\mathcal}{K}$ may be assigned a barycentre $v_a$ for which:
1. Each closed cell $a$ of ${\mathcal}{K}$ is realised as the geometric simplicial complex of simplices the convex hulls of $\{v_{a_1},v_{a_2},\ldots,v_a \}$ for $\{a_1,a_2,\ldots,a\} \in ({\mathscr}{F}({\mathcal}{K}))_\Delta$.
2. Whenever $\Phi$ is a cellular rigid motion between closed cells $a,b$ of ${\mathcal}{K}$, then $\Phi$ maps $v_a$ to $v_b$.
For example, if the cells of ${\mathcal}{K}$ are convex then ${\mathcal}{K}$ is polytopal, one may choose the barycentre $v_c$ to be the centre of mass of $c$ in its supporting hyperplane.
Suppose that our hierarchical system ${\mathfrak}{T}_\infty$ is based on CW posets ${\mathscr}{T}$ and ${\mathscr}{T}_\alpha$ of polytopal CW decompositions of $\mathbb{R}^d$, that the allowed cellular isomorphisms $L_\alpha$ are based upon rigid motions and that:
1. There exists some $r$ for which each cell $a \in {\mathscr}{T}$ is fully contained in some $r$–ball.
2. For each $R >0$, for sufficiently large $\alpha \in {\mathcal}{I}$ the barycentres $v_a$ of cells $a \in {\mathscr}{T}_\alpha$ satisfy $|v_a - v_b| \geq R$ for $a \neq b$.
Then we say that *${\mathfrak}{T}_\infty$ is based on polytopal and growing cells*. In this case, the hierarchical system is expansive. The polytopal condition allows one to use the barycentric subdivision to define regions surrounding the cells which deformation retract to lower dimensional skeleta. The sets $N^{\alpha,k}(x)$ may be defined using these regions, and any potential fringe effects associated to making the process cellular may be avoided by choosing slightly larger regions about cells as $k$ decreases. In the case of non-trivial isotropy, one may need to invoke the divisibility of the coefficient group in the construction of the boundary chains $\tau$, analogously to in the proof of Proposition \[prop: PE BCS\].
The Method of Computation {#subsec: method}
-------------------------
Fix a coefficient ring $G$ which, as previously stated, we will assume to have division by the order of isotropy of allowed cellular isomorphisms of each $L_\alpha$. For $\alpha \in {\mathcal}{I}$ we define the *approximant chain complex* $A_\bullet^\alpha$ as follows. Let $A_n^\alpha \leq C_n({\mathscr}{T}_\alpha)$ consist of the Borel–Moore $n$–chains of ${\mathscr}{T}_\alpha$ which assign coefficients to cells in a way which depends only locally on stars of cells of ${\mathfrak}{T}_\alpha$. That is, $\sigma \in A_n^\alpha$ if and only if for any cellular isomorphism $\Phi \co \overline{\text{St}(a)} \to \overline{\text{St}(b)}$ permitted by $L_\alpha$ sending $a^n$ to $b^n$, we have that $\sigma(\omega_a) = \sigma(\Phi_*(\omega_a))$ for an orientation $\omega_a$ of $a$. The cellular boundary map makes each $A_\bullet^\alpha$ a chain complex. We call the corresponding homology $H_\bullet A_\alpha \coloneqq H(A_\bullet^\alpha)$ an *approximant homology*.
We now wish to define, for each $\alpha \leq \beta$ in ${\mathcal}{I}$, a homomorphism $s_\alpha^\beta \co H_\bullet A_\alpha \to H_\bullet A_\beta$ so that the approximant homologies fit into a directed system over ${\mathcal}{I}$. So let $\sigma$ be a cycle of $A_n^\alpha$. We may identify $\sigma$ with a cycle of ${\mathscr}{T}$, supported on the subcomplex corresponding to the $n$–skeleton of ${\mathscr}{T}_\alpha$. Since this chain assigns values to cells based only immediate local data in ${\mathfrak}{T}_\alpha$, it still only depends on local data in ${\mathfrak}{T}_\beta$ by our assumption of the tilings forming a hierarchical system. Unfortunately, $\sigma$ may not be supported on the $n$–skeleton of ${\mathscr}{T}_\beta$.
The idea is to now push $\sigma$ back to the $n$–skeleton using consistent choices for each equivalent cell. Suppose the the maximal dimension of cell is $d$. If $n = d$, then $\sigma$ is already supported on the $n$–skeleton. Otherwise, for each equivalence class of $d$–cell of ${\mathscr}{T}_\beta$, choose a representative $a$. Restrict $\sigma$ to $a_{\mathscr}{T}$ and choose an $(n+1)$–chain $\tau_a$ supported on the interior of $a$ for which $\sigma \restriction a_{\mathscr}{T} + \partial(\tau_a)$ is supported on the boundary of $a$. One may now copy the chain $\tau_a$ to each equivalent cell using the cellular isomorphisms permitted by $L_\beta$. In the case of non-trivial cell isotropy, one needs to invoke the divisibility of the coefficient group to ‘average’ the boundary chain over the cells’ symmetries, as in the proof of Proposition \[prop: PE BCS\]. By repeating this operation for each equivalence class of cell one defines a chain $\tau$, which only depends locally on stars of cells in ${\mathfrak}{T}_\beta$, for which $\sigma + \partial(\tau)$ is supported on the $(d-1)$–skeleton. Continue this operation iteratively down the skeleta until the original chain is seen to be homologous to one supported on the $n$–skeleton of ${\mathscr}{T}_\beta$. We denote the corresponding cycle of $A_n^\beta$ by $s_\alpha^\beta(\sigma)$. It shall follow from the proof of Theorem \[thm: method\] that this procedure induces well-defined homomorphisms, called *connecting maps*, between the approximant homology groups, making them a directed system over ${\mathcal}{I}$.
To summarise, the construction goes as follows:
1. For each $\alpha \in {\mathcal}{I}$, enumerate the list of star-patches of cells of ${\mathscr}{T}_\alpha$, up to equivalence.
2. Define the approximant chain groups $A_n^\alpha$ to be the groups freely generated by the equivalence classes of star-patches of $n$–cells of ${\mathscr}{T}_\alpha$ (which do not possess self-symmetries reversing orientations).
3. Compute the corresponding approximant homology $H_\bullet A_\alpha$.
4. For $\alpha \leq \beta$, connecting maps $s_\alpha^\beta \co H_\bullet A_\alpha \to H_\bullet A_\beta$ are defined as follows. Given a cycle $\sigma \in A_n^\alpha$, firstly canonically identify it with the cycle of the finer complex ${\mathscr}{T}$. Iteratively push $\sigma$ down the skeleta of ${\mathscr}{T}_\beta$ by making a choice of $(n+1)$–chain for each equivalence class of star-patch of ${\mathscr}{T}_\beta$ (which is invariant under any potential non-trivial cell isotropy). The resulting cycle of $H_\bullet A_\beta$ is defined to be $s_\alpha^\beta(\sigma)$.
5. Compute the direct limit of the directed system of approximant homologies $H_\bullet A_\alpha$ and substitution homomorphisms $s_\alpha^\beta$ between them.
Notice that for a Borel–Moore chain of ${\mathscr}{T}$ which is PE to radius $n$ in ${\mathfrak}{T}_\alpha$, the same is true of its boundary. Denote by $B_\bullet^\alpha$ the chain complex of chains which are PE to radius $0$ in ${\mathfrak}{T}_\alpha$, that is, the complex of chains which depend only on the cells of their immediate neighbourhood in ${\mathfrak}{T}_\alpha$. By the assumption of the ${\mathfrak}{T}_\alpha$ forming a hierarchical system, we have an inclusion of chain complexes $$\iota_\alpha^\beta \co B_\bullet^\alpha \hookrightarrow B_\bullet^\beta$$ for $\alpha \leq \beta$. The corresponding union of chain complexes shall be denoted $$B_\bullet \coloneqq \bigcup_{\alpha \in {\mathcal}{I}} B_\bullet^\alpha \cong \varinjlim_{\mathcal}{I}(B_\bullet^\alpha,\iota_\alpha^\beta),$$ which is the subcomplex of $C_\bullet({\mathfrak}{T}_\infty)$ of chains which are PE to radius $0$ in some ${\mathfrak}{T}_\alpha$.
\[thm: method\] There exist canonical isomorphisms $H_\bullet A_\alpha \cong H(B_\bullet^\alpha)$ for $\alpha \in {\mathcal}{I}$ which together define an isomorphism between the diagrams $(H_\bullet A_\alpha,s_\alpha^\beta)$ and $(H(B_\bullet^\alpha),(\iota_\alpha^\beta)_*)$. If the hierarchy is expansive, then the canonical inclusion of chain complexes $$i \co B_\bullet \to C_\bullet({\mathfrak}{T}_\infty)$$ is a quasi-isomorphism. In particular, in this case (such as when ${\mathfrak}{T}_\infty$ is Euclidean and based upon on polytopal and growing cells), the method outlined above computes the PE homology of ${\mathfrak}{T}_\infty$: $$H_\bullet({\mathfrak}{T}_\infty) \cong \varinjlim_{\mathcal}{I} (H_\bullet A_\alpha,s_\alpha^\beta).$$
The isomorphisms $H_n A_\alpha \cong H_n(B_\bullet^\alpha)$ are induced from the canonical chain isomorphisms $C_\bullet({\mathscr}{T}_\alpha) \cong H_\bullet({\mathscr}{T}^\bullet_\alpha,{\mathscr}{T}^{\bullet-1}_\alpha)$. To ensure that this induces a quasi-isomorphism upon restricting to chains which are PE to radius $0$ in each ${\mathfrak}{T}_\alpha$, one needs to implement the assumption of the coefficient group having division by the order of the isotropy of cells, analogously to the proof of Proposition \[prop: PE BCS\].
So now suppose that the hierarchical system is expansive. We shall firstly show that the inclusion of chain complexes $i$ induces a surjective map on homology. Let $\sigma$ be a cycle of $C_n({\mathfrak}{T}_\infty)$, so $\sigma$ is PE to some radius $m$ in some ${\mathfrak}{T}_\alpha$. By assumption, there exists $\beta \geq \alpha$ and choices $N^{\beta,k}(x)$ of neighbours to cells $x$ for which conditions $1$–$3$ of Definition \[def: expansive\] are satisfied. Condition $1$ implies that $\sigma \in N_n^{\beta,d}$. With repeated application of condition $3$, we may find a PE $(n+1)$–chain $\tau$ for which $\sigma + \partial(\tau) \in N_n^{\beta,n}$. By condition $2$, for each $n$–cell $a \in {\mathscr}{T}_\alpha$ there exists an $n$–cell $x \in a_{\mathscr}{T}$ for which $N^{\beta,n}(x) \subseteq \overline{\text{St}(a)}$. Since the value of $\sigma + \partial(\tau)$ on $x$ determines the value of $\sigma + \partial(\tau)$ on $a_{\mathscr}{T}$, and $\sigma + \partial(\tau) \in N_n^{\beta,n}$, we see that $\sigma +\partial(\tau) \in B_\bullet^\beta$. Hence, the inclusion homomorphism $i$ induces a surjective map on homology.
To show surjectivity, suppose that $\sigma \in B_n$ and that $\sigma = \partial(\tau)$ for $\tau \in C_{n+1}({\mathfrak}{T}_\infty)$. There exists some $\alpha \in {\mathcal}{I}$ and $m \in \mathbb{N}_0$ for which $\sigma$ is PE to radius $0$ in ${\mathfrak}{T}_\alpha$ and $\tau$ is PE to radius $m$ in ${\mathfrak}{T}_\alpha$. By expansivity, we may pick $\beta \geq \alpha$ for which conditions $1$–$3$ of Definition \[def: expansive\] are satisfied. Note that there exists some chain $\tau' \in B_{n+1}^\beta$ for which $\sigma' \coloneq \sigma + \partial(\tau')$ is supported on ${\mathscr}{T}_\beta^m$. Since $\sigma' \simeq \sigma$ in $H_n(B_\bullet^\beta)$ and $\tau + \tau'$ is still PE to radius $m$ in ${\mathfrak}{T}_\beta$, it is sufficient to show that $\tau + \tau'$ is homologous to an element of $B_{n+1}$. Similarly to the above, we have that $\tau + \tau' \in N_{n+1}^{\beta,d}$ and, since $\partial(\tau+\tau') = \sigma'$ is supported on the subcomplex of ${\mathscr}{T}$ corresponding to the $n$–skeleton of ${\mathscr}{T}_\beta$, we may make repeated use of condition $3$ to find an element $\tau'' \in N_{n+1}^{\beta,n+1}$ which is homologous to $\tau + \tau'$. Again, by condition $2$ of Definition \[def: expansive\], it follows that $\tau'' - \tau' \in B_{n+1}^\beta$. Since $\partial(\tau'' - \tau') = \partial(\tau) = \sigma$, the inclusion of chain complexes $i$ is injective on homology.
Discussion of Method, and its Relation to Others
------------------------------------------------
When one has a handle on the local combinatorics for each ${\mathfrak}{T}_\alpha$, and the rules connecting them between neighbouring levels of the hierarchy, the above theorem allows one to compute the PE homology of ${\mathfrak}{T}_\infty$. For example, in the case that ${\mathfrak}{T}_\infty$ corresponds to the hierarchy of a Euclidean substitution tiling, the combinatorics at each level and the rules for passing between them are constant, and so there is essentially only one approximant homology group and one connecting map to be determined. More generally (see Example \[ex: expansive Euc\]), the method will apply to any (recognisable) Euclidean hierarchical tiling based on polytopal and growing cells, such as for a polytopal mixed substitution tiling.
The method of computation seems to be quite efficient. Indeed, without determining that the tiling ‘forces the border’ (see [@AndPut98 §4]), the list of star-patches is the minimal amount of collaring information required to determine the immediate neighbourhood of a point of the tiling. This list of star-patches, and their incidences, define the approximant chain complexes in a conveniently direct way.
The approach seems to be closely related to that of Barge, Diamond, Hunton and Sadun [@BDHS10], although making the connection precise seems to be somewhat technical. The argument that the two methods produce the same direct limits would go as follows. Take the ‘dual stratification’ $S_0 \subset S_1 \subset \cdots \subset K_t$ of the BDHS approximant (where $t$ is small with respect to the size of the tiles) where each $S_i$ corresponds to the union of $k$–cell flaps with $k \geq d-i$ (see [@BDHS10 §4]). Then, using a Poincaré duality type argument, so long as the ‘cell-flaps’ and their incidences match the combinatorics of the tiling, one should be able to show that the (regraded) approximant chain complex $A^0_{d-\bullet}$ is isomorphic to the relative complex $H^\bullet(S_{\bullet},S_{\bullet-1})$, and that this has cohomology isomophic to $H^{d-\bullet}(K_t)$, by the vanishing of the relative groups $H^i(S_j,S_{j-1})$ for $i \neq j$.
These computations also seem to be related to a method developed in [@Gon11], where Gonçalves used the duals of the approximant complexes $A^\alpha_\bullet$ defined here to determine approximant groups to the $K$–theory associated to the stable equivalence relation of a substitution tiling. It was observed there that there appears to be a duality between the resulting direct limit groups and the $K$-theory of the tiling space. For a concrete relation one would also need to compare the connecting maps between approximant complexes defined in [@Gon11] to those constructed here.
Finally, we wish to discuss the computations in the presence of rotational symmetries. The method outlined above will compute the PE homology $H_\bullet({\mathfrak}{T}_\infty)$, but over non-divisible coefficients these groups need not be Poincaré dual to the PE cohomology (see Example \[ex: triangle\] and §\[Penrose\]). However, one may modify the method, by restricting the value of chains at cells whose star-patches possess non-trivial symmetries. This allows one to compute the modified PE homology $H_\bullet^\dagger({\mathfrak}{T}^0)$ (see §\[subsec: mod\]), which is Poincaré dual to the PE cohomology, see the computation of §\[Penrose\].
Example Computations {#subsec: Example Computations}
--------------------
### Fibonacci Tilings {#Fibonacci}
![Fibonacci Tiling[]{data-label="fig: Fibonacci"}](Fibonacci.pdf)
Fibonacci tilings are tilings of $\mathbb{R}^1$ of two prototiles, which we shall call here $0$ and $1$, which are intervals of lengths the golden ratio $\varphi$ and $1$, respectively. They are examples of Sturmian words, but may also be constructed via the recognisable substitution $0 \mapsto 01$, $1 \mapsto 0$. A tiling $T$ admitted by this substitution defines a cellular decomposition ${\mathscr}{T}$ of the real line and an infinite sequence of tilings $T = T_0$, $T_1$, …for which the substitution rule subdivides $T_{i+1}$ to $T_i$. The tiles of $T_i$ are based on inflations of the original tiles of $T_0$ by a scaling factor of $\varphi^i$.
We shall compute the PE homology groups associated to the translational hull of Fibonacci tilings using the method outlined above. To determine the approximant chain complexes $A_n^i$, we firstly enumerate the equivalence classes of star-patches up to translation. They are given by the ‘vertex types’ $0.1$, $1.0$ and $0.0$, and the ‘edge types’ are given by the equivalence classes associated to the two distinct prototiles $0$ and $1$. So each approximant chain complex is of the form $$0 \leftarrow \mathbb{Z}^3 \xleftarrow{\partial_1} \mathbb{Z}^2 \leftarrow 0.$$ Orient the $1$–cells to point to the right and define the $0$–chains ${\text{\usefont{U}{bbm}{m}{n}1}}(0.1)_i$, ${\text{\usefont{U}{bbm}{m}{n}1}}(1.0)_i$, ${\text{\usefont{U}{bbm}{m}{n}1}}(0.0)_i$ and $1$–chains ${\text{\usefont{U}{bbm}{m}{n}1}}(0)_i$ and ${\text{\usefont{U}{bbm}{m}{n}1}}(1)_i$ to be the canonical indicator chains associated to each of the edge and vertex types in the level $i$ tiling. Then the degree one boundary maps are given by: $$\begin{gathered}
\partial_1({\text{\usefont{U}{bbm}{m}{n}1}}(0)_i) = {\text{\usefont{U}{bbm}{m}{n}1}}(0.1)_i - {\text{\usefont{U}{bbm}{m}{n}1}}(1.0)_i\\
\partial_1({\text{\usefont{U}{bbm}{m}{n}1}}(1)_i) = {\text{\usefont{U}{bbm}{m}{n}1}}(1.0)_i - {\text{\usefont{U}{bbm}{m}{n}1}}(0.1)_i\end{gathered}$$ So $H_0A_i \cong \mathbb{Z}^2$ and is freely generated by the indicator chains $a_i \coloneqq {\text{\usefont{U}{bbm}{m}{n}1}}(0.1)_i$ and $b_i \coloneqq {\text{\usefont{U}{bbm}{m}{n}1}}(0.0)_i$. The degree one approximant homology group $H_1A_i \cong \mathbb{Z}$ is generated by the fundamental class $\Gamma_i \coloneqq {\text{\usefont{U}{bbm}{m}{n}1}}(0)_i + {\text{\usefont{U}{bbm}{m}{n}1}}(1)_i$.
We now calculate the connecting maps $s_i^{i+1}$ between these approximant groups. Recall that to define these homomorphisms, given a cycle $\sigma \in H_nA_i$ we consider it as a chain depending only on its immediate neighbourhood in the tiling $T_{i+1}$ next up the hierarchy. One then pushes the chain back to the $n$–skeleton to define $s_i^{i+1}(\sigma)$. Clearly the fundamental class $\Gamma_i$ is mapped to $\Gamma_{i+1}$. To calculate the connecting map in degree zero, firstly consider the generator $a_i$ of $H_0A_i$. The $0.1$ vertices of $T_i$ are interior points of the $0$ tiles of $T_{i+1}$, which subdivides to $01$ in $T_i$. We may push this chain to the $0$–skeleton of $T_{i+1}$ in a local way by, for example, moving it to the right-hand endpoint of each $0$ tile of $T_{i+1}$ (see Figure \[fig: Fibonacci\]), so we see that $s_i^{i+1}(a_i) \simeq {\text{\usefont{U}{bbm}{m}{n}1}}(0.1)_{i+1} + {\text{\usefont{U}{bbm}{m}{n}1}}(0.0)_{i+1} = a_{i+1}+b_{i+1}$. For the generator $b_i$, note that each $0.0$ vertex of $T_i$ is found precisely on a $1.0$ vertex of $T_{i+1}$. So we see that $s_i^{i+1}(b_i) = {\text{\usefont{U}{bbm}{m}{n}1}}(1.0)_{i+1} \simeq {\text{\usefont{U}{bbm}{m}{n}1}}(0.1)_{i+1} = a_{i+1}$. Then the connecting maps $s_i^{i+1} \co \mathbb{Z}^2 \to \mathbb{Z}^2$ send the generators $a_i \mapsto a_{i+1} + b_{i+1}$ and $b_i \mapsto a_{i+1}$, which are isomorphisms. So we have confirmed that: $$\begin{gathered}
\check{H}^0(\Omega^1) \cong H^0({\mathfrak}{T}^1) \cong H_1({\mathfrak}{T}^1) \cong \varinjlim(\mathbb{Z},s_i^{i+1}) \cong \mathbb{Z}\\
\check{H}^1(\Omega^1) \cong H^1({\mathfrak}{T}^1) \cong H_0({\mathfrak}{T}^1) \cong \varinjlim(\mathbb{Z}^2,s_i^{i+1}) \cong \mathbb{Z}^2\end{gathered}$$ The first isomorphisms come from the fact that the Čech cohomology of the translational hull $\Omega^1$ is isomorphic to the PE cohomology (see [@Sad07] or §\[subsec: Tiling Spaces\]), the second from the PE Poincaré duality of Theorem \[thm: PEPD\] and the penultimate isomorphism from Theorem \[thm: method\] and the recognisability of the substitution.
### Thue–Morse Tilings {#Thue--Morse}
The Thue–Morse tilings are produced via the recognisable substitution $0 \mapsto 01$ and $1 \mapsto 10$. In this case, the vertex types are given by $0.0$, $0.1$, $1.0$ and $1.1$, and the edge types by $0$ and $1$. The boundary maps have the same description as the above example, so the approximant homology groups are $H_0A_i \cong \mathbb{Z}^3$ and $H_1A_i \cong \mathbb{Z}$. Of course, $H_1A_i$ is generated by the fundamental class $\Gamma_i = {\text{\usefont{U}{bbm}{m}{n}1}}(0)_i + {\text{\usefont{U}{bbm}{m}{n}1}}(1)_i$ and $s_i^{i+1}(\Gamma_i) = \Gamma_{i+1}$. For the degree zero calculation, firstly note that $H_0A_i$ is generated by the $0$–chains $a_i \coloneqq {\text{\usefont{U}{bbm}{m}{n}1}}(0.1)_i$, $b_i \coloneqq {\text{\usefont{U}{bbm}{m}{n}1}}(0.0)_i$ and $c_i \coloneqq {\text{\usefont{U}{bbm}{m}{n}1}}(1.1)_i$. The $0.1$ vertices of $T_i$ are found precisely at the centres of $0$ tiles and the $1.1$ vertices of $T_{i+1}$. Shifting the chain from the centres of the $0$ tiles to the right and back onto the $0$–skeleton, we see that $s_i^{i+1}(a_i) = ({\text{\usefont{U}{bbm}{m}{n}1}}(0.1)_{i+1} + {\text{\usefont{U}{bbm}{m}{n}1}}(0.0)_{i+1}) + {\text{\usefont{U}{bbm}{m}{n}1}}(1.1)_{i+1} = a_{i+1} + b_{i+1} + c_{i+1}$. The $0.0$ vertices of $T_i$ lie precisely on the $1.0$ vertices of $T_{i+1}$ so $s_i^{i+1}(b_i) = {\text{\usefont{U}{bbm}{m}{n}1}}(1.0)_{i+1} \simeq a_{i+1}$. Similarly, the $1.1$ vertices of $T_i$ lie precisely on the $0.1$ vertices of $T_{i+1}$, so $s_i^{i+1}(c_i) = {\text{\usefont{U}{bbm}{m}{n}1}}(0.1)_{i+1} = a_{i+1}$. In summation, the connecting maps $s_i^{i+1} \co \mathbb{Z}^3 \to \mathbb{Z}^3$ are given by: $$\begin{aligned}
a_i &\mapsto a_{i+1} + b_{i+1} + c_{i+1}\\
b_i &\mapsto a_{i+1}\\
c_i &\mapsto a_{i+1}\end{aligned}$$ This linear map has eigenvectors with eigenvalues $0$, $-1$ and $2$, although they only span an index $3$ sublattice of $\mathbb{Z}^3$. With some further calculation one may evaluate the direct limit as being isomorphic to $\mathbb{Z} \oplus \mathbb{Z}[1/2]$, so: $$\begin{gathered}
\check{H}^0(\Omega^1) \cong H_1({\mathfrak}{T}^1) \cong \varinjlim(\mathbb{Z},s_i^{i+1}) \cong \mathbb{Z}\\
\check{H}^1(\Omega^1) \cong H_0({\mathfrak}{T}^1) \cong \varinjlim(\mathbb{Z}^3,s_i^{i+1}) \cong \mathbb{Z} \oplus \mathbb{Z}[1/2]\end{gathered}$$
### Penrose Kite and Dart Tilings {#Penrose}
We consider now Penrose’s famous kite and dart tilings of the plane. Just as for the Fibonacci tilings, this example may be produced via a cut-and-project scheme or via a substitution rule. Since the substitution rule for the kite and dart tiles does not perfectly decompose the support of a supertile into its constituent tiles, we take the underlying complex ${\mathscr}{T}$ to be given by a tiling of Robinson triangles.
For this example, we shall allow patches to be compared using orientation preserving rigid motions, so that a Borel–Moore chain $\sigma \in H_n^{\text{BM}}({\mathscr}{T})$ is PE in this setup if and only if, for sufficiently large radius $r$, the value of $\sigma$ at an oriented $n$–cell $c$ depends only on the patch of tiles within radius $r$ of $c$, up to equivalence of rigid motion. The homology calculations in the case where we compare patches only up to translation are determined by Poincaré duality, which we provide here for reference (see [@AndPut98]): $$\begin{gathered}
\check{H}^0(\Omega^1) \cong H^0({\mathfrak}{T}^1) \cong H_2({\mathfrak}{T}^1) \cong \mathbb{Z}\\
\check{H}^1(\Omega^1) \cong H^1({\mathfrak}{T}^1) \cong H_1({\mathfrak}{T}^1) \cong \mathbb{Z}^5\\
\check{H}^2(\Omega^1) \cong H^2({\mathfrak}{T}^1) \cong H_0({\mathfrak}{T}^1) \cong \mathbb{Z}^8\end{gathered}$$
![Vertex and Edge Types of the Penrose Kite and Dart Tilings[]{data-label="fig: VaETypes"}](VaETypes.pdf)
To begin calculation, we must firstly enumerate the list of star-patches of cells in the tiling up to rigid motion. It turns out that there are $7$ such star-patches at vertices (named sun, star, ace, deuce, jack, queen and king in Conway’s notation) and there are $7$ ways for tiles to meet along an edge, which we shall denote by $E1$–$E7$; see Figure \[fig: VaETypes\] where the vertex and edge types are listed in these respective orders. Of course, the two face types are given by the two types of tiles, kites and darts. So the approximant chain complexes are of the form $$0 \leftarrow \mathbb{Z}^7 \xleftarrow{\partial_1} \mathbb{Z}^7 \xleftarrow{\partial_2} \mathbb{Z}^2 \leftarrow 0.$$ The $\partial_1$ boundary map, with respect to the bases of vertex and edge types ordered as in Figure \[fig: VaETypes\], is represented as a matrix by: $$\left( \begin{array}{ccccccc}
5 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & -5 & 0 & 0 & 0 & 0 & 0\\
-1 & 0 & -1 & 1 & 0 & 0 & 0\\
0 & 1 & 1 & -1 & 0 & 0 & 1\\
1 & 0 & 1 & -1 & -1 & -1 & 0\\
-1 & 0 & 0 & 0 & 1 & 1 & -2\\
0 & -2 & 0 & 0 & 1 & 1 & -1 \end{array} \right)$$ The first column, for example, is $(5,0,-1,0,1,-1,0)^T$ since at any sun vertex there are $5$ incoming $E1$ edges, at an ace there is an outgoing $E1$, at a jack there is an incoming $E1$ and at a queen there is an outgoing $E1$.
![Torsion Element $t_0$ with $5t_0+\partial_1(-{\text{\usefont{U}{bbm}{m}{n}1}}(E1)_0+{\text{\usefont{U}{bbm}{m}{n}1}}(E2)_0 - {\text{\usefont{U}{bbm}{m}{n}1}}(E4)_0 -2 \cdot {\text{\usefont{U}{bbm}{m}{n}1}}(E7)_0)=0$.[]{data-label="fig: S0TorBdy"}](S0TorBdy.pdf)
Some simple calculations using the Smith normal form show that the approximant groups $H_0A_i \cong \mathbb{Z}^2 \oplus \mathbb{Z}/5\mathbb{Z}$, with basis of the free part generated by $a_i$ as the indicator of sun vertex types and $b_i$ the indicator of star vertex types. The $5$–torsion is generated by the element $t_i = a_i + b_i - c_i$, where $c_i$ is the indicator of queen vertex types. The torsion element $t_0$ is illustrated in Figure \[fig: S0TorBdy\], where it is shown pictorially that $5t_0$ is nullhomologous via the boundary of $-{\text{\usefont{U}{bbm}{m}{n}1}}(E1)_0+{\text{\usefont{U}{bbm}{m}{n}1}}(E2)_0 - {\text{\usefont{U}{bbm}{m}{n}1}}(E4)_0 -2 \cdot {\text{\usefont{U}{bbm}{m}{n}1}}(E7)_0$.
To calculate the degree zero connecting maps, note that the sun vertices of $T_i$ lie precisely at the star, queen and king vertices of $T_{i+1}$, the star vertices at the sun vertices of $T_{i+1}$ and the queen vertices at the deuce vertices of $T_{i+1}$. Some simple calculations then show that: $$\begin{aligned}
a_i &\mapsto 3a_{i+1} -b_{i+1} + 2t_{i+1} \\
b_i &\mapsto a_{i+1} \\
t_i &\mapsto t_{i+1}\end{aligned}$$ Since each connecting map is an isomorphism, we conclude that $$H_0({\mathfrak}{T}^0) \cong \varinjlim(\mathbb{Z}^2 \oplus \mathbb{Z}/5\mathbb{Z},s_i^{i+1}) \cong \mathbb{Z}^2 \oplus \mathbb{Z}/5\mathbb{Z}.$$ To calculate using the modified complexes, so as to restore Poincaré duality, one uses the approximant chain complexes $$0 \leftarrow 5\mathbb{Z} \oplus 5\mathbb{Z} \oplus \mathbb{Z}^5 \xleftarrow{\partial_1} \mathbb{Z}^7 \xleftarrow{\partial_2} \mathbb{Z}^2 \leftarrow 0,$$ where the degree zero group is the subgroup of $A_0^i$ which restricts the coefficients on the sun and star vertices to multiples of $5$, since these vertices have $5$–fold rotational symmetry and the other vertices have trivial rotational symmetry. We calculate the modified approximant homology groups in degree zero as $\mathbb{Z}^2$ and the connecting maps as isomorphisms, which confirms that $$\check{H}^2(\Omega^0) \cong H^2({\mathfrak}{T}^0) \cong H_0^\dagger({\mathfrak}{T}^0) \cong \varinjlim(\mathbb{Z}^2,s_i^{i+1}) \cong \mathbb{Z}^2.$$ The inclusion of chain complexes $C_\bullet^\dagger({\mathfrak}{T}^0) \hookrightarrow C_\bullet({\mathfrak}{T}^0)$ induces the following short exact sequence $$(0 \to [H_0^\dagger({\mathfrak}{T}^0) \cong \check{H}^2(\Omega_T^0)] \to H_0({\mathfrak}{T}^0) \to (\mathbb{Z}/5\mathbb{Z})^2 \to 0) \cong$$ $$0 \to \mathbb{Z}^2 \underset{f}{\rightarrow} \mathbb{Z}^2 \oplus \mathbb{Z}/5\mathbb{Z} \underset{g}\to (\mathbb{Z}/5\mathbb{Z})^2 \to 0,$$ where $f(a,b)=(a+2b,a-3b,[4a+3b]_5)$ and $g(a,b,[c]_5)=([a+c]_5,[b+c]_5)$, exhibiting the failure of PE Poincaré duality in this example.
For degree one the situation is illustrated in Figure \[fig: Penrose\]. One may calculate that each degree one approximant group $H_1A_i \cong \mathbb{Z}$ is generated by the $1$–chain $e_i \coloneqq {\text{\usefont{U}{bbm}{m}{n}1}}(E3)_i + {\text{\usefont{U}{bbm}{m}{n}1}}(E4)_i$. That is, the degree one approximant groups are freely generated by the $1$–cycles which trail the bottoms of the level $i$ dart supertiles. Such a cycle is illustrated in red in Figure \[fig: Penrose\], along with the analogous green cycle next up the hierarchy (although we have removed any indication of orientations to decrease clutter). As shown in the figure, the two chains are related by the boundary of the indicator $2$–chain associated to the dart tiles of the lower hierarchy. So we see that $s_i^{i+1}(e_i) = -e_{i+1}$ (note the reversal of orientations) and, in particular, the connecting maps in degree one are isomorphisms, which confirms that $$\check{H}^1(\Omega^0) \cong H^1({\mathfrak}{T}^0) \cong H_1({\mathfrak}{T}^0) \cong \mathbb{Z}.$$
![[]{data-label="fig: Penrose"}](Penrose.pdf)
### Arnoux–Rauzy Words {#Arnoux--Rauzy}
Since the previous examples were based upon purely substitutive systems, the approximant homology groups and connecting maps between them were canonically isomorphic between adjacent levels of the hierarchy. We consider now a *mixed* substitution system, where one has a set of substitutions and an infinite sequence in which to apply them.
Our examples here will be the Arnoux–Rauzy words, originally introduced in [@ArnRau91] as a generalisation of Sturmian words. Let $k \in \mathbb{N}_{\geq 2}$. The Arnoux–Rauzy substitutions are defined over the alphabet ${\mathcal}{A}_k = \{1,2,\ldots,k\}$ and the $k$ different substitutions $\rho_i$ are given by $\rho_i(j) = ji$ for $i \neq j$ and $\rho_i(i)=i$. Fix an infinite sequence $(n_i)_i = (n_0,n_1,\ldots) \in {\mathcal}{A}_k^{\mathbb{N}_0}$ for which each element of ${\mathcal}{A}_k$ occurs infinitely often. Then there exist bi-infinite *Arnoux–Rauzy words* for which every finite sub-word is contained in some translate of a ‘supertile’ $\rho_{n_0} \circ \rho_{n_1} \circ \cdots \circ \rho_{n_l}(i)$. We may consider such a word as defining a tiling of labelled unit intervals of $\mathbb{R}^1$. Everything is recognisable, so for such a tiling $T_0$, one may uniquely group the tiles to a tiling $T_1$ of tiles of labelled intervals (although of different lengths) so that the substitution $\rho_{n_0}$ decomposes $T_1$ to $T_0$. The process may be repeated, so one in fact obtains an infinite hierarchy of tilings, the supertiles of which become arbitrarily large as one passes up the hierarchy.
The two letter words of $T_i$ are the elements of ${\mathcal}{A}_k \times {\mathcal}{A}_k$ with at least one occurrence of $n_i$. So it is easy to see that $H_0A_i \cong \mathbb{Z}^k$ is freely generated by the indicator $0$–chains of vertices of the form $n_i . j$ where $j \in {\mathcal}{A}_k$ is arbitrary. A simple calculation shows that, with this choice of basis, the connecting map $s_i^{i+1}$ is the unimodular matrix $M_i$ given by the identity matrix with a column of $1$’s down the $n_i^{\text{th}}$ column which, incidentally, is the incidence matrix of the substitution.
It follows that the degree one Čech cohomology of the tiling space of the Arnoux–Rauzy words associated to any given sequence $(n_i)_i \in {\mathcal}{A}_k^{\mathbb{N}_0}$ is $$\check{H}^1(\Omega^1) \cong H^1({\mathfrak}{T}_\infty) \cong H_0({\mathfrak}{T}_\infty) \cong \varinjlim(\mathbb{Z}^k \xrightarrow{M_0} \mathbb{Z}^k \xrightarrow{M_1} \mathbb{Z}^k \xrightarrow{M_2} \cdots) \cong \mathbb{Z}^k.$$ It is interesting to note that the matrices above are related to continued fraction algorithms. For example, for the $k=2$ case, the Arnoux–Rauzy words are precisely the Sturmian words. To an irrational $\alpha$, the sequence $(n_i)_i$ is chosen according to the continued fraction algorithm for $\alpha$ (see [@Dur00 §3.2]) and the sequence of matrices $M_i$ of the above direct limit determine the partial quotients of $\alpha$. Whilst the isomorphism classes of the first Čech cohomology groups do not distinguish these tiling spaces, their order structure, which is determined by the above direct limit, is a rich invariant.
### Solenoids as Hierarchical Tilings {#solenoid}
Let $S_k^d$ be the periodic tiling of $\mathbb{R}^d$ of hypercubes with side-length $k \in \mathbb{N}$ and vertices on the lattice $k \mathbb{Z}^d$. These tilings define cellular decompositions ${\mathscr}{S}_k^d$ of $\mathbb{R}^d$. Define a partial ordering on $\mathbb{N}$ by setting $m$ less than or equal to $n$ if and only if $m$ divides $n$; note that $(\mathbb{N},\divides)$ is a directed set. Comparing patches via translation, we have that two cells $a,b \in {\mathscr}{S}_1^d$ are equivalent in the tiling $S_k^d$ (to any given radius $n$) if and only if $a$ and $b$ are related by a vector of $k\mathbb{Z}^d$. Then, with the partial order on $\mathbb{N}$ given above, this defines a hierarchical system of tilings. Indeed, for $m \divides n$ we have that local patches of $S_n^d$ determine those of $S_m^d$ by subdividing each of the hypercubes of side-length $n$ into $(n/m)^d$ hypercubes of side-length $m$.
Define ${\mathfrak}{S}^d_\infty$ to be the SIS associated to this hierarchical system of tilings, where we compare patches via translation. Then a Borel–Moore chain is PE in ${\mathfrak}{S}^d_\infty$ if and only if it is invariant under translation by some full-rank sublattice. For $d=1$, for example, it is easy to see that $H_0({\mathfrak}{S}_\infty^1) \cong \mathbb{Q}$, the isomorphism identifies $p/q \in \mathbb{Q}$ with the homology class of $0$–chain which evaluates to $p$ on each vertex of the tiling $S_q^1$ of intervals of length $q$. The group $H_1({\mathfrak}{S}_\infty^1)$ is freely generated by a fundamental class for $\mathbb{R}^1$. So for this example: $$\begin{gathered}
\check{H}^0(\Omega({\mathfrak}{S}_\infty^1)) \cong H^0({\mathfrak}{S}_\infty^1) \cong H_1({\mathfrak}{S}_\infty^1) \cong \mathbb{Z}\\
\check{H}^1(\Omega({\mathfrak}{S}_\infty^1)) \cong H^1({\mathfrak}{S}_\infty^1) \cong H_0({\mathfrak}{S}_\infty^1) \cong \mathbb{Q}\end{gathered}$$ The tiling space $\Omega({\mathfrak}{S}_\infty^1)$ is the inverse limit of circles $S^1$ over the directed set $(\mathbb{N},\divides)$, where for $m \divides n$ the map $\pi_{m,n} \co S^1 \to S^1$ is given by the degree $(n/m)$ covering map. Note that the sequence $n_i \coloneqq i!$ is linearly ordered and cofinal in $(\mathbb{N},\divides)$, so: $$\Omega({\mathfrak}{S}_\infty^1) \cong \varprojlim(S^1 \xleftarrow{\times 2} S^1 \xleftarrow{\times 3} S^1 \xleftarrow{\times 4} S^1 \xleftarrow{\times 5} \ldots)$$ Restricting the constructions above to the sequence $n_i \coloneqq 2^i$, we realise Example \[ex: dyadic\]. In this case, the tiling space is homeomorphic to $$\mathbb{D}_2^1 \coloneqq \varprojlim(S^1 \xleftarrow{\times 2} S^1 \xleftarrow{\times 2} S^1 \xleftarrow{\times 2} \ldots),$$ the dyadic solenoid, and a Borel–Moore chain of $\mathbb{R}^1$ is PE in this setup if and only if it is periodic with some period $2^k$ for $k \in \mathbb{N}$.
### Bowers and Stephenson’s “Regular” Pentagonal Tilings {#Pentagonal}
Our final example will be a non-Euclidean one. Bowers and Stephenson introduced in [@BowSte97] a combinatorial substitution of a pentagons which, analogously to the Euclidean case, through repeated iteration produces larger and larger patches of tiles. These finite patches may be used to define combinatorial tilings with a certain hierarchical structure. Of course, there is no tiling of Euclidean space by regular pentagons. However, one may define a supporting metric space for which each $2$–cell is isometric to a regular Euclidean pentagon. The resulting tilings are supported on spaces conformally equivalent to the complex plane.
Despite not being tilings of Euclidean space, our methods here are essentially blind to the distinction, and will be as applicable to this example as to any Euclidean substitution tiling. A Bowers–Stephenson pentagonal tiling defines a CW poset ${\mathscr}{T}$, and for each such tiling there is a uniquely defined ‘supertiling’ which subdivides to it; the substitution is recognisable.
There is no natural notion of translation on the supporting spaces of these tilings, although there is of orientation, so we shall define ${\mathfrak}{T}$ by comparing patches of tiles using cellular isomorphisms which preserve the orientations of the $2$–cells. There exist self-similarities of star-patches which are non-trivial on the source cells i.e., $2$–cells have $5$–fold rotational symmetry and the $1$–cells have local $2$–fold symmetry. So to compute homology using the original CW decomposition of the tiling, the method will only work in general over divisible coefficients. To compute over $\mathbb{R}$ coefficients, we note that there are two equivalence classes of $0$–cells (associated to valence $3$ and valence $4$ vertices), there is one equivalence class of $1$–cell and one equivalence class of $2$–cell. However, the $1$–cells possess local symmetries reversing orientations on the $1$–cells, so the approximant complexes over $\mathbb{R}$ coefficients are $$0 \leftarrow \mathbb{R}^2 \xleftarrow{\partial_1} 0 \xleftarrow{\partial_2} \mathbb{R} \leftarrow 0.$$ It follows that the approximant homologies over $\mathbb{R}$ coefficients are $H_0A_i({\mathfrak}{T};\mathbb{R}) \cong \mathbb{R}^2$, $0$, $\mathbb{R}$ for $i=0$, $1$, $2$, respectively. We have that $H_2({\mathfrak}{T};\mathbb{R}) \cong \mathbb{R}$ is generated over $\mathbb{R}$ by a fundamental class, and it follows from the approximant homologies in degree one being trivial that $H_1({\mathfrak}{T};\mathbb{R}) \cong 0$. Using the procedure outlined in §\[subsec: method\] one finds the connecting maps in degree zero to be isomorphisms, so $H_0({\mathfrak}{T};\mathbb{R}) \cong \mathbb{R}^2$.
To compute homology over integral coefficients, we pass to a barycentric subdivision of the setup. Now we have four vertex types: two of them corresponding to the two vertex types of the original CW decomposition ${\mathscr}{T}$, one corresponding the the barycentre of each edge and one corresponding the the barycentre of each pentagon. There are three edge types and two face types. So the approximant chain complexes over $\mathbb{Z}$ coefficients are $$0 \leftarrow \mathbb{Z}^4 \xleftarrow{\partial_1} \mathbb{Z}^3 \xleftarrow{\partial_2} \mathbb{Z}^2 \leftarrow 0.$$ One computes that $H_0A_i \cong \mathbb{Z}^2$, $H_1A_i \cong 0$ and $H_2A_i \cong \mathbb{Z}$. So $H_1({\mathfrak}{T}_\Delta) \cong 0$, and of course $H_2({\mathfrak}{T}_\Delta) \cong \mathbb{Z}$ is generated by a fundamental class. One may calculate the connecting map in degree zero as having eigenvectors which span $\mathbb{Z}^2$ and have eigenvalues $1$ and $6$, and so $H_0({\mathfrak}{T}_\Delta) \cong \mathbb{Z} \oplus \mathbb{Z}[1/6]$.
To compute the PE cohomology of ${\mathfrak}{T}_\Delta$ (and hence the Čech cohomology of the associated tiling space), one may use the modified PE chain complexes and implement Poincaré duality. At the approximant stage, this amounts to using instead the chain complexes $$0 \leftarrow 2\mathbb{Z} \oplus 3\mathbb{Z} \oplus 4\mathbb{Z} \oplus 5\mathbb{Z} \xleftarrow{\partial_1} \mathbb{Z}^3 \xleftarrow{\partial_2} \mathbb{Z}^2 \leftarrow 0$$ since the vertices of ${\mathscr}{T}_\Delta$ possess local isotropy of orders $2$, $3$, $4$ and $5$. After computing the connecting maps and corresponding direct limits, we obtain: $$\begin{gathered}
\check{H}^0(\Omega({\mathfrak}{T}_\Delta)) \cong H^0({\mathfrak}{T}_\Delta) \cong H_2^\dagger({\mathfrak}{T}_\Delta) \cong \mathbb{Z}\\
\check{H}^1(\Omega({\mathfrak}{T}_\Delta)) \cong H^1({\mathfrak}{T}_\Delta) \cong H_1^\dagger({\mathfrak}{T}_\Delta) \cong 0\\
\check{H}^2(\Omega({\mathfrak}{T}_\Delta)) \cong H^2({\mathfrak}{T}_\Delta) \cong H_0^\dagger({\mathfrak}{T}_\Delta) \cong \mathbb{Z} \oplus \mathbb{Z}[1/6]\end{gathered}$$ The space $\Omega({\mathfrak}{T}_\Delta)$ corresponds to the moduli space of pointed regular pentagonal tilings taken modulo rotational symmetries, the analogue of the space $\Omega^0 = \Omega^{\text{rot}}/\text{SO}(d)$ (see [@BDHS10]) associated to a Euclidean tiling.
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[^1]: But also, at least under some mild restrictions on the CW complex, as ‘locally finite chains’ or ‘infinite chains’, and the corresponding homology may go by ‘homology with closed/compact supports’.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In order to induce a family of mixing patterns of leptons which accommodate the experimental data with a simple mathematical construct, we construct a novel object from the hybrid of two elements of a finite group with a parameter $\theta$. This construct is an element of a mathematical structure called group-algebra. It could reduce to a generator of a cyclic group if $\theta/2\pi$ is a rational number. We discuss a specific example on the base of the group $S_{4}$. This example shows that infinite cyclic groups could give the viable mixing patterns for Dirac neutrinos.'
author:
- 'Shu-jun Rong'
title: A novel mathematical construct for the family of leptonic mixing patterns
---
Introduction
============
How to explain the mixing pattern of leptons is a challenging question in neutrino physics. One of the most popular strategies is resorting to a discrete flavor group [@1; @2; @3; @4; @5; @6; @7; @8; @9]. A special group $G_{f}$ is chosen to design the Lagrangian of the general theory. Then after the spontaneous symmetry breaking of $G_{f}$, the residual symmetries in the leptonic sector could determine the mixing pattern of leptons [@10]. However, the mixing pattern accommodating the experimental data is not unique. There are a set of patterns which could not be discriminated by experiments. Furthermore, considering the results of the scan of finite groups, the order of the flavor group which gives a viable pattern is large [@11; @12; @13]. Accordingly, the dynamic model based on these groups is complex in techniques. So we confront an important question whether there is a simple mathematical construct to induce a family of mixing patterns which satisfy the experimental constraints. In this paper, we introduce a mathematical object inspired by the idea in quantum mechanics (QM). As we know, a general state of a system in QM could be decomposed with the eigenstates of some observables . With the free coefficients of superposition, we could obtain a complete set of the states. Similarly, we speculate that the mathematical construct which gives a family of mixing patterns could be composed of simple group elements which predict special patterns. Its expression is as follow $$\label{eq:1}
X(\theta)=\cos\theta A + i\sin\theta B,$$ where A, B are elements of a small finite group $G$ in the 3-dimensional unitary representation and $\theta$ is a parameter to mark the hybrid of them, $i$ is the imaginary factor. In modern mathematics, $X(\theta)$ is an element of the group-algebra $F[G]$ constructed by introduction of addition in the group $G$. To keep that $X(\theta)$ is unitary, A, B satisfy the condition $$\label{eq:2}
AB^{+}=BA^{+},~~ A^{+}B=B^{+}A.$$ This condition is equivalent to the following one $$\label{eq:3}
A=C_{1}B,~~ A=BC_{2},~~ with~~ C^{2}_{1}=C^{2}_{2}=I,$$ where $I$ is the identity matrix.
We suppose that the neutrino sector satisfies the symmetry expressed by $X(\theta)$. So the mass matrix of neutrinos follows the relation $$X^{T}(\theta)M_{\nu}X(\theta)=M_{\nu},$$ for Majorana neutrinos or $$X^{+}(\theta)M^{+}_{\nu}M_{\nu}X(\theta)=M^{+}_{\nu}M_{\nu},$$ for Dirac neutrinos. Then employing the leptonic mixing matrix, $X(\theta)$ could be diagonalized as $$U^{+}X(\theta)U=diag(\pm1,~~ \pm1,~~ \pm1),~~ for~~ Majorana~~ neutrinos,$$ $$U^{+}X(\theta)U=diag(e^{i\alpha},~~ e^{i\beta},~~ e^{i\gamma}),~~ for~~ Dirac~~ neutrinos,$$ where $U$ is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix. Here we work in the basis where the mass matrix of charged leptons is diagonal. In other words, the symmetry in this sector is trivial which could only affect the mixing matrix through the permutation of rows and nonphysical phases. We note that $X(\theta)$ is an element of the group $Z_{2}$ in the case of Majorana neutrinos. It has been widely studied in published literature. We focus on the case of Dirac neutrinos in this paper. In the following section, we give an specific realisation of our construct.
An example
==========
Once the group elements $A$, $B$ are given, we could obtain a set of mixing patterns with the eigenvectors of the symmetry $X(\theta)$. As an illustrative example, we choose $A$, $B$ from the group $S_{4}$ generated by 3 elements which observe the following relations [@14; @15]: $$S^{2}=V^{2}=(SV)^{2}=(TV)^{2}=E,~~
T^{3}=(ST)^{3}=E, ~~(STV)^{4}=E,$$ where $E$ is the identity element. The 3-dimensional representations of the generators are expressed as [@14; @15] $$\ S=\frac{1}{3}\left(
\begin{array}{ccc}
-1 & 2 & 2 \\
2 & -1 & 2 \\
2 & 2 & -1 \\
\end{array}
\right),\ T=\left(
\begin{array}{ccc}
1 & 0 & 0 \\
0 & \omega^{2} & 0 \\
0 & 0 & \omega\\
\end{array}
\right),\ V=\mp\left(
\begin{array}{ccc}
1 &0 & 0 \\
0 & 0 & 1 \\
0 & 1 & 0 \\
\end{array}
\right),$$ where $\omega=e^{i2\pi/3}$, the sign $\mp$ denotes the representation $\mathbf{3}$ and $\mathbf{3}^{\prime}$ respectively. Considering the unitary condition in the form of Eq. (\[eq:2\]) or Eq. (\[eq:3\]), a viable realization of $A$, $B$ reads $A=TV$, $B=STV$. With the minus sign for the representation of $V$, $X(\theta)$ is expressed as $$\label{eq:10}
X(\theta)=\left(
\begin{array}{ccc}
\frac{i}{3} \sin \theta -\cos \theta & \frac{2}{3} e^{i \pi /6} \sin \theta & -\frac{2}{3}e^{-i\pi/6} \sin \theta \\
-\frac{2i}{3} \sin \theta & \frac{2}{3}e^{i \pi /6} \sin \theta & \frac{1}{3} e^{-i\pi/6}\sin \theta -e^{-i2\pi/3} \cos \theta \\
-\frac{2i}{3} \sin \theta & -\frac{1}{3} e^{i\pi/6}\sin \theta-e^{i 2\pi/3}\cos \theta & -\frac{2}{3}e^{-i \pi /6} \sin \theta \\
\end{array}
\right).$$ Note that the sign of $V$ is not important in this case. The replacement $V\rightarrow-V$ is equivalent to the transformation $X(\theta)\rightarrow-X(\theta)$. The eigenvalues and corresponding eigenvectors of $X(\theta)$ are $$\lambda_{1}=-1, ~~\lambda_{2}=-e^{-i\theta},~~ \lambda_{3}=1,$$ $$u_{1}=\frac{1}{N_{1}}\left(
\begin{array}{c}
\frac{e^{i\theta}+1}{e^{i\theta}+1-\sqrt{3}e^{-i\pi/6}} \\
\frac{e^{i\pi/3}-e^{i\theta}}{e^{-i\pi/3}-e^{i\theta}} \\
1 \\
\end{array}
\right),~~u_{2}=\frac{1}{\sqrt{3}}\left(
\begin{array}{c}
\omega\\
\omega^{2} \\
1\\
\end{array}
\right),~~u_{3}=\frac{1}{N_{3}}\left(
\begin{array}{c}
\frac{e^{i\theta}-1}{e^{i\theta}-1+\sqrt{3}e^{-i\pi/6}} \\
\frac{e^{i\pi/3}+e^{i\theta}}{e^{-i\pi/3}+e^{i\theta}} \\
1 \\
\end{array}
\right),$$ where $N_{1}$ and $N_{3}$ are normalization factors. Then the mixing matrix of leptons reads $$\label{eq:13}
U(\theta) =(u_{1}~~ u_{2}~~ u_{3})$$ up to permutations of rows or columns and nonphysical phases. We give some comments here:\
1. Although $X(\theta)$ could be transformed to the diagonal form, i.e., $diag(-1, ~e^{-i\theta},~1)$, in general, it is not equivalent to a $U(1)$ group because $U(\theta)$ depends on the parameter $\theta$. Furthermore, $X(\theta)$ could not denotes an infinite group in general cases, because neither the identity element nor the inverse of $X(\theta)$ could be written in the form of $X(\theta^{\prime})$ .\
2. If $\theta/2\pi $ equals $\pm j/k$, where natural number $i, j$ are coprime , $X(\theta)$ reduces to a generator of the cyclic group $Z_{2k}$ or $Z_{k}$ when $k$ is odd or even. Furthermore, we note that if a special value, i.e., $2(\pm j/k)\pi$ of $\theta$, could make the mixing matrix $U(\theta)$ accommodate the experimental data, infinite ones in the form $2(\pm j/k\pm j^{\prime}/k^{\prime})\pi$ could also hold when $j^{\prime}/k^{\prime}$ is small enough. In other words, if a finite group survives in the experimental constraints, so do infinite ones.\
3. Because the translation $\theta\rightarrow\theta+\pi$ is equivalent to the replacement $X(\theta)\rightarrow-X(\theta)$, so the independent range of the parameter $\theta$ is $[0, \pi]$.
Now we give the phenomenological results of $X(\theta)$. Compared with the standard parametrization of the mixing matrix, i.e., $$U=
\left(
\begin{array}{ccc}
c_{12}c_{13} & s_{12}c_{13} & s_{13}e^{-i\delta} \\
-s_{12}c_{23}-c_{12}s_{13}s_{23}e^{i\delta} & c_{12}c_{23}-s_{12}s_{13}s_{23}e^{i\delta} & c_{13}s_{23} \\
s_{12}s_{23}-c_{12}s_{13}c_{23}e^{i\delta} & -c_{12}s_{23}-s_{12}s_{13}c_{23}e^{i\delta} & c_{13}c_{23}
\end{array}
\right),$$ where $s_{ij}\equiv\sin{\theta_{ij}}$, $c_{ij}\equiv\cos{\theta_{ij}}$, $\delta$ is the Dirac CP-violating phase, we could obtain the mixing angles with the following relations: $$\sin^{2}\theta_{13}=|U_{e3}|^{2}, ~~\sin^{2}\theta_{23}=|U_{\mu3}|^{2}/(1-|U_{e3}|^{2}),~~\sin^{2}\theta_{12}=|U_{e2}|^{2}/(1-|U_{e3}|^{2}).$$ Employing the $\chi^{2}$ function defined as $$\chi^{2}=\sum_{ij=13,23,12}(\frac{\sin^{2}\theta_{ij}-(\sin^{2}\theta_{ij})_{exp}}{\sigma_{ij}})^{2},$$ where $(\sin^{2}\theta_{ij})_{exp}$ are best global fit values from Ref. [@16], $\sigma_{ij}$ are 1$\sigma$ errors, the best fit data of the parameter $\theta$ and mixing angles are shown in Table \[tab:1\].
[|c|c|c|c|c|c|]{} $Ordering$ & $\chi^{2}_{min}~~~~~$ & $\theta_{bf1}$ & $(\sin^{2}\theta_{13})_{bf}$ & $(\sin^{2}\theta_{23})_{bf}$ & $(\sin^{2}\theta_{12})_{bf}$ \
Normal & 11.9396 & 0.11496$\pi$ & 0.021504 & 0.39575 & 0.34066 \
Inverted & 9.0379 & 0.11546$\pi$ & 0.02169 & 0.6047 & 0.34072 \
Note that because of the freedom of permutations of rows or columns of the mixing matrix in Eq. (\[eq:13\]), the best fit value of $\theta$ in the range \[0, $\pi$\] is not unique. The complete set of the best fit values for either mass ordering is listed as $$\begin{array}{c}
\theta_{bf1},~\theta_{bf2}=\pi/3-\theta_{bf1}, ~\theta_{bf3}=\pi/3+\theta_{bf1}, \\
\theta_{bf4}=2\pi/3-\theta_{bf1}, ~\theta_{bf5}=2\pi/3+\theta_{bf1}, \theta_{bf6}=\pi-\theta_{bf1}.
\end{array}$$ In the case of normal mass ordering, these best fit data correspond to the following mixing matrix respectively: $$U_{1}=S_{23}U,~U_{2}=S_{23}S_{12}US_{13},~U_{3}=S_{12}US_{13},~U_{4}=S_{13}S_{12}U,~U_{5}=S_{13}U,~U_{6}=US_{13},$$ where $U$ is written as Eq (\[eq:13\]), the permutation matrices $S_{ij}$ are expressed as $$S_{23}=\left(
\begin{array}{ccc}
1 & 0 & 0 \\
0 & 0 & 1 \\
0 & 1 & 0 \\
\end{array}
\right), ~~S_{13}=\left(
\begin{array}{ccc}
0 & 0 & 1 \\
0 & 1 & 0 \\
1 & 0 & 0 \\
\end{array}
\right),~~S_{12}=\left(
\begin{array}{ccc}
0 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 1 \\
\end{array}
\right).$$ The mixing matrices in the case of inverted mass ordering are $U^{\prime}_{j}=S_{23}U_{j}$, with $j=1,~2,~3,~4,~5,~6$. Either in the case of normal or inverted mass ordering, the six best fit values of $\theta$ correspond to the same $\chi^{2}_{min}$ and the same mixing angles. This observation could seen from the function $\chi^{2}$ shown in Figure \[fig:1\]. And some comments are given as follows:\
1. Begin with the curve of the function $\chi^{2}$ of $U_{1}$ or $U^{\prime}_{1}$, other curves of $\chi^{2}$ could be obtained by the space inversion with the axis $\theta=k\pi/6$ step by step, with $k=1,~2,~3,~4,~5.$ Accordingly, the range of the parameter $\theta$ at the level of $3\sigma$ of any mixing matrix could be derived from that of a special mixing matrix. For example, the range of $\theta$ at $3\sigma$ level for $U_{3}$ and $U^{\prime}_{3}$ is \[$0.4422\pi, ~0.4545\pi$\], \[$0.4422\pi, ~0.4546\pi$\] respectively. So that for $U_{2}$ and $U^{\prime}_{2}$ is \[$(2/3-0.4545)\pi, ~(2/3-0.4522)\pi$\], \[$(2/3-0.4546)\pi, ~(2/3-0.4522)\pi$\] respectively. The range of $\theta$ at $3\sigma$ level for any mixing matrix is narrow, which results from the stringent experimental constraint of $\sin\theta_{13}$. This observation could seen from the the correlation of two mixing angels such as those of $U_{3}$. See Figure \[fig:2\] for example, where the mixing angle $\theta_{13}$ covers its full range at $3\sigma$ level, while $\theta_{23}$ or $\theta_{12}$ covers only several percents of the measure of its $3\sigma$ range.\
2. As the parameter $\theta$ varies in the range at $3\sigma$ level, we could obtain a family of mixing patterns. Specially, the values which could be expressed as $2(j/k)\pi$ are in the range of $3\sigma$ level for any mixing matrix. For example, $4\pi/9$ is in the range of $\theta$ for the matrix $U_{3}$ or $U^{\prime}_{3}$. $X(\theta=4\pi/9)$ is a generator of the group $Z_{18}$. We note that $Z_{18}$ have been obtained as a residual symmetry in the neutrino sector by the scan of thousands of finite groups in the literature [@12]. According to the aforementioned observation, there are infinite cyclic groups $Z_{n}$ which could accommodate the experimental data. Furthermore, as an interesting phenomenon, the value $\theta=\pi/3$ which is a axis of the reflection of two curves of $\chi^{2}$ corresponds to the well known tri-bi-maximal (TBM) mixing pattern [@17; @18], i.e., $U(\theta=\pi/3)=U_{TBM}$.\
3. The Dirac CP phase could obtained through the Jarlskog invariant expressed as [@19] $$J_{cp}\equiv\mathrm{Im}[U_{22}U^{*}_{23}U^{*}_{32}U_{33}]=\frac{1}{8}\sin2\theta_{13}\sin2\theta_{23}\sin2\theta_{12}\cos\theta_{13}\sin\delta.$$ Substitute $J_{CP}$ with the elements of $U(\theta)$, we find it equals 0 independent of the parameter $\theta$. Hence the Dirac CP phase is trivial, i.e., $\sin\delta=0$.
\[fig:1\] ![\[fig:1\] The function $\chi^{2}$ for the viable mixing matrices. The left panel is for the case of normal mass ordering and the right one is for the inverted case. The black dashed curve is the function $\chi^{2}$ of $U_{1}(U^{\prime}_{1})$, the blue curve is for $\chi^{2}$ of $U_{2}(U^{\prime}_{2})$, the blue dashed curve is for $\chi^{2}$ of $U_{3}(U^{\prime}_{3})$, the red dashed curve is for $\chi^{2}$ of $U_{4}(U^{\prime}_{4})$, the red curve is for $\chi^{2}$ of $U_{5}(U^{\prime}_{5})$, the black curve is for $\chi^{2}$ of $U_{6}(U^{\prime}_{6})$.](1.pdf "fig:"){width=".48\textwidth"} ![\[fig:1\] The function $\chi^{2}$ for the viable mixing matrices. The left panel is for the case of normal mass ordering and the right one is for the inverted case. The black dashed curve is the function $\chi^{2}$ of $U_{1}(U^{\prime}_{1})$, the blue curve is for $\chi^{2}$ of $U_{2}(U^{\prime}_{2})$, the blue dashed curve is for $\chi^{2}$ of $U_{3}(U^{\prime}_{3})$, the red dashed curve is for $\chi^{2}$ of $U_{4}(U^{\prime}_{4})$, the red curve is for $\chi^{2}$ of $U_{5}(U^{\prime}_{5})$, the black curve is for $\chi^{2}$ of $U_{6}(U^{\prime}_{6})$.](2.pdf "fig:"){width=".48\textwidth"}
\[fig:2\] ![\[fig:2\] Correlations of the mixing angles of the mixing matrix $U_{3}$.](3.pdf "fig:"){width=".48\textwidth"} ![\[fig:2\] Correlations of the mixing angles of the mixing matrix $U_{3}$.](4.pdf "fig:"){width=".48\textwidth"}
Conclusion
==========
Inspired by the ideal in QM, as a generalization of application of finite groups, we construct a mathematical structure called group-algebra to give a family of mixing patterns of leptons. This construct with a parameter $\theta$ could reduce to a generator of a cyclic group when $\theta/2\pi$ is a rational number. A specific example made from the group $S_{4}$ is given, which shows that a set of mixing patterns for Dirac neutrinos could accommodate the experimental data at the $3\sigma$ level. And infinite cyclic groups could work as the symmetry in the neutrino sector, which supplements the observation where several ones are found by the scan of finite groups up to order 2000 in the literature. Construct of a dynamical model on the base of this mathematical structure is intriguing, which will be considered in our future work.
This work was supported by the National Natural Science Foundation of China under the Grant No. 11405101 and the research foundation of Shaanxi University of Technology under the Grant No. SLGQD-13-10. The author declares that there is no conflict of interest regarding the publication of this paper.
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|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- Kohei Shinohara
- Atsuto Seko
- Takashi Horiyama
- Masakazu Ishihata
- Junya Honda
- Isao Tanaka
bibliography:
- 'references.bib'
title: Derivative structure enumeration using binary decision diagram
---
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: '[Isotropic gases irradiated by long pulses of intense IR light can generate very high harmonics of the incident field. It is generally accepted that, due to the symmetry of the generating medium, be it an atomic or an isotropic molecular gas, only odd harmonics of the driving field can be produced. [Here we show how the interplay of electronic and nuclear dynamics can lead to a marked breakdown of this standard picture: a substantial part of the harmonic spectrum can consist of *even* rather than odd harmonics. We demonstrate the effect using ab-initio solutions of the time-dependent Schrödinger equation for H$_{2}^{+}$ and its isotopes in full dimensionality. By means of a simple analytical model, we identify its physical origin, which is the appearance of a permanent dipole moment in dissociating homonuclear molecules, caused by light-induced localization of the electric charge during dissociation. The effect arises for sufficiently long laser pulses and the region of the spectrum where even harmonics are produced is controlled by pulse duration.]{} Our results (i) show how the interplay of femtosecond nuclear and attosecond electronic dynamics, which affects the charge flow inside the dissociating molecule, is reflected in the nonlinear response, and (ii) force one to augment standard selection rules found in nonlinear optics textbooks by considering light-induced modifications of the medium during the generation process.]{}'
author:
- 'R. E. F. Silva'
- 'P. Rivière'
- 'F. Morales'
- 'O. Smirnova'
- 'M. Ivanov'
- 'F. Martín'
title: Even harmonic generation in isotropic media of dissociating homonuclear molecules
---
Introduction\[sec:Introduction-1\] {#introductionsecintroduction-1 .unnumbered}
==================================
[Attosecond technology originated in nonlinear optics, with high harmonic generation (HHG) being the fundamental physical process underlying the generation of attosecond pulses [@RevModPhys.81.163; @scrinzi2006attosecond]. In the two decades since its inception, attosecond science has grown rapidly [@murmane_atto_1; @sansone2006isolated; @murmane_atto_2] with]{} applications in physics, chemistry [@lepine2014attosecond; @Nature_jonh; @smirnova2009high], materials science [@cavalieri2007attosecond; @schiffrin2013optical; @ghimire2011observation] and even biology [@calegari2014ultrafast; @kraus2015measurement]. High harmonic emission results from nonlinear response of a medium to an intense laser field. Its basic mechanism was first described in [@HHG_CORKUM; @KULANDER_HHG; @HHG_QUANTUM] (see also [@kuchiev1987atomic]). After the intense laser field frees an electron from the ionic core, the electron gains energy from the field and revisits the parent ion. Radiative recombination converts the gained energy into high-frequency radiation.
In isotropic atomic gases irradiated by long IR pulses, [the electron round-trips between ionization and recombination are launched during successive laser half-cycles. Mirror symmetry of the driving electric field implies that these round trips are mirror images of each other. Electrons revisiting the parent ion from opposite directions yield emission bursts]{} with the same amplitude but opposite signs. As a consequence, even-order harmonics interfere destructively and vanish, while the odd-order harmonics interfere constructively [@dahlstrom2011quantum], [leading to the spectral peaks at odd multiples of the fundamental frequency, $\Omega_{n}=(2n+1)\hbar\omega_{{\rm IR}}$.]{}
A similar behavior is commonly expected for any isotropic medium, such as an isotropic distribution of homonuclear diatomic molecules. However, the physical picture underlying high harmonic generation suggests that the expectation of odd-only harmonics requires that the inversion symmetry holds during the whole interaction. This is indeed the case if the molecular nuclei do not, or barely, move. However, as predicted in recent theoretical work on H$_{2}^{+}$ [@lein2005attosecond; @morales2014high; @riviere2014time; @bian2014probing], mostly using low dimensionality models [@lein2005attosecond; @morales2014high; @riviere2014time], new features may arise in the high harmonic spectrum if the nuclei move significantly during [one or several laser half cycles.]{} These include the reduction of the maximum (cutoff) energy in the harmonic spectrum [@lein2005attosecond], a modest red shift of the harmonic peak positions [@bian2014probing], suppression of specific odd harmonics [@riviere2014time] and, as found more recently, the appearance of weak even harmonics between the strong odd harmonic peaks [@morales2014high]. Common wisdom suggests that these features should disappear for long driving laser pulses, restoring the expected symmetry of the overall process and balancing contributions from the adjacent laser half-cycles.
{width="2\columnwidth"}
[However, results of our study of high harmonic generation from H$_{2}^{+}$, D$_{2}^{+}$ and T$_{2}^{+}$, using a full dimensionality model for the electronic and vibrational degrees of freedom, contrast with this expectation even for rather long laser pulses. We find that the nuclear motion has an even more dramatic effect than anticipated in the previous work. For sufficiently long pulses, the HHG spectrum of the lighter molecules can exclusively consist of *even* harmonics in the plateau region. Furthermore, by changing the pulse duration, one can control the region of the plateau where the even harmonics appear. We unambiguously link the appearance of even harmonics to electron localization on one side of the molecule. This induces a permanent dipole during dissociation, even for pulses containing tens of laser cycles. The linear Stark effect associated with the permanent dipole introduces a relative phase between the two consecutive electron round trips initiated in the adjacent laser half-cycles. As this phase approaches $\pi$, the odd harmonics are replaced by the even ones. ]{}
HHG spectra\[sec:HHG-spectra\] {#hhg-spectrasechhg-spectra .unnumbered}
==============================
Fig. \[fig:HHG3D\_5\_cycles\] shows the calculated HHG spectra for the three different isotopes of the ${\mbox{H$_{2}^{+}$}}$ molecule and different pulse durations. For the shortest pulse, 5 optical cycles, our results are [very close to our earlier results for low dimensionality models [@morales2014high; @riviere2014time].]{} [The spectra from different isotopes are generally very similar, with several broad even harmonic lines between orders 36-40. The effect is independent of the isotope and hence of the nuclear motion; it is associated with the frequency chirp induced by the changing laser intensity between successive half-cycles in an ultra-short pulse. As expected, the effect disappears for the longer, 10-cycle pulse, and for the heaviest isotope, where the nuclear motion is negligible. However, for the lightest isotope even harmonics remain very prominent for 10, 14, and 20-cycle pulses. As one moves to higher orders, the harmonic peaks in ${\mbox{H$_{2}^{+}$}}$ experience a red shift, up to a point that only even harmonics are observed for high enough orders. The spectral region where even harmonics dominate shifts to higher orders with increasing pulse duration. For the 20-cycle pulse even harmonics also appear for the D$_{2}^{+}$ isotope. Additionally, for the longer pulses, the harmonics from D$_{2}^{+}$ and T$_{2}^{+}$ are strongly suppressed in the plateau region compared to H$_{2}^{+}$, with rather dramatic modifications of the shape of the harmonic lines. What could be the origin of even harmonics in H$_{2}^{+}$, their shift with the pulse duration, and the emergence of two spectral regions where they are seen for 20-cycle pulse (the plateau and the cutoff)? What could be the origin of the combination of harmonic suppression and the dramatic harmonic line shape modifications for T$_{2}^{+}$?]{}
[Two effects may be responsible. The first is electron localization, prominent in molecular dissociation [@kling2006control; @Nature_jonh]. Electron localization breaks the symmetry of the system and hence can lead to even harmonics, [@morales2014high; @riviere2014time; @smirnova2007anatomy]. The second is the asymmetry introduced by molecular dissociation between the raising and descending parts of the IR pulse [@bian2014probing], in particular due to the shift in the characteristic ionization potential $I_{p}$. As pointed out in Ref. [@bian2014probing], this asymmetry leads to the red shift. The analysis of the harmonic spectra for different pulse durations and isotopes allows us to distinguish the contributions of these two effects. We argue that both are important but have quite different impact in multi-cycle pulses. While electron localization controls the interference of the emission bursts from successive laser half-cycles, the front-back asymmetry pertains to the longer time-scale and hence leads to finer-scale modifications in the harmonic spectrum, for long pulses.]{}
[We first analyze the impact of dissociation-induced asymmetry between the raising and the falling edges of the pulse. As discussed above, the harmonic spectrum is formed by the interference of emission bursts produced during successive half-cycles. If the system response during several successive half-cycles is nearly identical, odd harmonics will form during the raising part of the pulse. The same would apply at the falling edge of the pulse, even if the system has changed due to dissociation. The interference of odd harmonics generated at the front and at the back of the pulse will then lead to modification of harmonic lineshapes, within their width. However, it will not turn odd harmonics into even. If the contributions from the raising and the falling parts of the pulse are phase-shifted by $\pi$ in some spectral window, strong reshaping and suppression in this spectral window may result. This is precisely what we find for the heaviest isotope, T$_{2}^{+}$, for a 20-cycle pulse, between harmonics $33-45$: overall suppression, strong modification of line-shapes, but prominent odd harmonics nevertheless. To shift the harmonic lines on the scale of $\hbar\omega_{{\rm IR}}$, where $\omega_{{\rm IR}}$ is the laser frequency, one has to modify the interference between successive half-cycles. We now show that the generation of a permanent dipole moment due to electron localization is the origin of such modification and the appearance of even harmonics.]{}
Electron localization\[sec:Electron-localization\] {#electron-localizationsecelectron-localization .unnumbered}
==================================================
![\[fig:HHG3D\_Process\]Sketch of the dynamics leading to electron localization. Full curves: potential energy curves of H$_{2}^{+}$. Dashed lines: initial nuclear wave packet. Vertical lines indicate transitions between molecular states. The meaning of processes 1, 2 and 3 is explained in the text.](fig3){width="0.9\columnwidth"}
The dissociative dynamics leading to eventual electron localization is well understood [@kling2006control]. [A sketch of this dynamics is shown in Fig. \[fig:HHG3D\_Process\]. During the first few cycles, the IR field, which is not yet intense enough to significantly ionize the molecule, induces Rabi-type oscillations between the $1s\sigma_{g}$ and $2p\sigma_{u}$ states, which lie much closer in energy than the $1s\sigma_{g}$ state and the ionization continuum (process 1). As a result, excited vibrational states associated with the $1s\sigma_{g}$ electronic state can be efficiently populated [@PRL_SILVA_2013], thus creating a vibrational wave packet. A similar vibrational wave packet is formed in the $2p\sigma_{u}$ state. These wave packets then move towards larger internuclear distances (process 2), until they reach a region of internuclear distances where the $1s\sigma_{g}$ and $2p\sigma_{u}$ electronic states are very close in energy and are strongly coupled by the IR field. Therefore they mix, leading to localized states $1s\sigma_{g}\pm2p\sigma_{u}$. The characteristic value of the internuclear distance $R=R_{C}$ where the onset of localization occurs can be estimated by using the criterion from [@ivanov1993coherent], $R_{C}E_{0}\omega_{{\rm IR}}\simeq\omega_{gu}^{2}(R_{C})$, where $E_{0}$ is the field amplitude and $\omega_{gu}$ the energy difference between the $1s\sigma_{g}$ and $2p\sigma_{u}$ states at $R=R_{C}$. This criterion yields $R_{C}\simeq3.7$ a.u., shown in Fig.\[fig:HHG3D\_Process\] with a pink vertical line. Localization opens the barrier for dissociation, through the process called bond softening [@bucksbaum1990softening]. By then, the IR field has reached (or nearly reached) its peak intensity and tunnel ionization becomes prominent (process 3). This enhancement of tunnel ionization in the long $R$ region is caused by both the reduced ionization potential (compared to the equilibrium geometry), which decreases with $R$, and the enhanced ionization caused by electron localization [@zuo1995charge; @seideman1995role]. It is important to stress that localization of the electron on one or the other side of the molecule depends on the phase of the laser field [@kelkensberg2011semi] and, therefore, can be controlled by the carrier-envelope phase [@kling2006control]. Thus, the ulterior recombination process, i.e., harmonic generation, is acutely sensitive to the window of $R$ where the medium symmetry is broken.]{}
![\[fig:HHG3D\_Rdependent\]$R$-dependent HHG spectrum obtained from equation (\[R-dep\]) for a pulse of $800$ nm, $I=3\times10^{14}$W/cm$^{2}$ and 14 optical cycles for ${\mbox{H$_{2}^{+}$}}$. The horizontal lines indicate the position of odd harmonics.](fig4){height="0.8\columnwidth"}
{width="0.7\columnwidth"}{width="0.7\columnwidth"}
To quantify this picture and confirm our explanation, we have evaluated an $R$-dependent HHG spectrum, $\ddot{d}(R,t)$, as the Fourier transform defined as $$\ddot{d}(R,t)=\intop\intop\psi^{*}\left(R,z,\rho,t\right){\cal O}(z)\psi\left(R,z,\rho,d\right)\rho d\rho dz,\label{R-dep}$$ [where $\psi$ is the solution of the time-dependent Schrödinger equation (TDSE), $\rho$ and $z$ are cylindrical coordinates describing the position of the electron (see Methods), $R$ is the internuclear distance, and ${\cal O}$ is the dipole acceleration operator.]{} The calculated $R$-dependent HHG spectrum is shown in Fig. \[fig:HHG3D\_Rdependent\] for the 14-cycles pulse. [As can be seen, even harmonics appear near the localization distance $R\gtrsim R_{C}$ a.u., far beyond the equilibrium distance $R_{eq}$=1.9 a.u., in agreement with our arguments. This also explains the lack of even harmonics for heavier isotopes, which dissociate slower and do not reach the localization region before the pulse is over.]{}
Fig. \[fig:HHG3D\_GABOR\] shows the time-windowed Fourier transforms (Gabor profiles) and the time evolution of the nuclear wave packets in ${\mbox{H$_{2}^{+}$}}$ for 10 and 20 cycle pulses. For each pulse duration we select the time, $t_{C}$, at which the density around the critical internuclear distance, $R_{C}$=3.7 a.u., is largest. By looking at the Gabor profile at $t>t_{C}$ and at the harmonics that are emitted at $t_{C}$, we can predict the location of the even harmonics in the HHG spectra. For the 10-cycles pulse, $R_{C}$ is reached when the intensity of the laser pulse is already decreasing. This leads to even harmonics in the lower region of the HHG spectrum. In contrast, for the 20-cycles pulse, $R_{C}$ is already reached when the intensity of the laser pulse is still increasing. Consequently, even harmonics appear at higher energies in the spectrum. [Note that even harmonics arise for sufficiently long pulses, when the nuclear wave packet has had enough time to reach $R_{C}$. By controlling the pulse duration one thus controls the region of the plateau where even harmonics appear.]{}
![\[fig:traj\]Sketch of the trajectories followed by the electron in two consecutive half cycles when it is initially delocalized (a) and localized (b).](fig6){height="0.7\columnwidth"}
[This physical picture is further substantiated by analyzing the phase-shift in the emission bursts during successive half-cycles, accumulated due to the induced dipole moment. Fig. \[fig:traj\] shows a sketch of the typical trajectory followed by an electron starting in a delocalized and localized initial bound state. In the second case, the localized bound state experiences a linear Stark shift, leading to additional phase difference accumulated between the ionization $t_{i}$ and recombination $t_{r}$ times, $$S_{t_{i}}^{R}=-\int_{t_{i}}^{t_{r}}\frac{R}{2}E\left(t\right)dt.$$ For trajectories generated in consecutive half-cycles, the accumulated phase difference is twice as large, $$\begin{aligned}
\Delta S_{t_{i}} & = & S_{t_{i}}^{L}-S_{t_{i}}^{R}=\int_{t_{i}}^{t_{r}}RE\left(t\right)dt=Rv(t_{r})\end{aligned}$$ In the last equality we have used the three-step model of HHG: for the electron starting with zero velocity at $t_{i}$, the integral of the laser field between $t_{i}$ and $t_{r}$ gives its instantaneous electron velocity $v(t_{r})$ at the return time $t_{r}$. Thus, we can rewrite this phase difference in terms of the electron recombination energy $E_{{\rm kin}}=v^{2}(t_{r})/2$, or the emitted photon energy, $N\omega_{{\rm IR}}$, $$\Delta S_{t_{i}}=R\sqrt{2E_{{\rm kin}}}=R\sqrt{2[N\omega_{{\rm IR}}-I_{p}(R)]}.\label{St}$$ Fig. \[fig:phase\] shows this phase difference as a function of the harmonic order and the internuclear distance. We have checked, by solving the classical equations of motion numerically, that the inclusion of the molecular potential adds an extra phase, which is however much smaller than that shown in Fig. \[fig:phase\], so that the global picture remains unchanged]{}. In the vicinity of $R_{C}\simeq3.7$ a.u., where localization takes place, there is a large range of harmonic orders where the additional phase difference between the right and the left trajectories is $\approx\pi$, thus leading to even harmonic generation. [According to Fig. \[fig:phase\], this occurs approximately between harmonic orders 20 and 40. Remarkably, Fig. \[fig:phase\] predicts a phase difference of $\approx3\pi$, hence a revival of even harmonic emission, between harmonic orders 60 and 80, in excellent agreement with the appearance of even harmonics in the cut-off region predicted by the ab initio calculations for H$_{2}^{+}$ (see Fig. \[fig:HHG3D\_5\_cycles\]c,d). Finally, Fig. \[fig:phase\] also shows that the positions of the harmonic lines experience a slow frequency shift across the spectrum as $R_{C}$ is varied, again in agreement with our ab-initio observations.]{}
Conclusion\[sec:Conclusion\] {#conclusionsecconclusion .unnumbered}
============================
Using the example of one-electron homonuclear diatomic molecules, we have shown how dynamics induced in the molecule can lead to a dramatic breakdown of the standard selection rules in high harmonic generation, including the nearly complete suppression of odd and the appearance of even harmonics for multi-cycle laser pulses, in a broad window of the harmonic spectrum. Our analysis links strong shifts of the harmonic lines with the appearance of a permanent dipole moment in dissociating homonuclear molecules, caused by electron localization. This dipole moment introduces phase shifts between the emission bursts during successive laser half-cycles, which can approach and exceed $\pi$. The ultimate origin of symmetry breaking is the sensitivity of the overall process to the carrier envelope phase of the laser pulse. The fact that minute changes in the driving laser field, associated with the carrier-envelope phase of a multi-cycle (20-cycle) laser pulse, can lead to strong effects in the harmonic spectrum, reflects very strong sensitivity of the underlying dissociation – localization dynamics to the details of the driving field. In classical systems, such extreme sensitivity is characteristic of dynamical chaos. Thus, our results suggest that high harmonic generation might also be a sensitive probe for the onset of dynamical chaos in light-driven systems.
![\[fig:phase\]Additional phase difference between two consecutive half cycles for an electron initially localized on one side of the molecule (in units of $\pi$). The region between the vertical dashed lines indicated where localization actually takes place.](fig7){width="1\columnwidth"}
Methods\[sec:Theoretical-Method\] {#methodssectheoretical-method .unnumbered}
=================================
Our theoretical method has been described in detail in [@Thomas_H2_plus]. Briefly, we solve the three-dimensional (3D) time-dependent Schrödinger equation (TDSE) in cylindrical coordinates, $\rho$ and $z$ for the electron, $R$ for the internuclear distance, where $z$ coincides with the linear polarization direction of the electric field. [We assume that the molecules are aligned with the linearly polarized driving IR field, neglecting their rotations. The electron azimuthal coordinate $\phi$ is removed by the cylindrical symmetry of the problem. ]{} The TDSE reads $$i\frac{\partial\psi{(\rho,z,R,t)}}{\partial t}=[{{H}_{el}(\rho,z,R)+{T}(R)}+{V}(z,t)]\psi{(\rho,z,R,t)},$$ where ${H}_{el}={T}_{el}+{V}_{eN}+1/R$ is the electronic Hamiltonian of H$_{2}^{+}$ or its molecular isotopes, ${T}$ is the nuclear kinetic energy operator, ${V}={z}{E}(t)$ [describes the interaction with the laser field]{} in the length gauge, ${T}_{el}$ is the [nuclear]{} kinetic energy operator, [and the Coulomb potential ${V}_{eN}$ describes the electron – nuclei interaction. Atomic units are used throughout unless stated otherwise]{}. The external laser field is $E\left(t\right)=E_{0}f(t)\sin\left(\omega t\right)$, where $$f\left(t\right)=\begin{cases}
\cos^{2}\left(\frac{\pi t}{T}\right) & \left|t\right|\leq\frac{T}{2}\\
0 & \left|t\right|>\frac{T}{2}\ \ ,
\end{cases}$$ [$E_{0}$ the electric field amplitude, $\omega$ the central frequency corresponding to the wavelength $\lambda=800$ nm, and $T$ the total pulse duration ranging between 5 and 20 optical cycles ($13.34-53.36$ fs). The peak intensity in all calculations is $3\times10{}^{14}$ W/cm$^{2}$.]{}
We have [used a non-equidistant cubic 3D grid with]{} $\left|z\right|<55$, $\rho<50$ and $R<30$, and grid spacings $\Delta z$=0.1, $\Delta\rho$=0.075 and $\Delta R$=0.05 a.u. at the center of the grid. The grid spacings increase gradually from the origin to the box boundaries [[@Thomas_H2_plus]]{}. [The Crank-Nicolson propagator with a split-operator method was used for time-propagation, with a time step $\Delta t_{elec}$=0.011 a.u. for the electrons and $\Delta t_{nuc}$=0.11 a.u. for the nuclei.]{} The convergence of these parameters was checked. [The initial, ground, state of the ${\mbox{H$_{2}^{+}$}}$ molecule]{} was obtained by diagonalizing the unperturbed Hamiltonian [using the SLEPc routines [@SLEPc_manual]]{}. Absorbers were placed at $\left|z\right|>35$ a.u. and $\rho>30$ a.u. to avoid [artificial reflections from the boundaries]{}.
At each time step, we have calculated the time-dependent dipole $\ddot{d}(t)$ as $$\ddot{d}(t)=\left<\psi(\rho,z,R,t)|{\cal O}(z)|\psi(\rho,z,R,t)\right>_{\rho,z,R},$$ where ${\cal O}$ is the dipole acceleration operator and integration is performed over electronic ($\rho$,$z$) and nuclear ($R$) coordinates. The harmonic spectrum $|\ddot{d}(\omega)|^{2}$ is given by the square of the Fourier transform (FT) of $\ddot{d}(t)$.
We gratefully acknowledge X. B. Bian and A. D. Bandrauk for fruitful discussions and for sharing their data. This work was accomplished with an allocation of computer time from Mare Nostrum BSC and CCC-UAM and was partially supported by the European Research Council Advanced Grant No. XCHEM 290853, MINECO Project No. FIS2013-42002-R, ERA-Chemistry Project No. PIM2010EEC-00751, European Grant No. MC-ITN CORINF, European COST Action XLIC CM1204, and the CAM project NANOFRONTMAG. R.E.F.S. acknowledges FCT - Fundação para a Ciência e Tecnologia, Portugal, Grant No. SFRH/BD/84053/2012. M.I. acknowledges support of the EPSRC programme Grant No. EP/I032517/1 and the Voronezh State University.
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---
abstract: 'There is an elegant relation [@FB2] among the double tangent lines, crossings, inflections points, and cusps of a singular curve in the plane. We give a new generalization to singular curves in $\mathbb{R}P^2$. We note that the quantities in the formula are naturally dual to each other in $\mathbb{R}P^2$, and we give a new dual formula.'
address: |
Mathematics Department\
University of California\
Davis, CA 95616\
USA
author:
- Abigail Thompson
bibliography:
- 'link.bib'
title: 'Invariants of curves in $\mathbb{R}P^2$ and $\mathbb{R}^2$'
---
There is an elegant relation found by Fabricius-Bjerre \[Math. Scand 40 (1977) 20–24\] among the double tangent lines, crossings, inflections points, and cusps of a singular curve in the plane. We give a new generalization to singular curves in RP\^2. We note that the quantities in the formula are naturally dual to each other in RP\^2, and we give a new dual formula.
There is an elegant relation found by Fabricius-Bjerre \[Math. Scand 40 (1977) 20–24\] among the double tangent lines, crossings, inflections points, and cusps of a singular curve in the plane. We give a new generalization to singular curves in $\mathbb{R}P^2$. We note that the quantities in the formula are naturally dual to each other in $\mathbb{R}P^2$, and we give a new dual formula.
There is an elegant relation found by Fabricius-Bjerre \[Math. Scand 40 (1977) 20–24\] among the double tangent lines, crossings, inflections points, and cusps of a singular curve in the plane. We give a new generalization to singular curves in <b>R</b>P<sup>2</sup>. We note that the quantities in the formula are naturally dual to each other in <b>R</b>P<sup>2</sup>, and we give a new dual formula.
Introduction {#sec1}
============
Let $K$ be a smooth immersed curve in the plane. Fabricius-Bjerre [@FB1] found the following relation among the double tangent lines, crossings, and inflections points for a generic $K$: $$T_1-T_2=C+(1/2)I$$ where $T_1$ and $T_2$ are the number of exterior and interior double tangent lines of $K$, $C$ is the number of crossings, and $I$ is the number of inflection points. Here “generic” means roughly that the interesting attributes of the curve are invariant under small smooth perturbations. Fabricius-Bjerre remarks on an example due to Juel which shows that the theorem cannot be straightforwardly generalized to the projective plane. A series of papers followed. Halpern [@H] re-proved the theorem and obtained some additional formulas using analytic techniques. Banchoff [@B] proved an analogue of the theorem for piecewise linear planar curves, using deformation methods. Fabricius-Bjerre gave a variant of the theorem for curves with cusps [@FB2]. Weiner [@W] generalized the formula to closed curves lying on a 2–sphere. Finally Pignoni [@P] generalized the formula to curves in real projective space, but his formula depends, both in the statement and in the proof, on the selection of a base point for the curve. Ferrand [@F] relates the Fabricius-Bjerre and Weiner formulas to Arnold’s invariants for plane curves. Note that any formula for curves in $\mathbb{R}P^2$ is more general than one for curves in $\mathbb{R}^2$, since one can specialize to curves in $\mathbb{R}^2$ by considering curves lying inside a small disk in $\mathbb{R}P^2$.
There are two main results in this paper. The first is a generalization of the theorem in [@FB2] to $\mathbb{R}P^2$, with no reference to a basepoint on the curve. The original theorem is transparently a special case of this result, which is not surprising as the techniques used to prove it are a combination of those found in [@FB1] and in [@W]. The difficulties encountered in the generalization are due to the problems in distinguishing between two “sides” of a closed geodesic in $\mathbb{R}P^2$. These are overcome by a careful attention to the natural metric on the space inherited from the round 2–sphere of radius one.
The main results are tied together by the observation that, in the version of the original formula which includes cusps [@FB2], the quantities in the formula are naturally dual to each other in $\mathbb{R}P^2$. This leads to the second, more surprising, main result, which is a dual formula for generic curves in $\mathbb{R}P^2$. This specializes to a new formula for generic smooth curves in the plane. This new formula has the interesting property that it reveals delicate geometric distinctions between topologically similar planar curves, for example quantifying some of the differences between the two curves shown in .
\[twocurves\]
The outline of the paper is as follows: in we state and prove the generalization of [@FB2] to curves in $\mathbb{R}P^2$. In we describe the duality between terms of the formula. In we state and prove the dual formulation, and give its corollaries for planar curves.
A Fabricius-Bjerre formula for curves in $\mathbb{R}P^2$ {#sec2}
========================================================
Let $\mathbb{R}P^2$ be endowed with the spherical metric, inherited from its double cover, the round 2–sphere of radius one. With this metric, a simple closed geodesic (or projective line) in $\mathbb{R}P^2$ has length $\pi$. The figures will use a standard disk model for $\mathbb{R}P^2$, in which the boundary of the disk twice covers a closed geodesic. Let $K$ be a generic oriented closed curve in $\mathbb{R}P^2$, which is smoothly immersed except for cusps of type 1, that is, cusps at which locally the two branches of $K$ coming into the cusp are on opposite sides of the tangent geodesic. We postpone the definition of [*generic*]{} until the end of section 3. We will need some definitions.
[**Definitions**]{}
Let [*$\tau_p$*]{} be the geodesic tangent to $K$ at $p$, with orientation induced by $K$.
Let [*$a_p$*]{}, the [*antipodal point to $p$*]{}, be the point on $\tau_p$ a distance $\pi/2$ from $p$.
$\tau_p$ is divided by $p$ and $a_p$ into two pieces. Let [*${\tau_p}^+$*]{} be the segment from $p$ to $a_p$ and [ *${\tau_p}^-$*]{} the segment from $a_p$ to $p$. At cusp points ${\tau_p}^+$ and ${\tau_p}^-$ are not well-defined.
Let [*$\nu_p$*]{} be the normal geodesic to $K$ at $p$.
Let [*$c_p$*]{} (which lies on $\nu_p$) be the center of curvature of $K$ at $p$, that is, the center of the osculating circle to $K$ at $p$.
We orient [*$\nu_p$*]{} so that the length of the (oriented) segment from $p$ to $c_p$ is less than the length of the segment from $c_p$ to $p$. This orientation is well-defined except at cusps and inflection points.
There is a natural duality from $\mathbb{R}P^2$ to itself. Under this duality simple closed geodesics, or projective lines, in $\mathbb{R}P^2$ are sent to points and vice versa. This duality is most easily described by passing to the 2–sphere $S$ which is the double cover of $\mathbb{R}P^2$; in this view a simple closed geodesic in $\mathbb{R}P^2$ lifts to a great circle on $S$. If this great circle is called the equator, the dual point in $\mathbb{R}P^2$ is the image of the north (or south) pole.
Under this duality the image of $K$ is a [*dual curve*]{} $K'$. To describe $K'$ we need only observe that a point on $K$ comes equipped with a tangent geodesic, $\tau_p$. The [*dual point to $p$*]{}, called $p'$, is the point dual to the tangent geodesic $\tau_p$.
Another useful description is that $p'$ is the point a distance $\pi/2$ along the normal geodesic to $K$ at $p$. Notice that $\nu_p=\nu_{p'}$ and $c_{p}=c_{p'}$.
An ordered pair of points $(p,q)$ on $K$ is an [*antipodal pair*]{} if $q=a_p$.
Let [*$Y_p$*]{} be the geodesic dual to the point $c_p$.
Let $(p,q)$ be an antipodal pair. Then $Y_p$ and $\tau_p$ intersect at $q$ and divide $\mathbb{R}P^2$ into two regions, $R_1$ and $R_2$. One of the regions, say $R_1$, contains $c_p$. The geodesic $\tau_q$ lies in one of the two regions. An antipodal pair $(p,q)$ is of type 1 if $\tau_q$ lies in $R_1$, type 2 if $\tau_q$ lies in $R_2$. Let $A_1$ be the number of type 1 antipodal pairs of $K$, $A_2$ the number of type 2.
$T$ is a [*double-supporting geodesic*]{} of $K$ if $T$ is either a double tangent geodesic, a tangent geodesic through a cusp or a geodesic through two cusps. The two tangent or cusp points of $K$ divide $T$ into two segments, one of which has length less than $\pi/2$. We distinguish two types of double supporting geodesics, depending on whether the two points of $K$ lie on the same side of this segment (type 1) or opposite sides (type 2). Let [*$T_1$*]{} be the number of double supporting geodesics of $K$ of type 1, [*$T_2$*]{} the number of type 2 (see ).
![[]{data-label="double tangent types"}](\figdir/fbfig2a "fig:")![[]{data-label="double tangent types"}](\figdir/fbfig2b "fig:")
\[fig2\]
The tangent geodesics at a crossing of $K$ define four angles, two of which, $\alpha$ and $\beta$, are less than $\pi/2$. In a small neighborhood of a crossing there are four segments of $K$. The crossing is of type 1 if one of these segments lies in $\alpha$ and another in $\beta$, type 2 if two lie in $\alpha$ or two lie in $\beta$. Let $C_1$ be the number of type 1 crossings of $K$, $C_2$ the number of type 2 (see ).
![[]{data-label="fig3"}](\figdir/fbfig3 "fig:")\[crossing types\]
Let [*$I$*]{} be the number of inflection points of $K$.
Let [*$U$*]{} be the number of (type 1) cusps of $K$.
We are now ready to state the first main theorem, which is a generalization of the main theorem of [@FB1] to the projective plane. We note that (unlike [@P]) we do not need to choose a base-point for $K$.
\[main1\] Let $K$ be a generic singular curve in $\mathbb{R}P^2$ with type 1 cusps. Then $$T_1-T_2=C_1+C_2+(1/2)I+U-(1/2)A_1+(1/2)A_2$$
The proof proceeds as in [@FB2], with some caution being required at antipodal pairs and at cusp points. We choose a starting point $p$ on $K$. Let ${M_p}^+$ be the number of times $K$ intersects ${\tau_p}^+$, ${M_p}^-$ be the number of times $K$ intersects ${\tau_p}^-$, and $M_p={M_p}^+- {M_p}^-$. We keep track of how $M_p$ changes as we traverse the knot. Double-supporting geodesics, crossings, cusps and inflection points all behave as in [@FB2]. Suppose $p$ is a point of an antipodal pair $(p,q)$. Let $p_1$ be a point immediately before $p$ on $K$, $p_2$ a point immediately after. If $(p,q)$ is of type 1 then $\tau_{p_1}$ intersects the arc of $K$ containing $q$ on ${\tau_ {p_1}}^-$ and $\tau_ {p_2}$ intersects the arc of $K$ containing $q$ on ${\tau_ {p_2}}^+$, hence $M_p$ increases by 2 as we pass through $p$. If $(p,q)$ is of type 2 then $\tau_
{p_1}$ intersects the arc of $K$ containing $q$ on ${\tau_ {p_1}}^+$ and $\tau_ {p_2}$ intersects the arc of $K$ containing $q$ on ${\tau_{p_2}}^-$, hence $M_p$ decreases by 2 as we pass through $p$. This is easiest to see by approximating $K$ near $p$ by a circle centered at $c_p$. As we traverse a piece of this circle from $p_1$ through $p$ to $p_2$, the antipodal points to $p_1$ and $p_2$ lie on the geodesic $ Y_p$. Hence the critical distinction to be made at $q$ is where the tangent geodesic at $q$ lies in relation to $Y_p$ and $\tau_p$. This is the exactly the distinction between type 1 and type 2 antipodal pairs.
Duality in $\mathbb{R}P^2$ {#sec3}
==========================
We describe the dual relations between crossings and double tangencies, cusps and inflection points, and antipodal points and normal-tangent pairs (defined below).
The points $p$ and $c_p$ divide $\nu_p$ into two pieces, [ *${\nu_p}^+$*]{} from $p$ to $c_p$ and [*${\nu_p}^-$*]{} from $c_p$ to $p$. An ordered pair of points $(p,q)$ on $K$ is a [ *normal-tangent pair*]{} if $\tau_q=\nu_p$. A normal-tangent pair $(p,q)$ is of type 1 if $q$ lies on ${\nu_p}^-$, type 2 if $q$ lies on ${\nu_p}^+$ (). Let $N_1$ be the number of type 1 normal-tangent pairs of $K$, $N_2$ the number of type 2 .
$q$ \[t\] <0pt, -2pt> at 92 332 $c(p)$ \[t\] at 155 332 $p$ \[tl\] <0pt, -2pt> at 212 332 at 146 254 $q$ \[t\] <0pt, -2pt> at 185 156 $c(p)$ \[t\] at 155 156 $p$ \[tl\] <0pt, -2pt> at 212 156 at 146 87 ![[]{data-label="fig4"}](\figdir/fbfig4 "fig:")\[normal-tangent\]
\[dualcor\] Let $K$ be a generic curve in $\mathbb{R}P^2$, with dual curve $K'$. Let $i=1,2$. Then:
1. A crossing of type $i$ in $K$ is dual to a double supporting geodesic of type $i$ in $K'$.
2. A cusp in $K$ is dual to an inflection point in $K'$.
3. An antipodal pair of type $i$ in $K$ is dual to a normal-tangent pair of type $i$ in $K'$.
As the dual of $K'$ is again $K$, these correspondences work in both directions.
The proof is by construction in $\mathbb{R}P^2$.
This correspondence breaks down slightly when we consider double supporting geodesics between cusps and tangents, or cusps and cusps. Fabricius-Bjerre suggests a small local alteration of $K$ to understand his argument at a cusp point, replacing the cusp point by a small “bump". Just as the dual to a small round circle in $\mathbb{R}P^2$ is a (long) curve that is close to a geodesic, his local change at cusps induces a more global change at inflection points, and so in order to incorporate curves with inflection points we need to add [*inflection geodesics*]{} to our picture of $K$.
Let $p$ be an inflection point of $K$, with tangent geodesic $\tau_p$. Endow $\tau_p-p$ with a normal direction at each point (except the inflection point) by the convention shown in .
![[]{data-label="fig5"}](\figdir/fbfig5 "fig:")\[inflectiongeo\]
Call this the [*inflection geodesic to $K$ at $p$*]{}.
For crossings between $K$ and an inflection geodesic $\tau_p$, or between two inflection geodesics, the piece of $\tau_p$ in the neighborhood of the crossing should be construed as bending slightly towards its normal direction for the purposes of classifying the crossing type. This convention preserves the correct duality between crossing type in $K$ and double supporting geodesic type in $K'$. A point on the inflection geodesic has center of curvature a distance $\pi/2$ in the normal direction, at $p'$. For $\alpha$ a point on an inflection geodesic $\tau_p$, $\nu_{\alpha}$ is the geodesic through $\alpha$ and $p'$.
Denote by $\wbar{K}$, $K$ together with all its inflection geodesics. Crossings and normal tangencies are counted as described above. The inflection points of $K$ (where the inflection geodesic intersects the curve) will still be counted as simply inflection points in $\wbar{K}$, not as new crossing points.
If $K$ is a generic curve with dual $K'$, then double supporting geodesics in $K$ involving cusp points correspond to crossings in $\wbar{K}'$ involving inflection geodesics, and an antipodal pair $(p,q)$ with $p$ a cusp point will correspond to a normal-tangent pair $(p',q')$ with $p'$ a point on an inflection geodesic in $\wbar{K}'$.
We end this section with the definition of what it means for $K$ to be generic:
$K$ is [*generic*]{} if:
- $K$ has a finite number of crossings, double tangent lines, cusps, inflection points, antipodal pairs, and normal-tangent pairs.
- Tangent geodesics at self-intersections of $K$ are neither parallel nor perpendicular.
- The tangent geodesic through an inflection point or at a cusp is everywhere else transverse to $K$.
- A geodesic goes through at most two tangent points or cusps of $K$.
- No crossings occur at inflection points.
- A geodesic normal to $K$ at one point is tangent to $K$ at at most one point and everywhere else transverse to $K$.
- The distance between two points on a double-supporting geodesic is not $\pi/2$.
- If $(p,q)$ is an antipodal pair, let $ Y_p$ be the geodesic dual to $c_p$. Then $\tau_q$ should be neither $\tau_p$ nor $ Y_p$.
- If $(p,q)$ on $K$ is a normal-tangent pair, $q$ is not $c_p$.
A dual formula, with applications {#sec4}
=================================
The simplest version of the dual theorem applies to curves with no inflection points.
\[dualthm\] Let $K$ be a generic singular curve in $\mathbb{R}P^2$ with type 1 cusps and no inflection points. Then $$C_1-C_2=T_1+T_2+(1/2)U-(1/2)N_1+(1/2)N_2$$
Since inflection points are dual to cusps, we also have:
Let $K$ be the dual in $\mathbb{R}P^2$ of a smooth singular curve. Then holds for $K$.
The theorem follows directly from duality on $\mathbb{R}P^2$, but it is illuminating to consider the dual of the proof of Theorem 1, as it provides a direct proof for curves in the plane. In Theorem 1 we count the number of intersections between the curve $K$ and $\tau_p$, with appropriate signs, as the curve is traversed once. Hence the main technical point is to understand the dual to $M_p$.
Assign an orientation to $K$. The geodesics $\tau_p$ and $\nu_p$ intersect in a single point (at $p$) and divide $\mathbb{R}P^2$ into two regions. We first define the [*tangent-normal frame $F_p$*]{} to $K$ at p as follows: $F_p$ is the union of $\tau_p$ and $\nu_p$ together with a black-and-white coloring of the two regions of $\mathbb{R}P^2$. We color them by the convention that if we think of $\tau_p$ and $\nu_p$ at $P$ as being analogous to the standard $x- $ and $y-$ axes, the region corresponding to the quadrants where $x$ and $y$ have the same sign is colored white, the complementary region black (see ). The frame and its coloring are well-defined at points that are neither cusps nor inflection points. At cusps, the orientations of $\tau_p$ and $\nu_p$ [*both*]{} reverse as we traverse $K$, with the happy effect that the coloring of the normal-tangent frame is well-defined as we pass through a cusp point (notice that this is not true if we allow type 2 cusps).
$p$ \[tl\] at 109 428 $p$ \[tl\] <0pt,-1pt> at 104 143 ![[]{data-label="fig6"}](\figdir/fbfig6 "fig:")
We now describe how the tangent-normal frame for the dual curve $K'$ is related to $M_p$ for the curve $K$.
Let $p$ be a point on $K$ with tangent geodesic $\tau_p$. Let $r$ be a point of intersection between $K$ and $\tau_p$. $r$ contributes either $+1$ or $-1$ to $M_p$, depending on where it lies relative to the antipodal point $a_p$. What does $r$ correspond to in the dual picture? Under duality $p$ is sent to the point $p'$, and $\nu_p=\nu_p'$. The point $r$ is mapped to a geodesic $g_r$ through $p'$. A small neighborhood of $r$ in $K$ is mapped to an arc of $K'$ tangent to $g_r$. If $r$ contributes $+1$ to $M_p$, $g_r$ lies in the white region of the tangent-normal frame at $p'$. If $r$ contributes $-1$ to $M_p$, $g_r$ lies in the black region of the tangent-normal frame at $p'$. This leads us to the following definition.
At a given point $p$ on $K$, we define $W_p$ to be the number of geodesics through $p$ and tangent to $K$ which lie in the white region and $B_p$ to be the number of geodesics through $p$ and tangent to $K$ which lie in the black region as defined by the tangent-normal frame at $p$. Let $V_p=W_p-B_p$. The proof consists of tracking how $V_p$ changes as we traverse $K$ once; each type of singularity contributes to $V_p$ according to the following table:
[*Contribution*]{}
------- --------------------
$C_1$ $+4$
$C_2$ $-4$
$T_i$ $-4$
$U$ $-2$
$N_1$ $+2$
$N_2$ $-2$
We can use the natural duality directly for the general case:
Let $K$ be a generic singular curve in $\mathbb{R}P^2$ with type 1 cusps. Then for $\wbar{K}$, $$C_1-C_2=T_1+T_2+(1/2)U+I-(1/2)N_1+(1/2)N_2$$
If $K$ is a curve with no cusps, inflection points, or antipodal pairs (or for a smooth immersed curve in $\mathbb{R}^2$ with no inflection points), then the pair of formulas: $$\begin{aligned}
T_1-T_2 & = & C_1+C_2\\
C_1-C_2 & = & T_1+T_2-(1/2)N_1+(1/2)N_2\end{aligned}$$ applies, and combining them we can obtain:
\[tcurves\]
For $K$ a curve with no cusps, inflection points, or antipodal pairs (or for a smooth immersed curve in $\mathbb{R}^2$ with no inflection points): $$\begin{aligned}
4T_1-4C_1=N_1-N_2\\
4T_2+4C_2=N_1-N_2\end{aligned}$$
![[]{data-label="fig7"}](\figdir/fbfig7 "fig:")\[twocurves\*\]
Note that for the two curves shown in (redrawn in ) , we obviously have the values $T_1=1$, $T_2=0$. For the right-hand curve, $C_1=1$ and $C_2=0$, while for the left, $C_1=0$ and $C_2=1$. By observation, the right curve has no normal-tangent pairs, and the two equations in are easily seen to be satisfied. Applying to the left-hand curve, however, we obtain $$4=N_1-N_2$$ and we can locate four normal-tangent pairs of type 1 ().
[**Acknowledgement**]{}Research supported in part by an NSF grant and by the von Neumann Fund and the Weyl Fund through the Institute for Advanced Study.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We investigate the disorder dependence of the static density, amplitude and current correlations within the attractive Hubbard model supplemented with on-site disorder. It is found that strong disorder favors a decoupling of density and amplitude correlations due to the formation of superconducting islands. This emergent granularity also induces an enhancement of the density correlations on the SC islands whereas amplitude fluctuations are most pronounced in the ’insulating’ regions. While density and amplitude correlations are short-ranged at strong disorder we show that current correlations have a long-range tail due to the formation of percolative current paths in agreement with the constant behavior expected from the analysis of one-dimensional models.'
author:
- 'G. Seibold'
- 'L. Benfatto'
- 'C. Castellani'
- 'J. Lorenzana'
title: 'Amplitude, density and current correlations of strongly disordered superconductors'
---
Introduction
============
More than $50$ years ago Anderson has discussed the behavior of a superconductor in the presence of strong disorder. [@and59] According to his analysis (and under the restriction to elastic scattering from non-magnetic impurities) the BCS wave-function, build from Bloch-type wave functions with opposite momenta, can be generalized to pairs made from the exact single-particle wave-functions of the disordered system plus their time-reversed partner. As a result one would expect a gradual dependence of the superconducting transition temperature on the presence of non-magnetic impurities caused mainly by a modification of single-electron properties as density of states etc. While this picture is certainly correct for weak disorder, experiments on thin films of strongly disordered superconductors [@hav89; @heb90; @shahar92; @adams04; @steiner05; @stew07; @sac08; @sac10; @mondal11; @chand12; @kaml13; @noat13] have revealed a much more interesting behavior than suggested by Ref. . In particular, the observation of a superconductor-insulator transition (SIT) with increasing disorder provides evidence for an interesting interplay between localization of Cooper pairs and long-range superconducting (SC) order. [@fm10] Moreover, the observation of a pseudogap in strongly disordered SC films [@sac08; @sac10; @mondal11; @chand12; @kaml13; @noat13] bears some resemblance to similar experimental findings in high-temperature superconductors [@timusk; @fischer07; @rullier11] that may suggest a common mechanism in some regions of the phase diagram.
Theoretical investigations of disordered superconductors are either based on bosonic or fermionic approaches. In case of s-wave superconductivity the latter typically start from attractive Hubbard models where disorder is usually implemented via a shift of onsite energy levels. [@triv96; @ghosal98; @scal99; @ghosal01; @dubi2007; @dey2008; @boua11; @sei12; @ghosh13] These hamiltonians then are either treated within a standard Bogoljubov-de Gennes approximation [@ghosal98; @ghosal01; @dubi2007; @dey2008; @sei12; @ghosh13] or with more sophisticated approaches like Monte-Carlo methods. [@triv96; @scal99; @boua11] Bosonic models are then obtained from a large-U expansion, as e.g. the pseudospin $XY$ model in a transverse field[@ma_prb85], where the hopping of Cooper pairs (corresponding to pseudospins aligned in the $XY$ plane) competes with localization due to random fields (corresponding to pseudospins aligned in the perpendicular direction). Further simplifications, as e.g. an Ising model in a random transverse field, are also introduced since they allow for analytical treatments. [@mez10]
In recent years both approaches have lead to a coherent picture of the SIT: With increasing disorder the system starts to break up into “puddles” with finite SC order parameter $|\Delta| > 0$ and intermediate regions with $|\Delta| \approx 0$ although the spectral gap remains finite. The order parameter distribution shows a universal scaling behavior, in agreement with experiment, where the relevant scaling variable is the logarithm of the order parameter distribution normalized to its variance. [@lem13; @mayoh15] The phases of different puddles are weakly coupled, so that the system bears some resemblance with a granular superconductor. Upon applying a vector potential the system accommodates the phase twist in the regions with $|\Delta| \approx 0$ so that the associated energy, and thus the superfluid stiffness, are strongly reduced. Moreover, calculations within the BdG approach of the attractive Hubbard model have shown that the induced current flows along a quasi one-dimensional percolative path or “superconducting backbone” which connects the puddles. [@sei12] This result has its counterpart in the analysis of the bosonic approach which has revealed a regime of broken-replica symmetry where the partition function is determined by a small number of paths. [@mez10] For both, fermionic and bosonic models, there exists a critical value for the disorder strength above which the system becomes insulating. The SIT is characterized by a vanishing of the superfluid stiffness, however, the single-particle gap persists across the transition[@boua11].
A still open issue is the nature of the spatial correlations in such granular SC state arising near the SIT. In the classical Ginzburg-Landau-Abrikosov-Gorkov theory [@glag] there is a single scale $\xi_0\sim v_F/\Delta$, whose reduction by disorder is mainly governed by the mean-free path $\ell$ via $\xi\sim \sqrt{\xi_0\ell}$. On the other hand, in the vicinity of the Anderson localization transition the coherence length is also controlled by the localization length. [@kapi85; @kotliar86] Concerning the disordered attractive Hubbard model with a fragmented SC ground state as mentioned above, there is only limited knowledge about amplitude, density and current correlations. Previous Quantum Monte-Carlo studies [@scal99] yield only limited information on the spatial dependence of the correlations due to the small ($8\times 8$) lattice sizes. On the other hand investigations of response functions on larger clusters within the BdG approach where so far restricted to mean-field studies. In the present paper we evaluate the density, amplitude and current correlations by including fluctuations on top of the BdG solution thus generalizing the approach of Refs. to the case with disorder. In particular we are interested in the question of how the physics is governed by different length scales in different channels and how the formation of SC islands for strong disorder reflects in the corresponding correlation lengths.
The paper is organized as follows: The model is introduced in Sec. \[sec:form\] where we also outline the computation of correlation functions on the basis of the BdG ground state. Results are presented in Sec. \[sec:res\] for amplitude, density and current correlations. We finally conclude our discussion in Sec. \[sec:conc\].
Formalism {#sec:form}
=========
BdG equations
-------------
Our starting point is the attractive Hubbard model with local disorder $$H=\sum_{ij\sigma}t_{ij}c^\dagger_{i\sigma}c_{j\sigma} -|U|\sum_{i}n_{i\uparrow}n_
{i\downarrow} +\sum_{i\sigma}V_i n_{i\sigma}$$ which we solve in mean-field using the BdG transformation $$c_{i\sigma}=\sum_k\left[u_i(k)\gamma_{k,\sigma}-\sigma v_i^*(k)\gamma_{k,-\sigma
}^\dagger\right]$$ $$\begin{aligned}
%\lb{uk}
\omega_k u_n(k)&=\sum_{j}t_{nj} u_j(k) + [V_n-\frac{|U|}{2}\langle n_n\rangle -
\mu] u_n(k) \nonumber \\ &+\Delta_n v_n(k)
\label{eq1}\\
%\lb{vk}
\omega_k v_n(k)&=-\sum_{j}t^*_{nj} v_j(k) -
[V_n-\frac{|U|}{2}\langle n_n\rangle -\mu] u_n(k)\nonumber \\
&+\Delta^*_n u_n(k)\label{eq2}\,.\end{aligned}$$
For simplicity only nearest-neighbor hopping $t_{ij}=-t$ is considered in this work. The disorder variables $V_i$ are taken from a flat, normalized distribution ranging from $-V_0$ to $+V_0$.
In the following $u_i(k)$ and $v_i(k)$ are taken to be real. Starting from an initial distribution of the gap $\Delta_i$ and density $\langle n_i\rangle$ values we diagonalize the system of equations (\[eq1\],\[eq2\]), compute the new values ($T=0$) $$\begin{aligned}
\label{op}
\Delta_i&=&|U|\sum_n u_i(n)v^*_i(n) \\
\langle n_i\rangle &=& 2\sum_n|v_i(n)|^2\end{aligned}$$ and iterate the obtained values, say $K$, (including also the chemical potential) up to a given accuracy $\delta K/K \le \epsilon$, typically $\epsilon=10^{-6}$. For the disordered systems studied in Sec. \[sec:res\] clusters with up to $24\times 24$ sites have been diagonalized. We mostly show results with filling $n=0.875$, but in some cases we also discuss the outcomes for smaller filling in order to avoid the proximity to half-filling, where specific effects can arise due to the tendency of the system to form a charge-density-wave (CDW) state as well.
Amplitude and Charge Correlations
---------------------------------
We denote correlation functions by $$\chi^{O,R}_{nm}(\omega)= i\int\!dt e^{i\omega t}
\langle {\cal T} \hat{O}_n(t) \hat{R}_m(0)\rangle$$ where in the following $\hat{O}$,$\hat{R}$ correspond to either amplitude $\delta A_i$ or density $\delta \rho_i$ fluctuations $$\begin{aligned}
\delta A_i &\equiv & (\delta\eta_i+\delta\eta^\dagger_i)/\sqrt{2} \\
\delta \rho_i &\equiv& \sum_{\sigma}\left(c^\dagger_{i\sigma}c_{i\sigma} - \langle c^\dagger_{i\sigma}c_{i\sigma} \rangle\right)\,,\end{aligned}$$ and we have defined the pair fluctuation operators $$\begin{aligned}
\delta\eta_i^\dagger &\equiv& c^\dagger_{i\uparrow}c^\dagger_{i\downarrow} -
\langle c^\dagger_{i\uparrow}c^\dagger_{i\downarrow} \rangle \\
\delta\eta_i &\equiv& c_{i\downarrow}c_{i\uparrow} - \langle c_{i\downarrow}c_{i\uparrow} \rangle \,.\end{aligned}$$
It is then convenient to define $2\times 2$ matrices for the bare mean-field susceptibility
$$\underline{\chi^0}_{ij}=\left(
\begin{array}{cc}
\chi^{AA}_{ij} & \chi^{A\rho}_{ij} \\
\chi^{\rho A}_{ij} & \chi^{\rho\rho}_{ij}
\end{array}\right) .$$
and the interaction
$$\underline{V}=\left(
\begin{array}{ccc}
-|U| & 0 \\
0 & -|U|/2
\end{array}\right)$$
which can be combined into “large” matrices according to
$$\underline{\underline{\chi^0_{ij}}} = \left(
\begin{array}{cccc}
\underline{\chi^0_{11}} & \underline{\chi^0_{12}} & \cdots & \underline{\chi^0_{1N}} \\
\underline{\chi^0_{21}} & \underline{\chi^0_{22}} & \cdots & \underline{\chi^0_{2N}} \\
\vdots & \vdots & \ddots & \vdots \\
\underline{\chi^0_{N1}} & \underline{\chi^0_{N2}} & \cdots & \underline{\chi^0_{NN}}
\end{array}\right)$$
and $$\underline{\underline{V_{ij}}} = \left(
\begin{array}{cccc}
\underline{V} & \underline{0} & \cdots & \underline{0} \\
\underline{0} & \underline{V} & \cdots & \underline{0} \\
\vdots & \vdots & \ddots & \vdots \\
\underline{0} & \underline{0} & \cdots & \underline{V}
\end{array}\right) .$$
The RPA resummation can then be written as $$\underline{\underline{\chi}}=\underline{\underline{\chi^0}}+\underline{\underline{\chi^0}}\;
\underline{\underline{V}}
\;
\underline{\underline{\chi}}\;$$ which is solved by $$\underline{\underline{\chi}}=\left\lbrack \underline{\underline{1}} -
\underline{\underline{\chi^0}}\; \underline{\underline{V}} \right\rbrack^{-1}
\underline{\underline{\chi^0}} .$$
Note that in this paper we will focus on static correlations. Since at Gaussian level the coupling between the phase fluctuations and the density/amplitude ones is proportional to the frequency[@benf04], they are decoupled in the static limit. On the other hand, the phase fluctuations enter in a crucial way in the calculation of the current fluctuations, as will be outlined in the next subsection.
Current Correlations
--------------------
The current response $J^{\alpha}_n(\omega)$ to a vector potential $A_x(n,\omega)$ (which we fix along the $x$-direction of our square lattice) is the sum of the diamagnetic and paramagnetic contribution[@scala] $$\label{eq:ja}
J^{\alpha}_n = \sum_m \left\lbrack \delta_{\alpha,x}\delta_{n,m} t_x(n)
+\chi(j^\alpha_n,j^x_m)\right\rbrack A_x(m)$$ where $t_x(n)=-t\sum_\sigma\langle c_{n,\sigma}^\dagger c_{n+x\sigma}
+ c_{n+x,\sigma}^\dagger c_{n\sigma}\rangle < 0$ denotes the kinetic energy on the bond between sites $R_n$ and $R_n+a_x$ and $j_n^\alpha=-it\sum_\sigma (c^\dagger_{n+\alpha,\sigma}c_{n\sigma}-h.c.)$ is the operator of the paramagnetic current flowing from site $R_n$ to $R_{n+\alpha}$. Note that the notation for the current correlation function $\chi(j^\alpha_n,j^\beta_m)$ is slightly different from the correlations defined in the previous subsection. At frequency $\omega=0$ the current only couples to phase fluctuations $\delta \Phi_i\equiv i(\delta\eta_i-\delta\eta^\dagger_i)/\sqrt{2}$ via the vertices $\Lambda_{nm}^\alpha=\chi^0(j^{\alpha}_n,\delta\Phi_m)$ and $\overline\Lambda_{nm}^\alpha=\chi^0(\delta\Phi_n,j^{\alpha}_m)$. Thus, the full (gauge invariant) current correlation function is then obtained from $$\begin{aligned}
\chi(j^\alpha_n,j^\beta_m)
&=& \chi^0(j^\alpha_n,j^\beta_m)\nonumber \\
\label{full}
&+& \Lambda_{nm}^\alpha V_{mk}
\left\lbrack \underline{\underline{1}} - \underline{\underline{\chi^0}}\underline{\underline{V}}\right\rbrack^{-1}_{kl}\overline{\Lambda}_{lm}^\beta ,\end{aligned}$$ with $\chi^0$ in the second term denoting the bare phase-phase correlation function and $V_{mk}=-U\delta_{mk}$.
For the Fourier transform of the configurational average one finally obtains $$\label{eq:jd}
J^{\alpha}_{\bf q} = -D^{\alpha,x}_{\bf q} A_x({\bf q})$$ where $D^{\alpha,x}_{\bf q} = -\langle T_x\rangle\delta_{\alpha,x} -
\langle \chi_{\bf q}(j^\alpha,j^x)\rangle$. For $J^{\alpha}_{\bf q}\equiv J^{x}_{\bf q}$ and taking ${\bf q}$ along the $y$ direction the limit $\lim_{q_y\rightarrow 0 } D^{xx}_{q_y}\equiv D_s $ corresponds to the superfluid stiffness and coincides with the quantity evaluated in Ref. from an expansion of the mean-field free energy up to quadratic order in the vector potential.
![(Color online) Top to bottom: amplitude ($\chi^{AA}({\bf q})$), density ($\chi^{\rho\rho}({\bf q})$), and off-diagonal ($\chi^{A \rho}({\bf q})$) correlation functions in the superconducting state for parameter $|U|/t=2$ and in the clean limit ($V_0/t=0$).[]{data-label="fig1"}](fig1.pdf){width="8cm"}
Results {#sec:res}
=======
Correlations in the homogeneous system
--------------------------------------
We start our considerations by a brief resume of the homogeneous case for which amplitude and density correlations have been analyzed in Ref. and which are in agreement with our following finite cluster analysis. Fig. \[fig1\] shows the amplitude $\chi^{AA}({\bf q})$, density $\chi^{\rho\rho}({\bf q})$ and mixed $\chi^{A,\rho}({\bf q})$ correlation function for filling $n=0.875$ and $|U|/t=2$ without disorder.
For these parameters the maximum of the amplitude correlations is at ${\bf q}=0$ where it can be approximated as $$\label{eq:fitsmall}
\chi^{AA}({\bf q})\approx \frac{1}{m^2+c q^2}$$ with the mass $m$ and a parameter $c$ characterizing the dispersion of excitations. The quantity $\xi_0=\sqrt{c/m^2}$ can then be interpreted as a length scale for the decay of the amplitude correlations. On the other hand the density response is dominated by the contribution at ${\bf q}={\bf Q}\equiv (\pi,\pi)$ and around this wave-vector can be described by $$\label{eq:fitq}
\chi^{\rho\rho}({\bf q}\approx {\bf Q})\approx
\frac{1}{m_Q^2+c_Q ({\bf q}-{\bf Q})^2}\,.$$ In real space this corresponds to a staggered decay of the density correlations with length scale $\xi_Q =\sqrt{c_Q/m^2_Q}$.
The mixed susceptibility $\chi^{A,\rho}({\bf q})$ is negative (positive) for densities $n<1$ ($n>1$) since the anomalous correlations $\langle c_{i\downarrow} c_{i\uparrow}\rangle$ are negative with a maximum of their absolute value at half-filling. Therefore a positive fluctuation in density $\delta\rho$ for $n<1$ will lower (i.e. enhance the magnitude) the anomalous correlations.
![(Color online) Distribution of the superconducting gap parameter $\Delta_i$ (displayed on a linear scale by circles) and superconducting currents (arrows) computed from Eq. (\[eq:ja\]) for constant vector potential $A_x$ and a specific disorder configuration. Parameters $|U|/t=5$, $V_0/t=2$. The dashed line (blue arrow) indicates the cut which is analyzed in Fig. \[fig:cutchi\]. []{data-label="fig2"}](fig2.pdf){width="8cm"}
For the present model, in the absence of disorder and at half-filling, there is an “accidental” symmetry[@yan90] which allows the superconducting order to be continuously rotated into the charge density wave (CDW) order at ${\bf
q}={\bf Q}$ without energy change, promoting the charge density mode to a Goldstone mode. The enhancement of $\chi^{\rho\rho}({\bf Q})$ at $n=0.875$ is a remainder of this CDW instability at half-filling which is transfered to the amplitude correlations via the mixed susceptibility $\chi^{A,\rho}({\bf q})$ shown in the bottom panel of Fig. \[fig1\]. Increasing $|U|/t$ enhances the CDW correlations so that at some point the ${\bf q}={\bf Q}$ amplitude correlations also dominate with respect to the ${\bf q}=0$ response. On the other hand the CDW correlations are suppressed in the dilute limit (not shown) so that upon reducing filling the maximum density response is first shifted away from ${\bf Q}$ along the Brillouin zone boundary and finally, below some concentration and depending on the value of $|U|/t$, the ${\bf q}=0$ response starts to dominate . A more detailed discussion on the filling dependence of the amplitude and density response in the clean case can be found in Ref. .
Disordered system: Real space analysis {#sec:decoup}
--------------------------------------
### Mean-field solution {#sec:mean-field-solution}
For sizeable disorder the density varies on the scale of the lattice constant and correlates with the strongly spatially fluctuating disorder potential. Further on, it has been shown in Refs. \[\] that for strong disorder the system disaggregates into SC islands with sizeable SC gap $\Delta_i$ which are embedded in regions with $\Delta_i\approx 0$. Fig. \[fig2\] shows a map of the order parameter encoded on the size of the red circles showing the formation of the superconducting islands. This island structure leads to a very weak superfluid stiffness. Indeed, upon applying a transverse vector potential, as done in Ref. , the current flows through an optimum percolative path or “superconducting backbone” which determines the global stiffness. The latter not only depends on the volume fraction of the superconducting island, but also on the connectivity of these islands to the superconducting backbone. Thus one may have a moderate superconducting fraction and a very small global stiffness if the connectivity is poor. Fig. \[fig2\] shows an example of the superconducting backbone for current circulation. Notice that it does not necessarily involve all significantly superconducting sites. For example, sites $(1,7)$ and $(3,12)$ in Fig. \[fig2\], where $\Delta_i$ is large, are left out which therefore are examples for poorly connected islands. Whereas connected islands determine the superfluid stiffness the disconnected islands dominantly contribute to the subgap absortpion in the optical conductivity. [@cea14]
Analyzing the mean-field solutions for several configurations of disorder we find that there is a strong tendency to form superconducting dimers. For example, for $V_0/t= 2\sim 4$ we find that the average number of strongly superconducting neighbors of a strongly superconducting site is in the range $0.7 \sim
0.8$. Here a strongly superconducting site is defined as a site with a local parameter $\Delta_i\geq 0.5\Delta_{max}$ where $\Delta_{max}$ is the largest value of $\Delta_i$ in the system (which is close to the maximal value $\Delta_{max}=|U|/2$) . Examples of dimers can be seen in Fig. \[fig2\] at sites $(1,6)-(1,7)$, $(12,15)-(13,15)$ and $(8,15)-(9,15)$. One also observes that dimers can act as seeds of more extended islands as in sites $(14,3)-(14,4)$.
![(Color online) (a): Distribution of the superconducting gap parameter $\Delta_i$ (circles), the local density correlation function $\chi^{\rho\rho}_{ii}$ (squares), and nearest-neighbor density correlations $\chi^{\rho\rho}_{\langle ij\rangle}$ (bars on the bonds). (b): The distribution of the local $\chi^{AA}_{ii}$ (squares) and nearest-neighbor $\chi^{AA}_{\langle ij\rangle}$ (bars on the bonds) amplitude correlations together with the SC gap (circles). (c): Magnitude of local off-diagonal amplitude-density correlations $|\chi^{\rho A}_{ii}|$ (squares) together with the SC gap (circles). The disorder configuration and parameters are the same as in Fig. \[fig2\]. The symbol size for the correlations is displayed on a logarithmic scale whereas the SC gap is plotted on a linear scale.[]{data-label="fig7"}](fig3.pdf){width="7.5cm"}
In previous work [@ghosal98; @ghosal01] it has been found that the preferable sites for the SC islands are those with the Hartree potential $H_i=-|U|\langle n_i\rangle +V_i$ being close to the chemical potential $\mu$, since this allows for strong particle-hole mixing. This would imply that the ’good’ SC sites are already encoded in the normal state since there exists a strong correlation between the local Hartree potentials in the SC and normal state. On the other hand the correlation between $H_i$ and the size of $\Delta_i$ weakens with increasing disorder, i.e. a small $|H_i-\mu|$ not necessarily correlates with a large $\Delta_i$ whereas a large $\Delta_i$ always implies a small $|H_i-\mu|$. A similar conclusion has been drawn in Ref. where the relation of order parameter variations and the shell effect has been investigated.
![(Color online) Cut of the order parameter distribution (full line, circles) and local density susceptibility $\chi_{ii}^{\rho\rho}$ in the superconducting state (dashed line, squares) and in the normal state (dot dashed line, diamonds). The cut is done along the row with $y=15$ of Fig. \[fig7\]b and is indicated by an arrow in Fig. \[fig2\]. []{data-label="fig:cutchi"}](fig4.pdf){width="8cm"}
### Real space structure of responses {#secrr}
The largest contribution to the density and amplitude correlations comes from the diagonal elements $\chi_{ii}^{AA}$ and $\chi_{ii}^{\rho\rho}$ that are shown as a logarithmic map in Fig. \[fig7\]a and b, respectively. Here the disorder realization is the same as in Fig. \[fig2\] and the local SC gap is shown with circles, whose size is proportional to the gap magnitude. Panel (a) shows also the nearest-neighbor density-density correlation $\chi^{\rho\rho}_{ij}$ encoded in the size of the bars on the bonds.
One finds that the strong superconducting sites coincide with sites which have a large charge density susceptibility. Also the dominant nearest neighbor density correlations $\chi^{\rho\rho}_{\langle ij\rangle}$ are attached to the SC islands and become particularly enhanced among the sites forming a SC dimer. We find that the bare local density correlations $\chi^{0,\rho\rho}_{ii}$ in the SC state show a similar structure (not shown) but with smaller absolute value ($\sim 1/20$).
This rises the “chicken and egg” question if sites are favorable for superconductivity because they have a large susceptibility already in the normal state or if the large susceptibility is due to the local superconducting correlations. To answer this question we have computed the charge density susceptibility in the absence of superconductivity. Although there is a tendency for sites with charge density susceptibility larger than the average in the normal state to become superconducting, there is an enormous enhancement of the charge density susceptibility on the superconducting sites. This can be seen in the cut of the local susceptibilities and order parameter shown in Fig. \[fig:cutchi\]. We see that on the superconducting sites the local susceptibility can be enhanced by two orders of magnitude. The inset shows a zoom of the intensity scale showing that the superconducting sites tend to have a charge density susceptibility larger than the average in the normal state but which does not explain the enhancement seen in the superconducting state. It also shows that on the sites with small order parameter the charge susceptibility remains the same in the superconducting and normal state. Clearly this behavior is due to the almost incompressible character of the phase without superconducting correlations which becomes instead highly compressible in the superconducting state. This physics is similar to that in the clean half-filled Hubbard model where a rotation between the two competing states, CDW and SC, essentially induces a transition from zero to very large compressibility $\kappa$.
![(Color online) Plot of the points $(\Delta_i,\chi_{ii}^{\rho\rho})$ (green) for the normal (left panels a, c) and SC (right panels b, d) system where $\Delta_i$ refers to the value in the SC state. The lines and errorbars have been obtained by collecting data in $10$ bins of $\Delta$. $|U|/t=2$, $V/t=1$ (upper panels a, b), $V/t=3$ (lower panels c, d).[]{data-label="corrgap"}](fig5ab.pdf "fig:"){width="8cm"} ![(Color online) Plot of the points $(\Delta_i,\chi_{ii}^{\rho\rho})$ (green) for the normal (left panels a, c) and SC (right panels b, d) system where $\Delta_i$ refers to the value in the SC state. The lines and errorbars have been obtained by collecting data in $10$ bins of $\Delta$. $|U|/t=2$, $V/t=1$ (upper panels a, b), $V/t=3$ (lower panels c, d).[]{data-label="corrgap"}](fig5cd.pdf "fig:"){width="8cm"}
The correlation between SC gap and local charge density susceptibility is summarized in Fig. \[corrgap\] which shows the distribution of $(\Delta_i, \chi_{ii}^{\rho\rho})$ points from $200$ samples for the normal and SC state and two values of disorder at $|U|/t=2$. Here $\Delta_i$ always refers to the value in the SC state whereas $\chi_{ii}^{\rho\rho}$ is evaluated in both normal and SC state. In the normal state and for weak disorder $V/t=1$ one observes a positive correlation between the local $\chi_{ii}^{\rho\rho}$ and the gap $\Delta_i$ which would develop in the SC state. This correlation gets sharper in the SC state (panel b) but extends over the same range of $\chi_{ii}^{\rho\rho}$ values than in the normal state. In contrast, for larger disorder $V/t=3$ there is almost no correlation between local charge density susceptibility and SC gap in the normal state while this correlation is strongly enhanced in the SC state and pushed to values of $\chi_{ii}^{\rho\rho}$ which are one order of magnitude larger than in the normal state.
The behavior of the amplitude fluctuations is also very interesting. We find that local amplitude fluctuations are significantly enhanced when the SC gap displays strong variations as a function of disorder strength. This feature is exemplified in Fig. \[fig8\] which, for fixed disorder realization (the same as used in Fig. \[fig2\] and Fig. \[fig7\]), shows the dependence of $\chi_{ii}^{AA}$ on $V_0$ for selected sites. One basically observes two kinds of behavior. First there are ’weak’ SC sites, as $(1,1)$ or $(10,5)$, whose order parameter immediately decreases with the onset of disorder. Besides there are ’strong’ SC sites, as $(3,12)$ or $(12,15)$ which initially resist disorder and where $\Delta_i$ can even get enhanced with respect to its $V_0=0$ value. The drop of $\Delta_i$ on the strong SC sites at a given $V_0/t$ is then accompanied by a peak in $\chi_{ii}^{AA}$ resembling the behavior close to a second order phase transition. However, the order parameter does not vanish on the disordered site of the transition but acquires a small finite value due to the proximity effect of other SC islands. For a given disorder strength only few sites are close to this regime and their number decreases with increasing $V_0/t$ due to the decrease of SC islands. There are also few sites, as $(10,5)$, where the SC order parameter reemerges at a large value of the disorder strength and stays finite over some range of $V_0$. In the appendix the behavior of $\Delta_i$ and $\chi_{ii}^{AA}$ for all sites of the sample is analyzed in more detail.
![(Color online) Disorder dependence of the SC gap (solid, black) and of the local amplitude correlations (red, dashed) for selected sites of the disorder configuration used in Figs. \[fig2\],\[fig7\]. $|U|/t=5$.[]{data-label="fig8"}](fig6.pdf){width="8cm"}
The present real space analysis reveals that in the strongly disordered regime, density correlations are dominant on the SC islands whereas the amplitude correlations are large in the other part of the system, i.e. where the SC gap is almost completely suppressed by disorder. As shown in Fig. \[fig7\]c there are only few sites with significant off-diagonal correlations $\chi_ {ii}^{\rho,A}$. Besides on the ’marginal’ sites $(3,12)$ and $(6,5)$, which are at the transition $\Delta\to 0$, the mixing of amplitude and density correlations is only observed on some of the SC sites. Clearly this decoupling of amplitude and density correlations will be even more pronounced in the average momentum dependent correlations which will be analyzed in the next subsections.
Disordered system: Fourier space analysis
-----------------------------------------
For a particular disorder configuration the Fourier transform of the correlation functions is given by $$\chi({\bf q}, {\bf q}')=\frac{1}{N}\sum_{ij}e^{i\left({\bf q}{\bf R}_i
-{\bf q}'{\bf R}_j\right)}\chi_{ij}$$ where $N$ denotes the number of lattice sites. Clearly, if $\chi_{ij}$ only depends on the distance between lattice sites ${\bf R}_i-{\bf R}_j$ then $\chi({\bf q}, {\bf q}')$ is diagonal in momenta. In the following we perform averages of $\chi_{ij}$ over different disorder realizations up to $n_d=200$ for lattice sizes up to $24\times 24$. This procedure restores translational invariance in the correlation functions so that $\langle \chi({\bf q}, {\bf q}') \rangle_{conf.}
\equiv \delta({\bf q}, {\bf q}') \chi({\bf q})$. In Figs. \[fig6\], \[fig4\] the errorbars in the compressibility and mass reflect the variance of $\chi({\bf q})$ at ${\bf q}=0$ and ${\bf q}={\bf Q}$, respectively. Although it increases with disorder and $|U|/t$ the mean-values exceed the variances for the ’worst’ cases by a factor $\sim 3$.
We fit the correlation function $\chi({\bf q})$, which is peaked at ${\bf q}={\bf Q}$, to the function
$$\label{eq:fitqc}
\chi({\bf q})=\lambda_0+\frac{\lambda_3}{1
+2 \lambda_1 \gamma_1({\bf q}-{\bf Q})
+2 \lambda_2 \gamma_2({\bf q}-{\bf Q})}$$
with $$\begin{aligned}
\gamma_1({\bf q}) &=& 2-\cos(q_x)-\cos(q_y) \\
\gamma_2({\bf q}) &=& 1-\cos(q_x)\cos(q_y)\,.\end{aligned}$$
Although Eq. (\[eq:fitqc\]) yields a good account of the correlations over the whole Brillouine zone (BZ) the fit is restricted to an area of $\approx 5\%$ of the BZ around the peak at ${\bf Q}$ in order to extract the parameters in Eqs. (\[eq:fitsmall\],\[eq:fitq\]). Expanding Eq. (\[eq:fitqc\]) around ${\bf Q}$ yields $$\begin{aligned}
m^2&=&\frac{1}{\lambda_0+\lambda_3} \label{eq:fitm}\\
c&=&\frac{\lambda_1+\lambda_2}{(\lambda_0+\lambda_3)} \label{eq:fitc}\\
\xi^2&=& c/m^2 = \lambda_1+\lambda_2 \,.\label{eq:fitxi}\end{aligned}$$
In the following we analyze the momentum structure of the averaged density-, off-diagonal and amplitude correlations. The various fitting parameters will be distinguished by (a) the reference momentum in the expansion, i.e. ${\bf q}=0$ or ${\bf Q}\equiv (\pi,\pi)$, and (b) a superscript which indicates the correlation function. For example, $\xi_0^{A}$ will denote the correlation length for amplitude fluctuations derived from an expansion of $\chi^{AA}({\bf q})$ around ${\bf q}=0$.
![(Color online) Average (number of samples $=200$) of Fourier transformed density correlations for parameter $|U|/t=2$, $V_0/t=3$.[]{data-label="fig5"}](fig7.pdf){width="8cm"}
### Momentum structure of $\chi^{\rho\rho}({\bf q})$ {#secrr}
We start with the analysis of the momentum dependence of the averaged density correlation function which is shown in Fig. \[fig5\] for parameters $|U|/t=2$ and $V_0/t=3$. Disorder induces an overall suppression of the response as compared to the clean case in Fig. \[fig1\]b. This is most pronounced for the CDW correlations at ${\bf q}={\bf Q}$ which for $V_0/t=3$ are reduced by a factor $1/20$ with respect to the clean case correlations. At ${\bf q}=0$ this reduction is only $1/2$ so that in Fig. \[fig5\] one observes a relative enhancement of the zone center correlations. For $|U|/t=2$ the crossover from dominant CDW to ${\bf q}=0$ correlations occurs at $V/t \approx 4$ whereas for larger values ($|U|/t=5$) $\chi^{\rho\rho}({\bf q})$ has a minimum at ${\bf q}=0$ up to the largest disorder investigated. Note that also for smaller filling disorder shifts the dominant correlations from incommensurate momenta in the clean case to ${\bf Q}=(\pi,\pi)$ so that the following analysis is representative for a wide doping range and disorder values.
Fig. \[fig6\] shows the parameters $(m_Q^\rho)^2$ and $c_Q^\rho$ obtained from the fit to Eq. \[eq:fitqc\] with ${\bf Q}=(\pi,\pi)$ as a function of disorder together with the compressibility $\kappa=\chi^{\rho\rho}_{{\bf q}=0}$.
![(Color online) Disorder dependence of the fit parameters $(m_Q^\rho)^2$ (circles), $c_Q^\rho$ (squares), and $\xi_Q^\rho$ (diamonds) for the staggered density correlations extracted from Eqs. (\[eq:fitqc\] - \[eq:fitxi\]). The disorder dependence of the compressibility is shown by the triangles. The dashed-dotted line in panel d) indicates the correlation length in the normal state. In panel b) the normal state $\xi_Q^\rho$ is numerically identical to the result in the SC state.[]{data-label="fig6"}](fig8.pdf){width="8cm"}
In the strong coupling limit (small $2t/|U|$) the clean case compressibility scales as $\kappa\approx |U|/8t^2$. [@lara] The enhancement of $\kappa$ with $|U|/t$ can also be observed in Fig. \[fig6\] for $V_0/t=0$ although the parameters $|U|/t=2,5$ are rather in the intermediate coupling regime so that the agreement with the above estimate is only qualitative. Upon increasing $V_0/t$ there is first a decrease of $\kappa$, in agreement with the results of Refs. . At large disorder one observes a tendency of the average compressibility $\kappa$ to saturate to a value that is weakly dependent on $U$. Since in this regime the dominant contribution to $\kappa$ comes from the (real space) diagonal elements $\chi_{ii}^{\rho\rho}$ on the SC islands there exists an apparent inverse correlation between the number of SC islands (which decreases with $V_0/t$) and the local compressibility $\chi_{ii}^{\rho\rho}$ (which gets enhanced with increasing $V_0/t$).
We now turn to the analysis of the CDW correlation length in the disordered SC system. For weak disorder $V_0/t=0.5$ there is a strong difference in the density distribution obtained for the two values of $|U|/t=2, 5$ which we have investigated. In fact, for $|U|/t=2$ we find that the difference in the density distribution between normal and SC state is small for each value of the disorder potential $V_0/t$. As a consequence the decrease of the CDW correlation length with $V_0/t$ (Fig. \[fig6\]b) is the same in the normal and SC state within the numerical accuracy. On the other hand, for $|U|/t=5$ we find that already for $V_0/t=0.5$ sites in the normal state system are either almost empty or doubly occupied. As already discussed above, the SC state induces a redistribution of charge density which in this case leads to a significant rearrangement with a more homogeneous distribution between $n \approx 0.2$ and $n\approx 1.7$. As a consequence of this effectively less disordered SC state one observes in panel (d) of Fig. \[fig6\] an enhancement of the correlation length at $V_0/t=0.5$ from $\xi_Q^\rho\approx 0.3$ in the normal state to $\xi_Q^\rho\approx 1$ in the SC system.
The behavior of fit parameters in the SC system, as shown in Fig. \[fig6\], can then be qualitatively understood from the evolution toward the bimodal charge density distribution, where the low (high) density peak approaches $n_L=0$ ($n_H=2$) with increasing disorder. We also adopt the result from a strong coupling expansion of $\chi^{\rho\rho}({\bf q})$ for the homogeneous system [@lara; @note1] which for the mass parameter yields $$\label{m2sc}
m_Q^2=\frac{8t^2}{|U|}\frac{\delta^2}{1-\delta^2}$$ and $\delta=1-n$ denotes the doping measured from half-filling. Averaging Eq. \[m2sc\] over the bimodal distribution. yields $\langle m_Q^2\rangle
\sim (\delta n)^2/(1-(\delta n)^2)$ with $\delta n =n_H - n_L$. The grow of $\delta n$ with $V_0/t$ then accounts for the increase of $m_Q^2$ with disorder as shown in Fig. \[fig6\].
In the strong-coupling clean case the parameter $c_Q$ is given by [@lara] $$c_Q=\frac{t^2}{|U|}\frac{1-2\delta^2}{1-\delta^2}=\frac{t^2}{|U|}-\frac{m_Q^2}{8}$$ and is thus expected to decrease with disorder proportional to the increase of $m_Q^2$. Within the numerical error this is in fact the behavior observed in Fig. \[fig6\] and also accounts for the decrease of the correlation length $\xi_Q$ with disorder.
### Momentum structure of amplitude correlations {#seccaa}
We proceed by analyzing the amplitude correlations $\chi^{AA}({\bf q})$ on top of the BdG solution whose momentum dependence is reported in Fig. \[fig3\] for $|U|/t=2$, $V_0/t=3$.
![(Color online) Average (number of samples $=200$) of Fourier transformed amplitude correlations for parameter $|U|/t=2$, $V_0/t=3$.[]{data-label="fig3"}](fig9.pdf){width="8cm"}
It turns out that disorder removes the enhancement of amplitude correlations at ${\bf Q}=(\pi,\pi)$, which were dominating in the clean case for this value of $|U|/t$. An interesting result is the concomitant enhancement of the ${\bf q}=0$ response by a factor of $\sim 5/2$ which therefore dominates the amplitude correlations for large disorder. As we have seen in the previous section, the density correlations are still peaked at ${\bf Q}=(\pi,\pi)$ for these parameters which indicates the decoupling of density and amplitude fluctuations with increasing disorder. Note that in contrast to the density correlations, the amplitude fluctuations in the normal state will always be unstable.
![(Color online) Disorder dependence of the fit parameters $(m0^A)^2$ (circles), $c_0^A$ (squares) and $\xi_0^A=\sqrt{c_0^A/(m_0^A)^2}$ (diamonds, right inset) as extracted from Eqs. (\[eq:fitqc\] - \[eq:fitxi\]) for $|U|/t=2$ (a) and $|U|/t=5$ (b). The left inset reports the average superconducing gap (circles) and average spectral gap (squares). The right insets also show the gap autocorrelation length $\lambda_{ac}$ (circles) computed from Eqs. (\[eq:autocorr\],\[fit2d\]).[]{data-label="fig4"}](fig10a.pdf "fig:"){width="8cm"} ![(Color online) Disorder dependence of the fit parameters $(m0^A)^2$ (circles), $c_0^A$ (squares) and $\xi_0^A=\sqrt{c_0^A/(m_0^A)^2}$ (diamonds, right inset) as extracted from Eqs. (\[eq:fitqc\] - \[eq:fitxi\]) for $|U|/t=2$ (a) and $|U|/t=5$ (b). The left inset reports the average superconducing gap (circles) and average spectral gap (squares). The right insets also show the gap autocorrelation length $\lambda_{ac}$ (circles) computed from Eqs. (\[eq:autocorr\],\[fit2d\]).[]{data-label="fig4"}](fig10b.pdf "fig:"){width="8cm"}
The latter are again characterized by the mass $(m_0^A)²$ and $c_0^A$ parameter obtained from the fit of $\chi^{AA}({\bf q})$ to Eq. \[eq:fitqc\] around ${\bf q}=(0,0)$. Fig. \[fig4\] reports the fit parameters as a function of disorder, again for values of the onsite attraction $|U|/t=2$ and $|U|/t=5$. Note that for the larger interaction $|U|/t=5$ and small disorder the correlations show the dominant peak at ${\bf Q}=(\pi,\pi)$ for which reason the fit parameters are only reported for $V_0/t \ge 0.5$.
The aforementioned enhancement of the ${\bf q}=(0,0)$ amplitude correlations with $V_0/t$ now results in the decrease of the mass $m_0^A$ with disorder with tendency to saturate at large $V_0/t \gtrsim 2$. Also the parameter $c$ decreases with the disorder strength so that the resulting correlation length $\xi_0^A=c_0^A/(m_0^A)^2$ (right insets to Fig. \[fig4\]) crucially depends on the relative change of $c_0^A$ and $(m_0^A)^2$ with $V_0/t$.
For $|U|/t=2$ the correlation length is almost constant up to $V_0/t=2.5$ and then starts to decrease with disorder. For larger $|U|/t$ one even observes an enhancement for small $V_0/t$ so that $\xi_0^A$ acquires a maximum around $V_0/t=2.5$. We note that this is not an effect of competing CDW order since the same result is observed in the low-density regime where such correlations are absent.
In the limit of small $V_0/t$ one can adopt the usual expression for the correlation length in dirty superconductors given by $\xi_0=\sqrt{\xi_{BCS} l}$ with the mean free path $l$ and the correlation length of the clean system $\xi_{BCS} \sim v_F/\Delta^{SC}$. The behavior of $\xi_0(V_0)$ therefore crucially depends on the depletion of the density of states, which lowers the superconducting $\Delta^{SC}$ gap, and the reduction of the mean free path $l$ with disorder. As noted in Ref. the situation in the strongly disordered system is more interesting since one has to distinguish between the average superconducting order parameter $\langle\Delta^{SC}\rangle$ and the spectral gap. As shown in the left insets to Fig. \[fig4\] $\langle\Delta^{SC}\rangle$ continuously decreases with disorder due to the increase of the ’non-SC’ area. On the other hand the spectral gap first shrinks with disorder due to the depletion of the density of states but grows again for strong disorder, signaling the formation of local boson pairs that get progressively localised as the SIT is approached. One can then argue that at strong disorder the BCS correlation length tends to scale as the inverse of the spectral gap, that acts as a cut-off to the increase of $\xi_0$ associated to the suppression of the SC order parameter. Alternatively one can relate the disorder dependence of the correlation length to the behavior of the nearest-neighbor amplitude correlations as shown in the appendix.
![(Color online) Average of Fourier transformed off-diagonal correlations $\chi^{A\rho}({\bf q})$ for $V_0/t=2.0$ and $|U|/t=2$.[]{data-label="fig2a"}](fig11.pdf){width="8cm"}
Recently[@kaml13] the spatial dependence of the STM spectra in strongly disordered NbN films has been analysed in terms of the autocorrelation function for the order parameter, i.e. $$\label{eq:autocorr}
\langle C(\bR)\rangle=\frac{1}{N}\langle \sum_i
\left( \Delta_i-\langle\Delta\rangle \right)
\left( \Delta_{i+\bR}-\langle\Delta\rangle \right)\rangle.$$ By performing an average over several disorder configurations we can extract the corresponding correlation length $\lambda_{ac}$ from a fit to the function $$\label{fit2d}
F(\bR)=a_0 + a_1 \mbox{e}^{-R/\lambda_{ac}\left( 1+a_2\sin^2(2\phi)
+a_3\sin^2(4\phi)\right)}\,.$$ Here $\phi$ is the polar angle related to $\bR$ which incorporates anisotropies in the correlations and we restrict the fit to $|\bR|>2$ in order to isolate the long-distance behavior. The resulting length $\lambda_{ac}$ as a function of disorder is shown by circles in the right inset to Fig. \[fig4\] and it is close to the correlation length $\xi_0^A$ extracted from the amplitude correlations. For $V_0\rightarrow 0$ one can apply linear response theory on the disorder and show that the two lengths coincide. In the strongly disordered regime the situation is more complex. We find numerically that both lengths are close to each other. Notice that for $|U|/t=5$ we observe that both $\lambda_{ac}$ and $\xi_0^A$ increase in the regime where the separation between the order parameter and the spectral gap starts to develop, while they collapse in the regime where the spectral gap tends to increase again. In Monte Carlo simulations[@boua11] the latter regime corresponds to the SIT, not captured by the present Bogoliubov-de-Gennes approach. This same tendency is observed in the experimental estimate of $\lambda_{ac}$ given in Ref. , done for samples in the so-called “pseudogap” region of the phase diagram, where the spectral gap is much larger than $T_c$.
### Momentum structure of off-diagonal correlations
Off-diagonal correlations $\chi^{A\rho}({\bf q})$ mix the density and amplitude sector and are shown in Fig. \[fig1\] for the clean case and in Fig. \[fig2a\] for the disordered system.
Upon coupling an external field in the density sector $H_1=\sum_{\bf q} \lambda_{\bf q}\rho_{\bf -q}$ the correlation function $\chi^{A\rho}({\bf q})$ yields the corresponding response for the gap amplitude. In particular, for ${\bf q}=0$ a spatially constant (and positive) $\lambda_{{\bf q}=0}$ induces an effective reduction of the chemical potential. Consider now the clean case where for the attractive Hubbard model with nearest-neighbor hopping the gap amplitude as a function of density has a maximum at half-filling and continuously decreases towards $n=0$ and $n=2$. Therefore off-diagonal correlations are negative for $n<1$ (where a positive $\lambda$ shifts the effective chemical potential away from half-filling) in agreement with Fig. \[fig1\] and positive for $n>1$. Similar arguments can be made for finite momenta. In particular, the strong enhancement of $|\chi^{A\rho}({\bf q})|$ at ${\bf q}={\bf Q}_{CDW}$ observed in Fig. \[fig1\] is due to the strong competition between CDW and SC correlations close to half-filling.
In the doped system Fig. \[fig2a\] reveals a strong suppression for the off-diagonal correlations due to the spatial separation of density- and amplitude fluctuations as demonstrated in Sec. \[sec:decoup\]. Naturally this is again most pronounced for the CDW momentum due to the removal of particle-hole symmetry by disorder. It is worth noting that in the dynamic limit (${\bf q}=0, \omega$ finite) the off-diagonal correlations show instead the opposite behavior. More specifically, as it has been recently discussed in Ref. \[\], the coupling between the amplitude and density/phase correlation at finite frequency is strongly enhanced by disorder, leading to a strong mixing between the amplitude and phase spectral functions at zero momentum.
Current correlations
====================
To conclude our analysis of the SC correlations we shall discuss now the change in the current-current correlation function induced upon entering the superconducting state. In particular we want to explore the consequences of the percolative current formation (cf. Fig. \[fig2\]) on the behaviour of the current correlation function $\chi^{jj}$ entering the definition (\[eq:ja\]) of the superfluid stiffness.
In order to obtain the intrinsic superconducting response, we have to subtract the contribution which is already present in the normal state (at finite momenta), and which can be either diamagnetic or paramagnetic depending on the filling of the system.
This is illustrated by the dashed line marked with diamonds in Fig. \[fig9a\] for the homogeneous non-superconducting system. Clearly, the current response of Eq. , $D_{q_y} = -\langle T_x\rangle + \langle \chi^{jj}({q_y})\rangle$ vanishes at $q_y=0$ (i.e. the SC stiffness) when the system is in the normal state, however, it becomes non-zero for finite momenta.
![(Color online) Transverse current response $D_{q_y} = -\langle T_x\rangle + \langle \chi^{jj}({q_y})\rangle $ for the non-SC (blue dashed, diamonds) and the sc homogeneous system at $|U|/t=5$ (red dashed, triangles). The normal state response ($\Delta_{sc}=0$) is independent of $|U|/t$. The solid lines report the difference $\Delta D_s(q_y)$ between $D_{q_y}$ for the sc- and normal system for $|U|/t=5$ (squares) and $|U|/t=2$ (circles). Filling $n=0.325$ (a) and $n=0.875$ (b). []{data-label="fig9a"}](fig12a.pdf "fig:"){width="8cm"} ![(Color online) Transverse current response $D_{q_y} = -\langle T_x\rangle + \langle \chi^{jj}({q_y})\rangle $ for the non-SC (blue dashed, diamonds) and the sc homogeneous system at $|U|/t=5$ (red dashed, triangles). The normal state response ($\Delta_{sc}=0$) is independent of $|U|/t$. The solid lines report the difference $\Delta D_s(q_y)$ between $D_{q_y}$ for the sc- and normal system for $|U|/t=5$ (squares) and $|U|/t=2$ (circles). Filling $n=0.325$ (a) and $n=0.875$ (b). []{data-label="fig9a"}](fig12b.pdf "fig:"){width="8cm"}
In particular at low density (cf. Fig. \[fig9a\]a) one recovers the finite-[**q**]{} diamagnetic response ($D_{q_y}>0$) related to Landau diamagnetism in agreement with the transverse current response of a Fermi liquid. [@pinesbook] In contrast, larger filling (cf. Fig. \[fig9a\]b) supports a finite-[**q**]{} paramagnetic current response which would even diverge at ${\bf
q}=(\pi,\pi)$ for $n=1$ (not shown). This feature is the starting point for the exploration of circulating current phases as possible candidates for the pseudogap in cuprate superconductors. [@schulz89]
As shown by the triangle symbols in Fig. \[fig9a\] a finite SC gap shifts up the curves in order to yield a diamagnetic $D_{q_y}$ independently on doping. In order to extract what is due to superconductivity we take the difference with respect to the normal state response $D^{normal}_{q_y}$ and the corresponding curves are shown by square symbols ($|U|/t=5$) and circles ($|U|/t=2$) in Fig. \[fig9a\] for $n=0.325$ and $n=0.875$, respectively. In the weak coupling limit the difference $\Delta D_s(q_y)=D^{SC}_{q_y} - D^{normal}_{q_y}$ is always strongly peaked at $q_y=0$ and the underlying normal state response does not influence the curvature of the peak which determines the SC coherence length. On the other hand, it turns out that for large filling and strong coupling (cf. squares in Fig. \[fig9a\]) $\Delta D_s(q_y)$ can even acquire a maximum at the zone boundary. Thus in this limit the SC diamagnetic response is largest on short length scales and corresponds to an oscillatory decay of the SC induced current correlations in real space.
Fig. \[fig9\] shows the transverse current response $\Delta D_s(q_y)$ for various disorder strength and interaction $|U|/t=2$ with the normal state result substracted. The latter has obtained for the same disorder configurations and by setting $\Delta^{SC}_i=0$.
![(Color online) Main panel: Transverse current correlations $\Delta D_s(q_y)$ measured with respect to the normal system for $|U|/t=2$, $n=0.875$, and various disorder strengths. Upper left inset: Correlation length extracted from Eq. (\[eq:xi\]). Upper right inset: Superfluid stiffness. []{data-label="fig9"}](fig13.pdf){width="8cm"}
We parametrize the long-wavelength structure as $$\label{eq:xi}
\Delta D_s(q_y)=D_s\left[1-(\xi_D q_y)^2\right]$$ which defines a SC coherence length related to the diamagnetic response and allows us to extract the stiffness $D_s$ as a function of disorder. Both quantities are shown in the insets to Fig. \[fig9\].
As discussed previously [@sei12] (see also Sec. \[sec:mean-field-solution\]) $D_s$ gets rapidly suppressed with disorder but since the BdG approach does not capture the SC-insulator transition it does not vanish even for large $V_0/t$. Also the coherence length (cf. left inset to Fig. \[fig9\]) is strongly suppressed by disorder. Above $V_0/t\approx 2$, $\Delta D(q_y)$ is essentially independent on the transverse momentum $q_y$ and $\xi_D \approx 0$ within the numerical accuracy.
However, due to the average over disorder configurations the above analysis does not capture the long-range current correlations which exist along the percolative path (cf. Sec. \[sec:decoup\]) and which we will analyze separately in the following.
First we identify the superconducting backbone. The criterion to decide which sites belong to the percolative path is chosen as follows: For the vector potential ${\bf A}$ along the $x$ direction, we determine the maximum current through a bond $j_x^{max}$ in the system and select all sites which have currents larger than $\alpha j_x^{max}$. We find that usually a value of $\alpha=1/3$ is appropriate in order to selecting the sites which are visited by the path. An example is shown in the inset to Fig. \[fig11\] where the squares indicate sites with $j_x(R_n) > j_x^{max}/3$. Clearly, there are sites (e.g. in the upper right corner) which are traversed by a minor current but are left out by the ’$\alpha=1/3$’ criterion. Reducing further the value of $\alpha$ would also include these sites, however, we note that the following results do not depend sensitively on the value of $\alpha$. The effect of a larger (smaller) $\alpha$ is to add sites with larger (smaller) current to the path which concomitantly slightly increases (decreases) the long-distance correlations which are calculated below.
![(Color online) Main panel: $\langle \Delta D_{nm}\rangle $ for both sites (black) and only one $R_n$ (blue) on the percolative path shown in the inset. The squares indicate sites $R_n$ with $j_x(R_n) > j_x^{max}/3$. $|U|/t=2$, $n=0.875$, $V/t=3$.[]{data-label="fig11"}](fig14.pdf){width="8cm"}
We proceed by evaluating the non-local stiffness $D_{n,m}$ between sites $R_n$ and $R_m$ $$\label{eq:Dsl}
D^{xx}_{nm} = \left\lbrack -\delta_{n,m} t_x(n)
- \chi_{nm}(j^x_n,j^x_m)\right\rbrack$$ and compute the difference between sc and normal state $\Delta D_{nm} = D^{sc}_{nm}-D^{nl}_{nm}$. Two cases are considered: (a) both sites $R_n$ and $R_m$ belong to the percolative path and (b) only one of the sites $R_n$, $R_m$ is on the path. The result for $D_{nm}$ in both cases is shown in Fig. \[fig11\] for the particular percolative path displayed in the inset. The ’errorbars’ indicate the variance due to the fact that different sites $R_n$ and $R_m$ have the same distance $|R_n-R_m|$ but different values for $D_{nm}$.
As can be seen the current correlations rapidly decay away from the percolative path and are practically ’zero’ for $|R_n-R_m|>3$. On the other hand correlations [*on*]{} the path stay finite up to the largest distances available in the system.
![(Color online) Current correlations $\langle \Delta D_{nm}\rangle $ averaged over 200 samples for disorder strength $V/t=3$ and $n=0.875$. Full symbols: $|U|/t=2$; Open symbols: $|U|/t=5$.[]{data-label="fig12"}](fig15.pdf){width="8cm"}
Finally, Fig. \[fig12\] shows the on- and off-path current correlations averaged over $200$ disorder configurations for $|U|/t=2$ and $|U|/t=5$, respectively. As for the specific sample shown in Fig. \[fig11\] the off-path correlations get rapidly suppressed while on-path correlations stay finite up to large $|R_n-R_m|$. Upon comparing the on-path correlations between the two $|U|/t$-values one finds, besides a reduction by a factor of $\approx 10$, that the decay of $\Delta D_{nm}$ with distance for $|U|/t=5$ is significantly smaller than for $|U|/t=2$ while still staying finite for the largest possible separation in the system ($\approx \sqrt{2}\times 10$ for a $20\times 20$ lattice when the percolative path is along the diagonal).
The persistance of the current correlations along the percolative path resembles closely the expected behavior for a one-dimensional chain, where it simply follows from the current conservation. This can be easily seen at $\omega=0$ by using the following classical phase-only action: $$\label{eq:gaussian}
S=\frac{1}{2} \sum_i J_i (\delta \Phi_i)^2$$ where $J_i$ are the local (random) stiffnesses (in units of the temperature $T$) and $\delta\Phi_i$ represents the local phase gradient $\delta \Phi_i\equiv (\theta_{i+1}-\theta_i)$, $\theta_i$ being the local SC phase. Eq. (\[eq:gaussian\]) can be obtained for example by expanding at Gaussian level a classical $XY$ model with random couplings $J_i$, that is the prototype model for the phase degrees of freedom of a superconductor. Eq. (\[eq:gaussian\]) is also obtained[@cea14] by mapping[@ma_prb85] at large $U$ the disordered Hubbard model into the pseudospin model. In this mapping the superconductivity corresponds to a spontaneous in-plane magnetization, i.e. to the usual $XY$ model with a coupling $J\sim t^2/U$, and disorder maps into a random out-of-plane field, that leads in turn to the disorder in the local couplings $J_i$ after a Holstein-Primakoff expansion around the mean-field solution. [@cea14]
The local current $I_i$ for the model (\[eq:gaussian\]) can be written, after minimal coupling substitution $\delta \Phi_i\to \delta \Phi_i-2A_i$ in Eq. (\[eq:gaussian\]) as: $$\label{eq:i}
I_i=2J_i(\delta\Phi_i-2A_i)\,.$$ In the one-dimensional case the current conservation implies that $I_i$ is independent on the site index, i.e. $(\delta\Phi_i-2A_i)=c/2J_i$, where $c$ is a constant. By summing over the site index and using the boundary condition $\sum_i \delta\Phi_i=0$ one then gets $c=-4(\sum_i A_i)/\sum_i(1/J_i)$. Since the superfluid stiffness is defines as usual (see Eq. (\[eq:jd\])) as $D_s=-I(q=0)/ A(q=0)$ one also deduces that $$D_{s}=4\left(\frac{1}{N}\sum_i\frac{1}{J_i}\right)^{-1}$$ so that $I_i= c=-(1/N)\sum_j D_s A_j$. By comparing this with Eq. (\[eq:Dsl\]) above we then recover that $D_{ij}=D_s/N$ for all pairs of sites $i,j$ along the chain. It is interesting to note that this result also implies that the paramagnetic contribution to the current must cancel out the local diamagnetic term $4J_i$ of Eq. (\[eq:i\]). This can be seen by computing explicitly the average current value from Eq. (\[eq:i\]) in linear response theory, in analogy with the expression (\[eq:Dsl\]) introduced above: $$\label{eq:dsnl}
\langle I_i\rangle =-4\sum_jJ_i(\delta_{ij}-X_{ij}J_j)A_j\equiv -\sum_j D_{ij}A_j$$ where $X_{ij}=\langle \delta\Phi_i\delta\Phi_j\rangle$ is easily determined from Eq. (\[eq:gaussian\]) as: $$\begin{aligned}
& &\langle \delta\Phi_i\delta\Phi_j\rangle=\nonumber\\
& &\frac{\int d\lambda
{\cal D}\delta \Phi \exp\left[-\frac{1}{2} \sum_k J_k (\delta \Phi_k)^2+i\lambda \sum_k \delta\Phi_k \right]
\delta \Phi_i\delta \Phi_j}{\int d\lambda
{\cal D}\delta \Phi \exp\left[-\frac{1}{2} \sum_k J_k (\delta \Phi_k)^2+i\lambda \sum_k \delta\Phi_k \right]}
\nonumber\\\end{aligned}$$ where the $\lambda$ integration accounts for the periodicity constraint. By making the change of variables $\delta\Phi_k\to \delta\Phi_k-i\lambda/J_k$ one immediately sees that $$X_{ij}=\frac{\delta_{ij}}{J_i}-\frac{D_s}{NJ_iJ_j}$$ that inserted into Eq. (\[eq:dsnl\]) gives $D_{ij}\equiv D_s/N$, as anticipated before. We note also that the independence of $D_{ij}$ in Eq. (\[eq:dsnl\]) on both site indexes can be also derived as a consequence of charge conservation and gauge invariance in one dimension. Indeed the independence of $D_{ij}$ on the site index $i$ is a consequence of a constant current $I_i$ on each site, while the independence of $D_{ij}$ on the second index $j$ is a consequence of the fact that at $\omega=0$ only to the $q=0$ component of the gauge field $A$ leads to a finite response.
Going back to our 2D system, we clearly see in Fig. \[fig12\] that for a fixed disorder strength the percolative path becomes more ’1D’-like with increasing $|U|/t$, which accounts for the crossover to a more constant $\Delta D_{nm}$ for $|U|/t=5$. Indeed, a larger $|U|/t$ corresponds to a smaller $J$ in the mapping into the $XY$-like bosonic model, with an enhanced influence of disorder and with smaller effective local stiffnesses ${J_i}$. This in turn is in agreement with the strong reduction of $\Delta D_{nm}$ from $|U|/t=2$ to $|U|/t=5$, as shown in Fig. \[fig12\].
Discussion and Conclusions {#sec:conc}
==========================
As we discussed in the introduction, it has been now established in several theoretical models that when the SIT is approached a granular SC state emerges, with SC puddles embedded in a non-SC background. Thanks to the enormous progresses made in the experimental techniques able to probe the systems in real space, it has been also established that such an emergent granularity is observed in disordered films of conventional superconductors, like e.g. NbN, InO$_x$ and TiN. [@sac08; @sac10; @mondal11; @chand12; @kaml13; @noat13] It is then crucial to assess how this inhomogeneous SC state affects the behavior of the amplitude, density and current correlations, in order to interpret the results of the various experimental probes.
In the present manuscript we analyzed this issue within the fermionic Hubbard model with on-site disorder. We presented a detailed study of the correlation functions both in real space, for a specific disorder configuration, and in momentum space, after the average over several disorder configurations. The momentum-space analysis allows us to extract the correlation length of each physical quantity in close analogy with the usual approach for homogeneous systems. As a first result, one then sees that while in the homogeneous case at low temperature amplitude and current correlation lengths coincide up to a numerical factor[@lara], in the presence of strong disorder this is no more true as can be seen in the summarizing figure \[figsum\]. By means of a simultaneous analysis of the real-space correlations we can then disentangle how the properties of the fragmented SC ground state influence the various correlation lengths. As we discussed in the manuscript, these two approaches give complementary informations, that we will summarize below. In this respect, even though our results are based on a RPA approximation, they have the advantage to allow for larger system sizes than Monte Carlo simulations, as e.g. those reported in Ref. . The use of large clusters is in turn crucial to trace back the behavior of different response functions to the inhomogeneous structure of the ground state and to perform a momentum-space analysis.
![(Color online) Summary of results for the various correlation lengths as a function of disorder for $|U|/t=2$ and $n=0.875$. At very small disorder the autocorrelation length $\lambda_{ac}$ cannot be properly defined since the approximated formula (\[fit2d\]) does not reproduce accurately the data, see also $C(R)$ in Fig. \[figcr\] below.[]{data-label="figsum"}](fig16.pdf){width="8cm"}
[*Amplitude and density correlations*]{}.
We find that in general the strength of the amplitude response $\sim 1/(m_0^A)^2$ increases with disorder while the charge response $\sim 1/(m_Q^\rho)^2$ gets suppressed by disorder (cf. Figs. \[fig6\] and \[fig4\]). This is similar to a previous Monte Carlo study[@scal99] which found that superconducting correlations are much more robust to disorder than charge correlations. Here, due to the larger system size, we could explore in detail the origin of this behavior.
The suppression of the charge response is easily understood by the tendency of disorder to localize the pairs and render the system incompressible almost everywhere except in the superconducting islands. The increase of the superconducting response is more subtle. For strong disorder the region in between the islands contains “marginal” sites where the order parameter is small but very susceptible to become large by small variations of the disorder \[see Fig. \[fig7\](b) and Fig. \[fig8\] for site (3,12)\] yielding a large overall pair susceptibility and resembling the behavior close to a second order phase transition. The decoupling of density and amplitude correlations in real space is reflected in the momentum-space structure of the susceptibilities. Thus, while in the homogeneous case[@lara] the maximum or $\chi_{\rho\rho}({\bf q})$ at the CDW vector ${\bf Q}=(\pi,\pi)$ leads to an enhancement of the amplitude correlations $\chi_{AA}({\bf q})$ at the same wavevector (see Fig. \[fig1\]), in the disordered case this effect disappears (Fig. \[fig3\]).
The resulting amplitude correlation length $\xi^A_0$, shown in Figs. \[fig4\], \[figsum\] has an interesting disorder dependence. Indeed, it stays constant or it is even enhanced at intermediate disorder levels, before then being ultimately suppressed as the SIT is approached. In the latter regime we argued that the decay of the correlation length is ruled by the behavior of the spectral gap, which increases as pairs become localized with disorder.
![Comparison between the experimental estimate (left panel) of the autocorrelation function, defined in Eq. (\[eq:autocorr\]), and the numerical computations (right panel). The experiments data are taken from Ref. \[\] and refer to three NbN films at different disorder level (labeled by the different critical temperatures $T_c$). The theoretical data are obtained for $|U|/t=2$, $n=0.875$, and disorder values $V_0/t=0.5$ (solid) and $V_0/t=3.$ (dashed). Considering that the typical size of the SC islands in these NbN films range between 20-40 nm, and it is one-two lattice spacings in our simulations, the length scales in the experiments and simulations are approximately comparable.[]{data-label="figcr"}](fig17.pdf){width="8cm"}
In Figs. \[fig4\], we also compared $\xi_0^A$ with the autocorrelation length $\lambda_{ac}$, that can be directly extracted experimentally from the STM maps of the SC ground state. This has recently been done for disordered NbN films [@kaml13] and we show for convenience the corresponding data in Fig. \[figcr\]a. In this work the SC islands are identified by the regions with a large SC coherence-peak height, that is usually taken[@boua11; @lem13] to be a measure of the local order parameter $\Delta_i$, i.e. the local gap solution in the BdG equations. By analyzing the spatial correlations between good SC sites the authors of Ref. \[\] found that the autocorrelation length $\lambda_{ac}$ becomes [*larger*]{} as disorder is increased. This is shown in Fig. \[figcr\]a where we report the experimental data for the autocorrelation function $C(R)$ defined in Eq. (\[eq:autocorr\]) above. A similar trend can be observed also in our simulations, see Fig. \[figcr\]b, where $C(R)$ shows first a rapid suppression over a length scale of the order of the SC island, followed by a long-tail decay that can be eventually fitted with the approximated formula (\[fit2d\]) in order to extract $\lambda_{ac}$. Since this tail can be thought as the response of the system to the fluctuations that created the island we expect that $\lambda_{ac}$ is close to $\xi_0^A$, as indeed we find numerically, see Fig. \[fig4\] and Fig. \[figsum\].
In contrast to the autocorrelation length, a direct estimate of the amplitude correlation length $\xi_0^A$ from the experiments is not so straightforward. Indeed, while within a Ginzburg-Landau approach, where a single length scale exists, $\xi_0$ can be estimated from the upper critical field at $T=0$ as $H_{C2}=\Phi_0/(2\pi\xi_0^2)$, at strong disorder this connection is not obvious. In particular when the superfluid stiffness $D_s$ is the lowest energy scale in the problem one would expect that $T_c\propto D_s$, so that also the upper critical field will scale with $D_s$, as suggested for example by a recent analysis of the microwave conductivity at finite magnetic field in disordered InO$_x$. [@armitage_prl13] In this sense, even though at intermediate disorder the decrease of $H_{c2}$ measured experimentally [@pratap11] can be interpreted as an increase of $\xi_0$ due to the weakening of the SC order parameter, as the SIT is approached one should not attribute the vanishing of $H_{c2}\propto T_c$ to a divergence of $\xi_0^A$ discussed above.
[*Current correlations*]{} The behavior of the current correlations is also strongly influenced by the formation of a fragmented SC state. Indeed, as already noticed before,[@sei12] the superfluid response is mainly determined by a few percolative paths that connect the good SC regions. As a consequence, the decay of the current correlations depends on the position of the initial and final sites with respect to this SC ’backbone’. If both sites belong to a percolative path the current correlations are long-ranged (essentially constant, see Fig. \[fig11\]), in agreement with what one expects for a truly one-dimensional system, like e.g. the one-dimensional $XY$-model. On the other hand, this long-range behavior is easily missed when the transverse current correlations are extracted from the response in momentum space after average over several disorder configuration. Indeed, the current-current correlation length $\xi_D$ is rapidly suppressed (cf. inset to Fig. \[fig9\] and Fig. \[figsum\]a), in analogy with the overall superfluid response. This behavior has to be contrasted to the one of the amplitude correlation length $\xi_0^A$, that is strongly suppressed only at the SIT. On the other hand, the persistence of current correlation along the percolative paths suggests that the existence of the SC backbone can be deduced in principle by the measurements of the space-dependent current susceptibilities, without having to evaluate explicitly the current pattern at finite applied field. The experimental study of these issues is of course challenging, but it should be accomplishable with four-point atomic force microscopy when the electrode spacing reaches the nanometer separation. Its observation would certainly contribute significantly to our understanding of the basic mechanisms leading to the formation of the inhomogeneous SC state as the SIT is approached in real systems.
This work has been supported by Italian MIUR under projects FIRB-HybridNanoDev-RBFR1236VV, PRINRIDEIRON-2012X3YFZ2 and Premiali-2012 ABNANOTECH, and by the Deutsche Forschungsgemeinschaft under SE806/15-1.
![(Color online) Top panel: disorder dependence of the SC gap value for the same disorder configuration used in Figs. \[fig2\], \[fig7\]. The site index for the $16\times 16$ lattice is obtained from $i_x+16(i_y-1)$. Lower panel: disorder dependence of the local amplitude correlations $\chi^{AA}_{ii}$ normalized to their maximum value at each site. $|U|/t=5$, $n=0.875$.[]{data-label="figapp1"}](fig18.pdf){width="8cm"}
\[apa\]
Disorder dependence of local SC gap and local correlations
==========================================================
Fig. \[fig8b\] reports the disorder dependence of the SC gap value and local amplitude correlations on each site for the same disorder configuration and parameters used in Figs. \[fig2\], \[fig7\]. Note that the amplitude correlations are normalized to their maximum value at each site. Clearly, the SC order parameter on the majority of sites drops to a small value around $V_0/t\approx 2$ but there are also singular sites where $\Delta_i$ extends up to $V_0/t\approx 4$ or where $\Delta_i$ reemerges at large disorder values.
In the clean system the onset of a finite SC gap below $T_c$ is accompanied by a divergence in the amplitude correlations, both the local and non-local ones. The pronounced enhancement of the amplitude correlations around $V_0/t \approx 2$ in Fig. \[figapp1\]b suggests a similar feature as a function of disorder with the difference that $\Delta_i$ does not vanish but becomes small beyond some value of $V_0$. To analyze this feature in more detail we plot in Fig. \[fig8b\] the probability density $P(\Delta<\epsilon)$ as a function of the disorder strength $V_0$. Here $P(\Delta<\epsilon)dV_0$ is the probability that the order parameter of a given site will fall below the threshold $\epsilon$ for the first time when the disorder is increased from $V_0$ to $V_0+dV_0$. Also shown are the probability distributions for the maximum in the local \[$P(\chi^{AA}_{ii}=max)$\] and nearest neighbor \[$P(\chi^{AA}_{\langle ij\rangle}=max)$\] amplitude correlations where, for example, $P(\chi^{AA}_{ii}=max) dV_0$ is the probability that $\chi^{AA}_{ii}$ for a given site $i$, attains its maximum value as a function of disorder in the interval $V_0$, $V_0+dV_0$. Clearly, for $|U|/t=5$ (right panel of Fig. \[fig8b\]) $P(\Delta<0.01t)$ has a pronounced peak around $V_0/t\approx 2 \dots 2.5 $ and one finds that for about $50\%$ of all sites $\Delta_i<0.01t$ between $1.5 < V_0/t < 2.5$. Concomitantly also the probability distributions for the local and non-local amplitude correlations are peaked at a somewhat lower value of $V_0/t \approx 1.5$. For smaller $|U|/t=2$ these distributions are broader and in particular the nearest-neighbor amplitude correlations are no longer characterized by a significant enhancement.
![(Color online) Black (full) steps: Probability distribution $P(\Delta<\epsilon)$ that the order parameter of a given site will fall below the threshold $\epsilon$ for the first time, upon increasing the disorder with $\epsilon=0.01t$ ; Red dashed step: Probability distribution $P(\chi^{AA}_{ii}=max)$ that the local amplitude correlation of a given site will attain its maximum value as a function of disorder strength. Blue thin step: Probability distribution $P(\chi^{AA}_{\langle ij\rangle}=max)$ that the nearest-neighbor amplitude correlations of a given bond will attain its maximum value as a function of disorder strength. Left panel: $|U|/t=2$, Right panel: $|U|/t=5$.[]{data-label="fig8b"}](fig19.pdf){width="8.5cm"}
This finding offers an alternative perspective for understanding the disorder dependence of the amplitude correlation length $\xi_0$ shown in Fig. \[fig4\]. Since $\xi_0$ is of the order of one lattice spacing the nearest-neighbor correlations yield the dominant contribution to the correlation length which accounts for the enhancement around $V_0/t= 2$. On the other hand, the distributions as a function of $V_0/t$ are significantly broader for $|U|/t=2$ (cf. Fig. \[fig8b\]a) which agrees with the behavior of $\xi_0$ shown in Fig. \[fig4\]a.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The use of sub-wavelength diameter tapered optical fibers (TOF’s) in warm rubidium vapor has recently been identified as a promising system for realizing ultra-low-power nonlinear optical effects. However, at the relatively high atomic densities needed for many of these experiments, rubidium atoms accumulating on the TOF surface can cause a significant loss of overall transmission through the fiber. Here we report direct measurements of the time-scale associated with this transmission degradation for various rubidium density conditions. Transmission is affected almost immediately after the introduction of rubidium vapor into the system, and declines rapidly as the density is increased. More significantly, we show how a heating element designed to raise the TOF temperature can be used to reduce this transmission loss and dramatically extend the effective TOF transmission lifetime.'
address: 'Physics Department, University of Maryland Baltimore County, Baltimore, MD 21250 USA'
author:
- 'M.M. Lai, J.D. Franson, and T.B. Pittman'
title: Transmission degradation and preservation for tapered optical fibers in rubidium vapor
---
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Introduction {#sec:introduction}
============
The desire for nonlinear optics using ultra-low-power fields is motivated by applications ranging from nanophotonic switching [@almeida04] to quantum computing with single-photon qubits [@milburn89]. One promising system for realizing these effects is the use of sub-wavelength diameter Tapered Optical Fibers (TOF’s) suspended in Rubidium vapor. These systems have recently been used to demonstrate nW-level saturated absorption and EIT [@spillane08], two-photon absorption [@hendrickson10], and all-optical modulation [@salit11]. Related work includes the use of hollow-core photonic band-gap fibers filled with Rb vapor [@ghosh06; @bajcsy09; @londero09; @saha11; @venkataraman11], and various integrated waveguides in contact with Rb vapor [@yang07; @wu10; @hendrickson12; @stern12]. In all of these systems, low-power nonlinearities are enabled by the high intensities achieved through compression of the optical mode-area interacting with the Rb atoms.
One of the major technical issues in these systems is the accumulation of Rb on the waveguide surfaces [@stern12; @slepkov08; @hendrickson09]. This can be particularly problematic for the case of TOF’s, where surface contaminants are known to cause scattering of the guided evanescent mode and, consequently, a degradation of transmission through the fiber [@fujiwara12]. Because high transmission is essential for low-power nonlinear optics applications, the loss due to Rb accumulation is currently one of the main limiting factors in these kinds of experiments. In order to move towards more practical systems, a better understanding of transmission degradation for TOF’s in Rb vapor is important at the present time.
In this paper, we describe an experiment which allowed real-time observations of TOF transmission degradation as the density of the surrounding Rb vapor was increased. We also show that a heating element used to raise the surface temperature of the TOF higher than the surrounding environment can reduce the accumulation of Rb atoms, thereby reducing the amount of transmission loss. Our results indicate that it is possible to achieve relatively high TOF transmission, even in the presence of the relatively high Rb vapor densities needed for many low-power nonlinear optics applications. This systematic study of the potential effectiveness of TOF heating was motivated by the use of basic heating elements in some of the original papers [@spillane08; @hendrickson10].
The main results of our study are summarized in Fig. \[fig:mainresults\]. Sub-wavelength diameter TOF’s were mounted on a custom heating unit, and installed in a vacuum system. Transmission through the TOF’s was then monitored as Rb vapor was introduced into the system. The rapidly increasing Rb density was monitored using the absorption of an auxiliary free-space resonant probe beam. The red curve in Fig. \[fig:mainresults\] shows the results of a run with the TOF heating unit turned off. A rapid drop in overall TOF transmission is seen within several minutes after the introduction of significant Rb density. In contrast, the blue curve shows the results of a virtually identical run with the TOF heating unit turned on. In this case the transmission drop is significantly smaller, leaving the TOF in a much more usable condition for the experiments of interest. The details of these TOF “heater off” vs. “heater on” measurements will be discussed in the following sections.
The remainder of the paper is organized as follows: in Section \[sec:setup\] we describe the design of the TOF heating unit and the overall experimental setup, and confirm the system’s suitability for low-power nonlinear optics applications by demonstrating Rb saturation with nW-level fields [@spillane08]. In Section \[sec:results\] we present detailed measurements of transmission degradation and preservation for three different scenarios, and in Section \[sec:TOFfailures\] we briefly describe suspected TOF failure mechanisms that prevented indefinite use of the system.
![A comparison of two runs measuring transmission degradation for TOF’s as Rb vapor is rapidly introduced into the system. The solid black line shows the absorption of an auxiliary free-space resonant probe beam; a substantial increase in Rb density occurs roughly 300 seconds into the run. The red curve shows the transmission of a TOF with the heating unit turned off; a significant loss of TOF transmission occurs with a time-scale on the order of several minutes. A second run done in virtually identical conditions (blue curve) used a TOF with the heating element turned on (the dashed black line shows the resonant probe for this run). The TOF transmission degradation is seen to be much less severe with the heater turned on. The vertical axis is normalized so the average transmission values are 1 before the introduction of Rb vapor.[]{data-label="fig:mainresults"}](fig1.eps){width="4in"}
TOF heating unit and experimental setup {#sec:setup}
=======================================
Our TOF’s were produced from standard single-mode fiber using the well-known “flame-brush” technique with an air-propane flame [@birks92]. The TOF’s typically had a minimum diameter of roughly 300 nm, and a sub-500 nm diameter over a length of about 5 mm. The overall transmission of freshly produced TOF’s was typically in the range of 40 - 70%.
Figure \[fig:heater\] shows an overview of the TOF heating unit used in our experiments. The TOF’s were mounted in an aluminum “canyon shaped” heating fixture using small dabs of low-outgassing UV curable epoxy. The TOF region was suspended above the canyon floor, and radiatively heated by the walls and floor on three sides; Rb vapor entered the canyon from the open top side. Conduction heating through the nanofiber itself also contributed to the elevated TOF surface temperature.
The aluminum heating fixture was placed on top of a $75 \times 25$ mm ceramic heater with an integrated thermocouple (Watlow CER-1-01-00007), and held in an aluminum holding jig that was mounted on the stainless steel tubes of a 2.75$''$ ConFlat (CF) feedthrough flange with exterior Swagelok fittings. These Swagelok fittings used Teflon feedthroughs for coupling the fiber in and out of the flange [@abraham98]. The flange also had power and thermocouple feedthroughs for controlling the ceramic heater.
![Overview of the TOF heating unit used to elevate the surface temperature of the TOF. Installation of the TOF involved 3 main steps: in Step , a freshly prepared TOF is mounted in an aluminum “canyon shaped” heating fixture using UV curable epoxy. In Step , the heating fixture is placed on top of a ceramic heating element, and in Step both are pressed tightly into an aluminum holding jig and secured using set-screws. The holding jig is mounted on a 2.75$''$ ConFlat (CF) feedthrough flange that allows the fiber to pass into and out of the vacuum system [@abraham98]. Power and thermocouple (tc) feedthroughs are used to control the TOF heating.[]{data-label="fig:heater"}](fig2.eps){width="4.25in"}
An overview of the vacuum system is shown on the left side of Fig. \[fig:setup\]. The feedthrough flange was installed on the top side of a standard 4.5$''$ CF six-way cube that served as the main vacuum chamber. The base pressure (measured by an Ion gauge near the pumps) was typically $\sim 10^{-7}$ Torr after baking out the system. Two large ovens were used to independently control the temperature of the main chamber, and a metallic Rb source (Alpha-Aesar 10315). In order to rapidly introduce Rb vapor into the chamber, we developed a repeatable procedure where the chamber was held at $\sim$ 60$^{o}$C, and the Rb source oven temperature was quickly increased to a value of $\sim$ 200$^{o}$C at the beginning of a run. In our system, this particular heating sequence caused a rapid increase in Rb density followed by fairly steady-state conditions lasting longer than the duration of the data runs of interest.
As shown in the lower right side of Fig. \[fig:setup\], the optical part of the system was driven by a tunable external cavity diode laser at 780 nm (New Focus Velocity 6312). A series of three fiber couplers split the laser into four signals: one to a wavelength meter and Rb reference cell, a second to the TOF, a third was used as a laser power reference (detector $D_{1}$), and the fourth was used as the free-space probe beam that passed through windows on the sides of the vacuum chamber. The TOF and free-space probe transmissions were monitored using detectors $D_{2}$ and $D_{3}$, respectively. A variable attenuator between two of the fiber couplers allowed us to reduce and control the TOF input powers to the desired nW power-levels.
![Overview of the experimental setup. In the optical part of the setup, the thin black lines represent single-mode fiber patch cords and couplers, while the thicker red arrows denote free-space beams. The TOF was installed inside the vacuum chamber with the external bare-fiber leads connected to the optical setup using two bare-fiber adapters (BFA’s). One of the standard fiber patch cord connectors (labelled FC$_{1}$) is identified due to its importance in the measurements. Additional details can be found in the main text.[]{data-label="fig:setup"}](fig3.eps){width="4.5in"}
Using this setup we were able to simultaneously perform Rb absorption spectroscopy using three different systems: (1) the auxiliary reference cell signal, (2) the free-space probe beam signal, and (3) the TOF signal. The laser frequency was slowly swept (5 seconds) over a 20 GHz range centered on the Rb D$_{2}$ absorption line at 780 nm. Figure \[fig:Psat\](a) shows the familiar absorption dips resulting from the four ground state hyperfine levels of $^{85}$Rb and $^{87}$Rb [@steck12]. The upper panel shows the absorption spectra obtained with the auxiliary reference cell, while the lower panel shows spectra obtained with the TOF for seven different power levels ranging from 4 - 267 nW.
The decreasing depth of the TOF dips with increasing power shows the ability to saturate the Rb vapor with extremely low powers, as first shown in the pioneering paper by Spillane et.al. [@spillane08]. Figure \[fig:Psat\](b) shows a Log-Linear plot of the “dip 2” ($^{85}$Rb, $F_{g}=3$) transmission data vs. estimated power in the TOF. The data is fit (blue line) by a simple transmission model $T = e^{-\alpha_{NL} L}$, with a nonlinear absorption coefficient defined as $\alpha_{NL} = \alpha/(1+P/P_{sat})$, and a best-fit saturation power of $P_{sat} = $ 72 nW. The ability to saturate the Rb vapor at such low powers arises from the reduced optical mode areas in the TOF waist region (on the order of 1 um$^{2}$ [@tong04]), and agrees with calculated saturation powers (on the order of 10 - 100 nW [@spillane08]) for these kinds TOF in Rb vapor systems.
We note that our ability to accurately determine the TOF powers interacting with the Rb vapor was hindered somewhat by non-ideal transmission [@frawley12] of the TOF’s themselves (67% in the present example), and simple fiber connector losses in the optical system. To avoid repeatedly disconnecting the fragile bare-fiber adapters (BFA’s, Fig. \[fig:setup\]), we measured the overall system transmission by comparing the input powers at the more robust connector FC$_{1}$ with the output powers at detector $D_{2}$. The TOF powers were then estimated by dividing the output power by an average BFA connector throughput value of 75% (determined by auxiliary tests).
![Spectroscopy and ultra-low power saturation using TOF’s in Rb vapor [@spillane08]. The upper panel in part (a) shows the absorption spectrum (with dips labelled 1-4) of the Rb D$_{2}$ line at 780 nm obtained in a standard reference cell, while the lower panel shows the corresponding absorption spectrum in the TOF signal for seven different TOF power levels (different color traces). Part (b) shows the TOF “dip 2” transmission values as a function of power; the data is fit by a simple nonlinear absorption model with a saturation power of only 72 nW. The main point of this data is to confirm that we have the kind of TOF in Rb vapor system needed for low-power nonlinear optics experiments [@spillane08; @hendrickson10; @salit11]. []{data-label="fig:Psat"}](fig4.eps){width="5in"}
Transmission degradation and preservation measurements {#sec:results}
======================================================
The behavior of the TOF system was studied by continuously sweeping the laser frequency back and forth across the same 20 GHz range (red sawtooth in Fig. \[fig:Psat\](a)) while Rb vapor density was introduced into the vacuum chamber. Monitoring the absorption dips and overall transmission of the TOF signal as a function of time provided a clear picture of transmission degradation and preservation during a run. The results of three key runs are shown in Figures \[fig:TOF1\] - \[fig:TOF3\]. For all of these runs, the TOF power was kept at $< 1$ nW to avoid any complications due to saturation effects.
Fig. \[fig:TOF1\](a) shows the results of a run with the TOF heating unit intentionally turned off. The gray trace in the background shows the transmission of the free-space probe (FSP) beam passing through the chamber; this had a long optical path length of 12 cm and was therefore very sensitive to the introduction of Rb vapor at the beginning of a run. The full 2700 second run corresponds to 540 consecutive sets of absorption dips (five second sweep), so the displayed FSP data appears as a thick gray shaded region. The thick red curve in the foreground is the TOF signal recorded at the same time. The overall TOF transmission rapidly degrades roughly 600 seconds into the run due to Rb accumulation on the TOF surface.
![Detailed results of a run with the TOF heater unit turned off. The laser is swept back and forth across the Rb D$_{2}$ line as Rb vapor is introduced into the system. In part (a), the 2700 second run represents 540 consecutive five-second sweeps through the Rb absorption dips; the grey trace is the free-space probe beam (FSP) signal, and the red trace is the TOF (TOF-1) signal. Parts (b)-(d) zoom-in on individual scans through the Rb absorption lines at three points during the run. The TOF-1 transmission degrades to 1.5% of its initial value by the end of the run, but still shows strong Rb absorption dips (part (d)).[]{data-label="fig:TOF1"}](fig5.eps){width="5in"}
Figures \[fig:TOF1\](b)-(d) show zoom-ins of individual five-second sweeps at three points during the run. At 413 seconds (part (b)) significant Rb vapor has entered the system; the sensitive FSP begins to show absorption dips, while dips are not yet apparent in the TOF (which has a much shorter effective interaction length on the order of 1 cm). By 593 seconds (part (c)) the Rb density has substantially increased; the TOF is showing deep absorption dips, but its overall transmission has already degraded to 60% of its initial value. At 2246 seconds (part (d)), the overall TOF transmission has dropped to 1.5% of its initial value. Remarkably, the remaining TOF signal still shows the Rb absorption dips.
In comparison, Figure \[fig:TOF2\] shows the results of a second run with the TOF heating unit turned on at a nominal value of 200$^{o}$C (measured by the internal tc of the ceramic heater). The idea was to raise the TOF surface temperature higher than the chamber (60$^{o}$C). In Fig. \[fig:TOF2\](a), transmission degradation is seen to occur on a similar time-scale as Fig. \[fig:TOF1\](a), but with substantially less overall loss and a final TOF transmission at 30% of its initial value. The zoom-ins in Figs. \[fig:TOF2\](b)-(d) show that the heating unit enables the preservation of relatively high steady-state TOF transmission (30%) in a relatively high Rb vapor density, as indicated by the nearly 100% TOF dips in part (d).
![Analogous run to Fig. \[fig:TOF1\], but with the TOF heater turned on at a nominal value of 200$^{o}$C (much higher than the vacuum chamber temperature of 60$^{o}$C). In this case, the TOF (TOF-2) transmission degradation is much less severe, with a final TOF-2 transmission of 30% of its initial value. Comparing the TOF-2 dip depths in part(d) with the TOF-1 dips in Fig. \[fig:TOF1\](d) suggests that the TOF heating unit can enable relatively high transmission while maintaining sufficient Rb density.[]{data-label="fig:TOF2"}](fig6.eps){width="5in"}
The primary “heater off” vs. “heater on” summary data in Fig. \[fig:mainresults\] was extracted from these two data sets (Figs. \[fig:TOF1\](a) vs. \[fig:TOF2\](a)). The solid and dashed black lines in Fig. \[fig:mainresults\] track the FSP beam while on resonance with the strong “dip 2” transition, while the red and blue curves correspond to TOF transmission data extracted away from any Rb resonance. This off-resonant TOF data provides the best indication of the overall TOF transmission vs. time for these runs.
Experimentally ensuring approximately the same Rb density vs. time profiles enabled a fair comparison between these “heater off” vs. “heater on” runs. This was confirmed using the depth of the FSP absorption dips to measure the rapidly changing relative density in the chamber at the beginning of a run; and the depths of the TOF dips once they became apparent after $t \sim 500$ seconds. Because of the difficulties in accurately fitting the absorption profiles of 100% dips [@siddons08], we simply verified that the optical depths (OD’s) of the smallest TOF dip (dip 4) were roughly the same for the latter parts of the runs. The OD’s of this particular signal raised to maximum values near 2.0 at $t \sim 1200$ seconds, and then slowly dropped towards values near 1.5 at the end of each run.
Fig. \[fig:TOF3\] shows the results of a third run which is not directly comparable to the first two runs. For convenience, the same Rb heating procedure was used, but the vacuum system was not thoroughly baked-out prior to the run. Consequently, the Rb vapor density was substantially lower during the run. In addition, the temperature of the TOF heating unit was raised from the nominal value of 200$^{o}$C to 265$^{o}$C. As shown in Fig. \[fig:TOF3\](a), this combination of lower density and higher TOF temperature allowed preservation of 100% transmission through the TOF. For this run, the OD (of the TOF dip-4 signal) only raised to maximum value of roughly 1.0 at $t \sim 900$ seconds, and then dropped to a steady-state value roughly 0.5 in the second half of the run. The zoom-ins in Figs. \[fig:TOF3\](b)-(d) show the corresponding increase and decrease of TOF dip depths. Although not conclusive, these results suggest the possibility of achieving 100% TOF transmission in even higher Rb densities by further TOF heating.
![Detailed results of a third run performed with lower Rb density, and a higher TOF heating unit temperature (265$^{o}$C). These results show the ability to preserve 100% TOF transmission (TOF-3) in relatively low Rb density. Comparing the TOF-3 data in parts (c) and (d) suggests the possibility of preserving high transmission in higher Rb densities by using an even hotter TOF heating unit.[]{data-label="fig:TOF3"}](fig7.eps){width="5in"}
TOF failure mechanisms {#sec:TOFfailures}
======================
The ability to indefinitely maintain relatively high overall TOF transmission in a standard “heater on” run (such as Fig. \[fig:TOF2\]) was primarily limited by a failure mechanism in which the TOF transmission would instantaneously drop to zero. Fig. \[fig:instantbreak\] shows the data capturing one of these TOF failures. In analogy with Figs. \[fig:TOF1\]-\[fig:TOF3\], the TOF data (purple trace) is plotted on top of the FSP data (grey trace). At $t = 511$ seconds into an otherwise stable run (part (a)), the TOF transmission suddenly drops to zero; the zoom-in (part(b)) shows that the time-scale of the drop is much less than 1 second.
![(a) Data showing an otherwise successful run in which the TOF transmission (purple trace) suddenly drops to zero at $t= 511$ seconds. (b) A zoom-in shows that the complete transmission drop occurs on time-scale of much less than 1 second, suggestive of a “fracture” in the TOF. The TOF data is normalized to 1 to highlight the transmission drop. These sudden transmission drops were the primary failure mechanism that prevented us from indefinitely using our heated-TOF’s in Rb vapor.[]{data-label="fig:instantbreak"}](fig8.eps){width="5in"}
This rapid time-scale is suggestive of some type of “fracture” within the TOF itself. Subsequent examination of the TOF’s with visible red-light usually showed large scattering out in the tapered region. The idea of a fracture is further supported by the fact that a smaller number of the failed runs resulted in the TOF being physically broken into two pieces. We note that these “fractures” were also observed while maintaining a TOF under vacuum without Rb vapor.
It is possible, however, that these rapid transmission drops were due to large contaminant particles “landing” on the TOF surface. Fujiwara et.al. have recently shown rapid transmission drops (albeit not to zero) due to dust particles [@fujiwara12], and similar effects have been seen by placing polystyrene clusters on TOF’s [@gregor09].
In contrast, transmission degradation due to Rb accumulation in our system (summarized in Fig. \[fig:mainresults\]) occurred on a time-scale of several minutes. We note that similar time-scale transmission degradation was also observed during initial vacuum system bake-outs with the TOF heating unit turned off; presumably water vapor and other contaminants accumulating on the TOF surface degraded transmission in a similar manner. From a practical point of view, keeping the TOF heating unit turned on at 200$^{o}$C entirely eliminated transmission degradation during bake-outs.
Similar slow time-scale transmission degradation was also observed when we attempted to raise the TOF heating unit temperature above 265$^{o}$C. We suspect outgassing of the epoxy (and/or other materials) at these higher temperatures contaminated the TOF surfaces. This effect prevented us from using higher TOF temperatures in an attempt to demonstrate preservation of TOF transmission at higher Rb densities.
Summary and conclusions {#sec:summary}
=======================
Transmission loss due to Rb accumulation on the surface of a TOF is currently one of the main limiting factors in the utility of these systems for low-power nonlinear optics [@spillane08; @hendrickson10; @salit11]. In this paper we have described real-time observations of TOF transmission degradation as the density of the surrounding Rb vapor was increased. By comparing the case of non-heated TOF’s with heated TOF’s, we showed that a heating unit can be effectively used to preserve relatively high TOF transmission, while maintaining the relatively high atomic densities needed for many low-power nonlinear optics applications. We achieved steady-state high TOF transmission conditions (like Figs. \[fig:TOF2\] and \[fig:TOF3\]) that lasted from several hours to several days in some cases, with suspected spontaneous “fractures” of the TOF (like Fig. \[fig:instantbreak\]) being the primary failure mechanism.
The interaction of Rb with silica surfaces is a complex process [@ma09]. In analogy to heating the windows in Rb vapor cell experiments, the basic idea of raising the TOF surface temperature higher than the surrounding environment is simply to prevent the initial accumulation of Rb on the tapered waist region. Although not as successful, we also found that our TOF heater could be used to remove Rb once accumulated, resulting in a partial recovery of TOF transmission. The preservation and recovery of TOF transmission has also been observed using a (relatively) high-power internally guided laser beam [@hendrickson09], and the possibility of external LIAD techniques with TOF systems may also be useful [@slepkov08]. The results presented here are encouraging for the advancement of the basic “TOF in Rb system” towards practical low-power nonlinear optical devices.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported in part by the DARPA ZOE program (Contract No. W31P4Q-09-C-0566). The authors thank D. Dubbel for assistance with the design and fabrication of the TOF heating system, and acknowledge fruitful discussions with S.M Hendrickson and T. Gougousi.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
It is well known that data scientists spend the majority of their time on preparing data for analysis. One of the first steps in this preparation phase is to load the data from the raw storage format. Comma-separated value (CSV) files are a popular format for tabular data due to their simplicity and ostensible ease of use. However, formatting standards for CSV files are not followed consistently, so each file requires manual inspection and potentially repair before the data can be loaded, an enormous waste of human effort for a task that should be one of the simplest parts of data science. The first and most essential step in retrieving data from CSV files is deciding on the dialect of the file, such as the cell delimiter and quote character. Existing dialect detection approaches are few and non-robust. In this paper, we propose a dialect detection method based on a novel measure of data consistency of parsed data files. Our method achieves
overall accuracy on a large corpus of real-world CSV files and improves the accuracy on messy CSV files by almost
compared to existing approaches, including those in the Python standard library. [\
.25***Keywords —*** Data Wrangling, Data Parsing, Comma Separated Values]{}
author:
- 'Gerrit J.J. van den Burg'
- 'Alfredo Naz[á]{}bal'
- Charles Sutton
bibliography:
- 'references.bib'
---
*CSV is a textbook example of how *not* to design a textual file format.*
— The Art of Unix Programming, .
Introduction
============
The goal of data science is to extract valuable knowledge from data through the use of machine learning and statistical analysis. Increasingly however, it has become clear that in reality data scientists spent the majority of their time importing, organizing, cleaning, and wrangling their data in preparation for the analysis . Collectively this represents an enormous amount of time, money, and talent. As the role of data science is expected to only increase in the future, it is important that the mundane tasks of data wrangling are automated as much as possible. It has been suggested that one reason that data scientists spent the majority of their time on data wrangling issues is due to what could be called the *double Anna Karenina principle*: “every messy dataset is messy in its own way, and every clean dataset is also clean in its own way” .[^1] Because of the wide variety of data quality issues and data formats that exist (“messy in its own way”), it is difficult to re-use data wrangling scripts and tools, perhaps explaining the manual effort required in data wrangling.
This problem can be observed even in the earliest and what might be considered the simplest stages of the data wrangling process, that of loading and parsing the data in the first place. In this work, we focus as an example on comma-separated value (CSV) files, which despite their deceptively simple nature, pose a rich source of formatting variability that frustrates data parsing. CSV files are ubiquitous as a format for sharing tabular data on the web; based on our data collection, we conservatively estimate that GitHub.com alone contains over 19 million CSV files. Open government data repositories make increasingly more datasets available and often present their data in CSV format.[^2] Advantages of CSV files include their simplicity and portability, but despite some standardization effort [RFC 4180; @shafranovich2005common], a wide variety of subtly incompatible variations of CSV files exist, including the 39 different dialects among the CSV files in our data. For example, we observe that values can be separated by commas, semicolons, spaces, tabs, or any other character, and can be surrounded by quotation marks or apostrophes to guard against delimiter collision or for no reason at all. Because so many variations of the CSV file format exist, manual inspection is often required before a file can be loaded into an analysis program.
![Illustration of some of the variations of real-world CSV files. See the main text for a description of each of the files. \[fig:csv\_weird\]](weird_csv){height="8.9cm"}
Every messy CSV file is indeed messy in its own way, as we see from analysing a corpus of over 18,000 files in our study (Section \[sec:comparison\]). Figure \[fig:csv\_weird\] illustrates a few of the variations and problems that real-world CSV files exhibit by presenting simplifications of real-world files encountered during this study. Figure \[fig:csv\_weird\](a) illustrates a normal CSV file that uses comma as the delimiter and has both empty and quoted cells. Figure \[fig:csv\_weird\](b) shows a variation encountered in this study that uses the caret symbol as delimiter and the tilde character as quotation mark. Next, Figure \[fig:csv\_weird\](c) illustrates an *ambiguous* CSV file: the entire rows are surrounded with quotation marks, implying that the correct interpretation is a single column of strings. However, if the quotation marks are stripped a table appears where values are separated by the comma. Figure \[fig:csv\_weird\](d) illustrates a file with comment lines that have the hash symbol as a prefix. Figure \[fig:csv\_weird\](e) is adapted from and illustrates a file where multiple choices for the delimiter result in the same number of rows and columns (the semicolon, space, comma, and pound sign all yield three columns). Finally, Figure \[fig:csv\_weird\](f) illustrates a number of issues simultaneously: quote escaping both by using an escape character and by using double quotes, delimiter escaping, and multi-line cells.
In this paper, we present a method for automatically determining the formatting parameters, which we call the *dialect*, of a CSV file. Our method is based on a novel consistency measure for parsed data files that allows us to search the space of dialects for one under which the parsed data is most consistent. By *consistency* here we consider primarily (a) the shape of the parsed data, which we capture using an abstraction called *row patterns*, and (b) the data types of the cells, such as integers or strings. This aims to capture how a human analyst might identify the dialect: searching for a character that results in regular row patterns and using knowledge of what real data “looks like”.
It may surprise the reader that CSV parsing is an open problem. However, even though almost every programming language provides functionality to read data from CSV files, very few are robust against the many issues encountered in real-world CSV files. In nearly every language the user is responsible for correctly setting the parameters of the parsing procedure by manually providing function arguments such as the cell delimiter, the quote character, whether or not headers exist, how many lines to skip before the data table starts, etc. To the best of our knowledge, Python is the only programming language whose standard library supports automatically detecting some of these parsing parameters through a so-called dialect “sniffer”. In all other languages the user is responsible for correctly choosing the parameters when the CSV file doesn’t adhere exactly to the format in the RFC 4180 specification, or otherwise risks that the file is loaded incorrectly or not at all. Moreover, we shall see below that even the Python dialect sniffer is not robust against many real-world CSV files. This means that in practice almost every file requires manual inspection before the data can be loaded, since it may contain a non-standard feature. Automatic detection of the dialect of CSV files aims to alleviate this issue.
In the formalism for CSV parsing we present in this paper, we distinguish two aspects of what makes the format of a file “messy”. First, CSV files have a dialect that captures how tabular data has been converted to a text file. This dialect contains the delimiter, the quote character, and the escape character. Knowing this dialect is the first step to importing the data from the file as it provides all the information necessary to convert the raw text file into a matrix of strings. Second, CSV files may contain headers, have additional comment text (potentially indicated by a prefix character), contain multiple tables, encode numeric data as strings, or contain any of the myriad other issues that affect how the tabular data is represented. Thus while the dialect is sufficient to recover a matrix of strings from the file, the full CSV parsing problem contains additional steps to recover the original data.
The problem of dialect detection, which is our focus here, is a subproblem of CSV parsing in general. The broader CSV parsing problem includes file encoding detection, dialect detection, table detection, header detection, and cell type detection. We restrict ourselves to dialect detection for two reasons. First, while encoding detection for CSV files may be slightly different than the general encoding detection problem due to different character frequencies, we consider this problem sufficiently solved for our purposes and employ the method of . With this in mind, the next problem to solve is that of dialect detection. Second, the vast majority of CSV files have only a single table and thus solving dialect detection is a more pressing issue than table detection.
A CSV file is created by some CSV “formatter” (either a program or a human) using a particular dialect. To load the file and decide on the dialect a human analyst would use their understanding of the CSV format and its many variations, an understanding of what the data embedded in the file represents, and potentially years of experience of dealing with messy CSV files. Automating this process is non-trivial: we receive a text file from an unknown source, created with an unknown formatter using an unknown dialect, that contains unknown data, and are asked to choose the parameters that allow a faithful reconstruction of the original data. Considering the large number of unknowns in this process, it may not be surprising that many programming languages do not attempt to automate this process at all and instead leave the user to deal with the issue. It may also explain why only very little attention has been paid to this problem in the literature.
This paper is structured as follows. In Section \[sec:related\_work\] we present an overview of related work on both table detection and CSV parsing. Next, Section \[sec:problem\_statement\] gives a formal mathematical description of CSV parsing that presents dialect detection as an inverse problem. Our proposed data consistency measure for dialect detection is presented in Section \[sec:consistency\]. Results of a thorough comparison of our method with the few existing alternatives are presented in Section \[sec:comparison\]. Section \[sec:conclusion\] concludes the paper.
Related Work {#sec:related_work}
============
Only very few publications have paid any attention to the problem of CSV parsing. explore a large collection (200K) of CSV files extracted from open data platforms of various governments. The authors use a number of heuristics to explore and parse the files. Dialect detection is done with the Python built-in CSV sniffer mentioned above, but the accuracy of this detection is not evaluated. Despite the extensive analysis of this large corpus of CSV files, the authors do not present a novel CSV parser or dialect detection method based on their heuristics.
A recent paper that does present a novel CSV parser and specifically addresses the problem of messy CSV files is that of . They present HypoParsr, a so-called “multi-hypothesis” CSV parser that builds a tree of all possible parser configurations and subsequently scores each configuration with a heuristic method. The authors evaluate their parser on 64 files with known ground truth and a corpus of 15,000 files from the UK open government portal without ground truth. While they achieve a high parsing “accuracy” on the latter corpus of CSV files, this does not necessarily reflect a correct parsing of the CSV files due to the absence of ground truth. In our preliminary analysis of this parser we observed that its implementation is rather fragile and does not return a parsing result for many messy CSV files. Moreover, on several files from the UK open government portal we observed that while the parser returned without error, it did in fact quietly drop several rows from the result. This illustrates the difficulty of the CSV parsing problem: without a large enough set of real-world messy CSV files it is hard to create a parser that is robust against all the problems and variations that can occur. A closely related subject to CSV parsing and dialect detection is that of detecting tables in free text. Early work on this topic includes the work of who address the problem of identifying the *location* of a table in free text. They use a decision tree classifier based on features extracted from the content of the text, such as whether or not a character is next to whitespace, or a character is a symbol, etc. Later work by applies a similar strategy for constructing features but instead uses conditional random fields and expands the problem by also identifying the semantic role of each row in the identified table (i.e. header, data row, etc.). However, CSV parsing differs from identifying tables in free text because CSV files have more structure and flexibility than free text tables. CSV files use a specific character to delimit cells and can employ the quoting mechanism to let cells span multiple lines and guard against delimiter collision. This explains why the methods mentioned above for detecting tables in free text cannot be readily applied to CSV parsing.
Other work related to CSV parsing is the work on DeExcelerator by [@eberius2013deexcelerator]. This work focuses mainly on spreadsheets in Excel format, but can also handle CSV files when the correct dialect is provided by the user. The DeExcelerator program then extracts tables from the spreadsheet and in the process performs header recognition, data type recognition, and value extrapolation, and other operations based on heuristic rules. In follow-up work, present a method for classifying the role of each cell in the spreadsheet (i.e. attribute, data, header, etc.) using surface-level features derived from the formatting of the text and classification methods such as decision trees and support vector machines. While the DeExcelerator package offers no method for detecting the dialect of CSV files, the methods for table and header detection are suitable to the general CSV parsing problem outlined above.
More broadly, there is some work on data wrangling in particular that can be considered here. present the PADS system for retrieving structured data from ad-hoc data sources such as log files of various systems. However, the authors explicitly mention that CSV files are not a source of ad-hoc data. Well-known work on the more general data wrangling problem is that of on the Wrangler system. Here the human-machine interaction of data wrangling is considered and a system is proposed to assist the human analyst. In follow-up work, provide a method for automatically suggesting data transformations to the analyst and additionally provide a “table suitability metric” that quantifies how well a table corresponds to a relational format. The relational format is also known as attribute-value format or “tidy data” and it is often the goal of the data wrangling process. In our experiments we compare our method to the table suitability metric of .
Finally, it is worth mentioning the few efforts that have aimed to solve the problem of CSV parsing at the source by proposing extensions or variations on the format that address some of the known issues with CSV files. A study on the use and future of CSV files on the web was performed by a working group of the World Wide Web Consortium . One of the results of this working group is a proposal to provide metadata about a CSV file through an accompanying JSON description file . A similar proposal is the CSV Dialect specification by that recommends storing the dialect of the CSV file in a separate JSON object. While these recommendations could certainly address some of the issues of CSV parsing, it requires users to specify a second file and maintaining it next to the CSV file itself. Moreover, it does not address the issues of the existing messy CSV files. Alternatives such as the CSVY format propose to add a YAML header with metadata. While this does combine the metadata and tabular data in a single file, it requires special import and export tools that may limit the adoption of these formats.
Problem Statement {#sec:problem_statement}
=================
We present a formal mathematical definition of CSV file parsing based on a generative process for how the file was constructed from the underlying data. This formal framework will define the language we subsequently use to present our methodology and will clarify the problem of CSV parsing in general. Within this framework dialect detection is presented as an inverse problem. This is a natural framing of the problem since we are interested in identifying the unknown dialect that was used to produce an observed outcome.
Let $\Sigma$ denote a finite alphabet of characters in some encoding $E$ and let $\Sigma^*$ be all the possible strings in this alphabet.[^3] Then a CSV file is an element ${\mathbf{x}} \in \Sigma^*$. A CSV file contains $N$ tuples ${\mathbf{t}}_i$ that represent elements of a product set, thus $${\mathbf{t}}_i \in \mathcal{V}_1 \times \mathcal{V}_2 \times
\cdot\cdot\cdot \times \mathcal{V}_{L_i}.$$ where the sets $\mathcal{V}_j$ are the domains of values in the tuples . These domains represent the sets that the values belong to, i.e. that floating point numbers have the domain $\mathcal{V} =
\mathbb{R}$, for instance. Note that the length of a tuple is given by $L_i$. Since CSV files can contain comments or multiple tables we cannot assume that the length of tuples is constant throughout the file. The collection of all tuples in the file is given by the array ${\mathbf{T}} = [{\mathbf{t}}_1, \ldots,
{\mathbf{t}}_N]$.
Next we define the *dialect* of CSV files. Conceptually, we can think of a CSV file as being generated by a *formatter* that takes the original data and converts it to its CSV file representation. The formatter has knowledge of the dialect ${\boldsymbol{\theta}}$ that has been used to create the CSV file, and it consists of two stages. In Stage 1 the formatter converts the data ${\mathbf{T}}$ to a string-only representation ${\mathbf{C}}$ where the domains in the tuples are the string domain (i.e. $\Sigma^*$). In Stage 2 this string-only representation is converted to a CSV file ${\mathbf{x}}$. When we wish to retrieve the data from the CSV file the information about both the formatter and the dialect has been lost. This leads us to the following definition.
\[def:dialect\] Given a CSV file ${\mathbf{x}} \in \Sigma^*$, the *dialect* ${\boldsymbol{\theta}}$ represents all parameters needed for a one-to-one mapping from an array of tuples of strings ${\mathbf{C}} = [{\mathbf{c}}_1,
\ldots, {\mathbf{c}}_N]$ to the file ${\mathbf{x}}$, thus $$f_{{\boldsymbol{\theta}}}({\mathbf{C}}) = {\mathbf{x}}.$$ Here $f_{{\boldsymbol{\theta}}}$ represents Stage 2 of the formatter. The inverse operation $f_{{\boldsymbol{\theta}}}^{-1}({\mathbf{x}}) = {\mathbf{C}}$ corresponds to converting the file ${\mathbf{x}}$ to an array of tuples of strings.
We may further define a function $g$ that is used in Stage 1 of the formatter to convert the data to the string representation, $$g({\mathbf{T}}) = {\mathbf{C}}.$$ The inverse of this function, $g^{-1}({\mathbf{C}})$, returns the elements in the tuples to their original representation and is in practice often achieved through type casting. In practice one may additionally wish to specify an inverse operation $h^{-1}({\mathbf{T}})$ that extracts from ${\mathbf{T}}$ the actual *tabular data* by identifying the location of rectangular tables in ${\mathbf{T}}$, detecting the presence of column headers, removing unnecessary comments, etc. This leads us to the following general definition.
Let ${\mathbf{x}} \in \Sigma^*$ be a CSV file. The problem of *CSV parsing* then reduces to evaluating $$h^{-1}(g^{-1}(f_{{\boldsymbol{\theta}}}^{-1}({\mathbf{x}}))).$$
Thus we see that the first step in parsing CSV files is indeed the detection of the correct dialect ${\boldsymbol{\theta}}$. In the remainder of this paper we will therefore focus our attention on this step. In the process we will often use the parsing function $f^{-1}_{{\boldsymbol{\theta}}}$ to parse the file for a given dialect. Although the exact definition of this function is not the main focus of the paper, it is worth mentioning that we base our implementation on the CSV parser in the Python standard library. Minor modifications of this code were made to simplify the dialect and handle certain edge cases differently, see Appendix \[app:parser\] for more details.
In this work we consider a dialect of three components: the delimiter ($\theta_d$), the quote character ($\theta_q$), and the escape character ($\theta_e$). The *delimiter* is used to separate cells (i.e. values in the tuple), the *quote character* is used to enclose cells that may contain the delimiter, a newline character, or neither, and the *escape character* can be used to achieve nested quotation marks. Each of these parameters can be absent in a CSV file, thus $\theta_d, \theta_q,
\theta_e \in \Sigma \cup \{ \varepsilon \}$ with $\varepsilon$ the empty string (note that $\theta_d = \varepsilon$ for a file with a single column of data).[^4] While some existing parsers include other components of the dialect, such as whether or not nested quotes are escaped with double quotes, we found that it was possible to formulate $f^{-1}_{{\boldsymbol{\theta}}}$ with only the three components given above. Moreover, some of these other components are more accurately described as parameters to the functions $g^{-1}$ or $h^{-1}$.
A Consistency Measure for Dialect Detection {#sec:consistency}
===========================================
Having illustrated above the difficulty of dialect detection in general, we present here our solution based on two components: row length patterns and data type inference. The main idea that if a CSV file is parsed with the correct dialect rather than an incorrect one, the resulting tuples in the parsed data will appear more consistent. By *consistency* here we consider primarily two aspects: (a) the parsed rows should have similar length, and (b) cells within the tuples should have the same data type, such as integers or strings. To that end, we propose a measure of consistency over parsed data, which has two components, one that measures each of these two kinds of consistency. Then we search the space of possible dialects for the dialect in which the parsed file is most consistent. This aims to capture both aspects of how a human analyst would identify the dialect: searching for a character that results in regular row patterns and using knowledge of what real data “looks like”.
More formally, we associate with each dialect ${\boldsymbol{\theta}}$ a consistency measure $Q({\mathbf{x}}, {\boldsymbol{\theta}}) = P({\mathbf{x}}, {\boldsymbol{\theta}}) \cdot T({\mathbf{x}},
{\boldsymbol{\theta}})$, where $P$ is a *pattern score* and $T$ is a *type score*. The estimate of the correct dialect is then obtained as $$\label{eq:problem}
{\boldsymbol{\hat{\theta}}} = \operatorname*{arg\,max}_{{\boldsymbol{\theta}} \in \Theta_{{\mathbf{x}}}}
Q({\mathbf{x}}, {\boldsymbol{\theta}}).$$ Algorithm \[alg:heuristic\] shows pseudocode for our search algorithm. The algorithm uses the fact that the type score $T({\mathbf{x}}, {\boldsymbol{\theta}})$ is between 0 and 1 to speed up the search. In specific instances multiple dialects can receive the same value for $Q({\mathbf{x}}, {\boldsymbol{\theta}})$ due to limitations in the type score. However, some of these ties can be broken reliably and we will expand on this below as well.
$\Theta_{{\mathbf{x}}} \gets$ $\mathcal{H} \gets \emptyset$ $Q_{max} \gets -\infty$ $P({\mathbf{x}}, {\boldsymbol{\theta}}) \gets$ **continue** $T({\mathbf{x}}, {\boldsymbol{\theta}}) \gets$ $Q({\mathbf{x}}, {\boldsymbol{\theta}}) \gets P({\mathbf{x}},
{\boldsymbol{\theta}})T({\mathbf{x}}, {\boldsymbol{\theta}})$ $\mathcal{H} \gets \{ {\boldsymbol{\theta}} \}$ $Q_{max} \gets Q({\mathbf{x}}, {\boldsymbol{\theta}})$ $\mathcal{H} \gets \mathcal{H} \cup \{
{\boldsymbol{\theta}} \}$ ${\boldsymbol{\theta}} \in \mathcal{H}$
Pattern Score
-------------
![Illustration of the data consistency measure for different dialect on a constructed example. The figure shows how different dialects can result in different row patterns and pattern scores, as well as different type scores. The difference between ${\boldsymbol{\theta}}_2$ and ${\boldsymbol{\theta}}_3$ is due to the fact that the string [`N`/A]{} belongs to a known type, but the string [“N/A”]{} does not.[]{data-label="fig:score_ill"}](score_illustration){width="\textwidth"}
The pattern score is the main driver of the data consistency measure. It is based on the observation that since CSV files generally contain tables, we expect to find rows with the same number of cells when we select the correct dialect. For a given dialect we therefore parse the file and determine the number of cells in each row. This parsing step takes nested quotes and escape characters into account, and returns for each row a so-called *row pattern*. See Figure \[fig:score\_ill\] for an illustration. The row patterns capture the repeated pattern of cells and delimiters and interpret quotes and escape characters where needed while abstracting away the content of the cells.[^5] Notice how in Figure \[fig:score\_ill\] the parsing result for ${\boldsymbol{\theta}}_1$ gives different row patterns than for ${\boldsymbol{\theta}}_2$ and ${\boldsymbol{\theta}}_3$. Each distinct row pattern $k$ has a length $L_k$, that is exactly one higher than the number of delimiters in the pattern. The number of times row pattern $k$ occurs in the file is called $N_k$ and the total number of distinct row patterns is denoted by $K$.
These properties are used to define the pattern score, as follows $$\label{eq:pattern_score}
P({\mathbf{x}}, {\boldsymbol{\theta}}) = \frac{1}{K} \sum_{k=1}^K N_k \frac{L_k -
1}{L_k}.$$ The pattern score is designed to favour row patterns that occur often and those that are long, and also to favour fewer row patterns. The reasoning behind this is that when the correct row pattern is chosen, we expect to observe few distinct row patterns, and that the ones that we do observe occur often (i.e. in tables). Figure \[fig:score\_ill\] illustrates this aspect, as the correct delimiter yields a single row pattern that occurs throughout the entire file. By including the row length ratio we favour longer row patterns over shorter ones. This is included because a long pattern indicates a regular pattern of delimiters and cells, whereas a short pattern might indicate an incorrectly chosen delimiter.
While this function works well for CSV files that contain tables, it gives a value of $0$ when the entire file is a single column of data. To handle these files in practice, we replace the numerator by $\max\{\alpha, L_k - 1\}$ with $\alpha$ a small constant. The value of $\alpha$ must be chosen such that single-column CSV files are detected correctly, while avoiding false positive results that assume regular CSV files are a single column of messy data. It was found empirically that $\alpha = 10^{-3}$ achieves this goal well.
Type Score
----------
While the pattern score is the main component of the data consistency measure, Figure \[fig:score\_ill\] shows that the type score is essential to obtaining state-of-the-art results. The goal of the type score is to act as a proxy for understanding what the cells of the file represent, capturing whether a dialect yields cells that “look like real data”. To do this, the type score measures the proportion of cells in the parsed file that can be matched to a data type from a set $\mathcal{T}$.
In our implementation, $\mathcal{T}$ includes empty cells, numbers in various formats, URLs, email addresses, percentages, currency values, times, dates, and alphanumeric strings (see Appendix \[app:types\]). We use regular expressions to detect these types, and denote the mapping from a string ${\mathbf{z}}$ to a type by $\mu({\mathbf{z}})$. Then the type score is defined as $$\label{eq:type_score}
T({\mathbf{x}}, {\boldsymbol{\theta}}) = \frac{1}{M} \sum_{i=1}^N \sum_{j=1}^{L_i}
I[\mu(c_{i,j}) \in \mathcal{T}]$$ where $M$ is the total number of cells in the parsing result ${\mathbf{C}}$, $c_{i,j}$ is the $j$-th value of tuple ${\mathbf{c}}_i$, and $I[\cdot]$ denotes the indicator function that returns 1 if its argument is true and 0 otherwise. While the type detection algorithm can identify many data types, it is infeasible to encapsulate all possible data types. Thus, the situation may arise where no type can be detected for any cell in the parsing result. This would result in a consistency measure $Q({\mathbf{x}}, {\boldsymbol{\theta}}) = 0$, even though the pattern score may give high confidence for a dialect. To avoid this problem in practice, we replace $T({\mathbf{x}}, {\boldsymbol{\theta}})$ by $\tilde{T}({\mathbf{x}}, {\boldsymbol{\theta}}) = \max\{\beta, T({\mathbf{x}}, {\boldsymbol{\theta}})\}$, with $\beta = 10^{-10}$.
Tie breaking
------------
Unfortunately the value of $Q({\mathbf{x}}, {\boldsymbol{\theta}})$ can be the same for different dialects even in the presence of the type score, due to the fact that the latter is necessarily incomplete. However in some cases these ties can be broken reliably. For example, if the same score is returned for two dialects that only differ in the quote character, then we can check whether or not the quote character has any effect on the parsing result. If it doesn’t, then the quote character only occurs *inside* cells and does not affect the parsing outcome ${\mathbf{C}} = f_{{\boldsymbol{\theta}}}^{-1}({\mathbf{x}})$. The correct solution is then to ignore the quote character. Similar tie breaking rules can be constructed for ties in the delimiter or the escape character.
Potential Dialects
------------------
In (\[eq:problem\]) we select the best dialect from a set of dialects $\Theta_{{\mathbf{x}}}$. It’s worthwhile to expand briefly on how $\Theta_{{\mathbf{x}}}$ is constructed. While in general it is the product set of all potential delimiters, quote characters, and escape characters in the file, there are small optimizations that can be done to shrink this parameter space. Doing this speeds up dialect detection and reduces erroneous results.
${\mathbf{\tilde{x}}} \gets$ $\mathcal{D} \gets$ $\mathcal{Q} \gets$ $\mathcal{E}_{d,q} \gets \{\varepsilon\} \quad \forall
\theta_d, \theta_q \in \mathcal{D}\times\mathcal{Q}$ $u, v \gets {\mathbf{\tilde{x}}}[i], {\mathbf{\tilde{x}}}[i+1]$ $\mathcal{E}_{d,q} \gets \mathcal{E}_{d,q} \cup \{u\}$ $\Theta_{{\mathbf{x}}} \gets \emptyset$ $\Theta_{{\mathbf{x}}} \gets
\Theta_{{\mathbf{x}}} \cup \{(\theta_d,
\theta_q, \theta_e)\}$ $\Theta_{{\mathbf{x}}}$
Algorithm \[alg:potential\_dialects\] presents pseudocode for the construction of $\Theta_{{\mathbf{x}}}$. The functions used in the algorithm are described in more detail in Appendix \[app:additional\_algorithms\]. The algorithm proceeds as follows. As a preprocessing step, URLs are filtered from the file to remove potential delimiters that only occur in URLs. Next, potential delimiters are found with [<span style="font-variant:small-caps;">GetDelimiters</span>]{} by comparing the Unicode category of each character with a set of allowed categories and filtering out characters that are explicitly blocked from being delimiters. Next, potential quote characters are selected by matching with a set of allowed quote characters. Subsequently, the escape characters are those that occur at least once before a delimiter or quote character and that belong to the Unicode “Punctuation, other” category . Finally, $\Theta_{{\mathbf{x}}}$ is the product set of the potential delimiters, quote characters, and escape characters, with the exception that dialects with delimiters that never occur outside the quote characters in the dialect are dropped. This latter step removes more false positives from the set of potential dialects.
Experiments {#sec:comparison}
===========
In this section we present the results of an extensive comparison study performed to evaluate our proposed method and existing alternatives. Since variability in CSV files is quite high and the number of potential CSV issues is large, an extensive study is necessary to thoroughly evaluate the robustness of each method. Moreover, since different groups of users apply different formats, it is important to consider more than one source of CSV files.
Our method presented above was created using a *development set* of CSV files from two different corpora. The experimental section presents results from a comparison on an independent *test set* that was unknown to the authors during the development of the method. This split aims to avoid overfitting of our method and report the accuracy of our method accurately.
In an effort to make our work transparent and reproducible, we release the full code to replicate the experiments through an online repository.[^6] This will also enable other researchers to easily build on our work and allow them to use our implementations of alternative detection methods.
Data
----
Data was collected from two sources: the UK government open data portal ([data.gov.uk](data.gov.uk)) and GitHub ([github.com](github.com)). These represent different groups of users (government employees vs. programmers) and it is expected that we find differences in both the format and the type of content of the CSV files. Data was collected by web scraping in the period of May/June 2018, yielding tens of thousands of CSV files. From these corpora of CSV files we randomly sampled a development set (3776 files from UKdata and 4536 files from GitHub). These were used to develop and fine-tune the consistency measure presented above, and in particular were used to fine-tune the type detection engine.
When development of the method was completed, an independent test set was sampled from the two sources (5000 files from each corpus). This test set is similar to the development data, with one exception. During development we noticed that the GitHub corpus often contained multiple files from the same code repository. These files usually have the same structure and dialect, thus representing essentially a single example. Therefore, during construction of the test set a limit of one CSV file per GitHub repository was put in place. Thus we expect that the test set has greater variability and difficulty than the development set. It is worth emphasizing that the test set was not used in any way during the development of the method.
Ground Truth
------------
To evaluate the detection method, we needed to obtain ground truth for the dialects of the CSV files. This is done through both automated and manual ways. The automated method is based on very strict functional tests that allow only simple CSV files with elementary cell contents. For instance, in one test we require that a file has a constant number of cells per row, no missing values, no nested quotes, etc. These automatic tests are sufficient to accurately determine the dialect of about a third of the CSV files. For the remainder of the files manual labelling was employed using a terminal-based annotation tool. Files that could not reasonably be considered CSV files were removed from the test set (i.e. HTML, XML, or JSON files, or simple text files without any tabular data). The same holds for files for which no objective ground truth could be established, such as files formatted similarly to the example in Figure \[fig:csv\_weird\](c). After filtering out these cases the test set contained
files from GitHub.com and
files from the UK government open data portal.
Alternatives
------------
Since the dialect detection problem has not received much consideration in the literature, there are only a few alternative methods to compare to. We briefly present them here.
### Python Sniffer
Python’s built-in CSV module contains a so-called “Dialect Sniffer” that aims to automatically detect the dialect of the file[^7]. This method detects the delimiter, the quote character, whether or not double quotes are used, and whether or not whitespace after the delimiter can be skipped. There are two methods used to detect these properties. The first method is used when quote characters are present in the file and detects adjacent occurrence of a quote character and another character (the potential delimiter). In this method the quote character and delimiter that occur most frequently are chosen. The second method is used when there are no quote characters in the file. In this case a frequency table is constructed that indicates how often a potential delimiter occurs and in how many rows (i.e. comma occurred $x$ times in $y$ rows). The character that most often matches the expected frequency is considered the delimiter, and a fallback list of preferred delimiters is used when a tie occurs. The detector also tries to detect whether or not double quoting is used within cells to escape a single quote character. This is done with a regular expression that can run into “catastrophic backtracking” for CSV files that end in many empty delimited rows. Therefore we place a timeout of two minutes on this detection method (normal operation never takes this long, so this restriction only captures this specific failure case). As mentioned, this method tries to detect when whitespace following the delimiter can be stripped. We purposefully do not include this in our method as the CSV specification states, “Spaces are considered part of a field and should not be ignored.” .
### HypoParsr
HypoParsr is the first dedicated CSV parser that takes the problem of dialect detection and messy CSV files into account.[^8] The method uses a hierarchy of possible parser configurations and a set of heuristics to try to determine which configuration gives the best result. Unfortunately it is not possible use the HypoParsr R package to detect the dialect without running the full search that also includes header, table, and data type detection. Therefore, we run the complete program and extract the dialect from the outcome. This means however that both the running time and any potential failure of the method are affected by subsequent parsing steps and not just by the dialect detection. This needs to be kept in mind when reviewing the results. As the method can be quite slow, we add a timeout of 10 minutes per file. Finally, the quote character in the dialect is not always reported faithfully in the final parsing result, since the underlying parser can strip quote characters automatically. We developed our own method to check what quote character was actually used during parsing.
### Wrangler
In a table suitability metric is presented that balances consistency of cell types against the number of empty cells and cells with potential delimiters in them. This can therefore be used to detect the dialect of CSV files by selecting the dialect that does best on this metric. The suitability metric uses the concept of column type homogeneity, i.e. the sum of squares of the proportions of each data type in a column. Since the exact type detection method used in the paper is not available, we use our type detection method instead.
### Variations
In addition to our complete data consistency measure, we also consider several variations to investigate the effect of each component. Thus, we include a method that only uses the pattern score and one that only uses the type score. We also include a variation that does not use tie-breaking.
Evaluation
----------
The methods are evaluated on the accuracy of the full dialect as well as on the accuracy of each component of the dialect. Note that a method can either fail by detecting the dialect incorrectly or it can fail by not returning a result at all. The latter case can happen due to a timeout or an exception in the code (for the Python Sniffer or HypoParsr), or due to a tie in the scoring measure (for the Wrangler suitability metric or our method). Both types of failure are considered to represent an incorrect dialect detection.
Results
-------
We describe the results by focusing on dialect detection accuracy, robustness of the methods, accuracy on messy CSV files, and runtime. Unless explicitly stated otherwise, all results reflect performance on the test dataset.
### Detection Accuracy
[@c@]{}
\
\
[**(a)** GitHub corpus]{}
[@c@]{}
\
\
[**(b)** UKdata corpus]{}
The accuracy of dialect detection is shown in Tables \[tab:accuracies\](a) and \[tab:accuracies\](b) respectively for the GitHub and UKdata corpora. We see that for both corpora and for all properties, our full data consistency method outperforms all alternatives. One exception occurs for the UKdata corpus, where the pattern-only score function yields a marginally higher accuracy on detecting the escape character. It is furthermore apparent from these tables that the GitHub corpus of CSV files is more difficult than the UKdata corpus. This is also reflected in the number of dialects observed in these corpora:
different dialects were found in the UKdata corpus vs.
in the GitHub corpus. We postulate that this difference is due to the nature of the creators of these files. CSV files from the UK government open data portal are often created using standard tools such as Microsoft Excel or LibreOffice, and therefore are more likely to adhere to the CSV format . On the other hand, the creators of the files in the GitHub corpus are more likely to be programmers and data scientists who may use non-standard or custom-made tools for importing and exporting their CSV files and may therefore use different formatting conventions. It is interesting to note that even though the files in the UKdata corpus can be considered more “regular”, we still achieve a considerable increase in detection accuracy over standard approaches.
Regarding the different variants of our method, we observe that the pattern score is in many cases almost as good as the full consistency measure. This confirms our earlier statement that the pattern score is the main driver of the method and that the type score serves mainly to further improve the accuracy. It is also clear that the type score alone does not suffice to accurately detect the dialect. The variant of our method that does not use tie-breaking yields a lower overall accuracy on both corpora, indicating the importance of tie-breaking in our method. Additional result tables are presented in Appendix \[app:results\] that show the accuracy separated by human vs. automatic detection of ground truth. Unsurprisingly, all methods perform better on files for which the dialect could be detected through automatic means than on those that required human annotation.
### Failure
![Percentage of failure for each of the detection methods on both corpora. Note that this shows how often the detection methods completely failed, not when it detected the parameters incorrectly. \[fig:failures\]](./results/test/analysis/figures/fail_combined){width="\textwidth"}
Analysing the failure cases for the detection methods provides insight into their robustness. Figure \[fig:failures\] shows the proportion of files where no detection was made either due to a failure of the program or due to unbreakable ties in the scoring metric. This figure shows that HypoParsr fails significantly more often on files from the GitHub corpus than files from the UKdata corpus. This can be due to the fact that HypoParsr was developed using files from the UKdata corpus, and because these files are generally easier to parse. However, since it is not possible to only use HypoParsr for dialect detection, it could also be the case that failures occur in a later stage of the parsing process. In fact, investigating the failure cases for this method further we observe that no result was obtained in
of the failures and that the timeout needed to be applied in
. Similarly, for the Python Sniffer these values are
and
respectively. Note further that our proposed method fails on no files in the UKdata corpus and on only
of files in the GitHub corpus. Failure in our method is only possible when ties occur, and the addition of both the type score and tie-breaking procedure ensures that this is rare. The high failure rate for the suitability metric from is exclusively due to ties in the scoring metric. Tie-breaking was not used here, as this is a feature of our method.
As Table \[tab:accuracies\](a) shows, an incorrect detection in our method occurs mostly due to an incorrect detection of the delimiter. Analysing these failure cases in more detail reveals that of the files where the delimiter was detected incorrectly, the majority of files had the comma as the delimiter. In these cases, an incorrect detection occurred mostly because our method predicted the space as the delimiter. Some of these files indeed had a regular pattern of whitespace in the cells and received a high type score as well, causing the confusion. Other files had the comma as true delimiter, but had only one column of data. In these cases the true comma delimiter could be deduced by the human annotator from a header or because certain cells that contained the comma were quoted, but this type of reasoning is not captured by the data consistency measure. In other failure cases, the pattern score predicted the correct delimiter, but the type score gave a low value, resulting in a low value of the consistency measure. Some of these failure cases can certainly be addressed by improving the type detection procedure.
### Messy CSV Files
[@c@]{}
\
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[**(a)** GitHub corpus]{}
[@c@]{}
\
\
[**(b)** UKdata corpus]{}
Separating the files into those that are messy and those that follow the CSV standard further illustrates how our method improves over existing methods (see Table \[tab:std\_messy\]). CSV files are considered “standard” when they use the comma as the delimiter, use either no quotes or the `"` character as quote character, and do not use the escape character. The table also highlights the difference in the amount of non-standard files in the different corpora and reiterates that the GitHub corpus contains more non-standard CSV files. This table also emphasises the significance of our contribution, with an increase of
over the Python Sniffer for messy files averaged over both corpora. Note that on both corpora we also improve over current methods for *standard* files.
### Runtime
Evaluating the runtime of each method is also relevant. Figure \[fig:runtimes\] shows violin plots for each method for both corpora. While the figures show that HypoParsr is the slowest detection method, this is not completely accurate. Since there is no way to use the method to only detect the dialect, the reported runtime is the time needed for the entire parsing process. The Python dialect sniffer is by far the fastest method and this can be most likely attributed to its simplicity in comparison to the other methods. Finally, all variations of our method have similar runtime characteristics and slightly outperform the Wrangler suitability metric. We note that our method has not been explicitly optimised for speed, and there are likely to be improvements that can be made in this aspect. Currently our method exhibits $O(|{\mathbf{x}}||\Theta_{{\mathbf{x}}}|)$ complexity. Note however that the mean of the computation time for our method still lies well below one second, which is acceptable in almost all practical applications.
![Runtime violin plots for both corpora. The whiskers show the minimum and maximum values and the dashed lines show the median. See the note on HypoParsr in the main text. \[fig:runtimes\]](./results/test/analysis/figures/violin_combined){width="\textwidth"}
Conclusion {#sec:conclusion}
==========
We have presented a data consistency measure for detecting the dialect of CSV files. This consistency measure emphasizes a regular pattern of cells in the rows and favours dialects that yield identifiable data types. While we apply the consistency measure only to CSV dialect detection in this paper, it is conceivable that there are other applications of this measure outside this area. For instance, one can imagine identifying unstructured tables in HTML documents or in free text, or use the measure to locate the tables within CSV or Excel files.
The main challenge for today’s data scientists is the inordinate amount of time spent on preparing data for analysis. One of the difficulties they face is importing data from messy CSV files that often require manual inspection and reformatting before the data can be loaded from the file. In this paper we have presented a method for automatic dialect detection of CSV files that achieves near-perfect accuracy on a large corpus of real-world examples, and especially improves the accuracy on messy CSV files. This represents an important step towards automatically loading structured tabular data from messy sources. As such, it allows many data scientists to spend less time on mundane data wrangling issues and more time on extracting valuable knowledge from their data.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors would like to acknowledge the funding provided by the UK Government’s Defence & Security Programme in support of the Alan Turing Institute. The authors thank Chris Williams for useful discussions.
Data Type Detection {#app:types}
===================
As mentioned in the main text, we use a regular expression based type detection engine. Below is a brief overview of the different types we consider and the detection method we use for that type. The order of the types corresponds to the order in which we evaluate the type tests, and we stop when a matching type is found.
### Empty Strings {#empty-strings .unnumbered}
Empty strings are considered a known type.
### URLs and Email Addresses {#urls-and-email-addresses .unnumbered}
For this we use two separate regular expressions.
### Numbers {#numbers .unnumbered}
We consider two different regular expressions for numbers. First, we consider numbers that use “digit grouping”, i.e. numbers that use a period or comma to separate groups of thousands. In this case we allow numbers with a comma or period as thousands separator and allow for using a comma or period as radix point, respectively. Numbers in this form can not have E-notation, but can have a leading sign symbol. The second regular expression captures the numbers that do not use digit grouping. These numbers can have a leading sign symbol (`+` or `-`), use a comma or period as radix point, and can use E-notation (i.e. `123e10`). The exponent in the E-notation can have a sign symbol as well.
### Time {#time .unnumbered}
Times are allowed in `HH:MM:SS`, `HH:MM`, and `H:MM` format. The AM/PM quantifiers are not included.
### Percentage {#percentage .unnumbered}
This corresponds to a number combined with the `%` symbol.
### Currency {#currency .unnumbered}
A currency value is a number preceded by a symbol from the Unicode `Sc` category .
### Alphanumeric {#alphanumeric .unnumbered}
An alphanumeric string can follow two alternatives. The first alternative consists of first one or more number characters, then one or more letter characters, and then zero or more numbers, letters, or special characters. An example of this is the string `3 degrees`. The second alternative first has one or more letter characters and then allows for zero or more numbers, letters, or special characters. An example of this is the string `NW1 2DB`. In both alternatives the allowed special characters are space, period, exclamation and question mark, and parentheses, including their international variants.
### N/A {#na .unnumbered}
While `nan` or `NaN` are caught in the alphanumeric test, we add here a separate test that considers `n/a` and `N/A`.
### Dates {#dates .unnumbered}
Dates are strings that are not numbers and that belong to one of forty different date formats. These date formats allow for the formats `(YY)YYx(M)Mx(D)D`, `(D)Dx(M)Mx(YY)YY`,\
`(M)Mx(D)Dx(YY)YY` where `x` is a separator (dash, period, or space) and parts within parentheses can optionally be omitted. Additionally, the Chinese/Japanese date format\
[UTF8]{}[gbsn]{}`(YY)YY年(M)M月(D)D日`
and the Korean date format
[UTF8]{}[mj]{}`(YY)YY년(M)M월(D)D일`
are included.
### Combined date and time {#combined-date-and-time .unnumbered}
These are formats for joint date and time descriptions. For these formats we consider\
`<date> <time>` and `<date>T<time>` as well as those with a time zone offset appended.
Additional Algorithm Descriptions {#app:additional_algorithms}
=================================
Below we explicitly provide algorithms for functions mentioned in the main text that were omitted due to space constraints.
Selecting Potential Delimiters
------------------------------
Algorithm \[alg:potential\_delims\] gives the function for selecting potential delimiters of the file ${\mathbf{x}}$. The function [ <span style="font-variant:small-caps;">FilterURLs</span>]{} replaces URLs in the file with a single letter, to avoid URLs generating spurious potential delimiters. We use $\mathcal{B} =
\{\verb+.+, \verb+/+, \verb+"+, \textrm{\textquotesingle}\}$ and $\mathcal{C} = \{\verb+Lu+, \verb+Ll+, \verb+Lt+, \verb+Lm+, \verb+Lo+,
\verb+Nd+, \verb+Nl+, \verb+No+, \verb+Ps+, \verb+Pe+, \verb+Cc+, \verb+Co+
\}$.
${\mathbf{\tilde{x}}} \gets$ $\mathcal{D} \gets \{\varepsilon \}$ $c \gets $ $\mathcal{D} \gets \mathcal{D} \cup
\{x\}$ $\mathcal{D}$
Selecting Potential Quote Characters
------------------------------------
The function [<span style="font-variant:small-caps;">FilterURLs</span>]{} is as described in the previous section.
${\mathbf{\tilde{x}}} \gets$ $\mathcal{Q} \gets \{\textrm{\textquotesingle},
\verb+"+, \verb+~+\} \cap {\mathbf{\tilde{x}}}$ $\mathcal{Q} \gets \mathcal{Q} \cup \{\varepsilon \}$ $\mathcal{Q}$
Selecting Potential Escape Characters
-------------------------------------
Potential escape characters are those in the Unicode `Po` (punctuation, other) category that are not explicitly blocked from being considered. For the latter we use a set\
$\mathcal{W} = \{ \verb+!+, \verb+?+, \verb+"+,
\textrm{\textquotesingle}, \verb+.+, \verb+,+, \verb+;+, \verb+:+, \verb+\%+,
\verb+*+, \verb+&+, \verb+#+ \}$.
$c \gets $ `false`
Masked by Quote Character
-------------------------
The function [<span style="font-variant:small-caps;">MaskedByQuote</span>]{} in Algorithm \[alg:potential\_dialects\] is used to prune the list of potential dialects by removing those where the delimiter never occurs outside a quoted environment. This function is straightforward to implement: it iterates over all characters in the file and keeps track of where a quoted section starts and ends, while taking quote escaping into account. Given this it is straightforward to check whether the given delimiter always occurs inside a quoted section or not.
Parser {#app:parser}
------
The code we use for our CSV parser $f^{-1}_{{\boldsymbol{\theta}}}$ borrows heavily from the CSV parser in the Python standard library, but differs in a few small but significant ways. First, our parser only interprets the escape character if it proceeds the delimiter, quote character, or itself. In any other case the escape character serves no purpose and is treated as any other character and is not dropped. Second, our parser only strips quotes from cells if they surround the entire cell, not if they occur within cells. This makes the parser more robust against misspecified quote characters. Finally, when we are in a quoted cell we automatically detect double quoting by looking ahead whenever we detect a quote, and checking if the next character is *also* a quote character. This enables us to drop double quoting from our dialect and only marginally affects the complexity of the code.
Additional Results {#app:results}
==================
[@c@]{}
\
\
[(a) GitHub – human annotated]{}
[@c@]{}
\
\
[(b) GitHub – automatically annotated]{}
[@c@]{}
\
\
[(c) UKdata – human annotated]{}
[@c@]{}
\
\
[(d) UKdata – automatically annotated]{}
[^1]: This problem is related to the principle of the fragility of good things .
[^2]: survey 200,000 CSV files from open government data portals.
[^3]: The set $\Sigma^*$ is known as the Kleene closure .
[^4]: Thus we restrict ourselves to CSV files where these parameters are all a single character. CSV files that use multi-character delimiters do exist, but are extremely rare.
[^5]: This process interprets the quotes as defined by the dialect, but it can occur that a spurious quote character remains in the row pattern when an incorrect dialect is chosen. In this case, we consider patterns such as `CDCQCDC` – where `Q` denotes the quote character, `D` the delimiter, and `C` any other character – to be distinct from `CDCDC`.
[^6]: See <https://github.com/alan-turing-institute/CSV_Wrangling>.
[^7]: The dialect sniffer was developed by Clifford Wells for his Python-DSV package and was incorporated into Python version 2.3.
[^8]: An R package for HypoParsr exists, but it was retracted from the R package repository on the request of the package maintainer. We nonetheless include the method in our experiments using the last available version.
|
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---
author:
- 'Yu.N. Uzikov, [V.I. Komarov]{}, F.Rathmann,'
- 'H. Seyfarth'
date: 'Received: date / Revised version: date'
title: ' Short-range NN-properties in the processes $pd\to dp$ and $pd\to (pp)n$'
---
[leer.eps]{} gsave 72 31 moveto 72 342 lineto 601 342 lineto 601 31 lineto 72 31 lineto showpage grestore
Introduction {#intro}
============
The structure of the lightest nuclei at short distances in the nucleon overlap region $r_{NN}< 0.5 $ fm, [*i. e.*]{} at high relative momenta $q_{NN}\sim 1/r_{NN}> 0.4 $ GeV/c between the nucleons, is a fundamental problem of nuclear physics. The structure can be tested by electromagnetic probes at high transferred momenta. However, a self-consistent picture of electro- and photo-nuclear processes is not yet developed due to the unknown strength of the meson-exchange currents. Hadron-nucleus collisions can give important independent informations. The theoretical analysis of hadron processes is complicated by initial and final state interactions and the excitation/de-excitation of nucleons in intermediate states. For instance, a large contribution of the double $pN$ scattering with excitation of the $\Delta(1232)$- resonance was found in proton-deuteron backward elastic scattering $pd\to dp$ at $\sim 0.5$ GeV [@cwilkin69; @kls; @bdillig; @imuz88; @uz98]. At higher beam energies the role of heavier baryon resonances is expected to increase. Unfortunately, these contributions are not well controlled in theory due to the rather poor information about the $pN\rightleftharpoons NN^*$and $pN\rightleftharpoons N\Delta$ amplitudes. To some extent these effects are taken into account in the one-pion exchange (OPE) model [@cwilkin69] with a virtual subprocess $pp\to d\pi^+$, but another important contribution, i.e. the one-nucleon-exchange (ONE) amplitude, cannot be included in this model.
To minimize these complicating effects, it was proposed [@imuz90] to study the deuteron breakup reaction $pd\to (pp)n$ in the kinematics of backward elastic $pd$ scattering. For low excitation energies $E_{pp} \leq 3$ MeV, the final $pp$ pair can be assumed to be in the $^1S_0$ spin singlet (isotriplet) state. This feature, in contrast to the $pd\to dp$ process, results in a considerable suppression of the $\Delta-$ (and $N^*)-$ excitation amplitudes by the isospin factor 1/3 in comparison with the one-nucleon exchange (ONE). Recently it was shown [@uzzhetf] that the same suppression factor acts for a broad class of diagrams with isovector meson–nucleon rescattering in the intermediate state including the excitation of any baryon resonance. Furthermore, the node in the half-off-shell $pp(^1S_0)$ scattering amplitude at the off-shell momentum $q\sim 0.4$ GeV/c results in remarkable irregularities in the spin observables and leads to a dip in the unpolarized cross section for the ONE mechanism [@imuz90; @uz2002]. In the $pd\to dp$ and $pd\to pX$ processes, the node in the deuteron S wave is hidden by the large contribution of the D wave. The irregularities of the observables allow new studies of (i) the commonly used potentials of the NN interaction at short distances and (ii) possible contributions from $N^*$- exchanges [@kk] and exotic three-baryon states [@kls].
The first data on the reaction $pd\to (pp)n$ at high beam energies $T_p=0.6 - 1.9$ GeV with forward emission of a fast proton pair of low excitation energy $E_{pp}$ were obtained at ANKE-COSY [@vikomarov]. Using a largely model-independent approach based on the Migdal-Watson and the Fäldt-Wilkin FSI theory, we discuss the relative strength of the measured singlet in comparison to the triplet channel. We present also the results of a calculation of the deuteron breakup cross section performed within the main mechanisms of the $pd\to dp$ process [@uz98].
The FSI theory {#sec:1}
===============
At excitation energies around 1 MeV, the $pd\to(pn)p$ cross section is strongly influenced by the $np$ FSI. The resulting peak is well described by the Migdal-Watson formulae [@Watson; @GW] which take into account the nearby poles in the FSI triplet ([*t*]{}) and singlet ([*s*]{}) $pn-$scattering amplitudes $$\label{mwatson}
d\sigma_{s,t} =FSI_{s,t}(k^2)\, K\, |A_{s,t}|^2.$$ Here $A_{s,t}$ is the production matrix element for the singlet and triplet state, $K$ is the kinematical factor, and $FSI_{s,t}$ is the Goldberger-Watson factor [@GW] $$\label{gwfsi}
FSI_{s,t}(k^2)=\frac{k^2+\beta_{s,t}^2}{k^2+\alpha_{s,t}^2}.$$ Here k is the relative momentum in the $pn$ system at the excitation energy $E_{np}=k^2/m$, where $m$ is the nucleon mass. The parameters $\alpha$ and $\beta$ are determined by the known properties of the on-shell $NN$-scattering amplitudes at low energies: $\alpha_t=0.232$ fm$^{-1}$, $\alpha_s=-0.04$ fm$^{-1}$, $\beta_t=0.91$ fm$^{-1}$, $\beta_s=0.79$ fm$^{-1}$ [@machl]. Information on the $pd\to pnp$ mechanism and the off-shell properties of the [*NN*]{} system is contained in the matrix elements $A_{s,t}$ and their ratio [@ukrs] $$\label{zeta}
\zeta=\frac{|A_{s}|^2 }{|A_{t}|^2 }.$$ The $pd\to (pp)n$ data, obtained at ANKE/COSY [@vikomarov], are presented as cms cross sections $$\label{2fold}
{\overline {\frac{d\sigma}{ d\Omega_n}}}=
\frac{1}{\Delta \Omega_n}\int_0^{E_{max}}dE_{pp}\int\int
m\frac{d^3\sigma}{dk^2 d\Omega_n} d\Omega_n,$$ integrated over $E_{pp}$ from 0 to 3 MeV and averaged over the neutron cms angle $\theta_n^*=172^\circ-180^\circ$, where $$\label{3fold}
\frac{d^3\sigma}{dk^2 d\Omega_n}=\frac{1}{(4\pi)^5} \frac{p_n}{p_i}
\frac{k}{s\, \sqrt{m^2+k^2}}\frac{1}{2}\int\int d\Omega_{\bf k}
{\overline {|M_{fi}|^2}}$$ and $\Delta\Omega_n$ is the neutron solid angle. In Eq. (\[3fold\]) $p_i$ and $p_n$ are the cms momenta of the incident proton and the final neutron, respectively; $M_{fi}$ is the full matrix element of the reaction. Due to the isospin invariance the following relation holds in the singlet cannel $$\frac{d\sigma}{d\Omega^*}(pd\to {(np)}_sp)=\frac{1}{2}
\frac{d\sigma}{d\Omega^*} (pd\to {(pp)}_sn).
\label{isospin}$$ Using the Fäldt-Wilkin extrapolation [@FW1] for the bound and the scattering S wave functions in the triplet state at short [*pn*]{} distances $r<1$ [fm]{}, and taking into account the short-range character of the interaction mechanism, the triplet cross section $pd\to (pn)_tn$ is obtained as $$\label{triplet}
\frac{d\sigma_t}{d\Omega^*} =\frac{p_f}{p_i} \, f^2(k^2)
\frac{d\sigma}{d\Omega^*}(pd\to dp),$$ where $$\label{fsi}
f^2(k^2) = \frac{2\pi\,m}{\alpha_t(k^2+\alpha_t^2)}$$ and $d\sigma/ d\Omega^*$ is the $pd\to pd$ cms cross section. After integration over $E_{pp}$ the triplet cross section (\[triplet\]) takes the form $$\label{atriplet}
{\overline {\frac{d\sigma_t}{d\Omega^*}}}=
\frac{p_f}{p_i} Z \frac{ d\sigma}{d\Omega^*}(pd\to dp),$$ where $$\label{z}
Z=\frac{1}{2\pi}\frac{2}{\alpha_t}\left \{ k_{max}-\alpha_t
\arctan{\left(\frac{k_{max}}{\alpha_t}\right )} \right \}.$$ On the other side, the triplet (t) and singlet (s) cross sections can be obtained by integration of Eq. (\[mwatson\]) over $E_{pp}$. $K$, $|A_s|^2$, and $|A_t|^2$, being very smooth functions of $E_{pp}$, can be assumed as constant. A ratio of the integrals $R=y_s/y_t$ can be defined, where $$\label{yst}
y_{s,t}=\int_0^{k^2_{max}}\,FSI_{s,t}(k)\,k\, dk^2.$$ With this ratio one finally gets the singlet-to-triplet ratio $\zeta$ defined by Eq. (\[zeta\]) as $$\label{zetacosy}
\zeta= \frac{1}{2\,R\, Z}
\frac{\frac{d\sigma}{d\Omega^*}(pd\to(pp)_sn)}
{\frac{ d\sigma}{d\Omega^*}(pd\to dp)}.$$ It is obvious that $\zeta$ is not a direct ratio of the $pd\to (pp)n $ and $pd\to dp$ cross sections, but contains also the additional factor $1/({2\,R\,Z})$.
Results and discussion
======================
For the numerical calculations of $\zeta$, defined by Eq. (\[zetacosy\]), we use the experimental data on the $pd\to dp$ cross section [@berth] and the new ANKE/COSY data for the $pd\to (pp)n$ reaction [@vikomarov]. From the Eqs.(\[gwfsi\]) and (\[yst\]) and with the values for $\alpha_i$ and $\beta_i$, given above, we get $Z=0.101$ and $R=2.29$ for $E_{pp}^{max}=3$ MeV. With these numbers and assuming systematic uncertainties of 10% for both the $pd\to dp$ cross section and the Fäldt-Wilkin ratio (\[triplet\]), we obtain $\zeta=(2.3\pm 0.5)$% for 0.6, $\zeta=(1.6\pm 0.3)$% for 0.7, and $\zeta=(2.1\pm 1.2)$% for 1.9 GeV beam energy. We find the surprising fact that $\zeta$ is constant within the errors over the whole investigated beam-energy range from 0.6 to 1.9 GeV.
The present, model-independent small result for $\zeta$ can be compared with $\zeta <5$%, obtained recently [@ukrs] for the $pd\to (pn)p$ reaction data, measured exclusively at 585 MeV and cms angle $\theta^*=92^\circ$ [@witten]. A similar, small value for the singlet admixture of a few percent was found for the reaction $pp\to pn\pi^+$ in the FSI region at beam energies of 492 MeV [@abaev] and 800 MeV [@uzcw], too. The smallness of the singlet contribution in the $\Delta$-region of the reaction $pd\to (pn)p$ can be explained by dominance of the OPE mechanism with the subprocess $pp\to (pn)_s\pi^+$ [@ukrs].
The singlet-to-triplet ratio can be estimated as $\zeta_{th}=R_S\times R_I\times R_X$. Here $R_S=1/3$ is the spin statistical factor. The isospin ratio $R_I$ is 1 for the ONE and 1/9 for both the $\Delta$-mechanism and the vector meson–nucleon exchanges [@uzzhetf]. $R_X$ is the ratio of the spatial singlet and triplet amplitudes. For the $\Delta$-mechanism, it reflects the difference of the $^1S_0$ and $^3S_1$ wave functions at $r < 1 $ fm. Since $\zeta$ according to Eqs. (\[mwatson\]) and (\[gwfsi\]) does not contain the FSI factor, $R_X$ is $\approx 1$ for the $\Delta$ amplitude. For the ONE it is $\approx 0.5$ due to the contribution of the D wave in the triplet and its absence in the singlet state. The calculation using Eqs.(\[3fold\]) and (\[2fold\]) with the Reid-Soft-Core (RSC) NN-potential [@rsc] gives $\zeta_{ONE}=5-8$% for $T_p=1.4-1.9$ GeV for the ONE which is 3 to 4 times the experimental value. For the $\Delta$ mechanism this ratio $\zeta_{\Delta}=R_S\times R_I=1/27$ is in better agreement with the experimental value.
Calculations [@imuz88; @uz2002] with use of the RSC NN potential and the ONE+SS+$\Delta$ model [@kls; @uz98], including single pN scattering (SS), describe the $pd\to (pp)n$ cross section data for the beam energies 0.6 and 0.7 GeV as is seen in Fig. 1. In this region the $\Delta$ mechanism dominates due to the nearby minimum of the ONE cross section. At higher energies the strong disagreement with the data is obvious. The data show no indication of the dip around 0.8 GeV, and for $T_p>1.3 $ GeV they are by a factor 2 to 4 below the prediction. A very similar discrepancy is observed for $pd\to dp$ backward elastic scattering, as can be seen from the upper part of Fig. 1. Both the earlier $pd\to dp$ and the recent $pd\to (pp)n$ data show that in the until now used model the contribution by the ONE, dominating for $T_p>1.3$ GeV, is too strong. On the other side, a calculation, using only the $\Delta$ mechanism with a cutoff momentum $\Lambda_\pi=0.7$ GeV/c allows one to describe the $pd\to (pp)n$ data for $T_p>0.7$ GeV. This mechanism does not involve such high-momentum components of the NN wave function as the ONE. A possible conclusion would be that the deuteron and the $pp(^1S_0)$ system at short NN distances are softer than modeled by the RSC NN potential. As one should note, the assumption about softness of the deuteron at short NN-distances is supported by (i) rather successful description of the $pd\to dp$ cross section within the OPE-model only [@kolybas; @uz98], and (ii) the strong disagreement between the ONE calculations and the $T_{20}$ data [@t20]. Detailed analyses of this conjecture are in progress.
[**Acknowledgments.**]{} This work was supported by the BMBF project grant KAZ 99/001 and the Heisenberg-Landau program.
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Yad. Fiz. [**33**]{} (1981) 1208. A. Boudard, M.Dillig. Phys. Rev. [**C31**]{} (1985) 302. O. Imambekov and Yu.N. Uzikov, Yad. Fiz. [**47**]{} (1988) 1089; O. Imambekov, Yu.N. Uzikov, L.V.Schevchenko,\
Z. Phys. [**A332**]{} (1989) 349. Yu.N. Uzikov, Phys. Part. Nucl. [**29**]{} (1998) 583. O. Imambekov and Yu.N. Uzikov,\
Sov. J. Nucl. Phys. [**52**]{} (1990) 862. Yu.N. Uzikov, Pisjma v ZhETF [**75**]{} (2002) 7. Yu.N. Uzikov, J. Phys. G: Nucl. Part. Phys. [**28**]{} (2002) B13. A.K. Kerman and L.S. Kisslinger,\
Phys. Rev. [**180**]{} (1969) 1483. V.I. Komarov et al., arXiv:nucl-ex/0210017 v2,\
submitted for publication in Phys. Lett. [**B**]{}. K.M. Watson, Phys. Rev. [**88**]{} (1952) 1163;\
A.B. Migdal, Sov. Phys. - JETP [**28**]{} (1955) 3. M.L. Goldberger and K.M. Watson.\
*Collision Theory* (John Wiley, New York, 1964) R. Machleidt, Phys. Rev. [**C 63**]{} (2001) 024001. Yu.N. Uzikov, V.I. Komarov, F. Rathmann,\
H.Seyfarth, Phys. Lett. [**B524**]{} (2002) 303. G. Fäldt and C. Wilkin, Phys. Scripta [**56**]{} (1997) 566. P. Berthet [*et al.*]{}, J. Phys. G: Nucl. Phys. [**8**]{} (1982) L111. T. Witten [*et al*]{}., Nucl. Phys. [**A254**]{} (1975) 269. V.V. Abaev [*et.al.*]{}, Phys. Lett. [**B521**]{} (2001) 158. Yu.N.Uzikov, C.Wilkin, Phys. Lett. [**B511**]{} (2001) 191. J.R.V. Reid, Ann. Phys. (NY) [**50**]{} (1968) 411. V.M. Kolybasov, N.Ya. Smorodinskaya,\
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---
abstract: 'Owing to the fact that $m_c^2 \sim m_b \Lambda_\text{QCD}$, the endpoint region of the charged lepton energy spectrum in the inclusive decay $B \to X_c \ell \bar{\nu}_\ell$ is affected by the Fermi motion of the initial-state $b$ quark bound in the $B$ meson. This effect is described in QCD by shape functions. Including the mass of the final-state quark, we find that a different set of operators as employed in Ref. [@Bauer:2002yu] is needed for a consistent matching, when incorporating the subleading contributions in $B \to X_q \ell \bar{\nu}_\ell$ for both $q = u$ and $q = c$. In addition, we modify the usual twist expansion in such a way that it yields a description of the lepton energy spectrum which is not just valid in the endpoint region, but over the entire phase space.'
author:
- Thomas Mannel
- 'Frank J. Tackmann'
date: 'August 25, 2004'
title: 'Shape Function Effects in $B \to X_c \ell \bar{\nu}_\ell$'
---
Introduction
============
The theoretical machinery for the determination of $\lvert V_{cb} \rvert$ from semileptonic $B$-meson decays has reached a mature state over the last years. With the very precise data on the exclusive $B \to D^{(*)} \ell \bar{\nu}_\ell$ channels as well as on the inclusive decays $B \to X_c \ell \bar{\nu}_\ell$ from the $B$-meson factories the theoretical description has been improved so much that currently relative theoretical uncertainties for $\lvert V_{cb} \rvert$ of less than 2% are quoted [@Benson:2003kp; @Bauer:2004ve], yielding a total uncertainty of about 2% [@Aubert:2004aw].
The nonperturbative corrections to the inclusive decays are parametrically of order $1/m_b^2$ and hence are expected to be smaller than in the exclusive channels, where the corrections are of order $1/m_c^2$. However, the inclusive rate depends on the $b$-quark mass $m_b$, which needs to be determined in a suitable scheme as precisely as possible.
In addition, also the parameters of the heavy quark expansion are needed, such as $\lambda_1$ and $\lambda_2$ at order $1/m_b^2$ and the corresponding parameters appearing at higher orders. These parameters are obtained experimentally by taking moments of the various inclusive distributions (such as lepton energy spectra or hadronic invariant mass distributions), but the higher moments become more and more sensitive to higher-order corrections in $1/m_b$, since the leading contribution to the $n$th moment is roughly of the order $1/m_b^n$.
When determining the heavy quark parameters from the lepton energy spectrum, the higher moments become sensitive to the endpoint region of the spectrum. Using the $1/m_b$ expansion the leading term is the partonic rate and still a smooth function, but already the first non-vanishing nonperturbative contribution exhibits an irregular behavior which is unphysical. The situation is in fact very similar to the one in $b \to u$ transitions, where it is known that these singular contributions can be resummed into a shape function. For heavy to light decays, such as the decays $b \to u \ell \bar{\nu}_\ell$ and $b \to s \gamma$, the twist expansion, that is, the resummation of nonperturbative contributions, has been performed to the subleading level in the $1/m_b$ expansion [@BLM1; @Bauer:2002yu; @BLM21; @Burrell:2003cf].
It has already been noticed some time ago [@Mannel:1994pm] that the light-cone distribution of the $B$ meson also has a significant effect on the endpoint region of the lepton energy spectrum in $B \to X_c \ell \bar{\nu}_\ell$. This is due to the fact that numerically $m_c^2 \sim m_b \Lambda_\text{QCD}$. The charm quark mass thus has to be counted as $\sqrt{m_b \Lambda_\text{QCD}}$ when performing the power counting. The endpoint region is known to be determined by the light-cone distribution of the initial state and has the width $\sqrt{ m_b \Lambda_\text{QCD}}$, which happens to be of the same order as $m_c$. Therefore, it is useful to consider the effects of the light-cone distribution of the $B$ meson also in $B \to X_c \ell \bar{\nu}_\ell$.
In the present paper we perform this analysis and compare with the standard expansion. As a by-product, we suggest a modified twist expansion which can be applied over the full phase space, incorporating the twist expansion in the endpoint region as well as the usual local expansion in the rest of phase space.
Furthermore, performing the limit $m_c \to 0$ we discover an inconsistency in comparison with previous work [@Bauer:2002yu]. It turns out that at subleading order additional operators are needed which are formally of leading order, but have coefficients of subleading order. Expanding into the usual local expansion we obtain the correct result for the terms of order $1/m_b^3$, indicating that our result is consistent. As a consequence, compared to Ref. [@Bauer:2002yu] additional nonperturbative input in the form of new shape functions appears.
Inclusive Decay Rate
--------------------
Neglecting the masses of the leptons the energy spectrum of the charged lepton in the $B$ rest frame is given via the optical theorem by $$\label{dGqdy}
\frac{{\mathrm{d}}\Gamma_q}{{\mathrm{d}}y} = 4 \Gamma_0 y^2 \theta(y) {\langle T_q \rangle_{\! B}}
,$$ where $y = 2E_\ell/m_b$ denotes the rescaled lepton energy, $$\Gamma_0 = \frac{G_F^2 \lvert V_{qb} \rvert^2 m_b^5}{192\pi^3}
,$$ and $q$ stands for either $u$ or $c$. The “$B$ expectation value” is defined as ${\langle {\mathcal{O}}\rangle_{\! B}} = \langle B \lvert {\mathcal{O}}\rvert B \rangle / 2m_B$, where the QCD states are normalized to $2 m_B$. The operator $T_q$ has the form $$\label{T_def}
T_q = \frac{48\pi^2}{m_b^3 y}
\mathrm{Im}\, \frac{{\mathrm{i}}}{\pi} \sum_{s_\ell} \int{\mathrm{d}}^4x\, e^{-{\mathrm{i}}p_\ell\cdot x}
T \bigl[ W_q^\dagger(x) W_q(0) \bigr]
,$$ where $s_\ell$ and $p_\ell$ are the lepton spin and momentum. The effective weak current is $$\label{W_def}
W_q = (\bar{q} \gamma_\alpha P_L b) (\bar{u}_\ell \gamma^\alpha P_L \nu_\ell)
= (\bar{q} \gamma_\alpha P_L \nu_\ell) (\bar{u}_\ell \gamma^\alpha P_L b)
,$$ with $P_L = (1 - \gamma_5)/2$ and $u_\ell$ denoting the lepton spinor. Plugging this into Eq. yields
$$\begin{aligned}
T_q &= \frac{48\pi^2}{m_b^3 y}
\biggl( -\frac{1}{\pi} \mathrm{Im}\, \bar{b} L_{\alpha\beta} b \Pi_q^{\alpha\beta} \biggr)
\quad \text{with} \\
\label{Lab_def}
L_{\alpha\beta} &= \gamma_\alpha { {p\mspace{-7.5mu}/\mspace{-1.5mu}} }_\ell \gamma_\beta P_L
,\\
\label{Piab_def}
\Pi_q^{\alpha\beta} &= -{\mathrm{i}}\!\int{\mathrm{d}}^4x e^{{\mathrm{i}}(p_b - p_\ell)\cdot x}
T \bigl[ (\bar{\nu}_\ell \gamma^\alpha P_L q)(x) (\bar{q} \gamma^\beta P_L \nu_\ell)(0) \bigr]
.\end{aligned}$$
Here, $L_{\alpha\beta}$ is the leptonic tensor and $\Pi_q^{\alpha\beta}$ represents the inclusive $q$-quark-neutrino loop with the momentum transfer $p_b - p_\ell$. The $b$-quark momentum $p_b$ contains a large part $m_b v$, where $v$ is the velocity of the $B$ meson. As usual, assuming that $Q = m_b v - p_\ell$ sets a perturbative scale, large compared to $k = p_b - m_b v \sim {\mathcal{O}}(\Lambda_\text{QCD})$, we may perform an OPE of the transition operator $T_q$ in powers of $\Lambda \equiv \Lambda_\text{QCD}/m_b$ [@ManoharWise; @Bigiinclusive1; @Mannelinc].
Light-Cone Vectors and Power Counting
-------------------------------------
Using appropriate powers of $m_b$ we may work with dimensionless variables, which are denoted by a hat, for example ${\hat{Q}}= Q/m_b = v - {\hat{p}}_\ell$. We also define ${\Delta}= 1 - y$ and $\rho = m_c^2/m_b^2$.
As already noted, the $c$-quark mass satisfies $m_c^2 \sim m_b \Lambda_\text{QCD}$ and thus $\rho \sim {\mathcal{O}}(\Lambda)$. The kinematic endpoint in the local OPE is given by ${\Delta}= \rho$, the partonic endpoint. The endpoint region of the lepton energy spectrum is defined by ${\Delta}\sim {\mathcal{O}}(\Lambda)$, and it is well known [@ManoharWise] that in this region the local OPE breaks down. However, it has been shown that one may still perform a light-cone or twist expansion [@NeubertShape; @NeubertShape1; @BigiMotion].
To set up the light-cone expansion we use the velocity $v$ and the lepton momentum $p_\ell$ to define a basis of two light-cone vectors,
$$v = \frac{1}{2} (n + {\bar{n}}), \quad p_\ell = E_\ell {\bar{n}}= \frac{m_b}{2} y {\bar{n}},$$
satisfying $n^2 = {\bar{n}}^2 = 0$ and $n \cdot {\bar{n}}= 2$. The metric is decomposed accordingly $$\eta^{\mu\nu}
= \frac{1}{2} n^\mu {\bar{n}}^\nu + \frac{1}{2} {\bar{n}}^\mu n^\nu + \eta_\perp^{\mu\nu}
,$$ and a generic four-momentum $p$ can be written as $$p^\mu = \frac{1}{2} p_- n^\mu + \frac{1}{2} p_+ {\bar{n}}^\mu + p_\perp^\mu
,$$ where we defined $$p_+ = n \cdot p, \quad p_- = {\bar{n}}\cdot p,
\quad p_\perp^\mu = \eta_\perp^{\mu\nu} p_\nu
.$$
The power counting $$\label{usual-counting}
{\hat{Q}}_- = 1, \quad {\hat{Q}}_+ = {\Delta}\sim {\mathcal{O}}(\Lambda), \quad \rho \sim {\mathcal{O}}(\Lambda),
\quad {\hat{k}}\sim {\mathcal{O}}(\Lambda)
,$$ yields the standard twist expansion by expanding in powers of $\Lambda$, taking also into account that $\rho \sim {\mathcal{O}}(\Lambda)$.
Note that this power counting becomes wrong for small lepton energies, since ${\Delta}$ becomes of ${\mathcal{O}}(1)$. In this case the spectrum is described by the usual local OPE. However, as we shall discuss below, with a slight modification of the twist expansion it is possible to describe the spectrum also for small lepton energies.
The relevant kinematic variable in the OPE is ${\hat{p}}= {\hat{p}}_b - {\hat{p}}_\ell = {\hat{Q}}+ {\hat{k}}$, the light-cone components of which are $$\label{hp_lc}
{\hat{p}}_+ = {\Delta}+ {\hat{k}}_+, \quad {\hat{p}}_- = 1 + {\hat{k}}_-,
\quad {\hat{p}}_\perp = {\hat{k}}_\perp
.$$ Obviously all ${\hat{k}}_+$ dependence occurs in the combination ${\Delta}_+ = {\Delta}+ {\hat{k}}_+$. In the local OPE the complete ${\hat{k}}$ dependence is expanded. In particular, this produces terms of the form ${\hat{k}}_+/{\Delta}$, which become large, of ${\mathcal{O}}(1)$, near the endpoint. The twist expansion avoids these terms, since only the ${\hat{k}}_-$ and ${\hat{k}}_\perp$ dependences are expanded.
{width="0.7\columnwidth"} {width="0.7\columnwidth"}
Alternatively to Eq. , we may as well treat the complete dependence on ${\Delta}_+$ (and $\rho$) exactly. That is, we use the power counting $$\label{mod-counting}
{\hat{k}}_- \sim {\mathcal{O}}(\Lambda), \quad {\hat{k}}_\perp \sim {\mathcal{O}}(\Lambda)
,$$ and only expand in powers of ${\hat{k}}_-$ and ${\hat{k}}_\perp$ from the very beginning. The order in $\Lambda$ of a twist term given by Eq. corresponds to the order of the first local term it contains, once the usual local OPE is performed. Eq. defines what we call the modified twist expansion and holds for large *and* small lepton energies. Our results will therefore be valid over the whole lepton energy spectrum, providing a correct interpolation between the usual twist and local expansions.
Our modified expansion is a direct extension of the usual twist expansion. The latter does not expand $\Delta_+ = \Delta + {\hat{k}}_+$ because both $\Delta$ and ${\hat{k}}_+$ are considered ${\mathcal{O}}(\Lambda)$. Once part of the ${\hat{k}}_+$ dependence is left unexpanded, it is consistent to keep the complete ${\hat{k}}_+$ dependence unexpanded, as it just means to keep correct small terms, which can now be treated exactly. This is similar to the local expansion, where $\rho =m_c^2/m_b^2$, although being numerically small, is always treated exactly, because it can be treated exactly. Once we exclude ${\hat{k}}_+$ from the power counting and treat it exactly, there is no need to count $\Delta$ as ${\mathcal{O}}(\Lambda)$ anymore. Thus we can treat it exactly much like $\rho$. This in turn extends the validity of our expansion down to low lepton energies (where numerically $\Delta$ is of order one which caused the breakdown of the usual twist expansion).
In other words, starting from the full expression (including all ${\hat{k}}$-components) both expansions expand the ${\hat{k}}_-$ and ${\hat{k}}_\perp$ components. This is where we stop, while the usual expansion additionally neglects terms of second and higher twist order caused by $\Delta_+$, e.g., terms like $\Delta_+^2$. We keep all those “kinematic” twist terms, because they are of ${\mathcal{O}}(1)$ for low lepton energies. Note that we do *not* claim to include all second-order twist contributions, i.e., our spectrum is only correct to ${\mathcal{O}}(\Lambda)$ in the endpoint region, but it is correct to ${\mathcal{O}}(\Lambda^2)$ for low lepton energies.
Therefore, the difference to the usual twist expansion, defined by Eq. , is that the modified expansion automatically keeps all twist contributions that in the usual power counting are of higher order only because of additional factors of ${\Delta}_+$ or $\rho$. These contributions are purely kinematic and do not require additional operators in the twist expansion, but appear only as higher-order terms in the OPE coefficients. Taking them into account yields a consistent result valid over the full region of lepton energies.
Operator Product Expansion
==========================
To keep things simple and to exhibit the structure of the OPE we will perform it in terms of QCD light-cone operators. Schematically, it has the form $$T_q = \int{\mathrm{d}}{\omega}\sum_n C^{(n)}({\omega}) {\mathcal{O}}_n({\omega})
.$$ In a second step the $B$ expectation values of the ${\mathcal{O}}_n({\omega})$ are parametrized in terms of shape functions.
We expand to ${\mathcal{O}}(\Lambda^2)$ as defined by Eq. . With respect to the usual twist power counting this includes all contributions of leading and subleading twist, as well as some of second and even third order. In other words, we obtain the full coefficients of all operators with up to two covariant derivatives, which in particular retains all local terms up to local ${\mathcal{O}}(\Lambda^2)$. To do so, we need to evaluate the zero- and one-gluon matrix elements of $T_q$, depicted in Fig. \[fig:me\], which we do to leading order in $\alpha_s$.
Leading Twist
-------------
We first consider the case $q = c$ and later take the limit $m_c \to 0$. The zero-gluon matrix element of $T_c$ yields the well-known result $$\begin{gathered}
\label{Tc_bb}
\langle b \lvert T_c \rvert b \rangle
= \theta({\hat{p}}^2 - \rho) \Bigl( 1 - \frac{\rho}{{\hat{p}}^2} \Bigr)^2
\\ \times
\biggl[ \Bigl( 1 + 2\frac{\rho}{{\hat{p}}^2} \Bigr) 3\eta_{(\mu\nu} {\bar{n}}_{\alpha)} {\hat{p}}^\mu {\hat{p}}^\nu
- 3\rho {\bar{n}}_{\alpha} \biggr] \bar{u}_b \gamma^\alpha P_L u_b
.\end{gathered}$$ Indices in round brackets are completely symmetrized, $$\eta_{(\mu\nu} {\bar{n}}_{\alpha)} = \frac{1}{3} ( \eta_{\mu\nu} {\bar{n}}_\alpha
+ \eta_{\mu\alpha} {\bar{n}}_\nu + \eta_{\alpha\nu} {\bar{n}}_\mu )
.$$ Extracting the terms of ${\mathcal{O}}(1)$, $$\langle b \lvert T_c \rvert b \rangle
= \theta({\Delta}_+ - \rho)
\Bigl(C^{(0)}_\alpha(-{\hat{k}}_+) + {\mathcal{O}}(\Lambda) \Bigr) \bar{u}_b \gamma^\alpha P_L u_b
,$$ we obtain the leading term in the OPE of $T_c$, $$\label{Tc_to-0,0}
T_c = \int{\mathrm{d}}{\omega}\theta({\Delta}_{\omega}- \rho)
\Bigl( C_\alpha^{(0)}({\omega}) {\mathcal{O}}_0^\alpha({\omega}) + {\mathcal{O}}(\Lambda) \Bigr)
,$$ where ${\Delta}_{\omega}= {\Delta}- {\omega}$. The leading operator has the form $$\label{leading-op}
{\mathcal{O}}_0^\alpha({\omega}) = \bar{b} \delta({\mathrm{i}}\hat{{\mathcal{D}}}_+ + {\omega}) \gamma^\alpha P_L b
,$$ and its coefficient is $$C^{(0)}_\alpha({\omega})
= (1 - 3R_{\omega}^2 + 2R_{\omega}^3) n_\alpha + (2 - 3R_{\omega}+ R_{\omega}^3){\Delta}_{\omega}{\bar{n}}_\alpha
,$$ with ${\mathrm{i}}{\mathcal{D}}= {\mathrm{i}}D - m_b v$ and $R_{\omega}= \rho/{\Delta}_{\omega}$. Note that in the modified twist expansion we keep the contributions proportional to ${\bar{n}}_\alpha$, which would usually be considered as subleading twist due to the additional factor of ${\Delta}_{\omega}$.
The $B$ expectation value of ${\mathcal{O}}_0^\alpha({\omega})$ is given to leading order by the leading shape function, $$\begin{split}
{\langle {\mathcal{O}}_0^\alpha({\omega}) \rangle_{\! B}}
&= \frac{1}{4} v^\alpha \langle B_\infty \lvert \bar{b}_v \delta({\mathrm{i}}\hat{D}_+ + {\omega}) b_v
\rvert B_\infty \rangle + {\mathcal{O}}(\Lambda)
\\
&= \frac{1}{2} v^\alpha f({\omega}) + {\mathcal{O}}(\Lambda)
,\end{split}
\raisetag{4ex}$$ where $\lvert B_\infty \rangle $ denotes the $B$ meson state in the infinite-mass limit and the $b_v$ are the static heavy quark fields.
Together with Eqs. and we find the lepton energy spectrum at leading order
\[dGqdy\_to-0,0\] $$\frac{{\mathrm{d}}\Gamma_c}{{\mathrm{d}}y} = 2 \Gamma_0 y^2 \theta(y)
\int{\mathrm{d}}{\omega}\theta({\Delta}_{\omega}- \rho) \Gamma_p({\Delta}_{\omega}) f({\omega})
,$$ where ($R = \rho/{\Delta}$) $$\Gamma_p({\Delta}) = (1 - 3R^2 + 2R^3) + (2 - 3R + R^3){\Delta}$$ contains a purely kinematic ${\Delta}$ dependence determined by the parton model. Letting $\rho \to 0$ we obtain the result for $b \to u$ $$\frac{{\mathrm{d}}\Gamma_u}{{\mathrm{d}}y} = 2 \Gamma_0 y^2 \theta(y)
\int{\mathrm{d}}{\omega}\theta({\Delta}_{\omega}) (1 + 2{\Delta}_{\omega}) f({\omega})
.$$
Obviously, the leading-order result amounts to convoluting the kinematic ${\Delta}$ dependence of the parton model with $f({\omega})$. The overall factor of $y^2 \theta(y)$ is not convoluted, since it is a trivial phase space factor, unrelated to the OPE. Thus, in the modified expansion we keep the factor $y^2 \theta(y)$ and Eqs. are valid (to ${\mathcal{O}}(1)$) over the entire phase space.
Furthermore, from the first relation in Eq. it is apparent that the twist expansion will always yield a convolution of the kinematic ${\Delta}$ dependence rather than the $m_b$ dependence, as originally argued in Ref. [@Mannel:1994pm]. While a convolution of the $m_b$ dependence is correct to leading order in the usual twist expansion, it introduces spurious subleading corrections, as has been noted before. In the modified expansion it already fails at leading order.
Higher Twist Contributions
--------------------------
The light-cone operators needed to consistently match all contributions of ${\mathcal{O}}(\Lambda)$ and ${\mathcal{O}}(\Lambda^2)$ are given by
\[higher-ops\] $$\begin{aligned}
{\mathcal{O}}_1^{\alpha\mu}({\omega})
&= \iint{\mathrm{d}}{\omega}_1 {\mathrm{d}}{\omega}_2 \delta'({\omega};{\omega}_1,{\omega}_2)
\bar{b} \delta({\mathrm{i}}\hat{{\mathcal{D}}}_+ + {\omega}_2) {\mathrm{i}}\hat{{\mathcal{D}}}^\mu \delta({\mathrm{i}}\hat{{\mathcal{D}}}_+ + {\omega}_1) \gamma^\alpha P_L b
,\\
{\mathcal{O}}_2^{\alpha\mu\nu}({\omega})
&= \iiint{\mathrm{d}}{\omega}_1 {\mathrm{d}}{\omega}_2 {\mathrm{d}}\bar{{\omega}} \delta''({\omega};{\omega}_1,{\omega}_2,\bar{{\omega}})
\bar{b} \delta({\mathrm{i}}\hat{{\mathcal{D}}}_+ + {\omega}_2) {\mathrm{i}}\hat{{\mathcal{D}}}^{(\mu} \delta({\mathrm{i}}\hat{{\mathcal{D}}}_+ + \bar{{\omega}}) {\mathrm{i}}\hat{{\mathcal{D}}}^{\nu)} \delta({\mathrm{i}}\hat{{\mathcal{D}}}_+ + {\omega}_1) \gamma^\alpha P_L b
,\\
{\mathcal{O}}_3^{\alpha\mu\nu}({\omega})
&= \iint{\mathrm{d}}{\omega}_1 {\mathrm{d}}{\omega}_2 \delta'({\omega};{\omega}_1,{\omega}_2)
\bar{b} \delta({\mathrm{i}}\hat{{\mathcal{D}}}_+ + {\omega}_2) {\mathrm{i}}\hat{{\mathcal{D}}}^{(\mu} {\mathrm{i}}\hat{{\mathcal{D}}}^{\nu)} \delta({\mathrm{i}}\hat{{\mathcal{D}}}_+ + {\omega}_1) \gamma^\alpha P_L b
,\\
{\mathcal{P}}_4^{\alpha\mu\nu}({\omega})
&= - \frac{g}{2} \iint{\mathrm{d}}{\omega}_1 {\mathrm{d}}{\omega}_2 \delta'({\omega};{\omega}_1,{\omega}_2)
\bar{b} \delta({\mathrm{i}}\hat{{\mathcal{D}}}_+ + {\omega}_2) (\varepsilon\cdot\hat{G})^{\mu\nu}
\delta({\mathrm{i}}\hat{{\mathcal{D}}}_+ + {\omega}_1) \gamma^\alpha P_L b
.\end{aligned}$$
Here, $(\varepsilon\cdot G)^{\mu\nu} = \varepsilon^{\mu\nu}{}_{\lambda\kappa} G^{\lambda\kappa}$ (with $\varepsilon_{0123} = 1$), and the $\delta$-function factors are
$$\begin{aligned}
\delta'({\omega};{\omega}_1,{\omega}_2)
&= \frac{\delta({\omega}- {\omega}_1) - \delta({\omega}- {\omega}_2)}{{\omega}_1 - {\omega}_2}
,\\
\delta''({\omega};{\omega}_1,{\omega}_2,\bar{{\omega}}) &= \frac{\delta'({\omega};{\omega}_1,\bar{{\omega}}) - \delta'({\omega};{\omega}_2,\bar{{\omega}})}{{\omega}_1 - {\omega}_2}
.\end{aligned}$$
This operator basis differs from that introduced in Ref. [@BLM1] and used in previous applications [@Bauer:2002yu; @Burrell:2003cf] by the different ${\mathcal{O}}_1({\omega})$ and the additional operator ${\mathcal{O}}_2({\omega})$. We note that this is not an artefact of our modified expansion. With respect to the usual twist power counting Eq. both operators are formally of leading order. Nevertheless, their coefficients are of at least subleading order in this power counting, because during the matching procedure one effectively shifts orders from the operators to their coefficients by partial integration with respect to ${\omega}$, as will be illustrated later on. In turn, the $B$ expectation values of ${\mathcal{O}}_1({\omega})$ and ${\mathcal{O}}_2({\omega})$ will be parametrized in terms of derivatives of shape functions. In the final expression for the spectrum these derivatives are then shifted by partial integration to act on the OPE coefficients. This ensures that in the final result the coefficients of all shape functions (apart from kinematic twist terms) are of usual twist ${\mathcal{O}}(1)$. The same also holds for the contributions of ${\mathcal{O}}_3({\omega})$ and ${\mathcal{P}}_4({\omega})$ that are of usual subsubleading twist.
The light-cone OPE of $T_c$ now takes the form $$\label{Tc_to-1,2}
\begin{split}
T_c
&= \int{\mathrm{d}}{\omega}\theta({\Delta}_{\omega}- \rho)[
C^{(0,0)}\cdot {\mathcal{O}}_0 + C^{(1,1)} \cdot {\mathcal{O}}_1
\\ & \quad
+ (C^{(1,2)} + C^{(2,2)} ) \cdot {\mathcal{O}}_2 + (D^{(1,2)} + D^{(2,2)} ) \cdot {\mathcal{O}}_3
\\ & \quad
+ (E^{(1,2)} + E^{(2,2)} ) \cdot {\mathcal{P}}_4 + {\mathcal{O}}(\Lambda^3) ]({\omega})
,\end{split}
\raisetag{3ex}$$ where the dots denote the contraction of all Lorentz indices, and the coefficients are
$$\begin{split}
C^{(0,0)}_\alpha({\omega}) &= C^{(0)}_\alpha({\omega})
= (1 - 3R_{\omega}^2 + 2R_{\omega}^3) n_\alpha
\\ & \quad
+ (2 - 3R_{\omega}+ R_{\omega}^3){\Delta}_{\omega}{\bar{n}}_\alpha
\\
C'^{(1,1)}_{\alpha\mu}({\omega})
&= 2(1 - R_{\omega}^3) (n_\alpha + {\Delta}_{\omega}{\bar{n}}_\alpha) {\bar{n}}_\mu
\\ & \quad
+ 2(1 - 3 R_{\omega}^2 + 2R_{\omega}^3) \eta_{\perp\alpha\mu}
,\\
C'^{(1,2)}_{\alpha\mu\nu}({\omega})
&= 4(1 - R_{\omega})^3(n_\alpha + {\Delta}_{\omega}{\bar{n}}_\alpha) \eta_{\perp\mu\nu}
,\\
C''^{(2,2)}_{\alpha\mu\nu}({\omega})
&= \bigl(2(4 - 3R_{\omega}^2 + 2R_{\omega}^3) n_\alpha
\\ & \quad
+ 6(2 - 2R_{\omega}+ R_{\omega}^2) \rho {\bar{n}}_\alpha \bigr) {\bar{n}}_\mu {\bar{n}}_\nu
\\ & \quad
+ 8[(1 - R_{\omega})^3 + 1 - R_{\omega}^3] \eta_{\perp\alpha(\mu} {\bar{n}}_{\nu)}
,\end{split}$$
and $$\begin{split}
D^{(1,2)}_{\alpha\mu\nu}({\omega})
&= - 3(1 - R_{\omega})^2 (n_\alpha + {\Delta}_{\omega}{\bar{n}}_\alpha) \eta_{\perp\mu\nu}
,\\
D'^{(2,2)}_{\alpha\mu\nu}({\omega})
&= -3(1 - R_{\omega})\bigl((1 + R_{\omega}) n_\alpha + 2\rho {\bar{n}}_\alpha \bigr) {\bar{n}}_\mu {\bar{n}}_\nu
\\ & \quad
- 6(1 - R_{\omega})^2 \eta_{\perp\alpha(\mu} {\bar{n}}_{\nu)}
,\\
E^{(1,2)}_{\alpha\mu\nu}({\omega})
&= - 3(1 - R_{\omega})^2 (n_\alpha + {\Delta}_{\omega}{\bar{n}}_\alpha) n_\mu {\bar{n}}_\nu/2
,\\
E'^{(2,2)}_{\alpha\mu\nu}({\omega})
&= 3(1 - R_{\omega})^2 \eta_{\perp\alpha[\mu} {\bar{n}}_{\nu]}
.\end{split}
\raisetag{3ex}$$
The indices enclosed in round or square brackets are completely symmetrized or antisymmetrized, respectively. The first superscript denotes the order of the coefficient’s term in the OPE as it appears in the usual twist expansion (i.e., the order of the respective shape function once the $B$ expectation value is taken), while the second superscript denotes the order of the coefficients’s term in our modified expansion.
For the above reasons, we only quote the derivatives of the OPE coefficients, as these are the ${\mathcal{O}}(1)$ coefficients which will eventually enter the energy spectrum. The OPE coefficients are obtained by integrating over ${\omega}$, which increases their order in the usual twist power counting. The constants of integration are such that each integral vanishes at ${\Delta}_{\omega}= \rho$, that is, the kinematic $\theta$-function does not contribute to the partial integrations.
We emphasize that the OPE is valid to ${\mathcal{O}}(\Lambda)$ over the entire phase space and to ${\mathcal{O}}(\Lambda^2)$ away from the endpoint. When expanded into local operators, it correctly reproduces the full result to ${\mathcal{O}}(\Lambda^2)$ [@ManoharWise], as well as all local ${\mathcal{O}}(\Lambda^3)$ terms [@Gremm:1996df] corresponding to leading and subleading order in the usual twist expansion.
Remarks on the Matching Procedure
---------------------------------
It is worthwhile to point out a subtlety in the matching procedure leading to Eq. . The ${\mathcal{O}}(\Lambda)$ terms contained in Eq. are $$\theta({\Delta}_+ - \rho) C'^{(1,1)}_{\alpha\mu}(-{\hat{k}}_+) {\hat{k}}^\mu \bar{u}_b \gamma^\alpha P_L u_b
,$$ and can be written as a convolution in two ways, $$\begin{split}
C'^{(1,1)}_{\alpha\mu}(-{\hat{k}}_+) {\hat{k}}^\mu
&= \int{\mathrm{d}}{\omega}C'^{(1,1)}_{\alpha\mu}({\omega}) {\hat{k}}^\mu \delta({\hat{k}}_+ + {\omega})
\\
&= \int{\mathrm{d}}{\omega}C^{(1,1)}_{\alpha\mu}({\omega}) (-{\hat{k}}^\mu \delta'({\hat{k}}_+ + {\omega}))
,\end{split}$$ corresponding to the two possibilities for the matching $$\label{extr}
C'^{(1,1)}_{\alpha\mu}({\omega}) \{{\mathrm{i}}\hat{{\mathcal{D}}}^\mu, \delta({\mathrm{i}}\hat{{\mathcal{D}}}_+ + {\omega})\}/2
,\quad
C^{(1,1)}_{\alpha\mu}({\omega}) {\mathcal{O}}_1^{\alpha\mu}({\omega})
.$$ This ambiguity has to be resolved by studying the one-gluon matrix element, since ${\hat{k}}^\mu$ commutes with the $\delta({\hat{k}}_+ + {\omega})$ function, while the covariant derivative and the $\delta({\mathrm{i}}\hat{{\mathcal{D}}}_+ + {\omega})$ function do not. The gluon has momentum ${\hat{l}}\sim {\mathcal{O}}(\Lambda)$, a polarization vector ${\epsilon}= T^a {\epsilon}^a$, and we work in light-cone gauge, $A_+ = 0$. In accordance with Eq. , we treat ${\hat{l}}_+$ exactly and expand only in ${\hat{l}}_-$, ${\hat{l}}_\perp$, ${\hat{\epsilon}}_-$, and ${\hat{\epsilon}}_\perp$. To ${\mathcal{O}}(\Lambda)$ we find
$$\langle b \lvert T_c \rvert bg \rangle
= -g
\frac{\theta({\Delta}_+ - \rho) C^{(1,1)}_{\alpha\mu}(-{\hat{k}}_+) -
\theta({\Delta}_+ + {\hat{l}}_+ - \rho) C^{(1,1)}_{\alpha\mu}(-{\hat{k}}_+ - {\hat{l}}_+)}{{\hat{l}}_+}
{\hat{\epsilon}}^\mu
\bar{u}_b \gamma^\alpha P_L u_b + {\mathcal{O}}(\Lambda^2)
,$$
showing that we have to match onto ${\mathcal{O}}_1({\omega})$. Taking the massless limit (i.e., $\rho \to 0$, $R_{\omega}\to 0$) we note that this is in disagreement with the results of Ref. [@Bauer:2002yu], where the first possibility in Eq. has been chosen. Note that the equations of motion of heavy quark effective theory (HQET) cannot be used for the operator ${\mathcal{O}}_1({\omega})$, since the covariant derivative does not act directly on the heavy quark fields.
A similar problem occurs in the comparison of our ${\mathcal{O}}(\Lambda^2)$ contributions with the ones in Ref. [@Bauer:2002yu]. In our case these contributions are more complicated, requiring the two different operators ${\mathcal{O}}_2({\omega})$ and ${\mathcal{O}}_3({\omega})$. This is again in contrast with Ref. [@Bauer:2002yu], where only ${\mathcal{O}}_3({\omega})$ appears. In both cases the differences start at ${\mathcal{O}}(\Lambda^3)$ in the local expansion of the operators, and thus also in the final spectrum, as we will see below.
While this paper was in the review process, studies of $B \to X_u \ell \bar{\nu}_\ell$ based on “soft collinear effective theory” (SCET) appeared which shed some light on these differences [@Bosch:2004cb; @Lee:2004ja; @Beneke:2004in]. The SCET-based calculations show that the basis introduced originally in Ref. [@BLM1] is a complete basis of subleading operators, at least at tree level. However, in all these cases the light-cone vectors are defined based on the momentum $m_b v - q$, where $q$ is the total leptonic momentum. Here and in Ref. [@Bauer:2002yu] a different choice of light-cone vectors is used, which is based on $m_b v - p_\ell$, where $p_\ell$ is the momentum of the charged lepton only. It should be possible to relate the two choices by a coordinate transformation, i.e., by a reparametrization. We shall not go into any details here, but our results show that for the latter choice of light-cone coordinates the operator basis in Ref. [@Bauer:2002yu] is incomplete.
The Lepton Energy Spectrum
==========================
Shape Functions
---------------
In the last step we need to parametrize the $B$ expectation values of the operators and . To be consistent with our modified expansion we have to include all shape functions of leading and subleading order in the usual twist power counting, but also those of usual subsubleading twist with moments of local ${\mathcal{O}}(\Lambda^2)$.
The expansion of the QCD fields and states into HQET ones produces many additional operators \[e.g., the $O_1({\omega})$ and $P_2({\omega})$ of Refs. [@BLM1; @Bauer:2002yu]\] and shape functions. However, these higher-order shape functions always occur in particular combinations with those arising at leading order in the HQET expansion and can be suitably combined with them. We therefore take a different approach and directly parametrize the operators in QCD, which automatically combines the leading and higher-order HQET shape functions appropriately.
This is in fact similar to what is used in the context of the local expansion, where for example the matrix element corresponding to the kinetic energy operator $\mu_\pi^2$ is also defined using the states of full QCD, and thus this matrix element is equal to the kinetic energy matrix element $\lambda_1$ of HQET only to leading order in the $1/m_b$ expansion.
For the leading operator we have $$\begin{aligned}
{\langle 2{\mathcal{O}}_0^\alpha({\omega}) \rangle_{\! B}}
&= F_0({\omega}) v^\alpha + K_0({\omega}) (n - v)^\alpha
,\end{aligned}$$ which is exact and defines the two QCD shape functions $F_0({\omega})$ and $K_0({\omega})$. They may be expanded into the usual ones of HQET,
$$\begin{aligned}
F_0({\omega}) &= f({\omega}) + \frac{1}{2} t({\omega}) + {\mathcal{O}}(\Lambda^3) \delta'({\omega})
,\\
K_0({\omega}) &= {\omega}f({\omega}) + h_1({\omega}) + {\mathcal{O}}(\Lambda^3) \delta'({\omega})
.\end{aligned}$$
Alternatively, we can directly perform their moment expansions and use HQET to parametrize their moments,
$$\begin{aligned}
\begin{split}
F_0({\omega})
&= \delta({\omega}) - \frac{{\hat{\lambda}}_0}{2} \delta'({\omega}) - \frac{{\hat{\lambda}}_1 + {\hat{\tau}}_1}{6} \delta''({\omega})
\\ & \quad
- \frac{{\hat{\rho}}_1}{18} \delta'''({\omega}) + \dotsb
,\end{split}
\\
K_0({\omega})
&= \frac{{\hat{\lambda}}_0 - {\hat{\rho}}_0/2}{3} \delta'({\omega}) + \frac{{\hat{\rho}}_0}{6} \delta''({\omega})
+ \dotsb
,\end{aligned}$$
where we abbreviated $$\begin{gathered}
{\hat{\tau}}_1 = \hat{\mathcal{T}}_1 + 3\hat{\mathcal{T}}_2
, \quad
{\hat{\tau}}_2 = \hat{\mathcal{T}}_3/3 + \hat{\mathcal{T}}_4
,\\
{\hat{\lambda}}_0 = {\hat{\lambda}}_1 + {\hat{\tau}}_1 + 3({\hat{\lambda}}_2 + {\hat{\tau}}_2)
, \quad {\hat{\rho}}_0 = {\hat{\rho}}_1 + 3 {\hat{\rho}}_2
,\end{gathered}$$ and the $\rho_i$ and $\mathcal{T}_i$ are defined in [@Gremm:1996df].
We note that $F_0({\omega})$ and $K_0({\omega})$ are defined in QCD without any reference to the heavy quark limit. Nevertheless, heavy quark symmetry still tells us that $K_0({\omega})$ is suppressed by one power of $\Lambda$ with respect to $F_0({\omega})$. The normalization of $F_0({\omega})$ is exact to all orders in QCD due to $b$-quark number conservation, while all other moments receive further corrections of local ${\mathcal{O}}(\Lambda^4)$ and higher. The leading contribution to the neglected moments is also of local ${\mathcal{O}}(\Lambda^4)$.
For the higher-twist operators we find
$$\begin{aligned}
{\langle 2{\mathcal{O}}_1^{\alpha\mu}({\omega}) \rangle_{\! B}}
&= - [{\omega}F_0({\omega}) v^\alpha + {\omega}K_0({\omega}) (n - v)^\alpha]' (n - v)^\mu
- [F_1({\omega}) v^\alpha + K_1({\omega}) (n - v)^\alpha ]' n^\mu
- \frac{1}{2} L_1'({\omega}) \eta^{\perp\alpha\mu}
,\\
\begin{split}
{\langle 2{\mathcal{O}}_2^{\alpha\mu\nu}({\omega}) \rangle_{\! B}}
&= - \frac{1}{4} [G_2({\omega}) v^\alpha + M_2({\omega}) (n - v)^\alpha]' \eta^{\perp\mu\nu}
+ \frac{1}{2} [{\omega}^2 F_0({\omega}) v^\alpha + {\omega}^2 K_0({\omega}) (n - v)^\alpha ]'' (n - v)^\mu (n - v)^\nu
\\ & \quad
+ \frac{1}{2} [{\omega}F_1({\omega}) v^\alpha + {\omega}K_1({\omega}) (n - v)^\alpha ]'' 2n^{(\mu} (n - v)^{\nu)}
+ \frac{1}{2} [F_2({\omega}) v^\alpha + K_2({\omega}) (n - v)^\alpha ]'' n^\mu n^\nu
\\ & \quad
+ \frac{1}{2} \eta^{\perp\alpha(\mu} [{\omega}L_1({\omega}) (n - v)^{\nu)} + L_2({\omega}) n^{\nu)} ]''
\\
&= -\frac{1}{4} G_2'({\omega}) v^\alpha \eta^{\perp\mu\nu}
+ \frac{1}{2} [{\omega}^2 F_0({\omega})]'' v^\alpha (n - v)^\mu (n - v)^\nu + \dotsb
,\end{split}
\\
{\langle 2{\mathcal{O}}_3^{\alpha\mu\nu}({\omega}) \rangle_{\! B}}
&= \frac{1}{2} G_3({\omega}) v^\alpha \eta^{\perp\mu\nu}
- [{\omega}^2 F_0({\omega}) ]' v^\alpha (n - v)^\mu (n - v)^\nu + \dotsb
,\\
{\langle 2{\mathcal{P}}_4^{\alpha\mu\nu}({\omega}) \rangle_{\! B}}
&= [H_4({\omega})(n - v)^\alpha + N_4({\omega})v^\alpha] 2 v^{[\mu} n^{\nu]}
- \eta^{\perp\alpha[\mu} [R_4({\omega}) (n - v)^{\nu]} + S_4({\omega}) n^{\nu]} ]'
.\end{aligned}$$
These relations are again exact and define the respective shape functions. For the sake of completeness we give the full parametrization of ${\mathcal{O}}_2({\omega})$. In its second line we neglected all Lorentz structures whose shape functions are of higher order and not needed to the order we are working. The operator ${\mathcal{O}}_3({\omega})$ obeys a similar parametrization as ${\mathcal{O}}_2({\omega})$, but with different higher-order shape functions.
The moment expansions of the relevant shape functions are $$\label{moment-exps}
\begin{split}
F_1({\omega}) &= -\frac{{\hat{\lambda}}_0}{2} \delta({\omega}) + {\mathcal{O}}(\Lambda^4)\delta'({\omega}) - \frac{{\hat{\rho}}_1}{18} \delta''({\omega}) + \dotsb
,\\
L_1({\omega}) &= \frac{2{\hat{\lambda}}_0 - {\hat{\rho}}_0}{3} \delta({\omega}) + \frac{{\hat{\rho}}_0}{3} \delta'({\omega}) + \dotsb
,\\
G_2({\omega}) &= -\frac{2({\hat{\lambda}}_1 + {\hat{\tau}}_1)}{3} \delta'({\omega}) - \frac{2{\hat{\rho}}_1}{9} \delta''({\omega}) + \dotsb
,\\
G_3({\omega}) &= -\frac{2({\hat{\lambda}}_1 + {\hat{\tau}}_1)}{3} \delta'({\omega}) + {\mathcal{O}}(\Lambda^4) \delta''({\omega}) + \dotsb
,\\
H_4({\omega}) &= - ({\hat{\lambda}}_2 + {\hat{\tau}}_2) \delta'({\omega}) + {\mathcal{O}}(\Lambda^4) \delta''({\omega}) + \dotsb
,\\
R_4({\omega}) &= -2({\hat{\lambda}}_2 + {\hat{\tau}}_2) \delta({\omega}) - {\hat{\rho}}_2 \delta'({\omega}) + \dotsb
,\\
S_4({\omega}) &= 2({\hat{\lambda}}_2 + {\hat{\tau}}_2) \delta({\omega}) + {\mathcal{O}}(\Lambda^4) \delta'({\omega}) + \dotsb
.\end{split}
\raisetag{19ex}$$ The corrections to all moments shown, as well as the first neglected moments, are again of local ${\mathcal{O}}(\Lambda^4)$. All other shape functions are of at least subsubleading twist in the usual power counting and do not have moments of local ${\mathcal{O}}(\Lambda^2)$, for example
$$\begin{split}
K_1({\omega}) &= \frac{{\hat{\rho}}_0}{6} \delta'({\omega}) + {\mathcal{O}}(\Lambda^4) \delta''({\omega}) + \dotsb
,\\
N_4({\omega}) &= {\mathcal{O}}(\Lambda^4) \delta''({\omega}) + \dotsb
,\end{split}$$
and $$\begin{split}
F_2({\omega}) &= {\mathcal{O}}(\Lambda^4) \delta({\omega}) + {\mathcal{O}}(\Lambda^4) \delta'({\omega}) + {\mathcal{O}}(\Lambda^4) \delta''({\omega}) + \dotsb
,\\
K_2({\omega}) &= {\mathcal{O}}(\Lambda^4) \delta'({\omega}) + \dotsb
,\\
L_2({\omega}) &= \frac{{\hat{\rho}}_0}{3} \delta({\omega}) + {\mathcal{O}}(\Lambda^4) \delta'({\omega}) + \dotsb
,\\
M_2({\omega}) &= {\mathcal{O}}(\Lambda^4) \delta''({\omega}) + \dotsb
.\end{split}
\raisetag{8ex}$$
The Spectrum
------------
In order to write the spectrum in a compact way, it is useful to define $$\begin{split}
\Gamma_p^\pm({\Delta}) &= \pm (1 - 3R^2 + 2R^3) + (2 - 3R + R^3){\Delta},\\
\Gamma_1^\pm({\Delta}) &= 2(1 - R^3) (\pm 1 + {\Delta})
,\\
\Gamma_2^\pm({\Delta}) &= \pm (4 - 3R^2 + 2R^3) + 3(2 - 2R + R^2)\rho
,\\
\Gamma_3^\pm({\Delta}) &= -3(1 - R)[\pm(1 + R) + 2\rho]
.\end{split}$$ The lepton energy spectrum now takes the form
\[dGqdy\_to-2,3\] $$\begin{split}
\frac{{\mathrm{d}}\Gamma_c}{{\mathrm{d}}y}
&= 2\Gamma_0 y^2 \theta(y) \int{\mathrm{d}}{\omega}\theta({\Delta}_{\omega}- \rho)
\Big\{\Gamma_p^+({\Delta}_{\omega}) F_0({\omega}) + \Gamma_p^-({\Delta}_{\omega}) K_0({\omega})
+ \Gamma_1^+({\Delta}_{\omega}) [{\omega}F_0({\omega}) + 2 F_1({\omega}) ]
\\ & \quad
+ \Gamma_1^-({\Delta}_{\omega}) [{\omega}K_0({\omega}) + \dotsb ] + \Gamma_2^+({\Delta}_{\omega}) [{\omega}^2 F_0({\omega}) + \dotsb ]
+ \Gamma_3^+({\Delta}_{\omega}) [{\omega}^2 F_0({\omega}) + \dotsb ]
\\ & \quad
+ 2(1 - 3 R_{\omega}^2 + 2R_{\omega}^3) L_1({\omega}) + 2(1 - R_{\omega})^3(1 + {\Delta}_{\omega}) G_2({\omega})
\\ & \quad
- 3(1 - R_{\omega})^2 \big[ (1 + {\Delta}_{\omega}) G_3({\omega}) - (1 - {\Delta}_{\omega}) H_4({\omega})
- R_4({\omega}) - 2S_4({\omega}) \big] + {\mathcal{O}}(\Lambda^3) \Bigr\}
.\end{split}$$ Here, ${\omega}K_0({\omega})$ and ${\omega}^2 F_0({\omega})$ are of usual subsubleading twist, but have a first moment of local ${\mathcal{O}}(\Lambda^2)$. The ellipsis mean that the same coefficient has more shape functions, which are of the same order in the usual twist power counting, but have only higher-order moments, e.g., $K_1({\omega})$ or ${\omega}F_1({\omega})$. Taking the limit $\rho \to 0$ we obtain the $b \to u$ result $$\begin{split}
\frac{{\mathrm{d}}\Gamma_u}{{\mathrm{d}}y}
&= 2\Gamma_0 y^2 \theta(y) \int{\mathrm{d}}{\omega}\theta({\Delta}_{\omega})
\Big\{(1 + 2{\Delta}_{\omega}) F_0({\omega}) + (-1 + 2{\Delta}_{\omega}) K_0({\omega})
+ 2(1 + {\Delta}_{\omega}) [{\omega}F_0({\omega}) + 2 F_1({\omega}) ]
\\ & \quad
+ 2(-1 + {\Delta}_{\omega}) [{\omega}K_0({\omega}) + \dotsb ]
+ 4 [{\omega}^2 F_0({\omega}) + \dotsb ]
- 3 [{\omega}^2 F_0({\omega}) + \dotsb ]
\\ & \quad
+ 2 L_1({\omega}) + 2(1 + {\Delta}_{\omega}) G_2({\omega})
- 3 \bigl[(1 + {\Delta}_{\omega}) G_3({\omega}) - (1 - {\Delta}_{\omega}) H_4({\omega})
- R_4({\omega}) - 2S_4({\omega})\bigr] + {\mathcal{O}}(\Lambda^3) \Bigr\}
.\end{split}$$
As for the light-cone OPE , Eqs. are valid to ${\mathcal{O}}(\Lambda)$ for all lepton energies and to ${\mathcal{O}}(\Lambda^2)$ away from the endpoint region. They provide the correct interpolation between the two regimes of the local expansion and the usual twist expansion.
Employing the moment expansions of the various shape functions, our results reproduce the full result to local ${\mathcal{O}}(\Lambda^2)$ [@ManoharWise] and all local ${\mathcal{O}}(\Lambda^3)$ contributions [@Gremm:1996df] belonging to leading and subleading order in the usual twist power counting.
To compare our $b \to u$ result with that of Ref. [@Bauer:2002yu], we neglect all terms that are of subsubleading twist in the usual twist power counting and include the overall $y^2 = (1 - {\Delta})^2$ in the power counting. Also dropping the $\theta(y)$ in front and expanding $F_0({\omega})$ and $K_0({\omega})$ into HQET shape functions, we have $$\begin{split}
\frac{{\mathrm{d}}\Gamma_u}{{\mathrm{d}}y}
&= 2\Gamma_0 \int{\mathrm{d}}{\omega}\theta({\Delta}_{\omega}) \biggl\{
(1 - {\omega}) f({\omega}) + \frac{1}{2}t({\omega}) - h_1({\omega})
+ 4F_1({\omega}) + 2G_2({\omega}) - 3 G_3({\omega}) + 3 H_4({\omega}) \biggr\}
\\
&= \Gamma_0 \biggl\{2\theta({\Delta}) - \frac{{\hat{\lambda}}_1}{3} [\delta({\Delta}) + \delta'({\Delta})]
- 11{\hat{\lambda}}_2 \delta({\Delta})
- \frac{{\hat{\rho}}_1}{3} \Bigl[5\delta'({\Delta}) + \frac{1}{3}\delta''({\Delta}) \Bigr]
- {\hat{\rho}}_2 \delta'({\Delta}) - \frac{{\hat{\tau}}_1}{3} \delta'({\Delta}) \biggr\}
.\end{split}$$
The second line shows the expansion into local terms. Both expressions disagree with Ref. [@Bauer:2002yu]. The differences arise from the new shape functions $F_1({\omega})$ and $G_2({\omega})$, introduced by ${\mathcal{O}}_1({\omega})$ and ${\mathcal{O}}_2({\omega})$, and are explicit in the coefficient of the ${\hat{\rho}}_1 \delta'({\Delta})$ term, which is correctly contained in our results. In Ref. [@Bauer:2002yu] these new shape functions are effectively set to $F_1({\omega}) = 0$ and $G_2({\omega}) = G_3({\omega})$.
Conclusions
===========
Since $m_c^2 \sim m_b \Lambda_\text{QCD}$ the endpoint region in $B \to X_c \ell \bar{\nu}_\ell$ is affected by shape-function effects. In the present paper we have considered these effects to subleading order in the twist expansion.
The usual twist expansion is valid in the endpoint region only; however, this expansion can easily be modified to become valid over the full phase space, thereby yielding a smooth expression for differential rates for any value of the kinematical variables. We have given the relevant expression for the spectrum up to ${\mathcal{O}}(1/m_b^2)$ and to subleading order in the twist expansion. Furthermore, in a similar fashion as for the heavy quark expansion parameters, we suggest to define shape functions using the full field operators and the full QCD states.
Considering the limiting case $m_c \to 0$ we reveal an inconsistency in previous work concerning the matching onto subleading shape functions. It turns out that with our specific choice of light-cone coordinates additional operators are needed to obtain a complete set of subleading non-local operators. In this way, the number of functions needed to describe the subleading twist effects increases.
The results of this paper may be useful for the estimation of higher moments and for higher-order corrections to the lower moments of the lepton spectrum or the hadronic invariant mass spectrum in $B \to X_c \ell \bar{\nu}_\ell$. Furthermore, since $B \to X_c \ell \bar{\nu}_\ell$ has a sensitivity to the light-cone distribution functions of the $B$ meson one could also make an attempt to extract the shape functions from this process. However, for a consistent treatment one would need to include also radiative corrections, which could be considered in the framework of SCET.
F.T. likes to thank the BaBar group in Dresden for its kind hospitality while this work was completed. T.M. likes to thank I. Bigi, M. Kraetz, and N. Uraltsev for discussions related to this subject and acknowledges the support from the German Ministry for Education and Research (BMBF).
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Recent submillimeter observations show nonaxisymmetric brightness distributions with a horseshoe-like morphology for more than a dozen transition disks. The most-accepted explanation for the observed asymmetries is the accumulation of dust in large-scale vortices. Protoplanetary disks vortices can form by the excitation of Rossby wave instability in the vicinity of a steep pressure gradient, which can develop at the edges of a giant planet-carved gap or at the edges of an accretionally inactive zone. We studied the formation and evolution of vortices formed in these two distinct scenarios by means of two-dimensional locally isothermal hydrodynamic simulations. We found that the vortex formed at the edge of a planetary gap is short-lived, unless the disk is nearly inviscid. In contrast, the vortex formed at the outer edge of a dead zone is long-lived. The vortex morphology can be significantly different in the two scenarios: the vortex radial and azimuthal extensions are $\sim1.5$ and $\sim3.5$ times larger for the dead-zone edge compared to gap models. In some particular cases, the vortex aspect ratios can be similar in the two scenarios; however, the vortex azimuthal extensions can be used to distinguish the vortex formation mechanisms. We calculated predictions for vortex observability in the submillimeter continuum with ALMA. We found that the azimuthal and radial extent of the brightness asymmetry correlates with the vortex formation process within the limitations of $\alpha$-viscosity prescription.'
author:
- 'Zs. Regály$^{1}$[^1], A. Juhász$^{2,3}$, D. Nehéz$^{1,4}$'
title: |
Interpreting Brightness Asymmetries in Transition Disks:\
Vortex at Dead Zone or Planet-carved Gap Edges?
---
Introduction
============
Transition disks are believed to be the bridging phase between gas-rich primordial disks and gas-poor debris disks. The characteristic feature of transition disks is the deficit of infrared excess in their spectral energy distribution [@Stormetal1989; @Skrutskieetal1990], which is attributed to the formation of dust-depleted gaps, cavities. Such cavities depleted of dust by 5-6 orders of magnitude have already been confirmed by submillimeter imaging (see a review in [@Espaillatetal2014]).
Recent spatially resolved submillimeter observations of transition disks revealed nonaxisymmetric brightness distributions for more than a dozen cases (see, e.g., @Andrewsetal2009 [@Andrewsetal2011; @Brownetal2009; @Hughesetal2009; @Isellaetal2010; @Mathewsetal2012; @Tangetal2012; @Casassusetal2013; @Casassusetal2015; @Fukagawaetal2013; @vanderMareletal2013; @Perezetal2014; @Hashimotoetal2015; @Marinoetal2015; @Momoseetal2015; @Wrightetal2015]). The observed lopsided brightness distributions have in most cases a horseshoe-like shape. Since the disk is believed to be optically thin at submillimeter wavelengths, these horseshoe-shaped asymmetries reflect perturbation in the density and/or temperature structure of the disk. The observed asymmetries show remarkable agreement with the morphology of large-scale anticyclonic vortices predicted to form in protoplanetary disks [@Regalyetal2012]. The accumulation of millimeter- to centimeter-sized dust has now becOme the commonly accepted explanation for the observed asymmetries.
Formation of a large-scale vortex can be triggered in protoplanetary disks due to baroclinic instability [@KlahrBodenheimer2003; @LyraKlahr2011; @Raettigetal2013; @Lyra2014] or to the Rossby wave instability (RWI) via the coagulation of smaller-scale vortices [@Lovelaceetal1999; @Lietal2000; @Lietal2001; @Lyraetal2009b; @Meheutetal2010; @Meheutetal2012a; @Meheutetal2012b; @Meheutetal2012c; @Meheutetal2013; @Richardetal2013; @Flocketal2015].
@BargeSommeria1995 and @KlahrHenning1997 have already shown that particles tend to get trapped in anticyclonic vortices. Particles of about a centimeter to a meter in size drifting into the pressure maxima, i.e., to the core of the vortices [@Lyraetal2009b; @Dzyurkevichetal2010; @Katoetal2010] may form gravitationally bound clumps of solids, which can coalesce forming embryos between the masses of the Moon and Mars. The swarm of these embryos can evolve further by mutual collisions forming massive cores ($\sim10\,M_\oplus$) of giant planets in $\sim5\times10^5$yr [@Sandoretal2011].
The radial pressure gradient at the location of the vortex can halt not only dust grains but also larger bodies migrating in the type I regime. However, trapping of low-mass protoplanets (about $10M_\oplus$) is found to be only temporary [@Regalyetal2013]. Large-scale vortices are known to accelerate the formation of planetesimals and planetary embryos in the core accretion paradigm; thus, such vortices are extremely important for planet formation, helping to overcome the meter-size barrier problem [@BlumWurm2008], and the time-scale problem of oligarchic growth [@Thommesetal2003]. Moreover, @KlahrBodenheimer2006 have also shown that vortices can hasten the formation of giant planets by decreasing the time to form massive cores ($\sim10-15M_\oplus$) required for launching runaway gas accretion. From this perspective, these vortices can be regarded as “planetary cradles.”
While the presence of such vortices in disks seems to be widely accepted, less is known about their formation and lifetime. In this paper, we study the formation and evolution of vortices excited by RWI. The RWI can be excited in protoplanetary disks in various situations, e.g., due to a steep pressure gradient. Such pressure gradients can form, e.g. at the edges of a gap opened by an embedded planet [@Lietal2005], at the edges of the disk’s accretionally inactive dead zone [@Lovelaceetal1999; @Lietal2000], or in protostellar infall from the natal cloud near the centrifugal radius [@Baeetal2015]. However, so far, vortices have only been seen in class II disks, which no longer have any envelope, so this scenario surely does not apply to them.
Both the large-scale vortex formation scenarios have difficulties explaining the asymmetric brightness features of transition disks. Formation of a large-scale vortex at the edge of a giant planet-carved gap is found to be delayed or even suppressed by the disk self-gravity for a sufficiently high-mass ($M_\mathrm{disk}/M_*\gtrsim0.38$) disk [@LinPapaloizou2011]. In shearing sheet simulations, vortices are found to be transient structures due to the effect of self-gravity, inhibiting the formation of a single large-scale vortex [@MamatsashviliRice2009]. If RWI is excited, vortices may also have short lifetimes due to the disk viscosity [@deValBorroetal2007; @Ataieeetal2013; @Fuetal2014b]. Moreover, vortices can be destroyed due to the effect of dust feedback if the local dust-to-gas mass ratio approaches unity [@Johansenetal2004; @InabaBarge2006; @Lyraetal2009a; @Fuetal2014a]. However, @Raettigetal2015 showed that the streaming instability that causes vortex decay is only a temporary effect, and vortices can be re-established. Excitation of vortices at the dead-zone edge by RWI seems to require a sharper viscosity transition [@Lyraetal2009b; @Regalyetal2012] than it is expected to form at the outer dead-zone edge [@Dzyurkevichetal2013]. Note, however, that @Lyraetal2015 found that a smooth change in gas resistivity does not imply a smooth transition in turbulent stress, in which case a large-scale vortex can form, although the resistivity transition is smooth. Considering the abovementioned difficulties, to understand the origin of large-scale brightness asymmetries of transition disks, we must rely on observations and compare them to the predictions given by vortex formation models.
In this paper, we investigate whether or not we can infer the formation mechanism of the vortex from the morphology of the disk as seen in submillimeter ALMA observations. In Section2, we model long-term evolution of RWI-excited large-scale vortices by means of locally isothermal 2D hydrodynamical simulations for a wide range of disk parameters and a morphological comparison of vortex structures in the gas distribution. In Section3, we present submillimeter images calculated for ALMA observations based on hydrodynamical results. In Section4, we provide a discussion on observed versus synthetic submillimeter brightness asymmetries and a review of our model caveats. The paper closes (Section5) with a summary of our findings and concluding remarks.
Formation of Large-scale Vortex
===============================
{width="15cm"}
Hydrodynamic Models
-------------------
We investigated the formation and evolution of a large-scale vortex in two scenarios. In the first scenario, the vortex forms at the edges of a gap opened by a giant planet. Note that we only model the vortex formation at the outer gap edge.[^2] In the second scenario, the vortex is excited by a steep transition in the viscosity, e.g. at the edge of a dead zone. Two-dimensional grid-based, global hydrodynamic disk simulations have been used to study the properties of the vortices under these conditions. We used the GPU-supported version of the code [FARGO]{} [@Masset2000], which numerically solves the vertically integrated continuity and Navier–Stokes equations, using locally isothermal approximation in a cylindrical coordinate system centered on the star.
We used the $\alpha$ prescription for the viscosity [@ShakuraSunyaev1973] with $10^{-5}\leq\alpha\leq10^{-3}$. Initially, the gas density has a power-law profile $\Sigma(R)=\Sigma_0R^{-p}$, with $p=0.5,\,1$ and $1.5$ and $\Sigma_0=1.45\times10^{-5},\,5.3\times10^{-4}$, and $1.45\times10^{-4}$, respectively, for which cases the disk mass is $0.02\,M_*$ independent of $p$. As the Toomre $Q$ parameter [@Toomre1964] initially is higher than unity ($Q_\mathrm{min}\simeq13$ at the vortex radial distance) for the investigated density profiles or even in the center of the fully developed large-scale vortex ($Q_\mathrm{min}\simeq5$), the disk is gravitationally stable; therefore, we neglect the disk self-gravity.
The pressure scale height of the disk is assumed to have a power-law dependence on the radius, $H(R) = hR^{1+\gamma}$, where $h$ is the aspect ratio and $\gamma$ is the flaring index. For flat-disk assumption, we use $h=0.05$ and $\gamma=0$, and for flaring-disk models, we use $h=0.05$ and $\gamma=2/7$ [@ChiangGoldreich1997]. We define a fiducial disk setup for which case $p=0.5$, $\gamma=0$, and $h=0.05$ are used.
Assuming locally isothermal approximation, the disk’s temperature profile is $T(r)\sim R^{2(\gamma-0.5)}$. In this approximation, the equation of state of gas is $P=c_\mathrm{s}(R)\Sigma(R)$, where $c_\mathrm{s}(R)=\Omega(R) H(R)$ is the sound speed $\Omega(R)$ and $H(R)$ are the local Keplerian angular velocity and the local pressure scale height, respectively. The unit length is taken to be 1au and the unit mass is the stellar mass. Assuming that the unit time is the inverse of the Kepler frequency, the orbital period becomes 2$\pi$ and the gravitational constant is unity.
The spatial extension of the computational domain is $3\textrm{--}50$au, consisting of $N_R = 512$ logarithmically distributed radial and $N_\phi=1024$ equidistant azimuthal grid cells. At the inner and outer boundaries, a wave-damping boundary condition is applied [@deValBorroetal2006]. For this boundary condition, the disk mass is not conserved. Indeed, the disk mass increases with time in the simulation, but the total variation of the disk mass during the entire simulation is less than a percent. Note that setting the boundary conditions to open has no significant effect on the results but increase the computational time due to waves developed near the inner boundary. In planet-bearing models, we use $\epsilon H(a)$ as the smoothing of the gravitational potential of the planet with an appropriate value of $\epsilon=0.6$ [@Kleyetal2012].
In our simulations, the star is always at the center of the numerical domain. Therefore, the indirect potential is taken into account (see its importance in [@MittalChiang2015; @ZhuBaruteau2016]); i.e., two additional indirect potentials are added to the total potential due to (1) the shift of the barycenter by the disk material (i.e. large density asymmetry caused by the vortex) and (2) the shift of the barycenter due to the massive planet.
Simulations cover $200\times10^{3}$yr, which corresponds to $\sim6350$ orbits of the nonmigrating planet. Since the initial parameters are set such that the vortex forms at $\sim18$ au, the simulation time corresponds to $\sim2600$ Keplerian orbits at the distance of the vortex center.
Vortex Development at the Gap Outer Edge {#sect:GAP-model}
----------------------------------------
To study the effect of the planet mass on the formation and evolution of vortices developed at the gap edge, we set the planet-to-star mass ratio to $q=1.19,\,2.38,\,4.77,$ and $9.54\times10^{-3}$, corresponding to a planet mass of $1.25,\,2.5,\,5,$ and $10\,M_\mathrm{Jup}$ assuming a solar-mass central star. We assume constant semi major axis of $a_\mathrm{p}=10$au for the planet. We also run simulations in three different viscosity regimes for $q=4.77\times10^{-3}$ giant planet, where $\alpha=10^{-3},\,10^{-4}$, and $10^{-5}$ are assumed. In the following, we refer to these models as “GAP models”.
A gap is opened by the planet in all simulations because the gap-opening criterion is satisfied as long as $\alpha\lesssim0.01$ for the assumed disk aspect ratio and planet-to-star mass ratio [@Cridaetal2006]. The RWI is excited in all models in about 100 orbits, initially with mode number $m=5$ being the fastest growth mode, as predicted by analytic theory for relatively low viscosity ($\alpha\leq10^{-3}$; [@Lovelaceetal1999]).
As the vortices are subject to a coalescing process, a large-scale vortex develops in a couple of additional planetary orbits, which is due to the domination of lower mode numbers (large vortices) after the saturation of the initially linear growth of the $m=5$ mode [@GodonLivio1999]. However, the vortex decays due to the viscous evolution of the disk (see, e.g., [@Cridaetal2006]). For $\alpha=10^{-3}$ and $10^{-4}$, the lifetime of the vortex, $\tau_\mathrm{v}$, is shorter than $11\times10^{3}$yr and $110\times10^3$yr, corresponding to about 350 and 3500 planetary (or $144$ and $1440$ vortex) orbits, respectively.
We run several models to explore the effect of the disk parameters $p$, $\gamma$ or planetary mass on $\tau_\mathrm{v}$ in the $\alpha=10^{-4}$ model. The value of $\tau_\mathrm{v}$ decreases for steeper surface density distributions as it is found to be $\sim90\times10^{3}$yr and $\sim60\times 10^3$yr (corresponding to $\sim1100$ and $\sim785$ vortex orbits) for $p=1.0$ and $1.5$, respectively. It is known that the indirect potential (emerged due to the barycenter shift) has the side effect of increasing the vortex strength and therefore its lifetime [@MittalChiang2015; @ZhuBaruteau2016; @RegalyVorobyov2017]. Since the mass confined inside the vortex decreases with increasing $p$, if the disk mass is kept constant, the indirect potential being proportional to the mass of the asymmetry is also decreases.
For the flaring-disk model, where $\gamma=2/7$ and $p=0.5$, $\tau_\mathrm{v}\simeq6.5\times10^3$yr (corresponding to about 200 planetary and $\sim85$ vortex orbits). This can be explained by that the viscosity $\nu_\mathrm{flared}\simeq5\nu_\mathrm{flat}$ at $R\simeq18$au, which results in a weaker vortex and faster viscous evolution of the gap edge.
Assuming the $p=0.5$ flat-disk model, $\tau_\mathrm{v}$ is found to be correlated with the planetary mass, i.e. $\tau_\mathrm{v}\simeq30,\,55,\,110$, and $150\times10^3$yr (corresponding to $\sim393,\,720,\,1440$, and $1964$ vortex orbits) for a $1.25,\,2.5,\,5$, and $10\,M_\mathrm{Jup}$ planet, respectively. This can be explained by the fact that the density gradient, and therefore the pressure gradient, is sharper for a higher planet mass, which results in a stronger vortex and longer vortex decay time.
Contrary to the previous cases, the vortex lifetime is longer than the simulations for $\alpha=10^{-5}$ models with $p=0.5$ and $1$, while the vortex is dissipated by the end of the simulation for $p=1.5$. The left panel of Fig.\[fig:vortex-evol-2D\] shows the density distribution of the disk in the GAP model at the end of the simulation for $\alpha=10^{-5}$ assuming $p=0.5$. We emphasise that the vortex can survive for a sufficiently long time to be observed in the submillimeter (let us say $\gg100\times10^{3}$yr by the time the transitional disk phase presumably develops) only for a nearly inviscid assumption ($\alpha\leq10^{-5}$) in conjunction with the results of previous investigations [@GodonLivio1999; @deValBorroetal2007; @Ataieeetal2013; @Fuetal2014a].
Vortex Evolution at the Dead-zone Outer Edge
--------------------------------------------
To model the formation of vortices in a disk with a dead zone, we reduced the value of $\alpha$ smoothly within a certain radius such that $\alpha(R)=\alpha\delta_\alpha(R)$. The viscosity reduction factor is given by $$\label{eq:deltaalpha}
\delta_\alpha(R)=1-\frac{1}{2}\left(1-\alpha_\mathrm{mod}\right)\left[1-\tanh\left(\frac{R-R_\mathrm{dze}}{\Delta R_\mathrm{dze}}\right)\right],$$ where $\alpha_\mathrm{mod}=0.01$ is the depth of turbulent viscosity reduction. To quantify the dead-zone edge radius, $R_\mathrm{dze}=24$au was used, where we adopt the results of @MatsumuraPudritz2005, who found that $R_\mathrm{dze}$ lies between 12 and 36au, depending on the density of the disk. For the dead-zone edge width, we assumed $\Delta R_\mathrm{dze}=1$ and $1.5H_\mathrm{dze}$, where $H_\mathrm{dze}=R_\mathrm{dze}h$ (assuming a flat-disk model) is the disk scale-height at the outer edge of the dead zone. Although these parameters correspond to 1.8 and 2.7au assuming $h=0.05$, according to the Equation (\[eq:deltaalpha\]), the gross widths of the regions, where the viscosity transitions occur are 3.6 and 5.4au. Note that we did not study the effect of the inner edge of the dead zone. In the following, we refer to these models as “DZE models”.
Due to the sudden change in viscosity (i.e. in the accretion rate) at the outer dead-zone edge, density and pressure enhancements appear that are found to be unstable to RWI [@VarniereTagger2006; @Terquem2008]. The fastest-growing mode is $m=5$; thus, five anticyclonic vortices initially form that later coalesce [@Lovelaceetal1999; @Lietal2000]. As a result, a large-scale vortex develops inside the dead zone. Note that the large-scale vortex forms at about the same distance, $R\simeq18$au, as in the GAP models assuming $R_\mathrm{dze}=24$au (see Fig.\[fig:vortex-evol-2D\]). DZE vortices are also subject to slow decay due to the disk viscous evolution similarly to the GAP models. However, the lifetime of the large-scale vortices are found to be longer than the simulation time in all models; i.e., DZE vortices are long-lived structures for these cases [@Regalyetal2012; @Lin2014].
Comparison of GAP and DZE Vortices {#sect:hydro-morph}
----------------------------------
![Evolution of the vortex aspect ratio assuming the Kida model (Equation\[eq:Kida\]) as a function of the number of the vortex orbits in six different DZE and GAP models. Since the outer spiral wave excited by the planet enters to the vortex periodically, $\chi$ is smoothed by a running average for GAP models.[]{data-label="fig:chi"}](fig2){width="\columnwidth"}
{width="8.8cm"} {width="8.6cm"}
To compare the morphologies of the full-fledged vortices in the GAP and DZE models, we determine the vortex strengths characterized by the Rossby number, $$R_\mathrm{o}(R,\phi)=\frac{\nabla\times({\bf u}(R,\phi)-R\Omega{(R)})}{2\Omega(R_\mathrm{v})},
\label{eq:Rossby-number}$$ i.e. the $z$ component of the vorticity measured in the local frame of the vortex divided by the global vorticity of the Keplerian disk. The vortex strength, $R_\mathrm{o}$, defined by the mean value of $R_\mathrm{o}(r,\phi)$ inside the vortex (set by the contour level of density at $1/e$ times the maximum density measured at the vortex center), is found to be $-0.14$ and $-0.13$ for $p=0.5$ and $1$ in the GAP models and $-0.068$, $-0.076$, and $-0.041$ for $p=0.5$, $1$, and $1.5$ in the DZE models at the end of the simulations.
For a steady-state vortex with uniform vorticity, $$R_o^\mathrm{Kida}=-\frac{3}{4}\frac{\chi^2+1}{\chi(\chi-1)}+\frac{3}{4},
\label{eq:Kida}$$ where $\chi$ is the vortex aspect ratio defined by the ratio of the azimuthal and radial axes of the vortex [@Kida1981; @Chavanis2000]. For a vorticity field whose strength linearly depends on the distance from the vortex center, @Goodmanetal1987 proposed $$R_o^\mathrm{GNG}=-\frac{\sqrt{3}}{2}\frac{\chi^2+1}{\chi\sqrt{\chi^2-1}}+\frac{3}{4},
\label{eq:GNG}$$ while @SurvilleBarge2015 recently proposed a more elaborate Gaussian model, where $$R_o^\mathrm{Gauss}=-\frac{1}{2}\frac{\chi^2+1}{\chi^2-1}\left(\frac{3}{2}-\sqrt{3}\right)+\frac{3}{2}.
\label{eq:Gauss}$$ We emphasize that Equations (\[eq:GNG\]) and (\[eq:Gauss\]) give $R_o=(1/4)(3-2\sqrt{3})<-0.116$, when $\chi\rightarrow\infty$. To compare vortex aspect ratios for the GAP and DZE vortices, we use the Kida approximation, as $R_\mathrm{o}$ is found to be $\geq-0.1$ in all DZE models.
The evolution of $\chi$ calculated according to Equations (\[eq:Rossby-number\]) and (\[eq:Kida\]) is shown in Fig. \[fig:chi\] for the DZE and GAP models. In the $\alpha=10^{-5}$ GAP models, $\chi$ is found to be $6\lesssim\chi\lesssim8$ and shows only weak evolution in time. However, the vortex lifetime is shorter than the simulation in the $\alpha=10^{-4}$ GAP models, and $\chi\gtrsim10$. Note that $\chi$ shows no clear trends with the planet mass, as we found that the strongest vortex, $\chi\simeq6$, formed in models where a $q=2.5\,M_\mathrm{Jup}$ planet is assumed (dark blue curve in Fig. \[fig:chi\]). In contrast, $\chi$ evolves strongly in time for the DZE models: initially, the vortex strengthens, i.e. $\chi$ decreases up to $\sim500$ orbits; then it starts to weaken, i.e., $\chi$ increases, $\chi_\mathrm{max}\simeq14$ and $10$ for $\alpha_\mathrm{dz}=10^{-4}$ and $10^{-5}$, respectively. Note that $\chi$ does not saturate by the end of the simulation for the DZE models. However, as we showed earlier, a quasi-stationary vortex mode can be reached after $\sim5000$ orbits of the vortex (see Figure5 in [@Regalyetal2012]). In summary, $\chi\simeq8$ in the GAP models where the vortex is maintained until the end of the simulations assuming $\alpha\leq10^{-5}$. For the DZE models, the vortex weakens by the end of the simulation and thus has a larger aspect ratio: $\chi\simeq14$ and $\chi\simeq10$ for $\alpha_\mathrm{dz}=10^{-4}$ and $\alpha_\mathrm{dz}=10^{-5}$, respectively.
In order to compare the spatial extensions of the vortices in the GAP and DZE models, the radial (left) and azimuthal (right) density profiles taken across the vortex center are shown in Fig.\[fig:dens-profiles\]. The radial and azimuthal extensions of the vortices, $\Delta_\mathrm{rad}$ and $\Delta_\mathrm{azim}$, are determined by measuring the spatial extension of the vortex at the $1/e$th of the density maximum. For the GAP model, $\Delta_\mathrm{rad}\simeq2.7$au and $\Delta_\mathrm{azim}\simeq25$au (corresponding to $\sim244^\circ$ angular width), while for the DZE model, $\Delta_\mathrm{rad}\simeq4.2$au and $\Delta_\mathrm{azim}\simeq80$au (corresponding to $\sim95^\circ$ angular width).
Although the vortex excitation model (GAP or DZE) could be distinguished by measuring the vortex aspect ratios, the precise determination of $\chi$ would require a very high resolution (i.e. $\sim0.01\arcsec$ beam size for our model assuming a 100pc source distance) to resolve the vortices radially. Moreover, in some particular cases, the vortex strengths are similar in the DZE and GAP models: $\chi\simeq10$ for the $\alpha=10^{-5}$ GAP and $\alpha_\mathrm{dz}=10^{-5}$ DZE models, respectively. Therefore, vortex aspect ratio is not an ideal parameter to use for distinguishing the two formation scenarios. However, vortices are about an order of magnitude wider in the azimuthal than in the radial direction, easing the determination of the vortex azimuthal width. The azimuthal extensions of the DZE vortices are found to be exceed $\sim180^\circ$, while they are only $\sim90^\circ$ for the GAP vortices, in agreement with the literature [@Lyraetal2015; @Mirandaetal2016; @Mirandaetal2017; @ZhuBaruteau2016]. Note that @LyraMacLow2012 and @Flocketal2015 presented azimuthally concentrated DZE vortices presumably caused by MHD effects, but see [@ZhuStone2014]. Note also that the vortex azimuthal extension in the GAP models that includes disk self-gravity is found to be in the range of $90^\circ-180^\circ$ [@LinPapaloizou2011]. However, the stretching requires a more massive disk than what we assumed. Therefore, we suggest that the vortex azimuthal extension is a key parameter that can be used to infer the vortex excitation mechanism as long as disk self-gravity is negligible.
Here we have to note that vertical shear instability [@Nelsonetal2013] is effective in transporting of angular momentum in the disk causing $\alpha\simeq10^{-4}$ [@StollKley2014]. Thus, the magnitude of viscosity, $\alpha=10^{-5}$, required for the long-term vortex in the GAP models is unreasonably small. However, the $\alpha_\mathrm{dz}=10^{-4}$ is a reasonable value for the disk viscosity in the dead zone. For this reason, we compare the observational properties of vortices formed in the $\alpha=10^{-5}$ GAP and $\alpha_\mathrm{dz}=10^{-4}$ models.
Synthetic ALMA Images
=====================
The differences in the density enhancement and the spatial extent of the vortices between the DZE and GAP models may provide a way to observationally study the origin of the vortices. Submillimeter continuum observation is a straightforward choice to observationally study the vortex morphology, as most protoplanetary disks are thought to be optically thin at these wavelengths; thus, the submillimeter continuum emission probes the total surface density of the dust. [^3] In this section, we present synthetic observations for the Atacama Large Millimetre and Submillimeter Array (ALMA) and investigate whether or not we can infer the formation mechanism of a large-scale vortex (gap outer edge vs. dead-zone outer edge) from the morphologies seen in the synthetic observations.
In order to study the morphology of the vortices in submillimeter observations, we assume that the mm-sized dust distribution follows that of the gas given by the hydrodynamic simulations. However, the dust inside the planetary orbit is artificially removed in the GAP models, and complete dust clearing of mm-size dust is assumed inside the dead zone in the DZE models. In the following section, we present the plausibility of this approximation.
Dust Depletion in the Inner Disk {#sect:dust-clearing}
--------------------------------
In the GAP model, it is assumed that the dust is completely removed because the planet clears all the material inside its gap. However, a giant planet opens only a narrow gap whose width is $\Delta R_\mathrm{GAP}\simeq5-6R_H$, where $R_H=a(q/3)^{1/3}$ is the radius of the planetary Hill sphere [@Cridaetal2006]. Therefore, a system of multiple planets is required to completely clear the inner disk [@Dodson-RobinsonSalyk2011; @Zhuetal2011]. In the hydrodynamic simulations of @Isellaetal2013, three equal-mass giant planets ($q=1.9\times10^{-3}$) orbiting at $a=21,\,34$ and $55$au were assumed to explain the observed brightness asymmetry of LkH$\alpha$330. They found complete gas clearing in their simulations due to overlapping of planetary gaps. We repeated their calculations with $\alpha=2\times10^{-3}$ and $2\times10^{-4}$ (we use the same $\alpha$ assumptions for the viscosity as @Isellaetal2013 are used), and confirmed the complete gas clearing. Note, however, that the vortex lifetime exceeds $10^{5}$yr only for the nearly inviscid case, where $\alpha$ is on the order of $10^{-5}$.
We emphasize that, according to our simulations, the observed vortex structures in the gas, i.e. the azimuthal extension and contrast, for multiple- and single-planet setups are similar. This can be explained by the fact that RWI excitation at the outer edge of a common gap is driven by the outermost giant planet only, while the inner planets do not affect the vortex formation there. Although our hydrodynamic models with a single giant planet (presented in Section\[sect:GAP-model\]) cannot explain the inner disk clearing, assuming complete dust clearing inside the planetary orbit fairly models the formation and evolution of the large-scale vortex for multiple-planet systems.
In the DZE model, however, the inner disk is not cleared in the gas; therefore, to test the dust removal assumption, we investigated the dust drift in a viscously evolving 1D disk model. The radial drift of the solid particles in the viscously evolving dead zone is modelled in a representative particle approach, in which dust coagulation and fragmentation are not included. We followed the dust drift for $3\times10^{5}$yr with $10^{5}$ dust particles. In this simple 1D model, we assume that the disk is axisymmetric, thus the evolution of the surface density of the gas ($\Sigma_g(R,t)$) can be given by $$\frac{\partial \Sigma_g(R,t)}{\partial t} = \frac{3}{R} \frac{\partial}{\partial R} \left[ R^{1/2} \frac{\partial}{\partial R} \left( \nu(R) \Sigma_g(R,t) R^{1/2} \right) \right],
\label{eq:ido}$$ where $\nu(R) = \alpha \delta_\alpha(R) c_s^2(R)$ is the kinematic viscosity of the gas in our dead-zone edge model, where $\delta_\alpha(R)$ is defined by equation(\[eq:deltaalpha\]).
{width="\columnwidth"} {width="\columnwidth"}
The coupling of the dust to the gas can be described by the Stokes number ($St$), which in the Epstein regime (where the interaction between particles and a single gas molecule becomes important, as the particle size is comparable to the mean free path of molecules) is $$St = \frac{a \rho_s }{\Sigma_g(R,t)}\frac{\pi}{2},
\label{eq:stokes}$$ where $a$ is the size (assumed to be mm and $\mu$m in two simulations, respectively) and the intrinsic density of the solid particle is set to $\rho_s=1.6\,\mathrm{g\,cm^{-3}}$. The radial velocity of the solid particles ($u_\mathrm{dust,r}$) is given by $$u_\mathrm{dust,r} = \frac{u_\mathrm{gas,r}}{1+St^2}+\frac{2}{St+St^{-1}}u_\mathrm{drift},
\label{eq:drift}$$ where $$u_\mathrm{gas,r} = - \frac{3}{\Sigma_g(R,t) R^{1/2}} \frac{\partial}{\partial R} \left( \Sigma_g(R,t) \nu(R) R^{1/2} \right)
\label{eq:vrg}$$ is the radial velocity of the gas and $$u_\mathrm{drift} = \frac{c_s^2(R)}{2 \Omega(R)R} \frac{d\ln{P}}{d\ln{R}}
\label{eq:drift}$$ is the maximum drift velocity of a particle, where $\Omega(R)=R^{-3/2}$ (see, e.g.,[@Weidenschilling1977; @Nakagawaetal1986; @YoudinLithwick2007] and [@Birnstieletal2012b]).
In our dust drift model, the the disk extends from 0.1 to 50au with 1500 radial cells. Similar to our fiducial 2D DZE hydrodynamic model, the outer dead-zone edge is set to $R_\mathrm{dze}=24$au, $\Delta R_\mathrm{dze}=1.5H_\mathrm{dze}$, and the initial surface mass density of the gas is $\Sigma_\mathrm{gas}(R)=\Sigma_0 R^{-0.5}$, where $\Sigma_\mathrm{g}=1.45316\times10^{-5} M_\odot\,\mathrm{au}^{-2}$ is used.
Fig.\[fig:radial-drift\] shows the evolution of the semimajor axis of mm-sized (left panel) and $\mu$m-sized (right panel) dust particles in the fiducial DZE model. A negative pressure gradient results in sub-Keplerian gas velocity and causes rapid inward drift of dust particles, while the gas is super-Keplerian at positive pressure gradients, which causes outward drift of dust particles (see Fig.\[fig:dze-feeding\]). As a result, most of the mm-sized dust particles initially orbiting inside the dead zone drift toward the central star, while the dust initially orbiting outside the dead-zone edge are trapped at the pressure maximum. As a result, mm-sized dust particles are accumulated at the local pressure maximum. However, $\mu$m-sized dust particles are well coupled to the gas; therefore, the dead zone remains well populated with $\mu$m-sized dust.
The radial extension of the feeding zones for the pressure maxima evolves in time, which can affect their dust accumulation efficiency. Due to the viscous evolution of the disk, the pressure maximum shifts inward (see e.g. [@Regalyetal2012]). We found that the mm-sized dust particles are completely absent in the dead zone by $\sim250\times10^3$yrs, while the dead zone is still populated by $\mu$m-sized dust. Note that if we would take into account the dust coagulation, the average dust size increases in time. Since the dust drift velocity increases with the particle size, the dust clearing would be accelerated.
![Azimuthally averaged density, pressure, and pressure gradient profiles in the dust drift model. The pressure maximum impedes to refill the dead zone with dust. Inside $\sim15$au, the dust is subject to drift toward the central star.[]{data-label="fig:dze-feeding"}](fig5){width="0.9\columnwidth"}
Note that @Birnstieletal2012a found that the presence of the dead zone fails to explain the dust clearing of the inner disk observed in transition disks using a more elaborate model that also takes into account dust coagulation and fragmentation. However, they did not take into account the effect of the pressure maximum formed at the dead-zone outer edge. Since particles (less than several meters in size) strongly coupled to the gas cannot cross the pressure bump (Fig.\[fig:dze-feeding\]), the dead zone cannot be replenished with mm-sized dust from the outer disk.
In short, it is a plausible assumption that the inner disk is completely cleared of mm-sized dust within several $10^5$yr for the DZE models assuming our fiducial disk parameters, while multiple planets are required to completely clear the inner disk of mm-sized dust for the GAP models.
Radiative transfer setup for synthetic image calculation
--------------------------------------------------------
Synthetic images are calculated by the 3D radiative transfer code RADMC-3D[^4]. Our imaging simulations contain three steps. First, the dust temperature is calculated in a thermal Monte Carlo simulation, then images at 880$\mu$m are calculated using the ray-tracing module of RADMC-3D, snd finally, ALMA observations are simulated using the Common Astronomy Software Applications (CASA) package[^5] v4.2.2.
Since FARGO solves dimensionless equations, the simulations can be scaled in terms of spatial extent and also in mass in the locally isothermal approximation. Therefore, to make our synthetic images comparable to observations, we can scale the hydrodynamic simulations such that the disk extends from 9 to 150au with the planetary orbital radius being at 30au. In this case, the vortex is located at about 30–45au from the star, similar to the observations (see, e.g., [@Brownetal2009]), and the simulation time corresponds to $\sim1$Myr. For both the DZE and GAP models, we run two set of simulations, where the core of the vortex is optically thin and thick (by appropriately scaling the density).
In the RADMC-3D simulations, we use a 3D spherical mesh ($R, \phi, \theta$). The radial grid extends between 9 and 150au, and contains $N_R=256$ grid cells. Along the azimuthal and poloidal angular coordinates, we use $N_\phi=512$ and $N_\theta=180$ grid cells, respectively. While the azimuthal grid is equidistant, the grid cells in the poloidal coordinate are distributed such that we use 10, 160, and 10 grid points in the \[0, $\pi/2-0.25$\], \[$\pi/2-0.25$, $\pi/2+0.25$\], and \[$\pi/2+0.25$, $\pi$\] intervals, respectively. In each poloidal interval, we use a uniform, equidistant grid.
Hydrodynamic frames are interpolated to the RADMC-3D mesh and converted to volume density assuming a Gaussian vertical density distribution assuming a constant aspect ratio of $h=0.05$, similar to the 2D hydrodynamic simulations. In the radiative transfer model, we use the optical constants of the astronomical silicate from @WeingartnerDraine2001 for the dust particles and calculate the absorption and scattering cross-sections using Mie theory. Dust grains in the model have a size distribution of $n(a)\propto a^{-3.5}$ between $0.1$ and 1000$\mu$m, corresponding to the steady-state mass distribution of the collisional fragmentation cascade [@Dohnanyi1969]. The central radiation source in the model has parameters representative of that of a HerbigAe star: R$_\star$=2.5$R_\odot$, T$_\star$=9500K, M$_\star$=2.0$M_\odot$.
To determine the dust temperature of individual dust grains, we run a thermal Monte Carlo simulation using 1.2$\times10^8$ photon packets. The continuum images at 880$\mu$m are calculated with ray tracing, consisting of 800 by 800 pixels, with a pixel size of 0.89mas assuming a source distance of 140pc.
Simulated ALMA observations are calculated with the CASA [simobserve]{} task to generate synthetic visibilities and the [clean]{} task for imaging. The full 12m array was used for the simulated observations, in three different configurations resulting in a spatial resolution of approximately 0.25", 0.18, and 0.1. We assume the full 7.5GHz continuum bandwidth of the ALMA correlator and calculate the visibilities for a total integration time of 30 minutes. The simulations include thermal noise arising from a water vapor column of 0.913mm representative of Band7 observations.
In both optically thin and thick cases, we calculated images at two inclination angles, $30^\circ$ and $60^\circ$ and five different azimuth angles for the vortex between 0$^\circ$ and 180$^\circ$ in 45$^\circ$ steps. To study the effect of the shape of the synthesized beam on the morphology of the images, we simulated observations assuming a source decl. of 25$^\circ$ and 70$^\circ$, and the times of the observations are centered on 0, -1, and -2hr angles.
Analysis of Synthetic Images
----------------------------
{width="19cm"}
The most striking difference between the vortices excited at the edge of a planet-carved gap and at the edge of a dead zone is their azimuthal extent (see Sect.\[sect:hydro-morph\]). Our hydrodynamic simulations show that vortices formed at the edge of a planetary gap are significantly more compact in the azimuthal direction than their counterparts at the edge of a dead zone. In the following, we show that this property can be used in observations to infer the formation mechanism of the vortices.
To test the applicability of this technique, we determine the azimuthal extent of the vortices in the synthetic ALMA observations in the following way. As a first step, we deproject the images with a known inclination and position angle. In real observations of well-studied objects, both the inclination and the position angle are known to a reasonable (several degrees) accuracy from gas-line observations. Then, we converted the image from Cartesian to polar coordinates centered on the stellar position. Finally, we measured the azimuthal size of the region where the intensity is 66% of that of the peak intensity in the core of the vortex. The measured sizes of the vortices are presented in Fig.\[fig:vortex\_size\] for different position angles of the vortex, hour angles the observations are centered on, and optically thick and thin vortex core assumptions.
For low inclination angles ($30^\circ$; panels (a), (c), and (e) of Fig.\[fig:vortex\_size\]), the DZE and GAP models can be separated in most cases based on the measured azimuthal size of the vortex. While for the lowest resolution ($\sim$0.25; see panel (a)), some confusion may occur between the two models, at the highest resolution (0.1; see panel (e)) the DZE and GAP models are clearly separated. For high inclination angles ($60^\circ$; panels (b), (d), and (f) of Fig.\[fig:vortex\_size\]), separating the two models becomes more challenging, especially for the lowest spatial resolution.
Based on our results presented in Fig.\[fig:vortex\_size\], we conclude that if (1) the disk is optically thin, (2) the inclination is low (probably 45$^\circ$ or less) and (3) the resolution is sufficiently high (the beam size is at least twice as small as the projected distance from the star to the vortex core), the DZE and GAP models can be distinguished. As we show in the following, the morphology of the vortex in the two different models can be very similar for higher optical depths and inclination angles and lower spatial resolutions.
### Effect of optical depth
![Synthetic ALMA observations of optically thin DZE (panels (a), (c)) and GAP (panels (b), (d)) models at two different resolutions (0.25 on panels (a), (b) and, 0.1 in panels (c), (d)). The arc of the vortex is significantly more extended in the azimuthal direction for a DZE model compare to a GAP model.[]{data-label="fig:clean_alma_prediction_i30"}](fig7){width="9cm"}
![Synthetic ALMA observations of an optically thin DZE (panels (a), (c)) and an and optically thick GAP (panels (b), (d)) models at two different resolutions (0.25 in panels (a), (b) and 0.1 in panels (c), (d)). Once the disk becomes optically thick at the edge of the gap, the arc of the vortex also appears azimuthally extended in a GAP model; however, the models can be distinguished at high resolution.[]{data-label="fig:optical_depth_effect_i60"}](fig8){width="9cm"}
The true azimuthal extent of the vortex can only be measured in submillimeter continuum images if the disk is optically thin, in which case the DZE and GAP models can easily be distinguished (Fig.\[fig:clean\_alma\_prediction\_i30\]). Once the disk becomes optically thick, the surface brightness is determined by the temperature instead of the surface density. Since the highest surface density is in the core of the vortex, the disk optical thickness becomes the highest there. The size of the region where the observed intensity drops below a certain fraction of the peak intensity (i.e. the azimuthal extension of the horseshoe-shaped asymmetry) also increases with the optical depth. Therefore, the azimuthal extension of the vortex is overestimated if the disk is optically thick at the vortex core. For the optically thick case, the azimuthal contrast across the vortex is also decreased by the saturation of the emission at the vortex core. The combination of these effects can make the appearance of a vortex at a planetary gap edge very similar to that of a vortex at the edge of a dead zone, but only for low resolution (Fig.\[fig:optical\_depth\_effect\_i60\]).
### Effect of beam shape
![Same as Fig.\[fig:optical\_depth\_effect\_i60\] but for an inclination angle of 30$^\circ$. If the synthesized beam is elongated and the major axis of the beam ellipse aligns with the position of the vortex, the azimuthal extent of the vortex in the GAP model (panel (b)) can look similar to that of a vortex in the DZE model (panel (a)). Increasing the spatial resolution can break the degeneracy in the apparent vortex morphology, and the two models can be distinguished (panels (c), (d)).[]{data-label="fig:optical_depth_effect_i30"}](fig9){width="9cm"}
In interferometric observations, the synthesized beam is not generally circular, but rather elliptic due to the combination of the elevation of the target above the horizon and the antenna configuration. The ellipticity of the beam causes the structures in the image to look more elongated along the direction of the beam major axis. Such elliptical beams can cause a characteristic changes to the apparent morphology of the observed vortices.
The azimuthal extent of the vortex in the GAP models depends on the position angle of the beam ellipse. The apparent azimuthal extent of the vortex is the highest if the beam major axis is perpendicular to the vortex. For the optically thick case, the vortex in the GAP model can look azimuthally as much extended as a DZE vortex (panels (a) and (b) of Fig.\[fig:optical\_depth\_effect\_i30\]).
At low inclination angles, the peak of the emission is located in the middle of the extended horseshoe-shaped structure, i.e. at the vortex center. At high inclination angles, however, the peak of the brightness can shift from the vortex center, depending on the relative position of the beam major axis and the vortex (panels (b) and (d) in Fig.\[fig:optical\_depth\_effect\_i60\]). We note that the horseshoe-shaped asymmetry is always azimuthally less extended in the GAP models than in the DZE models in the optically thin case (Fig.\[fig:optical\_depth\_effect\_i60\]).
In contrast to the GAP models, the vortex in the DZE models can split into two bright blobs if the beam is elongated. For optically thick DZE vortices, this is always the case. If, however, a DZE vortex is optically thin, the number of bright blobs and/or the azimuthal extent of the observed horseshoe depends strongly on the position angle of the beam ellipse with respect to that of the vortex. If the beam major axis is perpendicular to the vortex, an azimuthally extended horseshoe appears with two distinct bright peaks (panel (a) of Fig.\[fig:beamsize\_effect\]). In contrast, if the beam major axis is parallel to the vortex, a single-peaked asymmetry appears with a significantly smaller azimuthal extension (panel (b) of Fig.\[fig:beamsize\_effect\]).
We note that this direction-dependent “smearing” effect also depends on the ratio of the structure size to the beam size. Beam-shape effects cannot change the morphology of structures in the image that are larger than a few beam sizes. Therefore, increasing the angular resolution in the observations can in most cases break the degeneracy, making it possible to distinguish DZE and GAP models based on the vortex azimuthal extent.
Discussion
==========
![Effect of beam shape on the image of a DZE vortex. The same radiative transfer image was used to generate the synthetic observations. To change the shape of the beam, we changed decl. of the source in the simulated observations from -20$^\circ$ (panels (a), (c)) to -70$^\circ$ (panels (b), (d)). The inclination of the disk and the position angle of the vortex were chosen to match those of SR21 [@Perezetal2014]. For elongated beams at low spatial resolution, a DZE vortex can show one or two distinct peaks, depending on whether the major axis of the beam aligns with the position of the vortex (panel (a)) or perpendicular to it (panel (b)). At high spatial resolution, this effect is also present (panels (c), (d)). []{data-label="fig:beamsize_effect"}](fig10){width="9cm"}
Comparison to observations
--------------------------
While there is now observational evidence for nonaxisymmetric surface brightness distribution in the submillimeter for more than a dozen transitional disks, the precise morphology of the asymmetry is rather uncertain in many cases due to insufficient spatial resolution and/or the low signal-to-noise ratio of the observation. However, in a few cases, if ALMA observations are available and the structure of the disk is known well enough, that we can qualitatively compare the observed morphology of the disk — most importantly, the azimuthal extent of the observed vortices — to our models.
There are four well-known transitional disks with ALMA observations and evidence for a vortex in the disk: OphIRS48 [@vanderMareletal2013], HD142527 [@Casassusetal2013], HD135344B and [@Perezetal2014], and SR21 [@Brownetal2009]. @vanderMareletal2013 explained the extreme strong asymmetry in the disk of IRS48 with a vortex excited by a massive ( $>10\,M_{\rm Jup}$) giant planet. @vanderMareletal2013 et al. presented a numerical model of a gap-edge vortex showing remarkable similarities with the observations.
The azimuthal extent of the asymmetries in the other three sources, HD142527, HD135344B, and SR21, seems to be significantly larger than that in the disk of OphIRS48 and thus may not necessarily be consistent with a planet-induced vortex. To fit the observed images, @Perezetal2014 assumed a superposition of a planet-induced vortex and a full ring for both HD135344B and SR21. For canonical disk viscosity values ($10^{-3}\leq\alpha\leq10^{-2}$), only a symmetric dust ring is expected to form at a planet-carved gap edge. Therefore, such azimuthally extended asymmetries can be better explained by a sudden jump in the viscosity (proposed by @VarniereTagger2006 and later compared to observations by @Regalyetal2012, rather than by putative giant planet(s).
Our models of a DZE vortex bear especially remarkable similarities with the images of SR21. In @Brownetal2008 the image of SR21, obtained with the Sub-Millimeter Array (SMA), shows a horseshoe-shaped structure with more than $270^\circ$ in azimuthal extent and with two bright blobs on the two opposite sides of the arc. In contrast, the ALMA image in @Perezetal2014 shows only a single maximum along the horseshoe-shaped asymmetry at the position between the two blobs seen in the SMA image. In Fig.\[fig:beamsize\_effect\], we show that an azimuthally extended DZE vortex can reproduce the morphology of both the SMA and the ALMA images remarkably well if we take into account the differences in the shapes of the synthesized beams in the two observations. Note that since our goal was to show a qualitative comparison of our models with the observations, we did not use the exact same uv-coverage as in the real SMA and ALMA observations of SR21. Instead, we changed the decl. of our target in the synthetic observations such that the resulting synthesized beam will have a similar shape to that in the real observations.
So far, horseshoe-shaped asymmetries in protoplanetary disks have been interpreted with vortices induced by planets. Here we suggest that in a few cases where the vortex is azimuthally extended, this may not be the case, and the vortex could be induced by a viscosity jump [@Regalyetal2012]. Unfortunately, the images obtained so far do not have high enough spatial resolution that an azimuthally extended vortex could be distinguished from other models, e.g., a planet-induced vortex + ring proposed by @Perezetal2014. A significantly higher spatial resolution (by a factor of about 2-3 at least) compared to already existing one is needed to unambiguously identify the presence of such extended vortices.
Outlook and Caveats
-------------------
Here we have to mention some caveats of our model. Regarding the hydrodynamical simulations, the most obvious caveat is that we run simulations in 2D. However, @Meheutetal2012c found that the vortex formation and development in 3D are indeed very similar to that of 2D: the disk vertical stratification only slightly decreases the growth rate of vortices compared to that of 2D.
We neglect the effect of disk self-gravity, while @LinPapaloizou2011 found that in self-gravitating disks, the gap-edge vortex coagulation ends at mode $m=2$ for relatively high disk masses ($\geq0.031M_*$). As a result, two vortices may present simultaneously in the disk. Thus, in the GAP models, we should see a double-peaked submillimeter brightness distribution similar to that of in the DZE models. @LovelaceHohlfeld2013 found that the disk self-gravity could be important regarding the vortex formation if $Q<R/h$. Recently, @ZhuBaruteau2016 found that taking into account the disk’s self-gravity results in weakened vortex, especially for massive disks. Note, however, that they artificially imposed a density jump on the disk to form a large-scale vortex. Therefore, DZE or GAP vortex formation in self-gravitating disks assuming modest disk mass is worth investigating (see, e.g. [@RegalyVorobyov2017b]).
We neglect the planet growth in the GAP models. Recently, @Hammeretal2017 found that the vortex azimuthal extension generated by a slowly growing giant planet ($T_\mathrm{growth}$ being on the order of several 1000 planetary orbits) can be larger by about a factor of two, similar to that of the DZE models. Note, however, that they prescribed planet growth by an artificial sinusoidal function (see their Equation (1)) neglecting the viscous dynamics inside the planet Hill sphere. Thus, it is worth modeling RWI excitation with a more elaborate accretion prescription, like what is described in @Kley1999.
We assume a locally isothermal disk, which is valid only if the cooling time of the gas is fast compared to the local orbital time-scale. This assumption, however, can be plausible for a several-million-years-old transition disk. Nevertheless, it is worth investigating vortex formation and survival in adiabatic disks, as @Lyraetal2009b found stronger RWI excitation in adiabatic disks, which may have consequences for the vortex evolution.
We completely neglect the MHD effects in our simulations. However, @ZhuStone2014 showed that 2D simulations assuming a kinematic viscosity (approximated to the stresses of their 3D MHD models) show vortex formation that is very similar to their full MHD simulations. Note also that a long vortex lifetime in the 3D MHD simulations of @ZhuStone2014 was only observed if a nonideal effect, such as the ambipolar diffusion, was taken into account. The nonideal effect of ambipolar diffusion results in suppression of MRI turbulence, i.e. low disk viscosity, which helps to maintain the vortex.
In our dead-zone models, we assume quite sharp viscosity transitions to excite RWI. The simulations of @Lyraetal2009b and @Regalyetal2012 agree in that RWI excitation occurs only for sharp dead-zone edges in which the width of the viscosity transition is smaller than twice the local pressure scale height. However, @Dzyurkevichetal2013 have shown that resistivity changes very smoothly (therefore, MRI generated viscosity parameter $\alpha$ too) in the outer dead-zone edge. As a result, the dead-zone edge in an $\alpha$ disk is about five times thicker than the required threshold value for RWI excitation. However, replacing the $\alpha$ prescription with resistive MHD, a smooth change in gas resistivity, is found not to imply a smooth transition in turbulent stress [@Lyraetal2015]. As a result, a large-scale vortex is stable even in smooth resistivity transition, and the abovementioned requirements for exciting RWI is only an inherent feature of $\alpha$ models.
Regarding the calculation of synthetic images, there are also some caveats in our model. The simple assumption of that the dust density distribution follows that of the gas (i.e. the dust-to-gas ratio is constant throughout the disk) neglects the earlier results of @Johansenetal2006, who observed strong enhancement of the dust-to-gas mass ratio in the presence of anticyclonic vortices. @Birnstieletal2013 also found that the dust-to-gas mass ratio could become higher than the global value if the viscosity parameter was larger than the particle’s Stokes number. Moreover, we neglect the dust feedback to the vortex evolution too. Recently, [@Fuetal2014b] investigated the dust feedback on the vortex excited by a giant planet having a $q=5\times10^{-3}$ planet-to-star mass ratio and found that the dust feedback can destroy the vortex if a large amount of dust ($10\,M_\oplus$) is accumulated inside the vortex. Note that @Regalyetal2013 investigated the migration trapping of low-mass planets ($10\,M_\oplus$) in the presence of dead-zone edge vortex and found similar vortex destruction as the planet enters to the vortex. However, since the trapping of the low-mass planet is found to be only temporary, the vortex redevelops due to an existing sharp density jump formed at the dead-zone edge. Therefore, we think that the gap-edge vortex is vulnerable to and the dead-zone edge vortex is stable against dust feedback. In any case, to give a more reliable model for the evolution of gap-edge and dead-zone-edge vortices, both the dust and gas dynamics should be incorporated in hydrodynamic models.
Summary and Conclusion
======================
The conventional explanation for the large-scale brightness asymmetries of transition disks is the accumulation of dust grains in the core of anticyclonic vortices. Such vortices can form by the excitation of RWI in the vicinity of steep pressure gradients formed, e.g. at the edges of a giant planet-carved gap (GAP model) or the outer edge of a disk’s dead zone (DZE model). We investigated by means of 2D hydrodynamical simulations whether or not we can infer the formation mechanism of the vortex from the morphology of the disk as seen in submillimeter ALMA observations.
We studied GAP and DZE vortex formation mechanisms by means of 2D locally isothermal hydrodynamical simulations with an $\alpha$-type prescription for the disk’s viscosity. We modeled the vortex formation at the outer edge of an embedded giant planet-carved gap ($R_p=10$au, $q=1.25,\,2.5,\,5,$ and $10\,M_\mathrm{Jup}$) in three different viscosity regimes ($\alpha=10^{-3},\, 10^{-4},\, 10^{-5}$) and in a disk that has a sharp dead-zone edge ($\alpha=10^{-2}$, $\alpha_\mathrm{dz}=10^{-4}$, $R_\mathrm{dze}=24$au, $\Delta R_\mathrm{dze}=1H_\mathrm{dze}$ and $1.5H_\mathrm{dze}$). We tested the vortex lifetime against the steepness of the initial density profiles ($\Sigma\sim R^{-p}$, where $p=0.5,\,1$, and $1.5$) and disk geometry that is flat and flared.
Although RWI is excited and a large-scale vortex develops in both models, its survival for a sufficiently long time to observe requires nearly inviscid GAP models, $\alpha\leq10^{-5}$, while an order of magnitude lower disk viscosity, $\alpha_\mathrm{dz}\leq10^{-4}$, is sufficient for DZE models. We emphasize that 3D hydrodynamical models showed that $\alpha\gtrsim10^{-4}$ in protoplanetary disks due to the vertical shear instability [@StollKley2014]. Long-lasting vortex development favors a relatively smooth initial density slope, i.e. $p\leq1$ in the GAP models. The GAP and DZE vortices have distinct geometric features, which is suitable to differentiate the two formation scenarios. Our main findings based on synthetic ALMA images calculated from the hydrodynamic simulations are as follows.
\(1) We showed that the GAP vortices are generally stronger than the DZE vortices; the aspect ratios are $\sim8/10$ and $\sim10/14$ for $\alpha=10^{-4}/10^{-5}$, respectively. Vortices formed in the DZE models are spatially more extended ($\gtrsim180^\circ$) than in the GAP models ($\sim90^\circ$), which can be used to distinguish the two formation scenarios.
\(2) Following the dust drift in a 1D model, the disk inside the dead zone is found to be cleared of mm-sized dust within $\sim2.5\times10^5$yr, due to the efficient dust collection of pressure maxima formed at the dead-zone edge. Contrarily, the inner disk is still populated with $\mu$m dust.
\(3) In the submillimeter, the brightness asymmetries are significantly different for the GAP and DZE models assuming optically thin disk emission. The brightness asymmetry is azimuthally more concentrated for the GAP than the DZE models: the azimuthal brightness contrast is $\sim4$ in GAP and $\sim2$ in DZE models. In the DZE models, the brightness distribution shows multiple peaks for low disk inclination angles ($i\simeq60^\circ$), due to the relatively lower azimuthal brightness contrast and projection effect.
In summary, we found a tentative evidence that the shape of the surface brightness asymmetries in submillimeter wavelengths correlates with the vortex formation process, within the limitations of the $\alpha$ viscosity approximation. One needs, however, to resolve the azimuthal extent of the vortex in the GAP models (i.e. by 4–5 resolution elements/beams). The azimuthal extent of the gap-edge vortex is found to be approximately 2 rad (see Sec\[sect:GAP-model\]). If the vortex is located at 40au from the star, and the star is at a distance of 140pc, this translates to a spatial resolution requirement of about 0.1–0.15.
Assuming that the $\alpha$-viscosity prescription of @ShakuraSunyaev1973 is adequate for transition disks, our analysis of synthetic images (Figs.\[fig:clean\_alma\_prediction\_i30\]-\[fig:optical\_depth\_effect\_i30\]) suggests that the dead-zone-edge vortex scenario might be more plausible than the gap-edge vortex scenario for sources that show double-peaked or horseshoe-like brightness asymmetries (e.g. HD142527, HD135344B, and SR21). However, the existence of an RW unstable viscosity transition in protoplanetary disks has not yet been confirmed. For sources that show single-peaked strong brightness asymmetries (e.g. OphIRS48), the gap-edge scenario could give a better explanation, although for conventional viscosity values ($\alpha\simeq10^{-3}-10^{-2}$), the shortlifetime of the gap-edge vortex challenges our current understanding.
Acknowledgments {#acknowledgments .unnumbered}
===============
This project was supported by the Hungarian National Research, Development and Innovation Office grant No. 119993. ZsR acknowledges the support of the Momentum grant of the MTA CSFK Lendület Disk Research Group. AJ acknowledges the support of the DISCSIM project, grant agreement 341137, funded by the European Research Council under ERC-2013-ADG. ZsR gratefully acknowledges the support of NVIDIA Corporation with the donation of Tesla GPUs and NIIF for access to computational resource based in Hungary at Debrecen. A report by an anonymous referee further improved the quality of the manuscript.
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[^1]: E-mail:[email protected]
[^2]: Since the vortex evolution at the inner gap edge can be strongly affected by the simulation domain’s inner boundary, we do not follow the evolution of vortices at the inner gap edge.
[^3]: Note that the vortex center can be optically thick due to significant dust accumulation. This is the reason why we investigate not only optically thin but also thick emission.
[^4]: http://www.ita.uni-heidelberg.de/dullemond/software/radmc-3d
[^5]: http://casa.nrao.edu/index.shtml
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'Embodiment is an important characteristic for all intelligent agents (creatures and robots), while existing scene description tasks mainly focus on analyzing images passively and the semantic understanding of the scenario is separated from the interaction between the agent and the environment. In this work, we propose the *Embodied Scene Description*, which exploits the embodiment ability of the agent to find an optimal viewpoint in its environment for scene description tasks. A learning framework with the paradigms of imitation learning and reinforcement learning is established to teach the intelligent agent to generate corresponding sensorimotor activities. The proposed framework is tested on both the AI2Thor dataset and a real world robotic platform demonstrating the effectiveness and extendability of the developed method.'
author:
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title: Towards Embodied Scene Description
---
introduction
============
When a visually impaired person enters a new room, he can easily take pictures of his surroundings using the smartphone and the built-in advanced computer vision modules are able to provide some scattered semantic information of these pictures. For example, the smartphone can detect certain classes of objects in the image with the help of an object detector and speak them out to the visually impaired person. However, such information is likely to make people confusing and uncomfortable due to its disorder and disorganization. A better way is to generate higher level semantic description such as natural language sentences or even paragraphs to describe the image. At present, great progress has been made in the areas of *Image Captioning*[@vinyals2015show][@xu2015show], *Dense Captioning*[@johnson2016densecap], and *Image Paragraphing* [@krause2017hierarchical], and it has been becoming more and more mature with the booming of deep learning techniques[@he2016deep]. See Fig.\[fig:Description\_Tasks\] for some typical scene description tasks. It is believed that such semantic description will be an indispensable approach for the visually impaired people to perceive the environment[@seeingAI]. In this case, a further question arouses – what is the next step?
In fact, no matter how accurate the semantic description is, it can only provide information that exists in the current image, but not tell the user what to do next. The semantic understanding of the scenario is separated from the interaction between the agent and the environment. When a visually impaired person enters a room, the first photo captured is likely to be a bare wall or window. At this time, he usually has to move the smartphone randomly expecting to capture a more meaningful image from another viewpoint. In this situation, it is more useful to tell him where to look next rather than just to provide him the vague description of the current scene (e.g. *there is window on the wall*). On the other hand, a notorious problem of the semantic description is that it is very sensitive to the camera viewpoint[@park2019robust]. Although the content of the captured image may seem good, a deviation in camera viewpoint will lead to a totally wrong semantic description result. Under this circumstance, it is important to tell the visually impaired person how to adjust the position of the camera and even his body (e.g. move left, right, forwards, backwards) to get more meaningful and accurate scene description for the current scenario. Unfortunately, existing scene description work[@anderson2018bottom][@johnson2016densecap][@krause2017hierarchical] and the free APP software[@seeingAI] do not take this point into account. The reason is that all of them ignore the embodiment which is a very important characteristic of all intelligent agents (creatures and robots). The embodiment concept asserts that the intelligence emerges by interactions between the agent and the environment. Without embodiment, the semantic understanding of the scenario is separated from actions. It is a difficult problem which mainly involves the key issues of semantic scene description, description evaluation, and action instruction generation.
![Typical scene description tasks: *Image Captioning*, *Dense Captioning*, and *Image Paragraphing*.[]{data-label="fig:Description_Tasks"}](Description_Tasks.png){width="2.8in"}
![An intuitive *Embodied Scene Description* demonstration. Here we take the *Image Captioning* as an example, while the idea is applicable to other tasks such as *Dense Captioning*, *Image Paragraphing*, etc. At first glance, the agent captures the initial image (rendered with *pink*). Since this image is non-informative, the generated caption provides very limited information about the scene. However, the agent may explore the environment by itself to find a better viewpoint to capture a new image (rendered with *blue*). The generated caption yields more informative and suitable results.[]{data-label="fig:demo"}](demo.jpg){width="3in"}
In this work, we propose the *Embodied Scene Description* problem, which exploits the embodiment ability of the agent to find an optimal viewpoint in its environment for scene description tasks (e.g. *Image Captioning*, *Dense Captioning*, *Image Paragraphing*, etc). The main idea is illustrated in Fig.\[fig:demo\]. In addition to the visually impaired person, this problem is also extensively applicable to mobile robots. For example, it can facilitate the robot with many tasks such as actively exploring the unknown environment, quickly acquiring meaningful scene, and automatic photo taking.
To tackle this newly proposed problem, we establish a framework that makes use of existing image description models to guide the agent to explore an embodied environment. We encourage the agent to actively explore the environment and capture scenarios with good semantic description. It is noted that we consider the following two aspects when defining a good semantic description: (1) there should be sufficient visual objects detected in the scene and (2) these visual objects are able to compose a complete and reasonable semantic description. Since both the object detector and the semantic description may make mistakes, the combination of the two aspects is supposed to yield more reliable results. Having the definition of a good scene, we can build a learning framework with the paradigms of imitation learning and reinforcement learning to teach the intelligent agent to generate corresponding sensorimotor activities to explore the environment actively. It is worth noting that this work is different from a type of the embodied QA task[@gordon2018iqa][@das2018embodied], which is driven by finding answer to the question. In our work, the agent implements the task of environment exploration entirely with intrinsic motivation.
The main contributions are summarized as follows:
1. We propose a new framework for the *Embodied Scene Description* problem, which exploits the embodiment characteristic of the intelligent agent to explore the environment to find the best viewpoint for scene description in an embodied environment.
2. We develop a learning framework with the paradigms of imitation learning and reinforcement learning to help the agent to acquire the intelligence to generate sensorimotor activities.
3. We testify the proposed method on AI2Thor dataset and evaluate its effectiveness using the quantitative and qualitative performance indexes.
4. We implement the proposed method on a robotic platform, which shows promising experimental results in real physical environment.
Related Work
============
The deep learning methods have brought great success in many computer vision tasks such as object recognition[@he2016deep] and detection[@ren2015faster]. Moreover, many research studies have began to investigate a higher level task of semantic scene description with natural language. The proposed work focuses on the embodiment task of finding an optimal viewpoint for these scene description tasks.
Refs.[@donahue2015long][@vinyals2015show] are some early-stage works that propose to use a combination of CNN and LSTM model to generate image captions. These image caption models are further improved by integrating different visual and semantic attention mechanisms[@xu2015show][@you2016image][@anderson2018bottom]. Due to the fact that information expressed in a single sentence is limited when describing an image[@song2018deterministic], researchers begin to investigate some more complex models to bridge the gap between images and human language. Therefore, *Dense Captioning*[@johnson2016densecap] is proposed, which describes an image with multiple sentences. Each sentence is corresponding to an area within a bounding box in the image. It is further improved by *Image Paragraphing*[@krause2017hierarchical], which is able to generate a long paragraph to describe an image instead of a single sentence. Ref.[@liang2017recurrent] proposes a better model for *Image Paragraphing*, which utilizes the attention and copying mechanisms, as well as the adversarial training technique.
With the recent rapid development in computer vision and many traditional computer vision problems being addressed, the problem of embodied exploration has gradually emerged[@chen2019behavioral]. In the embodied exploration, an embodied agent actively explores the environment to have a better understanding of the scene[@li2019deep]. Contrary to traditional computer vision tasks, which mainly focus on analyzing static images passively, embodied exploration requires the agent both understands the content of the current image and takes proper actions accordingly to explore the environment. In most cases, the agent needs to make decisions based on observed image sequences instead of a single image[@sadeghi2019divis].
Ref.[@zhu2017target] develops the target-driven visual navigation, where the agent tries to find an object that is given by an RGB image in an indoor scenario. The model is improved in [@ye2019gaple] by incorporating the semantic segmentation information. The embodied visual recognition task proposed in [@yang2019embodied] aims to address the problem of navigating in an embodied environment to find an object which might be occluded at first glance. Refs.[@jayaraman2018learning][@ramakrishnan2019emergence] investigate the look-around behavior through active observation completion. Recently, language understanding and active vision are tightly coupled. In [@anderson2018vision], the authors propose the task of visual-and-language navigation, where the agent is expected to follow the given language instructions, and use the collected vision information to navigate through the indoor scene. Refs.[@gordon2018iqa][@das2018embodied] develop embodied question answering and interactive question answering tasks, where an agent is spawned at a random location in a 3D environment and explore to answer a given question. Such tasks have attracted many attentions from the computer vision communities[@yu2019multi][@wijmans2019embodied][@wu2019revisiting][@das2018neural]. Although more and more work has taken the embodiment into consideration, the investigated tasks mainly focus on object search, scene recognition, and question answering. The problem of scene description in an embodied environment has not be investigated yet.
To solve the embodied perception problem, the deep reinforcement learning has become the most popular method for its ability to integrate the perception and action modules seamlessly. However, many scholars have pointed out that the end-to-end training for such complex tasks is rather difficult to converge[@jayaraman2018end]. To tackle this problem, some hybrid learning methods are proposed, such as sidekick policy learning which allows the agent to learn via an easier auxiliary task[@ramakrishnan2018sidekick]. In addition, some work prefers to use the imitation learning method[@wang2019reinforced][@li2018oil] for pre-training and use reinforcement learning for fine-tuning[@haarnoja2018learning]. In this work, we resort to such methodology to solve the proposed *embodied scene description* task.
Problem Formulation
===================
The goal of this work is to develop a method to help the agent to rapidly find a proper viewpoint to capture a scene for generating the high-quality semantic scene description. Concretely speaking, we denote the image captured by the agent as $\bm{I}_t$ and the corresponding description as $\mathcal{U}(\bm{I}_t)$, where $t$ is the time instant. Please note that the operator $\mathcal{U}(\cdot)$ denotes the description generation procedure, which can be easily implemented by existing work, such as *Image Captioning*, *Dense Captioning*, *Image Paragraphing*, and so on.
Fig.\[fig:demo\] gives an intuitive introduction of the *Embodied Image Description* problem. At time instant $t=0$, the captured $\bm{I}_0$ may contain *wall* only and the produced caption *a white wall with a white wall with a wall in a wall* is non-informative. Then the agent exploits its embodied capability to select an action to explore the room and get a new image. Such procedure is iterated until the agent captures an image containing plenty of objects and produces the informative caption *A living room with a couch and a table*. The problem is therefore formulated as to develop an appropriate policy $\pi$ to help the agent to search a high-quality scene description about the scene. At each step $t$, the developed policy is used for the agent to take action $a_t$ to acquire the observed image $\bm{I}_t$.
Though our general idea is to learn action policies for an agent to locate a target scene in indoor environments using only visual inputs, the target scene is not specified by the user. This significantly differs from the work in [@zhu2017target][@ye2019gaple] which requires a pre-specified target image.
Navigation Model
================
{width="6in"}
![Demonstration of the visual input feed into the ResNet.[]{data-label="fig:Input_Channels"}](Input_Channels.jpg){width="2.9in"}
The proposed navigation model is shown in Fig.\[fig:Network\_Architecture\]. The action the agent would take in one step can be relevant to all its previous actions and observations. Therefore, we model it using the LSTM network, which is very commonly used for sequence modeling [@vinyals2015show][@xu2015show]. With the learned policy, the agent is expected to take as few steps as possible to approach the target scene from a random starting position.
State Representation
--------------------
In our implementation, we use a small ResNet-18 [@he2016deep] as a feature extractor, which is trained from scratch, jointly with the navigation model.
Since the image semantic segmentation results can improve the generalization performance[@ye2019gaple], we also use the class segmentation map to help image description generation. To this end, we modify the number of input channels of the original ResNet-18 from 3 to 6. The added 3 channels are used to deal with the class segmentation map. Fig.\[fig:Input\_Channels\] shows how those 6 channels of the input fed to ResNet are generated. We use PSPNet [@zhao2017pyramid] to predict the class segmentation map for a given image.
Furthermore, since we need to train the model with thousands of images in one batch (100 scenes times approximately 30 steps at most for the demonstration trajectories of those scenes, which means about 3000 images in one batch), we shrink the original ResNet-18 to a smaller network with 10 convolution layers. There are 4 kinds of residue building blocks in the original ResNet18, and each of them is repeated twice, leading to 16 convolution layers in residue blocks with parameters (and there are another 2 convolution layers in ResNet18). We use each kind of those residue building blocks only once, yielding a model with only $18 - (16 - 8) = 10$ weighted layers. Besides, the output channels for all convolution layers are also halved (e.g. 512 output channels are shrunk to 256 channels for the final output layer).
Since the description result $\mathcal{U}(\bm{I}_t)$ can directly show how well the scene description model performs for the current frame, we extract the Bag-of-Words (BoW) feature $\bm{L}_{t}$ for all appeared words (after removing stop words) of the output of the 2D image understanding model.
Finally, we combine those multiple features to form the state representation. Denoting the class segmentation map of $\bm{I}_t$ as $\tilde{\bm{I}}_t$, the state vector can be represented as $$\bm{s}_t = [ResNet(\bm{I}_t, \tilde{\bm{I}}_t); W_L \bm{L}_{t}]$$ where $ResNet$ denotes the feature extraction module mentioned above. $W_L$ is a trainable parameter for language embedding.
![A representative action space. Please note that some actions (such as *move left*) can be easily implemented in the simulation environment, but cannot be realized by some mobile agents, due to the non-holonomic constraints.[]{data-label="fig:Action_Space"}](Action_Space.jpg){width="3in"}
Action Space
------------
As illustrated in Fig.\[fig:Action\_Space\], for one step, we permit the agent to perform the following two kinds of discrete actions in the plane:
1. *Move:* The agent can take nine basic actions which correspond to 8 directions and *no move*. The move step is set to a fixed value $\Delta_m$ and the set of the movement actions is denoted as $\mathcal{A}_M$. In this work, we set $\Delta_m = 0.25$m.
2. *Rotation:* The agent can rotate for a fixed interval of $\Delta_r$. In this work, we set $\Delta_r = 45^\circ$ and therefore the set of the rotation actions $\mathcal{A}_R$ contains 8 action atoms.
For each step, the complete action space of the agent is $\mathcal{A} = \mathcal{A}_M \times \mathcal{A}_R$ and the agent is permitted to take the move actions firstly and the rotation action secondly. Please note that if the agent selects the action *no move* from $\mathcal{A}_M$ and $0$ from $\mathcal{A}_R$, then the exploration is completed and the obtained image with description is reported as the final result.
In practical environment, the agent has motion limitation and may encounter obstacles or dead corner. Thus the selected action may not be realized. To solve this problem, the agent may use its sensors to detect the feasible region and construct the available action set $\mathcal{A}_t \subset \mathcal{A}$ for the $t$-th step.
Matching between scene image and description
============================================
![Demonstration of the proposed scoring function.[]{data-label="fig:MatchingReward"}](Matching_Reward){width="3in"}
Since the goal of the navigation module is to guide the agent to find the scene which is good for both the image itself and the semantic description, we should design a matching score between the scene image and its description. This is indeed not a trivial task because the visual object detection results may contain noises and the image-text translator is far from perfectness. On one hand, an object-rich image is preferred but may lead to poor or non-informative description. On the other hand, a good description may include some wrong objects which do not appear in the image at all. See Fig.\[fig:Image\_Captioning\_Failed\_Cases\] for some examples.
![Some examples which show the mis-match between images and the corresponding description.[]{data-label="fig:Image_Captioning_Failed_Cases"}](Image_Captioning_Failed_Cases){width="3in"}
To tackle this difficulty, we apply the off-the-shelf object detectors on the image $\bm{I}$ to find the visual objects, and extract the object nouns in the text description $\mathcal{U}(\bm{I})$. The matching score $score(\bm{I})$ is designed according to their connections. By matching those words with all of the detected objects for one image, we can quantitatively measure how “good” a viewpoint is.
We denote all appeared words in the category labels of all detected objects in the image $I$ as $\mathcal{O}(\bm{I}) = \{o_1, o_2, \cdots, o_n\}$, and all appeared noun words in the output of the description model $\mathcal{U}(\bm{I})$ as $\mathcal{W}(\mathcal{U}(\bm{I})) = \{w_1, w_2, \cdots, w_m\}$, where $n$ and $m$ are the numbers of the detected visual objects in the image $\bm{I}$ and the extracted noun words in the description $\mathcal{U}(\bm{I})$. Since the vocabularies adopted by the visual object detector and the semantic description may be different, the same object may be expressed by different words (such as *desk* in the image and *table* in the description). We resort to the Word2Vec [@mikolov2013distributed] embedding to semantically vectorize these words. For the object category label $o_i$ in $\mathcal{O}(\bm{I})$ and the noun word $w_j$ in $\mathcal{W}(\mathcal{U}(\bm{I}))$, we can define their similarly as $$R(o_i, w_j) = k(o_i, w_j) cos\langle o_i, w_j \rangle$$ where $\cos \langle o_i, w_j \rangle $ is the cosine similarity between the word vectors of the two words. The value $k(o_i, w_j)$ is related to the confidence score of the word and the bounding box. Fig.\[fig:MatchingReward\] is an intuitive demonstration of the matching-based score function.
The determination of the confidence $k(o_i,w_j)$ is dependent of the adopted description model. For example, if the adopted scene description task is *Image Captioning* or *Image Paragraphing*, since there is no specific information provided for the confidence score by the image description model, we just set $k(o_i, w_j) = 1$. For *Dense Captioning* task, we have the confidence score and bounding box provided by the dense captioning model, therefore we can set $k(o_i, w_j) = IoU({BB}(o_i), {BB}({w_j}))\cdot C(w_j)$, where $BB(\cdot)$ is the corresponding bounding box and $C(\cdot)$ is the confidence score for the bounding box provided by the dense captioning model.
Based on the definition of $R(o_i, w_j)$, we can formulate the calculation of the similarity $sim(\bm{I}, U(\bm{I}))$ as the maximum matching problem between the sets $\mathcal{O}(\bm{I})$ and $\mathcal{W}(\mathcal{U}(\bm{I}))$. Such a problem can be easily solved using the Hungarian algorithm. Please note that this similarity value is normalized to \[0,1\].
Finally, we combine the similarity between image-description and the richness of objects to define the following viewpoint scoring function: $$score(\bm{I}) = sim(\bm{I}, \mathcal{U}(\bm{I})) + \lambda \frac{ |\mathcal{O}(\bm{I})|}{N}
\label{eq:score}$$ where $\lambda$ is a penalty parameter; the symbol $|\cdot|$ denotes the number of atoms in a set and $N$ represents the number of all possible objects. The second term encourages the agent to search the object-rich scene. It is very useful to some description tasks such as *Image Captioning*, which usually contains few words.
Learning for Embodied Scene Description
=======================================
A natural method to train the model presented in the previous section is the reinforcement learning algorithm. However, training such a complex model using end-to-end reinforcement learning from scratch is very hard to converge[@ramakrishnan2018sidekick]. Therefore, we first use demonstrations to develop imitation learning method to train the embodied scene description model from scratch, and then fine-tune this model with reinforcement learning. Such methodology has been extensively used for several difficult tasks[@das2018embodied].
Imitation Learning
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![Demonstration of selecting the target locations.[]{data-label="fig:GeneratedTargetPoints"}](TargetLocation){width="3in"}
![Demonstration of the generated shortest path.[]{data-label="fig:GeneratedShortestPath"}](ShortestPath){width="3in"}
{width="6in"} {width="6in"} {width="6in"} {width="6in"}
The goal of imitation learning for sequential prediction problems is to train the agent to mimic expert behavior for some tasks. To develop the imitation learning algorithm, we have to annotate some scenes with the pre-trained caption model and generate demonstrations for the agent. Therefore, for a specific scene $\mathcal{S}$, we discretize it with grids of a fixed size of $\Delta_m$, and fixed angle $\Delta_r$ as is stated in the *Action Space* section. For each possible position $(x,y)$ with rotation $\phi$, the corresponding viewpoint can be represented as the tuple $(x, y, \phi)$. We denote all these possible discrete viewpoints in the given scene $\mathcal{S}$ as $\mathcal{S}_D$. With some abuse of notation, we use $score(x,y,\phi)$ to represent the score of the image which is captured at this viewpoint.
To produce the demonstration trajectories for a scene, we first find a special viewpoint $(x^*, y^*, \phi^*)$: $$(x^*, y^*, \phi^*) = \mathop {\arg \max }\limits_{(x,y,\phi)\in \mathcal{S}_D} {score(x,y,\phi)}$$ which achieves the highest score $s_{max} = score(x^*, y^*, \phi^*)$. Then we randomly sample one item from the set of candidate locations of which the score is in the interval of $[\gamma s_{max}, s_{max}]$ as the target location (demonstrated in Fig.\[fig:GeneratedTargetPoints\]). The parameter $\gamma$ is set to 0.95 to prevent over-fitting. Finally, the shortest path between one randomly selected initial point and the target point, demonstrated in Fig.\[fig:GeneratedShortestPath\], can be obtained using the all-pairs shortest path table generated by the Floyd-Warshall algorithm[@hougardy2010floyd]. This path is used as the demonstration trajectory. Using the multiple demonstrations from various scenes, we can develop supervised imitation learning to train the feature extractor and the navigation model together. The loss function is defined as follows $$\mathcal{L}_{\theta} = \sum_{k=1}^{K}{\sum_{t=1}^{T_k}{-\log{ \pi_\theta(\hat{a}_{k, t} | \hat{s}_{k, 0}, \hat{a}_{k, 0}, \hat{s}_{k, 1}, \hat{a}_{k, 1}, \cdots, \hat{s}_{k, t}) }}},$$ where $K$ is the number of demonstration trajectories used for training in one batch, $T_k$ is the length of the $k$-th trajectory, $\hat{s}_{k, t}$ and $\hat{a}_{k, t}$ are the annotated observation and action, and $\theta$ denotes all of the parameters to be optimized. During the training phase, we assume a map of the environment is available and give the agent access to information about the shortest paths to some targets.
NoS $SoL^*$ $SoL$ BLEU-1 BLEU-2 BLEU-3 BLEU-4 Meteor ROUGE\_L CIDEr
------------------ ------- ------------ ------------ ------------ ------------ ------------ ------------ ------------ ------------ ------------ --
Random 26.78 0.3015 0.3017 0.6176 0.5088 0.4252 0.3686 0.2598 0.6086 1.7135
IL (RGB) 21.38 0.7398 0.7430 0.8471 0.7997 0.7537 0.7144 0.4633 0.8293 4.8382
IL (Segm.) 19.76 0.7524 0.7525 0.8607 0.8113 0.7611 0.7154 0.4653 0.8368 4.6627
IL (RGB+Segm.) 15.43 0.7777 **0.7734** 0.8741 0.8334 0.7902 0.7502 0.4910 0.8625 4.9376
RL (RGB+Segm.) 18.38 0.4490 0.4531 0.7228 0.6399 0.5682 0.5139 0.3401 0.7106 2.8099
IL+RL(RGB+Segm.) 15.10 **0.7813** 0.7724 **0.8752** **0.8345** **0.7906** **0.7502** **0.4913** **0.8626** **4.9482**
\* denotes the evaluations on the validation set.
\[table:Result\]
Fine-Tuning with Reinforcement Learning
---------------------------------------
After pre-training the navigation model with imitation learning, we then try to further improve its performance using the REINFORCE algorithm. The key to fine-tune the model is to design the reward function for the generated trajectory. Generally speaking, we hope to get high-quality caption within a short period of time and therefore a score can be designed as $p_t = score(\bm{I}_t) - \rho t$, where $\rho$ is used to balance the scales of the two terms and is set to 0.01.
According to the above definition, the immediate reward is set as the incremental of the score $r(s_t, a_t) = p_t - p_{t-1}$ and the cumulative reward which is used to fine-tune the model is as $$R(s_t, a_t) = r(s_t, a_t) + \sum_{t'=t+1}^{T} \beta^{t'-t} r(s_t, a_t),$$ where the discounted parameter $\beta$ is set to 0.99 and $T$ is the prescribed maximum steps and is set to 40.
Based on the above-defined reward function, we use REINFORCE algorithm to fine-tune the policy network. To reduce the variance of the reward and improve the stability, we record the moving average of the reward and minus the reward by that moving average in practical training. We use SGD optimizer with a learning rate of $10^{-3}$.
Experiment Results
==================
The proposed framework is able to generalized to various semantic description tasks such as *Image Captioning*, *Dense Captioning*, *Image Paragraphing*, and so on. Considering that the *Image Captioning* provides a single-sentence description, which is more intuitive and convenient for practical applications, we focus on the scene description task of *Image Captioning* in this section for performance evaluations. The experimental results on the tasks of *Dense Captioning* and *Image Paragraphing* are illustrated in the supplementary materials.
Dataset
-------
For there isn’t any existing dataset for the proposed *Embodied Scene Description* task, we generate a new dataset with the AI2Thor dataset. A pretrained caption model is used to generate captions for each scene from different viewpoints.
The AI2Thor dataset contains 120 scenarios belonging to four categories: *Living Room*, *Kitchen*, *Bedroom* and *Bathroom*. Each category has 30 rooms. For each category, we use 25 rooms as the training set, and 5 rooms as the validation/test set. The layout of the room is discretized with grids. For those rooms used as validation/test set, one fixed point in every $4 \times 4$ grid is regarded as a point in validation set and the rest 15 points belong to the test set. The image caption model proposed in [@anderson2018bottom] is adopted for its satisfying performance. The model is trained with the MSCOCO captioning dataset[@vinyals2016show].
![The annotation for the scene using selected three representative viewpoints.[]{data-label="fig:scene_annotation"}](Generated_Caption.jpg){width="3in"}
Evaluation Metrics
------------------
To evaluate the performance of the *embodied scene description* task, we resort to the score function defined in Eq.(\[eq:score\]) to calculate the score for each location in the room. Concretely speaking, for each scene, we select all of the locations whose scores are in the interval of $[\gamma s_{max}, s_{max}]$ and generate their corresponding captions. The generated captions are combined together to act as the ground truth annotation of the scene (Fig.\[fig:scene\_annotation\]). Based on this ground truth annotation, the following metrics are designed:
1. *Number of Steps (NoS):* The number of steps the agent takes before stopping.
2. *Score of the Last Image(SoL):* The score of the location which triggers the *Stop* action.
3. *Natural Language Metrics:* With the generated ground truth annotation for each scene, metrics for natural language tasks can be used for evaluating the proposed task. We select several metrics including *BLEU-1*, *BLEU-2*, *BLEU-3*, and *BLEU-4* which are based on the $n$-gram precision[@papineni2002bleu], *Meteor* which considers the word-level alignment, *ROUGE\_L* which is based on the longest common sub-sequence[@denkowski2014meteor], and CIDEr[@vedantam2015cider].
Result Analysis
---------------
The performance of the proposed framework for the scene description of image captioning is illustrated in Table \[table:Result\]. Comprehensive comparisons are conducted with several different settings. The full implementation of the proposed framework is denoted as *IL+RL (RGB+Segm.)*. *IL (RGB)* and *IL (Segm.)* only use the imitation learning for the RGB image and segmentation map respectively. *IL (RGB+Segm.)* uses both the RGB image and semantic segmentation map under the same imitation learning framework without reinforcement learning. We also investigate the performance of the reinforcement learning only framework which is trained from scratch and it is denoted as *RL (RGB+Segm.)*. The baseline method is to randomly select each action.
{width="6in"}
The detailed results over all test samples are summarized in Table \[table:Result\], from which we have the following observations:
- *The combination of IL+RL demonstrates good performance:* The results show that IL+RL method which contains both the pre-training process using imitation learning and fine-tuning process using reinforcement learning achieves the best performance according to all of the language-related metrics including BLEU, Meteor, ROUGE\_L and CIDEr. This verifies that the proposed method indeed helps the agent to find good viewpoints to get high-quality captions with an average of 15.10 steps that is satisfying among all the methods.
- *The RL only method works poorly.* The RL only method, though using the same information with IL+RL, yields very poor results, which is just better than the baseline random method. It demonstrates that the pre-training process using the imitation learning is helpful.
- *Both RGB and semantic segmentation information are important.* It can be seen that with the same imitation learning framework, using both the RGB image and semantic segmentation information has better performance than that with single modal information. The main reason is that RGB image is supposed to provide more details, while the segmentation map provides higher-level semantic information. In addition, IL(RGB) and IL(Segm.) methods take more steps before the stop action is triggered.
Although the obtained results are promising, we notice that the fine-tuning process using reinforcement learning only slightly improves the performance. This is in accordance with the results shown in existing literature [@das2018embodied][@wang2019reinforced]. However, we believe the fine-tuning step could play more important roles when some more sophisticated strategies are utilized.
Representative Examples
-----------------------
In Fig.\[fig:Sample\_Success\], we list four representative examples for different scenario categories. The agent is able to navigate in the room and finally find a good viewpoint to describe the scene. For example, for a living room which is illustrated in the second row of Fig.\[fig:Sample\_Success\], the agent can only see the television initially, and then it continuously explores in the room until it reaches a position where a good view of the living room is obtained. For a bathroom which is illustrated in the last row of Fig.\[fig:Sample\_Success\], the agent starts at a location where only the wall is visible. With the help of the navigation model, it gradually discovers the mirror and the sinks. Finally, it generates a description that contains major objects in the bathroom.
![LEFT: The developed robotic platform. A Kinect is equipped on the top of it and we carefully adjust its position to ensure the optical center of the camera in Kinect is aligned with the center of the platform. RIGHT: Two representative real working scenes for the agent. For each scene, we show the the-third-person view (left) and the first-person view (right).[]{data-label="fig:robot"}](robot_platform.jpg){width="3in"}
It is noted that the performance of *embodied scene description* is also strongly dependent of the scene description task (*Image Captioning* in this task). In Fig.\[fig:Sample\_Failed\_1\] we show a failure case. One possible reason is that the caption model generates improper caption for the scene even though the agent is actually find a good viewpoint for the scene description.
We also perform extensive experimental validation on some other typical scene description tasks such as *Dense Captioning* and *Image Paragraphing*. The results are shown in the Appendix and the video.
Real-World Experiments
----------------------
Our model is trained on the AI2Thor dataset, rendered with Unity in high quality real-time realistic computer graphics, which makes the difference between the real world and simulation environments minimal. Therefore, it is possible for a simulation-to-real transfer and applying our model to real-world robots and scenes. The learned policy is able to provide action instructions to the robot.
As shown in Fig.\[fig:robot\], a mobile robot equipped with a Kinect camera is used in the real world experiment. The mobile robot is able to rotate 360 degrees around itself and move forwards/backwards flexibly, which allows for implementing actions in the action space. The Kinect camera is mounted on the top of the mobile robot and is used to collect egocentric images in real time. The robot is placed in an unseen hotel room for a simulation-to-real experiment. Although the layout of the room and the viewpoint of the camera are significantly different from those in the simulation environment, promising results are obtained to validate the effectiveness of the trained model. In Real Scene 1 (Fig.\[fig:Sample\_Real\_1\]), the robot firstly faces to a corner of the room. With the generated instructions, the robot moves around the room until it recognizes that it is a bedroom scene. In Real Scene 2 (Fig.\[fig:Sample\_Real\_2\]), the robot starts with a scene where it faces to the door of a cabinet and it is difficult to obtain much useful semantic description. Then, with generated action instructions, the robot adjusts its positions gradually and stops at a position where it can get a full view of the room. It reflects that the learned model is capable of transferring the semantic knowledge learned in simulation environment to real world environment. However, it can be seen that although the robot moves in a reasonable path, the captions generated are indeed wrong. It is because that the caption model used is not robust and accurate enough. More details can be found in the attached video.
![Real Scene 1: The robot turns around from the wall corner to the bed and finally correctly recognizes the bedroom scene.[]{data-label="fig:Sample_Real_1"}](Robot_Scene_1.jpg){width="3in"}
![Real Scene 2: The robot adjusts its position to observe the room. But the caption model mistakenly recognizes the scene as a bathroom.[]{data-label="fig:Sample_Real_2"}](Robot_Scene_2.jpg){width="2.8in"}
Conclusions
===========
In this work, we propose the new *Embodied Scene Description* problem, in which the agent exploits its embodiment ability to find an optimal viewpoint in its environment for scene description tasks. A learning framework with the paradigms of imitation learning and reinforcement learning is established to teach the agent to generate corresponding sensorimotor activities. The trained model is evaluated in both the simulation and real world environment demonstrating that the agent is able to actively explore the environment for good scene description.
This work only takes one single frame into consideration for each step. Image sequences collected during the exploration process is believed to reveal more information for a better scene description. It will be also useful to leverage the attention, preference, and 3D relationship between objects to further actively understand the scenario. In the future, we plan to integrate this feature into smartphones and intelligent glasses, which are supposed to assist visually impaired person for a better living.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'An impurity in a Tomonaga-Luttinger liquid leads to a crossover between short- and long-distance regime which describes many physical phenomena. However, calculation of the entire crossover of correlation functions over different length scales has been difficult. We develop a powerful numerical method based on infinite DMRG utilizing a finite system with infinite boundary conditions, which can be applied to correlation functions near an impurity. For the $S=1/2$ chain, we demonstrate that the full crossover can be precisely obtained, and that their limiting behaviors show a good agreement with field-theory predictions.'
author:
- 'Chung-Yu Lo (羅中佑)'
- 'Yoshiki Fukusumi (福住吉喜)'
- 'Masaki Oshikawa (押川正毅)'
- 'Ying-Jer Kao (高英哲)'
- 'Pochung Chen (陳柏中)'
bibliography:
- 'Ref.bib'
date:
-
-
title: 'Crossover of Correlation Functions near a Quantum Impurity in a Tomonaga-Luttinger Liquid'
---
[UTF8]{}
In one-dimensional (1D) systems, even weak Coulomb interactions have dramatic effects and the Fermi liquid theory describing their higher dimensional counterparts breaks down. This results in a Tomonaga-Luttinger liquid (TLL) [@Tomonaga:1950hq; @*Luttinger:1963pt; @Giamarchi:2004ix], which is nothing but a relativistic free boson field theory. The TLL behaviors have been experimentally demonstrated in carbon nanowires [@Laroche:2014qf; @*Ishii:2003vn; @*Yao:1999fj; @*Kim:2007gf; @*Postma:2000ul], allowing for further studies of the electron transport in 1D quantum wires. On the theory side, there exists a plethora of powerful analytical and numerical methods available to study the behavior of 1D systems. Analytical tools such as the bosonization, conformal field theory (CFT) and the renormalization group (RG) can be employed to analyze the physical properties of TLL [@Giamarchi:2004ix; @Cardy:2010zl; @Affleck:2010cr].
An important class of problems is the effects of a quantum impurity on a TLL [@Kane:1992kx; @*Kane:1992gj; @Furusaki:1993tw; @Matveev:1993xq; @Wong:1994zt; @Rylands:2016eu; @Shi:2016rm]. In the simplest setting, Kane and Fisher have shown that a single quantum impurity affects the transport property of the TLL in an essential way [@Kane:1992kx; @*Kane:1992gj; @Furusaki:1993tw]: when the interaction is attractive the system renormalizes to a fixed point corresponding to a single fully connected wire. When the interaction is repulsive, however, the system renormalizes to two disconnected wires. An equivalent problem was also studied in a context of quantum spin chains [@EggertAffleck1992]. In terms of CFT, a RG fixed point of the impurity problem is associated to a conformally invariant boundary condition (CIBC) [@Saleur-Lecture1998]. Thus the first question in the impurity problem is the classification of CIBCs. While nontrivial CIBCs appear in various settings [@Affleck:1991pz; @Oshikawa:2006sw], only the simple Dirichlet and Neumann boundary conditions of the free boson field theory are relevant for the original Kane-Fisher problem (a single quantum impurity) in a spinless single-channel interacting TLL. For each CIBCs, correlation functions can be calculated with boundary CFT techniques. However, the system is renormalized to the low-energy/large-distance (infrared, IR) fixed point only asymptotically. In order to describe various observable properties, such as finite-temperature properties, we need to describe the RG flow towards the IR fixed point, not just the CIBC corresponding to the IR fixed point. The system is often renormalized close to a high-energy/short-distance (ultraviolet, UV) fixed point first, before flowing towards the IR fixed point. In such a case, the finite-energy/finite-distance properties can be described as a crossover between the UV and IR fixed points. The crossover phenomena cannot be dealt with the boundary CFT techniques alone. In some cases, the crossover of a physical quantity can be exactly obtained in terms of an integrable boundary RG flow [@FendleyLudwigSaleur]. Nevertheless, for more general quantities, and for other settings, numerical approach is indispensable to describe the crossover.
In general it is difficult to simulate 1D (boundary) critical systems, of which the TLL is an example, because large system sizes are required to capture the asymptotic behavior. Lo *et al.*[@Lo:2014vr] use a scale-invariant tensor network to directly extract scaling operators and scaling dimensions for both bulk and boundary CFTs. Although the method can successfully describe the physics at the IR fixed point, it can not probe the UV to IR RG flow. Rahmani *et al.* [@Rahmani:2010jr; @Rahmani:2012bq] perform a conformal mapping of the wire junction to a finite strip so that a finite-size DMRG calculation can be carried out. However, an [*ad hoc*]{} mirror boundary condition has to be added. Furthermore, the conformal mapping makes it necessary to use of the chord distance, instead of the direct site distance. It is therefore difficult to probe short-distance and crossover behavior using this approach. An improved numerical method is hence called for.
In this Letter, we present a numerical method based on an infinite DMRG (iDMRG) scheme that allows us to directly simulate the junction of semi-infinite TLL wires via infinite boundary condition (IBC) and study the crossover from the impurity site to the long length scale. We are able to obtain various correlation functions which are in agreement, at both short and long length scale, with those obtained by the boundary perturbation theory based on bosonization.[@Fateev:1997nn; @Fendley:1998pq; @Fendley:1998sq; @PhysRevLett.75.4492; @PhysRevB.58.5529]
We start from two semi-infinite wires of spinless electrons. The two wires are connected by a link of strength $t$ to form the junction as sketched in Fig.\[fig:sketch\](a). Using Jordan-Wigner transformation, the wire and the link Hamiltonians can be written in spin language as $$H_{\text{wires}} = \sum_{\substack{\mu=\alpha,\beta,\\ i \in \mathcal{Z}^+ + \frac{1}{2} }}
-\left( S^{+\mu}_i S^{- \mu}_{i+1} +S^{- \mu}_i S^{+ \mu }_{i+1} \right) + V S^{z \mu}_i S^{z \mu}_{i+1},$$ and $$H_{\text{link}} = -t ( S^{+\alpha}_{1/2} S^{-\beta}_{1/2} +S^{-\alpha}_{1/2} S^{+\beta}_{1/2} ),
\label{eq.link}$$ respectively, where $\alpha$, $\beta$ are wire indices. The junction Hamiltonian is then defined as $H_{\text{junc}}= H_{\text{wires}}+H_{\text{link}}$. We also define a bulk Hamiltonian for a single infinite TLL wire as $$H_{\text{bulk}} = \sum_{i \in \mathcal{Z} + \frac{1}{2} }
-\left( S^x_i S^x_{i+1} + S^y_i S^y_{i+1} \right) + V S^z_i S^z_{i+1}.
\label{eq:bulk}$$ It differs from a junction with strength $t=1$ only by the interaction across the junction: $V S^{z\alpha}_{1/2} S^{z\beta}_{1/2}$. Both the semi-infinite wires of the junction and the bulk wire are described by the TLL theory with Luttinger parameter $g=\pi/(2\arccos(-V/2))$. We will consider three inter-wire correlation functions: $\langle S^{\alpha + }_i S^{\beta -}_i \rangle$, $\langle S^{\alpha z }_i S^{\beta z}_i \rangle$, and $\langle J^{\alpha}_i J^{\beta}_i \rangle$. Here the current operator is defined as $J^\mu_i \equiv -i ( S^{\mu +r}_{i-1/2} S^{\mu -}_{i+1/2} - S^{\mu +}_{i+1/2} S^{\mu -}_{i-1/2} )$.
![(Color online) (a) Sketch of the junction with a link of strength $t$. (b) Infinite matrix product state diagram for $|\Psi\rangle_{\text{junc}}^{G, 2L}$. (c) Matrix product operator diagram for $H_{\text{junc}}^{2L}$.[]{data-label="fig:sketch"}](sketch3.eps){width="0.9\columnwidth"}
To find the ground state of $H_\text{junc}$, we use the IBC [@Michel:2010uo; @*Phien:2012dr; @*Phien:2013kh] to construct an effective junction Hamiltonian for a finite-size window that contain the impurity. The effective Hamiltonian is then used to optimized the ground state within the window. This results in an optimized ground state in the form of an infinite matrix product state (iMPS), from which correlation functions within the window can be easily calculated. Specifically we start from the translationally invariant bulk Hamiltonian $H_{\text{bulk}}$. We assume that its ground state is described by an one-site (or two-site) translationally invariant iMPS: $$\begin{aligned}
|\Psi\rangle_{\text{bulk}}^{G}& = \sum_{ \{s_i\} }
\cdots \lambda \Gamma^{s_{i-1}} \lambda \Gamma^{s_i} \lambda \Gamma^{s_{i+1}} \lambda \Gamma^{s_{i+2}} \cdots
|\mathbf{s}\rangle,\nonumber\\
&= \sum_{s_i} \cdots A^{s_{i-1}} A^{s_i} \lambda B^{s_i} B^{s_{i+1}} \cdots |\mathbf{s}\rangle,\end{aligned}$$ where $ |\mathbf{s}\rangle=|\dots s_{i-1}, s_i, s_{i+1}, s_{i+2} \dots \rangle$ and $s_i$ are local spin basis. Furthermore, $\Gamma^{s}=\Gamma$ are site-independent $d\times D\times D$ tensors and $\lambda$ is a $D\times D$ diagonal matrix, where $d$ and $D$ are physical dimension and bond dimension respectively. The second line corresponds to the mixed canonical form with $A^s=A=\lambda \Gamma$ and $B^s=B=\Gamma \lambda$. Here $A$ and $B$ satisfy the left and right canonical form constraints respectively. They can be obtained by optimizing $ |\Psi\rangle_{\text{bulk}}^{G}$ with the bulk Hamiltonian $H_{\text{bulk}}$ via any conventional iDMRG algorithm. Due to the presence of the quantum impurity, the translational invariance is broken and a translational invariant iMPS is no longer a good ansatz for the ground state of $H_{\text{junc}}$. On the other hand, since $H_{\text{junct}}$ differs from $H_{\text{bulk}}$ only at the impurity sites, we expect that far away from these sites the ground states of $H_{\text{junct}}$ and $H_{\text{bulk}}$ should resemble each other locally. We hence assume that there is a finite window of size $2L$ with sites $ i \in [-L-\frac{1}{2}, \cdots, L+\frac{1}{2} ]$ within which the ground states of $H_{\text{junc}}$ and $H_{\text{bulk}}$ differ, while outside this window they are locally described by the same matrices. This leads to the following iMPS ansatz for the ground state of $H_{\text{junc }}$: $$|\Psi\rangle_{\text{junc}}^{G, 2L} =
\sum_{s_i} \cdots A^{s_{-L-1}} \left ] \right [ M^{s_{-L}} \cdots M^{s_L} \left ] \right [ B^{s_{L+1}} \cdots |\mathbf{s}\rangle,$$ as sketched in Fig.\[fig:sketch\](b). Here $L$ is an adjustable parameter that can be easily enlarged. And the $M$ matrices are optimized with an effective Hamiltonian as described below.
Starting from $H_{\text{bulk}}$ in the form of matrix product operators (MPOs) [@Schollwock:2011gl; @*McCulloch:2007gi]: $$H_{\text{bulk}} = \cdots W_{-L-\frac{3}{2}} W_{-L-\frac{1}{2}} \cdots W_{-\frac{1}{2}} W_{\frac{1}{2}} \cdots W_{L+\frac{1}{2}} W_{L+\frac{3}{2}} \cdots$$ where the matrix $W_i=W$ is site independent. The effective Hamiltonian of the finite window can be expressed as $$H_{\text{junc}}^{2L} = \widetilde{W}_{\mathcal{L}} W_{-L-\frac{1}{2}} \cdots \widetilde{W}_{-\frac{1}{2}} W_{\frac{1}{2}} \cdots W_{L+\frac{1}{2}} \widetilde{W}_{\mathcal{R}}$$ as sketched in Fig.\[fig:sketch\](c). Here $W_{-1/2}$ are replaced by $\widetilde{W}_{-1/2}$ to represent $H_{\text{link}}$. Furthermore, the left and right IBCs, $\widetilde{W}_{\mathcal{L}}$ and $\widetilde{W}_{\mathcal{R}}$, are constructed from the left and right dominant eigenvectors of the generalized transfer matrices $T_\mathcal{L} = \sum_{ss^\prime} \langle s | W | s^\prime\rangle A^{s^\prime \dagger} A^{s}$ and $T_\mathcal{R} = \sum_{ss^\prime} \langle s | W | s^\prime\rangle B^{s^\prime} B^{s \dagger}$ respectively [@Phien:2012dr]. Here the IBCs are used to represent the semi-infinite extensions of the bulk system to the left and right. In this way, we reduce an infinite-size system to an effective finite-size one [@SM]. Once $H_{\text{junc}}^{2L}$ is obtained, one can use any conventional finite-size MPS/DMRG algorithm to optimize the $M$ matrices. Also, with the left and right-canonical conditions satisfied by $A$ and $B$ matrices, correlation functions within the window can be calculated using only $M$ matrices.
------------------------------------------------- ---------------- ---------------- ------------------ ---------------- -------------------- ----------------
Bulk IR IR UV Bulk IR IR UV
$\langle S^{+,\alpha }_i S^{-,\beta}_i \rangle$ $\frac{1}{2g}$ $\frac{1}{2g}$ $\frac{3}{2g}-1$ $\frac{1}{2g}$ $\frac{3}{2g}-1$ $\frac{1}{2g}$
$\langle S^{z,\alpha }_i S^{z,\beta}_i \rangle$ 2 2 $\frac{2}{g}$ $2g$ $2g+\frac{2}{g}-2$ $2g$
$\langle J^{\alpha}_i J^{\beta}_i \rangle$ 2 2 $\frac{2}{g}$ $2$ $\frac{2}{g}$ $2$
------------------------------------------------- ---------------- ---------------- ------------------ ---------------- -------------------- ----------------
: Dominant exponents for each correlation function
\[table:exponents\]
![(Color online) $\langle S^{\alpha, + }_i S^{\beta, -}_i \rangle$, $\langle S^{\alpha, z }_i S^{\beta, z}_i \rangle$, and $\langle J^{\alpha}_i J^{\beta}_i \rangle$ correlation functions for $g=1.5$. Data for the bulk and junctions with $t=0.1, 0.3, 0.8$ are plotted. Solid lines are power law fitting to the bulk data with bulk exponents from bosonization. (Cf. Table.\[table:exponents\].)[]{data-label="fig:g>_IR"}]({corr_u1_g1.5_tAll_m800}.eps){width="0.9\columnwidth"}
When $g>1$, at long distance (IR limit) the system is renormalized to a single wire with the same Luttinger parameter. For a weaker link (smaller $t$), we expect that it would take longer distance for the system to heal from the perturbation due to the impurity. Figure \[fig:g>\_IR\] shows spin-spin and current-current correlation functions between two leads: $\langle S^{+,\alpha }_i S^{-,\beta}_i \rangle$, $\langle S^{z,\alpha }_i S^{z,\beta}_i \rangle$, and $\langle J^{\alpha}_i J^{\beta}_i \rangle$ for junctions with $g=1.5$ and various $t$ as well as a bulk wire. We observe that at IR limit all correlation functions merge into their bulk counterparts. This confirms that asymptotically the system is renormalized to a single defect-free wire.
When $g<1$, in contrast, we expect that in the IR limit the system is renormalized into two disconnected semi-infinite wires. For non-zero $t$, we still expect that the correlation functions to decay as a power law, but with exponents that are larger than the bulk counterparts. In Fig.\[fig:g<\], we plot the same correlation functions for $g=0.6$ and $t=0.1, 0.01$ as well as the single bulk wire. A scaling prefactor of $t^{-1}$ or $t^{-2}$ is also included in order to collapse the curves with different $t$’s. It is clear that the correlators in the IR limit decay faster than their bulk counterparts, supporting the picture that at IR limit the system is renormalized into broken wires.
To further understand the behavior of these correlation functions in both the UV and the IR limits, we use boundary perturbation theory to determine the exponents of the power laws. To the leading order, we derive the exponents of uniform and staggered part of the correlation functions respectively. In the bosonization framework, the system is described by the TLL with the Lagrangean density$$\mathcal{L} = \frac{1}{2\pi g} (\partial_\mu \phi)^2 .$$ In the leading orders, the spin and current operators are expressed as [@Giamarchi:2004ix; @Lukyanov:2002fg] $$\begin{aligned}
S^{-}_j & \sim e^{- i \theta }\left(b +c\left(-1\right)^{j} \cos{(2\phi)}) \right) , \nonumber\\
S^z_j & \sim - \frac{1}{\pi}\frac{\partial\phi}{\partial x}+a(-1)^j \sin{(2\phi)} , \nonumber\\
J_j & =i\left(S^{+}_{j+1}S^{-}_{j}-S^{+}_{j}S^{-}_{j+1}\right) \sim \frac{gv}{\pi} \frac{\partial \theta}{\partial x},\end{aligned}$$ where $a$, $b$, and $c$ are constants. Here $\theta$ is the dual field of $\phi$ defined by $\theta \equiv \frac{1}{g} (\phi_L - \phi_R)$, where $\phi = \phi_L(\bar{z}) + \phi_R(z)$ is the chiral decomposition into left/right-movers with the complex coordinate $z=x+it$. The correlation functions of the chiral fields on the full complex plane (without an impurity) read $$\begin{aligned}
\langle \phi_R(0) \phi_R(z) \rangle & \sim
- \frac{g}{4} \log{z} + \mbox{const.} , \nonumber\\
\langle \phi_L(0) \phi_L(\bar{z}) \rangle & \sim
- \frac{g}{4} \log{\bar{z}} + \mbox{const.}.\end{aligned}$$ In this convention, $\phi$ and $\theta$ are compactified respectively as $\phi \sim \phi + \pi$ and $\theta \sim \theta + 2\pi$.
![(Color online) Rescaled $\langle S^{\alpha, + }_i S^{\beta, -}_i \rangle$, $\langle S^{\alpha, z }_i S^{\beta, z}_i \rangle$, and $\langle J^{\alpha}_i J^{\beta}_i \rangle$ correlation functions for $g=0.6$. Data for the bulk and junctions with $t=0.1, 0.01$ are plotted. Solid (dotted) lines are power law fitting to the long (short) distance data with IR (UV) exponents from bosonization. (Cf. Table.\[table:exponents\].) []{data-label="fig:g<"}]({corr_u1_g0.6_tAll_m800_v2}.eps){width="0.9\columnwidth"}
The geometry is half plane with the interaction on the line $x=0$. In the limit $t=0$, the system is two decoupled half-chains. The end of each half-chain is renormalized to the Dirichlet boundary condition $\phi^\gamma=0$ for $\gamma=\alpha,\beta$. We then introduce the link Eq. between the two decoupled chains through the boundary. Let us consider the correlation function $\left\langle S^{\alpha + }_{i + \frac{1}{2}} S^{\beta -}_{i + \frac{1}{2}} \right\rangle$, across the link. Obviously it vanishes when $t=0$. In the first order of $t$, the correlation function is given as $$t \int d\tau \;
\left[ \frac{1}{2}
\langle S^{+\alpha}_{i+\frac{1}{2}}(0) S^{\alpha-}_{\frac{1}{2}}(\tau)\rangle_{D}
\langle S^{\beta +}_{\frac{1}{2}}(\tau) S^{-\beta}_{i+\frac{1}{2}}(0) \rangle_{D}
\right].
\label{eq.SpSm_1st}$$ The problem is thus reduced to calculation of the correlation functions with the Dirichlet boundary condition [@PhysRevB.58.5529]. The Dirichlet boundary condition can be solved by an analytical continuation of $\phi^{\gamma,R}$ to $x<0$ and $\phi^{\gamma,L}(x,\tau) \equiv - \phi^{\gamma,R}(-x,\tau).$ Using this, the correlation function is given as $$\label{eq:SS_D}
\left\langle S^{\alpha + }_{i + \frac{1}{2}} S^{\beta -}_{i + \frac{1}{2}}\right \rangle = t \left[
C_0 r^{ -\left(\frac{3}{2g} -1 \right) } +
C^\prime_0 (-1)^{r} r^{ -\left(\frac{3}{2g} +2g-1 \right) } \right],$$ where $i \in \mathcal{Z}$, $r=2i+1$, and $C_0, C^\prime_0$ are constants. Similarly we find $$\label{eq:ZZ_D}
\left\langle S^{\alpha z }_{i + \frac{1}{2}} S^{\beta z}_{i + \frac{1}{2}} \right\rangle = t^2 \left[
C_0 r^{ -\left( \frac{2}{g} \right) } +
C^\prime_0 (-1)^{r} r^{ -\left( 2g + \frac{2}{g} -2 \right) } \right],$$ where $r=2i+1$ and $$\label{eq:JJ_D}
\left\langle J^{\alpha}_i J^{\beta}_i \right\rangle = t^2 C_0 r^{-\frac{2}{g} },$$ where $r=2i$. These results describe the IR behavior for $g<1$ and the UV behavior for $g>1$. We find that for $g>1$, the uniform part always dominates and the UV exponents are $3/2g-1$, $2/g$, and $2/g$ respectively. In contrast, for $g<1$ the staggered part of the $\langle S^{z,\alpha }_i S^{z,\beta}_i \rangle$ becomes dominant and the IR exponents become $3/2g-1$, $2g+2/g-2$, and $2/g$ respectively.
The case of a weak barrier (small $1-t$), on the other hand, corresponds to the free boundary condition. For this case we regard the junction as a defect in CFT with $$H_{\text{barrier}}
= (1-t) ( S^{\alpha +}_{1/2} S^{\beta -}_{1/2} +S^{\alpha -}_{1/2} S^{\beta +}_{1/2} ) - V S^{z\alpha}_{1/2} S^{z\beta}_{1/2}.
$$ We evaluate this defect by using operator product expansion for CFT. By the usual perturbation theory, we find $$\label{eq:SS}
\left\langle S^{\alpha + }_{i + \frac{1}{2}} S^{\beta -}_{i + \frac{1}{2}}\right \rangle =
C_0 r^{ -\frac{1}{2g} } + C^\prime_0 (-1)^r r^{ -\frac{1}{2g}+2g },$$ $$\label{eq:ZZ}
\left\langle S^{\alpha z }_{i + \frac{1}{2}} S^{\beta z}_{i + \frac{1}{2}}\right \rangle =
C_0 r^{ -2 } + C^\prime_0 r^{ -2g },$$ where $i \in \mathcal{Z}$ and $ r=2i+1 $. $$\label{eq:JJ}
\left\langle J^{\alpha}_i J^{\beta}_i \right\rangle = C_0 r^{ -2 }$$ where $ r=2i $. These results describe the IR behavior for $g>1$ and the UV behavior for $g<1$. We find that for $g>1$, the uniform part always dominate and the IR exponents are $1/2g$, $2$, and $2$ respectively. In contrast, for $g<1$ the staggered part of the $\langle S^zS^z\rangle$ dominates and the UV exponents are $1/2g$, $2g$, and $2$. These results also describe the IR behavior for a single bulk TLL wire. In Table \[table:exponents\], we summarize the dominant exponent for each correlation function [@SM].
We now compare the numerical results against these power laws. Figure \[fig:g>\_IR\] shows the fit of the bulk correlation functions to Eqs. (\[eq:SS\]), (\[eq:ZZ\]), and (\[eq:JJ\]) for the case of an attractive interaction $g=1.5$. The numerical results confirm that in the IR limit all correlation functions decay with corresponding bulk exponents. Also, we see that for a weaker link (smaller $t$), it takes longer distance for the system to reach the IR limit, indicating it needs to more steps to renormalize away the impurity.
In Fig. \[fig:g<\] we compare the numerical results against the bosonization results for the case of a repulsive interaction $g=0.6$. At the IR limit the system is renormalized into two disconnected semi-infinite wires, which corresponds to the Dirichlet boundary condition. Very weak links with $t=0.1, 0.01$ are used to probe the IR behavior. We observe at long distance not only the exponents of the rescaled correlation function agree with Eqs. (\[eq:SS\_D\]), (\[eq:ZZ\_D\]), and (\[eq:JJ\_D\]), but also $t$-dependent prefactors agree. This suggests that the perturbative calculation indeed captures the correct physics of the junction. On the other hand, while small $1-t$ is assumed in the derivation of Eqs. (\[eq:SS\]), (\[eq:ZZ\]), and (\[eq:JJ\]), the exponents describe well the numerical results at short distances even when $t$ is small (and thus $1-t$ is not small). Furthermore the same scaling prefactors also result in data collapse at short distance. Interestingly, this indicates that near the junction, the system does not know which fixed point it should renormalize into, and the correlation in the short distance looks like a bulk wire, scaled with the junction strength $t$.
![(Color online) Rescaled $\langle S^{\alpha, + }_i S^{\beta, -}_i \rangle$, $\langle S^{\alpha, z }_i S^{\beta, z}_i \rangle$, and $\langle J^{\alpha}_i J^{\beta}_i \rangle$ correlation functions for $g=1.2$. Data for junctions with $t=0.1, 0.01, 0.001$ are plotted. Solid (dotted) lines are power law fitting to the long (short) distance data with IR (UV) exponents from bosonization. (Cf. Table.\[table:exponents\].)[]{data-label="fig:g>_UV"}]({corr_u1_g1.2_tAll_m800}.eps){width="0.9\columnwidth"}
Finally, we analyze the UV behavior for the case of $g>1$. Figure \[fig:g>\_UV\] plots rescaled correlation functions for the case of $g=1.2$ and various extremely small $t$ in order to expose the UV regime. In this limit, we are able to fit the numerical results to the boundary perturbation theory results with Dirichlet boundary condition before crossing over to the long distance behavior. Both the exponents and scaling prefactor agree well. In Fig. \[fig:g>\_UV\] we also show the power law fitting in the IR limit and crossover from UV to IR exponents are clearly observed.
In summary we present a robust and powerful numerical method to study a junction between two quantum wires using IBC with finite-size DMRG. This method allows, for the first time, to study numerically the crossover of correlation functions near a quantum impurity between the short- and long-distance regimes, as demonstrated by the perfect fit of the UV and IR behaviors between the numerical and bosonization results. This may lead to further exploration of the crossover behavior from UV to IR [@PhysRevB.13.316]. We also emphasize that this method is also applicable to a more general class of interesting problems, such as the Y-junction for TLL leads [@Oshikawa:2006sw; @Rahmani:2010jr; @Rahmani:2012bq], TLL leads with different Luttinger parameters[@Hou:2012hz], and junctions with spin-1/2 interacting fermion leads [@Hou:2008do].
This work was supported by the Ministry of Science and Technology (MOST) of Taiwan under Grants No. 105-2112-M-002-023-MY3, 104-2628-M-007-005-MY3, 104-2112-M-002-022-MY3, and by MEXT/JSPS KAKENHI Grants JP16K05469 and JP17H06462 of Japan. CYL thanks Shuai Yin for helpful discussions. YJK thanks the hospitality of ISSP, University of Tokyo, where part of the work was done. Numerical calculation was done using Uni10 tensor network library (https://uni10.gitlab.io/) [@uni10].
Supplemental Material for Crossover of Correlation Functions near Quantum Impurity in a Tomonaga-Luttinger Liquid
=================================================================================================================
In this Supplemental Material, we provide additional information on the the construction and the explicit form of the matrix product operators (MPO) and the bosonization derivation of correlation functions.
Infinite boundary conditions and the effective Hamiltonian
==========================================================
We follow the approaches used in Refs. [@Michel:2010uo; @Phien:2012dr; @Phien:2013kh; @Milsted:2013iq; @Zauner:2015jl] to obtain the effective Hamiltonian for the finite-size window in the MPO form. We first express the ground state of the bulk in the form of an infinite matrix product state (iMPS). $$\begin{aligned}
|\Psi\rangle_{\text{bulk}}^{G}& = \sum_{ \{s_i\} }
\cdots \lambda \Gamma^{s_{i-1}} \lambda \Gamma^{s_i} \lambda \Gamma^{s_{i+1}} \lambda \Gamma^{s_{i+2}} \cdots
|\mathbf{s}\rangle,\nonumber\\
&= \sum_{s_i} \cdots A^{s_{i-1}} A^{s_i} \lambda B^{s_i} B^{s_{i+1}} \cdots |\mathbf{s}\rangle.\end{aligned}$$ On the other hand the MPO of the bulk Hamiltonian reads: $$H_{\text{bulk}} = \cdots W_{-L-\frac{3}{2}} W_{-L-\frac{1}{2}} \cdots W_{-\frac{1}{2}} W_{\frac{1}{2}} \cdots W_{L+\frac{1}{2}} W_{L+\frac{3}{2}} \cdots,$$ where $$W_i =
\left[ \begin{array}{ccccc}
\mathbbm{1}_i & 0 & 0 & 0 & 0 \\
S_i^x & 0 & 0 & 0 & 0 \\
S_i^y & 0 & 0 & 0 & 0 \\
S_i^z & 0 & 0 & 0 & 0 \\
0 & -S_i^x & -S_i^y & \Delta S_i^z & \mathbbm{1}_i
\end{array} \right].$$ We assume that the ground state of the junction Hamiltonian has the following iMPS form: $$|\Psi\rangle_{\text{junc}}^{G, 2L} =
\sum_{s_i} \cdots A^{s_{-L-1}} \left ] \right [ M^{s_{-L}} \cdots M^{s_L} \left ] \right [ B^{s_{L+1}} \cdots |\mathbf{s}\rangle,$$ where $L$ is the size of the window. The MPO of the junction Hamiltonian reads: $$H_{\text{junc}} = \cdots W_{-L-\frac{3}{2}} W_{-L-\frac{1}{2}} \cdots \widetilde{W}_{-\frac{1}{2}} W_{\frac{1}{2}} \cdots W_{L+\frac{1}{2}} W_{L+\frac{3}{2}} \cdots,$$ which is obtained from $H_{\text{bulk}}$ by replacing $W_{-1/2}$ with $\widetilde{W}_{-1/2}$ to represent the link, where $$\widetilde{W}_{-1/2} =
\left[ \begin{array}{ccccc}
\mathbbm{1}_{-\frac{1}{2}} & 0 & 0 & 0 & 0 \\
S_{-1/2}^x & 0 & 0 & 0 & 0 \\
S_{-1/2}^y & 0 & 0 & 0 & 0 \\
S_{-1/2}^z & 0 & 0 & 0 & 0 \\
0 & -t S_{-1/2}^x & -t S_{-1/2}^y & 0 & \mathbbm{1}_{-1/2}
\end{array} \right].$$
![(Color online) Illustration of the process to find the effective Hamiltonian of a finite window. (a) Diagrams for the ground state iMPS $|\Psi\rangle^{G,2L}_{junc}$ and the junction Hamiltonian $H_{junc}$. (b) Diagram for the expectation value $\langle \Psi|H_{junc}|\Psi \rangle^{G,2L}_{junc}$. (Framed) Left and right generalized transfer matrices. (c) Removing the total energy outside the window and attaching IBCs $\widetilde{W}_{\mathcal{L}}$ and $\widetilde{W}_{\mathcal{R}}$ to represent the left and right semi-infinite extensions respectively. (d) The MPO of the effective Hamiltonian $H_{\text{junc}}^{2L}$.[]{data-label="fig:h_window"}](h_window_v1.eps){width="0.9\columnwidth"}
In Fig.\[fig:h\_window\](a) we sketch the tensor network diagram for $|\Psi\rangle_{\text{junc}}^{G, 2L}$ and $H_{\text{junc}}$, while in Fig.\[fig:h\_window\](b) we sketch the tensor network diagram for $\langle \Psi | H_{\text{junc}} |\Psi\rangle_{\text{junc}}^{G, 2L}$. Since in the thermodynamics limit the total energy $\langle \Psi | H_{\text{junc}} |\Psi\rangle_{\text{junc}}^{G, 2L}$ is divergent, one has to remove the total energy of the sites outside the windows. This results in $$\langle \Psi | H_{\text{junc}} |\Psi\rangle_{\text{junc}}^{G, 2L} \sim \langle \Psi | H_{\text{junc}}^{2L} |\Psi\rangle_{\text{junc}}^{G, 2L} + e_0 (n_L+n_R),$$ where $e_0$ is the energy per site of the infinite chain. This leads to the tensor network diagram for $\langle \Psi | H_{\text{junc}}^{2L} |\Psi\rangle_{\text{junc}}^{G, 2L}$ as sketched in Fig.\[fig:h\_window\](c) and the MPO form of the $H_{\text{junc}}^{2L}$ $$H_{\text{junc}}^{2L} = \widetilde{W}_{\mathcal{L}} W_{-L-\frac{1}{2}} \cdots \widetilde{W}_{-\frac{1}{2}} W_{\frac{1}{2}} \cdots W_{L+\frac{1}{2}} \widetilde{W}_{\mathcal{R}},$$ as sketched in Fig.\[fig:h\_window\](d). Here $\widetilde{W}_{\mathcal{L}}$ and $\widetilde{W}_{\mathcal{R}}$ are the left and right IBC respectively.
![(Color online)(a) Generalized transfer matrices $T_{\mathcal{L, R}}$. (b) Recursion relations of the generalized eigenvectors.[]{data-label="fig:eig_vec"}](eig_vec.eps){width="0.9\columnwidth"}
To identify $\widetilde{W}_{\mathcal{L, R}}$ we consider the generalized transfer matrices $T_{\mathcal{L, R}}$ as sketched in Fig.\[fig:eig\_vec\](a). Due to the lower triangle structure of $\mathcal{W}$, the transfer matrices $T_{\mathcal{L, R}}$ are not diagonalizable. However, one can use the recursion relations as sketched in Fig.\[fig:eig\_vec\](b) to show that the left and right generalized eigenvectors $E_{L}(n) $ and $E_R(n)$ have the form $$E_L(n) =
\left[ \begin{array}{ccccc}
\widetilde{H}_L + e_0 n \widetilde{\mathbbm{1}}_L & -\widetilde{S}_L^x & -\widetilde{S}_L^y & \Delta\widetilde{S}_L^z & \widetilde{\mathbbm{1}}_L
\end{array} \right],$$ $$E_R(n) =
\left[ \begin{array}{c}
\widetilde{\mathbbm{1}}_R \\
\widetilde{S}_R^x \\
\widetilde{S}_R^y \\
\widetilde{S}_R^z \\
\widetilde{H}_R + e_0 n \widetilde{\mathbbm{1}}_R
\end{array} \right],$$ where $$\widetilde{S}^{x,y,z}_L = \sum_{ss^\prime} \langle s |S^{x,y,z}| s^\prime \rangle A^{\dagger s^\prime} A^s,$$ and $$\widetilde{S}^{x,y,z}_R = \sum_{ss^\prime} \langle s |S^{x,y,z}| s^\prime \rangle B^s B^{\dagger s^\prime}.$$ Furthermore $\widetilde{H}_{L,R}$ is obtained by solving a linear equation as sketched in Fig.\[fig:hb\_eqn\]. Finally we obtain the left and right IBC $\widetilde{W}_{L,R}$ by dropping the the divergent energy of the semi-infinite extension outside the window to get
![(Color online) The linear equation to solve for $\widetilde{H}_{L}$. $\widetilde{H}_{R}$ is solved in a similar manner, where the tensors are contracted from the right.[]{data-label="fig:hb_eqn"}](hb_eqn.eps){width="0.9\columnwidth"}
$$\widetilde{W}_L =
\left[ \begin{array}{ccccc}
\widetilde{H}_L & -\widetilde{S}_L^x & -\widetilde{S}_L^y & \Delta\widetilde{S}_L^z & \widetilde{\mathbbm{1}}_L
\end{array} \right] \text{and }
\widetilde{W}_R =
\left[ \begin{array}{c}
\widetilde{\mathbbm{1}}_R \\
\widetilde{S}_R^x \\
\widetilde{S}_R^y \\
\widetilde{S}_R^z \\
\widetilde{H}_R
\end{array} \right].$$
Bosonization
============
In this section, we explain the detail of the analytical calculation we have done in the main text.
First we used the bosonization for the spin chain by Lukyanov[@Lukyanov:2002fg], $$\begin{aligned}
S^{-}_j & \sim e^{- i \theta }\left(b+c\left(-1\right)^j\mbox{sin}\left(2\phi\right)\right) , \\
S^z_j & \sim \frac{1}{\pi }\frac{\partial\phi}{\partial x}+a(-1)^j \sin{2\phi} , \\
J_r & =i\left(S^{+}_{j+1}S^{-}_{j}-S^{+}_{j}S^{-}_{j+1}\right)\sim \frac{gv}{\pi} \frac{\partial \theta}{\partial x}.\end{aligned}$$ Here $g$ is the Luttinger parameter and $v$ is the spin-wave velocity. The Lagrangian density and the definition of the bosonic field are, $$\mathcal{L} = \frac{1}{2\pi g} (\partial_\mu \phi)^2 .
\label{eq:Lag}$$ $$\begin{aligned}
\langle \phi_R(0) \phi_R(z) \rangle & \sim
- \frac{g}{4} \log{z} + \mbox{const.} , \\
\langle \phi_L(0) \phi_L(\bar{z}) \rangle & \sim
- \frac{g}{4} \log{\bar{z}} + \mbox{const.}. \\\end{aligned}$$ We used the complex coordinate $z=x+it$ and assumed the geometry of the theory without junction is a full complex plane.
The field $\phi$ is subject to the compactification $$\phi = \phi + \pi .$$ Using this, the dual field $\theta$ is defined as $$\theta \equiv (1/g) ( \phi_L - \phi_R ) ,$$ where $\theta$ is subject to the compactification $$\theta \sim \theta +2\pi.$$
Then we used the boundary perturbation procedure. For Dirichlet boundary condition, we gradually introduce interaction to two decoupled chains through the boundary. The geometry is two half plane with the interaction on the line $x=0$. We note the calculation for $S_{r}^{\alpha +}S_{r}^{\beta -}$ as an example. The first order perturbation leads to, $$t \int d\tau \;
\left[ \frac{1}{2}
\langle S^{+\alpha}_{i+1/2}(0) S^{-\alpha}_{1/2}(\tau)\rangle_{D}
\langle S^{+\beta }_{1/2}(\tau) S^{-\beta}_{i+1/2}(0) \rangle_{D}
\right],
\label{eq.SpSm_1st}$$ The Dirichlet boundary condition for bosonic field is, $$\phi^L_\gamma(0,\tau) + \phi^R_\gamma(0,\tau) = 0, \gamma=\alpha, \beta.$$ This can be resolved by an analytical continuation of $\phi^R_2$ to $x<0$ and $$\phi^L_\gamma(x,\tau) \equiv - \phi^R_\gamma(-x,\tau).$$ Using this, the correlation function is given as $$\begin{aligned}
& t \int d\tau \;
\left(\frac{1}{r^2+\tau^2}\right)^{1/g} (2r)^{1/2g} \\
\sim &
t r^{-(2/g-1)} r^{1/2g} = t r^{-(3/2g-1)} .\end{aligned}$$ For free boundary condition, we think of the junction as defect in CFT. We evaluated this defect by using operator product expansion for CFT. $$\begin{split}
H_{\text{link}} = (1-t) ( S^{+\alpha}_{1/2} S^{-\beta}_{1/2} +S^{-\alpha}_{1/2} S^{+\beta}_{1/2} ) \\
\sim \mu_{B}\int ^{\infty}_{-\infty}d\tau \mbox{cos}\left( 4\phi (i\tau) \right).
\end{split}$$ We have introduced the parameter $\mu_B$ for the boundary perturbation parameter. For Dirichlet boundary condition, it is proportional to $t$. For the connected wire, it is proportional to $1-t$.
Bulk perturbation effect
========================
In this section, we show the results of the bulk perturbation. The field theoretic bulk action for XXZ spin chain is written as $$S=S_{CFT}+\mu\int dx'd\tau '\cos \left( 4\phi\right).$$ By adding the contribution of this term, we can obtain the results in the main section. The $\mu$ is determined by $V$ and it is explicitly determined without a junction.
For example, we show the correction of the results in the previous section. For Dirichlet boundary condition, the stagger part of the field gives the following correction of the term for each wire, $$\langle S^{+\alpha}_{r+1/2}(0)S^{-\alpha}_{1/2}(\tau)\rangle_{\rm uniform}=r^{-\frac{3}{4g}}C'_{0}+\mu r^{-4g-\frac{3}{4g}+2}C'_{1},$$ where $C'_0(\tau/r)$ and $C'_1(\tau/r)$ are functions of $\tau/r$. The second term is obtained from the following integral, $$\begin{split}
&\mu \int dx' d\tau ' b^2 e^{i\theta (r) R} e^{-i\theta (i\tau) R} \cos \left( 4\phi(x', \tau ')\right) \\
&=\mu r^{-4g-\frac{3}{4g}+2}C'_{1} \left(\frac{\tau}{r}\right).
\end{split}$$ The stagger term of the field gives, $$\langle S^{+\alpha}_{i+1/2}(0)S^{-\alpha}_{1/2}(\tau)\rangle_{\rm stagger}=r^{-\frac{3}{4g}-g}C_{0}+\mu r^{-5g-\frac{3}{4g}+2}C_{1}.$$ Then by composing contribution of each wire and the integration of $\tau$, we can obtain the desired correlation function. However, we do not know the effect of $(\textrm{stagger})\times(\textrm{uniform})$ terms and the precise prefactor of them. The results for $(\textrm{stagger})\times(\textrm{stagger})$ are, $$\begin{aligned}
\langle S^{+ \alpha}_{r+1/2} S^{- \beta}_{r+1/2} & \rangle=\left( -1\right)^{r}\mu_B \nonumber \\
&\left( r^{-\frac{3}{2g}-2g+1}
+\mu r^{-\frac{3}{2g}-6g+3}+
\mu^{2} r^{-\frac{3}{2g}-10g+5}\right),\end{aligned}$$ $$\begin{aligned}
\langle S^{z \alpha}_{r+1/2} S^{z \beta}_{r+1/2} &\rangle=\left( -1 \right)^{r} \mu_{B}^{2} \nonumber \\
&\left( r^{-\frac{2}{g}-2g+2}+\mu r^{-\frac{2}{g}-6g+4}+\mu^{2} r^{-\frac{2}{g}-10g+6} \right).\end{aligned}$$ The results for $(\textrm{uniform})\times(\textrm{uniform})$ are, $$\begin{aligned}
\begin{split}
\langle S^{+ \alpha}_{r+1/2} S^{- \beta}_{r+1/2}\rangle=\mu_{B}
\left( r^{-\frac{3}{2g}+1}+\mu r^{-\frac{3}{2g}-4g+3}+\mu^2 r^{-\frac{3}{2g}-8g+5} \right),
\end{split}
\\
\begin{split}
\langle S^{z \alpha}_{r+1/2} S^{z \beta}_{r+1/2}\rangle=\mu^{2}_{B}
\left( r^{-\frac{2}{g}}+\mu r^{-\frac{2}{g}-4g+2}+\mu^{2}r^{-\frac{2}{g}-8g+4}\right),
\end{split}
\\
\begin{split}
\langle J^{\alpha}_{r+1/2} J^{\beta}_{r+1/2}\rangle=\mu_{B}^{2} \left( r^{-\frac{2}{g}}+\mu r^{-\frac{2}{g}-4g+2} +\mu^{2}r^{-\frac{2}{g}-8g+4}\right).
\end{split}\end{aligned}$$ Here and the following discussion we omit the constants for simplicity.
For the free boundary condition, we can get the correction by the same procedure, but it does not cause the problem of the mixing of stagger and uniform terms. The lowest order of the boundary perturbation can be changed by the effect of the bulk perturbation in this case. The results are $$\begin{aligned}
\begin{split}
\langle S^{+\alpha}_{r+1/2} S^{-\beta}_{r+1/2} \rangle&=r^{-\frac{1}{2g}}+\mu^{2}r^{-\frac{1}{2g}-8g+4}+\mu\mu_{B}r^{-\frac{1}{2g}-8g+3} \\
&+\mu^{2}_{B}r^{-\frac{1}{2g}-8g+2}+\left(-1\right)^{r}r^{-\frac{1}{2g}-2g} \\
&+\left(-1\right)^{r}\mu r^{-\frac{1}{2g}-6g+2}+\left(-1\right)^{r}\mu_{B}r^{-\frac{1}{2g}-6g+1}.
\end{split}
\\
\begin{split}
\langle S^{z\alpha}_{r+1/2} S^{z\beta}_{r+1/2} \rangle&=r^{-2}+\mu^{2}r^{-8g+2}+\mu\mu_{B}r^{-8g+1} \\
&+\mu^{2}_{B}r^{-8g}+\left(-1\right)^{r}r^{-2g} \\
&+\left(-1\right)^{r}\mu r^{-6g+2}+\left(-1\right)^{r}\mu_{B}r^{-6g+1}.
\end{split}
\\
\begin{split}
\langle J_{r+1/2} J_{r+1/2} \rangle&=r^{-2}+\mu^{2}r^{-8g+2}+\mu\mu_{B}r^{-8g+1} \\
&+\mu^{2}_{B}r^{-8g}.
\end{split}\end{aligned}$$ In any case, all of subleading terms resulted from the correction of the bulk perturbation decay faster than leading order terms we as have explained in the previous section. Hence that verifies the validity of the results in the main text.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
We present measurements of $B^+\to \bar{D}^{*0} \tau^+ \nu_{\tau}$ and $B^+\to \bar{D}^{0} \tau^+ \nu_{\tau}$ decays in a data sample of $657 \times 10^6$ $B\bar{B}$ pairs collected with the Belle detector at the KEKB asymmetric-energy $e^+e^-$ collider. We find $446^{+58}_{-56}$ events of the decay $B^+\to \bar{D}^{*0} \tau^+ \nu_{\tau}$ with a significance of 8.1 standard deviations, and $146^{+42}_{-41}$ events of the decay $B^+\to \bar{D}^{0} \tau^+ \nu_{\tau}$ with a significance of 3.5 standard deviations. The latter signal provides the first evidence for this decay mode. The measured branching fractions are $\mathcal{B}(B^+\to \bar{D}^{*0} \tau ^+
\nu_{\tau})=(2.12^{+0.28}_{-0.27} ({\rm stat}) \pm 0.29 ({\rm syst})) \% $ and $\mathcal{B}(B^+\to \bar{D}^{0} \tau ^+
\nu_{\tau})=(0.77\pm 0.22 ({\rm stat}) \pm 0.12 ({\rm syst})) \% $.
author:
- 'A. Bozek'
- 'M. Rozanska'
- 'I. Adachi'
- 'H. Aihara'
- 'K. Arinstein'
- 'V. Aulchenko'
- 'T. Aushev'
- 'T. Aziz'
- 'A. M. Bakich'
- 'V. Bhardwaj'
- 'M. Bischofberger'
- 'A. Bondar'
- 'M. Bračko'
- 'T. E. Browder'
- 'Y. Chao'
- 'A. Chen'
- 'B. G. Cheon'
- 'I.-S. Cho'
- 'K.-S. Choi'
- 'Y. Choi'
- 'J. Dalseno'
- 'Z. Doležal'
- 'Z. Drásal'
- 'A. Drutskoy'
- 'W. Dungel'
- 'S. Eidelman'
- 'P. Goldenzweig'
- 'B. Golob'
- 'H. Ha'
- 'K. Hara'
- 'Y. Hasegawa'
- 'H. Hayashii'
- 'T. Higuchi'
- 'Y. Horii'
- 'Y. Hoshi'
- 'W.-S. Hou'
- 'H. J. Hyun'
- 'T. Iijima'
- 'K. Inami'
- 'M. Iwabuchi'
- 'Y. Iwasaki'
- 'N. J. Joshi'
- 'J. H. Kang'
- 'P. Kapusta'
- 'H. Kawai'
- 'T. Kawasaki'
- 'H. Kichimi'
- 'C. Kiesling'
- 'H. O. Kim'
- 'J. H. Kim'
- 'M. J. Kim'
- 'S. K. Kim'
- 'Y. J. Kim'
- 'B. R. Ko'
- 'S. Korpar'
- 'P. Križan'
- 'P. Krokovny'
- 'T. Kuhr'
- 'T. Kumita'
- 'A. Kuzmin'
- 'Y.-J. Kwon'
- 'S.-H. Kyeong'
- 'M. J. Lee'
- 'S.-H. Lee'
- 'J. Li'
- 'D. Liventsev'
- 'R. Louvot'
- 'A. Matyja'
- 'S. McOnie'
- 'H. Miyata'
- 'Y. Miyazaki'
- 'R. Mizuk'
- 'G. B. Mohanty'
- 'E. Nakano'
- 'M. Nakao'
- 'H. Nakazawa'
- 'S. Neubauer'
- 'S. Nishida'
- 'O. Nitoh'
- 'T. Nozaki'
- 'S. Ogawa'
- 'T. Ohshima'
- 'S. Okuno'
- 'S. L. Olsen'
- 'W. Ostrowicz'
- 'P. Pakhlov'
- 'G. Pakhlova'
- 'C. W. Park'
- 'H. K. Park'
- 'R. Pestotnik'
- 'M. Petrič'
- 'L. E. Piilonen'
- 'H. Sahoo'
- 'Y. Sakai'
- 'O. Schneider'
- 'J. Schümann'
- 'C. Schwanda'
- 'A. J. Schwartz'
- 'K. Senyo'
- 'J.-G. Shiu'
- 'B. Shwartz'
- 'R. Sinha'
- 'P. Smerkol'
- 'A. Sokolov'
- 'E. Solovieva'
- 'M. Starič'
- 'J. Stypula'
- 'T. Sumiyoshi'
- 'G. N. Taylor'
- 'Y. Teramoto'
- 'I. Tikhomirov'
- 'K. Trabelsi'
- 'S. Uehara'
- 'Y. Unno'
- 'S. Uno'
- 'G. Varner'
- 'K. E. Varvell'
- 'K. Vervink'
- 'C. H. Wang'
- 'M.-Z. Wang'
- 'P. Wang'
- 'Y. Watanabe'
- 'R. Wedd'
- 'E. Won'
- 'B. D. Yabsley'
- 'Y. Yamashita'
- 'V. Zhulanov'
- 'T. Zivko'
- 'A. Zupanc'
title: |
\
Observation of $ B^+ \to \bar{D}^{*0} \tau^+
\nu_{\tau}$ and Evidence for $ B^+ \to \bar{D}^{0} \tau^+
\nu_{\tau}$ at Belle.
---
Measurements of leptonic and semileptonic decays of $B$ mesons to the $\tau$ lepton can provide important constraints on the Standard Model (SM) and its extensions. Due to the large mass of the lepton in the final state, these decays are sensitive probes of models with extended Higgs sectors [@Itoh]. Semileptonic modes with $b \to c \tau^- {\bar\nu}_{\tau}$ [@CC] transitions provide more observables sensitive to new physics than purely leptonic $B^+\to \tau^+ \nu_{\tau}$ decays. Of particular interest is $\tau$ polarization. The effects of new physics are expected to be larger in $B\to \bar{D} \tau ^+ \nu_{\tau}$ than in $B\to \bar{D}^* \tau ^+ \nu_{\tau}$. We note that decays to the vector meson offer the interesting possibility of studying correlations between the D\* polarization and other observables [@Garisto].
The predicted branching fractions, based on the SM, are around 1.4% and 0.7% for $B^0 \to {D}^{*-} \tau ^+ \nu_{\tau}$ and $B^0 \to {D}^- \tau ^+ \nu_{\tau}$, respectively (see [*e.g.*]{}, [@hwang]). Despite relatively large branching fractions, multiple neutrinos in the final states make the search for semi-tauonic $B$ decays very challenging. Inclusive and semi-inclusive branching fractions have been measured in LEP experiments [@lep] with an average branching fraction of $\mathcal{B}(b \to \tau \nu_{\tau} X)=(2.48\pm
0.26)\%$ [@PDG]. The exclusive decay was first observed by Belle [@Matyja] in the $B^0\to {D}^{*-} \tau ^+ \nu_{\tau}$ mode. Other modes have also been measured by BaBar [@BaBar-1] and Belle [@Kozakai]. The results are still statistically limited. In particular, the Belle prelimnary result [@Kozakai] is the only evidence to date for $B^+\to \bar{D}^0 \tau ^+ \nu_{\tau}$. Further improvements in precision could tightly constrain theoretical models.
Decays of $B$ mesons to multi-neutrino final states can be studied at $B$-factories via the recoil of the accompanying $B$ meson ($B_{\rm tag}$). Reconstruction of the $B_{\rm tag}$ allows one to calculate the missing four-momentum in the $B_{\rm sig}$ decay; this helps separate signal events from copious backgrounds. At the same time the presence of a $B_{\rm tag}$ strongly suppresses the combinatorial and continuum backgrounds. The disadvantage is the low $B_{\rm tag}$ reconstruction efficiency. To increase statistics, we reconstruct the $B_{\rm tag}$ “inclusively” from all the remaining particles after the $B_{\rm sig}$ selection (see Ref. [@Matyja]). A data sample consisting of $657 \times 10^6$ $B\bar{B}$ pairs is used in this analysis. It was collected with the Belle detector [@Belle] at the KEKB asymmetric-energy $e^+e^-$ (3.5 on 8 GeV) collider [@KEKB] operating at the $\Upsilon(4S)$ resonance ($\sqrt{s}=10.58$ GeV).
We use Monte Carlo (MC) simulations to estimate signal efficiencies and background contributions. Large samples of the signal $B^+\to \bar{D}^{(*)0} \tau ^+ \nu_{\tau}$ decays are generated with the EvtGen package [@evtgen] using the ISGW2 model [@isgw2]. Radiative effects are modeled using the PHOTOS code [@photos]. We use large MC samples of continuum $q\bar{q}$ ($q=u,d,s,c$) and inclusive $B\bar{B}$ events to model the background. The sizes of these samples are, respectively, six and nine times that of the data.
Primary charged tracks are required to have impact parameters consistent with an origin at the interaction point (IP), and to have momenta above 50 MeV/$c$ in the laboratory frame. $K^0_S$ mesons are reconstructed using pairs of charged tracks satisfying $482 ~{\rm MeV}/c^2 < M_{\pi^+\pi^-} < 514 ~{\rm MeV}/c^2$ with a vertex displacement from the IP consistent with the reconstructed momentum vector. Muons, electrons, charged pions, kaons and protons are identified using information from particle identification subsystems [@PID]. The momenta of particles identified as electrons are corrected for bremsstrahlung by adding photons within a 50 mrad cone along the lepton trajectory.
The $\pi^0$ candidates are reconstructed from photon pairs having 118 MeV/$c^2<M_{\gamma\gamma}<150\ {\rm MeV}/c^2$. For candidates that share a common $\gamma$, we select the one with the smallest $\chi^2$ value resulting from a $\pi ^0$ mass-constrained fit. To reduce the combinatorial background, we require that the photons from the $\pi^ 0$ have energies greater than 60 MeV - 120 MeV, depending on the photon’s polar angle. Photons that are not associated with a $\pi^0$ are accepted if their energy exceeds a polar-angle dependent threshold ranging from 100 MeV to 200 MeV.
The $\bar{D}^0$ candidates are reconstructed in the $K^+\pi^-$ and $K^+\pi^-\pi^0$ final states. We accept $\bar{D}^0$ candidates having an invariant mass in a $3\sigma$ window of the nominal $M_{D^0}$ mass.
The $\bar{D}^{*0}$ candidates are reconstructed from $\bar{D}^0 \pi^0$. We require that the mass difference $\Delta M = M_{D^*}-M_{D^0}$ is in a 3$\sigma$ window around its nominal value. We also accept $\bar{D}^0 \gamma$ pairs that do not fulfill the requirement on $\Delta M$ if they are kinematically consistent with the hypothesis that $\bar{D}^0$ and $\gamma$ come from the decay $\bar{D}^{*0}\to \bar{D}^0 \pi^0$ with one undetected photon ($\gamma_{\rm miss}$) from the $\pi^0$ decay (“partial reconstruction” of $\bar{D}^{*0}$). For this purpose $\cos(\theta_{\gamma,\gamma_{\rm miss}})$, the cosine of the angle between two photons from the $\pi^0~$ is calculated in the $\bar{D}^{*0}$ rest frame taking the nominal $\bar{D}^{*0}$ and $\pi^0$ masses. We require $|\cos(\theta_{\gamma,\gamma_{\rm miss}})|\:<\:1.1$ (taking into account experimental precision) and that the energy of the detected photon exceeds 120 MeV. The partial reconstruction of $\bar{D}^{*0}$ increases the reconstruction efficiency by a factor of about four, but due to higher background it is only used in the subchannels with $\bar{D}^0 \to K^+ \pi^-$ decay.
To reconstruct the $\tau$ lepton candidates, we use the $\tau^+ \to e^+\nu_e\bar{\nu}_{\tau}$, $\tau^+ \to \mu^+\nu_{\mu}\bar{\nu}_{\tau}$, and $\tau^+ \to \pi^+\bar{\nu}_{\tau}$ modes. In the latter case, we also take into account the contribution from the $\tau^+ \to \rho^+\bar{\nu}_{\tau}$ channel. The $\tau^+\to \pi^+\bar{\nu}_{\tau}$ mode has a sensitivity similar to the $\tau^+ \to e^+\nu_e\bar{\nu}_{\tau}$ or $\tau^+ \to \mu^+\nu_{\mu}\bar{\nu}_{\tau}$ mode, and can be used to study $\tau$ polarization. For this channel, due to the higher combinatorial background, we analyze only the decay chains with the $\bar{D}^0 \to K^+ \pi^-$ mode. In total, we consider 13 different decay chains, eight with $\bar{D}^{*0}$ and five with $\bar{D}^0$ in the final states.
The signal candidates are selected by combining a $\bar{D}^{(*)0}$ meson with an appropriately charged electron, muon or pion. In the sub-channels with the $\tau^+ \to \pi^+\bar{\nu}_{\tau}$ decay, the large combinatorial background is suppressed by requiring the pion energy $E_{\pi} > 0.6$ GeV. From multiple candidates we select a ($\bar{D}^{(*)0}d^+_{\tau}$) pair (throughout the paper $d_{\tau}$ stands for the charged $\tau$ daughter: $e$, $\mu$ or $\pi$) with the best ${\bar{D}^{(*)0}}$ candidate, based on the value of $\Delta M$ (for subchannels where $\Delta M$ is available) or $M_{D^0}$. For the pairs sharing the same ${\bar{D}^{(*)0}}$ candidate, we select the candidate with the largest vertex probability fit on the tagging side. Once a $B_{\rm sig}$ candidate is found, the remaining particles that are not assigned to $B_{\rm sig}$ are used to reconstruct the $B_{\rm tag}$ decay. The consistency of a $B_{\rm tag}$ candidate with a $B$-meson decay is checked using the beam-energy constrained mass and the energy difference variables: $M_{\rm tag} = \sqrt{E^2_{\rm beam} - {\bf p}^2_{\rm tag}},~~
{\bf p}_{\rm tag} = \sum_i {\bf p}_i$, and $\Delta E_{\rm tag} = E_{\rm tag} - E_{\rm beam}, ~~ E_{\rm tag}
= \sum_i E_i$, where $E_{\rm beam}$ is the beam energy and ${\bf p}_i$ and $E_i$ denote the 3-momentum vector and energy of the $i$’th particle. All quantities are evaluated in the $\Upsilon(4S)$ rest frame. The summation is over all particles that are assigned to $B_{\rm tag}$. We require that the candidate events have $M_{\rm tag}> 5.2 ~{\rm GeV}/c^2$ and $-0.3 ~{\rm GeV} <\Delta E_{\rm tag}<0.05 ~{\rm GeV}$. With this requirement the $M_{\rm tag}$ distribution of the signal peaks at the $B^+$ mass with about 80% of the events being contained in the signal-enhanced region $M_{\rm tag}>$ 5.26 GeV/$c^2$.
To suppress background and improve the quality of the $B_{\rm tag}$ selection, we impose the following requirements: zero total event charge; no charged leptons in the event (except those coming from the signal side); zero net proton/antiproton number; residual energy in the electromagnetic calorimeter ([*i.e.*]{}, the sum of energies that are not included in the $B_{\rm sig}$ nor $B_{\rm tag}$) should be less than 0.35 GeV (0.30 GeV or 0.25 GeV in sub-channels with higher backgrounds); the number of neutral particles on the tagging side $N_{\pi^0}+N_{\gamma}<6$, $N_{\gamma} <3$, and less than four tracks that do not satisfy the requirements imposed on the impact parameters. For decay modes with higher background, we impose further constraints on the total event strangeness and require no $K^0_L$ in the event. These criteria, which we refer to as “the $B_{\rm tag}$-selection”, reject events in which some particles were undetected and suppress events with a large number of spurious showers. In the samples of the ($\bar{D}^{(*)0}l^+$) pairs ($l=e, \mu$), the dominant background comes from semileptonic $B$ decays, $B^+\to \bar{D}^{(*)0} X l^+\nu_{l}$, whereas in the case of the ($\bar{D}^{(*)0}\pi^+$) pairs, the combinatorial background from hadronic $B$ decays dominates. Further background suppression exploits observables that characterize the signal decay: missing energy $E_{\rm miss} =
E_{\rm beam}-E_{\bar{D}^{(*)0}}-E_{d_{\tau}^+}$; visible energy $E_{\rm vis}$, [*i.e.*]{}, the sum of the energies of all particles in the event; the square of missing mass $M_{\rm miss}^2 = E_{\rm miss}^2 -
({\bf p}_{\rm sig} -
{\bf p}_{\bar{D}^{(*)0}} -
{\bf p}_{d_{\tau}^+})^2$ and the effective mass of the ($\tau^+ \nu_{\tau}$) pair, $q^2 = (E_{\rm beam} -
E_{\bar{D}^{(*)0}})^2 -
({\bf p}_{\rm sig} -
{\bf p}_{\bar{D}^{(*)0}})^2$ where ${\bf p}_{\rm sig} = -{\bf p}_{\rm tag}$ (all kinematical variables are in the $\Upsilon(4S)$ rest frame). The most useful variable for separating signal and background is obtained by combining $E_{\rm miss}$ and ($\bar{D}^{(*)0} d_{\tau}^+$) pair momentum: $X_{\rm miss} = (E_{\rm miss} - |{\bf p}_{\bar{D}^{(*)0}} +
{\bf p}_{d_{\tau}^+}|)/
\sqrt{E_{\rm beam}^2 -m_{B^+}^2}$ where $m_{B^+}$ is the $B^+$ mass. The $X_{\rm miss}$ variable is closely related to the missing mass in the $B_{\rm sig}$ decay but does not depend on the $B_{\rm tag}$ reconstruction [@Matyja]. The signal selection criteria are optimized individually in each decay chain, by maximizing the expected significance $N_S/\sqrt{N_S+N_B}$, where $N_S$ and $N_B$ are the number of signal and background events in the signal-enhanced region, assuming the SM prediction [@hwang] for the signal branching fractions. The expected background $N_B$ is evaluated using generic MC samples. We require $E_{\rm vis}< 8.3 ~{\rm GeV}$ – $8.5 ~{\rm GeV}$, $E_{\rm miss}> 1.5 ~{\rm GeV}$ – $1.9 ~{\rm GeV}$ and $X_{\rm miss}>$ 2.0 – 2.75 for leptonic $\tau$ decays, or $X_{\rm miss}>$ 1.0 – 1.5 for the modes with $\tau \to \pi \nu_{\tau}$. In the latter case, where the $\tau$ decays to a final state with a single neutrino, we further require $\cos(\theta_{\nu_1\nu_2})$ to be in the range $[-1,1]$, where $\theta_{\nu_1\nu_2}$ denotes the angle between the two neutrinos in the ($\tau^+\nu_{\tau}$) rest frame and is calculated from the $M_{\rm miss}^2$ and $q^2$ variables. In the sample with ($\bar{D}^0d^+_{\tau}$) pairs, to suppress the cross-feeds from the $B\to \bar{D}^*\tau^+ \nu_{\tau}$ modes, we impose a loose requirement on $q^2< 9.5 ~{\rm GeV}^2/c^4$.
The above requirements result in flat $M_{\rm tag}$ distributions for most background components, while the distribution of the signal modes remains unchanged. The main sources of the peaking background are the semileptonic decays $B^+ \to \bar{D}^{*0}l^+\nu_l$ and $B \to \bar{D}^{(*)}\pi l^+\nu_l$ (including $\bar{D}^{**}l^+\nu_l$). In order to estimate the peaking background reliably, in particular from poorly known semileptonic modes of the type $B\to \bar{D}^{**}l\nu_l$, we divide the MC sample into the following categories: $B\to\bar{D}^*l^+\nu_l$, $B\to\bar{D}l^+\nu_l$, $B\to\bar{D}^{**}l^+\nu_l$, other $B$ decays, $c\bar{c}$ and ($u\bar{u}+d\bar{d}+s\bar{s}$) continuum. The normalizations of these components are determined from simultaneous fits to experimental distributions of $M_{\rm tag}$, $\Delta E_{\rm tag}$, $E_{d_{\tau}}$, $X_{\rm miss}$, $E_{\rm vis}$, $q^2$, and $R_2$, the ratio of the second and zeroth Fox-Wolfram moments [@FW]. These fits are performed separately for the subsamples defined by the ($\bar{D}^{(*)0}d^+_{\tau}$) pairs, excluding the region $M_{\rm tag}>5.26 ~{\rm GeV/c}^2$ and $X_{\rm miss}>2.0$, where we expect enhanced signal contributions. The signal and combinatorial background yields are extracted from an extended unbinned maximum likelihood fit to the $M_{\rm tag}$ and $P_{D^0}$ (momentum of $D^0$ from $B_{\rm sig}$ measured in the $\Upsilon(4S)$ frame) variables. The $M_{\rm tag}$ variable allows us to separate the combinatorial background from the signal, while $P_{D^0}$ helps to distinguish between the two signal modes. Correlations between these variables are found to be small.
Parameterizations of two-dimensional probability density functions (PDFs) are determined from the MC samples. They are expressed as the product of one-dimensional PDF’s for each variable. The PDF’s for $M_{\rm tag}$ of the signal and peaking background components are described using an empirical parameterization introduced by the Crystal Ball collaboration [@CB], while combinatorial backgrounds are parameterized by the ARGUS function [@ARGUS]. It has been empirically found that the PDF’s for $P_{D^0}$ are well modeled as a sum of two Gaussian distributions. The fits are performed in the range $M_{\rm tag}>5.2$ GeV/$c^2$, simultaneously to all data subsets. In each of the subchannels, we describe the data as the sum of four components: signal, cross-feed between $\bar{D}^{*0}\tau^+\nu_{\tau}$ and $\bar{D}^{0}\tau^+\nu_{\tau}$, combinatorial and peaking backgrounds. The common signal branching fractions $\mathcal{B}(B^+\to \bar{D}^{*0}\tau^+\nu_{\tau})$ and $\mathcal{B}(B^+\to \bar{D}^{0}\tau^+\nu_{\tau})$, and the numbers of combinatorial background in each subchannel are free parameters of the fit, while the normalizations of peaking background contributions are fixed to the values obtained from the rescaled MC samples (as described above). The signal yields and branching fractions for $B^+\to \bar{D}^{(*)0}\tau^+\nu_{\tau}$ decays are related using the following formula, which assumes equal fractions of charged and neutral $B$ meson pairs produced in $\Upsilon(4S)$ decays: $\mathcal{B}(B^+\to\bar{D}^{(*)0}\tau^+\nu_{\tau}) =
N_s^{D^{(*)}}/(N_{B\bar{B}}\times \sum_{k}\epsilon_{k}\mathcal{B}_{k})$, where $N_{B\bar{B}}$ is the number of $B\bar{B}$ pairs and the index $k$ runs over the 13 decay chains; $\epsilon_{k}$ denotes the reconstruction efficiency of the specific subchannel and $\mathcal{B}_{k}$ is the product of intermediate branching fractions. All the intermediate branching fractions are taken from the PDG compilation [@PDG]. The efficiencies of the signal reconstruction, as well as the expected combinatorial and peaking backgrounds are given in Table \[tab-yields\].
The signal extraction procedure has been tested by fitting ensembles of simulated experiments containing all signal and background components. These pseudo-experiments are generated using the shapes of the fitted PDF’s for the signal and background components and with the number of events are Poisson-distributed around the expected yields. The pull distributions of the extracted signal branching fractions are consistent with standard normal distributions. The small biases in the mean values are included in the final systematic uncertainties.
Mode $N_{s}$ $N_b$ $N_b^{\rm MC}$ $N_p$ $ \epsilon (10^{-6})$ $\mathcal{B}(\%)$ $\Sigma (\sigma)$
-------------------------------- ------------------------- -------------------- ---------------- ----------------- ----------------------- ------------------------ ------------------- -- --
$\bar{D}^{*0}\tau^+\nu_{\tau}$ $446^{+58}_{-56} (226)$ $1075^{+37}_{-35}$ $1029\pm 20$ $31.0\pm 17.7$ $32.6\pm 0.2 (16.3)$ $2.12^{+0.28}_{-0.27}$ $8.8$
$\bar{D}^{0}\tau^+\nu_{\tau}$ $146^{+42}_{-41} (15)$ $1245^{+40}_{-39}$ $1310\pm 19$ $78.2\pm 12.6 $ $30.0\pm 0.4 (3.2)$ $0.77\pm 0.22$ $3.6$
\[tab-yields\]
The procedure established above is applied to the data. The $M_{\rm tag}$ and $P_{D^0}$ distributions for the $\bar{D}^{*0}\tau^+\nu_{\tau}$ and $\bar{D}^{0}\tau^+\nu_{\tau}$ samples in data are shown in Fig. \[pic-fit\]. The overlaid histograms represent the expected background, scaled to the data luminosity. A clear excess of events over background is visible in the signal-enhanced region. The branching fractions extracted from the fit are $\mathcal{B}(B^+\to \bar{D}^{*0} \tau ^+
\nu_{\tau})=(2.12^{+0.28}_{-0.27} ({\rm stat})) \% $ and $\mathcal{B}(B^+\to \bar{D}^{0} \tau ^+
\nu_{\tau})=(0.77 \pm 0.22 ({\rm stat})) \% $. The signal yields are $446^{+58}_{-56}$ $B^+\to \bar{D}^{*0} \tau^+ \nu_{\tau}$ events and $146^{+42}_{-41}$ $B^+\to \bar{D}^{0} \tau^+ \nu_{\tau}$ events. The statistical significances, defined as $\Sigma =
\sqrt{-2{\ln}(\mathcal{L}_{\rm 0}/\mathcal{L}_{\rm max})}$, corespond to 8.8 and 3.6 standard deviations ($\sigma$), respectively. Here $\mathcal{L}_{\rm max}$ denotes the maximum likelihood value and $\mathcal{L}_{\rm 0}$ is the likelihood for the zero signal hypothesis. The fitted yields of combinatorial background in the individual submodes are consistent within statistical uncertainties with the MC-based expectations. The fit results are summarized in Table \[tab-yields\]. The fit projections in $M_{\rm tag}$ and $P_{D^0}$ are shown in Fig. \[pic-fit\].
![The fit projections to $M_{\rm tag}$, and $P_{D^0}$ for $M_{\rm tag}>5.26 ~{\rm GeV}/c^2$ (a,b) for $\bar{D}^{*0}\tau^+\nu_{\tau}$, (c,d) for $\bar{D}^{0}\tau^+\nu_{\tau}$. The black curves show the result of the fits. The solid dashed curves represent the background and the dashed dotted ones show the combinatorial component. The dot-long-dashed and dot-short-dashed curvess represent, respectively, the signal contributions from $B^+\to\bar{D}^{*0}\tau^+\nu_{\tau}$ and $B^+\to\bar{D}^{0}\tau^+\nu_{\tau}$. The histograms represent the MC-predicted background. []{data-label="pic-fit"}](fig_bdtau_1a.eps "fig:"){width="22.00000%"} ![The fit projections to $M_{\rm tag}$, and $P_{D^0}$ for $M_{\rm tag}>5.26 ~{\rm GeV}/c^2$ (a,b) for $\bar{D}^{*0}\tau^+\nu_{\tau}$, (c,d) for $\bar{D}^{0}\tau^+\nu_{\tau}$. The black curves show the result of the fits. The solid dashed curves represent the background and the dashed dotted ones show the combinatorial component. The dot-long-dashed and dot-short-dashed curvess represent, respectively, the signal contributions from $B^+\to\bar{D}^{*0}\tau^+\nu_{\tau}$ and $B^+\to\bar{D}^{0}\tau^+\nu_{\tau}$. The histograms represent the MC-predicted background. []{data-label="pic-fit"}](fig_bdtau_1b.eps "fig:"){width="22.00000%"} ![The fit projections to $M_{\rm tag}$, and $P_{D^0}$ for $M_{\rm tag}>5.26 ~{\rm GeV}/c^2$ (a,b) for $\bar{D}^{*0}\tau^+\nu_{\tau}$, (c,d) for $\bar{D}^{0}\tau^+\nu_{\tau}$. The black curves show the result of the fits. The solid dashed curves represent the background and the dashed dotted ones show the combinatorial component. The dot-long-dashed and dot-short-dashed curvess represent, respectively, the signal contributions from $B^+\to\bar{D}^{*0}\tau^+\nu_{\tau}$ and $B^+\to\bar{D}^{0}\tau^+\nu_{\tau}$. The histograms represent the MC-predicted background. []{data-label="pic-fit"}](fig_bdtau_1c.eps "fig:"){width="22.00000%"} ![The fit projections to $M_{\rm tag}$, and $P_{D^0}$ for $M_{\rm tag}>5.26 ~{\rm GeV}/c^2$ (a,b) for $\bar{D}^{*0}\tau^+\nu_{\tau}$, (c,d) for $\bar{D}^{0}\tau^+\nu_{\tau}$. The black curves show the result of the fits. The solid dashed curves represent the background and the dashed dotted ones show the combinatorial component. The dot-long-dashed and dot-short-dashed curvess represent, respectively, the signal contributions from $B^+\to\bar{D}^{*0}\tau^+\nu_{\tau}$ and $B^+\to\bar{D}^{0}\tau^+\nu_{\tau}$. The histograms represent the MC-predicted background. []{data-label="pic-fit"}](fig_bdtau_1d.eps "fig:"){width="22.00000%"}
As a cross-check, we extract the signal yields from an extended unbinned maximum likelihood fit to one-dimensional distributions in $M_{\rm tag}$ and obtain consistent results with the two-dimensional fit. We also examine the distributions of variables used in the signal selection, applying all requirements except those that are related to the considered variable. In all cases the distributions are well-reproduced by the sum of signal and background components with normalizations fixed from the fit to the $(M_{\rm tag},P_{D^0})$ distribution.
The systematic uncertainties in the branching fractions are summarized in Table \[tab-sys\]. They include uncertainties in the total number of $B\bar{B}$ pairs, the effective efficiencies $\sum_{k}\epsilon_{k}\mathcal{B}_{k}$, and the signal-yield extractions. The systematic uncertainties associated with the effective efficiencies include errors in determination of the efficiencies for $B_{\rm tag}$ reconstruction and ($\bar{D}^{(*)0}d^+_{\tau}$) pair selection, coming from efficiencies of tracking, neutral particle reconstruction, particle identification, and from imperfect modeling of real processes. The uncertainty in the $B_{\rm tag}$ and part of the $B_{\rm sig}$ reconstruction efficiency is evaluated from data control samples with $B^+\to\bar{D}^{*0}\pi^+$ and $B^+\to\bar{D}^{0}\pi^+$ decays on the signal-side. The absolute normalizations of the data and MC control samples agree to within 13%. The difference, as well as uncertainties in the relative amounts of $D^{*0}-D^{0}$ cross-feeds are included in the systematic uncertainty of $B_{\rm tag}$ and $B_{\rm sig}$ reconstruction. The remaining uncertainties in the lepton identification and signal selection are estimated separately. The latter are determined by comparing MC and data distributions in the variables used for signal selection. The uncertainties due to the partial branching fractions $\mathcal{B}_k$ are taken from the errors quoted by the PDG [@PDG].
The systematic uncertainties in the signal yield originate from the background evaluation and from the PDF parameterizations of the signal and background components. The resulting error is evaluated from changes in the signal yields obtained from fits where the PDF parameters and the relative contributions of the background components are varied by $\pm 1\sigma$. All of the above sources of systematic uncertainties are combined together taking into account correlations between different decay chains. The combined systematic uncertainty is 13.9% for the $B^+ \to \bar{D}^{*0}\tau^+\nu_{\tau}$ mode and 15.2% for $B^+\to\bar{D}^{0}\tau^+\nu_{\tau}$.
Source $\bar{D}^{*0}\tau^+\nu_{\tau}$ $\bar{D}^{0}\tau^+\nu_{\tau}$
--------------------------------------------------- -------------------------------- -------------------------------
$N_{B\bar{B}}$ $\pm 1.4$% $\pm 1.4$%
Reconstruction of $B_{\rm tag}$ and $B_{\rm sig}$ $\pm 12.9$% $\pm 12.8$%
Lepton-id and signal selection $^{+1.5}_{-1.6}$% $^{+4.4}_{-4.5}$%
Shape of the signal PDF’s $\pm 2.5$% $\pm 6.0$%
Comb. and peaking backgrounds $\pm 3.3$% $\pm 2.7$%
Fitting procedure $\pm 0.8$% $\pm 1.5$%
Total $\pm 13.9$% $\pm 15.2$%
: Summary of the systematic uncertainties.
\[tab-sys\]
We include the effect of systematic uncertainties in the signal yields on the significances of the observed signals by convolving the likelihood function from the fit with a Gaussian systematic error distribution. The significances of the observed signals after including systematic uncertainties are 8.1$\sigma$ and 3.5$\sigma$ for the $B^+\to \bar{D}^{*0} \tau^+ \nu_{\tau}$ and $B^+\to \bar{D}^{0} \tau^+ \nu_{\tau}$ modes, respectively.
In conclusion, in a sample of 657$\times 10^6~B\bar{B}$ pairs we measure branching fractions $\mathcal{B}(B^+\to \bar{D}^{*0}\tau^+\nu_{\tau}) = (2.12^{+0.28}_{-0.27}
({\rm stat}) \pm 0.29({\rm syst}))$%, and $\mathcal{B}(B^+\to \bar{D}^{0}\tau^+\nu_{\tau}) = (0.77 \pm 0.22
({\rm stat}) \pm 0.12({\rm syst}))$%, which are consistent within experimental uncertainties with SM expectations [@hwang]. The result on $B^+\to \bar{D}^0\tau ^+\nu_{\tau}$ is the first evidence for this decay mode. We thank the KEKB group for excellent operation of the accelerator, the KEK cryogenics group for efficient solenoid operations, and the KEK computer group and the NII for valuable computing and SINET3 network support. We acknowledge support from MEXT, JSPS and Nagoya’s TLPRC (Japan); ARC and DIISR (Australia); NSFC (China); MSMT (Czechia); DST (India); MEST, NRF, NSDC of KISTI, and WCU (Korea); MNiSW (Poland); MES and RFAAE (Russia); ARRS (Slovenia); SNSF (Switzerland); NSC and MOE (Taiwan); and DOE (USA).
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|
{
"pile_set_name": "ArXiv"
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|
---
abstract: 'We consider the problem of quantifying information flow in interactive systems, modelled as finite-state transducers in the style of Goguen and Meseguer. Our main result is that if the system is deterministic then the information flow is either logarithmic or linear, and there is a polynomial-time algorithm to distinguish the two cases and compute the rate of logarithmic flow. To achieve this we first extend the theory of information leakage through channels to the case of interactive systems, and establish a number of results which greatly simplify computation. We then show that for deterministic systems the information flow corresponds to the growth rate of antichains inside a certain regular language, a property called the width of the language. In a companion work we have shown that there is a dichotomy between polynomial and exponential antichain growth, and a polynomial time algorithm to distinguish the two cases and to compute the order of polynomial growth. We observe that these two cases correspond to logarithmic and linear information flow respectively. Finally, we formulate several attractive open problems, covering the cases of probabilistic systems, systems with more than two users and nondeterministic systems where the nondeterminism is assumed to be innocent rather than demonic.'
author:
-
bibliography:
- 'mybib.bib'
title: |
Quantifying information flow in interactive systems\
[^1]
---
Quantified information flow, automata theory
Introduction
============
The notion of ‘noninterference’ was introduced by Goguen and Meseguer in [@goguen1982security]. It has long been recognised, however, that this condition—that no information can reach Bob about the actions of Alice—may in some circumstances be too strong. The field of quantitative information flow therefore aims to compute the *amount* of information that can reach Bob about Alice’s actions.
The contributions of this work are in two main parts. In the first part we extend the theory of information flow through channels developed by Smith, Palamidessi and many others to the case of interactive systems. In addition to basic definitions, we establish a number of results which greatly simplify computation. In particular, we show that it suffices to consider probability distributions over deterministic strategies for the two parties and that one of them may be assumed to adopt a pure deterministic strategy. We also show that if the system itself is deterministic then there is a possiblistic characterisation of the information flow which avoids quantifying over probability distributions altogether; this will be essential for the work of the second part.
In the second part we study determinstic interactive systems modelled as finite-state transducers in the style of Goguen and Meseguer. We define the information-flow capacity of such systems, before addressing the formidable technical problem of computing it. The key idea is to show that this can be reduced to a certain combinatorial problem on partially ordered sets. This problem is solved in a companion work [@mestel2018widths], with the consequence that we are able to show (Theorem \[thm:entcap\]) that for such systems there is a dichotomy between logarithmic and linear information flow, and a polynomial-time algorithm to distinguish the two cases. These two cases are naturally interpreted as ‘safe’ and ‘dangerous’ respectively, so we have shown that it is possible to distinguish genuinely dangerous information flow. We thereby accomplish a goal proposed by Ryan, McLean, Millen and Gligor at CSFW’01 in [@ryan2001noninterference].
Overview {#overview .unnumbered}
--------
In Section \[sec:channels\] we first recall some relevant theory on the information-flow capacity of channels, and improve a result of Alvim, Chatzikokolakis, McIver, Morgan, Palamidessi and Smith giving an upper bound on the ‘Dalenius leakage’ of a channel to an exact formula (Theorem \[thm:delan\]). We then consider interactive channels, where both parties may be required to make choices. We define leakage and information-flow capacity in this setting, and show that Bob’s strategy may be assumed to be deterministic (Corollary \[cor:bobdet\]). We show (Theorem \[thm:aliceposs\]) that in the case of deterministic channels we may take a possbilistic view of Alice’s actions, which we will find simplifies calculation considerably. Finally we show (Theorem \[thm:probstrat\]) that for systems which may involve multiple rounds of interaction it suffices to consider probability distributions over deterministic, rather than probabilistic, strategies.
In Section \[sec:detint\] we model deterministic interactive systems as finite-state transducers, and define their information-flow capacity. We then show how to reduce the problem of computing this to a problem involving only nondeterministic finite automata, and then to the combinatorial problem of computing the ‘width’ of the languages generated by the relevant automata. We observe that this problem is solved in [@mestel2018widths], and consequently conclude (Theorem \[thm:entcap\]) that there is a dichotomy between logarithmic and linear information flow, and a polynomial-time algorithm to distinguish the two cases. The structure of the sequence of reductions leading to this theorem is summarised in Figure \[fig:chapstruct\], and we illustrate the theory by applying it to a simple scheduler.
In Section \[sec:future\] we discuss some generalisations of the systems studied in main part of this work: namely nondeterministic systems, systems with more than two agents (which we observe encompasses the case of nondeterministic systems), and probabilistic systems. For the latter two we define the information-flow capacity and formulate the open problems of computing it. Finally in Section \[sec:relatedwork\] we discuss related work and in Section \[sec:conclusion\] we conclude.
Information-theoretic preliminaries {#sec:channels}
===================================
Leakage through channels {#sec:simpchan}
------------------------
We consider first the case of leakage through a channel from a space ${\mathcal{X}}$ of inputs to a space ${\mathcal{Y}}$ of outputs, corresponding to a situation in which the attacker is purely passive: Alice selects an input according to a known prior distribution $p_X$ and Bob (the attacker) receives an output according to the conditional distribution $p_{Y|X}$, which specifies the channel. How much information should we say that Bob has received?
The first work on quantified information flow adopted the classical information-theoretic notion of *mutual information* introduced by Shannon in the 1940s [@shannon1948communication]. However, Smith observed in [@smith2009foundations] the problems with this consensus definition. The essential problem is that mutual information represents in some sense the average number of bits of information leaked by the system. This is appropriate for the noisy coding theorem, where we are interested in the limit of many uses of the channel, but not for the case of information leakage where we assume that the adversary receives only one output (or a small number of outputs).
This means that, in the example used by Smith, a system which leaks the whole secret 1/8 of the time is seen as largely secure (because $H(X|Y)=\tfrac{7}{8}H(X)$), although it allows (for instance) a cryptographic key to be guessed 1/8 of the time. Smith addresses this by adopting the *min-entropy leakage*, defined[^2] as the expected value of the increase in the probability of guessing the input upon observing the output $y$: $${\mathcal{L}}_\infty(X,Y) = \log \mathbb{E}_{y\sim Y} \frac{\sup_{x\in{\mathcal{X}}}
p_{X|Y}(x|y)}{\sup_{x\in{\mathcal{X}}} p_X(x)}.$$
Given a channel ${\mathcal{C}}$ specified by a matrix of conditional probabilities $p_{Y|X}$, we may be interested in its *capacity*, which is the maximum value of the leakage over all possible priors $p_X$: $${\mathcal{L}}_\infty({\mathcal{C}}) = \sup_{(X,Y)\sim {\mathcal{C}}} {\mathcal{L}}_\infty(X,Y),$$ where the notation $(X,Y)\sim {\mathcal{C}}$ means that $X$ and $Y$ are random variables compatible with ${\mathcal{C}}$; that is, that the conditional probabilities $p_{Y|X}$ (where defined) correspond to the matrix defining ${\mathcal{C}}$.
In [@alvim2012measuring], Alvim, Chatzikolakis, Palamidessi and Smith generalise this definition to the notion of *$g$-leakage*, in which Bob makes a guess drawn from a set ${\mathcal{W}}$, and receives a payoff according to the function $g:{\mathcal{W}}\times{\mathcal{X}}\rightarrow [0,1]$. The leakage with respect to $g$ is then $${\mathcal{L}}_g(X,Y) = \log \mathbb{E}_{y\sim Y} \frac{\sup_{w\in {\mathcal{W}}} \sum_{x\in {\mathcal{X}}}
p_{X|Y}(x|y)g(w,x)}{\sup_{w\in {\mathcal{W}}} \sum_{x\in {\mathcal{X}}} p_{X}(x)g(w,x)}.$$
Once again, we can define the capacity of a channel ${\mathcal{C}}$: $${\mathcal{L}}_g({\mathcal{C}}) = \sup_{(X,Y)\sim {\mathcal{C}}} {\mathcal{L}}_g(X,Y).$$
In Theorem 5.1 of [@alvim2012measuring], the authors prove the so-called ‘miracle’ theorem, which states that for any channel ${\mathcal{C}}$ and any gain function $g$ we have that the $g$-capacity is at most the min-entropy capacity: $${\mathcal{L}}_g({\mathcal{C}}) \leq {\mathcal{L}}_\infty({\mathcal{C}}).$$
However, it may be the case that the secret which Bob is trying to guess is not Alice’s input but some other secret value (a cryptographic key, say) which is related to $x$ in some known but unspecified way. We may be interested in bounding the possible gain for Bob for any possible secret and any (probabilistic) relationship to the choice of $x$; this is sometimes known as the ‘Dalenius leakage’, after a desideratum attributed to T. Dalenius by Dwork in [@dwork2006differential]. We may therefore define $${\mathcal{L}}_D(X,Y) = \sup_{Z \in \mathcal{D}} {\mathcal{L}}_\infty(Z,Y),$$ where ${\mathcal{D}}$ is the collection of random variables $Z$ such that $Z\rightarrow X
\rightarrow Y$ forms a Markov chain (that is, $p_{X,Y,Z}(x,y,z) =
p_Z(z)p_{X|Z}(x|z) p_{Y|X}(y|x)$).
In [@alvim2014additive], Alvim, Chatzikokolakis, McIver, Morgan, Palamidessi and Smith give an upper bound for the Dalenius leakage: they show in Corollary 23 that for any Markov chain $Z\rightarrow X\rightarrow Y$ we have that $$\sup_g {\mathcal{L}}_g(Z,Y) \leq \sup_g {\mathcal{L}}_g(Y,X),$$ where the suprema are taken over gain functions $g$. Hence in particular we have that ${\mathcal{L}}_\infty(Z,Y) \leq \sup_g {\mathcal{L}}_g(Y,X)$. But ${\mathcal{L}}_g(Y,X) \leq
{\mathcal{L}}_g({\mathcal{C}})$, where ${\mathcal{C}}$ is any channel such that $(X,Y)\sim {\mathcal{C}}$, and by the miracle theorem we have that ${\mathcal{L}}_g({\mathcal{C}}) \leq {\mathcal{L}}_\infty({\mathcal{C}})$, and hence we have that $${\mathcal{L}}_D(X,Y) \leq {\mathcal{L}}_\infty({\mathcal{C}}).$$
We are able to improve this to a precise formula for the Dalenius leakage between two random variables.
\[thm:delan\] Let $X,Y$ be any discrete random variables. Then $$\begin{aligned}
{\mathcal{L}}_D(X,Y) &= \log \mathbb{E}_{y\sim Y} \sup_{x\in {\mathcal{X}}} \frac{p_{Y|X}(y|x)}
{p_Y(y)}\\
&= \log \sum_{y\in {\mathcal{Y}}^+} \sup_{x \in {\mathcal{X}}} p_{Y|X} (y|x),\end{aligned}$$ where ${\mathcal{Y}}^+\subseteq {\mathcal{Y}}$ is the set of $y\in {\mathcal{Y}}$ such that $p_Y(y) > 0$.
We may assume without loss of generality that $p_Y(y) > 0$ for all $y\in Y$ (otherwise redefine ${\mathcal{Y}}$ to be the set of values on which $p_Y$ is supported).
For the upper bound, we recall that Braun, Chatzikokolakis and Palamidessi observe in Proposition 5.11 of [@braun2009quantitative] that there is a simple formula for the min-entropy capacity of a channel ${\mathcal{C}}$ defined by matrix $p_{Y|X}$: $$\label{eq:minentcap} {\mathcal{L}}_\infty({\mathcal{C}}) = \log \sum_{y\in {\mathcal{Y}}}
\sup_{x\in {\mathcal{X}}} p_{Y|X}(y|x).$$
This is proved in [@braun2009quantitative] for random variables with finite image. For general discrete random variables, the upper bound on ${\mathcal{L}}_\infty({\mathcal{C}})$ is obtained by replacing $\max$ with $\sup$ as appropriate, but the lower bound requires a little more care since it is given by considering the uniform distribution on ${\mathcal{X}}$. However, the lower bound can be recovered for infinite ${\mathcal{X}}$ by considering the uniform distribution on the first $k$ elements of ${\mathcal{X}}$ and taking the limit as $k\rightarrow \infty$.
The upper bound on ${\mathcal{L}}_D(X,Y)$ is immediate from by taking ${\mathcal{C}}$ to be any channel such that $(X,Y)\sim {\mathcal{C}}$.
For the lower bound, suppose that ${\mathcal{X}}=\{x_1,x_2,x_3,\ldots\}$, and define the function $f:[0,1)\rightarrow {\mathcal{X}}$ by $f(\xi) = x_k$ if $$\sum_{i=1}^{k-1} p_X(x_i) < \xi \leq \sum_{i=1}^k p_X(x_i).$$ Note that for each $x\in {\mathcal{X}}$ we have that $p_X(x) = \mu(f^{-1}(x))$, where $\mu$ is the Borel measure.
For each positive integer $n$, let ${\mathcal{Z}}_n = \{0,1,2,\ldots,2^n-1\}$ and let $Z_n$ be a random variable taking values in ${\mathcal{Z}}_n$, with $$p_{X,Z_n} (x,z) = \mu\left(f^{-1}(x)\cap \left[\frac{z}{2^n},\frac{z+1}{2^n}
\right]\right).$$ Note that by the previous observation we have $\sum_{z\in {\mathcal{Z}}_n} p_{X,Z_n}(x,z)
= \mu(f^{-1}(x)) = p_X(x)$ as required. Note also that we have that $p_Z(z) =
\sum_{x\in {\mathcal{X}}} p_{X,Z_n}(x,z) = 2^{-n}$.
Now we have $$\begin{aligned}
{\mathcal{L}}_\infty(Z_n,Y) &= \log\sum_{y\in {\mathcal{Y}}} p_Y(y) \frac{\max_z
p_{Z_n|Y}(z|y)}{\max_z p_{Z_n}(z)} \\
&= \log\sum_{y\in {\mathcal{Y}}} p_Y(y) 2^n \max_z p_{Z_n|Y} (z|y).\end{aligned}$$ We claim that $$\label{eq:limzy}
\lim_{n\rightarrow \infty} p_Y(y) 2^n \max_{z} p_{Z_n|Y}(z|y)
\geq \sup_x p_{Y|X} (y|x)$$ for all $y\in {\mathcal{Y}}$.
Indeed, by Bayes’ theorem we have $$\begin{aligned}
p_Y(y)p_{Z_n|Y}(z|y)&=p_{Z_n}(z)p_{Y|Z_n}(y|z) \\
&= 2^{-n} \sum_{x\in {\mathcal{X}}} p_{Y|X}(y|x) p_{X|Z_n}(x|z).\end{aligned}$$ Let $x\in {\mathcal{X}}$ be arbitrary. For sufficiently large $n$ we have that $p_{X|Z_n}(x|z) = 1$ for some $z\in {\mathcal{Z}}_n$, and hence for this $z$ we have that $p_Y(y)2^np_{Z_n|Y}(z|y) \geq p_{Y|X}(y|x)$, proving the claim. Summing (\[eq:limzy\]) over all $y\in {\mathcal{Y}}$ and rearranging gives $$\lim_{n\rightarrow \infty} {\mathcal{L}}_\infty(Z_n,Y) \geq \log \sum_{y\in {\mathcal{Y}}} \max_x
p_{Y|X} (y|x),$$ as required.
Interactive channels {#sec:intchan}
--------------------
More generally, we will be interested in *interactive channels*, where an input is chosen by both Alice and Bob, and the system then produces an output to Bob. This means that the space ${\mathcal{X}}$ is of the form ${\mathcal{X}}_A\times {\mathcal{X}}_B$, where the spaces ${\mathcal{X}}_A$ and ${\mathcal{X}}_B$ are the spaces of inputs for Alice and Bob respectively, and the interactive channel ${\mathcal{C}}$ is defined by the matrix of conditional probabilites $p_{Y|X_A,X_B}$.
Note that if the system involves a sequence of outputs and actions by Alice and Bob then the ‘inputs’ $x_A$ and $x_B$ will in fact represent *strategies* for Alice and Bob, determining their actions on the basis of the outputs they have seen so far (in general these may be probabilistic, but we will see in Section \[sec:probstrat\] that in fact it is sufficient to consider only deterministic strategies). We will write $((X_A,X_B),Y)\sim {\mathcal{C}}$ to mean that the random variables $X_A,X_B$ and $Y$ are consistent with the channel ${\mathcal{C}}$: that is, that $X_A$ and $X_B$ are independent and the matrix $p_{Y|X_A,X_B}$ corresponds with the matrix defining ${\mathcal{C}}$.
We can once again define the min-entropy leakage as the expected increase in Bob’s probability of guessing the value of the input based on having seen the output: $$\begin{aligned}
&{\mathcal{L}}_\infty ((X_A,X_B),Y) \\
&\qquad= \log \mathop{{\mathbb{E}}}_{x_B\sim X_B, y\sim Y} \frac{\sup_{x_A\in
{\mathcal{X}}_A} p_{X_A|X_B,Y}(x_A|x_B,y)} {\sup_{x_A\in {\mathcal{X}}_A} p_{X_A}(x_A)}\\
&\qquad= \log {\mathbb{E}}_{x_B\sim X_B} 2^{{\mathcal{L}}_\infty(X_A,Y|X_B=x_B)}.\end{aligned}$$ Again the capacity of the channel is defined as the maximum leakage over all possible priors $p_{X_A}$ and $p_{X_B}$. $${\mathcal{L}}_\infty({\mathcal{C}}) = \sup_{((X_A,X_B),Y)\sim {\mathcal{C}}} {\mathcal{L}}_\infty((X_A,X_B),Y).$$
It appears at first glance that calculating ${\mathcal{L}}_\infty({\mathcal{C}})$ may in general be highly intractible: we have to quantify over mixed strategies (that is over probability distributions on strategies) for Alice and Bob. However, it turns out that we may assume without loss of generality that Bob chooses a pure strategy.[^3] Indeed, this holds not only for the choices we have made but for all reasonable such choices.
Specifically, we chose a leakage measure, namely ${\mathcal{L}}_\infty$, and a method of averaging the leakage over different values of $x_B$, namely taking $\log
{\mathbb{E}}_{x_B} 2^{\mathcal{L}}$. The following proposition shows that we may assume a pure strategy for Bob for any choice of leakage measure, and any method of averaging which is ‘reasonable’ in the sense that if the distribution of leakage is constant with value $x$ then the value is $x$, and also that the value of a weighted sum of leakage distributions cannot be more than the maximum value of the distributions making up the sum (this last property is known as ‘quasiconvexity’).
\[prop:genpure\] Let ${\mathcal{L}}:\mathbb{D}({\mathcal{X}}_A\times {\mathcal{Y}}) \rightarrow \mathbb{R}$ (the ‘leakage function’) be any function and let $\phi:\mathbb{D}(\mathbb{R}) \rightarrow
\mathbb{R}$ (the ‘averaging function’) be any function such that if $X\in
\mathbb{D}(\mathbb{R})$ is constant $x$ then $\phi(X) = x$ and for any $X_1,X_2,\ldots \in \mathbb{D}(\mathbb{R})$ and any $\rho_1,\rho_2,\ldots$ with $\sum_i \rho_i = 1$ we have $$\label{eq:phinice}\phi\left(\sum_i \rho_i X_i\right) \leq \sup_i
\phi(X_i).$$ Let $${\mathcal{L}}_\phi({\mathcal{C}}) = \sup_{((X_A,X_B),Y)\sim {\mathcal{C}}} \phi({\mathcal{L}}(X_A,Y|X_B)).$$ Then we have $${\mathcal{L}}_\phi({\mathcal{C}}) = \sup_{x_B\in {\mathcal{X}}_B} \sup_{(X_A,x_B,Y) \sim {\mathcal{C}}}
{\mathcal{L}}(X_A,Y),$$ where the notation $(X_A,x_B,Y)$ means the distribution with $p_{X_B}(x_B) = 1$, and in the above $\mathbb{D}({\mathcal{X}})$ means the space of probability distributions over the set ${\mathcal{X}}$.
Suppose that $(X_A,X_B,Y)\sim {\mathcal{C}}$. We have $$\phi({\mathcal{L}}(X_A,Y|X_B)) = \phi\left(\sum_{x_B} p_{X_B}(x_B) {\mathcal{L}}(X_A,Y|X_B=x_B)
\right).$$ Hence for any $\epsilon > 0$, by (\[eq:phinice\]) there exists some $x_B$ such that $$\begin{aligned}
\phi({\mathcal{L}}(X_A,Y|X_B=x_B)) &= {\mathcal{L}}(X_A,Y|X_B=x_B) \\
&\geq \phi({\mathcal{L}}(X_A,Y|X_B)) - \epsilon.\end{aligned}$$ Hence we have $$\sup_{x_B\in{\mathcal{X}}_B} {\mathcal{L}}(X_A,Y|X_B=x_B) = \phi({\mathcal{L}}(X_A,Y|X_B)),$$ establishing the result.
The min-entropy capacity is a special case of this result, with ${\mathcal{L}}={\mathcal{L}}_\infty$ and $\phi(X) = \log{\mathbb{E}}_{x\sim X} 2^x$.
\[cor:bobdet\] Let ${\mathcal{C}}$ be an interactive channel. Then we have $${\mathcal{L}}_\infty({\mathcal{C}}) = \sup_{x_B\in{\mathcal{X}}_B} {\mathcal{L}}_\infty({\mathcal{C}}|X_B=x_B).$$
Deterministic channels
----------------------
For the channels we have considered above, once the inputs from Alice and Bob are fixed we obtain a probability distribution on outputs. However, for some systems it may be that the output is not probabilistic, but is determined by the values of the inputs; we will call such a channel deterministic. More concretely, an interactive channel ${\mathcal{C}}$ defined by the matrix $p_{Y|X_A,X_B}$ is *deterministic* if for all $x_A,x_B,y$ we have $$p_{Y|X_A,X_B}(y|x_A,x_B) \in \{0,1\}.$$
If ${\mathcal{C}}$ is deterministic then the computation of ${\mathcal{L}}_\infty({\mathcal{C}})$ simplifies considerably, because it turns out that we can take a purely possibilistic view of Alice’s actions and avoid any quantification over probability distributions.
\[thm:aliceposs\] Let ${\mathcal{C}}$ be a deterministic interactive channel. Then $$\begin{gathered}
{\mathcal{L}}_\infty({\mathcal{C}}) = \sup_{x_B\in {\mathcal{X}}_B} \log \left| \left\{y\in{\mathcal{Y}}\middle\vert \exists
x_A\in {\mathcal{X}}_A:\right.\right.\\
\left.\left. p_{Y|X_A,X_B}(y|x_A,x_B)=1\right\}\right|.\end{gathered}$$
By Corollary \[cor:bobdet\], it suffices to prove that $$\begin{gathered}
{\mathcal{L}}_\infty({\mathcal{C}}|X_B=x_B) = \log \left| \left\{y\in{\mathcal{Y}}\middle\vert \exists
x_A\in {\mathcal{X}}_A:\right.\right.\\
\left.\left. p_{Y|X_A,X_B}(y|x_A,x_B)=1\right\}\right|.\end{gathered}$$ By the formula for ${\mathcal{L}}_\infty({\mathcal{C}})$ from [@braun2009quantitative] (recalled as (\[eq:minentcap\]) in the proof of Theorem \[thm:delan\]) we have $$\begin{gathered}
{\mathcal{L}}_\infty({\mathcal{C}}|X_B=x_B) = \log \sum_{y\in {\mathcal{Y}}} \max_{x_A\in {\mathcal{X}}_A}
p_{Y|X_A,X_B} (y|x_A,x_B) \\
= \log \left| \left\{y\in{\mathcal{Y}}\middle\vert \exists
x_A\in {\mathcal{X}}_A: p_{Y|X_A,X_B}(y|x_A,x_B)=1\right\}\right|,\end{gathered}$$ since ${\mathcal{C}}$ is deterministic and so $p_{Y|X_A,X_B}(y|x_A,x_B) \in \{0,1\}$.
Theorem \[thm:aliceposs\] essentially says that it suffices to count the maximum number of outputs that can be seen by Bob, consistently with his choice of strategy. The corresponding result for non-interactive channels is Theorem 1 of [@smith2009foundations].
Probabilistic vs deterministic strategies {#sec:probstrat}
-----------------------------------------
We observed in Section \[sec:intchan\] that the ‘channel’ paradigm is able to model systems involving many rounds of interaction, because we can take Alice and Bob’s inputs to be strategies, determining the actions they will take at each step of the interaction. At each step, Alice (respectively Bob) will have observed a trace of the interaction thus far drawn from a set $T$, and must select an action drawn from a set $\Sigma$. To specify a randomised strategy for Alice or Bob, we must therefore specify for each $t\in T$ a probability distribution over $\Sigma$, so the set of strategies is the set of maps $T\rightarrow {\mathbb{D}}\Sigma$.
In this section we will show that in fact it suffices to consider only deterministic strategies for Alice and Bob. The intuition behind this is fairly straightforward: given a probabilistic strategy, we could imagine that any necessary coins are tossed before the execution begins, which gives a probability distribution over deterministic strategies. This changes nothing except that it allows Bob to see how the random choices made by his strategy were resolved, but this only gives him more information and so does not affect the information flow capacity. To avoid technical measurability issues we will assume that the sets $T$ and $\Sigma$ are finite.
Let $T$ be a finite set of *traces* and $\Sigma$ a finite set of *actions*. A *strategy* over $T$ and $\Sigma$ is a function $f:T\rightarrow
{\mathbb{D}}(\Sigma)$. The set of strategies over $T$ and $\Sigma$ is denoted ${\mathcal{S}}_{T,\Sigma}$.
A strategy $f\in {\mathcal{S}}_{T,\Sigma}$ is *determinsitic* if we have $$f(t)(x) \in \{0,1\}$$ for all $t\in T$ and $x\in \Sigma$. We write ${\mathcal{D}}_{T,\Sigma} \subset
{\mathcal{S}}_{T,\Sigma}$ for the set of deterministic strategies over $T$ and $\Sigma$.
In the execution itself, these strategies will be executed and particular actions chosen. The output $y\in {\mathcal{Y}}$ displayed to Bob is then a function (which may be probabilistic) of the choices that were made; the system is defined by this function, which is a map from pairs of functions $T \rightarrow \Sigma$ (the choices made by Alice and Bob respectively) to distributions over ${\mathcal{Y}}$. Note that it may be that in some executions not all traces are actually presented to Alice and Bob for decision; this can be represented by the choices made in response to those traces being ignored, so no generality is lost by considering total functions $T\rightarrow
\Sigma$ (similarly the trace-sets relevant to Alice and Bob may be distinct, but this can be represented by ignoring the choices made by Alice on Bob’s traces and vice versa).
We write $\Sigma^T$ for the set of functions $T\rightarrow \Sigma$; the probability that a particular function is realised by a particular strategy can be computed by multiplying the probabilities for each decision (note that nothing is lost by assuming independence: if Alice and Bob are supposed to know about previous choices they have made then this can be encoded in the traces).
\[def:realise\] Let $f\in {\mathcal{S}}_{T,\Sigma}$ be any strategy and $g\in \Sigma^T$. The probability that $f$ *realises* $g$, written $f(g)$, is given by $$f(g) = \prod_{t\in T} f(t)(g(t)).$$
\[def:output\] Let $\phi:\Sigma^T\times \Sigma^T \rightarrow {\mathbb{D}}{\mathcal{Y}}$ be any map, and let ${\mathcal{X}}_A$ and ${\mathcal{X}}_B$ be any subsets of ${\mathcal{S}}_{T,\Sigma}$. The *interactive channel determined by $\phi, {\mathcal{X}}_A$ and ${\mathcal{X}}_B$*, denoted ${\mathcal{C}}_{\phi,{\mathcal{X}}_A,{\mathcal{X}}_B}$, is determined by the matrix of conditional probabilities $$p_{Y|X_A,X_B}(y|f_A,f_B) = \sum_{g_A,g_B\in \Sigma^T}
f_A(g_A)f_B(g_B) \phi(g_A,g_B)(y).$$
We observe that if the function $\phi$ defining the system is deterministic, and if Alice and Bob use only deterministic strategies, then the channel produced is a deterministic interactive channel in the sense of the previous section, such that Theorem \[thm:aliceposs\] applies to it.
Suppose that $\phi(g,g')(y)\in \{0,1\}$ for every $g,g'\in \Sigma^T$ and $y\in
{\mathcal{Y}}$. Then ${\mathcal{C}}_{\phi,{\mathcal{D}}_{T,\Sigma},{\mathcal{D}}_{T,\Sigma}}$ is a deterministic interactive channel.
If $f_A,f_B\in {\mathcal{D}}_{T,\Sigma}$ then $f_A(g),f_B(g) \in \{0,1\}$ for all $g\in
\Sigma^T$. Hence if $\phi(g_A,g_B,y) \in \{0,1\}$ for all $g_A,g_B,y$ then we have $p_{Y|X_A,X_B}(y|f_A,f_B) \in \{0,1\}$ for all $f_A,f_B,y$, as required.
The main theorem of this section is that in fact it suffices to conisder only deterministic strategies for Alice and Bob.
\[thm:probstrat\] Let $\Sigma$ and $T$ be any finite sets, ${\mathcal{Y}}$ any set and $\phi:\Sigma^T \times
\Sigma^T \rightarrow {\mathbb{D}}{\mathcal{Y}}$ be any map. Then we have $${\mathcal{L}}_\infty\left( {\mathcal{C}}_{\phi,{\mathcal{S}}_{T,\Sigma},{\mathcal{S}}_{T,\Sigma}} \right)
= {\mathcal{L}}_\infty\left( {\mathcal{C}}_{\phi,{\mathcal{D}}_{T,\Sigma},{\mathcal{D}}_{T,\Sigma}} \right).$$
The lower bound is immediate: since whenever $((X_A,X_B),Y)\sim
{\mathcal{C}}_{\phi,{\mathcal{D}}_{T,\Sigma},{\mathcal{D}}_{T,\Sigma}}$ then also $((X_A,X_B),Y) \sim
{\mathcal{C}}_{\phi,{\mathcal{S}}_{T,\Sigma},{\mathcal{S}}_{T,\Sigma}}$, we must have (writing ${\mathcal{C}}_{{\mathcal{S}}}$ and ${\mathcal{C}}_{{\mathcal{D}}}$ resectively for the two channels in the statement of the theorem) $$\begin{aligned}
{\mathcal{L}}_\infty({\mathcal{C}}_{{\mathcal{S}}}) &=
\sup_{((X_A,X_B),Y)\sim {\mathcal{C}}_{\mathcal{S}}}
{\mathcal{L}}_\infty((X_A,X_B),Y) \\
&\geq \sup_{((X_A,X_B),Y)\sim {\mathcal{C}}_{\mathcal{D}}}
{\mathcal{L}}_\infty((X_A,X_B),Y) \\
&= {\mathcal{L}}_{\infty}({\mathcal{C}}_{\mathcal{D}}).\end{aligned}$$
For the upper bound, let $X_A$ and $X_B$ be any independent ${\mathcal{S}}_{T,\Sigma}$-valued random variables. We will first show that without loss of generality we may assume that $X_B$ is supported only on ${\mathcal{D}}_{T,\Sigma}$. By Corollary \[cor:bobdet\] it suffices to show this where $X_B$ is a point distribution,[^4] so say that $X_B$ takes the value $f_B\in {\mathcal{S}}_{T,\Sigma}$.
Define the random variable $X_B'$ to be supported only on ${\mathcal{D}}_{T,\Sigma}$, and for $f \in {\mathcal{D}}_{T,\Sigma}$ let $$p_{X_B'}(f) = f_B(\widetilde{f}),$$ where $\widetilde{f}$ is the function $T\rightarrow \Sigma$ induced by $f$: that is, $\widetilde{f}(t)$ is the unique element $x$ of $\Sigma$ such that $f(t)(x)=1$.
Note that by Definitions \[def:realise\] and \[def:output\] we have that $(X_A,X_B)$ and $(X_A,X_B')$ induce the same output distribution $Y$, and so it suffices to prove that for each $y\in {\mathcal{Y}}$ we have $$\begin{gathered}
{\mathbb{E}}_{f'_B\sim X_B'} \sup_{f_A\in {\mathcal{S}}_{T,\Sigma}} p_{X_A|X_B',Y} (f_A|f_B',y)
\geq \\ \sup_{f_A\in {\mathcal{S}}_{T,\Sigma}} p_{X_A|X_B,Y} (f_A|f_B,y).\end{gathered}$$ Now on the one hand we have $$\begin{gathered}
\label{eq:xbbig}
{\mathbb{E}}_{f'_B\sim X_B'} \sup_{f_A\in {\mathcal{S}}_{T,\Sigma}} p_{X_A|X_B',Y} (f_A|f_B',y) = \\
\sum_{f_B'\in {\mathcal{D}}_{T,\Sigma}} f_B(\widetilde{f_B'}) \sup_{f_A \in {\mathcal{S}}_{T,\Sigma}}
p_{X_A|X_B',Y} (f_A|f_B',y).\end{gathered}$$ On the other hand we have $$\begin{gathered}
\label{eq:xbsmall}
\sup_{f_A\in {\mathcal{S}}_{T,\Sigma}} p_{X_A|X_B,Y}(f_A|f_B,y) = \\
\sup_{f_A\in {\mathcal{S}}_{T,\Sigma}} \sum_{f_B'\in {\mathcal{D}}_{T,\Sigma}} f_B(\widetilde{f_B'})
p_{X_A|X'_B,Y} (f_A|f_B',y).\end{gathered}$$ Plainly $\eqref{eq:xbbig} \geq \eqref{eq:xbsmall}$, establishing the result.
We now show that we may also assume that $X_A$ is supported only on ${\mathcal{D}}_{T,\Sigma}$, and again by Corollary \[cor:bobdet\] it suffices to show this where $X_B$ takes only a single value, say $f_B\in {\mathcal{D}}_{T,\Sigma}$. By the min-entropy capacity formula conditioned on $X_B=f_B$ it suffices to show that for every $y\in {\mathcal{Y}}$ we have $$\sup_{f_A\in {\mathcal{S}}_{T,\Sigma}} p_{Y|X_A,X_B}(y|f_A,f_B) \leq
\max_{f_A\in {\mathcal{D}}_{T,\Sigma}} p_{Y|X_A,X_B}(y|f_A,f_B).$$ But this is straightforward: indeed, for any $f_A\in {\mathcal{S}}_{T,\Sigma}$ we have $$\begin{aligned}
p_{Y|X_A,X_B}(y|f_A,f_B) &= \sum_{f_A'\in {\mathcal{D}}_{T,\Sigma}} f_A(\widetilde{f_A'})
\phi(\widetilde{f_A'},\widetilde{f_B},y) \\
&\leq \max_{f_A'\in {\mathcal{D}}_{T,\Sigma}} \phi(\widetilde{f_A'},\widetilde{f_B},y) \\
&= \max_{f_A'\in {\mathcal{D}}_{T,\Sigma}} p_{Y|X_A,X_B}(y|f_A',f_B,y),\end{aligned}$$ as required.
Deterministic interactive systems {#sec:detint}
=================================
Finite-state transducers
------------------------
We will model deterministic interactive systems as deterministic finite-state transducers. Whereas Goguen and Meseguer in [@goguen1982security] modelled such systems as ‘state-observed’ transducers, we will consider the more general notion of ‘action-observed’ transducers (see the work of van der Meyden and Zhang in [@vandermeyden2007comparison] for further discussion of the relationship between noninterference properties in these two models; this model is also essentially equivalent to the notion of ‘Input-Output Labelled Transition System’ used by Clark and Hunt in the non-quantitative setting in [@clark2009noninterference]).
A *deterministic finite-state transducer* (DFST) is a 7-tuple ${\mathcal{T}}=
(Q,q_0,F,\Sigma,\Gamma,\delta,\sigma)$, where $Q$ is a finite set of *states*, $q_0\in Q$ is the *initial state*, $F\subseteq Q$ is the set of *accepting states*, $\delta:Q\times \Sigma\rightarrow Q$ is the *transition function* and $\sigma:Q\times \Sigma\rightarrow \Gamma\cup
\{\epsilon\}$ is the *output function*.
A pair $(a_1a_2\ldots a_k, b_1b_2\ldots b_l)\in \Sigma^*\times \Gamma^*$ is accepted by ${\mathcal{T}}$ if there exists a sequence of states $q_1\ldots q_k\in Q^*$ such that $q_k\in F$, for every $0\leq i < k$ we have $q_{i+1} = \delta(q_i,
a_{i+1})$ and $b_1\ldots b_l = \sigma(q_0,a_{1}) \sigma(q_1,a_2) \ldots
\sigma(q_{k-1},a_k)$. We will write $L({\mathcal{T}})$ for the subset of $\Sigma^*\times \Gamma^*$ accepted by ${\mathcal{T}}$; such a set is a deterministic finite-state *transduction*, which we will also abbreviate by DFST.
This definition is not quite convenient for our purposes, because we assume that the agents are able to observe the passage of time. Hence even at a timestep where the machine does nothing, there should be a record in the trace of the fact that time has passed. We ensure this by requiring that there should be an output at each step, and apply the non-standard term ‘synchronised’ to describe this property (such a transducer is also sometimes called ‘letter-to-letter’).
A DFST ${\mathcal{T}}= (Q, q_0, F, \Sigma, \Gamma, \delta, \sigma)$ is *synchronised* if $\sigma(Q,\Sigma)\subseteq \Gamma$ (that is, we do not have $\sigma(q,a)=\epsilon$ for any $q\in Q$ and $a\in \Sigma$). In this case we say that ${\mathcal{T}}$ is a synchronised deterministic finite-state transducer (SDFST).
Note that this definition almost corresponds with the original definition of a *Mealy machine* ([@mealy1955method]), except that we allow for a set of final states $F\neq Q$. It is clear that if ${\mathcal{T}}$ is synchronised then $(a_1\ldots a_k,b_1\ldots b_l)\in \Sigma^*\times \Gamma^*$ is accepted by ${\mathcal{T}}$ only if $l=k$. We shall therefore apply the ‘zip’ operation and view ${\mathcal{T}}$ as accepting elements of $(\Sigma\times \Gamma)^*$.
We are interested in SDFSTs of a special kind, representing the fact that the system communicates separately with Alice and Bob. We will consider SDFSTs whose input and output alphabets $\Sigma$ and $\Gamma$ are of the form $\Sigma_A\times \Sigma_B$ and $\Gamma_A\times \Gamma_B$ respectively. The pairs $(\Sigma_A,\Gamma_A)$ and $(\Sigma_B,\Gamma_B)$ represent the input and output alphabets used for communication with Alice and Bob respectively.
A simple example of such a transducer is the system which simply relays messages between the two agents (with $\Sigma_A=\Sigma_B=\{a,b\}$ and $\Gamma_A=\Gamma_B
=\{a',b'\}$). This is shown in Figure \[fig:relay\].
(q\_0) [$q_0$]{}; (q\_0) edge \[loop above\] node [$(x,y)|(y',x') \forall x,y\in \{a,b\}$]{} ();
Strategies and information flow
-------------------------------
In order to apply the framework of the previous section, we must define the spaces ${\mathcal{X}}_A, {\mathcal{X}}_B$ of strategies for Alice and Bob, the space $Y$ of outcomes visible to Bob, and the matrix $p_{Y|X_A,X_B}$ governing which outcomes occur. Since we are considering deterministic specifications, the matrix $p_{Y|X_A,X_B}$ will be 0-1-valued.
Alice and Bob must each decide on an action based on the trace they have seen thus far, so a strategy for Alice is a function $$x_A:(\Sigma_A\times\Gamma_A)^* \rightarrow \Sigma_A,$$ and similarly a strategy for Bob is a function $x_B:(\Sigma_B\times \Gamma_B)^*
\rightarrow \Sigma_B$.
Recall that by Theorem \[thm:probstrat\] it suffices to consider deterministic strategies for Alice and Bob: in the language of Section \[sec:probstrat\], we have $T=(\Sigma_A\times\Gamma_A)^*\cup (\Sigma_B\times
\Gamma_B)^*$ and $\Sigma = \Sigma_A\cup \Sigma_B$. We will have that the function $\phi(g_A,g_B,y)$ ignores the values of $g_A$ on $(\Sigma_B\times
\Gamma_B)^*$ and the values of $g_B$ on $(\Sigma_A\times \Gamma_A)^*$, and treats all elements of $\Sigma_B$ in the image of $g_A$ as equivalent to some fixed $a\in \Sigma_A$ and similarly all elements of $\Sigma_A$ in the image of $g_B$ as equivalent to some fixed $b\in \Sigma_B$. By Theorem \[thm:probstrat\] it suffices to consider deterministic strategies for Alice and Bob and so it is more convenient to refer to the sets of deterministic strategies directly as ${\mathcal{X}}_A$ and ${\mathcal{X}}_B$, and to $\phi(x_A,x_B)(y)$ directly as the channel matrix $p_{Y|X_A,X_B}(y|x_A,x_B)$.
Given an SDFST ${\mathcal{T}}$, and strategies $x_A$ and $x_B$ for Alice and Bob respectively, what output or outputs can be shown to Bob? We consider first the case where $F=Q$, postponing for later the issues that arise when $F\subsetneq Q$.
\[def:cons\]We will say that a word $$\begin{gathered}
w=((a_1,a_1'),(b_1,b_1'))\ldots ((a_k,a_k'),(b_k,b_k')) \\ \in (\Sigma\times
\Gamma)^* = \left((\Sigma_A\times \Sigma_B)\times (\Gamma_A \times
\Gamma_B)\right)^*\end{gathered}$$ (so $a_i\in \Sigma_A, a_i'\in \Sigma_B, b_i\in \Gamma_A$ and $b_i'\in \Gamma_B$) is *consistent* with SDFST ${\mathcal{T}}$ and strategies $x_A, x_B$ if
(i) $w\in L({\mathcal{T}})$, and
(ii) \[it:strats\] for every $1\leq i \leq k$ we have $$a_i =
x_A((a_1,b_1),\ldots, (a_{i-1}, b_{i-1})),$$ and $$a_i' = x_B((a_1',b_1'),\ldots,
(a'_{i-1}, b'_{i-1})).$$
A word $(a'_1,b'_1)\ldots (a'_k,b'_k)\in (\Sigma_B\times \Gamma_B)^*$ is consistent with ${\mathcal{T}}, x_A$ and $x_B$ if there exist $a_1,\ldots,a_k \in
\Sigma_A$ and $b_1,\ldots,b_k\in \Gamma_A$ such that $(((a_1,a_1'),(b_1,b_1'))\ldots \allowbreak ((a_k,a_k'),(b_k,b_k')))$ is consistent with ${\mathcal{T}}, x_A$ and $x_B$.
We will sometimes refer to limb (\[it:strats\]) of the above Definition as ‘being consistent with $x_A, x_B$’; then being consistent with ${\mathcal{T}}, x_A, x_B$ means being an element of $L({\mathcal{T}})$ and being consistent with $x_A,x_B$.
Could we choose to have $Y=(\Sigma_B\times \Gamma_B)^*$, and say that $p_{Y|X_A,X_B}(y|x_A,x_B)=1$ if $y$ is consistent with ${\mathcal{T}}, x_A$ and $x_B$?
No, because such a $y$ may not be unique, and so the matrix $p_{Y|X_A,X_B}(y|x_A,x_B)$ would not in general be stochastic. For example, if ${\mathcal{T}}$ is the identity transduction and $x_A$ and $x_B$ are both constant $a$, we have that $(a,a)^k$ is consistent with ${\mathcal{T}}, x_A$ and $x_B$ for all $k$. But prefixes are the only way this can happen.
\[prop:uniqword\]Let ${\mathcal{T}}$ be an SDFST as above and let $x_A,x_B$ be strategies for Alice and Bob. Then there exists some $w_0\in
(\Sigma\times \Gamma)^{\omega}$ such that for any $w\in L({\mathcal{T}})$ we have that $w$ is consistent with $x_A$ and $x_B$ if and only if $w\leq w_0$.
Define the infinite word $$w_0 =
((a_1,a_1'),(b_1,b'_1))((a_2,a_2'),(b_2,b_2'))\ldots \in
(\Sigma\times\Gamma)^{\omega}$$ by $$\begin{aligned}
a_i &= x_A((a_1,b_1)\ldots
(a_{i-1},b_{i-1})), \\
a_i' &= x_B((a'_1,b'_1)\ldots(a'_{i-1},b'_{i-1})), \text{ and} \\
(b_i,b'_i) &= \sigma(q_{i-1},(a_i,a'_i)),\end{aligned}$$ where $q_0$ is the initial state and the sequence $q_0q_1\ldots$ is defined by $q_i = \delta(q_{i-1},(a_i,a'_i))$ for $i\geq 1$.
Clearly if $w \leq w_0$ then $w$ satisfies limb (\[it:strats\]) of Definition \[def:cons\], and so if also $w\in L({\mathcal{T}})$ then $w$ is consistent with ${\mathcal{T}},
x_A$ and $x_B$.
Conversely suppose that $w\not\leq w_0$. Then we have that $w = w'(a,b)w''$ for some $w'=((a_1,a'_1),(b_1,b'_1)) \ldots \allowbreak ((a_k,a'_k),(b_k,b'_k)) \leq w_0$, some $w''\in
\Sigma\times\Gamma^*$ and some $(a,b)\in \Sigma\times \Gamma$ with $(a,b)\neq
((a_{k+1},a'_{k+1}),(b_{k+1},b'_{k+1}))$. But if $a\neq (a_{k+1},a'_{k+1})$ then without loss of generality we have $\operatorname{fst}(a)\neq a_{k+1} =
x_A((a_1,b_1)\ldots(a_k,b_k))$ and so $w$ is not consistent with $x_A,x_B$.
On the other hand if $b\neq (b_{k+1},b'_{k+1}) = \sigma(q_k, (a_k,a'_k))$ then $w \notin L({\mathcal{T}})$. Either way we have that $w$ is not consistent with ${\mathcal{T}},
x_A$ and $x_B$.
The intuition here is that having fixed $x_A$ and $x_B$, these uniquely determine the actions of Alice and Bob at each step given the outputs they are shown, and ${\mathcal{T}}$ determines those outputs uniquely based on the actions up to the current time.
Projecting $w_0$ onto $(\Sigma_B\times \Gamma_B)^{\omega}$ gives
\[cor:uniqpref\] Let ${\mathcal{T}}, x_A$ and $x_B$ be as above. There exists some $w_0\in (\Sigma_B\times \Gamma_B)^{\omega}$ such that if $w\in (\Sigma_B\times
\Gamma_B)^*$ is consistent with ${\mathcal{T}}, x_A$ and $x_B$ then $w \leq w_0$.
So can we have $Y=(\Sigma_B\times \Gamma_B)^{\omega}$, and $p_{Y|X_A,X_B}(y|x_A,x_B) = 1$ for $y=w_0$ as in Corollary \[cor:uniqpref\]?
One reason why not is that this is not at all realistic: it corresponds to Bob being able to conduct an experiment lasting for an infinite time. Moreover it would allow Bob to acquire an infinite (or at least unbounded) amount of information, and it is not clear how this should be interpreted.
For this reason we will consider Bob’s interaction with the system not as a single experiment, but as a *family* of experiments, parametrised by the amount of time allowed; that is, by the length of traces which we consider as outcomes. Assuming for the moment that $F=Q$, we then have that the matrix $p_{Y|X_A,X_B}$ is stochastic.
\[prop:stochmat\] Let ${\mathcal{T}}$ be an SDFST with $F=Q$, and let $Y=(\Sigma_B\times \Gamma_B)^k$ for some $k\in {\mathbb{N}}$. Let the matrix $p_{Y|X_A,X_B}$ be defined by $p_{Y|X_A,X_B}(y|x_A,x_B)=1$ if $y$ is compatible with ${\mathcal{T}}, x_A$ and $x_B$, and 0 otherwise. Then $p_{Y|X_A,X_B}$ is stochastic; that is, we have $$\sum_{y\in Y} p_{Y|X_A,X_B}(y|x_A,x_B) = 1$$ for all $x_A\in {\mathcal{X}}_A$ and $x_B\in {\mathcal{X}}_B$.
By Corollary \[cor:uniqpref\], we have that for fixed $x_A, x_B$ there is at most one $y\in (\Sigma_B\times \Gamma_B)^k$ which is consistent with ${\mathcal{T}}, x_A$ and $x_B$. On the other hand it is clear from the definitions that if $F=Q$ then all prefixes of the infinite word $w_0$ from Proposition \[prop:uniqword\] are accepted by ${\mathcal{T}}$. Hence projecting $w_0$ onto $(\Sigma_B\times \Gamma_B)^k$ gives a suitable $y$.
Truncating at length $k$ also means that strategies $x_A, x_B$ can be viewed as drawn from the spaces of functions $(\Sigma_A\times \Gamma_A)^{<k}\rightarrow
\Sigma_A$ and $(\Sigma_B\times \Gamma_B)^{<k}\rightarrow \Sigma_B$ respectively. This means that the spaces ${\mathcal{X}}_A$ and ${\mathcal{X}}_B$ of possible strategies for Alice and Bob are also finite.
We can now apply Theorem \[thm:aliceposs\] to calculate the information flow as the size of the largest possible set of outcomes that can consistently be seen by Bob, and for convenience we will adopt this as a definition.
\[def:intinf\] Let ${\mathcal{T}}$ be an SDFST over input and output alphabets $\Sigma_A\times \Sigma_B$, and let ${\mathcal{X}}_A, {\mathcal{X}}_B$ be the spaces of functions $(\Sigma_A\times
\Gamma_A)^*\rightarrow \Sigma_A$ and $(\Sigma_B\times \Gamma_B)^* \rightarrow
\Sigma_B$ respectively. Define $$\begin{gathered}
{\mathcal{L}}_k({\mathcal{T}}) = \max_{x_B\in {\mathcal{X}}_B} \log\left| \left\{ y\in (\Sigma_B\times
\Gamma_B)^k \middle\vert \exists x_A\in {\mathcal{X}}_A :\right.\right. \\
\left.\left. \text{$y$ is consistent with ${\mathcal{T}},
x_A$ and $x_B$}\right\} \right|.\end{gathered}$$
Observe that if $F=Q$ then by Theorem \[thm:aliceposs\] we have that ${\mathcal{L}}_k({\mathcal{T}})={\mathcal{L}}_\infty({\mathcal{C}})$, where ${\mathcal{C}}$ is the interactive channel defined by the matrix of conditional probabilities in the statement of Proposition \[prop:stochmat\].
What about the case where $F \subsetneq Q$? The treatment of this depends on what we consider to be the meaning of a run ending in a non-accepting state. One interpretation is that it represents a catastrophically bad outcome (say, the intruder being detected) which must be avoided. By Corollary \[cor:bobdet\] we may assume that Bob is employing a pure (i.e. non-random) strategy, and so Alice can ensure that non-accepting runs are avoided by avoiding particular $x_A$. This means that Definition \[def:intinf\] is exactly right for this interpretation.
Another possible interpretation is that a run ending in a non-accepting state produces some kind of ‘error’ output, where all errors are indistinguishable. This essentially increases the number of possible observations by Bob by either 1 or 0, depending on whether or not the extremal $x_B$ allows for non-accepting runs. This means that the amount of information is either ${\mathcal{L}}_k({\mathcal{T}})$ or $\log (1 +
2^{{\mathcal{L}}_k({\mathcal{T}})})$, which we consider to be a trivial difference.
A third possiblity of course is that we reject the very notion of a non-accepting run, and consider only SDFSTs with $F=Q$. Note that many kinds of behaviour which may involve the system going into an ‘error’ state and producing only a fixed ‘dummy’ output symbol can straightforwardly be modelled as an SDFST with $F=Q$.
Which of these three options the reader considers most satisfactory is, to some extent, a matter of personal taste. However, since as noted above all are modelled adequately by Definition \[def:intinf\], that is what we shall adopt as the basic definition for the remainder of this analysis.
Definition \[def:intinf\] is in some sense an intensional definition, in the sense that it involves directly considering all possible strategies for Alice and Bob. It will be helpful to have a more extensional version. Definition \[def:intinf\] can be recast as $${\mathcal{L}}_k({\mathcal{T}})=\max_{X\in{\mathcal{F}}} \log|X|,$$ where ${\mathcal{F}}\subseteq {\mathcal{P}}((\Sigma_B\times \Gamma_B)^k)$ is the family of sets $X$ such that there exists some $x_B\in{\mathcal{X}}_B$ such that $$\begin{gathered}
X=\left\{ y\in (\Sigma_B\times
\Gamma_B)^k \middle\vert \exists x_A\in {\mathcal{X}}_A :\right.\\
\left. \text{$y$ is consistent with ${\mathcal{T}}, x_A$ and $x_B$}\right\}.\end{gathered}$$ So having an extensional characterisation of ${\mathcal{L}}_k({\mathcal{T}})$ amounts to having a condition for a set $X$ to be a member of ${\mathcal{F}}$.
\[thm:firstdiff\] Let ${\mathcal{T}}, {\mathcal{X}}_A$ and ${\mathcal{X}}_B$ be as above. Let ${\mathcal{F}}\subseteq {\mathcal{P}}\left((\Sigma_B\times \Gamma_B)^*\right)$ be defined by $Y\in {\mathcal{F}}$ if and only if there exists some $x_B\in {\mathcal{X}}_B$ such that $$\begin{gathered}
Y = \left\{ y\in (\Sigma_B\times \Gamma_B)^* \middle\vert
\exists x_A\in {\mathcal{X}}_A :\right. \\
\left.\text{$y$ is consistent with ${\mathcal{T}}, x_A$ and $x_B$}\right\}.\end{gathered}$$
Let $X\subseteq (\Sigma_B\times \Gamma_B)^*$ be arbitrary. Then $X\subseteq X'$ for some $X'\in {\mathcal{F}}$ if and only if
(i) $X\subseteq \left. L({\mathcal{T}}) \right\rvert_{(\Sigma_B\times \Gamma_B)^*}$, and
(ii) $X$ does not contain two elements which first differ by an element of $\Sigma_B$. That is, we do not have $w_1,w_2\in X$ such that $w_1 = w (a,b) w'$ and $w_2 = w (a',b') w''$ with $w,w',w'' \in (\Sigma_B\times \Gamma)^*, a,a'\in
\Sigma_B$ and $b,b'\in \Gamma_B$ with $a\neq a'$,
where the notation $\left. L({\mathcal{T}})\right\rvert_{(\Sigma_B\times \Gamma_B)^*}$ means the projection of $L({\mathcal{T}})\subseteq ((\Sigma_A\times \Sigma_B)\times
(\Gamma_A\times\Gamma_B))^*$ onto the set $(\Sigma_B\times\Gamma_B)^*$.
The ‘only if’ direction is straightforward. Part (i) is immediate from the definitions, and for part (ii) we must have $a = x_B(w) = a'$ (for the relevant $x_B$).
For the ‘if’ direction, suppose that $X$ satisfies the two conditions in the statement of the theorem. Define the partial function $x:(\Sigma_B\times
\Gamma_B)^* \rightharpoondown \Sigma_B$ by $x(w') = a$ whenever $w'(a,b) \leq w$ for some $w\in X$ and some $b\in \Gamma_B$. This is well-defined by condition (ii). Define $x_B:(\Sigma_B\times \Gamma_B)^*\rightarrow \Sigma_B$ to be $x$, extended arbitrarily where $x$ is undefined. We claim that $$\begin{gathered}
X\subseteq Y = \left\{ y\in (\Sigma_B\times \Gamma_B)^* \middle\vert \exists
x_A\in {\mathcal{X}}_A :\right. \\
\left. \text{$y$ is consistent with ${\mathcal{T}}, x_A$ and $x_B$}\right\}.\end{gathered}$$
Indeed, let $w\in X$ be arbitrary. Plainly $w$ is consistent with $x_B$. Since $w\in
\left.L({\mathcal{T}})\right\rvert_{(\Sigma_B\times\Gamma_B)^*}$, there exists some $w'\in L({\mathcal{T}})$ such that $\left. w'\right\rvert_{(\Sigma_B\times \Gamma_B)^*}
= w$. Define the partial function $x':(\Sigma_A\times
\Gamma_A)^*\rightharpoondown \Sigma_A$ by $x'(w'') = a$ whenever $w'' (a,b) \leq
w'$ for some $b\in \Gamma_A$. Let $x_A:(\Sigma_A\times \Gamma_A)^*\rightarrow
\Sigma_A$ be an arbitrary total extension of $x'$. Then plainly $w'$ is consistent with $x_A$, and is also consistent with $x_B$ since $w$ was. Hence $w$ is consistent with ${\mathcal{T}}, x_A$ and $x_B$, as required.
Truncating to length $k$, and observing that $$\max_{X\in {\mathcal{F}}} \log|X| = \max_{X\subseteq X' \in {\mathcal{F}}} \log|X|$$ gives
\[cor:firstdiff\]Let ${\mathcal{T}}, {\mathcal{X}}_A$ and ${\mathcal{X}}_B$ be as above. Then we have $${\mathcal{L}}_k({\mathcal{T}}) = \max_{X\in {\mathcal{F}}'_k} \log|X|,$$ where ${\mathcal{F}}'_k\subseteq {\mathcal{P}}\left(\left.L({\mathcal{T}})_{=k}
\right\rvert_{(\Sigma_B\times\Gamma_B)^k}\right)$ is the collection of sets which do not contain two words which first differ by an element of $\Sigma_B$ (and this has the same meaning as in part (ii) of Theorem \[thm:firstdiff\]).
Reduction to automata
---------------------
In this section, we show how to reduce the problem of computing ${\mathcal{L}}_k({\mathcal{T}})$ from a problem about transducers to a problem which mentions only automata. The first step is to produce an automaton whose language is in correspondence with Bob’s interface with ${\mathcal{T}}$.
\[def:transaut\]Let ${\mathcal{T}}= (Q, q_0, F, \Sigma_A\times \Sigma_B, \Gamma_A\times
\Gamma_B, \delta, \sigma)$ be an SDFST. Define the nondeterministic finite automaton ${\mathcal{A}}_{{\mathcal{T}}} = (Q\cup(Q\times \Gamma_B), q_0, F, \Sigma_B\cup\Gamma_B,
\Delta)$, where $$\Delta(q,a') = \left\{(\delta(q,(a,a')),\operatorname{snd}(\sigma(q,(a,a')))) \middle\vert a\in
\Sigma_A
\right\}$$ for all $q\in Q$ and $a'\in \Sigma_B$, $\Delta(q,b')=\emptyset$ for all $b'\in
\Gamma_B$, and $$\Delta((q,b'),x) = \begin{cases}\{q\} &\quad \text{if $x=b'$} \\
\emptyset &\quad \text{otherwise}\end{cases}$$ for all $(q,b')\in Q\times \Gamma_B$ and $x\in \Sigma_B\cup \Gamma_B$.
Informally, we introduce an auxiliary state for each pair $(q,b')\in Q\times
\Gamma_B$ to represent the behaviour ‘emit the event $b'$ and then go into state $q$’. For states $q,q'\in Q$ and events $a'\in \Sigma_B, b'\in \Gamma_B$ we have a transition from $q$ to $(q',b')$ if and only there exist some $a\in
\Sigma_A$ and $b\in \Gamma_A$ such that $\delta(q,(a,a')) = q'$ and $\sigma(q,(a,a')) = (b,b')$. In the language of Communicating Sequential Procceses, this corresponds to treating Alice’s behavious as nondeterministic and hiding all of her events: that is, the familiar *lazy abstraction* formulation of noninterference [@roscoe1994non].
(q\_0) [$q_0$]{}; (q\_0a) \[above right=of q\_0\] [$(q_0,a)$]{}; (q\_0b) \[below right=of q\_0\] [$(q_0,b)$]{}; (q\_0) edge \[bend left=25\] node \[above\] [$a,b$]{} (q\_0a) edge \[bend left=25\] node \[above\] [$a,b$]{} (q\_0b) (q\_0a) edge \[bend left=25\] node \[below\] [$a'$]{} (q\_0) (q\_0b) edge \[bend left=25\] node \[below\] [$b'$]{} (q\_0);
The following lemma is immediate from the definitions, and expresses the fact that the words accepted by ${\mathcal{A}}_{\mathcal{T}}$ are in precise correspondence with the words accepted by ${\mathcal{T}}$, projected onto $\Sigma_B\times \Gamma_B$.
\[lem:flat\] Let $f:(\Sigma_B\times \Gamma_B)^* \rightarrow (\Sigma_B\cup \Gamma_B)^*$ be the *flattening* operation defined by $f((a_1,b_1)\ldots(a_k,b_k)) =
a_1b_1\ldots a_kb_k$. Then we have $$L({\mathcal{A}}_{{\mathcal{T}}}) = f\left(\left. L({\mathcal{T}})
\right\rvert_{(\Sigma_B\times\Gamma_B)^*}\right).$$
Note that since elements of $\Sigma_B$ appear at odd-numbered positions in traces of ${\mathcal{A}}_{\mathcal{T}}$ and elements of $\Gamma_B$ appear at even-numbered positions, we may assume without loss of generality that $\Sigma_B$ and $\Gamma_B$ are disjoint. Then combining Lemma \[lem:flat\] with Corollary \[cor:firstdiff\] gives
\[thm:autdiff\]Let ${\mathcal{T}}$ be an SDFST as above such that $\Sigma_B$ and $\Gamma_B$ are disjoint. Then $${\mathcal{L}}_k({\mathcal{T}}) = \max_{X\in {\mathcal{F}}_k} \log |X|,$$ where ${\mathcal{F}}_k\subseteq {\mathcal{P}}\left(L\left({\mathcal{A}}_{\mathcal{T}}\right)_{=2k} \right)$ is the collections of sets which do not contain two words which first differ by an element of $\Sigma_B$; that is, for $X\in {\mathcal{F}}_{k}$ we do not have $w_1,w_2\in X$ with $w_1=w a w', w_2 = w a' w''$, with $w,w',w''\in (\Sigma_B\cup \Gamma_B)^*$ and $a\neq a' \in \Sigma_B$.
Note that an alternative notation for this theorem (and, *mutatis mutandis*, Corollary \[cor:firstdiff\]) would be to define a single family ${\mathcal{F}}\subseteq {\mathcal{P}}\left((\Sigma_B\cup \Gamma_B)^*\right)$ consisting of the sets which do not contain words first differing on an element of $\Sigma_B$, and then say that $${\mathcal{L}}_k({\mathcal{T}}) = \max_{X\in {\mathcal{F}}} \log\left|X\cap L({\mathcal{A}}_{\mathcal{T}})_{=2k}\right|.$$
We have therefore reduced computing the information flow permitted by a deterministic interactive system to an instance of a more general problem over finite automata, which we call the *$\Sigma$-deterministic subset growth* problem.
Let $\Sigma, \Gamma$ be disjoint finite sets. A set $X\subseteq (\Sigma\cup \Gamma)^*$ is *$\Sigma$-deterministic* if it does not contain two words which first differ by an element of $\Sigma$; that is, we do not have $w_1,w_2\in X$ with $w_1=waw', w_2=wa'w''$, with $w,w',w''\in (\Sigma\cup \Gamma)^*$ and $a\neq a'
\in \Sigma$.
For a nondeterministic finite automaton ${\mathcal{A}}$ over alphabet $\Sigma\cup \Gamma$, define $$D_k({\mathcal{A}}) = \max_{X\in {\mathcal{F}}_k} |X|,$$ where ${\mathcal{F}}_k$ consists of the $\Sigma$-deterministic subsets of $L({\mathcal{A}})_{=k}$.
\[prob:detgrowth\] Given a nondeterministic finite automaton ${\mathcal{A}}$ over $\Sigma\cup \Gamma$, determine the growth rate of $D_k({\mathcal{A}})$.
Of course, the statement of this problem is somewhat informal, in that the meaning of ‘determine the growth rate’ is not precisely specified. This is in some sense inevitable, considering that $D_k({\mathcal{A}})$ is an infinite collection of values, so many types of results are possible. Below we will obtain results on the asymptotic growth of $D_k({\mathcal{A}})$ as $k\rightarrow \infty$.
Antichains {#sec:autantichains}
----------
In this section we will see that Problem \[prob:detgrowth\] can be further reduced, to that of computing the ‘width’ of $L({\mathcal{A}})$.
Let $X$ be a set, and let $\leq$ be a partial order on $X$. Then the *lexicographic* order induced by $\leq$ on $X^*$, denoted $\preceq$, is defined by $$\begin{aligned}
\forall w\in X^*&: \epsilon \preceq w \text{ (and $w\not\preceq \epsilon$ if $w\neq
\epsilon$), and}\\
\forall x,y\in X, w,w' \in X^* &: xw \preceq yw' \text{ if and only if either
$x < y$} \\ &\qquad\qquad \text{or $x=y$ and $w\preceq w'$.}\end{aligned}$$
Observe that $\preceq$ defines a partial order. Indeed, suppose that $w_1,w_2\in
X^*$ are of minimum total length such that $w_1\preceq w_2, w_2\preceq w_1$ but $w_1 \neq w_2$. Trivially if $w_1 = \epsilon$ then also $w_2 = \epsilon = w_1$ (and vice versa). Otherwise we have $w_1 = xw, w_2 = yw'$, and either $x < y$ or $x=y$ and $w\preceq w'$, and on the other hand either $y < x$ or $y=x$ and $w'\preceq w$. Hence we have $y=x$ and both $w\preceq w'$ and $w' \preceq w$, so by induction $w=w'$. Hence $w_1=w_2$, a contradiction, so indeed $\preceq$ is antisymmetric.
Similarly suppose that $w_1, w_2, w_3\in X^*$ are of minimum total length such that $w_1\preceq w_2$ and $w_2\preceq w_3$ but $w_1\not\preceq w_3$. Since $w_1\not\preceq w_3$ we have $w_1\neq \epsilon$, and plainly $w_1\neq w_2$ and $w_2\neq w_3$ and so $w_2, w_3 \neq \epsilon$. Write $w_1 =
xw, w_2 = yw'$ and $w_3 = zw''$. If $x <y$ then (since $y\leq z$) we have $x <
z$ and so $w_1\preceq w_3$. Similarly if $y<z$ then (since $x\leq y$) we have $x<x$ so $w_1\preceq w_3$. Hence we have $x=y=z$ and $w\preceq w'$ and $w' \preceq
w''$. But then by induction we have $w\preceq w''$ and so $w_1\preceq w_3$, a contradiction. Hence indeed $\preceq$ is transitive and so (since we have also shown it is antisymmetric, and it is trivially reflexive) it is a partial order.
(info) [$\sup_{((X_A,X_B),Y)} {\mathcal{L}}_\infty((X_A,X_B),Y)$]{}; (uniform) [$\begin{aligned}&\sup_{x_B\in {\mathcal{X}}_B} \log \left| \left\{y\in{\mathcal{Y}}\middle\vert \exists x_A\in {\mathcal{X}}_A: \right.\right. \\
&\qquad\qquad\left. \left. p_{Y|X_A,X_B}(y|x_A,x_B)=1\right\}\right|\end{aligned}$]{}; (firstdiff) [$\begin{aligned}&\max \left\{\log |X|\middle\vert X\subseteq
\left.L({\mathcal{T}})\right\rvert_{(\Sigma_B\times \Gamma_B)^k},\right. \\ &\qquad\qquad \left.\text{ no $w,w'\in X$
first differ in $\Sigma_B$}\right\} \end{aligned}$]{}; (auto) [$\max \left\{\log|X| \middle\vert X\subseteq
L\left({\mathcal{A}}_{\mathcal{T}}\right)_{=2k} \text{ is $\Sigma_B$-deterministic}\right\}$]{}; (antichain) [$\max \left\{\log|X| \middle\vert X\subseteq
L\left({\mathcal{A}}_{\mathcal{T}}\right)_{=2k} \text{ is an antichain}\right\}$]{};
(info) edge node \[right\] [Theorem \[thm:aliceposs\]]{} (uniform) (uniform) edge node \[right\] [Corollary \[cor:firstdiff\]]{} (firstdiff) (firstdiff) edge node \[right\] [Theorem \[thm:autdiff\]]{} (auto) (auto) edge node \[right\] [Theorem \[thm:infanti\]]{} (antichain);
The study of partially ordered sets is often concerned with *chains* (sets wherein any two elements are comparable) and *antichains* (sets where no two elements are comparable).
Let $X$ be a partially ordered set, partially orderd by $\leq$. A set $Y\subseteq X$ is a *chain* if for any $x,y\in Y$ we have $x\leq y$ or $y\leq x$. $Y$ is an *antichain* if for any $x,y \in Y$ such that $x\leq y$ we have $x=y$. Let $Y \subseteq X$ be an antichain of maximum size. Then $|Y|$ is the *width* of $X$, denoted $w(X)$.
An example of the relevance of the width of a partially ordered set to its structure is given by the celebrated theorem of Robert Dilworth [@dilworth1950decomposition].
Let $X$ be a partially ordered set. Let $k$ be minimal such that $X=Y_1\cup\ldots \cup Y_k$ with each $Y_k$ a chain. Then $k=w(X)$.
The relevance of these ideas to Problem \[prob:detgrowth\] is established by the following theorem.
\[thm:infanti\]Let $\Sigma, \Gamma$ be disjoint sets. Define the partial order $\leq$ on $\Sigma\cup \Gamma$ by setting $\left.\leq\right \rvert_{\Sigma}$ to be an arbitrary linear order on $\Sigma$, and setting $x\not\leq y, y\not\leq x$ for all $x\in \Gamma$ and all $y\in \Sigma\cup \Gamma$ with $y \neq x$.
Let $X\subseteq (\Sigma\cup\Gamma)^k$ be arbitrary. Then $X$ is $\Sigma$-deterministic if and only if it is an antichain with respect to the lexicographic order induced by $\leq$.
If $w_1,w_2\in (\Sigma\cup \Gamma)^k$ first differ by an element of $\Sigma$, say $w_1 = w a w'$, $w_2 = w a' w''$ with $a\neq a' \in \Sigma$. Then without loss of generality $a < a'$, so $w_1 \preceq w_2$. Conversely, if $w_1 \preceq
w_2$, then write $w_1 = w x w', w_2 = w y w''$ for some $x\neq y \in \Sigma\cup
\Gamma$. But then we must have $x < y$, and hence $x,y\in \Sigma$ and so $w_1,w_2$ first differ by an element of $\Sigma$.
We have thus reduced Problem \[prob:detgrowth\] to the problem of calculating the growth rate of the width of a regular language, with respect to this partial order, a special case of the following problem.
\[prob:antgrowth\] Given a nondeterministic finite automaton ${\mathcal{A}}$ over a finite partially ordered set $(\Sigma,\leq)$, determine the growth rate of $w\left(L({\mathcal{A}})_{=k}\right)$, with respect to the lexicographic order.
The structure of the reductions in the preceding sections is shown in Figure \[fig:chapstruct\].
Problem \[prob:antgrowth\] is solved in [@mestel2018widths], the relevant results of which are summarised in Theorem \[thm:antgrowth\] (Theorems 16, 18, 25 and 28 of [@mestel2018widths]).
\[thm:antgrowth\] Let ${\mathcal{A}}$ be an NFA over a partially ordered set $(\Sigma,\leq)$. Then we have the following:
(i) The antichain growth of $L({\mathcal{A}})$ is either polynomial or exponential. That is, we have either $w(L({\mathcal{A}})_{=n}) = O(n^k)$ for some $k$ or $w(L({\mathcal{A}})_{=n}) = \Omega(2^{\epsilon n})$ for some $\epsilon > 0$.
(ii) There is a polynomial-time algorithm to determine whether a given ${\mathcal{A}}$ has polynomial or exponential antichain growth.
(iii) In the case of polynomial antichain growth, we have that $w(L({\mathcal{A}})_{=n})=\Theta(n^k)$ for some integer $k$, and there is a polynomial-time algorithm to compute $k$ for a given automaton.
Combining Theorem \[thm:antgrowth\] with the reduction shown in Figure \[fig:chapstruct\] yields the main theorem of this work, that any SDFST has either logarithmic or linear min-entropy capacity, and there is a polynomial-time algorithm to distinguish the two cases (and determine the constant for logarithmic capacity).
\[thm:entcap\] Let ${\mathcal{T}}= (Q,q_0,F,\Sigma_A\times\Sigma_B,\Gamma_A\times\Gamma_B,\delta,
\sigma)$ be an SDFST. Then we have the following:
(i) The min-entropy capacity ${\mathcal{L}}_n({\mathcal{T}})$ is either logarithmic or linear. That is, we have either ${\mathcal{L}}_n({\mathcal{T}}) = O(\log n)$ or ${\mathcal{L}}_n({\mathcal{T}}) = \Theta(n)$.
(ii) There is a polynomial-time algorithm to determine whether a given ${\mathcal{T}}$ has logarithmic or linear capacity growth.
(iii) In the case of logarithmic capacity, we have that ${\mathcal{L}}_n({\mathcal{T}}) \sim
k\log n$ for some integer $k$, and there is a polynomial-time algorithm to compute $k$ for a given SDFST.
Note in particular that the information flow capacity is bounded if and only if $w(L({\mathcal{A}})_{=n})$ has polynomial growth of order 0.
Returning to the relay system shown in Figure \[fig:relay\] at the beginning of this section, it is easy to see that the corresponding automaton shown in Figure \[fig:transaut\] has exponential antichain growth, since in particular its language contains the exponential antichain $(aa'+ab')^*$. We conclude that the system allows linear information flow, which is as expected since in $n$ steps Alice can transmit $n$ independent bits to Bob.
We claim that the cases of linear and logarithmic information flow can in some sense be interpreted as ‘dangerous’ and ‘safe’ respectively. That linear information flow is dangerous should require no explanation: it offers an adversary an exponential speedup over exhaustive guessing of a secret (for instance a cryptographic key). On the other hand, if the information flow in time $n$ is only proportional to $\log n$, then this offers the adversary at most a polynomial speedup over exhaustive guessing.
Of course it will not be appropriate in every situation to regard logarithmic antichain growth as ‘safe’, and for instance we may sometimes be more interested in the precise amount of information flow that can occur in a fixed time $n$. This is given by $w(L({\mathcal{A}}_{\mathcal{T}})_{=n})$, which can be computed by a straightforward dynamic programming algorithm at the cost of determinising ${\mathcal{A}}_{\mathcal{T}}$; see p.89 of the author’s PhD thesis [@mestelthesis] for details. Whether there is an algorithm which is polynomial in $n$ and the size of ${\mathcal{A}}_{\mathcal{T}}$ (as an NFA) is an open problem.
Example: a simple scheduler
---------------------------
We now illustrate the theory of the preceding two sections by applying it to analyse a simple scheduler. A resource is shared between Alice and Bob, and at each step Alice can transmit $a$, signifying that she wishes to use the resource, or $b$, signifying that she does not. She receives back either an $a'$, signifying that she was succesful, or a $b'$, signifying that she was not (if she did not ask to use the resource then she always receives a $b'$). The interface for Bob is similar but with primed and unprimed alphabets reversed.
Initially, Bob has priority over the use of the system, and for as long as Alice transmits $b$ he retains it. However, as soon as Alice seeks to use the system by transmitting an $a$ she obtains priority and retains it for as long as she uses it continuously. As soon as she transmits a $b$ priority shifts back to Bob, who retains it for the remainder of the execution.
The transducer ${\mathcal{T}}$ corresponding to this system is depicted in Figure \[fig:interrupt\] (where missing arguments mean that the input from that user is ignored).
We can now apply Definition \[def:transaut\] to construct the corresponding automaton ${\mathcal{A}}$, which is shown in Figure \[fig:interruptaut\]. By Theorems \[thm:autdiff\] and \[thm:infanti\] we have that ${\mathcal{L}}_n({\mathcal{T}}) = w(L({\mathcal{A}})_{=k})$, where $L({\mathcal{A}})$ is given the lexicographic order with the primed letters linearly ordered and the unprimed letters incomparable.
By the criteria in Theorems 16 and 28 of [@mestel2018widths], this automaton has polynomial antichain growth of order 2, and so the system has logarithmic information flow, with ${\mathcal{L}}_n({\mathcal{T}})\sim 2\log n$ (see Section 6.5.2 of [@mestelthesis] for a more detailed discussion). Note that this makes intuitive sense: Alice can choose when to start using the resource and when to stop, which she can do in $\binom{n}{2} = \Theta(n^2)$ ways.
Nondeterministic, multi-agent and probabilistic systems {#sec:future}
=======================================================
In this section we describe some open problems relating to various generalisations of the deterministic, two-agent systems considered in Section \[sec:detint\].
Nondeterministic systems
------------------------
In Section \[sec:detint\] we considered only deterministic systems. More generally, however, we may be interested in systems which are nondeterministic:
A *synchronised nondeterministic finite-state transducer* (SNDFST) is a 6-tuple ${\mathcal{T}}=(Q,q_0,F,\Sigma,\Gamma,\Delta)$, where $Q, q_0$ and $F$ are as in the definition of DFST, and $\Delta\subseteq Q\times
\Sigma \times Q \times \Gamma$ is the *transition relation*.
Similarly to before we say that $(a_1a_2\ldots a_k,b_1b_2\ldots b_k) \in
\Sigma^*\times \Gamma^*$ is accepted by ${\mathcal{T}}$ if there exists a sequence of states $q_1\ldots q_k\in Q^*$ such that $q_k\in F$ and for every $0 \leq i < k$ we have $(q_i, a_i, q_{i+1}, b_i) \in \Delta$. As before we will consider systems for which $\Sigma=\Sigma_A\times \Sigma_B$ and $\Gamma=\Gamma_A \times
\Gamma_B$, representing the inputs and outputs of Alice and Bob respectively.
The question then arises of how the nondeterminism in the system should be interpreted. One option is to consider it is essentially ‘demonic’—that is, under the control of Alice and available to be used to convey information to Bob. This precisely corresponds to Definition \[def:intinf\], which can be adopted wholesale, and a construction similar to that in Definition \[def:transaut\] can be used to produce an NFA ${\mathcal{A}}_{\mathcal{T}}$ such that the capacity of ${\mathcal{T}}$ is equivalent to the antichain growth of ${\mathcal{A}}$. We therefore have that Theorem \[thm:entcap\] holds also for nondeterministic systems interpreted in this way.
However, the assumption of demonic nondeterminism may in some circumstances be too pessimistic. In particular, it may sometimes be reasonable to assume that the way the nondeterminism is resolved depends only on the previous events, and not on Alice’s secret. Equivalently, we may imagine that the resolution of the nondeterminism is controlled by an ‘innocent’ third party who is isolated from both Alice and Bob (but is able to see their inputs and outputs). We thus have that if we can handle deterministic systems with multiple agents then we will be able to handle nondeterministic systems with this interpretation.
Multi-agent systems
-------------------
We will model multi-agent systems as SDSFTs, as before, but now with $$\begin{aligned}
\Sigma &=
\Sigma_A\times \Sigma_B \times \Sigma_1 \times \ldots \times \Sigma_k \text{
and}\\
\Gamma &= \Gamma_A \times \Gamma_B \times \Gamma_1 \times \ldots \times
\Gamma_k\end{aligned}$$ for some $k$, where the $\Sigma_i$ and $\Gamma_i$ represent the inputs and outputs respectively to the $i$th ‘innocent’ agent. We will call such a system a $k$-SDSFT.
We will now require that the $k$ innocent agents choose distributions over strategies. An argument similar to Proposition \[prop:genpure\] shows that we may assume that the innocent agents select deterministic strategies, and so we adopt a definition analagous to Definition \[def:intinf\].
Let ${\mathcal{T}}$ be a $k$-SDFST. We define $$\begin{gathered}
{\mathcal{L}}_n({\mathcal{T}}) = \max_{\substack{x_B\in {\mathcal{X}}_B, \\ x_1\in {\mathcal{X}}_1,\ldots,x_k \in {\mathcal{X}}_k}} \log \left|
\left\{ y\in (\Sigma_B\times \Gamma_B)^k \middle\vert \exists x_A \in {\mathcal{X}}_A:
\right.\right. \\ \left.\left. \text{$y$ is consistent with ${\mathcal{T}},x_A,x_B,x_1,\ldots,x_k$} \right\}
\right|,\end{gathered}$$ where ${\mathcal{X}}_i$ is the set of functions $(\Sigma_i\times \Gamma_i) \rightarrow
\Sigma_i$, and consistency is defined similarly to Definition \[def:cons\].
Our first open problem is to compute the min-entropy capacity of a $k$-SDFST. We conjecture that there should still be a dichotomy between polynomial and exponential growth.
Probabilistic systems
---------------------
We may also wish to handle systems whose behaviour is probabilistic. We model such systems as probabilistic finite-state transducers.
A *probabilistic finite-state transducer* is a tuple ${\mathcal{T}}=(Q,q_0,\Sigma,
\Gamma,\Delta)$, where $Q$ is a finite set of *states*, $q_0\in Q$ is the *initial state* and $\Delta:Q\times \Sigma\times Q \times \Gamma
\rightarrow \mathbb{R}^{\geq 0}$ is the *transition function*, such that for all $q\in Q$ and all $a\in \Sigma$ we have $$\sum_{b\in \Gamma, q'\in Q} \Delta(q,a,q',b) = 1.$$
We interpret $\Delta(q,a,q',b)$ as the probability that on receiving the input $a$ in state $q$, the system outputs $b$ and moves to state $q'$.
As before we require that $\Sigma$ and $\Gamma$ are of the form $\Sigma_A\times
\Sigma_B$ and $\Gamma_A\times \Gamma_B$ respectively (although of course it would also be possible to consider multi-agent probabilistic systems), and the sets ${\mathcal{X}}_A$ and ${\mathcal{X}}_B$ are as before. For fixed $x_A$ and $x_B$, the output $Y$ produced to Bob after $n$ steps is a sequence $y\in (\Sigma_B\times
\Gamma_B)^n$, where $y=((a'_1,b'_1),\ldots,(a'_n,b'_n))$ occurs with probability $$\sum_{((a_1,b_1),\ldots, (a_n,b_n))\in Z} \sum_{q_1,\ldots,q_n \in Q}
\prod_{i=1}^n \Delta(q_{i-1},(a_i,a'_i),q_i,(b_i,b'_i)),$$ where $Z$ is the set of $((a_1,b_1),\ldots,(a_n,b_n)) \in (\Sigma_A\times
\Gamma_A)^n$ such that $((a_1,a'_1),(b_1,b'_1))\ldots \allowbreak
((a_n,a'_n),(b_n,b'_n))$ is consistent with $x_A$ and $x_B$.
This defines an interactive channel ${\mathcal{C}}_n$, and so our second open problem is to compute the growth of ${\mathcal{L}}_\infty({\mathcal{C}}_n)$. This seems to be a rather formidable challenge since we lack a way to reduce to a possibilistic view of Alice’s actions, and so we may genuinely have to quantify over probability distributions for $X_A$.
Related work {#sec:relatedwork}
============
So far as we are aware this is the first quantitative study which is able to analyse interactive systems in full generality, that is to say where inputs may be provided by both parties, according to distributions chosen adversarially so as to maximise information flow, rather than being specified as part of the system.
Mardziel, Alvim, Hicks and Clarkson in [@mardziel2014quantifying] consider interactive systems in essentially the same model as we use in Section \[sec:detint\] of this paper: they represent the system by a probabilistic finite automaton, which is executed in a ‘context’ consisting of the strategy functions for the high and low users. They then employ probabilistic programming to analyse particular systems with respect to particular contexts, demonstrating for instance that allowing an adaptive adversary can greatly increase information flow. However, they acknowledge that they are not able to analyse the maximum leakage over all possible contexts, instead observing that ‘We consider such worst-case reasoning challenging future work’. The present work addresses this question for the case where the system is deterministic.
Köpf and Basin in [@kopf2007infomodel] show how to calculate information leakage for a particular model relating to side-channel attacks in which the attacker is repeatedly permitted to make queries drawn from some fixed set. They give an exhaustive algorithm to compute the maximum amount of information leakege after $n$ queries.
Boreale and Pampaloni in [@boreale2015quantitative] consider the case of repeated queries issued by the attacker (possibly adaptively) to a stateless system and show that under certain reasonable assumptions the problem of computing the maximum leakage after $n$ queries is NP-hard. In [@boreale2011asymptotic], the same authors together with Paolini study the asymptotics of the leakage resulting from $n$ independent uses of a single channel for large $n$. This is in some sense dual to the situation we have considered, of the asymptotics of a single, long execution of a stateful system.
In [@andres2010computing], Andr[é]{}s, Palamidessi, van Rossum and Smith compute the leakage of what they term ‘interactive information-hiding systems’ (IIHS), which are essentially automata over (secret) inputs and (observed) outputs. However, they assume an essentially passive attacker: apart from the values of the secret (whose distribution they sometimes allow to be chosen so as to maximise information flow), the system is assumed to follow known probabilistic behaviour. In follow-up work [@alvim2012quantitative], Alvim, Andr[é]{}s and Palamidessi demonstrate interesting connections between the mutual information capacity of such systems and the directed information capacity of channels with feedback, although this is of limited practical significance since it is now recognised that mutual information is not generally an appropriate measure of information flow.
An interesting alternative algorithmic approach is taken by Kawamoto and Given-Wilson in [@kawamoto2015quantitative], although for a completely different problem from that addressed in this work. In [@kawamoto2015quantitative], the authors consider a purely passive observer who is shown the outputs of two channels, interleaved according to some scheduler; the goal is to find a scheduler which minimises the information leakage. They show that this can be expressed as a linear programming problem, and therefore solved in time polynomial in the number of possible interleavings, which unfortunately is exponential in the number of possible traces.
Conclusions {#sec:conclusion}
===========
In [@ryan2001noninterference], Ryan, McLean, Millen and Gligor write the following:
> Even at a theoretical level where timings are not available, and a bit per millisecond is not distinguishable from a bit per fortnight or a bit per century, a channel that compromises an unbounded amount of information is substantially different from one that cannot. Characterization of unbounded channels is suggested as the kind of goal that would advance the study of this subject
In Theorem \[thm:entcap\] we have achieved this goal for deterministic systems, and in fact slightly more: we have shown that even among unbounded channels there is a dichotomy between ‘safe’ and ‘dangerous’ information flow, and this can be determined for a given system in polynomial time.
Having characterised the notion of safe versus dangerous information flow, one may ask about the question of enforcement of the safety criterion. In one sense this question is already answered by Theorem \[thm:entcap\], since it includes a polynomial-time algorithm to determine whether the condition is satisfied for a given system. However, the development of automated tools implementing this algorithm, which preferably would allow realistic systems to be specified using more convenient notation than the rather abstract mathematical formalism of finite-state transducers, is certainly an important area for future work.
Acknowledgements {#acknowledgements .unnumbered}
----------------
The author is grateful to Catuscia Palamidessi for helpful comments on an earlier version of this work, and to Dimiter Ostrev for comments on the final version.
[^1]: The author is supported by FNR under grant number 11689058 (Q-CoDe).
[^2]: Smith and subsequent authors generally define leakage only for random variables whose images are finite sets. However, their definitions are straightforwardly generalised to arbitrary discrete random variables by replacing $\max$ with $\sup$ where appropriate. Except where noted, the proofs of all quoted results remain valid after the same modification.
[^3]: Note that this means a pure strategy *over the set ${\mathcal{X}}_B$*, which in an interactive system may contain probabilistic strategies (although we will see in Theorem \[thm:probstrat\] that these may be ignored without loss of generality).
[^4]: Strictly speaking Corollary \[cor:bobdet\] was proved for discrete distributions, whereas ${\mathcal{S}}_{T,\Sigma}$ is a continuous subset of $\mathbb{R}^{|T|\cdot
|\Sigma|}$. The proof for this case is exactly the same, with sums over ${\mathcal{X}}_B$ replaced by integrals with respect to the Lebesgue measure.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We consider discrete one-dimensional Schrödinger operators with Sturmian potentials. For a full-measure set of rotation numbers including the Fibonacci case we prove absence of eigenvalues for all elements in the hull.'
author:
- |
David Damanik and Daniel Lenz\
Fachbereich Mathematik\
Johann Wolfgang Goethe-Universität\
60054 Frankfurt/Main\
Germany
title: |
Uniform spectral properties of one-dimensional quasicrystals,\
I. Absence of eigenvalues
---
\[section\] \[lemma\][Proposition]{}
Introduction
============
In this paper we consider discrete one-dimensional Schrödinger operators in $l^2({\bf Z})$ with Sturmian potentials, namely, $$\label{hlt}
(H_{\lambda,\alpha,\theta}u)(n)=u(n+1)+u(n-1)+\lambda v_{\alpha,\theta}(n)u(n),$$ where $$\label{stpot}
v_{\alpha,\theta}(n)=\chi_{[1-\alpha,1)}(n\alpha +\theta \, mod \, 1),$$ $\lambda \not= 0$, $\alpha \in (0,1)$ irrational and $\theta \in [0,1)$, along with the corresponding difference equation $$\label{eve}
H_{\lambda,\alpha,\theta} u=Eu.$$ The operator family (\[hlt\]) describes a standard one-dimensional quasicrystal model [@lp1; @s] and has been studied in many papers, e.g. [@bist; @bit; @d1; @irt; @k1; @r]. Moreover, the operators $H_{\lambda,\alpha,\theta}$ have attracted attention since they exhibit spectral properties such as zero-measure spectrum and purely singular continuous spectral measures that seemingly hold for the entire family in contrast to the almost Mathieu operator where similar properties were shown to hold for a strict subclass of parameter values [@j; @l1].\
\
To put our present study into perspective, let us consider the class of discrete one-dimensional Schrödinger operators with strictly ergodic (i.e., minimal and uniquely ergodic), aperiodic potentials taking finitely many values. Among these, potentials generated by primitive substitutions and circle maps have received particular interest [@bbg1; @bbg2; @bg1; @bg2; @d2; @d3; @dp1; @dp2; @h; @hks; @it; @k1; @s3; @s4]. Sturmian potentials (\[stpot\]) are circle map potentials sharing some crucial properties with potentials generated by primitive substitutions. The general belief is that, as a rule, the spectrum has zero Lebesgue measure and the spectral measures are purely singular continuous. By results of Kotani [@k2] and Last-Simon [@ls], absence of absolutely continuous spectrum holds in full generality, that is, for every such family of operators and for every element of the family. Zero-measure spectrum was proven for all Sturmian potentials by Bellissard [*et al.*]{} [@bist] and for a large class of primitive substitutions by Bovier-Ghez [@bg1; @bg2]. Again, the zero-measure property holds for all elements of the family since, by minimality, the spectrum is constant over the hull. On the other hand, absence of point spectrum has not yet been shown to hold in similar generality. Generic absence of eigenvalues for certain models was proven in [@bbg1; @bist; @bg1; @bg2; @d1; @dp2; @hks; @s3], while the works [@d2; @d3; @dp1; @k1] contain almost sure results. However, no uniform result, that is, absence of eigenvalues for an entire such family, was known yet. For the particular case of Sturmian potentials, generic absence of eigenvalues is essentially due to Bellissard [*et al.*]{} [@bist] (the paper does not state the result, see [@d1; @hks] for the result and proofs), whereas almost sure absence of eigenvalues was shown by Kaminaga [@k1], extending an argument of Delyon-Petritis [@dp1] who had already obtained a partial result.\
\
Our purpose here is to prove the following theorem.
\[theo\] Suppose $\limsup a_n \not= 2$, where the $a_n$ are the coefficients in the continued fraction expansion of $\alpha$. Then, for every $\lambda$ and every $\theta$, the operator $H_{\lambda,\alpha,\theta}$ has empty point spectrum.
[*Remarks.*]{}
1. The set of $\alpha$’s obeying the assumption of Theorem \[theo\] has full Lebesgue measure [@khin].
2. In particular, the Fibonacci case $\alpha = \frac{\sqrt{5}-1}{2}$ is included as all the $a_n$ are equal to $1$ in this case.
3. Some works associate a slightly different family to the parameters $\lambda,\alpha$ [@h; @hks] which is larger than the family parametrized by $\theta \in [0,1)$. The proof also works for the additional elements in that larger hull.
4. Our approach is based upon the two-block method [@g; @s3] and yields additional information about stability properties that will be discussed within a more general context in [@dl2].
5. Another key ingredient in our proof is an analogue to the hierarchical structures and the concept of (de-)composition in self-similar tilings [@gs]. This will be further exploited in [@dl1].
Thus, combining Theorem \[theo\] with the results of Bellissard [*et al.*]{} [@bist], the general picture proves to be correct for most parameter values.
Suppose $\alpha$ obeys the assumption of Theorem \[theo\]. Then, for every $\lambda$ and every $\theta$, the operator $H_{\lambda,\alpha,\theta}$ has purely singular continuous spectrum supported on a Cantor set of zero Lebesgue measure.
The organization of the paper is as follows. Section 2 recalls basic properties of Sturmian potentials as well as the two-block method. Hierarchical structures in Sturmian sequences are studied in Section 3, while Section 4 provides a proof of Theorem \[theo\].
Basic properties of Sturmian potentials
=======================================
Since our argument will be based upon the two-block method, our strategy will be to exhibit appropriate squares adjacent to a certain site $i \in
{\bf Z}$. This approach is entirely independent of the actual numerical values the potential takes, and we can, without loss of generality, restrict our attention to the particular case $\lambda = 1$. That is, we shall study the combinatorial properties of the sequences $v_{\alpha,\theta}$. Consider such a sequence as a two-sided infinite word over the alphabet $A \equiv \{0,1\}$. All these words share the property that their subword complexity is minimal among the aperiodic sequences. Infinite words with this property are called [*Sturmian*]{}. We shall recall some basic facts about these words that will be used in Sections 3 and 4. The material is taken from [@bist; @b]. Moreover, as a general reference on Sturmian words we want to mention the forthcoming monograph [@l2].\
\
Fix $\alpha$ and consider its continued fraction expansion (for general information on continued fractions see, e.g., [@khin]), $$\label{continuedfraction}
\alpha = \cfrac{1}{a_1+ \cfrac{1}{a_2+ \cfrac{1}{a_3 + \cdots}}}
\equiv [a_1,a_2,a_3,...]$$ with uniquely determined $a_n \in {\bf N}$. The associated rational approximants $\frac{p_n}{q_n}$ obey $$p_0 = 0, \; p_1 = 1, \; p_n = a_n p_{n-1} + p_{n-2},$$ $$q_0 = 1, \; q_1 = a_1, \; q_n = a_n q_{n-1} + q_{n-2}.$$ Define the words $s_n$ over the alphabet $A$ by
$$\label{recursive}
s_{-1} \equiv 1, \;\; s_0 \equiv 0, \;\; s_1 \equiv s_0^{a_1 - 1} s_{-1},
\;\; s_n \equiv s_{n-1}^{a_n} s_{n-2}, \; n \ge 2.$$
In particular, the length of the word $s_n$ is equal to $q_n$, $n \ge 0$.
\[palin\] There exist palindromes $\pi_n$, $n \ge 2$, such that $$\label{even}
s_{2n} = \pi_{2n}10,$$ $$\label{odd}
s_{2n+1} = \pi_{2n+1}01$$
[*Proof.*]{} See [@b], where also the recursive relations obeyed by the $\pi_n$ can be found.$\Box$\
\
By definition, for $n \ge 2$, $s_{n-1}$ is a prefix of $s_n$. Therefore, the following (“right”-) limit exists in an obvious sense, $$\label{standard}
c_\alpha \equiv \lim_{n \rightarrow \infty} s_n.$$ Similarly, from (\[recursive\]) we infer that, for $n \ge 1$, $s_{2n-2}$ is a suffix of $s_{2n}$, and hence, the following (“left”-) limit exists, $$\label{standleft}
d_\alpha \equiv \lim_{n \rightarrow \infty} s_{2n}.$$ The exceptional role played by the sequence $v_{\alpha,0}$ is demonstrated by the following proposition.
\[vsymm\] $v_{\alpha,0}$ restricted to $\{1,2,3,\ldots\}$ coincides with $c_\alpha$, $v_{\alpha,0}$ restricted to $\{\ldots,-2,-1,0\}$ coincides with $d_\alpha$.
[*Proof.*]{} The first claim was shown by Bellissard [*et al.*]{} in [@bist]. The second claim follows from the first combined with Proposition \[palin\] and the symmetry $v_{\alpha,0}(-k) =
v_{\alpha,0} (k-1)$, $k \ge 2$, also shown in [@bist].$\Box$\
\
For later use, we also want to note the following elementary formula.
\[wunderformel\] For each $n \ge 2$, $s_n s_{n+1}= s_{n+1} s_{n-1}^{a_n - 1} s_{n-2} s_{n-1}$.
[*Proof.*]{} $s_n s_{n+1} = s_n s_n^{a_{n+1}} s_{n-1} = s_n^{a_{n+1}}
s_n s_{n-1} = s_n^{a_{n+1}} s_{n-1}^{a_n} s_{n-2} s_{n-1} = s_{n+1}
s_{n-1}^{a_n - 1} s_{n-2} s_{n-1}.$$\Box$\
\
The point is that the word $s_n s_{n+1}$ has $s_{n+1}$ as a prefix. Note that the dependence of $a_n, p_n, q_n, s_n, \pi_n$ on $\alpha$ is left implicit. This, however, should not lead to any real confusion as $\alpha$ will always be fixed within a local context.\
\
We now turn to the study of generalized eigenfunctions, i.e. solutions of (\[eve\]). Recall the standard reformulation of (\[eve\]) in terms of transfer matrices, namely, $$U(n+1)=T_{\lambda,\alpha,\theta}(n,E)U(n),$$ where $u$ is a solution of (\[eve\]), $$U(n) \equiv \left( \begin{array}{c} u(n)\\u(n-1) \end{array}\right),$$ and $$T_{\lambda,\alpha,\theta}(n,E) \equiv \left(
\begin{array}{cc}
E-\lambda v_{\alpha,\theta}(n) & -1\\
1 & 0
\end{array}
\right).$$ Thus, the sequence $(u(n))_{n \in {\bf Z}}$ is determined by two consecutive values, say $u(0)$ and $u(1)$, and all other values can be determined by applying a matrix product of the form $M_{\lambda,\alpha,\theta}(n,E)
\equiv T_{\lambda,\alpha,\theta}(n,E) \times \cdots \times
T_{\lambda,\alpha,\theta}(1,E)$ to the vector $U(1)$ (for $n \ge 1$, the case $n \le 0$ is similar). Note that $\det(M_{\lambda,\alpha,\theta}(n,E))=1$. Hence, the characteristic equation of $M_{\lambda,\alpha,\theta}(n,E)$ takes the form
$$\label{char}
M_{\lambda,\alpha,\theta}(n,E)^2 - tr(M_{\lambda,\alpha,\theta}(n,E))
M_{\lambda,\alpha,\theta}(n,E) + I = 0.$$
Now, the spectrum of $H_{\lambda,\alpha,\theta}$ is independent of $\theta$ [@bist] and can thus be denoted by $\Sigma_{\lambda,\alpha}$.
\[tracebound\] For every $\lambda \not= 0$, there exists $C_\lambda$ such that, for every irrational $\alpha$, every $E \in \Sigma_{\lambda,\alpha}$ and every $n \in {\bf N}$, we have $| tr(M_{\lambda,\alpha,0}(q_n,E) | < C_\lambda$.
[*Proof.*]{} See [@bist].$\Box$\
\
Before we formulate our basic criterion for absence of eigenvalues, we introduce the following two concepts. Let $v$ be a two-sided sequence over $A$ (think of $v=v_{\alpha,\theta}$). A subword $x=x_1 \ldots x_l$ of $v$ is called [*adjacent to $i \in {\bf Z}$*]{} if $v_i \ldots
v_{i+l-1} = x$ or $v_{i-l+1} \ldots v_i = x$. Two finite words $x,y$ having the same length are called [*conjugate*]{} if $x$ is a subword of $yy$. This notion is easily seen to induce an equivalence relation on $A^l$, for any fixed $l$. Intuitively, the equivalence class of a word $x=x_1
\ldots x_l$ is given by the collection of all cyclic permutations $x_{j+1}
\ldots x_l x_1 \ldots x_j$ of $x$.
\[zweiblock\] Suppose $\alpha,\theta$ are such that $v_{\alpha,\theta}$ has infinitely many squares $u_k u_k$ adjacent to some $i \in {\bf Z}$, where $u_k$ is conjugate to some $s_{n_k}$. Then, for every $\lambda$, $\sigma_{pp}
(H_{\lambda, \alpha,\theta}) = \emptyset$.
[*Proof.*]{} In case $i=1$, the assertion follows by standard arguments [@g; @s3] from (\[char\]) together with Proposition \[tracebound\]. The general case can be reduced to this case by a suitably chosen shift of the sequence $v_{\alpha,\theta}$, which, of course, leaves the spectral type of the associated operators $H_{\lambda,\alpha,\theta}$ invariant.$\Box$\
\
We close this section by introducing the shift operator $T$ on functions on ${\bf Z}$, i.e. $$(T f) (n) \equiv f(n+1)$$ for arbitrary functions $f$ on ${\bf Z}$.
The partition lemma
===================
Consider for a fixed $\alpha$ the family (in $\theta$) of all the sequences of the form $(v_{\alpha,\theta}(k))_{k\in {\bf Z}}$. It is well known that in each sequence of the family the same words occur. Moreover any of these words occurs with a fixed frequency greater than zero which is independent of the sequence (cf. Appendix of [@h]). Thus, from a measure theoretical point of view, the family $\{(v_{\alpha,\theta}(k))_{k\in {\bf Z}}\,|\,\theta \in [0,1)\}$ behaves very uniformly. However, to prove the absence of eigenvalues for all $\theta$ we need a kind of uniform topological structure. In this section we provide this structure by showing that each sequence $(v_{\alpha,\theta}(k))_{k\in {\bf Z}}$ can be decomposed into blocks of the form $s_n$ and $s_{n-1}$ for all $n\in {\bf N}_0$. Here we denote by ${\bf N}_0$ the set of all integers together with $0$, i.e. ${\bf N}_0=\{0,1,2,\ldots\}$.
\[n-partition\]Let $n\in {\bf N}_0$ be given. An $(n,\alpha)$-partition of a sequence $(f_k)_{k\in{\bf Z}}$ with $f_k\in\{0,1\}$ is a sequence $(I_j, z_j)_{j\in{\bf Z}}$ of pairs $I_j=\{d_j,d_j+1,\ldots,d_{j+1}-1\}\subset {\bf Z}$ and $z_j\in \{s_n,
s_{n-1}\}$ with $0\in I_0$ s.t. $$f_{d_j}f_{d_j +1}...f_{d_{j+1}-1}=z_j$$ for all $j\in {\bf Z}$.\
An $(n,\alpha)$-partition of a function $f:{\bf Z}\longrightarrow \{ 0,1 \}$ is an $(n,\alpha)$-partition of the sequence $(f(n))_{n\in {\bf Z}}$.\
The $z_j$ are referred to as blocks in the $n$-partition or more specifically as blocks of the form $s_n$ if $z_j=s_n$, and as blocks of the form $s_{n-1}$ if $z_j=s_{n-1}$. The $I_j$ are referred to as positions of the blocks $z_j$.
In the sequel we will sometimes suppress the dependence on $\alpha$ if it is clear from the context to which $\alpha$ we refer. In particular, we will write $n$-partition instead of $(n,\alpha)$-partition.\
\
[*Remarks.*]{}
1. One can think of an $n$-partition as a tiling of ${\bf Z}$ by $s_n$ and $s_{n-1}$ generating $f$. So $f$ is composed out of the blocks $z_j$ at the positions $I_j$.
2. Our notion of $n$-partition is analogous to the notion of $n$th composition used in the study of self similar tilings [@lp]. This notion has been used by Hof in [@h]. There he studies the Lyapunov exponent and the integrated density of states of discrete Schrödinger operators with a potential generated by a primitive substitution.
As $s_0=0$ and $s_{-1}=1$ every $f:{\bf Z}\longrightarrow \{ 0,1 \}$ admits a $0$-partition. But for a general $f$ there does not necessarily exist an $n$-partition for $n>0$. However, it is crucial to our analysis of the eigenvalue problem for $H_{\lambda,\alpha,\theta}$ that for the sequences $v_{\alpha,\theta}$ there do exist unique $n$-partitions for all $n\in {\bf N}_0$. This is the content of the next lemma.
\[partition-lemma\]
- For every $n\in {\bf N}_0$, there exists a unique $n$-partition of $v_{\alpha,0}$.
- For every $n\in {\bf N}_0$ and every $\theta \in [0,1)$, there exists a unique $n$-partition of $v_{\alpha,\theta}$.
[*Proof.*]{} Let $\alpha=[a_1,a_2,...]$ be the continued fraction expansion of $\alpha$ (cf. equation \[continuedfraction\]).\
(i) [*Existence:*]{} Set $v \equiv v_{\alpha,0}$. We show that there are $n$-partitions of $(v(k))_{k\geq 1}$ and of $(v(k))_{k\leq 0}$. Here an $n$-partition of a one sided sequence is defined in the obvious way. By (\[recursive\]), (\[standard\]) and Proposition \[vsymm\], it is clear that there exists an $n$-partition of $ (v(k))_{k\geq 1}$ for all $n$.\
The existence of an $n$-partition for $ (v(k))_{k\leq 0}$ follows similarly by (\[recursive\]), (\[standleft\]) and Proposition \[vsymm\].\
[*Uniqueness:*]{} This follows by induction: As $s_0=0, s_{-1}=1$ uniqueness is clear for $n=0$. By (\[recursive\]), every $(n+1)$-partition gives rise to an $n$-partition and the positions of the $s_{n+1}$ in the $(n+1)$-partition are determined by the positions of $s_{n-1}$ in the $n$-partition. Thus, uniqueness of the $n$-partition implies uniqueness of the $(n+1)$-partition.\
\
(ii) Fix $\theta \in [0,1)$. As $\alpha$ is irrational there exists a sequence $(n_k)_{k\in {\bf N}}$, $n_k\in {\bf N}$, s.t. the sequence $(T^{n_k} v_{\alpha, 0})_{k\in {\bf N}}$ converges to $ v_{\alpha, \theta}$ in the product topology on $\{0,1\}^{{\bf Z}}$ for $k\to \infty$. By (i), it is clear that the $ T^{n_k} v_{ \alpha, 0}$ admit unique $n$-partitions for all $n\in {\bf N}_0$. To use this to conclude (ii) we introduce the following notion of convergence:\
\
Let $f_k,k\in {\bf N}$, and $f$ be functions for which there exist unique $n$-partitions denoted by $(I_j^k, z_j^k)$ and $(I_j,z_j)$, respectively. We say that the $f_k$ converge to $f$ in the $n$-sense for $k\to \infty$ if for all $C>0$ there exists a $k_0$ s.t. for $k\geq k_0$ $$(I_j^k, z_j^k)=(I_j,z_j) \mbox{ for all}\;\: I_j\subset (-C,C)$$ holds.\
Clearly, (ii) follows if we prove the following claim.\
\
[*Claim.*]{} For each $n$ there exists a unique $n$-partition of $v_{\alpha,\theta}$ and the sequence $(T^{n_k}v_{\alpha,0})_{k\in {\bf N}_0}$ converges to $v_{\alpha,\theta}$ in the $n$-sense for $k\to \infty$.\
\
[*Proof of the claim.*]{} This will be done by induction. We will consider two cases.\
[*Case 1:*]{} $a_1=1$.\
As $s_{-1}=s_{1}=1$ and $s_0=0$, the cases $n=0$ and $n=1$ are clear. So suppose the statement is true for $n\geq 1$ fixed. Let $(I_j,z_j)$ be the $n$-partition of $v_{\alpha,\theta}$. By $s_{n+1}=s_n^{a_{n+1}} s_{n-1}$ (cf. equation \[recursive\]), the existence of an $(n+1)$-partition of $v_{\alpha,\theta}$ will follow if we show that to the left of each block of the form $s_{n-1}$ in the $n$-partition of $v_{\alpha,\theta}$ there are at least $a_{n+1}$ blocks $s_n$. That is, we have to show that $z_j=s_{n-1}$ for $j\in {\bf Z}$ implies $z_k=s_n$ for $k= j - a_{n+1}, ..., j-1$. As $T^{n_k}v_{\alpha,0}$ admits a unique $n$-partition for each $n\in {\bf N}_0$, there are at least $a_{n+1}$ blocks $s_n$ to the left of each block of the form $s_{n-1}$ in the $n$-partition of $T^{n_k}v_{\alpha,0}$. As the $T^{n_k}v_{\alpha,0}$ converge to $v_{\alpha,\theta}$ in the $n$-sense, the corresponding statement is true for $v_{\alpha,\theta}$. This gives the existence of an $(n+1)$-partition of $v_{\alpha,\theta}$. The uniqueness of the $(n+1)$-partition follows from the uniqueness of the $n$-partition as in $(i)$. As the blocks $s_{n+1}$ in the $(n+1)$-partition of $v_{\alpha,\theta}$ arise from blocks $s_n^{a_{n+1}} s_{n-1}$ in the $n$-partition of $v_{\alpha,\theta}$, it is clear that the convergence of $T^{n_k}v_{\alpha,0}$ to $v_{\alpha,\theta}$ in the $n$-sense implies the convergence in the $(n+1)$-sense. This proves the claim in Case 1.\
[*Case 2:*]{} $a_1>1$.\
As $s_{-1}=1$ and $s_0=0$, the case $n=0$ is clear. So fix $n\geq 0$. If $n>0$ we can continue exactly as in Case 1. If $n=0$ we can continue as in Case 1 after replacing $a_{n+1}=a_1$ by $a_{1}-1\geq1$. This proves the claim in Case 2.\
The proof of the lemma is therefore finished.$\Box$
\[potenzen\] Let, for $n\in {\bf N}$, $(I^n_j,z^n_j)$ be the $n$-partition of $v_{\alpha,\theta}$. If $z^n_j=s_{n-1}$, then $z^n_{j-1}=z^n_{j+1}=s_n$. If $z^n_j=s_n$, then there is an interval $I=\{d,d+1,\ldots,d+l-1\}\subset {\bf Z}$ of length $l\in\{a_{n+1},a_{n+1}
+1\}$ with $j\in I$ and $z^n_i=s_n$ for all $i\in I$ and $z^n_{d-1}=z^n_{d+l}=s_{n-1}$.
[*Proof.*]{} By the existence part of the partition lemma, there exists an $(n+1)$-partition of $v_{\alpha,\theta}$. By the uniqueness part of the partition lemma and the formula $s_{n+1}= s_n^{a_{n+1}} s_{n-1}$, all the blocks of the form $s_{n-1}$ in the $n$-partition of $v_{\alpha,\theta}$ arise from blocks of the form $s_{n+1}$ in the $(n+1)$-partition. This shows that there is no $j\in {\bf Z}$ with $z_j^n=z_{j+1}^n=s_{n-1}$ and that there are at least $a_{n+1}$ blocks of the form $s_n$ between two blocks of the form $s_{n-1}$. That there are at most $a_{n+1}+1$ such blocks follows, as there are not two adjacent blocks of the form $s_n$ in the $(n+1)$-partition. This proves the corollary.$\Box$\
\
[*Remarks.*]{}
1. Define $\Omega$ to be the set of accumulation points of translates of $v_{\alpha,0}$ with respect to pointwise convergence, that is, $$\Omega(\alpha) \equiv \{ \omega \in A^{{\bf Z}} \; | \; \omega = \lim T^{n_i}
v_{\alpha,0}, \; n_i \rightarrow \infty\}.$$ Then the method of the previous lemma can easily be adopted to prove the existence of a unique $n$-partition for all $\omega\in
\Omega(\alpha)$. We refer the reader to [@dl2] for a discussion of the relationship between $\Omega(\alpha) $ and the set $\{v_{\alpha,\theta}\;|\; \theta \in [0,1)\}.$
2. The above lemma and a careful analysis of $v_{\alpha,0}$ allow to show that the blocks $s_n$ and $s_{n-1}$ in the $n$-partition of an $\omega \in \Omega(\alpha)$ occur with fixed frequency, see [@dl1] for details. This can be used together with Theorem 1 of [@gh] to show that indeed every word that occurs in some $\omega_0\in \Omega(\alpha)$ occurs in every $\omega\in \Omega(\alpha)$ with a fixed frequency greater than zero. Thus, the above partition lemma is indeed a stronger result than the results about the frequencies of words mentioned at the beginning of this section.
Absence of eigenvalues
======================
Let $\alpha=[a_1,a_2,...]$ be the continued fraction expansion of $\alpha$ (cf. equation \[continuedfraction\]).\
\
The proof of Theorem \[theo\] will be split into two parts.
\[drei\] If $\limsup_{n\to \infty} a_n\geq 3$, then, for every $\theta$ and every $\lambda$, the operator $H_{\lambda,\alpha,
\theta}$ has no eigenvalues.
\[fib\] If $\limsup_{n\to \infty} a_n=1$, then, for every $\theta$ and every $\lambda$, the operator $H_{\lambda,\alpha, \theta}$ has no eigenvalues.
[*Remark.*]{} In fact the proofs yield the absence of eigenvalues for all potentials in the respective hulls $$\Omega(\lambda,\alpha):=\{\lambda \omega: \omega\in \Omega(\alpha)\}.$$\
[*Proof of Proposition \[drei\].*]{} Fix $\theta\in [0,1)$. We will exclude eigenvalues of $H_{\lambda,\alpha,\theta}$ using Lemma \[zweiblock\], that is, by exhibiting many appropriate squares in $v_{\alpha,\theta}$ at zero. By the partition lemma \[partition-lemma\], there is for each $n\in {\bf N}_0$ an $n$-partition $$((I_j^n,z_j^n))_{j\in {\bf Z}}, \;I_j^n=\{d_j^n,\ldots, d_{j+1}^n-1\},
\;z^n_j\in\{s_n,s_{n-1}\}$$ of $v_{\alpha,\theta}$. We will consider two cases.\
[*Case 1*]{}: There are infinitely many $n\in {\bf N}_0$ with $z^n_0=s_{n-1}$.\
Consider such an $n$ with $n\geq 4$. Corollary \[potenzen\] yields $z_1^n=s_n$. As $s_{n-1}$ is a prefix of $s_n$ and $z_2^n\in
\{s_n,s_{n-1}\}$ we have $$z_0^n z_1^n z_2^n=s_{n-1} s_n s_{n-1} w$$ with a suitable word $w$. Using $s_n= s_{n-1}^{a_n} s_{n-2}$ (cf. equation \[recursive\]), we get $$z_0^n z_1^n z_2^n=s_{n-1} s_{n-1}^{a_n} s_{n-2} s_{n-1} w .$$ By $s_{n-2} s_{n-1}=s_{n-1} v$ with a suitable $v$ (cf. Proposition \[wunderformel\]), we finally arrive at $$z_0^n z_1^n z_2^n=s_{n-1} s_{n-1}^{a_n} s_{n-1} v w.$$ This implies $$v_{\alpha,\theta}(d_0^n) v_{\alpha,\theta}(d_0^n+1)... v_{\alpha,\theta}(d_0^n+ 3 |s_{n-1}| -1)= s_{n-1} s_{n-1} s_{n-1}
,$$ where $0\in \{d_0^n,..., d_0^n + |s_{n-1}| -1)$. Thus, there exists a square $x x$ with $x$ being a cyclic permutation of $s_{n-1}$ to the right of zero. Since this is true for infinitely many $n$, we can use Lemma \[zweiblock\] to exclude eigenvalues.\
[*Case 2*]{}: There is an $n_0\in {\bf N}_0$ s.t. $z_0^n=s_n $ for all $n\geq n_0$\
By $\limsup_{n\to \infty} a_n\geq 3$, there are infinitely many $n\geq
n_0$ s.t. $a_n\geq 3$. Fix such an $n$. As $a_n\geq 3$ and $z_0^n=s_n$, there are three cases by Corollary \[potenzen\].\
\
[*Subcase 1*]{}: $z_0^n=z_1^n=z_2^n=s_n$.\
In this case we have $z_0^n z_1^n z_2^n=s_n s_n s_n$.\
[*Subcase 2*]{}: $z_0^n=z_{-1}^n=z_{-2}^n=s_n$.\
In this case we have $ z_{-2}^n z_{-1}^n z_0^n=s_n s_n s_n$.\
[*Subcase 3*]{}: $z_{-1}^n=z_0^n=z_1^n= s_n$, $ z_{-2}^n=z_2^n=s_{n-1}$.\
Calculating as in Case 1 we get $z_0^n z_1^n z_2^n z_3^n=s_n s_n s_n w$ with a suitable word $w$.\
\
Thus, in all subcases we can conclude as in Case 1 that there exists a square $x x$, with $x$ being a cyclic permutation of $s_n$ either to the left or to the right of zero. Since this applies to infinitely many $n$, we can use Lemma \[zweiblock\] to exclude eigenvalues in Case 2 as well. This proves the proposition.$\Box$\
\
[*Proof of Proposition \[fib\].*]{} The proof is similar to the proof of Proposition \[drei\]. So fix $\theta\in [0,1)$. By Lemma \[partition-lemma\], there exists a unique $n$-partition $ ((I_j^n,z_j^n))_{j\in {\bf Z}}$ of $ v_{\alpha,\theta}.$ Again we will consider two cases.\
[*Case 1*]{}: There are infinitely many $n\in {\bf N}_0$ with $z^n_0=s_{n-1}$.\
This case can be treated as in the proof of Theorem \[drei\].\
[*Case 2*]{}: There is an $n_0\in {\bf N}_0$ s.t. $z_0^n=s_n $ for all $n\geq
n_0$.\
By $\limsup_{n\to \infty} a_n=1$, there is an $n_1\in {\bf N}_0$ s.t. $a_n=1$ for all $n\geq
n_1$. Let $c:=\max\{n_0,n_1\}$.\
As $z_0^n=s_n$ and $s_{n+1}=s_n s_{n-1}$ for all $n\geq c$, it follows easily by induction that $$d_0^n=d_0^c \;\:\mbox{for all} \;\:n \geq c.$$ Moreover $z_0^n=s_n$ and $s_{n+1}=s_n s_{n-1}$ imply $z_1^n=s_{n-1} $ for all $n\geq c$ and this in turn implies $z_2^n=s_n$ for all $n\geq
c$. We therefore have $$z_0^n z_1^n z_2^n=s_n s_{n-1} s_n= s_n s_n w,$$ where we used $s_{n-1} s_n= s_n w$ with a suitable word $w$ (cf. Proposition \[wunderformel\]). Putting all this together, we see $$v_{\alpha,\theta}(d_0^c)...v_{\alpha,\theta}(d_0^c+ 2|s_n| -1)= s_n s_n \;\:\mbox{for all} \;\:n \geq c.$$ Again, an application of Lemma \[zweiblock\] yields the desired absence of eigenvalues, concluding the proof. $\Box$\
\
[*Acknowledgment.*]{} One of us (D. L.) gratefully acknowledges financial support from Studienstiftung des deutschen Volkes (Doktorandenstipendium).
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abstract: 'We make use of $\mathcal{F}$-structures and technology developed by Paternain - Petean to compute minimal entropy, minimal volume, and Yamabe invariant of symplectic 4-manifolds, as well as to study their collapse with sectional curvature bounded from below. À la Gompf, we show that these invariants vanish on symplectic 4-manifolds that realize any given finitely presented group as their fundamental group. We extend to the symplectic realm a result of LeBrun which relates the Kodaira dimension with the Yamabe invariant of compact complex surfaces.'
address:
- |
Instituto de Matemáticas - Universidad Nacional Autónoma de México\
Circuito Exterior, Ciudad Universitaria\
Coyoacán, 04510\
Mexico City\
Mexico
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Scuola Internazionale Superiori di Studi Avanzati\
Via Bonomea 265\
34136\
Trieste\
Italy
author:
- 'Pablo Suárez-Serrato and Rafael Torres'
title: 'A note on collapse, entropy, and vanishing of the Yamabe invariant of symplectic 4-manifolds'
---
Introduction and main results
=============================
Let $(M, g)$ be a closed oriented Riemannian manifold. The minimal entropy $h(M)$ is defined as the infimum of the topological entropy of the geodesic flow of a smooth metric $g$ on $M$. The simplicial volume $||M||$ was defined by Gromov [@[G82]] as the infimum of $\Sigma_{i}|r_{i}|$ where $r_{i}$ are the coefficients of any *real* singular cycle representing the fundamental class of $M$. It was introduced in relation to the [*minimal volume*]{}, which Gromov defined as the infimum of volumes of metrics whose sectional curvature is bounded between -1 and 1. Other types of minimal volumes may be considered, for example, ${\rm Vol}_K(M)$, which is the infimum of volumes of metrics with sectional curvature bounded [*below*]{} by -1. Due to rescalling properties of these invariants and in order to obtain interesting data, one assumes the normalization $Vol((M, g)) = 1$ of the volume.\
The computation of these invariants can be challenging. The existence of a circle action, for example, implies that they all vanish. A generalization of a circle action is given by the notion of an $\mathcal{F}$-structure, as introduced by Cheeger and Gromov [@[CG1]; @[G82]] (Definiton \[Definition F\]). In a series of papers, Cheeger-Gromov [@[CG1]; @[CG2]] and Paternain-Petean [@[PP1]; @[PP2]; @[PP3]] have shown that the existence of an $\mathcal{F}$-structure implies that $h(M)$, $||M||$, and ${\rm Vol}_K(M)$ are all zero.\
Recall the minmax definition of the [*Yamabe invariant*]{} [@[Be]; @[Sc]]. Consider a conformal class $$\gamma:= [g] = \{ug : M\overset{u}{\rightarrow} {\mathbb{R}}^+\}$$ of Riemannian metrics on $M$. The Yamabe constant of $(M, \gamma)$ is defined as:$$\mathcal{Y}(M, \gamma):= \underset{g\in \gamma}{\inf} \frac{\int_M {\rm Scal}_g d{\rm vol}_g}{({\rm Vol} (M, g))^{2/n}}.$$
Here ${\rm Scal}_g$ stands for the scalar curvature and $d{\rm vol}_g$ for the volume form that is associated to the smooth metric $g$. The Yamabe invariant of $M$ is then given by: $$\mathcal{Y}(M):= \underset{\gamma}{\sup} \mathcal{Y}(M, \gamma).$$
LeBrun [@[LeB]] and Paternain-Petean [@[PP2]] studied the value of these fundamental invariants of Riemannian manifolds on complex surfaces. Given the fundamental role that symplectic manifolds play in our understanding of 4-manifold topology, a goal of the present paper is to study the value of these invariants on the much larger class of symplectic 4-manifolds. The principal technical result is to equip constructions of symplectic 4-manifolds with an $\mathcal{F}$-structure.\
In order to state our main results, we first recall the following definition [@[L1]]. Let $(M, \omega)$ be a minimal symplectic 4-manifold, and let $K_{\omega}$ denote its canonical bundle. The symplectic Kodaira dimension $kod(M, \omega)$ of $(M, \omega)$ is defined as:
$$kod(M, \omega) = \left\{
\begin{array}{ccccr}
-\infty & \quad{\rm if}\quad & K_{\omega} \cdot [\omega] < 0 & \quad{\rm or}\quad & K_{\omega} \cdot K_{\omega} < 0\\
0 & \quad{\rm if}\quad & K_{\omega} \cdot [\omega] = 0 & \quad{\rm and}\quad &K_{\omega} \cdot K_{\omega} = 0\\
1 & \quad{\rm if} \quad & K_{\omega} \cdot [\omega] > 0 & \quad{\rm and}\quad & K_{\omega} \cdot K_{\omega} = 0\\
2 & \quad{\rm if} \quad & K_{\omega} \cdot [\omega] > 0 & \quad{\rm and}\quad & K_{\omega}\cdot K_{\omega} > 0
\end{array}
\right.$$
The symplectic Kodaira dimension of a symplectic manifold is defined as the Kodaira dimension of its minimal model, and in the presence of a holomorphic structure, the holomorphic and the symplectic Kodaira dimensions coincide [@[L1]; @[DZ13]].
[\[Theorem Gen\]]{} All known examples of closed symplectic 4-manifolds of symplectic Kodaira dimension at most one admit an $\mathcal{F}$-structure.
No closed symplectic 4-manifold of symplectic Kodaira dimension two admits an $\mathcal{F}$-structure.
[\[Corollary Gen\]]{} Every known closed symplectic 4-manifold of Kodaira dimension at most one has zero minimal entropy, zero minimal volume, and it collapses with sectional curvature bounded from below.
Symplectic 4-manifolds of Kodaira dimension two do not collapse with bounded scalar curvature.
LeBrun [@[LeB]] found an explicit relation between the holomorphic Kodaira dimension and the Yamabe invariant of compact complex surfaces. We extend his result to the symplectic realm.
[\[Theorem Kod\]]{} Let $(M, \omega)$ be a closed symplectic 4-manifold whose existence is known at the time this paper was written. The following holds. $$\mathcal{Y}(M)= \left\{
\begin{array}{cccr}
> 0 & \quad {\rm if} \quad & kod(M, \omega) = & -\infty\\
0 & \quad {\rm if}\quad & kod(M, \omega) = & 0\\
0 & \quad {\rm if}\quad & kod(M, \omega) = & 1\\
<0 & \quad {\rm if}\quad & kod(M, \omega) = & 2
\end{array}
\right.$$
No symplectic 4-manifold of Kodaira dimension one realizes its Yamabe invariant, and neither do several symplectic 4-manifolds of Kodaira dimension zero. This has been previously observed for Kähler surfaces by LeBrun [@[LeB]], and on the homeomorphism type of the K3 surface by the second author of this note [@[To]]. A result of Hitchin [@[Hi]] is fundamental for these purposes.\
A classical result of Gompf [@[Go]] states that any finitely presented group can be the fundamental group of a symplectic 4-manifold. Building on a construction of Baldridge-Kirk [@[BK0]], our third main result addresses the value of these invariants for the manifolds that they constructed in their paper. The precise statement is as follows.
[\[Theorem Prescribed\]]{} Let $G$ be a finitely presented group. There exists a minimal symplectic 4-manifold $M(G)$ with fundamental group $\pi_1(M(G)) \cong G$ and such that $M(G)$ admits an $\mathcal{F}$-structure. Consequently, $$h(M(G)) = 0 = ||M(G)|| = Vol_K(M(G))$$ and $M(G)$ collapses with sectional curvature bounded from below.
Moreover, the Yamabe invariant satisfies $$\mathcal{Y}(M(G)) = 0$$ and it is not realized, i.e., there exist no scalar-flat Riemannian metrics on $M(G)$.
The organization of the paper is as follows. Section \[Section F\] contains the definition of our main technical tool and result. Interesting constructions of $\mathcal{F}$-structures are given in Section \[Section F\], Section \[Section Log\], Section \[Section Zero\], and Section \[Section One\]. The strategy and results used to show the vanishing and non-realization of the Yamabe invariant are described in Section \[Section YI\]. Theorem \[Theorem Prescribed\] is proven in Section \[Section Proof\]. Theorem \[Theorem Gen\], Corollary \[Corollary Gen\], and Theorem \[Theorem Kod\] are proven by cases in terms of Kodaira dimensions in Section \[Section ProofKod\], where each case is addressed in a subsection.
Acknowledgements
----------------
R. T. thanks Claudio Arezzo, Steven Frankel, Baptiste, Morin, and Stefano Vidussi for useful conversations. He gratefully acknowledges the Erwin Schrödinger International Institute for Mathematical Physics for its support and hospitality during the writing of the manuscript. PSS thanks CONACyT Mexico and PAPIIT UNAM for supporting various research activities. Both authors would like to thank Jimmy Petean for interesting conversations, and Weiyi Zhang for his interest in our paper.
Collapse, $\mathcal{F}$-structures, and Yamabe invariant
========================================================
$\mathcal{F}$-structures and collapse
-------------------------------------
[\[Section F\]]{} The concept of an $\mathcal{F}$-structure was introduced by Gromov [@[G82]] in terms of sheaves. We will work through out the manuscript with the equivalent definition given in Paternain-Petean [@[PP3] Section 2].
[\[Definition F\]]{}An *$\mathcal{F}$-structure* on a smooth closed manifold $M$ is given by
- a finite open cover $\{U_1, \ldots, U_N\}$ of $M$;
- a finite Galois covering $\pi_i: \widetilde{U_i}\rightarrow U_i$ with $\Gamma_i$ a group of deck transformations for $1\leq i \leq N$;
- a smooth effective torus action with finite kernel of a $k_i$-dimensional torus $$\phi_i: T^{k_i}\rightarrow Diff(\widetilde{U_i})$$ for $1\leq i \leq N$;
- a representation $\Phi_i: \Gamma_i \rightarrow Aut(T^{k_i})$ such that $$\gamma(\phi_i(t)(x)) = \phi_i(\Gamma_i(\gamma)(t))(\gamma x)$$ for all $\gamma \in \Gamma_i$, $t\in T^{k_i}$, and $x\in \widetilde{U}_i$;
- for any subcollection $\{U_{i_1}, \ldots, U_{i_l}\}$ that satisfies $U_{i_1\cdots i_l}:= U_{i_1} \cap \cdots \cap U_{i_l} \neq \emptyset$, the following compatibility condition holds: let $\widetilde{U}_{i_1\cdots i_l}$ be the set of all points $(x_{i_1}, \ldots, x_{i_l}) \in \widetilde{U}_{i_1}\times \cdots \times \widetilde{U}_{i_l}$ such that $\pi_{i_1}(x_{i_1}) = \cdots = \pi_{i_l}(x_{i_l})$. The set $\widetilde{U}_{i_1\cdots i_l}$ covers $\pi^{-1}(U_{i_1\cdots i_l}) \subset \widetilde{U}_{i_1\cdots i_l}$ for all $1\leq j \leq l$. It is required that $\phi_{i_j}$ leaves $\pi^{-1}_{i_j}(U_{i_1\cdots i_l})$ invariant, and it lifts to an action on $\widetilde{U}_{i_1\cdots i_l}$ such that all lifted actions commute.
- An $\mathcal{F}$-structure is called a *$\mathcal{T}$-structure* if the Galois coverings $\pi_i: \widetilde{U}_i \rightarrow U_i$ can be taken to be trivial for every $i$.
The reader is directed towards [@[CG1]; @[CG2]; @[PP1]; @[PP2]; @[PP3]; @[SS09]; @[PSS]; @[To]] for interesting constructions of $\mathcal{T}$-structures. The key ingredient that we use to prove our main results is the following theorem, which explains a motivation to equip manifolds with a $\mathcal{F}$-structure.
[\[Theorem PP\]]{} Paternain-Petean [@[PP1]]. If a closed n-manifold $M$ admits an $\mathcal{F}$-structure, then $$h(M) = 0 = ||M|| = Vol_K(M),$$ and it collapses with sectional curvature bounded from below.
If $n\geq 3$, then the Yamabe invariant of $M$ satisfies $$\mathcal{Y}(M) \geq 0.$$
Every $\mathcal{F}$-structure that is constructed in this manuscript is a $\mathcal{T}$-structure.
Logarithmic transformations and polarized $\mathcal{T}$-structures
------------------------------------------------------------------
[\[Section Log\]]{} Let $T$ be a 2-torus of self-intersection zero that is contained inside a 4-manifold $M$. The tubular neighborhood $\nu(T)$ is diffeomorphic to $T^2\times D^2$. A logarithmic transfomation is the procedure of deleting the submanifold $\nu (T)$ from $M$ and gluing back $T^2\times D^2$ $$(M - \nu(T)) \cup_{\varphi} (T^2\times D^2)$$ using a diffeomorphism $\varphi: T^2\times \partial D^2$ to identify the common boundaries (see [@[BK0]; @[CL]] for more details). In what follows, $\Sigma_g$ denotes a closed oriented surface of genus $g\in {\mathbb{N}}$.
[\[Proposition Luttinger\]]{} There exists a 4-manifold $M_g(i)$ that is obtained through an application of $i\in \{0, 1, \ldots, 2g\}$ logarithmic transformations to $T^2\times \Sigma_g$ along homologically essential Lagrangian tori taken with respect to the symplectic form $\pi^{\ast}\omega_{T^2}\oplus \pi^{\ast}\omega_{\Sigma_g}$, which admits a polarized $\mathcal{T}$-structure for all choices of $i$ and $g\in {\mathbb{N}}$. Moreover, $b_1(M_g(i)) = 2 + 2g - i$, and $c_2(M_g(i)) = 0 = \sigma(M_g(i))$.
The Lagrangian tori that are used in the logarithmic transformations are of the form $$\{x_1\}\times S^1\times S^1\subset S^1\times (S^1\times \Sigma_g) = T^2\times \Sigma_g$$ and $$S^1\times \{x_2\}\ \times S^1 \subset S^1\times S^1\times \Sigma_g = T^2 \times \Sigma_g.$$ Let $\alpha_{j}$ be a curve in $T^2$ that carries a generator of the group $\pi_1(T^2)$, and let $\beta_{k}$ be a curve in $\Sigma_g$ that carries a generator of $\pi_1(\Sigma_g)$. We denote the Lagrangian tori by $T_{j, k} :=\alpha_{j}\times \beta_{k}$. When handling several tori simultaneously, we will use the heavy notation $\alpha^{(i)}_j, \beta^{(i)}_k$, and $T^{(i)}_{j, k}$. The curves $\alpha^{(i)}_j$ are parallel push offs of $\alpha_j$, and the curves $\beta^{(i)}_k$ of $\beta_k$ (see [@[BK]] for details). The Lagrangian framings will be used for these curves on every logarithmic transformation, and we denote them by $S^1_{\alpha_i}$ and $S^1_{\beta_k}$. The meridian $\mu_{j, k}$ of $T_{j, k}$ in the complement of the tubular neighborhood of the torus inside $M$ is a curve within the same isotopy class of $\{t\}\times \partial D^2 \subset \partial \nu(T_{j, k})$. Recall that two homotopic loops inside a 4-manifolds are isotopic. The diffeomorphism $\varphi$ used to glue the pieces together satisfies $\varphi_{\ast}([\partial D^2]) = p [S^1_{\alpha_j}] + q[S^1_{\beta_k}] + r [\mu_{j, k}]$ in the homology group $H_1(M - \nu(T_{j, k}); {\mathbb{Z}}) = {\mathbb{Z}}^3$. Without loss of generality, we can assume that the tori $T_{j, k}$ are disjoint and the surgeries that will be performed require only one of the integers $p$ or $q$ to be non-zero.
Each such such logarithmic transformations reduces the first Betti number by one, and its second Betti number by two. The Euler characteristic is invariant under torus surgeries, and Novikov additivity implies that so is the signature. In particular, we have $c_2(M_g(i)) = c_2(T^2\times \Sigma_g) = 0$, and $\sigma(M_g(i)) = \sigma(T^2\times \Sigma_g) = 0$.
Fix arbitrary $g$ and $i$; these choices depend on the desired values of first Betti number of $M_g(i)$. We apply the logarithmic transformations simultaneously to $T^2\times \Sigma_g$ to obtain $$M_g(i):= (T^2\times \Sigma_g - \underset{i}{\bigsqcup} \nu(T_{j, k})) \bigcup_{\varphi_{i}} (\underset{i}{\bigsqcup} T^2 \times D^2),$$ where notation is being abused. Up to diffeomorphism of the resulting manifold, the gluing diffeomorphism $\varphi_{i}$ can be replaced by an affine transformation $A_{i}$ of the common $T^3$ boundary that is isotopic to $\varphi_{i}$.
The task at hand is to equip $T^2\times \Sigma_g$ and each copy of $T^2\times D^2$ with polarized $\mathcal{T}$-structures such that they commute with each other at their common $T^3$-boundaries. We will do so by equipping each piece in the decomposition of $M_g(i)$ with a free circle action. This yields a polarized $\mathcal{T}$-structure on $M_g(i)$, and we proceed to do so now. For each curve $\alpha^{(i)}_{j}$ there exists a free $S^1$-action $\sigma_{j}$ on $T^2\times \Sigma_g$ whose orbits are in the same homotopy class of $\alpha^{(i)}_{j}$, and these actions commute among them.
Now equip each copy of $T^2 \times D^2$ with a fixed point free $S^1$-action $\tau_{i}$ whose orbits are in the homotopy class of the image of $\alpha^{(i)}_j$ under the affine transformation $A_{i}\alpha^{(i)}_{j}$. Since the actions $\sigma_{i}$ and $\tau_{i}$ commute with respect to conjugation with the affine maps $A_{i}$$$A_{i}^{-1}\tau_{i} A_{i} \sigma_{i} = \sigma_{i} A_{i}^{-1}\tau_{i} A_{i},$$ (see [@[SS09]] for details). The local circle actions paste together to yield a polarized $\mathcal{T}$-structure on $M_g(i)$. Since the choices of values of $i$ and $g$ were arbitrary, this concludes the proof.
Vanishing of Yamabe invariant
-----------------------------
[\[Section YI\]]{} In this section we recall the results that we use to prove the vanishing of the Yamabe invariant for symplectic 4-manifolds of nonnegative Kodaira dimension, and its non-realization. In particular, we build greatly upon work of Hitchin, Kazdan-Wagner, LeBrun, Taubes, and Paternain-Petean. Theorem \[Theorem PP\] states that in the presence of an $\mathcal{F}$-structure, there is the collapse $Vol_K(M) = 0$, which implies $Vol_{Ric}(M) = 0 = Vol_{|Scal|}(M)$. The sectional curvature is denoted by $K$ and the Ricci curvature by Ric. If the dimension of $M$ is at least three, then $Vol_{|Scal|}(M) = Vol_{Scal}(M)$, and $Vol_{Scal}(M) = 0$ is equivalent to $\mathcal{Y}(M) \geq 0$ [@[LeB] Proposition 5].\
The non-vanishing of the Seiberg-Witten invariant of a symplectic 4-manifold [@[Ta]] is a known obstruction for the existence of a Riemannian metric of positive scalar curvature [@[W94]]. Moreover, $\mathcal{Y}(M) > 0$ if and only if there exists a Riemannian metric of positive scalar curvature on $M$. Hence, one obtains the first part of the following known result.
[\[Proposition ZeroYamabe\]]{} Let $M$ be a closed manifold of dimension greater than two, which collapses with scalar curvature bounded from below. If $M$ does not admit a Riemannian metric of positive scalar curvature, then its Yamabe invariant satisfies $$\mathcal{Y}(M) = 0,$$ and any scalar-flat Riemannian metric on $M$ is Ricci-flat.
The statement concerning the vanishing of the Ricci curvature is an instance of Kazdan-Wagner’s result on the trichotomy problem of scalar curvature of a Riemannian metric on a compact manifold [@[Be] 4.35 Theorem].\
To prove that the Yamabe invariant is not realized, we will use a beautiful theorem of Hitchin [@[Hit1]]. In the case of Kodaira dimension zero and one, we have $c_1^2(M) = 2c_2(M) + 3\sigma(M) = 0$. Hitchin’s result [@[Be] 6.37 Theorem] states that a Ricci-flat oriented 4-manifold has either zero sectional curvature, is the K3 surface, the Enriques surface, or a quotient of the later by a free antiholomorphic involution. Since the holomorphic Kodaira dimension coincides with the symplectic one, this immediately implies that no symplectic 4-manifold of Kodaira dimension one can realize a vanishing Yamabe invariant, and restricts those of Kodaira dimension zero that do realize it (see [@[LeB]; @[To]]).
Symplectic sums: proof of Theorem \[Theorem Prescribed\]
--------------------------------------------------------
[\[Section Proof\]]{} Let $M_1$ and $M_2$ be closed symplectic 4-manifolds, each containing an embedded symplectic torus $T_{M_1}\subset M_1$ and $T_{M_2}\subset M_2$ with trivial normal bundles. In particular their tubular neighborhoods $\nu(T_{M_1})$ and $\nu(T_{M_2})$ are diffeomorphic to $T^2\times D^2$. The *symplectic sum* of $M_1$ and $M_2$ along $T_{M_1}$ and $T_{M_2}$ is $$M_1\#_{T_{M_1} = T_{M_2}} M_2 := (M_1 - \nu(T_{M_1})) \cup_{\varphi} (M_2 - \nu(T_{M_2}))$$ where the gluing map $\varphi: \partial \nu(T_{M_1})\rightarrow \partial \nu(T_{M_2}) = T^2\times \partial D^2 = T^3$ is an orientation-reversing diffeomorphism. Moreover, any diffeomorphism of the 3-torus is isotopic to an affine transformation (see [@[SS09] Corollary 10] for a proof).\
We now proceed to prove Theorem \[Theorem Prescribed\]. We build upon the following result.
[\[Theorem BK\]]{} Baldridge-Kirk [@[BK0]]. Let $G$ be a group with a presentation that consists of $g$ generators and $r$ relations. There exists a minimal symplectic 4-manifold $M(G)$ with fundamental group $\pi_1(M(G)) = G$, and characteristic numbers $c_2(M(G)) = 12(g + r + 1)$ and $\sigma(M(G)) = - 8(g + r + 1)$.
The first claim of Theorem \[Theorem Prescribed\] is that the symplectic manifold $M(G)$ admits a $\mathcal{T}$-structure, and we proceed to construct it. Baldridge-Kirk built the manifold $M(G)$ as a symplectic sum of $g + r + 1$ copies of the elliptic surface $E(1)$ with a symplectic manifold of the form $X = Y\times S^1$, where $Y$ is a fibered 3-manifold. The group $G$ can be expressed in terms of the fundamental group of $X$ as $$G\cong \pi_1(X)/N(s, t, \gamma_1, \cdots, \gamma_{r +g}),$$contains classes where $\{s, t, \gamma_1, \cdots, \gamma_{r + g}\}$ are classes in the fundamental group of $X$ and $N(s, t, \gamma_1, \cdots, \gamma_{r+g})$ is the normal subgroup generated by such classes. The pieces are pasted together along regular fibers of the elliptic surfaces, and symplectic tori $\{T_0, T_1, \cdots, T_{g+r}\}$ contained in $X$. The two generators of $\pi_1(T_0)$ represent the class $s$ and $t$, and the two generators of $\pi_1(T_i)$ represent the classes $s$ and $\gamma_i$. All these tori have self-intersection zero, and their tubular neighborhoods are diffeomorphic to $T^2\times D^2$ (see [@[Go]; @[BK0]] for further details on the construction). The task at hand is to choose $\mathcal{F}$-structures on the elliptic surface and on the product 4-manifold, which commute with the gluing map at the common $T^3$-boundaries. This ensures that they paste well together into a global $\mathcal{F}$-structure on $M(G)$. In particular, all the $\mathcal{F}$-structures involved are $\mathcal{T}$-structures.\
The choices of $\mathcal{T}$-structures are the following. Take the canonical $S^1$-action on $X = Y\times S^1$ given by rotation on the second factor, and denote this polarized $\mathcal{T}$-structure by $\tau_0$. The $\mathcal{T}$-structure we use for $E(1)$ blocks was constructed by Paternain-Petean in [@[PP1] Proof of Theorem 5.10] using its structure of an elliptic fibration, and we proceed to describe it. Consider the diffeomorphism $J := (I, H): T^2\times S^2\rightarrow T^2\times S^2$ with eight fixed points given by $I:{\mathbb{R}}^2/{\mathbb{Z}}^2\rightarrow {\mathbb{R}}^2/{\mathbb{Z}}^2 = T^2$ as $z\mapsto - z$, and $H:S^2\rightarrow S^2$ as the 180 degrees rotation about the z-axis. Let $B\subset {\mathbb{C}}^2$ be a ball on which the circle acts by $\lambda\cdot (w_1, w_2) = (w_1, \lambda\cdot w_2)$. Consider $U:= \{(z, l)\in B\times \mathbb{CP}^1 : z\in l\}$, and the canonical projection $U\rightarrow B$. The aforementioned circle action on the ball $B$ commutes with the involution $J$, and hence it induces a circle action on $U$. Inside $T^2\times S^2$, identify a neighborhood of each fixed point with $B$ and replace it with $U$ to construct a complex surface $\tilde{S}$. The involution $J$ extends to an involution $\tilde{J}: \tilde{S}\rightarrow \tilde{S}$ whose fixed point set consists of eight spheres. In particular, one has $E(1) = \tilde{S}/\tilde{J}$ and it can be expressed as $E(1)\rightarrow S^2$ with $T^2$ as a generic fiber. Paternain-Petean extend the circle action on the $U$ piece to a circle action on the fibers corresponding to the pre images of the north and south poles of $E(1)\rightarrow S^2$. On the complement of these fibers, the elliptic surface is the total space of a fiber bundle with structure group $\{id, I\}$. Call this $\mathcal{T}$-structure $\tau_1$.\
The gluing diffeomorphism $\varphi$ that is used to construct the symplectic sum of $X$ and a copy of $E(1)$ can be assumed to be isotopic to an affine transformation of the common 3-torus boundary of the building blocks. With respect to this affine representative of the isotopy class of $\varphi$, the circle actions $\tau_0$ and $\tau_1$ commute after conjugation with the affine map. Therefore, they paste well together to a global $\mathcal{T}$-structure (the precise computations can be found in [@[SS09]]). Moreover, this extension property for $\mathcal{T}$-structures holds true for a finite arbitrary number of symplectic sums. Therefore, equipping each of the $g + r + 1$ copies of $E(1)$ with the $\tau_1$ structure and $X$ with the circle action $\tau_0$ yield a $\mathcal{T}$-structure on the symplectic manifold $M(G)$.\
Theorem \[Theorem PP\] says its minimal topological entropy and simplicial volume vanish, and that it collapses with sectional curvature bounded from below. Proposition \[Proposition ZeroYamabe\] now says that its Yamabe invariant is $\mathcal{Y}(M(G)) = 0$. Indeed, Taubes’ results [@[Ta]] imply that the Seiberg-Witten invariants of $M(G)$ are non-trivial, and therefore it does not admit a Riemannian metric of positive scalar curvature. Suppose the Yamabe invariant were achieved, that is, that there exists a Riemannian metric of zero scalar curvature on $M(G)$. Such a metric would have to be Ricci-flat (cf. Proposition \[Proposition ZeroYamabe\]), and in particular an Einstein metric [@[Be]]. The characteristic numbers of Theorem \[Theorem BK\] imply that the equality is achieved $|\sigma(M(G))| = \frac{3}{2} c_2(M(G))$. A well-known result of Hitchin [@[Be] 6.37 Theorem] implies that the holomorphic Kodaira dimension of such a manifold is zero. Recall that the symplectic Kodaira dimension of $M(G)$ is one. Since the holomorphic Kodaira dimension, and the symplectic Kodaira dimension coincide, we conclude this is a contradiction. In particular, the Yamabe invariant of $M(G)$ vanishes and it is not realized (cf. Section \[Section YI\]). This concludes the proof of Theorem \[Theorem Prescribed\].
Proof of Theorem \[Theorem Gen\], Corollary \[Corollary Gen\], and Theorem \[Theorem Kod\]
==========================================================================================
[\[Section ProofKod\]]{}
As it was mentioned in the introduction, the proofs of our main results are done case-by-case in terms of the symplectic Kodaira dimension. Each dimension is addressed in a subsection. Notice that it suffices to consider only minimal symplectic 4-manifolds in our proofs. Indeed, any non-minimal symplectic 4-manifold $N$ is diffeomorphic to a connected sum $N = M\# k \overline{\mathbb{CP}^2}$ for $k\in {\mathbb{N}}$, where $M$ is its symplectic minimal model. The manifold $\overline{\mathbb{CP}^2}$ admits a non-trivial $S^1$-action, i.e., a $\mathcal{T}$-structure. Paternain-Petean [@[PP1] Theorem 5.9] have shown that the existence of $\mathcal{F}$-structures is closed under connected sums.
Kodaira dimension $- \infty $
------------------------------
Liu [@[L96]] has classified minimal symplectic 4-manifolds of Kodaira dimension $- \infty$. Such a manifold is either rational or ruled. A rational symplectic 4-manifold is diffeomorphic to $S^2\times S^2$ or to $\mathbb{CP}^2$ blown up $k$ times. A ruled symplectic 4-manifold is diffeomorphic to an $S^2$-bundle over a Riemann surface blown up $k$ times. In particular, all these manifolds admit Kähler structure. Their collapse was studied by LeBrun in [@[LeB]]. Paternain-Petean [@[PP1]; @[PP2]] have equipped these manifolds with a $\mathcal{T}$-structure, yielding the following result.
LeBrun, Liu, Paternain-Petean. Every symplectic 4-manifold of Kodaira dimension $- \infty$ admits an $\mathcal{F}$-structure, has vanishing minimal entropy and simplicial volume, and collapses with sectional curvature bounded from below. Its Yamabe invariant is positive.
Kodaira dimension zero
----------------------
[\[Section Zero\]]{} The known examples of symplectic 4-manifolds of vanishing Kodaira dimension at the time of writing of this note can be summarized in the following list.
- K3 surface,
- Enriques surface,
- $T^2$-bundles over $T^2$,
- $S^1$-bundle over $Y$, where $Y$ is a 3-manifold that fibers over the circle,
- primary Kodaira surface, and
- cohomologically symplectic infrasolvmanifolds.
The reader is directed towards [@[L1]; @[L2]; @[FV13]] for further details. In particular, all known symplectic 4-manifolds of zero Kodaira dimension with positive first Betti number are infrasolvmanifolds (see [@[Hi]] for the definition).
[\[Kodaira Zero\]]{} Let $M$ be a minimal symplectic closed 4-manifold of symplectic Kodaira dimension zero that is included in the previous list. Then $M$ admits an $\mathcal{F}$-structure, and consequently $$h(M) = 0 = ||M|| = Vol_K(M),$$ and $M$ collapses with sectional curvature bounded from below.
The Yamabe invariant satisfies $$\mathcal{Y}(M) = 0,$$ and it is realized if and only if $M$ is diffeomorphic to the K3 surface, Enriques surface, a complex torus or a hyperelliptic complex surface.
Moreover, if $c_2(M) = 0$, the $\mathcal{F}$-structure is polarized and $$MinVol(M) = 0,$$ and $M$ collapses with bounded sectional curvature.
The two known examples with small fundamental group are the K3 surface and the Enriques surface. They are both complex elliptic surfaces, hence they admit an $\mathcal{F}$-structure [@[PP1] Theorem 5.10]. The manifolds in the list that have positive first Betti number are infrasolvmanifolds [@[Hi] Chapter 8], and a polarized $\mathcal{F}$-structure was constructed on them in [@[SS09]]. The claim $MinVol(M) = 0$ follows from [@[CG1]]. We add that this claim also follows from Proposition \[Proposition Luttinger\]. Let $G$ be the fundamental group of a minimal symplectic 4-manifold $M$ of symplectic Kodaira dimension zero, and suppose $b_1(G) \neq 0$. The work of Bauer [@[Ba]] and Li [@[L3]] implies $2\leq b_1(G) \leq 4$ and $c_2(M) = 0 = \sigma(M)$. Examples with these Betti and characteristic numbers can be constructed by performing Luttinger surgeries [@[Lu]; @[ADK]] to the 4-torus as in [@[BK]]. It was proven in [@[CL]] that this cut-and-paste operation preserves Kodaira dimension. Proposition \[Proposition Luttinger\] implies that these manifolds admit a polarized $\mathcal{F}$-structure.
Kodaira dimension one
---------------------
[\[Section One\]]{} As it was mentioned in the introduction, Gompf’s work [@[Go]] implies that symplectic 4-manifolds of positive kodaira dimension are not classifiable. Focus is hence shifted to illustrate their diversity in terms of the geography problem [@[Go]]. Baldridge-Li [@[BL]] have studied the geography of symplectic 4-manifolds with Kodaira dimension one, and our task in this section is to equipped the manifolds that they have constructed with a $\mathcal{T}$-structure. Notice that the symplectic manifolds of Theorem \[Theorem Prescribed\] have Kodaira dimension one. We begin by recalling the definitions that are needed to state the result.\
The degeneracy of a symplectic 4-manifold $(M, \omega)$ is the rank of the kernel of the map $$\cup[\omega]: H^1(M; {\mathbb{R}})\rightarrow H^3(M; {\mathbb{R}}).$$
A triple $(a, b, c)\in {\mathbb{Z}}^3$ is said to be admissible if and only if $a = 8k$ for a non-positive integer $k$, $$b\geq max\{0, 2 + a/4\}$$ and $$0\leq c \leq b,$$ and $b - c$ is an even number.
The integer $a$ is the signature $\sigma$ of the 4-manifold $(M, \omega)$, $b$ is the first Betti number, and $c$ its degeneracy.
[\[Proposition BL\]]{} The minimal symplectic 4-manifold $(M, \omega)$ of Kodaira dimension one that realizes any admissible triple $(a, b, c)$ in the sense that $$(a, b, c) = (\sigma(M), b_1(M), d(M, \omega))$$ admits a $\mathcal{T}$-structure. Consequently, every such symplectic 4-manifold has $$h(M) = 0 = ||M|| = Vol_K(M),$$ and $M$ collapses with sectional curvature bounded from below.
Moreover, the Yamabe invariant satisfies $$\mathcal{Y}(M) = 0$$ and it is not realized, i.e., there exist no scalar-flat Riemannian metrics on $M$.
We recall now Baldridge-Li’s [@[BL]] construction of these symplectic manifolds. Consider a surface $\Sigma$ of genus $g>0$, a diffeomorphism $\phi$ of $\Sigma$, and the mapping torus $Y=(\Sigma \times [0,1]) / ((x,1) \sim (\phi(x), 0))$.
In their work, Baldridge-Li use certain symplectic $S^1$-bundles over $Y$ that they call [*bundle manifolds*]{}. The symplectic manifolds that realize any admissible triple in Proposition \[Proposition BL\] are symplectic sums of bundle manifolds and simply connected elliptic surfaces. For simplicity we will focus on the aspect of their construction that yields a $\mathcal{T}$-structure, and the rest of our claimed results.
Any circle bundle over a 3-manifold $Y$ admits a free circle action, and hence a $\mathcal{T}$-structure. Let $t$ be a section of the projection $\pi : Y\to S^1$. The preimage $T:= \pi^{-1}(t)$ is a non-trivial torus contained inside $Y$. Call $s$ the loop spanned by the $S^1$-factor of $Y\times S^1$, and express the torus as $T=s \times t$. Define $\sigma$ as the obvious circle action on the $S^1$-factor of $Y\times S^1$.
Let $T'$ be a regular fiber of $E(n)$ in the neighborhood of a cusp fibre. Take a tubular neighborhood $\nu(T')$ for the elliptic fibration about $T'$, so that $\nu(T')=T^2\times D^2$. Define, for $\tau_{i}, \theta_{i}$ in $S^1$ and $z$ in $D^2$: $$\tau :T^2\times T^2 \times D^2 \to T^2 \times D^2 \quad ; \quad \tau(\tau_1, \tau_2, \theta_1, \theta_2, z)= (\tau_1\theta_1, \tau_2\theta_2, z)$$
Observe that $E(n)$ admits a $\mathcal{T}$-structure which includes the action $\tau$ on $\nu(T')$.
Next let $M = E(n)\#_{T=T'}(Y\times S^1)$ denote the symplectic sum [@[Go]] of $E(n)$ and $Y\times S^1$ along $T$ and $T'$. Then, the $\mathcal{T}$-structure on $E(n)$ and the circle action $\sigma$ on $Y\times S^1$ together form a $\mathcal{T}$-structure on M (as in the proof of Theorem \[Theorem Prescribed\]). The gluing map used in the symplectic sum construction can be replaced by an affine transformation $A$ of $T^3$ in the same isotopy class. The actions $\sigma$ and $\tau$ commute with respect to conjugation with $A$, i.e., $A^{-1}\tau A \sigma = \sigma A^{-1}\tau A$ [@[SS09]]. Therefore, we have equipped $M$ with a $\mathcal{T}$-structure.
Theorem \[Theorem PP\] implies that $h(M) = 0 = ||M|| = Vol_K(M)$ and $M$ collapses with curvature bounded below. The claims that concern the Yamabe invariant are proven using an argument verbatim to that one in the last paragraph of the proof of Theorem \[Theorem Prescribed\], as it was described in Section \[Section YI\].
Kodaira dimension two
---------------------
LeBrun computed the Yamabe invariant of a surface of general type [@[LeB0] Theorem 7] in terms of the square of its first Chern class, and he studied some of their minimal volumes and their collapse with bounded curvatures [@[LeB0] Theorem 8]. The basic ingredients of his proofs are the non-vanishing of certain Seiberg-Witten invariants and the canonical line bundle of the complex surface. Since a symplectic 4-manifold has a canonical line bundle as well, and the symplectic 4-manifolds studied in this section have non-vanishing Seiberg-Witten invariant, his results have broader generality. In particular, he has shown the following proposition [@[LeB0]].
LeBrun. Every symplectic 4-manifold $(M, \omega)$ with $kod(M, \omega) = 2$ satisfies$$\mathcal{Y}(M) < 0$$ and $$Vol_{Scal}(M) > 0.$$ In particular, such 4-manifolds do not admit $\mathcal{F}$-structures.
Let $\mathcal{M}_{Scal}$ be the set of Riemannian metrics on a closed 4-manifold that satisfy $Scal \geq - 12$. LeBrun defined the scalar minimal volume of $(M, g)$ as $$Vol_{Scal}(M): = \underset{g \in \mathcal{M}_{Scal}}{inf} \int_M d vol_g.$$
The claim regarding the Yamabe invariant follows from the inequalities $$\mathcal{Y}(M) = - \underset{\gamma}{inf}|\mathcal{Y}(M, \gamma)| = -\underset{g}{inf}\left[\int_M Scal_g^2 dvol_g \right]^{\frac{1}{2}} \leq - 4\pi \sqrt{2c_1^2(M)} < 0$$ that were studied by LeBrun. Such a symplectic manifold has a non-zero Seiberg-Witten invariant for every Riemannian metric [@[LeB0]], and every conformal class has negative Yamabe constant. This implies the first equality from left to right, and that the Yamabe invariant of symplectic 4-manifolds with Kodaira dimension two is negative. The second inequality from right to left follows from [@[LeB0] Theorem 4]. It is here where the symplectic canonical bundle is used. Since the Yamabe invariant of a 4-manifold that admits an $\mathcal{F}$-structure is non-negative by Theorem \[Theorem PP\], it follows that symplectic 4-manifolds of Kodaira dimension two do not admit $\mathcal{F}$-structures.
Regarding the positivity of the scalar minimal volume, we have the following chain of inequalities $$Vol_{Scal}(M) = \underset{g\in \mathcal{M}}{inf} \frac{(min Scal_g)^2}{144} \int_M dvol_g = \underset{\gamma\in \mathcal{C}}{inf}\underset{g\in \gamma}{inf} \frac{(min Scal_g)^2}{144} \int_M dvol_g =$$ $$= \underset{\gamma\in \mathcal{C}}{inf}\underset{g\in \gamma}{inf} \frac{1}{144} \int_M Scal_g^2 dvol_g = \underset{g\in \mathcal{M}}{inf}\frac{1}{144} \int_M Scal_g^2 dvol_g = \underset{g\in \mathcal{M}}{inf}\frac{|\mathcal{Y}(M, \gamma)|^2}{144} > 0.$$ The set of conformal classes of Riemmanian metrics on $M$ is denoted by $\mathcal{C}$, and $\mathcal{M}$ denotes the set of Riemannian metrics. The first equality from left to right follows from rescaling properties. The equality going from equation (29) to equation (30) follows from [@[LeB0] Lemma 2], since the non-vanishing of the Seiberg-Witten invariant implies that every conformal class has negative Yamabe constant, and therefore the hypothesis of the Lemma are satisfied. The last equality follows from [@[LeB0] Lemma 1].
Since the inequalities$$Vol_{|K|}(M) \geq Vol_K(M) \geq Vol_{Ric}(M) \geq Vol_{Scal}(M),$$ hold, where $K$ is the sectional curvature, and Ric is the Ricci curvature, Gromov’s minimal volume $Vol_{|K|}(M)$ is positive for symplectic 4-manifolds of Kodaira dimension two.
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'Jet quenching is observed at both RHIC and LHC energies. This suggests that partons lose energy as they traverse the medium. When the trigger jet is studied relative to the event plane, the path length dependence of medium modifications can be studied. We present measurements of the angular correlations relative to the event plane between reconstructed jets and charged hadrons in Pb–Pb collisions at $\sqrt{s_{\mathrm{NN}}}=$2.76 TeV in ALICE. A newly implemented, robust background subtraction method to remove the complex, flow dominated, combinatorial background is used in this analysis.'
address: 'Department of Physics, University of Tennessee, Knoxville, Tennessee 37996-1200, USA'
author:
- Joel Mazer
title: 'Jet-hadron correlations relative to the event plane at the LHC with ALICE'
---
Introduction {#Sec:Intro}
============
Jets are ideal probes of the Quark Gluon Plasma (QGP) because they originate from hard-scattered partons created early in the collision, prior to the formation of the medium. These partons are modified in the presence of a medium through collisional energy loss and induced gluon radiation. This modification is observed at both LHC and RHIC energies via the suppression of high-momentum particles [@Adare:2007vu; @Adams:2003kv; @Adare:2010mq; @Adam:2015ewa]. A suppression is also seen for high $\it{p}_{T}$ di-hadron correlations [@PhysRevLett.95.152301; @PhysRevC.80.064912; @PhysRevC.85.014903; @PhysRevC.82.024912].
By reconstructing jets, ideally, a more complete picture of how lost energy is redistributed and how the effects of jet quenching emerge can be obtained. This article will discuss the status of azimuthal correlations of reconstructed jets binned relative to the event plane with charged hadrons in ALICE.
Experiment {#Sec:ExpSetup}
==========
Clusters measured in the Electromagnetic Calorimeter (EMCal) and charged hadrons measured with the central tracking system allow ALICE to study fully reconstructed jets [@Adam:2015ewa]. Charged hadrons are reconstructed from their tracks using information from the Time Projection Chamber (TPC) and the Inner Tracking System (ITS).
Tracks are reconstructed over the full azimuthal range and come from mid-rapidity in a range of $|\eta_{lab}|<$0.7 when reconstructed in a jet, and $|\eta_{lab}|<$0.9 for the associated charged hadrons. The EMCal has an acceptance window of $|\eta_{lab}|<$0.7 and $|\Delta \phi|$=107${^{\rm o}}$. For a complete description of the ALICE detector, see [@Abelev:2014ffa; @Abeysekara:2010ze; @Abelevetal:2014dna; @Alme2010316].
Jet-hadron correlations {#Sect:jethadcorr}
=======================
Data sample {#subsect:Datasample}
-----------
The data used for the correlation analysis was collected by the ALICE Experiment in 2011 during the 2.76 TeV Pb–Pb collision data taking. This work uses events which fired the gamma trigger used by the ALICE EMCal [@Abeysekara:2010ze]. In addition, a cluster constituent of the trigger jet was required to be matched to the fired trigger patch of the event.
Jet Reconstruction {#subsect:JetReco}
------------------
Jets are reconstructed with a resolution parameter of R=0.2 using the anti-$k_{T}$ jet-finding algorithm from the FastJet package [@Cacciari:2011ma]. Jets reconstructed in this analysis require a leading cluster to have $E_{T}>$6.0 GeV, which exceeds the trigger threshold. The jets are reconstructed with constituent tracks of $\it{p}_{T}>$3.0 [GeV/$c$]{}and clusters of $E_{T}>$3.0 GeV. Since the primary goal of jet finding is to reconstruct the jet axis accurately, this high constituent cut limits the influence of background on jet finding.
Measurement {#subsect:Measurement}
-----------
We define the jet-hadron correlation function in heavy-ion collisions by Eq. \[correlation\_eqn\_PbPb\].
$$\frac{1}{N_{trig}} \frac{d^2N_{assoc,jet}}{d\Delta\phi d\Delta\eta} = \frac{1}{\epsilon a N_{trig}}
\frac{d^2 N^{same}_{pairs} }{d\Delta\phi d\Delta\eta}
-b_0(1+ \sum v^{trig}_n v^{assoc}_n \cos(n\Delta \phi))
\label{correlation_eqn_PbPb}$$
Here the first term represents the same event pairs which are divided by an acceptance correction, a, provided by mixed events. Mixed events are used to correct for the limited detector efficiency and acceptance for track pairs. The single track reconstruction efficiency of associated particles is denoted by $\epsilon$. The second term of Eq. \[correlation\_eqn\_PbPb\] is the combinatorial heavy-ion background where $b_{0}$ is the background level and the $v_n$ terms are the Fourier coefficients of the trigger jet and associated particles.
The trigger jets in this analysis are binned in angle relative to the event plane to explore the path length dependence of medium modifications. The orientations are defined such that in-plane is $0 < | \Delta \varphi | < \frac{\pi}{6}$, mid-plane is $\frac{\pi}{6} < | \Delta \varphi | < \frac{\pi}{3}$, and out-of-plane is $\frac{\pi}{3} < | \Delta \varphi | < \frac{\pi}{2}$, where $\Delta\varphi$ denotes the angular difference between the trigger jet and the reconstructed event plane.
Background subtraction {#subsect:backgroundsub}
----------------------
When the trigger jet is restricted relative to the event plane, both the background level and $v_{n}$ are modified and will contain a dependence on the reaction plane resolution $\Re$. The derivation of the event plane dependent background equations are given in [@Bielcikova:2003ku]. The reaction plane resolution corrects for the difference between the reconstructed event plane and the underlying symmetry plane, $\Psi_{n}$.
Various background subtraction methods were developed in [@Sharma:2015qra] and applied to the azimuthal correlation functions. The primary method is the reaction plane fit (RPF). The RPF method works under the assumption that the signal is negligible in the large $\Delta \eta$ and small $\Delta\phi$ region. The 2D correlations are projected over 0.8$<|\Delta\eta|<$1.2 to define the background dominated region, while the signal+background region is defined to be $|\Delta\eta|<$0.6. To ensure as much information as possible is going into the fit by constraining the shape and level of the background, the in-plane, mid-plane, and out-of-plane orientations are simultaneously fit up to fourth order in $v_{n}$ and required to have the same fit parameters.
The background dominated region is fit over the region $|\Delta\phi|<\pi/2$, shown in Fig. \[fig:RPFfitData1.5-2.0\]. We see from the blue band that the RPF fit models the data quite well, even at low $\it{p}_{T}$ where the background is large. The higher order Fourier coefficents clearly matter, as the background for trigger jets mid-plane has four peaks, consistent with a $v_4$ dependence. This method does not require independent measurements of $v_{n}$ and is able to extract the signal with smaller errors while requiring fewer assumptions and less bias than prior subtraction methods. This is especially useful since there are currently no $v^{jet}_3$ or $v^{jet}_4$ measurements.
![\[fig:RPFfitData1.5-2.0\] Signal+background region, background dominated region and RPF fit to background are shown for the 1.5-2.0 [GeV/$c$]{}associated $\it{p}_{T}$ bin for full jets in [Pb–Pb]{}collisions at [$\sqrt{s_{\mathrm{NN}}}$]{} = 2.76 TeV for the 30-50% most central events. The blue band shows the uncertainty of the background fit, which is non-trivially correlated point-to-point [@Sharma:2015qra; @Nattrass:2016cln].](2016-Oct-21-PWfitBgExtract_Ass1.5-2.0_Trig20-40_C30-50free_rbX2_FINAL.pdf){width="37pc"}
Results {#subsect:Results}
-------
We extract the signal by subtracting the large correlated background from the correlation function. Figure \[fig:CorrelationPlot2.0-3.0\] shows the signal for associated particles of 2.0-3.0 [GeV/$c$]{}. The uncertainties are dominated by statistics. With higher statistics, the uncertainties could be vastly reduced to allow for a more precise measurement.
![\[fig:CorrelationPlot2.0-3.0\] Corrected $\Delta \phi$ correlation function for 2.0$<\it{p}_{T}<$3.0 [GeV/$c$]{} associated hadrons for full jets in [Pb–Pb]{}collisions at [$\sqrt{s_{\mathrm{NN}}}$]{} = 2.76 TeV for the 30-50% most central events. The blue band corresponds to the correlated uncertainty and the grey band corresponds to the background uncertainty which is non-trivially correlated point-to-point [@Sharma:2015qra; @Nattrass:2016cln]. There is an additional 6% global scale uncertainty.](2016-Oct-21-dPhiCorrelationsJ20-40_C30-50_2.0-3.0rebinX2BG_FINAL.pdf){width="37pc"}
We define the yields by Eq. \[eqn:Yield\].
$$Y = \frac{1}{N_{trig}} \int_{c}^{d} \int_{a}^{b} \frac{d(N_{meas} - N_{bkgd})}{d\Delta\phi} d\Delta\phi d\Delta\eta
\label{eqn:Yield}$$
Where the integration limits a and b correspond to $\Delta \phi$ values of -1.047 and +1.047 on the near-side and +2.094 and +4.189 on the away-side respectively. In addition, the integration limits c and d correspond to $\Delta \eta$ values of -0.6 and 0.6 for both the near-side and away-side. The near-side (left) and away-side (right) jet yields for 1.0$<\it{p}^{assoc}_{T}<$10.0 [GeV/$c$]{}are shown in Fig. \[fig:Yield\]. The near-side yield is consistent with little or no modification. There is no clear dependence of the away-side peaks on orientation relative to the event plane. There are competing effects across different $\it{p}_{T}$ ranges. Jet quenching could cause a decrease in yield going from in-plane to out-of-plane, while gluon radiation could cause an increase.
![\[fig:Yield\] The near-side yield (left) and away-side yield (right) for full jets in [Pb–Pb]{}collisions at [$\sqrt{s_{\mathrm{NN}}}$]{} = 2.76 TeV for the 30-50% most central events. The colored bands correspond to the correlated uncertainties and the grey band corresponds to the background uncertainty which is non-trivially correlated point-to-point [@Sharma:2015qra; @Nattrass:2016cln]. There is an additional 6% global scale uncertainty. ](2016-Oct-21-NearSideYieldJ20-40_C30-50BGnewL_FINAL.pdf){width="18.5pc"}
![\[fig:Yield\] The near-side yield (left) and away-side yield (right) for full jets in [Pb–Pb]{}collisions at [$\sqrt{s_{\mathrm{NN}}}$]{} = 2.76 TeV for the 30-50% most central events. The colored bands correspond to the correlated uncertainties and the grey band corresponds to the background uncertainty which is non-trivially correlated point-to-point [@Sharma:2015qra; @Nattrass:2016cln]. There is an additional 6% global scale uncertainty. ](2016-Oct-21-AwaySideYieldJ20-40_C30-50BGnewL_FINAL.pdf){width="18.5pc"}
Summary and Outlook {#Sec:Summary}
===================
The jet-hadron correlation results were seen to have their uncertainties dominated by statistics. No significant event plane dependence was seen to within the current uncertainties on the extracted jet yield. The RPF background subtraction method applied to data was seen to have various advantages in that it does not require independent $v_n$ measurements, has less bias and fewer assumptions than prior background subtraction methods, and can extract the signal with great precision. The current results are consistent with the re-analysis of STAR data seen in [@Nattrass:2016cln].
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==========
[10]{} url \#1[[\#1]{}]{}urlprefix\[2\]\[\][[\#2](#2)]{}
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The feasibility of [*trust-free*]{} long-haul quantum key distribution (QKD) is addressed. We combine measurement-device-independent QKD (MDI-QKD), as an access technology, with a quantum repeater setup, at the core of future quantum communication networks. This will provide a quantum link none of whose intermediary nodes need to be trusted, or, in our terminology, a trust-free QKD link. As the main figure of merit, we calculate the secret key generation rate when a particular probabilistic quantum repeater protocol is in use. We assume the users are equipped with imperfect single photon sources, which can possibly emit two single photons, or laser sources to implement decoy-state techniques. We consider apparatus imperfection, such as quantum efficiency and dark count of photodetectors, path loss of the channel, and writing and reading efficiencies of quantum memories. By optimizing different system parameters, we estimate the maximum distance over which users can share secret keys when a finite number of memories are employed in the repeater setup.'
author:
- Nicoló Lo Piparo
- Mohsen Razavi
bibliography:
- 'Bibli28Sept12.bib'
- 'bib.bib'
title: 'Long-Distance Trust-Free Quantum Key Distribution'
---
Introduction
============
Future quantum communications networks will enable secure key exchange among remote users. They ideally rely on user friendly access protocols in conjunction with a reliable network of core nodes [@QInternet_Kimble; @QAccess_Toshiba; @Razavi_MulipleAccessQKD]. For economic reasons, they need to share infrastructure with existing and developing classical optical communication networks, such as passive optical networks (PONs) that enable fiber-to-the-home services [@Shields.PRX.coexist; @Townsend_QI_home_2011]. The first generation of quantum key distribution (QKD) networks are anticipated to rely on a [*trusted*]{} set of core nodes [@secoqc; @Sasaki:TokyoQKD:2011]. This approach, although the only feasible one at the moment, may suffer from security breaches over the long run. In the future generations of quantum networks, this trust requirement can be removed by relying on entanglement in QKD protocols [@Ekert_91; @Biham:ReverseEPR:1996]. This can be facilitated via using the recently proposed measurement-device-independent QKD (MDI-QKD) [@Lo:MIQKD:2012; @MXF:MIQKD:2012; @MDIQKD_finite_PhysRevA2012; @Pan_expMDIQKD_PRL2013] at the access nodes of a PON [@Razavi_IWCIT12] and quantum repeaters at the backbone of the network, as we consider in this paper. The former enables easy access to the network via low-cost optical sources and encoders, whereas the latter may rely on high-end technologies for quantum memories and gates. Both systems, however, rely on entanglement swapping, which makes them naturally merge together. More importantly, in neither systems would we need to trust the intermediary nodes that perform Bell-state measurements (BSMs). In this paper, we study the feasibility of such a [*trust-free*]{} hybrid scheme by finding the relationship between the achievable secret key generation rate as a function of various system parameters. We remark that this setup does not provide full device-independence but it removes the trust requirement from the intermediary network nodes that perform measurement operations. Our work provides insights into the feasibility of such systems in the future. The system proposed in [@Azuma:All_optical_QR_2013] combines MDI-QKD with quantum repeaters by using time reversed all photonic quantum repeaters. However, [@Azuma:All_optical_QR_2013] requires single photon sources as well as large cluster states. Instead, our scheme relies on conventional quantum repeaters, where entangled quantum memories are used to store qubits which are teleported to large distances through entanglement swapping. Moreover, users can use imperfect single-photon sources or lasers. MDI-QKD is an attractive candidate for the access part of quantum networks. First, it provides a means to secure key exchange without trusting measurement devices. This is a huge practical advantage considering the range of attacks on the measurement tools of QKD users [@Qi:TimeShift:2007; @Makarov:Bright:09; @Wiechers:AftergateAttack:2011; @Weier:DeadtimeAttack:2011]. Moreover, at the users’ ends, it only requires optical encoders driven by weak laser pulses. That not only makes the required technology for the end users much simpler, but also it implies that the costly parts of the network, including detectors and quantum memories, are now shared between all networks users, and are maintained by service providers. One final advantage of MDI-QKD is its reliance on entanglement swapping, which makes its merging with quantum repeaters, also relying on the same technique, straightforward. This will help us develop quantum networks in several generations, where the compatibility of older, e.g. trusted-node, and newer, e.g., our trust-free, networks can be easily achieved.In
Quantum repeaters are the key ingredients to trust-free networks. They traditionally rely on quantum memories (QMs) to store entangled states. In order to avoid the exponential decay of rate with channel length, in quantum repeaters, entanglement is first distributed over shorter distances and stored in QMs. Once we learn about the establishment of this initial entanglement, we can perform BSMs to extend entanglement over longer distances [@Razavi.Lutkenhaus.09]. Considering the complexity of joint operations needed for BSMs, as well as possible purification thereafter, quantum repeaters are anticipated to be developed in several stages. The first generation of quantum repeaters may rely on probabilistic approaches to BSMs, which can be implemented using linear optics devices [@DLCZ_01; @Razavi.Amirloo.10; @ProbReps:RevModPhys.2011; @RUS_arXiv]. These systems expect to cover moderately long distances up to around 1000 km without the need for purification. In order to go farther we need to develop efficient tools for purification and deterministic BSMs as was initially envisaged in [@Zoller_Qrepeater_98]. Such deterministic quantum repeaters will replace the probabilistic setups once their technology is sufficiently mature. Finally, the most advanced class of repeaters are the recently proposed no-memory ones [@Munro:NatPhot:2012; @Azuma:All_optical_QR_2013; @Liang:NoMemRep_PRL2014], in which, by using extensive error correction, one can literally transfer quantum states from one point to another.
In this paper, we focus on the probabilistic setups for quantum repeaters, and, among all possible options, we use the protocol proposed in [@Sangouard:single-photon:2007], which relies on single-photon sources (SPSs). In an earlier work [@LoPiparo:2013], we compared the performance of this protocol, which we refer to as the SPS protocol, in the context of QKD, with several other alternatives, once imperfections in the SPSs are accounted for. We found that under realistic assumptions, this protocol is capable of providing the best (normalized) key rate versus distance behavior as compared to other protocols considered in [@LoPiparo:2013]. The particular setup that we are going to consider in this paper is then a phase-encoded MDI-QKD setup, whose reach and rate are improved by incorporating a repeater setup, as above, in between the two users. It is worth noting that the easiest way to improve rate-vs-distance behavior is to add two quantum memories in the MDI-QKD setup [@Brus:MDIQKD-QM_2013; @Panayi_NJP2014; @Nicolo_paper2]. This approach will almost double the distance one can exchange secret keys without trusting middle nodes, but it is not scalable the same way that quantum repeaters are. It, nevertheless, provides a practical route toward building scalable quantum-repeater-based links.
The paper is structured as follows. In Sec. II, we describe the main ingredients of our setup including the phase encoding MDI-QKD and the SPS-based quantum repeaters. In Sec. III, we present our methodology for calculating the secret key generation rate for our hybrid system, followed by numerical results in Sec. IV. We draw our conclusions in Sec. V.
Setup description
=================
![\[fig:General-scheme\]A general scheme for trust-free QKD links. Entangled states are created between internal nodes of the core network using quantum repeaters. The two BSMs will then enable an end-to-end MDI-QKD protocol.](scheme){width="8.6cm"}
In this section we first introduce the general idea behind our trust-free architecture and, then, explain particular MDI-QKD and quantum-repeater protocols considered for its implementation. Let us first consider the ideal scenario considered in Fig. \[fig:General-scheme\]. In this scheme, by using quantum repeaters, we distribute (polarization) entanglement between two memories apart by a distance $L_{\rm rep}$. This operation is part of the core network and is facilitated by the service provider. On the users’ end, each user is equipped with a BB84 encoder, which sends polarization-encoded single photons to a BSM module at a short distance $L_s$ from its respective source. This resembles the access part of the network, where the BSM module is located at the nearest service point to the user. For each transmitted photon by the users, we need an entangled pair of memories to be read, i.e., their states need to be transferred into single photons. These photons will then interact with the users’ photons at the two BSMs in Fig. \[fig:General-scheme\].
The setup of Fig. \[fig:General-scheme\] effectively enables an enlarged MDI-QKD scheme. In MDI-QKD, the two photons sent by Alice and Bob are directly interacting at a BSM module [@Lo:MIQKD:2012]. Here, by the use of entangled memories, it is as if the Alice’s photon is being [*teleported*]{} to the other side, and will interact with the Bob’s photon at the second BSM. The overall effect is, nevertheless, the same, and once Alice and Bob consider the possible rotations in the memory states corresponding to the obtained BSM results, they can come up with correlated or anti-correlated bits for their sifted keys. Post processing is then performed to convert these sifted keys to secret keys.
![[\[fig:Diagram-for-MDI-QKD\]]{}Schematic diagram for a trust-free QKD link based on phase encoding. Memories are entangled using the SPS repeater protocol. Here, PBS stands for polarizing beam splitter and PM stands for phase modulator.](setup){width="8.6cm"}
The same idea as in Fig. \[fig:General-scheme\] can be implemented via phase-encoding techniques as shown in Fig. \[fig:Diagram-for-MDI-QKD\]. Here, for simplicity, we have considered the dual-rail setup. The equivalent, and more practical, single-rail setup can also be achieved by time multiplexing as shown in [@MXF:MIQKD:2012]. In Fig. \[fig:Diagram-for-MDI-QKD\], the quantum repeater ideally leaves memories $A_i$-$B_i$, for $i=1,2$, in the state $|\psi_{\rm ent}\rangle_{A_i B_i} = |0\rangle_{A_i}|1\rangle_{B_i} + |1\rangle_{A_i}|0\rangle_{B_i}$, where we have neglected normalization factors, and $|n\rangle_K$ represents $n$ excitations in memory $K$. The implicit assumption is that the memory is of ensemble type so that it can store multiple excitations [@Razavi.DLCZ.06]. The phase encoding that matches this type of entangled states is as follows. Alice and Bob encode their states either in the $z$ or in the $x$ basis. Alice encodes her bits in the $z$ basis by sending, ideally, a photon in the $r$ or in the $s$ mode. This can be achieved by sending horizontally or vertically polarized pulses to the polarizing beam splitter (PBS) at the encoder. The same holds for Bob and his $u$ and $v$ modes. As for the $x$ basis, we can send a $+45^\circ$-polarized signal through the PBS to generate a superposition of $r\, (u)$ and $s\, (v)$ modes for Alice (Bob) state. Alice (Bob) encodes her (his) bits by choosing the phase value of the phase modulator (PM), $\phi_A \, (\phi_B)$, to be either 0 or $\pi$.
The BSMs used in the scheme of Fig. \[fig:Diagram-for-MDI-QKD\] are probabilistic ones. They will be successful if exactly two detectors, one from the top branch, and one from the bottom one, click. We recognize two types of detection. For the Alice’s side (and, similarly, for the Bob’s side), type I refers to getting a click on $r_0$-$s_0$ or on $r_1$-$s_1$. Type II refers to the case when $r_0$-$s_1$ or $r_1$-$s_0$ click. In order to get one bit of sifted key, Alice and Bob must use the same basis and both BSMs in Fig. \[fig:Diagram-for-MDI-QKD\] must be successful. Depending on the results of these BSMs and the chosen basis by the two parties, Alice and Bob may end up with correlated or anti-correlated bits, where in the latter case, Bob will flip his bit. Table \[Tab:bitassign\] summarizes the bit assignment procedure for our scheme. Note that these BSMs can be performed by untrusted parties.
Basis Alice BSM Bob BSM Bit assignement
------- ------------- ------------- -------------------
$z$ type I/II type I/II Bob flips his bit
$x$ type I (II) type I (II) Bob keeps his bit
$x$ type I (II) type II (I) Bob flips his bit
: Bit assignment protocol depending on the results of the two BSMs. \[Tab:bitassign\]
The repetition rate for our scheme is a function of several factors. In order to do a proper BSM, for each photon sent by the users, there must be [*two*]{} entangled pairs of memories ready to be read. In principle, the fastest that we can repeat our scheme is the minimum of the maximum source repetition rate, $R_S$, and half the entanglement generation rate of the quantum repeater, $R_{\rm rep}/2$. The latter is a function of the number of memories in use [@Razavi_SPIE]. We therefore consider two regimes of operation. If $R_S > R_{\rm rep}/2$, we then run our encoders at a rate equivalent to $R_{\rm rep}/2$ and will look at the achievable key rate per QM used. If $R_S < R_{\rm rep}/2$, i.e., when for every photon sent, there will be more than two entangled pairs ready, then we run our scheme at the rate $R_S$ and will look at the key rate per transmitted pulse as a figure of merit.
In the following, we describe the quantum repeater protocol used in our scheme as well as different types of (imperfect) sources that users may use. Later, we look at the above achievable key rates once certain imperfections are considered in our setup.
Source Imperfections {#Sec:Source}
--------------------
In our work, we consider two types of sources for the end users. The first type, which we will use as a point of reference for comparison purposes, is an imperfect SPS, with the following output state $$\rho_{j}^{(\rm SPS)}=(1-p)\,|1\left\rangle _{jj}\right\langle 1|+p\,|2\left\rangle _{jj}\right\langle 2|,\quad j=A,\, B\label{eq:init_dens_fock_state}$$ where $p$ is the probability to emit two, rather than one, photons. In practical regimes of operation, $p \ll 1$, hence, in our analysis, we neglect the simultaneous emission of two photons by both sources. The second type of source considered is a phase-randomized coherent source, which will be used in the decoy-state version of the protocol. In this case, Alice (Bob) will send $\mu = |\alpha|^2$ ($\nu = |\beta|^2$) photons on average for her (his) main signal states. Other values will be used for decoy pulses. Our analysis here only considers the case when there are infinitely many decoy states in use, although in practice we expect to achieve the same performance by using just a small number of decoy states [@MDIQKD_finite_PhysRevA2012].
SPS Repeater Protocol
---------------------
![\[fig:SPS-protocol\]The SPS protocol for entanglement distribution.](SPS){width="7cm"}
The SPS protocol, proposed in [@Sangouard:single-photon:2007], attempts to reduce the contribution of multi-photon errors by using single-photon sources. The SPS setup for its initial entanglement distribution is shown in Fig. \[fig:SPS-protocol\]. In order to entangle two QMs at a distance $L_0$, corresponding to the shortest segment of the repeater setup, we send single photons through identical beam splitters with transmission coefficients $\eta$. The photons can be reflected and stored in the QM or go through the quantum channel and be coupled at a 50:50 beam splitter. If exactly one of the two photodetectors in Fig. \[fig:SPS-protocol\] clicks, the memories are left in a mixture of an entangled state and a spurious vacuum term, where the latter can be selected out in later stages. For the entanglement swapping stage, we again use the 50:50 beam splitter followed by two single-photon detectors to perform a partial BSM. In [@LoPiparo:2013], we calculate the secret key generation rate for the SPS protocol assuming that, instead of perfect SPSs, we are equipped with imperfect sources as in Eq. . This is particularly a fundamental source of error, if one uses ensemble-based memories and the partial readout technique for generating single photons [@Sangouard:single-photon:2007]. Without loss of generality, we assume ensamble-based QMs with $\Lambda-$ level configuration and infinite decoherence time. The effect due to a finite decoherence time has been already considered in a previous paper [@Nicolo_paper2]. By considering writing and reading efficiencies for the QMs in use, respectively, denoted by $\eta_w$ and $\eta_r$, here we use the results of [@LoPiparo:2013] to find the relevant density matrices, $\rho_{A_i B_i}$ for $i=1,2$, for memories entangled by the SPS protocol for different values of $p$ and for different nesting levels $n$. Other sources of imperfections considered throughout the paper are the path loss given by $\eta_{\rm ch}(l) = \exp(-l/L_{\rm att})$ with $L_{\rm att}$ being the attenuation length of the channel, photodetectors’ quantum efficiency, $\eta_{d}$, and photodetectors’ dark count per pulse given by $d_c$.
![[\[fig:(a)Multimemory-configuration-and\]Multi-memory configuration for quantum repeaters.]{} ](BSM_MM){width="8.6cm"}
In order to improve the entanglement generation rates in probabilistic quantum repeaters, it is essential to make use of multiple memories and/or multi-mode memories. Here, we assume a multi-memory structure as shown in Fig. \[fig:(a)Multimemory-configuration-and\] with $N$ memories per node, and employ the cyclic protocol proposed in [@Razavi.Lutkenhaus.09]. In this protocol, at each cycle of duration $L_{0}/c$ where $c$ the speed of light in the channel, we try to entangle, here using the SPS protocol, all the unentangled pairs of QMs at distance $L_{0}.$ At each cycle, we also perform as many BSMs as possible at the intermediate nodes. The main requirement for such a protocol is that, at the stations that we perform BSMs, we must be aware of establishment of entanglement over links of length $l/2$ before extending it to distance $l$ (informed BSMs). We use the results of [@Razavi.Lutkenhaus.09] to calculate the generation rate of entangled states [*per memory*]{} used, which is given by $$\begin{array}{c}
R_{\mathrm{ent}}(L)=NP_{S}(L_{0})P_{M}^{(1)}P_{M}^{(2)}...P_{M}^{(n)}/T_{0}N2^{n+1}\\
=P_{S}(L_{0})P_{M}^{(1)}P_{M}^{(2)}...P_{M}^{(n)}/(2L/c)
\end{array}\label{eq:Rent}$$ where $T_{0}$ is the duration of each cycle and $P_{S}\left(L/2^{n}\right)$ is the probability that the entanglement distribution protocol succeeds over a distance $L_0$, $P_{M}^{(i)},i=1...n$, is the BSM success probability at nesting level $i$ for a quantum repeater with $n$ nesting levels. In our analysis, we use the expressions for $P_{S}$ and $P_{M}^{(i)}$ up to two nesting levels as found in [@LoPiparo:2013]. Finally, the total generation rate of entangled states in the limit of $N R_{\mathrm{ent}}(L) L/c \gg 1$ is given by $$\label{eq:Rrep}
R_{\mathrm{rep}}(L)= N_{\rm QM} R_{\mathrm{ent}}(L),$$ where $N_{\rm QM} = 2^{n+1}N$ is the total number of logical memories in Fig. \[fig:(a)Multimemory-configuration-and\].
secret key generation rate
==========================
![\[fig:BSM-module\]BSM module with generic transmission coefficient represented by fictitious beam splitters. In our setup, $\eta_{a}$ is the path loss; $\eta_{b}$ is the reading efficiency and $\eta_{d}$ is the detection efficiency.](measurement_device){width="7.5cm"}
In this section, we find the secret key generation rate, $R_{\mathrm{QKD}}$, per logical memory used, for the scheme of Fig. \[fig:Diagram-for-MDI-QKD\] under the normal mode of operation when no eavesdropper is present. We consider two types of sources as discussed in Sec. \[Sec:Source\].
Imperfect SPSs
--------------
Here, Alice and Bob each use an SPS with the output state as given by Eq. in their encoder. In the limit of an infinitely long key and a sufficiently large number of QMs, their normalized secret key generation rate per employed memory is lower bounded by $$\begin{array}{c}
R_{\rm QKD} = \frac{\min(R_S,R_{\rm rep}/2)}{N_{\rm QM}}\\
\times \mathrm{max}\left\{ Q_{11}^{z}\left(1-h\left(e_{11}^{x}\right)\right)-Q_{pp}^{z}f\, h\left(E_{pp}^{z}\right),0\right\} \label{eq:Rss}
\end{array}$$ where $Q_{11}^{z} = (1-p)^{2}Y_{11}^{z}$, with $Y_{11}^{z}$ being the probability of a successful click pattern in the $z$ basis when Alice and Bob send exactly one photon each; $e_{11}^{x}$ is the quantum bit error rate (QBER) in the $x$ basis, provided that Alice and Bob are each sending exactly a single photon; $Q_{pp}^{z}$ is the probability of a successful click pattern in the $z$ basis when Alice and Bob use sources with outputs as in Eq. , with the corresponding QBER given by $E_{pp}^{z}$; $f$ is the error correction inefficiency, and $h\left(x\right)=-x\,\log_{2}\left(x\right)-(1-x)\,\log_{2}\left(1-x\right)$ is the Shannon binary entropy function.
Appendix \[AppA\] provides us with the full derivation of the relevant terms in Eq. . Our general approach to find these terms is as follows. For any basis $\Phi = x,z$ and any possible encoded state $\rho_{\rm enc}^\Phi = \rho_{rs} \otimes \rho_{uv}$ by Alice and Bob, the initial state of the system for memories $A_1$-$B_1$ and $A_2$-$B_2$ is given by $$\rho_{\rm in}^\Phi = \rho_{\rm enc}^\Phi \otimes \rho_{A_1B_1} \otimes \rho_{A_2B_2} \label{eq:initial_dens_mat}$$ where $\rho_{A_iB_i}$ has been obtained in [@LoPiparo:2013]. Once memories are read, their states will be transferred to photonic states, which we denote by the same label as their original memories. In that case, optical fields corresponding to modes $r$ and $A_1$, as well as the other three pairs of modes in Fig. \[fig:Diagram-for-MDI-QKD\], would undergo through the setup shown in Fig. \[fig:BSM-module\], where $\eta_a = \eta_r \eta_d$ and $\eta_b = \eta_{\rm ch}(L_s) \eta_d$. The equivalent sub-module in Fig. \[fig:BSM-module\] is what we refer to as an asymmetric butterfly module, whose operation is denoted by $B_{\eta_a \eta_b}^{ab}$ when it acts on two incoming modes $a$ and $b$. In [@Nicolo_paper2], we have derived the output states of a butterfly module for relevant number states at its input. Using those results, we can then find the pre-measurement state right before the photodetection at the BSM modules by $$\rho_{\rm out}^\Phi = B_{\eta_a \eta_b}^{rA_1} \otimes B_{\eta_a \eta_b}^{sA_2} \otimes B_{\eta_a \eta_b}^{uB_1}\otimes B_{\eta_a \eta_b}^{vB_2} (\rho_{\rm in}^\Phi).$$ Note that we have already accounted for the quantum efficiency of photodetectors in our butterfly modules. The probability for a particular pattern of clicks on detectors $r_i$, $s_j$, $u_k$, and $v_l$, for $i,j,k,l = 0,1$, is given by $$P_{r_{i}s_{j}u_{k}v_{l}}(\rho_{\rm enc}^\Phi)=\mathrm{tr}{\left(\mathit{\rho_{\rm out}^\Phi M_{r_{i}}M_{s_{j}}M_{u_{k}}M_{v_{l}}}\right),}\label{eq:Probability}$$ where for $x=r,s,u,v$ $$\begin{array}{c}
M_{x_{0}}=(1-d_{c})\left[\left(I_{x_{0}}-|0\rangle_{x_{0}x_{0}}\langle0|\right)\otimes|0\rangle_{x_{1}x_{1}}\langle0|\right.\\
\left.+d_{c}|0\rangle_{x_{0}x_{0}}\langle0|\otimes|0\rangle_{x_{1}x_{1}}\langle0|\right]\label{eq:measurement_operators-1}
\end{array}$$ is the measurement operator to get a click on detector $x_0$ but not on $x_1$. Here, $I_{x_0}$ denotes the identity operator for the mode entering detector $x_0$. One can define a similar operator $M_{x_1}$ by swapping subscripts 0 and 1 in the above equation. Hence, for example, looking at Fig. \[fig:Diagram-for-MDI-QKD\] the measurement operator corresponding to a click on detector $r_0$ and no click on $r_1$ is given by $$\begin{array}{c}
M_{r_{0}}=(1-d_{c})\left[\left(I_{r_{0}}-|0\rangle_{r_{0}r_{0}}\langle0|\right)\otimes|0\rangle_{r_{1}r_{1}}\langle0|\right.\\
\left.+d_{c}|0\rangle_{r_{0}r_{0}}\langle0|\otimes|0\rangle_{r_{1}r_{1}}\langle0|\right]
\end{array}$$ The relevant terms in Eq. can now be calculated by using Eq. as shown in Appendix \[AppA\].
Coherent sources
----------------
In this section we replace the SPSs with lasers sources and use the decoy-state technique to exchange secret keys. This is a more user friendly approach as the complexity of the required equipment for the end users would be minimized. In the limit of infinitely many decoy states, infinitely long key, and sufficiently large number of memories, the secret key generation rate per logical memory used is lower bounded by $$\begin{array}{c}
R_{\rm QKD} = \frac{\min(R_S,R_{\rm rep}/2)}{N_{\rm QM}}\\
\times \mathrm{max}\left\{ Q_{11}^{z}\left(1-H\left(e_{11}^{x}\right)\right)-Q_{\mu\nu}^{z}f\, H\left(E_{\mu\nu}^{z}\right),0\right\}, \label{eq:Rcc}
\end{array}$$ where $Q_{\mu\nu}^{z}$ is the probability of a successful click pattern in the $z$ basis when Alice and Bob send phase-randomized coherent pulses, respectively, with mean photon number $\mu = |\alpha|^{2}$ and $\nu = |\beta|^{2}$ and $E_{\mu\nu}^{z}$ is the QBER in the $z$ basis in the same scenario.
The procedure to find $Q_{\mu\nu}^z$ and $E_{\mu\nu}^z$ is the same as what we outlined in Eqs. -. The only difference here is that in our butterfly modules, we now need to know the output of the module to coherent states in one input port, for the signal coming from the users, and number states in the other, representing the state of QMs. Table \[tab:coherent\] in Appendix \[AppA\] provides us with the input-output relations for a range of relevant input states. We can then find the relevant terms of the key rate, as shown in Appendix \[AppA\].
Numerical Results
=================
In this section, we present numerical results for the secret key generation rate of our long-haul trust-free QKD link versus different system parameters. We look at two regimes of operation; the [*source-limited*]{} regime when memories are abundant and we are slowed down by source rates, i.e., $2 R_S < R_{\rm rep}$, versus the [*repeater-limited*]{} regime when the rate limitations come from the quantum repeater side, i.e., $2 R_S > R_{\rm rep}$. In the latter case, we should still satisfy the condition $N R_{\mathrm{ent}}(L) L/c \gg 1$ in order that Eqs. - remain valid. We have used Maple 15 to analytically derive expressions for Eqs. \[eq:Rss\] and \[eq:Rcc\]. Unless otherwise noted, we use the nominal values summarized in Table \[tab:Nominal-values-used\].
Memory writing efficiency, $\eta_{w}$ 0.78
--------------------------------------- ------------------------
Quantum efficiency, $\eta_{d}$ 0.93
Memory reading efficiency, $\eta_{r}$ 0.87
Dark count per pulse, $d_{c}$ $10^{-9}$
Attenuation length, $L_{\rm att}$ 25 km
Speed of light in optical fiber, $c$ $2 \times 10^{5}$ km/s
Double-photon probability, $p$ $10^{-4}$
Access network length, $L_s$ 5 km
Error correction inefficiency, $f$ 1.16
: [\[tab:Nominal-values-used\]Nominal values used in our numerical results.]{}
The first thing to obtain is the optimum intensity for our decoy-state scheme. Let us assume that in the symmetric scenario, as considered in this paper, Alice and Bob both use the same intensity value $\mu = |\alpha|^2 = \nu$ for their coherent signal states. Figure \[fig:R\_vs\_alpha\] shows the secret key generation rate per pulse versus $|\alpha|$ for (a) different values of $d_{c}$ and (b) different values of $p$ of the quantum repeater at $L_{\rm rep}=100$ km. We assume that $2 R_S < R_{\rm rep}$ and the plotted curves represent $R_{\rm QKD}N_{\rm QM}/R_S$ in Eq. . It can be seen in both figures that $|\alpha| = 1$ almost gives us the maximum rate in most scenarios. The optimal value is to some extent a function of $d_c$ as can be seen in Fig. \[fig:R\_vs\_alpha\](a). By increasing $d_{c}$, the optimal intensity slightly decreases. Dark count represents the main source of error in the $z$ basis, therefore, when $d_{c}$ increases, the tolerance for the multiple-photon terms in a coherent state decreases, hence the maximum allowed value of $|\alpha|$ will go down as well. This leads to a slightly shifted curve and therefore lower values for the optimal values of $|\alpha|$. On the contrary, $E_{\mu\nu}^{z}$ is not affected much by the double-photon probability $p$ and there is not much difference in the optimal intensity when $p$ increases as shown in Fig. \[fig:R\_vs\_alpha\](b). We also obtain the same optimal values of $|\alpha|$ for nesting levels one and two in the repeater-limited regime. Throughout this section, we then use $|\mu| = |\nu| = 1$ in our calculations.
![\[fig:R\_vs\_alpha\] Secret key generation rate per pulse versus $|\alpha| = |\beta|$ for different values of (a) the dark count and (b) the repeater’s double photon probability. Here, $L_{\rm rep}$ = 100 km and the other values are as in Tab. \[tab:Nominal-values-used\].](vs_alpha_beta){width="8.6cm"}
Rate versus distance
--------------------
Figures \[fig:R vs dist inf mem\] and \[fig: R vs dist fin mem\] show the secret key generation rate, at the optimal value of intensity, versus the total distance, $L = 2 L_s + L_{\rm rep}$, between Alice and Bob. In both figures, we assume $L_s$ is a fixed short distance resembling the length of the access network. We vary $L_{\rm rep}$ then to effectively increase the link distance. Figure \[fig:R vs dist inf mem\] shows the secret key generation rate per transmitted pulse in the source-limited regime, whereas Fig. \[fig: R vs dist fin mem\] represents the key rate per logical memory used in the repeater-limited regime. In both cases we consider SPSs at $p=10^{-4}$ as well as coherent decoy states. The difference in the performance of the systems relying on these sources, as expected, is low, and that again confirms the possibility, and practicality, of using the decoy-state technique for end-user devices. The cut-off security distance, i.e., the distance beyond which secure key exchange is not possible, almost doubles every time we increase the nesting level so long as memories decoherence rates are correspondingly low. This distance at $n=0$ is about 800 km, similar to the no-memory case for the parameter values used and at $n=1$ and $n=2$, respectively, reaches around 1500 km and 2500 km. Security distances are slightly higher for the SPSs than coherent-state sources.
![[\[fig:R vs dist inf mem\]Secret key generation rate per transmitted pulse, in the source-limited regime, versus distance when (a) imperfect SPSs and (b) decoy coherent states are used.]{}](comparison_infinite_memories){width="8.6cm"}
The slope of the curves in Fig. \[fig:R vs dist inf mem\] is different than that of Fig. \[fig: R vs dist fin mem\]. In Fig. \[fig:R vs dist inf mem\] curves are almost flat until they reach their cut-off distances. That has two reasons. First, in the source-limited regime, $R_{\rm QKD}$ is proportional to the constant $R_S$, whereas, it scales with $R_{\rm ent}$, which exponentially decays with $L_0$ [@LoPiparo:2013], in the repeater-limited regime. Second, and this is common in both figures, in the absence of the decoherence, the fidelity of the entangled states generated by our probabilistic repeater effectively reaches a constant value once we increase the distance [@Razavi.Amirloo.10]. That means that the double-photon-driven error terms in the key rate are almost fixed until dark count becomes significant and the rate goes down.
![[\[fig: R vs dist fin mem\]$R_{\mathrm{QKD}}$, in the repeater-limited regime, versus distance when (a) imperfect SPSs and (b) decoy coherent states are used. ]{}](comparison_finite_memories){width="8.6cm"}
The implications on the achievable key rate is also different in the two figures. In Fig. \[fig:R vs dist inf mem\], at a nominal distance of $L = 1000$ km and a source rate of $R_S = 1$ GHz, the key rate is in the region of Mb/s. The assumption $2R_S < R_{\rm rep}$, however, implies that we need something on the order of $10^{15}$ QMs in our core network to work in the source-limited regime, which seems, at the moment, quite impractical. In the repeater-limited regime, we still need many memories to obtain a decent rate. For instance, at $L=1000$ km, we would need around 1 billion QMs to get a key rate on the order of kb/s. This is still a huge number of resources for the current technology of QMs. This is in fact the same number of memories in use in our classical computers, which was perhaps inconceivable a few decades ago. Progress in solid-state QMs is much needed to meet the above requirements.
Crossover distance
------------------
 Crossover distance versus QM’s recall efficiency in the repeater-limited regime. (b) Optimum spacing $L_{0}$ between adjacent nodes of a quantum repeater at $\eta_r = 0.3$. ]{}](cutoff_p_and_crossover){width="8.6cm"}
The different slopes in Figs. \[fig:R vs dist inf mem\] and \[fig: R vs dist fin mem\] result in appreciably different values for crossover distances, i.e., the distances where one nesting level outperforms its previous one. In the source-limited regime, in Fig. \[fig:R vs dist inf mem\], the curve for $n=1$ outperforms that of $n=0$ for $L$ greater than around 750 km. The crossover distance to nesting level 2 is then around 1400 km. These are quite large distances, which imply that $L_0$, the spacing between adjacent nodes in our quantum repeater, could be as large as 700 km. This sparse location of memories in the system has some advantages in the sense that resources are more or less centralized, rather than distributed, but at the same time it imposes harder conditions on maintaining phase and polarization stability over such long distances. In the repeater-limited regime of Fig. \[fig: R vs dist fin mem\], the nodes are much closer as now the crossover distance is around/below 500 km. This implies that the optimum architecture of our core network relies on, among other things, how many QMs are available at the time of development.
The crossover distance is also a function of the efficiency of various system parameters. In Fig. \[fig:(a)-Crossover-distance\](a), we have looked at the crossover distance as a function of the recall efficiency, $\eta_r$, in the repeater-limited regime. This is particularly important, because $\eta_r$ implicitly accounts for the amplitude decay in memories. As expected, the crossover distance decreases with the recall efficiency as there would be less of rate reduction because of the BSM operation. Figure \[fig:(a)-Crossover-distance\](b) shows this effect on the optimal value of $L_0$. It can be seen that at $\eta_r = 0.3$ the optimal spacing is much wider than what can be obtained from Fig. \[fig: R vs dist fin mem\] at $\eta_r = 0.87$. It can be seen that the curve for optimal $L_0$ is non-continuous as we have limited our study to the case when the number of segments in a repeater setup is a power of 2. By developing new repeater protocols for arbitrarily number of segments, one can get a smoother curve for optimal $L_0$. At $\eta_r = 0.3$, $L_0$ is on average around 250 km for the set of parameters as in Table \[tab:Nominal-values-used\].
Conclusions
===========
In this paper we combined MDI-QKD with a quantum repeater setup in order to obtain a long-distance key exchange scheme without the need to trust any of the intermediate nodes or measurement tools. This trust-free network could be used in future generations of quantum networks, where the easy cost-efficient access to the network would be facilitated by laser-based encoders and the repeater technology, at the backbone, would be maintained by the service provider. We considered a particular entanglement distribution scheme for our quantum repeater, which relied on imperfect single-photon sources. We merged memories entangled by this probabilistic repeater setup with photons sent and phase encoded by the two users via two BSM modules. We showed that it would be possible to exchange secret keys up to over 2500 km using repeaters with two nesting levels. It turned out that in order to get a key rate on the order of 1 kb/s, one may need to employ and control billions of memories at the core network. We also showed that the network architecture depends on the number of memories at stake. In the limit of infinitely many memories, the repeater nodes would be sparsely located, although each node may contain a large number of memories. Our results showed how challenging it would be to build trust-free quantum communication networks.
Derivation of key rate terms {#AppA}
============================
In this Appendix, we derive the key rate terms in Eqs. and under the normal mode of operation when no eavesdropper is present. We use the formulation developed in Eqs. - to obtain $\Gamma_{11}^z = Y_{11}^z$, $\epsilon_{11}^x = e_{11}^x$, $\Gamma_{pp}^z = Q_{pp}^z$, $\epsilon_{pp}^z = E_{pp}^z$, $\Gamma_{\mu \nu}^z = Q_{\mu \nu}^z$, and $\epsilon_{\mu \nu}^z = E_{\mu \nu}^z$, where new unifying notations $\Gamma$ and $\epsilon$ are used in this section.
$\rho_{AB}$ ${\rm tr}\left(M_{x_{0}}B_{\eta_{a}\eta_{b}}^{AB}\left(\rho_{AB}\right)\right)$
------------------------------------------------------------ -------------------------------------------------------------------------------------------------------------------------------------------
$|\alpha0\left\rangle \left\langle \alpha0|\right.\right.$ $(1-d_{c})\left[e^{-\frac{\eta_{a}}{2}\mu}\left(1-e^{-\frac{\eta_{a}}{2}\mu}\right)+d_{c}e^{-\eta_{a}\mu}\right]$
$|\alpha1\left\rangle \left\langle \alpha1|\right.\right.$ $(1-d_{c})\left[\frac{\eta_{b}}{2}e^{-\frac{\eta_{a}}{2}\mu}\left(1+\frac{\eta_{a}}{2}\mu\right)\right.$
$+e^{-\frac{\eta_{a}}{2}\mu}\left(1-\eta_{b}\right)\left(1-e^{-\frac{\eta_{a}}{2}\mu}\right)$
$\left.+d_{c}\left(1-\eta_{b}\right)\left(1-e^{-\eta_{a}\mu}\right)\right]$
$|\alpha2\left\rangle \left\langle \alpha2|\right.\right.$ $(1-d_{c})\left\{ \frac{\eta_{b}^{2}}{4}e^{-\frac{\eta_{a}}{2}\mu}\left[1+\right.\right.$
$+\frac{\eta_{a}^{2}}{4}\mu^{2}\left(\frac{1}{2}-8\, e^{-\frac{\eta_{a}}{2}\mu}\right)\left.+\eta_{a}\mu\right]$
$+\eta_{b}e^{-\frac{\eta_{a}}{2}\mu}\left(1-\eta_{b}\right)\left(1+\frac{\eta_{a}}{2}\mu\right)$
$+e^{-\frac{\eta_{a}}{2}\mu}\left(1-\eta_{b}\right)^{2}\left(1-e^{-\frac{\eta_{a}}{2}\mu}\right)$
$\left.+d_{c}\left[\frac{\eta_{a}^{2}\eta_{b}^{2}}{2}e^{-\eta_{a}\mu}\mu^{2}+e^{-\eta_{a}\mu}\left(1-\eta_{b}\right)^{2}\right]\right\} $
$|\alpha1\left\rangle \left\langle \alpha0|\right.\right.$ $(1-d_{c})\left(\frac{1}{2}\sqrt{\eta_{a}\eta_{b}}\alpha e^{-\frac{\eta_{a}}{2}\mu}\right)$
$|\alpha0\left\rangle \left\langle \alpha1|\right.\right.$ $(1-d_{c})\left(\frac{1}{2}\sqrt{\eta_{a}\eta_{b}}\alpha e^{-\frac{\eta_{a}}{2}\mu}\right)$
$|\alpha1\left\rangle \left\langle \alpha2|\right.\right.$ $(1-d_{c})\left(\sqrt{\frac{\eta_{a}\eta_{b}}{2}}\alpha\left(\frac{\eta_{b}}{2}-\frac{\eta_{a}\eta_{b}}{8}-1\right)\right)$
$|\alpha2\left\rangle \left\langle \alpha1|\right.\right.$ $(1-d_{c})\left(\sqrt{\frac{\eta_{a}\eta_{b}}{2}}\alpha\left(\frac{\eta_{b}}{2}-\frac{\eta_{a}\eta_{b}}{8}-1\right)\right)$
: \[tab:coherent\][The input-output relationship for a butterfly module with coherent states in one input and number states in the other. The column on the right represents the probability that the output state causes a click on detector $x_0$, but not $x_1$, assuming that detector $x_0$ measures the left output port and $x_1$ the right one. The expression ${\rm tr}\left(M_{x_{1}}B_{\eta_a,\eta_b}^{AB}\left(\rho_{AB}\right)\right)$ will give the same results as above for symmetrical input states; a minus sign correction is needed for asymmetrical input states. Here, $\mu = |\alpha|^2$.]{}
Let $\rho_{\rm enc}^\Phi(mn)$ denote the output state of Alice and Bob’s encoders for, respectively, sending bits $m$ and $n$, for $m,n = 0,1$, in basis $\Phi$. With the above notation, the probability that an acceptable click pattern occurs in basis $\Phi$, $\Gamma_{\gamma \delta}^{\Phi}$, is given by $$\Gamma_{\gamma \delta}^{\Phi}= \sum_{i,j,k,l,m,n=0,1}{P_{r_{i}s_{j}u_{k}v_{l}}(\rho_{\rm enc}^\Phi(mn))/4},$$ where $\gamma = \delta = 1$ refers to the case when Alice and Bob are sending exactly one photon each; when $\gamma =\delta =p$, imperfect SPSs are used and when $\gamma=\mu$ and $\delta=\nu$ coherent states with mean photon number $\mu$ and $\nu$, are, respectively, in use. In above, some of the successful click patterns would result in errors in the end, while the other in correct sifted key bits. By separating these two components, we obtain $$\Gamma_{\gamma \delta}^{\Phi}=\Gamma_{\gamma \delta;C}^{\Phi} + \Gamma_{\gamma \delta;E}^{\Phi},
\label{eq:y_z}$$ where $\Gamma_{\gamma \delta;C(E)}^{\Phi}$ represents the click terms that result in correct (erroneous) inference of bits by Alice and Bob. In the $z$ basis, $$\Gamma_{\gamma \delta;C}^{z}= \sum_{i,j,k,l, m,n=0,1; m+n =1}{P_{r_{i}s_{j}u_{k}v_{l}}(\rho_{\rm enc}^z(mn))/4}$$ and $\Gamma_{\gamma \delta;E}^{\Phi} = \Gamma_{\gamma \delta}^{\Phi} - \Gamma_{\gamma \delta;C}^{\Phi}$. In the $x$ basis, $$\begin{array}{c}
\Gamma_{\gamma\delta;C}^{x}={\displaystyle \sum_{i,k,m,n=0,1;m\oplus n=0}\left(P_{r_{i}s_{i}u_{k}v_{k}}(\rho_{{\rm enc}}^{x}(mn))/4\right.}\\
\left.+P_{r_{i}s_{i\oplus1}u_{k}v_{k\oplus1}}(\rho_{{\rm enc}}^{x}(mn))/4\right)\\
+{\displaystyle \sum_{i,k,m,n=0,1;m\oplus n=1}}\left(P_{r_{i}s_{i}u_{k}v_{k\oplus1}}(\rho_{{\rm enc}}^{x}(mn))/4\right.\\
\left.+P_{r_{i}s_{i\oplus1}u_{k}v_{k}}(\rho_{{\rm enc}}^{x}(mn))/4\right),
\end{array}$$ where $\oplus$ denotes addition modulo two. Finally, all QBER terms can be obtained from the following. $$\epsilon_{\gamma \delta}^{\Phi}=\frac{\Gamma_{\gamma \delta;E}^{\Phi}}{\Gamma_{\gamma \delta}^{\Phi}}.$$
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We study the effect of time-odd components of the Skyrme energy density functionals on the ground state of finite nuclei and in nuclear matter. The spin-density dependent terms, which have been recently proposed as an extension of the standard Skyrme interaction, are shown to change the total binding energy of odd-nuclei by only few tenths of keV, while the time-odd components of standard Skyrme interactions give an effect that is larger by one order of magnitude. The HFB-17 mass formula based on a Skyrme parametrization is adjusted including the new spin-density dependent terms. A comprehensive study of binding energies in the whole mass table of 2149 nuclei gives a root mean square (rms) deviation of 0.575 MeV between experimental data and the calculated results, which is slightly better than the original HFB-17 mass formula. From the analysis of the spin instabilities of nuclear matter, restrictions on the parameters governing the spin-density dependent terms are evaluated. We conclude that with the extended Skyrme interaction, the Landau parameters $G_0$ and $G_0^\prime$ could be tuned with a large flexibility without changing the ground-state properties in nuclei and in nuclear matter.'
address:
- '$^1$Institut de Physique Nucléaire, Université Paris-Sud, IN2P3-CNRS, F-91406 Orsay Cedex, France'
- '$^2$Institut d’Astronomie et d’Astrophysique, CP-226, Université Libre de Bruxelles, 1050 Brussels, Belgium'
- '$^3$Dipartimento di Fisica, Università degli Studi and INFN Sez. di Milano, Via Celoria 16, 20133 Milano, Italy'
- '$^4$Center for Mathematics and Physics, University of Aizu, Aizu-Wakamatsu, 965-8580 Fukushima, Japan'
author:
- 'J. Margueron$^1$, S. Goriely$^2$, M. Grasso$^1$, G. Colò$^3$ and H. Sagawa$^4$'
title: |
Extended Skyrme interaction (II):\
ground state of nuclei and of nuclear matter
---
Introduction
============
Despite many theoretical and experimental investigations, the spin and the spin-isospin channels in either the ground and the excited states of nuclei are still widely open for future study [@bor84; @ost92; @suz99; @eng99; @mar02; @frac07]. It is indeed difficult to probe the spin and the spin-isospin channels of nuclear interaction since the ground states of nuclei are non-spin polarized in the case of even-even nuclei and at most polarized by the last unpaired nucleons in odd nuclei.
The analysis of spin and spin-isospin collective modes such as magnetic dipole (M1) and Gamow-Teller (GT) states gives access to the nuclear interaction in these channels. The Landau parameter $G_0^\prime$ has been deduced from the analysis of the GT mode: a model based on Woods-Saxon single-particle states plus one-pion and rho meson exchange interactions gives $G_0^\prime=1.3\pm0.2$ (see Ref. [@ost92; @suz99] and references therein). A slightly different value $G_0^{\prime} = 1.0\pm0.1$ was derived from observed GT and M1 strength distributions using the phenomenological energy density functionals DF3 [@bor84; @bor06]. Anyway, in both cases empirical single-particle energies are used (that is, for states close to the Fermi energy the effective mass $m^*/m$ is $\approx$ 1).
Self-consistent Hartree-Fock (HF) plus Random Phase Approximation (RPA) calculations constitute a somewhat different framework, in which the density of states around the Fermi energy is lower (or, equivalently, the effective mass is about 30% smaller). As compared with the empirical case, the unperturbed particle-hole transitions have larger energies and as a consequence one needs a smaller residual repulsive effect to fit the observed GT peak. It is not surprising, therefore, that self-consistent calculations of the GT resonance, performed using different Skyrme interactions in Ref. [@frac07], point to $G_0^\prime \sim 0.6$. Skyrme interactions are characterized by a spin-isospin $G_1^\prime$ parameter as well, and specific terms of the effective mean field, like the spin-orbit potential, may also break the simple correlation between the GT properties and the parameter $G_0^\prime$. However, as widely used interaction like SLy5 have unrealistic (negative) values of $G_0^\prime$, it is undeniable that adding more flexibility to the spin-isospin part of Skyrme forces is useful.
The spin and spin-isospin component of the nuclear interaction is also reflected into the time-odd component of the mean field. The properties of even nuclei give constraints to the time-even component of the mean field while very little is known about properties of the time-odd mean fields. Time-odd components of the mean field compete with pairing correlations in determining the odd-even mass staggering [@dug01]. Fast rotation induces time-odd components in the mean field [@doba95] which could be probed from the measurement of the dynamical moments of superdeformed bands.
In Ref. [@marg08], new spin-density dependent terms have been introduced on top of standard Skyrme forces in order to remove the ferromagnetic instability associated with all Skyrme parameterizations [@mar02]. The new terms retain the simplicity and the good properties of Skyrme interactions for nuclear matter and the ground states of even-even nuclei. However, these new terms slightly change the properties of odd systems. This work aims at studying quantitatively the effects of these new terms on the ground state properties of odd nuclei, in particular the total binding energy and the density distribution. Since these terms contribute only in odd nuclei, both odd-even and odd-odd nuclei are considered in the present study. To provide an approximate maximal estimate of the effects while keeping our model simple, we perform HF calculations with the following approximation to treat odd nuclei. We use the equal filling approximation, so that the time-reversal symmetry is not broken, but with an additional ansatz. As within this scheme the spin-densities would be by definition equal to zero, when constructing the spin-densities with the wave function of the odd nucleon, we assume that the spin-up state is completely filled while the spin-down state is empty between the two possible spin orientations (or, equivalently, the opposite). We call this procedure for the construction of spin-densities the *one-spin polarized approximation* (OSPA). The OSPA gives an upper value of the contribution of the new terms.
The article is organized as follows: in Sec. II we remind the newly proposed spin-density dependent terms as in Ref. [@marg08]. In Sec. III, we estimate how much these new spin-density dependent terms affect the total binding energy of odd nuclei. In Sec. IV, it will be shown that, introducing these terms in the most predictive HF-Bogoliubov (HFB) mass formula [@BSK17], the quality of the mass fit can be recovered with an optimal renormalization of the Skyrme parameters. In Sec. V we will analyze the ground state properties of infinite nuclear matter with respect to spin-polarization and give a range for the parameter of newly introduced spin-density $x_3^s$. Finally, conclusions and outlook are given in Sec. VI.
Extended Skyrme interactions
============================
As described in Ref. [@marg08], the new spin-density dependent terms added to the conventional Skyrme force are of the following form: $$\begin{aligned}
V^\mathrm{s,st}(\mathbf{r}_1,\mathbf{r}_2)=
\frac{1}{6}t_3^s(1+x_3^sP_\sigma)[\rho_s(\mathbf{R})]^{\gamma_s}\delta(\mathbf{r})%\nonumber\\
+\frac{1}{6}t_3^{st}(1+x_3^{st}P_\sigma)[\rho_{st}(\mathbf{R})]^{\gamma_{st}}\delta(\mathbf{r})
\nonumber\\
\label{eq:spin}\end{aligned}$$ where $P_\sigma=(1+\sigma_1\cdot\sigma_2)/2$ is the spin-exchange operator, $\mathbf{r}=\mathbf{r}_1-\mathbf{r}_2$ and $\mathbf{R}=(\mathbf{r}_1+\mathbf{r}_2)/2$. In Eq. (\[eq:spin\]), we have introduced the spin-density $\rho_s \equiv \rho_\uparrow-\rho_\downarrow$ and the spin-isospin-density $\rho_{st} \equiv \rho_{n\uparrow}-\rho_{n\downarrow}-\rho_{p\uparrow}+\rho_{p\downarrow}$. Spin symmetry is satisfied if the power of the density-dependent terms $\gamma_s$ and $\gamma_{st}$ is even.
The total energy $E_\mathrm{TOT}$ in finite nuclei is related to the (local) energy density $\mathcal{H}(r)$ through $$E_\mathrm{TOT}\equiv \int d^3r \; \mathcal{H}(r) \; .
\label{eq:etot}$$ In the following, we adopt the notation of Ref. [@chaba] where $\mathcal{H}$ is expressed as the sum of a kinetic term $\mathcal{K}$, a zero-range term $\mathcal{H}_0$, a density-dependent term $\mathcal{H}_3$, an effective-mass term $\mathcal{H}_\mathrm{eff}$, a finite-range term $\mathcal{H}_\mathrm{fin}$, spin-orbit and spin-gradient terms ($\mathcal{H}_\mathrm{so}$ and $\mathcal{H}_\mathrm{sg}$), and eventually the Coulomb term $\mathcal{H}_\mathrm{Coul}$. However, the expression for the energy density provided in [@chaba] holds in the case of time-reversal symmetry; in odd nuclei, the energy density $\mathcal{H}$ acquires also a dependence on the spin-densities $\rho_{s}$ and $\rho_{st}$, hereafter named $\mathcal{H}^\mathrm{odd}$, even without additional terms in the force depending on these densities. Expressions for $\mathcal{H}^\mathrm{odd}$ have been derived for instance in Refs. [@doba95; @bend2002] or in Appendix I of Ref. [@flocard]. For the reader’s convenience we repeat here, in Appendix A, the expression for $\mathcal{H}^\mathrm{odd}$.
The additional terms (\[eq:spin\]) modify the standard $\mathcal{H}_3$ contribution to be $\mathcal{H}_3+\mathcal{H}_3^{s}+\mathcal{H}_3^{st}$, where the last two terms read $$\begin{aligned}
\mathcal{H}_3^s&=&\frac{t_3^s}{12} \rho_s^{\gamma_s}
\Big[ (1+\frac{x_3^s}{2})\rho^2+\frac{x_3^s}{2}\rho_s^2 %\nonumber \\
-(x_3^s+\frac{1}{2})(\rho_n^2+\rho_p^2)-
\frac{1}{2}(\rho_{sn}^2+\rho_{sp}^2) \Big] ,
\label{eq:h3s} \\
\mathcal{H}_3^{st}&=&\frac{t_3^{st}}{12}\rho_{st}^{\gamma_{st}}
\Big[ (1+\frac{x_3^{st}}{2})\rho^2+\frac{x_3^{st}}{2}\rho_s^2 %\nonumber \\
-(x_3^{st}+\frac{1}{2})(\rho_n^2+\rho_p^2)-
\frac{1}{2}(\rho_{sn}^2+\rho_{sp}^2) \Big] ,
\label{eq:h3st}\end{aligned}$$ with $\rho_{sn}=\rho_{n\uparrow}-\rho_{n\downarrow}$ and $\rho_{sp}=\rho_{p\uparrow}-\rho_{p\downarrow}$.
The additional terms to the mean field coming from $\mathcal{H}^{s}_3$ and $\mathcal{H}^{st}_3$ are $$\begin{aligned}
U_q^\mathrm{s,st}&=&\frac{t_3^s}{12}\rho_s^{\gamma_s}\Big[(2+x_3^s)\rho-(1+2x_3^s)\rho_q\Big]
%\nonumber \\
+\frac{t_3^{st}}{12}\rho_{st}^{\gamma_{st}}\Big[(2+x_3^{st})\rho-(1+2x_3^{st})\rho_q\Big].
\label{eq:uq}
\nonumber\\\end{aligned}$$ where $q=n,p$.
As already mentioned, the additional contributions to the mean field are zero in even-even nuclei. Since most of the Skyrme interactions are adjusted on (few) even-even nuclei, it is thus possible to add for these interactions the new terms (\[eq:spin\]) in a perturbative manner. The new four parameters $t_3^s$, $x_3^s$, $t_3^{st}$ and $x_3^{st}$ in Eq. (\[eq:spin\]) have been adjusted in Ref. [@marg08] in order to reproduce the Landau parameters extracted from a G-matrix calculation in uniform matter, while $\gamma_s=\gamma_{st}=2$ is imposed by spin symmetry.
Some other interactions, in particular those produced by the Brussels-Montréal group, are globally adjusted by fitting the properties (essentially the nuclear masses) of both even and odd nuclei. In this case, when including the new spin-density terms, a global re-adjustment of the interaction might be necessary.
In the following, we first analyze the effect of the new spin-density dependent terms for a few selected nuclei using the SLy5 Skyrme interaction for which the new spin-density dependent terms are added perturbatively. Later, on the basis of the latest BSk17 Skyrme parameter set, we study the global impact of our new terms on nuclear masses and show that the Skyrme parameters can be refitted to provide almost equivalent properties.
Ground states of Ca nuclei
==========================
According to the OSPA, we define the densities $\rho_s(r)$ and $\rho_{st}(r)$ in odd nuclei as $$\begin{aligned}
\rho_s(r)&=&\frac{1}{4\pi r^2}\sum_i \varphi_i^2(r) m_s(i) \; , \\
\rho_{st}(r)&=&\frac{1}{4\pi r^2}\sum_i \varphi_i^2(r) m_s(i) m_t(i),\end{aligned}$$ where $m_s(i)$ and $m_t(i)$ are the spin and isospin z-component for each single nucleon having the wave function $\varphi_i(r)$. The last occupied state fully contributes to the spin-density. It is then clear that the OSPA corresponds to maximizing the spin-density and its effects.
Note that if the time-reversal symmetry is not broken (in the filling approximation, for instance) both spin-up and spin-down states must be degenerate and the densities $\rho_s$ and $\rho_{st}$ are zero. Since in this work we aim at an approximate and maximal estimate and not at a precise prediction of the effects, we use the HF method with the OSPA to treat odd nuclei.
All calculations in this Section are performed with the SLy5 parameterization for the conventional Skyrme force and the additional parameters for the spin-density terms given in Table 2 of [@marg08]. We choose two systems: $^{41}$Ca and $^{49}$Ca. Being built on top of double magic nuclei, pairing is neglected for these nuclei in this section. Both nuclei are odd-even, then the proton spin-density is zero and the spin-isospin-density is equal to the spin-density for neutrons. For the nucleus $^{41}$Ca the spin-density is built with the neutron 1$f_{7/2}$ wave function while in $^{49}$Ca it is constructed with the neutron $2p_{3/2}$ wave function. The choice of these nuclei has been driven by the fact that one of the nuclei, $^{41}$Ca, has a spin-density which probes the external region of the nucleus, while for the other nucleus, $^{49}$Ca, the spin-density may probe more the central part. We could not use the OSPA if the single-particle wave functions changed when adding one or two nucleons. Therefore, as a necessary step to proceed with the evaluation of the impact of spin-density dependent terms within the OSPA, we have first checked that the wave function of the neutron state $1f_{7/2}$ does not change appreciably when passing from the even nucleus $^{40}$Ca to the next even isotope with $N+2$ neutrons, $^{42}$Ca. An analogous check has been made for the wave function of the proton state $1f_{7/2}$ in $^{40}$Ca and in the $Z+2$ isotone $^{42}$Ti. Also, the spin-density calculated with the OSPA for $^{41}$Ca has been compared with half the difference of the neutron densities of $^{42}$Ca and $^{40}$Ca, where effects coming from the rearrangement of the deeper state are also included. Very small differences are found, mainly in the central region. Analogous results are obtained for $^{48}$Ca, $^{49}$Ca and $^{50}$Ca.
The spin-densities in $^{41}$Ca and $^{49}$Ca are plotted in the two panels of Fig. \[fig1\] together with the neutron densities of both nuclei. For comparison, the neutron densities of the nearest even-even nuclei $^{42}$Ca and $^{50}$Ca are also displayed. The spin-densities and the neutron densities have been calculated in the two odd-even systems. Two independent calculations have been actually performed with and without the new spin-density dependent terms (\[eq:spin\]) and only negligible differences have been found so that they are not appreciable in the figures.
Since the contributions to the total energy $E_\mathrm{TOT}$ coming from the spin-density dependent terms (\[eq:h3s\]) and (\[eq:h3st\]) are proportional to the square of the spin densities represented in Fig. \[fig1\], these contributions are expected to be negligible as compared to the usual density dependent term $\mathcal{H}_3$. To be more quantitative, we have made several calculations for the ground-state energies that are summarized in Tables \[tab:etot\] and \[tab:particle\]. The partial contributions to the ground-state energy (\[eq:etot\]) are written $E_\mathrm{MF}\equiv\int d^3r \; (\mathcal{H}_\mathrm{0}+\mathcal{H}_\mathrm{3}+\mathcal{H}_\mathrm{eff}+\mathcal{H}_\mathrm{fin}+\mathcal{H}_\mathrm{sg})$, $E_\mathrm{so}\equiv\int d^3r \; \mathcal{H}_\mathrm{so}$, $E_\mathrm{Coul}\equiv\int d^3r \; \mathcal{H}_\mathrm{Coul}$ and $E_\mathrm{kin}\equiv\int d^3r \; \mathcal{H}_\mathrm{kin}$, using the notations of Ref. [@chaba]. In Table \[tab:etot\] the total energy, the mean field, the spin-orbit, the Coulomb and the kinetic contributions to the total energy and the single-particle energy (for the neutron states $1f_{7/2}$ or $2p_{3/2}$) are provided for the odd nuclei $^{41}$Ca and $^{49}$Ca and for the nearest even-even nuclei $^{42}$Ca and $^{50}$Ca. The calculation for the odd nuclei are performed either with or without the spin-density dependent interaction (\[eq:spin\]). The difference between the total energy, and its contributions, without and with the corrections $\Delta E$ is shown to be less than 50 keV in both odd nuclei $^{41}$Ca and $^{49}$Ca. It is interesting to notice that the kinetic energy contributes to reduce the impact of the spin-dependent interaction (\[eq:spin\]). The spin-orbit and the Coulomb energies are very weakly affected by the spin-density terms.
In Table \[tab:particle\] the total and separate contributions coming from the time-odd terms of the Skyrme interaction (see Eqs. (\[eq:a1\])-(\[eq:a5\]) in Appendix A) are provided, in the case of the odd nuclei $^{41}$Ca and $^{49}$Ca: these contributions are calculated perturbatively within the OSPA. These terms are classified according to the standard notations and labeled as $E_0^\mathrm{odd}$, $E_\mathrm{eff+fin+sg}^\mathrm{odd}$ and $E_3^\mathrm{odd}$ which are the contributions from the central, the momentum dependent and the density dependent terms, respectively. The corrections $E_0^\mathrm{odd}$ and $E_3^\mathrm{odd}$ give a dominant and repulsive contribution which increases the total energy while the correction $E_\mathrm{eff+fin+sg}^\mathrm{odd}$ is smaller and attractive. The total correction remains quite small, that is, of the order of 0.15-0.3 MeV for both nuclei. Note that the sign of these corrections could change from one Skyrme interaction to another, but such corrections in Table \[tab:particle\] remain larger than $\Delta E_\mathrm{TOT}$ in Table \[tab:etot\]. From the quantitative comparison shown in Tables \[tab:etot\] and \[tab:particle\], we can infer, as expected, that the new spin-density dependent terms (\[eq:spin\]) modify the ground state energies of odd nuclei much smaller than those coming from the time-odd terms of the standard Skyrme interaction.
Global adjustment on the nuclear chart
======================================
Some Skyrme interactions have been determined by fitting the parameters to essentially all of the available mass data and therefore are constrained to even as well as odd systems. In this case, the new spin-density dependent terms (\[eq:spin\]) added to the standard Skyrme interaction may modify the quality of the fit. To study the impact of the new terms on the prediction of nuclear masses, we consider now the latest and most accurate HFB-17 mass formula (with a rms deviation of 0.581 MeV on the 2149 measured masses of [@audi03]) obtained with the BSk17 Skyrme force [@BSK17].
If we consider the parameters of the additional spin-density dependent interaction (\[eq:spin\]) determined in Ref. [@marg08], namely $t_3^s=2\times10^4$ MeV fm$^4$, $t_3^{st}=1.5\times10^4$ MeV fm$^4$, $x_3^s=-2$, $x_3^{st}=0$ and $\gamma_s=\gamma_{st}=2$, we find that the impact of the new terms on nuclear masses are relatively small, as already discussed in Sect. III. Fig. \[fig2\] shows the mass difference obtained by a spherical HFB calculation when the spin terms are added or not. The nuclear masses of odd-$A$ and odd-odd nuclei are globally increased by a value of the order of 100 keV. For light nuclei this correction is the largest and can reach at most 350 keV. As already shown in Ref. [@marg08], the spin-density terms are repulsive and leads to an increase of the rms deviation with respect to all the 2149 measured masses from 0.581 MeV to 0.591 MeV, keeping the good quality of the mass fit. Deterioration can potentially be avoided if the force parameters are re-adjusted to the nuclear mass data.
We have determined a new BSk17st Skyrme force which essentially corresponds to the BSk17 force, but for which the Skyrme as well as the pairing parameters have been slightly renormalized by a new fit on the whole set of mass data. The new parameter set has been built with the idea of evaluating a maximum effect of the new spin-density dependent term (in the same spirit of the OSPA discussed above). The values adopted for the parameters are $t_3^s=4\times10^4$ MeV fm$^4$, $t_3^{st}=3\times10^4$ MeV fm$^4$, $x_3^s=-0.5$, $x_3^{st}=0$ and $\gamma_s=\gamma_{st}=2$. $t_3^s$ and $t_3^{st}$ are twice larger than in Ref. [@marg08] and this leads to Landau parameters $G_0=-0.03$ and $G_0^{\prime}=0.99$ in symmetric matter at saturation density, whereas the parameter set of Ref. [@marg08] is associated with lower values, namely $G_0=-0.36$ and $G_0^{\prime}=0.75$. As explained in the Introduction, the value of $G_0^\prime$ may be considered large in keeping with the effective mass 0.8 of BSk17, yet still quite acceptable for the purpose of the present study. With a value of $t_3^s$ twice larger than BSk16st, the value of $x_3^s=-2$ in Ref. [@marg08] has been modified consistently to be $-0.5$ to keep the contribution to the Landau parameter $G_0$ in spin-saturated infinite neutron matter identical to the one determined in Ref. [@marg08] \[see Eq. (11) of Ref. [@marg08]\]. As far as the parameter $x_3^{st}$ is concerned, there is so far no constraint that could guide us in fixing its value. For this reason the zero value was assumed in Ref. [@marg08]. However, as shown in Sec V, the value of $x_3^{st}$ influences the stability of the partially polarized neutron matter, i.e., the lower its value, the higher the barrier between the $S=0$ and $S=1$ configuration. For this reason, as discussed in Sec. V, the value of $x_3^{st}$ is set to $-3$.
The strategy of the mass fit is the same as the one described in Ref. [@BSK17; @cha08]. In particular, the mass model is given from deformed HFB and the pairing force is constructed from the microscopic pairing gaps of symmetric nuclear matter and neutron matter calculated from realistic two- and three-body forces, with medium-polarization effects included. To accelerate the fit, a first estimation of the energy gained by deformation is performed and subtracted to the experimental masses. A first series of parameters are then obtained from the comparison of spherical HFB mass model with the corrected experimental masses. The corrections due to deformation are then reevaluated and a new fit is performed. This fast procedure is repeated until convergence. The isoscalar effective mass $m^*_s/m$ is constrained to 0.80 and the symmetry energy at saturation $J$ to be 30 MeV in order to reproduce at best the energy-density curve of neutron matter [@fp81] from realistic two- and three-nucleon forces. Note that the parameters of the additional spin-density interaction are not fitted in this procedure.
The final force parameters labeled BSk17st and resulting from a fit to essentially all mass data are given in Table \[tab:BSK\]. It can be seen that there is little difference between the parameters of the BSk17 and BSk17st forces; note that the parameters of the rotational and vibrational corrections [@cha08], are identical for both forces. The same holds for the parameters of infinite matter with the incompressibility coefficient $K_v=241.7$ MeV, the volume energy coefficient $a_v=-16.053$ MeV and the isovector effective mass of $m^*_v/m=0.784$, as in Ref. [@BSK17].
The rms residuals for the BSk17 and BSk17st sets are compared in Table \[tab:rms\]. The inclusion of the new spin-density dependent terms (\[eq:spin\]) which slightly deteriorated the accuracy of the BSk17 force leads now even to a small improvement of the predictions by about 6 keV, with the BSk17st force parameters. Both forces have been used to estimate the mass of the 8508 nuclei with $8 \le Z \le 110$ and lying between the proton and neutron drip lines. Differences of no more than roughly $\pm 0.5$ MeV are found on the entire set.
Finally note that the nuclear binding energies remain extremely insensitive to the value adopted for the parameters $x_3^s$ and $x_3^{st}$. In particular, setting $x_3^s=-2$ instead of $-0.5$ decreases the binding energy by no more than 10 keV. Similarly, a change of $x_3^{st}$ from zero to $-3$ impact the masses by maximum 8 keV. Therefore, these two parameters should be constrained rather by stability conditions of polarized or non-polarized infinite nuclear matter, as shown in the next Section.
Ground state of infinite nuclear matter
=======================================
In Ref. [@marg08], it has been shown that the new spin-density dependent interaction (\[eq:spin\]) stabilize non-polarized matter with respect to spin-fluctuations. As shown in Fig. \[fig3\], with the new terms, the Landau parameters $G_0$ and $G_0^{\prime}$ remains larger than $-1$ at all densities for SLy5st, LNSst [@LNS] (which include the new terms as parametrized in Ref. [@marg08]) and the BSk17st forces
However, it has not been checked if the true ground state is really that of non-polarized matter. To do so, the energy for different spin-polarizations should be compared with that of non-polarized matter. This is done in the next two subsections for both symmetric nuclear matter and neutron matter
Symmetric nuclear matter
------------------------
The difference of the binding energy of spin-polarized matter to that of spin-symmetric matter, $E/A(\delta_S,\rho)-E/A(\delta_S=0,\rho)$, is represented in Fig. \[fig4\] as a function of the polarization $\delta_S=(\rho_\uparrow-\rho_\downarrow)/\rho$. In the left panel, we have represented the binding energy of the BSk17 Skyrme interaction without the new spin-density dependent terms (\[eq:spin\]). The instability occurs between $\rho$=0.18 and 0.2 fm$^{-3}$. At $\rho$=0.2 fm$^{-3}$ the minimum energy is obtained for a polarization $\delta_S$=0.76.
The binding energy of BSk17st which includes the spin-density dependent terms is represented in the right panel of Fig. \[fig4\]. As expected, the energy of non-polarized matter is convex around $\delta_S=0$, but there is a change of convexity for large values of $\delta_S\sim 0.8$. We have indeed observed a large influence of the parameter $x_3^s$ on the binding energy of fully polarized matter. In Fig. \[fig5\] are represented the binding energies $E/A(\delta_S,\rho)-E/A(\delta_S=0,\rho)$ for the three modified Skyrme interactions BSk17st, LNSst and SLy5st for which we changed the values of the parameter $x_3^s$. Its values are indicated in the legend of Fig. \[fig5\]. We fixed the density $\rho$=0.6 fm$^{-3}$ to be the highest value where the nuclear Skyrme interaction is applied. For values of the parameter $x_3^s$=-3 the ground state of nuclear matter is fully polarized ($\delta_S=1$) for the interactions BSk17st and LNSst. Increasing the value of the parameter $x_3^s$ from -3 to 0, the binding energy of fully polarized matter is going up in Fig. \[fig5\]. There is then a critical value above which non-polarized matter is the ground-state of nuclear matter.
One could obtain an estimate of this critical value by analyzing the contribution of the new spin-density terms (\[eq:spin\]) in spin-polarized symmetric matter. It reads $$\mathcal{H}_3^s(sym.)=\frac{t_3^s}{16}\rho^2 \rho_s^{\gamma_s}
\Big[1+\frac{2x_3^s-1}{3}\delta_S^2\Big],
\label{eq:Hs}$$ and $\mathcal{H}_3^{st}(sym.)=0$ since $\rho_{st}=0$. The term (\[eq:Hs\]) is zero for the spin-symmetric matter with $\rho_s=0$ and is always positive for $\delta_S=1$ if one chooses $x_3^s>-1$. It is thus clear that one necessary condition for the spin-symmetric matter to be the absolute ground state at all densities is $x_3^s>-1$. This is the case for the adopted value of $x_3^s=-0.5$ for BSk17st. Nevertheless, as shown in Fig. \[fig5\] for instance for SLy5st, the stability of spin-symmetric matter could be obtained even if $x_3^s<-1$ at the density $\rho$=0.6 fm$^{-3}$. At lower density, SLy5st is also stable, but not at higher density.
We remind that from the analysis of the Landau parameters the stability around spin-symmetric matter requires that $x_3^s<1$ (see Eq. (11) of Ref. [@marg08]). As a conclusion, one could adjust the parameter $x_3^s$ inside the range $-1\lesssim x_3^s<1$.
Neutron matter
--------------
The case of pure neutron matter is somehow very peculiar. The correction due to the spin-density dependent terms reads $$\mathcal{H}_3^s(neut.)=\frac{t_3^s}{24}\rho^2 \rho_s^{\gamma_s}(1-x^s_3)
\Big[1-\delta_S^2\Big].$$ It is then clear that the correction is zero for $\delta_S=0$ and also for $\delta_S=1$. This property is related to the anti-symmetrization of the interacting nucleons. Indeed, in fully polarized neutron matter, the quantum numbers for spin and isospin are $S=1$ and $T=1$ while the new spin-density dependent interaction (\[eq:spin\]) act in the $L=0$ channel. The new spin-density dependent interaction (\[eq:spin\]) have thus no effect at all in the purely spin-polarized neutron matter. Only odd $L$ terms could play a role in the fully polarized neutron matter. This property has been used to provide a necessary condition to remove the spin instabilities and lead to the condition $-5/4<x_2<-1$ [@kut94]. This condition has been used in the fitting procedure of SLy5 [@chaba] and it explains the robustness of the spin-symmetric ground state for this interaction. However if more flexibility in the Skyrme parameters is necessary, it might be interesting to introduce an interaction of the following form $$t_5^{s}(1+x_5^s P_\sigma) \; \mathbf{k}^\prime \rho_s(\mathbf{R}) \cdot
\delta(\mathbf{r}) \mathbf{k}\; .$$ This $L=1$ term will not contribute to spin-symmetric matter and could be adjusted to fit the energy of fully polarized matter.
In Fig. \[fig6\], it is shown that the new spin-density terms (\[eq:spin\]) contribute to the binding energy for partially polarized matter and tend to stabilize the state $\delta_S=0$. After the ferromagnetic transition, the state $\delta_S=0$ is not any more the absolute ground state in pure neutron matter. However, the new spin-density terms generate a potential barrier between the non-polarized and fully polarized states. The height of the barrier depends on the chosen parameters of the new spin terms (\[eq:spin\]) as well as on the neutron matter density. In the right panel of Fig. \[fig6\], the top of the barrier at the ferromagnetic transition is located around the polarization $\delta_S=1/\sqrt{2}$ with a height (in MeV per nucleon) of $$\frac{1}{96}\left( t_3^s(1-x_3^s)+t_3^{st}(1-x_3^{st}) \right) \rho^3 \;.
\label{eq:bar}$$ So, the higher the density, the higher the barrier. This barrier height is always positive since the curvature of the binding energy around $\delta_S=0$ is related to the Landau parameter $G_{0,NM}$ in neutron matter which is larger than -1 at all densities (see Fig. \[fig3\]). For the BSk17st force, at $\rho\simeq \rho_f=0.19$ fm$^{-3}$ this barrier amounts to about 10 MeV per nucleon.
From Eq. (\[eq:bar\]), it can also be seen that the parameter $x_3^{st}$ influences the height of the barrier. The lower $x_3^{st}$, the larger the barrier. For BSk17st, we set $x_3^{st}=-3$ to get a barrier above the non-polarized ground state of the order of 10 MeV per nucleon. This condition is chosen with respect to the theoretical predictions [@fan01; @vid02a; @vid02b; @bom06] that nuclear matter is spin-symmetric up to reasonable high densities (see for instance discussion in the introduction of Ref. [@marg08]). With a barrier height of the order of 10 MeV per nucleon, newly born neutron stars with typical temperature going from 1 to 5 MeV might not be the site of a ferromagnetic phase transition. The transition towards a non-polarized cold neutron star shall then be stable and the remaining neutron star spin-symmetric. Notice however that despite the theoretical predictions that dense matter is not spin-polarized [@fan01; @vid02a; @vid02b; @bom06], there are no strong evidences against the occurrence of ferromagnetic phase transition from observation of neutron stars. Indeed, a spin-polarized phase in the core of neutron stars might induce the very huge magnetic fields 10$^{15-16}$ G yet unexplained that have been proposed as the driving force for the braking of magnetars [@hae96].
Conclusions
===========
The occurrence of the spin instability beyond the saturation density is a common feature shared by different effective mean-field approaches such as Skyrme HF, Gogny HF [@marg01] or relativistic HF [@pilar]. The analysis of the spin component of the Skyrme interaction as well as its extensions might thus guide us to a wider understanding of the spin channel in general for nuclear interaction. There are many reason for looking at this channel. For instance for its competition with pairing correlations in the odd-even-mass staggering [@dug01], for rotating superdeformed nuclei [@doba95], for a better description of GT response, and for all applications in astrophysics such as for instance predictions of $\beta$-decay half-lives of very neutron rich nuclei produced during the *r*-process nucleosynthesis [@bor06], reliable calculation of neutron star crust properties such as ground-states and collective motion [@gra08], for 0$\nu-$ and 2$\nu$ double beta decay processes, for URCA fast cooling, and also for neutrino mean free path in proto-neutron stars.
In this paper we have carefully analyzed the ground state properties of finite nuclei and infinite matter obtained by the extended Skyrme interactions with the spin-density terms proposed in Ref. [@marg08]. In finite even nuclei, the new spin-density interaction (\[eq:spin\]) are simply zero and from the OSPA we have shown that these terms has only negligible contributions to the ground-state of odd nuclei. These results has been obtained either by introducing the new spin-density dependent terms in a perturbative way to existing Skyrme interactions such as SLy5st or performing a global adjustment of the parameter set on the nuclear chart. A new mass formula HFB-17st adjusted in the whole isotope chart (2149 nuclei) is obtained with the rms deviation of about 575 keV. From the analysis of the ground state of nuclear matter, a range for the parameter $x_3^s$ is restricted to $-1\lesssim x_3^s<1$ in order to stabilize the spin-symmetric matter. The case of neutron matter is also discussed and it is shown that the new terms (\[eq:spin\]) with the relative angular momentum $L=0$ have no contribution to fully polarized neutron matter. Thus, it has been shown that by using the extended Skyrme interactions [@marg08], the Landau parameters $G_0$ and $G_0^\prime$ could be tuned to realistic values without altering the ground-state properties in odd nuclei as well as of nuclear matter.
S.G. acknowledges financial support from FNRS. The work is partially supported by COMPSTAR, an ESF Research Networking Programme and the Japanese Ministry of Education, Culture, Sports, Science and Technology by Grant-in-Aid for Scientific Research under the program numbers (C) 20540277.
Time-odd components in the mean field of the Skyrme interaction
===============================================================
In odd nuclei, the energy density $\mathcal{H}$ acquires a dependence on the spin-densities $\rho_{s}$ and $\rho_{st}$ [@doba95; @bend2002]. Respecting the decomposition of the Skyrme energy functional proposed in Ref. [@chaba], the components of $\mathcal{H}^\mathrm{odd}$ are $$\begin{aligned}
\mathcal{H}_0^\mathrm{odd} &=& \frac{1}{4}t_0\Big[(x_0-\frac{1}{2})\rho_s^2-\frac{1}{2}\rho_{st}^2\Big] \label{eq:a2} \; ,\\
\mathcal{H}_3^\mathrm{odd} &=& \frac{1}{24}t_3\rho^\gamma\Big[(x_3-\frac{1}{2})\rho_s^2-\frac{1}{2}\rho_{st}^2\Big] \label{eq:a1} \; ,\\
\mathcal{H}_\mathrm{fin}^\mathrm{odd} &=&
\frac{1}{32}[3t_1(1-x_1)+t_2(1+x_2)]\Big(\rho_{sn}\nabla^2\rho_{sn}+\rho_{sp}\nabla^2\rho_{sp} \Big)
\nonumber\\
&&+\frac{1}{32}[t_2x_2-3t_1x_1]\Big(\rho_{sn}\nabla^2\rho_{sp}+\rho_{sp}\nabla^2\rho_{sn} \Big)
\label{eq:a3} \; ,\\
\mathcal{H}_\mathrm{eff}^\mathrm{odd} &=&
\frac{1}{16}[-t_1(1-2x_1)+t_2(1+2x_2)]\rho_{s}\tau_s
+\frac{1}{16}[t_2-t_1]\rho_{st}\tau_{st} \label{eq:a4} \; ,\\
\mathcal{H}_\mathrm{sg}^\mathrm{odd} &=&
\frac{1}{16}[t_1(1-2x_1)-t_2(1+2x_2)]\mathbf{j}^2_{s}
+\frac{1}{16}[t_1-t_2]\mathbf{j}^2_{st}, \label{eq:a5}\end{aligned}$$ where $\tau_{s}$ ($\tau_{st}$) is the spin (spin-isospin) kinetic density energy defined as $\tau_{s}=\tau_{\uparrow}-\tau_{\downarrow}$ ($\tau_{st}=\tau_{n\uparrow}-\tau_{n\downarrow}-\tau_{p\uparrow}+\tau_{p\downarrow}$) and $\mathbf{j}^2_{s}$ ($\mathbf{j}^2_{st}$) is the spin (spin-isospin) current.
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[crrrrrr]{} Nucleus & $E_\mathrm{TOT}$ & $E_\mathrm{MF}$ & $E_\mathrm{so}$& $E_\mathrm{Coul}$& $E_\mathrm{kin}$& s.p. energy\
& (MeV) & (MeV) & (MeV) & (MeV) & (MeV) & (MeV)\
$^{42}$Ca & -362.591 & -1111.434 & -9.173 & 72.023 & 685.993 & -9.66\
$^{41}$Ca & -352.942 & -1081.395 & -5.259 & 72.116 & 661.596 & -9.64\
$^{41}$Ca with (\[eq:spin\]) & -352.918 & -1081.359 & -5.259 & 72.115 & 661.584 & -9.64\
$\Delta E$ & 0.024 & 0.036 & 0.000 & -0.001 & -0.012 &\
$^{50}$Ca & -429.654 & -1326.381 & -33.958 & 70.905 & 859.779 & -5.84\
$^{49}$Ca & -423.876 & -1305.865 & -33.639 & 71.105 & 844.523 & -5.70\
$^{49}$Ca with (\[eq:spin\]) & -423.825 & -1305.754 & -33.634 & 71.102 & 844.461 & -5.70\
$\Delta E$ & 0.051 & 0.111 & 0.005 & -0.003 & -0.062 &\
[crrrr]{} Nucleus & $E_\mathrm{TOT}^\mathrm{odd}$ & $E_{0}^\mathrm{odd}$ & $E_\mathrm{eff+fin+sg}^\mathrm{odd}$ & $E_{3}^\mathrm{odd}$\
& (MeV) & (MeV) & (MeV) & (MeV)\
$^{41}$Ca & 0.329 & 0.196 & -0.007 & 0.140\
$^{49}$Ca & 0.151 & 0.187 & -0.176 & 0.140\
[ccc]{} & BSk17 & BSk17st\
$t_0$ & -1837.33 &-1837.19\
$t_1$ & 389.102 & 388.916\
$t_2$ & -3.1742 & -5.3076\
$t_3$ & 11523.8 & 11522.7\
$t_3^{s}$ &0 & 40000\
$t_3^{st}$ &0 & 30000\
$x_0$ & 0.411377 & 0.410279\
$x_1$ & -0.832102 & -0.834832\
$x_2$ & 49.4875 & 29.0669\
$x_3$ & 0.654962 & 0.655322\
$x_3^{s}$ & 0 & -0.5\
$x_3^{st}$ & 0 & -3\
$\gamma$ & 0.3 & 0.3\
$\gamma^{s}$ & - & 2\
$\gamma^{st}$ & - & 2\
$W_0$ & 145.885 & 146.048\
$f_{n}^+$ & 1.000 & 1.000\
$f_{n}^-$ & 1.044 & 1.045\
$f_{p}^+$ & 1.055 & 1.059\
$f_{p}^-$ & 1.050 & 1.059\
$V_W$ & -2.00 & -2.06\
$\lambda$ & 320 & 410\
$V_W^{\prime}$ & 0.86 & 084\
$A_0$ & 28 & 28\
[ccc]{} &HFB-17&HFB-17st\
$\sigma(2149~M)$ &0.581 &0.575\
$\sigma(M_{nr})$&0.729& 0.738\
$\sigma(S_n)$ &0.506&0.495\
$\sigma(Q_\beta)$ &0.583&0.585\
$\sigma(R_c)$ &0.0300&0.0302\
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We consider an extension of the conditional min- and max-entropies to infinite-dimensional separable Hilbert spaces. We show that these satisfy characterizing properties known from the finite-dimensional case, and retain information-theoretic operational interpretations, e.g., the min-entropy as maximum achievable quantum correlation, and the max-entropy as decoupling accuracy. We furthermore generalize the smoothed versions of these entropies and prove an infinite-dimensional quantum asymptotic equipartition property. To facilitate these generalizations we show that the min- and max-entropy can be expressed in terms of convergent sequences of finite-dimensional min- and max-entropies, which provides a convenient technique to extend proofs from the finite to the infinite-dimensional setting.'
author:
- Fabian Furrer
- 'Johan [Å]{}berg'
- Renato Renner
title: 'Min- and Max-Entropy in Infinite Dimensions'
---
Introduction
============
Entropy measures are fundamental to information theory. For example, in classical information theory a central role is played by the Shannon entropy [@Shannon] and in quantum information theory by the von Neumann entropy. Their usefulness partially stems from the fact that they have several convenient mathematical properties (e.g. strong subadditivity) that facilitate a ‘calculus’ of information and uncertainty. Indeed, entropy measures can even be characterized axiomatically in terms of such properties [@Renyi]. However, equally important for their use in information theory is the fact that they are related to operational quantities. This means that they characterize the optimal efficiency by which various information-theoretic tasks can be solved. One example of such a task is source coding, where one considers a source that randomly outputs data according to some given probability distribution. The question of interest is how much memory is needed in order to store and faithfully regenerate the data. Another example is channel coding, where the aim is to reliably transmit information over a channel. Here we ask how many bits (or qubits in the quantum case) one can optimally transmit per use of the channel [@Shannon; @ChannelCoding; @Schuhmacher].
The operational relevance of Shannon and von Neumann entropy is normally limited to the case when one considers the asymptotic limit over infinitely many instances of a random experiment, which are independent and identically distributed (iid) or can be described by a Markov process. In the case of source coding this corresponds to assuming an iid repetition of the source. In the limit of infinitely many such repetitions, the average number of bits one needs to store per output is given by the Shannon entropy of the distribution of the source [@Shannon]. In the general case, where we have more complicated types of correlations, or where we only consider finite instances, the role of the Shannon or von Neumann entropies appears to be taken over by other measures of entropy, referred to as the smooth min- and max-entropies [@RennerPhD]. For example, in [@ChannelCodingMaxEntropy; @RenesRenner] it was found that the smooth max-entropy characterizes one-shot data compression, i.e., when we wish to compress a single output of an information source. Furthermore, in [@ChannelCodingMinEntropy] it was proved that in one single use of a classical channel, the transmission can be characterized by the difference between a smooth min- and max-entropy. The von Neumann entropy of a state can be regained via the quantum asymptotic equipartition property (AEP) [@RennerPhD; @QuantumAEP], by applying these measures to asymptotically many iid repetitions of the state. This allows us to derive properties of the von Neumann entropy from the smooth min- and max-entropies; a technique that has been used for an alternative proof of the quantum reverse Shannon theorem [@Reverse], and to derive an entropic uncertainty relation [@Uncertainty]. The min- and max-entropies furthermore generalize the spectral entropy rates [@InformationSpectrum] (that are defined in an asymptotic sense) which themselves have been introduced as generalizations of the Shannon entropy [@Han; @HanVerdu]. Closely related quantities are the relative min- and max-entropies [@parent], which have been applied to entanglement theory [@entang1; @entang2] as well as channel capacity [@capacity].
So far, the investigations of the operational relevance and properties of the min- and max-entropy and their smoothed versions have been almost exclusively focused on quantum systems with finite-dimensional Hilbert spaces. Here we consider the min- and max-entropy in infinite-dimensional separable Hilbert spaces. Since the modeling in vast parts of quantum physics is firmly rooted in infinite-dimensional Hilbert spaces, it appears that such a generalization is crucial for the application of these tools. For example, it has recently been shown that the smooth min- and max-entropies are the relevant measures of entropy in certain statistical mechanics settings [@Oscar; @Lidia]. An extension of these ideas to, e.g., quantized classical systems, would require an infinite-dimensional version of the min- and max-entropy. Another example is quantum key distribution (QKD), where in the finite-dimensional case the smooth min-entropy bounds the length of the secure key that can be extracted from an initial raw key [@RennerPhD]. The generalization to infinite dimensions has therefore direct relevance for continuous variable QKD (for references see, e.g., Section II.D. 3 of [@Scarani]). In such a scheme one uses the quadratures of the electromagnetic field to establish a secret key (as opposed to other schemes that use, e.g., the polarization degree of freedom of single photons). Since such QKD methods are based on the generation of coherent states and measurement of quadratures, it appears rather unavoidable to use infinite-dimensional Hilbert spaces to model the states of the field modes. Beyond the obvious application to continuous variable quantum key distribution, one can argue that there are several quantum cryptographic tasks that today are analyzed in finite-dimensional settings, which strictly speaking would require an analysis in infinite-dimensions, since there is in general no reason to assume the Hilbert spaces of the adversary’s systems to be finite.
As indicated by the above discussion, an extension of the min- and max-entropies to an infinite-dimensional setting does not only require that we can reproduce known mathematical properties of these measures, but also that we should retain their operational interpretations. A complete study of this two-fold goal would bring us far beyond the scope of this work. However, here we pave the way for this development by introducing an infinite-dimensional generalization of the min- and max-entropy, and demonstrating a collection of ‘core’ properties and operational interpretations. In particular, we derive (under conditions detailed below) a quantum AEP for a specific choice of an infinite-dimensional conditional von Neumann entropy. On a more practical level we introduce a technique that facilitates the extension of results proved for the finite-dimensional case to the setting of separable Hilbert spaces. More precisely, we show that the conditional min- and max-entropies for infinite-dimensional states can be expressed as limits of entropies obtained by finite-dimensional truncations of the original state (Proposition \[p:reduction of Hmin to finite dim\]). This turns out to be a convenient tool for generalizations, and we illustrate this on the various infinite-dimensional extensions that we consider.
The $\epsilon$-smoothed min-and max-entropies are defined in terms of the ‘un-smoothed’ ($\epsilon=0$) min- and max-entropies (which we simply refer to as ‘min- and max-entropy’). In Section \[subsec:def\] we extend these ‘plain’ min- and max-entropies to separable Hilbert spaces. Section \[section:reduction of cond entropies\] contains the main technical tool, Proposition \[p:reduction of Hmin to finite dim\], by which the infinite-dimensional min- and max-entropies can be expressed as limits of sequences of finite-dimensional entropies. The proof of Proposition \[p:reduction of Hmin to finite dim\] is given in Appendix \[app:proofprop1\]. In Section \[section:properties of min- and max-entropy\] we consider properties of the min- and max-entropy, e.g., additivity and the data processing inequality. Section \[chapter:operational interpretation\] focuses on the generalization of operational interpretations. In Section \[chapter:smooth entropy\] we consider the extension of the $\epsilon$-smooth min- and max-entropies, for $\epsilon>0$. In Section \[AEP\] we bound the smooth min- and max-entropy of an iid state on a system $A$ conditioned on a system $B$ in terms of the conditional von Neumann entropy (Proposition \[prop:AEP lower bound\]). This result relies on the assumption that $A$ has finite von Neumann entropy. If $A$ furthermore has a finite-dimensional Hilbert space (but the Hilbert space of $B$ is allowed to be separable) we prove that these smooth entropies converge to the conditional von Neumann entropy (Corollary \[cor:AEP\]), which corresponds to a quantum AEP. The paper ends with a short summary and outlook in Section \[concl\].
\[chapter:conditioned entropies\] Min- and Max-Entropy
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\[subsec:def\]Definition of the conditional min- and max-entropy
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Associated to each quantum system is a Hilbert space $H$, which we assume to be separable in all that follows. We denote the bounded operators by $\mc{L}(H)=\{A:H \rightarrow H \ |\ \Vert A\Vert < \infty\}$, where $\Vert A\Vert = \sup_{\Vert \psi\Vert = 1}\Vert A|\psi\rangle\Vert $ is the standard operator norm. Among these, the trace class operators satisfy the additional feature of having a finite trace norm $\Vert T\Vert_1 := \operatorname{tr}|T| = \operatorname{tr}\sqrt{T^{\dagger}T}$. The set of trace class operators is denoted by $\tau_1(H):= \{ T \in \mc{L}(H)| \ \Vert T\Vert_1 < \infty \}$.
We consider states which can be represented as density operators, i.e., normal states [@Bratteli; @Robinson], and denote the set of all these states as $\mc{S}(H):=\{\rho \in \tau_1(H)| \ \rho \geq 0, \Vert \rho\Vert_1 =1 \}$. It is often convenient to allow non-normalized density operators, which form the positive cone $\tau^+_1(H)\subset\tau_1(H)$ consisting of all non-negative trace class operators.
We define the conditional min- and max-entropy of bipartite quantum systems analogously to the finite-dimensional case [@OperationalMeaning].[^1]
\[def:min/max-entropy\]Let $H_A$ and $H_B$ be separable Hilbert spaces and $\roo \in \tau_1^+(\hahb)$. The min-entropy of $\rho_{AB}$ conditioned on $\sigma_{B} \in \tau_1^+(\hb)$ is defined by $$\label{def,eq1:min/max-entropy}
H_{\mathrm{min}}(\rho_{AB}|\sigma_{B}) := -\log \inf \{\lambda \in \mathbb{R} | \lambda \operatorname{id}_{A} \otimes \sigma_{B} \geq \rho_{AB} \},$$ where we let $H_{\mathrm{min}}(\roo|\sii) := -\infty$ if the condition $\lambda \operatorname{id}_{A} \otimes \sigma_{B} \geq \rho_{AB}$ cannot be satisfied for any $\lambda \in \mathbb{R}$. Moreover, we define the min-entropy of $\roo$ conditioned on B by $$\label{def,eq2:min/max-entropy}
H_{\mathrm{min}}(\rho_{AB}|B) := \sup_{\sigma_{B} \in \mc{S}(\hb)} H_{\mathrm{min}}(\rho_{AB}|\sigma_{B}).$$ The max-entropy of $\rho_{AB}$ conditioned on B is defined as the dual of the min-entropy $$\label{def,eq3:min/max-entropy}
H_{\mathrm{max}}(\rho_{AB}|B) := -H_{\mathrm{min}}(\rho_{AC}|C),$$ where $\rho_{ABC}$ is a purification of $ \rho_{AB}$.
In the definition above, and in all that follows, we let “$\log$” denote the binary logarithm. The reduction of a state to a subsystem is indicated by the labels of the Hilbert space, e.g., $\rho_A = \operatorname{tr}_C\rho_{AC}$. Note that the max-entropy $ H_{\mathrm{max}}(\rho_{AB}|B)$ as defined in (\[def,eq3:min/max-entropy\]) is independent of the choice of the purification $\rho_{ABC}$, and thus well-defined. This follows from the fact that two purifications can only differ by a partial isometry on the purifying system, and the min-entropy $H_{\mathrm{min}}(\rho_{AC}|C)$ is invariant under these partial isometries on subsystem C.
The two optimizations in the definition of $H_{\mathrm{min}}(\rho_{AB}|B)$, in Eqs. (\[def,eq1:min/max-entropy\]) and (\[def,eq2:min/max-entropy\]), can be combined into $$\label{eq:equiv def of the min-entropy}
H_{\mathrm{min}}(\rho_{AB}|B) = -\log \big(\inf \{ \operatorname{tr}\siit \ | \ \siit \in \tau^+_1(H_B), \\ \operatorname{id}_A \otimes \siit \geq \roo\} \big).$$ For convenience we introduce the following two quantities: $$\begin{aligned}
\label{lambdadef1}
\Lambda(\roo|\sii) & := & 2^{- H_{\mathrm{min}}(\rho_{AB}|\sii)} =\inf \{\lambda \in \mathbb{R} | \lambda \operatorname{id}_{A} \otimes \sigma_{B} \geq \rho_{AB} \},\\
\label{lambdadef2} \Lambda(\roo|B) &:= & 2^{- H_{\mathrm{min}}(\rho_{AB}|B)} = \inf \{ \operatorname{tr}\siit \ | \ \siit \in \tau^+_1(H_B), \operatorname{id}_A \otimes \siit \geq \roo\}.\end{aligned}$$
\[section:reduction of cond entropies\] Finite-dimensional approximations of min- and max-entropies
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In this section we present the main result, Proposition \[p:reduction of Hmin to finite dim\], that provides a method to express the conditional min- and max-entropy as a limit of min- and max-entropies of finite-dimensional systems. The rough idea is to choose sequences $\{P^A_k\}_{k=1}^{\infty}$ and $\{P^B_k\}_{k=1}^{\infty}$ of projectors [^2] onto finite-dimensional subspaces $U_k^A \subset H_A$ and $U_k^B \subset H_B$, respectively, both converging to the identity. Then we define a sequence of non-normalized states as $\roo^k = (P^A_k \otimes P^B_k) \roo (P^A_k \otimes P^B_k)$. The min- or max-entropy of $\roo^k$ can now be treated as if the underlying Hilbert space would be $U_k^A \otimes U_k^B$ (Lemma \[I:id-P\]), and therefore finite-dimensional. Proposition \[p:reduction of Hmin to finite dim\] shows that, as $k\rightarrow \infty$, these finite-dimensional entropies approach the desired infinite-dimensional entropy. As we will see, this provides a convenient method to extend properties from the finite to the infinite setting.
When we say that an operator sequence $Q_{k}$ converges to $Q$ in the weak operator topology we intend that $\lim_{k\rightarrow 0} \langle \chi |Q-Q_{k}|\psi\rangle = 0$ for all $|\chi\rangle,|\psi\rangle\in H$. The sequence converges in the strong operator topology if $\lim_{k\rightarrow 0} \Vert (Q-Q_{k})|\psi\rangle\Vert = 0$ for all $|\psi\rangle\in H$.
\[def:projected states\] Let $\{P^A_k\}_{k\in\mathbb{N}} \subset \mc{L}(H_A)$, $\{P^B_k\}_{k\in\mathbb{N}} \subset \mc{L}(H_B)$ be sequences of projectors such that for each $k \in \mathbb{N}$ the projection spaces $U_k^A\subset H_A$, $U_k^B\subset H_B$ of $P_k^A$, $P_k^B$ are finite-dimensional, $P_k^A \leq P_{k'}^A$ and $P_k^B \leq P_{k'}^B$ for all $k \leq k'$, and $P_k^{A}$, $P_k^B$ converge in the weak operator topology to the identity. We refer to such a sequence $(P_k^A,P_k^B)$ as a generator of projected states. For $\roo \in \mc{S}(\hahb)$ we define the (non-normalized) states $$\label{defprojectedstates}
\roo^k := (P^A_k \otimes P^B_k) \roo (P^A_k \otimes P^B_k),$$ which we call the projected states of $\roo$ relative to $(P_k^A,P_k^B)$. Moreover, we refer to $$\label{snklv}
\rooh^k := \frac{\roo^k}{ \operatorname{tr}\roo^k}$$ as the normalized projected states of $\roo$ relative to $(P_k^A,P_k^B)$.
Note that a sequence of projectors that converges in the weak operator topology to the identity also converges in the strong operator topology to the identity. As a matter of convenience, we can thus in all that follows regard the generators of projected states to converge in the strong operator topology. One may also note that the sequence of projected states $\roo^k$ (as well as the normalized projected states $\rooh^k$) converges to $\roo$ in the trace norm (see Corollary \[dfnbkl\] in Appendix \[section:Technical Lemmas\]). The normalized projected states in Eq. (\[snklv\]) are of course only defined if $\operatorname{tr}\roo^k \neq 0$. However, this is true for all sufficiently large $k$ due to the trace norm convergence to $\roo$.
\[p:reduction of Hmin to finite dim\] For $\roo\in\mc{S}(\hahb)$, let $\{\roo^k\}_{k\in\mathbb{N}}$ be the projected states of $\roo$ relative to a generator $(P_k^A,P_k^B)$, and $\roh^k_{AB}$ the corresponding normalized projected states. Furthermore, let $\sii \in \mc{S}(\hb)$ and define the operators $\sigma_B^k := P^B_k\sigma_B P^B_k$ and $\sih_B^k :=\operatorname{tr}(\sigma_B^k)^{-1}\sigma^k_B$. Then, the following three statements hold. $$\label{p,eq1:reduction of Hmin to finite dim}
\Hmin = \lim_{k\rightarrow \infty} \Hmink =\lim_{k\rightarrow \infty} H_{\mathrm{min}}\big(\roh^k_{AB}|\sih^k_B\big),$$ and the infimum in Eq. (\[def,eq1:min/max-entropy\]) is attained if $\Hmin$ is finite. $$\label{p,eq2:reduction of Hmin to finite dim}
\HminB = \lim_{k\rightarrow \infty} H_{\mathrm{min}}(\roo^k|B_k) = \lim_{k\rightarrow \infty} H_{\mathrm{min}} \big(\roh^k_{AB}|B_k\big),$$ and the supremum in Eq. (\[def,eq2:min/max-entropy\]) is attained if $\HminB$ is finite. $$\label{p,eq3:reduction of Hmin to finite dim}
\HmaxB = \lim_{k \rightarrow \infty} H_{\mathrm{max}}(\roo^k|B_k) = \lim_{k\rightarrow \infty} H_{\mathrm{max}} \big(\roh^k_{AB}|B_k\big).$$ Here, $B_k$ denotes the restriction of system $B$ to the projection space $U_k^B$ of $P_k^B$.
The proof of this proposition is found in Appendix \[app:proofprop1\]. When we say that the infimum in (\[def,eq1:min/max-entropy\]) is attained, it means that there exists a finite $\lambda'$ such that $ \lambda' \operatorname{id}_{A} \otimes \sigma_{B} - \rho_{AB} \geq 0$ and $H_{\mathrm{min}}(\rho_{AB}|\sigma_{B}) = -\log \lambda'$. Similarly, that the supremum in (\[def,eq2:min/max-entropy\]) is attained, means that there exists a $\sigma'_{B} \in \tau_1^+(H_B)$ satisfying $\operatorname{id}\otimes \sigma'_B \geq \roo$ such that $H_{\mathrm{min}}(\rho_{AB}|B) = H_{\mathrm{min}}(\rho_{AB}|\sigma'_{B})$.
Given the above proposition, a natural question is if $\HminB$ and $\HmaxB$ are trace norm continuous in general. In the finite-dimensional case [@OnTheSmoothing] it is known that these entropies are continuous with a Lipschitz constant depending on the dimension of $H_A$. However, the following example shows that they are in general not continuous in the infinite-dimensional case. Let $\{|k\rangle\}_{k= 0,1,\ldots}$ be an arbitrary orthonormal basis of the Hilbert space $H_A$. For each $n= 1,2,\ldots$ let $$\rho_{n} = (1-\frac{1}{n})|0\rangle\langle 0| + \frac{1}{n^{2}}\sum_{k=1}^{n}|k\rangle\langle k|.$$ One can see that $\rho_n$ converges in the trace norm to $|0\rangle\langle0|$ as $n\rightarrow\infty$, while $\lim_{n\rightarrow\infty}H_{\mathrm{max}}(\rho_n) = 2$, and $H_{\mathrm{max}}(|0\rangle\langle 0|) = 0$. Hence, the max-entropy is not continuous. ($H_{\mathrm{max}}(\rho)$ without conditioning means that we condition on a trivial subsystem $B$. See Eq. (\[cor,eq1:unconditioned max-entropy\]).) The duality, Eq. (\[def,eq3:min/max-entropy\]), yields an example also for the min-entropy.
\[section:properties of min- and max-entropy\] Properties of Min- and Max-Entropy
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Additivity and the data processing inequality
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Proposition \[p:reduction of Hmin to finite dim\] can be used as a tool to generalize known finite-dimensional results to the infinite-dimensional case. A simple example is the ordering property [@QuantumAEP] $$\label{orderingHminHmax}
H_{\mathrm{min}}(\roo|B) \leq H_{\mathrm{max}}(\roo|B),$$ which is obtained by a direct application of Proposition \[p:reduction of Hmin to finite dim\]. Another example is the additivity, which in the finite-dimensional case was proved in [@RennerPhD]. A direct generalization of the proof techniques they employed appears rather challenging, while Proposition \[p:reduction of Hmin to finite dim\] makes the generalization straightforward.
\[p:additivity\] Let $\roo \in \mc{S}(\hahb)$ and $\rho_{A'B'} \in \mc{S}(H_{A'} \otimes H_{B'}) $ for $H_A, H_{A'}$, $H_B$, and $H_{B'}$ separable Hilbert spaces. Then, it follows that $$\begin{aligned}
\label{Hminadd} H_{\mathrm{min}}(\roo \otimes \rho_{A'B'}| BB') & = H_{\mathrm{min}}(\roo|B) + H_{\mathrm{min}}(\rho_{A'B'}|B'), \\
\label{Hmaxadd} H_{\mathrm{max}}(\roo \otimes \rho_{A'B'}|BB') & = H_{\mathrm{max}}(\roo|B) + H_{\mathrm{max}}(\rho_{A'B'}|B').\end{aligned}$$
The proof is a simple application of the approximation scheme in Proposition \[p:reduction of Hmin to finite dim\] combined with Lemma \[nvdakj\] and the finite-dimensional version of Proposition \[p:additivity\], and therefore omitted.
For the sake of completeness we note that the data processing inequalities [@RennerPhD] also hold in the infinite-dimensional setting. In this case, however, there is no need to resort to Proposition \[p:reduction of Hmin to finite dim\], as the proof in [@RennerPhD] can be generalized directly.
\[p:strsubadditivity\] Let $\rho_{ABC} \in \tau_{+}(\hahb\otimes H_C)$ for separable Hilbert spaces $H_A$, $H_B$ and $H_C$. Then, it follows that $$\begin{aligned}
\label{p,eq2:strong subadd of Hmin} & & H_{\mathrm{min}}(\rho_{ABC}|BC) \leq H_{\mathrm{min}}(\rho_{AB}|B), \\
\label{p:strong subadd of Hmax} & & H_{\mathrm{max}}(\rho_{ABC}|BC) \leq H_{\mathrm{max}}(\rho_{AB}|B).\end{aligned}$$
The data processing inequalities can be regarded as the min- and max-entropy counterparts of the strong subadditivity of the von Neumann entropy (and are sometimes directly referred to as “strong subadditivity”). One reason for this is that the standard formulation of the strong subadditivity of von Neumann entropy [@LiebRuskai1; @Lieb; @LiebRuskai2], $H(\rho_{ABC}) + H(\rho_{B})\leq H(\rho_{AB}) + H(\rho_{BC})$, can be recast in the same form.
\[subsect:entropy of pure states versus unconditioned entropy\] Entropy of pure states, and a bound for general states
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Here we briefly consider the fact that the min-entropy can take the value $-\infty$, and the max-entropy can take the value $+\infty$. For this purpose we discuss the special case of pure states, as well as the case of no conditioning (i.e., if there is no subsystem $B$). Based on this we obtain a general bound which says that the conditional min- and max-entropies of a state $\roo$ are finite if the operator $\sqrt{\rho_A}$ is trace class. Moreover it turns out that the min-entropy cannot attain the value $+\infty$, while the max-entropy cannot attain $-\infty$.
\[cor:min-entropy for purestates\] The min-entropy of $\roo=\vert \psi \rangle \langle \psi \vert$, where $\vert\psi\rangle \in \hahb$, is given by $$\label{cor,eq1:min-entropy for purestates}
H_{\mathrm{min}}(\roo|B) = - 2 \log\operatorname{tr}\sqrt{\rho_A}.$$
From this lemma we can conclude that $H_{\mathrm{min}}(\roo|B)$ is finite if and only if $\sqrt{\rho_{A}}$ is trace class. Otherwise $H_{\mathrm{min}}(\roo|B) = -\infty$. If the Schmidt decomposition [@Convertability] of $\psi$ is given by $\sum_{k=1}^{\infty} r_k \vert a_k\rangle\vert b_k\rangle$, we have $\operatorname{tr}\sqrt{\rho_A}=\sum_{k=1}^{\infty}r_k$, such that a finite Schmidt rank always implies that $H_{\mathrm{min}}(\roo|B)$ is finite. Recall that the Schmidt coefficients characterize the entanglement of a pure state, and, roughly speaking, that the more uniformly the Schmidt coefficients are distributed the stronger is the entanglement (see for instance [@Convertability]). This suggests that pure states with $H_{\mathrm{min}}(\roo|B) =-\infty$ are entangled in a rather strong sense.
Let $\vert \psi \rangle = \sum_{k=1}^{\infty} r_k \vert a_k\rangle\vert b_k\rangle$ be the Schmidt decomposition of $\vert \psi \rangle$, and $\tilde{\sigma}_{B} \in \tau^+_1(\hb)$ any operator that satisfies $\operatorname{id}_A \otimes \siit \geq \roo$. For each $n \in \mathbb{N}$ define $|\chi_{n}\rangle = \sum_{k=1}^{n}|a_{k}\rangle\vert b_{k}\rangle$. It follows that $$\operatorname{tr}\siit \geq \langle\chi_{n}|\operatorname{id}_A\otimes \siit|\chi_{n}\rangle \geq \langle\chi_{n}|\roo|\chi_{n}\rangle = \big(\sum_{k=1}^{n}r_{k}\big)^{2},$$ and thus, by taking the infimum over all $\tilde{\sigma}_{B}$ with $\operatorname{id}_A \otimes \siit \geq \roo$, as well as the supremum over all $n$, we find $\Lambda(\rho_{AB}|B) \geq (\operatorname{tr}\sqrt{\rho_A})^2$. Especially, we see that if $\operatorname{tr}\sqrt{\rho_A} = +\infty$ then $\Lambda(\roo|B) = +\infty$ (and thus $H_{\mathrm{min}}(\roo|B)=-\infty$). In the following we assume that $\operatorname{tr}\sqrt{\rho_A}< +\infty$, i.e., $\sqrt{\rho_A} \in \tau^+_1(\ha)$. We show that the lower bound $\Lambda(\roo|B) \geq (\operatorname{tr}\sqrt{\rho_A})^2$ is attained, by proving that $\siit := \operatorname{tr}(\sqrt{\rho_A})\sqrt{\rho_B}$ satisfies $id_A \otimes \siit \geq \roo$. By using the Schmidt decomposition of $\psi$ we compute for an arbitrary $\eta \in \hahb$ $$\begin{aligned}
\langle \eta \vert ( \operatorname{id}\otimes\siit-\roo) \vert \eta\rangle
= & \operatorname{tr}(\sqrt{\rho_A})\sum_{k,l=1}^{\infty}|c_{k,l}|^{2}r_{l} -\Big|\sum_{k=1}^{\infty}c_{k,k}r_{k}\Big|^{2} \\
\geq & \sum_{l=1}^{\infty}r_l \sum_{k=1}^{\infty}|c_{k,k}|^{2}r_{k} - \Big|\sum_{k=1}^{\infty}c_{k,k}r_{k}\Big|^{2} \geq 0 ,\end{aligned}$$ where $c_{k,l} = (\langle a_k\vert\langle b_l\vert)\vert \eta \rangle$, and the last step follows from the Cauchy-Schwarz inequality. Hence, $\operatorname{id}_A \otimes \siit - \roo$ is positive and therefore $ \operatorname{tr}(\siit) \geq \Lambda(\roo|B)$. Combined with $\Lambda(\roo|B) \geq (\operatorname{tr}\sqrt{\rho_A})^2$, we find $H_{\mathrm{min}}(\roo|B) = -\log \Lambda(\roo|B)=- 2 \log\operatorname{tr}\sqrt{\rho_A}$.
The duality (\[def,eq3:min/max-entropy\]) allows us to rewrite Lemma \[cor:min-entropy for purestates\] by using the unconditional max-entropy. For every $\rho \in \mc{S}(H)$ this yields the quantum $1/2$-Rényi entropy (cf. [@OperationalMeaning]), $$\label{cor,eq1:unconditioned max-entropy}
H_{\mathrm{max}}(\rho) = 2\log\operatorname{tr}\sqrt{\rho} = H_{\frac{1}{2}}(\rho),$$ if $\sqrt{\rho}$ is trace-class. Otherwise $H_{\mathrm{max}}(\rho)=+\infty$.
The unconditional min-entropy is obtained by conditioning on a trivial subsystem $B$. One can see that $$\label{cor,eq1:unconditioned min-entropy}
H_{\mathrm{min}}(\rho) = -\log \Vert \rho\Vert .$$ For a pure state $\roo = \vert \psi \rangle \langle \psi \vert\in \mc{S}(\hahb)$, the max-entropy is given by $$\label{cor,eq1:max-entropy of pure states}
H_{\mathrm{max}}(\roo|B) = \log \Vert \rho_A\Vert.$$ To see this one can apply the duality (\[def,eq3:min/max-entropy\]) where we purify the pure state $\roo$ with a trivial system $C$, and next use Eq. (\[cor,eq1:unconditioned min-entropy\]).
By combining these facts with the data processing inequality, $H_{\mathrm{min}}(\rho_{ABC}|BC) \leq \HminB \leq H_{\mathrm{min}}(\rho_A)$ and $H_{\mathrm{max}}(\rho_{ABC}|BC) \leq \HmaxB \leq H_{\mathrm{max}}(\rho_A)$, for $\rho_{ABC}$ a purification of $\roo$, we find the following bounds on the min- and max-entropy.
\[cor:upper bound for min/max-entropy\] For every state $\roo \in S(\hahb)$ it holds that $$\begin{aligned}
-2\log\operatorname{tr}\sqrt{\rho_A} \leq &\HminB & \leq -\log \Vert \rho_A\Vert, \label{cor,eq1:upper bound for min-entropy} \\
\log \Vert \rho_A\Vert \leq & \HmaxB & \leq 2\log\operatorname{tr}\sqrt{\rho_A}. \label{cor,eq2:upper bound for max-entropy}\end{aligned}$$ Hence, $\HminB$ and $\HmaxB$ are finite if $\sqrt{\rho_A}$ is trace-class.
\[chapter:operational interpretation\]Operational Interpretations of Min- and Max-Entropy
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Min- and max-entropy can be regarded as answers to operational questions, i.e., they quantify the optimal solution to certain information-theoretic tasks. Max-entropy $\HmaxB$ answers the question of how distinguishable $\roo$ is from states that are maximally mixed on A, while uncorrelated with B [@OperationalMeaning] (see also Definition \[def:decoupling accuracy\] below). This is a useful concept, e.g., in quantum key distribution, where one ideally would have a maximally random key uncorrelated with the eavesdropper’s state. Thus, the above distinguishability quantifies how well this is achieved. Min-entropy $\HminB$ is related to the question of how close one can bring the state $\roo$ to a maximally entangled state on the bipartite system AB, allowing only local quantum operations on the B system [@OperationalMeaning]. In the special case that A is classical (i.e., we have a classical-quantum state, see Eq. (\[eq:cq state\]) below) one finds that $\HminB$ is related to the guessing probability, i.e., our best chance to correctly guess the value of the classical system A, given the quantum system B. In the following sections we show that these results can be generalized to the case that $H_{B}$ is infinite-dimensional. These generalizations are for instance crucial in cryptographic settings, where there is a priori no reason to expect an eavesdropper to be limited to a finite-dimensional Hilbert space, while it is reasonable to assume the key to be finite. The operational interpretations of the min- and max-entropy exhibit a direct dependence on the dimension of the A system, which is why a naive generalization to an infinite-dimensional A appears challenging, and will not be considered here.
\[section:oper interp of max entropy\] Max-entropy as decoupling accuracy
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To define decoupling accuracy we use fidelity $F(\rho,\sigma) := \Vert \sqrt{\rho}\sqrt{\sigma}\Vert_1$ as a distance measure between states.
\[def:decoupling accuracy\] For a finite-dimensional Hilbert space $\ha$ and an arbitrary separable Hilbert space $\hb$, we define the decoupling accuracy of $\rho_{AB} \in \tau_1^+(\hahb)$ w.r.t. the system B as $$\label{def,eq1:decoupling accuracy}
d(\roo|B) := \sup_{\sii \in \mc{S}(\hb)} d_A F( \roo , \tau_A \otimes \sii )^2 .$$ Here, $d_A$ is the dimension of $\ha$, and $\tau_A := d_A^{-1} \operatorname{id}_A$ is the maximally mixed state on A.
Note that in infinite-dimensional Hilbert spaces there is no trace class operator which can be regarded as a generalization of the maximally mixed state in finite dimensions. We must thus require system A to be finite-dimensional in order to keep the decoupling accuracy well-defined. In [@OperationalMeaning], Proposition \[prop:operational interp of max entropy\] was proved in the case where $H_B$ is assumed to be finite-dimensional. Below we use Proposition \[p:reduction of Hmin to finite dim\] to extend the assertion to the infinite-dimensional case.
\[prop:operational interp of max entropy\] Let $\ha$ be a finite-dimensional and $\hb$ a separable Hilbert space. It follows that $$\label{prop,eq1:operational interp of max entropy}
d(\roo|B) = 2^{H_{\mathrm{max}}(\roo|B)},$$ for each $\roo \in \tau_1^+(\hahb)$.
In the following we will need to consider physical operations (channels) on states, i.e., trace preserving completely positive maps [@Kraus]. By $\operatorname{TPCPM}(\ha,\hb)$ we denote the set of all trace preserving completely positive maps $\mc{E}: \tau_1(H_A) \rightarrow \tau_1(H_B)$. Let $\mathcal{I}$ denote the identity map.
Let us take projected states $\roo^k$ relative to a generator of the form $(\operatorname{id}_A,P^B_k)$ (this is a proper generator since $\dim H_A<\infty$). Denote the space onto which $P^B_k$ projects by $U_k^B$ and set $P_k := \operatorname{id}_A \otimes P_k^B$. The finite-dimensional version of Proposition \[prop:operational interp of max entropy\] together with Proposition \[p:reduction of Hmin to finite dim\] yield $d(\roo^k|B_k)= 2^{H_{\mathrm{max}}(\roo^k|B_k)}\rightarrow 2^{H_{\mathrm{max}}(\roo|B)}$, as $k\rightarrow\infty$.
In order to prove $d(\roo|B) \leq 2^{H_{\mathrm{max}}(\roo|B)}$ we construct a suitable TPCPM and use the fact that the fidelity can only increase under its action [@NielsenChuang]. For each $k \in \mathbb{N}$ choose a normalized state $\vert \theta_k \rangle \in \hahb$ such that $P_k\vert \theta_k \rangle =0$. We define a channel $\mc{E}_k \in \operatorname{TPCPM}(\hahb,\hahb)$ as $\mc{E}_k(\eta) := P_k \eta P_k + q_k(\eta) \vert \theta_k \rangle \langle \theta_k \vert$, with $q_k(\eta) :=\operatorname{tr}[ \eta (\operatorname{id}-P_k)]$. Then, for all $\sii \in \mc{S}(\hb)$ we find $$\begin{aligned}
F(\roo, \tau_{A} \otimes \sii) & \leq F\big( \mc{E}_k(\roo) , \mc{E}_k(\tau_{A} \otimes \sii) \big)\\
& = \big\Vert \sqrt{\roo^k}\sqrt{\tau_{A} \otimes \sii^k} + \sqrt{q_k(\roo)q_k(\tau_{A} \otimes \sii)} \vert \theta_k \rangle \langle \theta_k \vert \, \big\Vert_1 \\
& \leq \big\Vert \sqrt{\roo^k}\sqrt{\tau_{A} \otimes \sii^k} \big\Vert_1 + \sqrt{q_k(\roo)} = F(\roo^k , \tau_{A} \otimes \sii^k ) + \sqrt{q_k(\roo)},\end{aligned}$$ where $\sii^k := P_k^B \sii P_k^B$. The second line is due to the fact that $\vert \theta_k \rangle $ is orthogonal to the support of both $\roo^k$ and $\tau_{A} \otimes \sii^k$. The last line follows by the triangle inequality and $q_k(\tau_{A} \otimes \sii) \leq 1$. By taking the supremum over all $\sii \in \mc{S}(\hb)$ we obtain $$\sqrt{d(\roo|B)} \leq \sqrt{d(\roo^k|B_k)} + \sqrt{d_A \operatorname{tr}[ \roo (\operatorname{id}-P_k)]} \rightarrow 2^{\frac{1}{2} H_{\mathrm{max}}(\roo|B)},$$ as $k\rightarrow\infty$. It remains to show $d(\roo|B) \geq 2^{H_{\mathrm{max}}(\roo|B)}$. For this purpose we use that the fidelity can be reformulated as $$\label{eq:Uhlmann}
F(\rho, \sigma) = \sup_{\vert \phi \rangle} F(\vert \psi\rangle, \vert \phi \rangle),$$ where $\vert \psi\rangle$ is a purification of $\rho$, and the supremum is taken over all purifications $\vert \phi \rangle$ of $\sigma$ [@Uhlmann1]. Let us fix an arbitrary $k\in \mathbb{N}$ and a $\sii \in \mc{S}(\hb)$. Assume $\vert \psi_{ABC} \rangle$ to be a purification of $\roo$, and note that $\vert \psi^k_{ABC} \rangle := \tilde{P}_k\vert \psi_{ABC} \rangle$, with $\tilde{P}_k=P_k \otimes \operatorname{id}_C$, is a purification of $\roo^k$. Let $\vert \phi \rangle\in H_{A}\otimes H_{B}\otimes H_{C}$ be an arbitrary purification of $\tau_{A} \otimes \sii$. According to (\[eq:Uhlmann\]) it follows that $$\begin{aligned}
F(\roo , \tau_{A} \otimes \sii) & \geq F(\vert \psi_{ABC} \rangle ,\vert \phi \rangle )= | \langle \psi_{ABC} \vert \phi \rangle | \\
& = | \langle \psi_{ABC} \vert \tilde{P}_k \vert \phi \rangle + \langle \psi_{ABC} \vert \operatorname{id}- \tilde{P}_k \vert \phi \rangle | \\
& \geq | \langle \psi^k_{ABC} \vert \phi \rangle | - \Vert (\operatorname{id}-\tilde{P}_k )|\psi_{ABC}\rangle\Vert,\end{aligned}$$ where the last line is obtained by the reverse triangle inequality and the Cauchy-Schwarz inequality. By taking the supremum over all the purifications $|\phi\rangle$ of $\tau_{A} \otimes \sii$ in the above inequality, Eq. (\[eq:Uhlmann\]) yields $F(\roo , \tau_{A} \otimes \sii) \geq F(\roo^k , \tau_{A} \otimes \sii) - \Vert (\operatorname{id}-\tilde{P}_k )|\psi_{ABC}\rangle\Vert$. As this holds for all $\sii \in \mc{S}(\hb)$ and all $k$, we obtain with the definition of the decoupling accuracy: $$d(\roo|B) \geq \lim_{k \rightarrow \infty} \Big(\sqrt{d(\roo^k|B_k)} - \sqrt{d_A} \ \Vert (\operatorname{id}-\tilde{P}_k )|\psi_{ABC}\rangle\Vert \Big)^2 = 2^{H_{\mathrm{max}}(\roo|B)}.$$
\[section:Min-entropy is maximum achievable quantum correlation\] Min-entropy as maximum achievable quantum correlation
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Assume a bipartite quantum system consisting of a finite-dimensional A system and an arbitrary B system. We can then define a maximally entangled state between the A and B system as $$\label{max entangled state}
\vert \Psi_{AB} \rangle := \frac{1}{\sqrt{d_A}}\sum_{k=1}^{d_A} \vert a_k \rangle\vert b_k \rangle.$$ Here, $d_A$ denotes the dimension of $\ha$, $\{a_k\}_{k=1}^{d_A}$ an arbitrary orthonormal basis of $H_A$ and $\{b_k\}_{k=1}^{d_A}$ an arbitrary orthonormal system in $H_B$, where we assume that $\dim(H_A) \leq \dim(H_B)$.
\[def:quantum correlation\] For $\ha$ a finite-dimensional and $\hb$ a separable Hilbert space (with $\dim H_A \leq \dim H_B$), we define the quantum correlation of a state $\rho_{AB} \in \mc{S}(\hahb)$ relative to B as $$\label{def,eq1:quantum correlation}
q(\roo|B) := \sup_{\mc{E}} d_A F\big(( \mathcal{I}_A \otimes \mc{E})\roo , \mcor\big)^2 ,$$ where the supremum is taken over all $\mc{E}$ in TPCPM($H_B$,$H_B$), and $\vert \Psi_{AB}\rangle$ is given by (\[max entangled state\]).
Due to the invariance of the fidelity under unitaries [@NielsenChuang], the definition of $q(\rho_{AB}|B)$ is independent of the choice of the maximally entangled state $\vert \Psi_{AB} \rangle$. The quantum correlation can be rewritten as $$\label{eq:quantum correlation equiv expression}
q(\roo|B) = \sup_{\mc{E}} d_A \langle \Psi_{AB} \vert (\mathcal{I}_A \otimes \mc{E})\roo \vert \Psi_{AB} \rangle.$$ The min-entropy is directly linked to the quantum correlation as shown in [@OperationalMeaning] for the finite-dimensional case. We extend this result to a B system with a separable Hilbert space.
\[prop:oper interpr of min entropy\] Let $H_A$ be a finite-dimensional and $H_B$ be a separable Hilbert space. It follows that $$q(\roo|B) = 2^{-H_{\mathrm{min}}(\roo|B)} ,$$ for each $\roo \in \mc{S}(\hahb)$.
Let $\{\roo^k\}_{k\in\mathbb{N}}$ be the projected states of $\roo$ relative to a generator of the form $(\operatorname{id}_A,P_k^B)$, and set $P_k := \operatorname{id}_A \otimes P_k^B$. Let us denote the projection space of $P_k^B$ by $U_k^B$ and assume that $\vert b_l\rangle \in U_k^B$, $l=1,...,d_A$, for all $k$, with $\vert b_l\rangle$ as in equation (\[max entangled state\]). By the already proved finite-dimensional version of Proposition \[prop:oper interpr of min entropy\] and Proposition \[p:reduction of Hmin to finite dim\], we obtain $q(\roo^k|B_k) = \Lambda(\roo^k|B_k) \rightarrow \Lambda(\roo|B)$.
We begin to prove $\Lambda(\roo|B) \leq q(\roo|B)$. Fix $k$ and choose $\mc{E}_k \in \operatorname{TPCPM}(U_k^B,U_k^B)$ such that $ q(\roo^k|B_k) = d_A \langle \Psi_{AB} \vert (\mathcal{I}_A \otimes \mc{E}_k)\roo^k \vert \Psi_{AB} \rangle$. Define $\tilde{\mc{E}}_k(\rho) = \mathcal{E}_k(P_k\rho P_k) + (\operatorname{id}_B-P_k^B)\rho (\operatorname{id}_B -P_k^B)$, which is a valid quantum operation in $\operatorname{TPCPM}(\hb,\hb)$. As $\tilde{\mc{E}}_k$ is just one possible TPCPM, it follows that $$q(\roo|B) \geq d_A \langle \Psi_{AB} \vert (\mathcal{I}_A \otimes \tilde{\mc{E}}_k)\roo \vert \Psi_{AB} \rangle \geq q(\roo^k|B_k).$$ We thus find $q(\roo|B) \geq \lim_{k \rightarrow \infty} q(\roo^k|B_k) = \Lambda(\roo|B)$.
We next prove $\Lambda(\roo|B) \geq q(\roo|B)$. Let $\mc{E}$ be an arbitrary $\operatorname{TPCPM}(\hb,\hb)$. As a special instance of Stinespring dilations we know that there exists an ancilla $H_R$ together with an unitary $U_{BR} \in \mc{L}(\hb \otimes H_R)$ and a state $\vert \theta_R \rangle \in H_R$, such that $\mc{E}(\sigma_B) = \operatorname{tr}_R[U_{BR} ( \sigma_B \otimes \vert \theta_R \rangle \langle \theta_R \vert )U^{\dagger}_{BR}]$ [@Kraus]. With $\vert \psi_{ABC} \rangle$ a purification of $\roo$, it follows according to (\[eq:Uhlmann\]) that $$\begin{aligned}
F \big(( \mathcal{I}_{A} \otimes \mc{E})\roo , \mcor\big)
& = \sup_{\eta_{CR}} F\big((\operatorname{id}\otimes U_{BR}) \vert \psi_{ABC} \rangle\vert\theta_R \rangle , \vert \Psi_{AB} \rangle\vert \eta_{CR} \rangle\big) \\
& \leq \sup_{\eta_{CR}} F\big(\rho_{AC} , \tau_A \otimes \operatorname{tr}_R(\vert \eta_{CR} \rangle \langle \eta_{CR} \vert)\big),\end{aligned}$$ where the last inequality is due to the monotonicity of fidelity under the partial trace and $\tau_A = d_A^{-1} \operatorname{id}_A = \operatorname{tr}_B(\mcor)$. The optimization over all pure states $\eta_{CR}$ can be replaced by the optimization over all density operators on $H_C$. Then, with Proposition \[prop:operational interp of max entropy\] it follows that $$\begin{aligned}
d_A F(( \mathcal{I}_{A} \otimes \mc{E})\roo , \mcor)^2 & \leq \sup_{\sigma_C} d_A F( \rho_{AC}, \tau_A \otimes \sigma_C)^2 = 2^{H_{\mathrm{max}}(\rho_{AC}|C)} \\
& = 2^{-H_{\mathrm{min}}(\roo|B)} = \Lambda(\roo|B).\end{aligned}$$ Since this holds for all $\mc{E} \in \operatorname{TPCPM}(\hb,\hb)$, we obtain $q(\roo|B) \leq \Lambda(\roo|B)$.
The quantum correlation and its relation to min-entropy applied to classical quantum states connects the min-entropy with the optimal guessing probability. Imagine a source that produces the quantum states $\roBx \in \mc{S}(H_B)$ at random, according to the probability distribution $P_X(x)$. The average output is characterized by the classical-quantum state, $$\label{eq:cq state}
\rho_{XB} = \sum_{x \in X} P_X(x) \vert x \rangle \langle x \vert \otimes \roBx,$$ where $X$ denotes the (finite) alphabet of the classical system describing the source and $\{ \vert x \rangle \}_{x\in X}$ is an orthonormal basis spanning $H_X$. We define the guessing probability $g(\rho_{XB}|B)$ as the probability to correctly guess $x$, permitting an optimal measurement strategy on subsystem $B$. Formally, this can be expressed as $$\label{eq:def of guessing prob}
g(\rho_{XB}|B) := \sup_{ \{M_x\}} \sum_{x\in X}P_X(x) \operatorname{tr}(\roBx M_x),$$ where the supremum is taken over all positive operator valued measures (POVM) on $H_B$. By POVM on $H_B$ we intend a set $\{M_x\}_{x \in X}$ of positive operators which sum up to the identity. For finite-dimensional $H_B$ it is known [@OperationalMeaning] that the guessing probability is linked to the min-entropy by $$\label{eq:guessin prob finite dim}
g(\rho_{XB}|B) = 2^{-H_{\mathrm{min}}(\rho_{XB}|B)}.$$ We will now use Proposition \[prop:oper interpr of min entropy\] to show that Eq. (\[eq:guessin prob finite dim\]) also holds for separable $H_B$.
Let $\rho_{XB}$ be a state as defined in Eq. (\[eq:cq state\]), and construct the state $|\Psi_{XB}\rangle := |X|^{-1/2}\sum_{x\in X}|x \rangle|x_B\rangle$, where $\{|x_B\rangle\}_{x\in X}$ is an arbitrary orthonormal set in $H_B$. We now define $Q(\rho_{XB},\mathcal{E}) := d_A \langle \Psi_{XB} \vert (\mathcal{I}_X \otimes \mc{E})\rho_{XB} \vert \Psi_{XB} \rangle$ (cf. Eq. (\[eq:quantum correlation equiv expression\])) and $G(\rho_{XB}, \{M_x\}) := \sum_{x\in X}P_X(x) \operatorname{tr}(\roBx M_x)$ (cf. Eq. (\[eq:def of guessing prob\])). Then, $$\label{nadflv}
Q(\rho_{XB},\mathcal{E}) = \sum_{x\in X}P_X(x)\operatorname{tr}[\mathcal{E}^{*}(|x_B\rangle\langle x_B|)\rho_B^{x}],$$ where $\mathcal{E}^{*}$ denotes the adjoint operation of $\mathcal{E}$. Let $\{M_x\}$ be an arbitrary $|X|$-element POVM on $H_B$. One can see that the TPCPM $\mathcal{E}(\rho) := \sum_{x\in X} \operatorname{tr}(M_x \rho) |x_B\rangle\langle x_B|$ satisfies $\mathcal{E}^{*}(|x_B\rangle\langle x_B|) = M_x$. Thus, by Eq. (\[nadflv\]), we find $Q(\rho_{XB},\mathcal{E}) = G(\rho_{XB}, \{M_x\})$. Since the POVM was arbitrary, it follows that $q(\rho_{XB}|B)\geq g(\rho_{XB}|B)$.
Next, let $\mathcal{E}$ be an arbitrary TPCPM on $H_B$. Define $P := \sum_{x\in X} |x_B\rangle\langle x_B|$ and $$M_{x} := \mathcal{E}^{*}(|x_B\rangle\langle x_B|) + \frac{1}{|X|}\mathcal{E}^{*}(\operatorname{id}_B-P),\quad x\in X.$$ One can verify that $\{M_x\}$ is a POVM on $H_B$. By using Eq. (\[nadflv\]) we can see that $G(\rho_{XB}, \{M_x\}) \geq Q(\rho_{XB},\mathcal{E})$. This implies $g(\rho_{XB}|B)\geq q(\rho_{XB}|B)$, and thus $g(\rho_{XB}|B) = q(\rho_{XB}|B)$.
\[chapter:smooth entropy\] Smooth Min- and Max-Entropy
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The entropic quantities that usually appear in operational settings are the smooth min- and max-entropies [@ChannelCodingMinEntropy; @ChannelCodingMaxEntropy; @OperationalMeaning]. They result from the non-smoothed versions by an optimization procedure over states close to the original state. The closeness is defined by an appropriate metric on the state space, and a smoothing parameter specifies the maximal distance to the original state. The choice of metric has varied in the literature, but here we follow [@OnTheSmoothing].
By $\mc{S}_{\leq}(H)$ we denote the set of positive trace class operators with trace norm smaller than or equal to 1. We define the generalized fidelity on $\mc{S}_{\leq}(H)$ by $\bar{F}(\rho,\sigma):= \Vert\sqrt{\rho}\sqrt{\sigma}\Vert_{1} + \sqrt{(1-\operatorname{tr}\rho)(1-\operatorname{tr}\sigma)}$, which induces a metric on $\mc{S}_{\leq}(H)$ via $$\label{purified distance}
P(\rho,\sigma) := \sqrt{1 - \bar{F}(\rho,\sigma)^2},$$ referred to as the purified distance.
\[def:smooth min-/max-entropy\] For $\epsilon >0$, we define the $\epsilon$-smooth min- and max-entropy of $\roo \in \mc{S}_{\leq}(\hahb)$ conditioned on $B$ as $$\label{def,eq1:smooth min-/max-entropy}
\sHmin := \ssup \HminBt,$$ $$\label{def,eq2:smooth min-/max-entropy}
\sHmax := \sinf \HmaxBt,$$ where the smoothing set $\mc{B}^{\epsilon}(\roo)$ is defined with respect to the purified distance $$\label{def,eq3:smooth min-/max-entropy}
\mc{B}^{\epsilon}(\roo) := \{ \root \in \mc{S}_{\leq}(H_A\otimes H_B)| P(\roo,\root) \leq \epsilon \}.$$
Closely related to this particular choice of smoothing set is the invariance of the smooth entropies under (partial) isometries acting locally on each of the subsystems. This can be used to show the duality relation of the smooth entropies, namely, for all states $\roo$ on $\hahb$ it follows that $$\label{eq:duality of smooth entropies}
\sHmin = - H_{\mathrm{max}}^{\epsilon}(\rho_{AC}|C),$$ where $\rho_{ABC}$ is an arbitrary purification of $\roo$ on an ancilla $H_C$. A proof for the finite-dimensional case can be found in [@OnTheSmoothing], which allows a straightforward modification to infinite dimensions.
A useful property of the smooth entropies is the data processing inequality.
\[p:strong subadditivity for smooth entropies\] Let be $ \rho_{ABC} \in \mc{S}_{\leq}(\hahb \otimes H_C)$, then it follows that $$\begin{aligned}
H_{\mathrm{min}}^{\epsilon}(\rho_{ABC}|BC) &\leq & \sHmin,\\
H_{\mathrm{max}}^{\epsilon}(\rho_{ABC}|BC) &\leq & \sHmax.\end{aligned}$$
Using the data processing inequality for the min-entropy, Eq. (\[p,eq2:strong subadd of Hmin\]), we obtain $$\begin{aligned}
H_{\mathrm{min}}^{\epsilon}(\rho_{ABC}|BC)=\sup_{\tilde{\rho}_{ABC} \in \mc{B}^{\epsilon}(\rho_{ABC})} H_{\mathrm{min}}(\rot_{ABC}|BC) \leq
\sup_{\tilde{\rho}_{ABC} \in \mc{B}^{\epsilon}(\rho_{ABC})} H_{\mathrm{min}}(\operatorname{tr}_C \rot_{ABC}|B).\end{aligned}$$ Thus, it is sufficient to show that $\operatorname{tr}_C( \ball(\rho_{ABC})) \subseteq \ball(\roo)$. But this follows directly from the fact that the purified distance does not increase under partial trace [@OnTheSmoothing], i.e., $P(\rho_{ABC},\rot_{ABC})\geq P(\rho_{AB},\rot_{AB})$.
The data processing inequality of the smooth max-entropy follows from the duality (\[eq:duality of smooth entropies\]), $$H_{\mathrm{max}}^{\epsilon}(\rho_{ABC}|BC) = -H_{\mathrm{min}}^{\epsilon}(\rho_{AD}|D) \leq- H_{\mathrm{min}}^{\epsilon}(\rho_{ACD}|CD) = \sHmax,$$ where $\rho_{ABCD}$ is a purification of $\rho_{ABC}$.
\[AEP\] An Infinite-Dimensional Quantum Asymptotic Equipartition Property
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In the finite-dimensional case the quantum asymptotic equipartition property (AEP) says that the conditional von Neumann entropy can be regained as an asymptotic quantity from the conditional smooth min- and max-entropy [@RennerPhD; @QuantumAEP]. (For a discussion on why the AEP can be formulated in terms of entropies, see [@Independent].) More precisely, $\lim_{\epsilon \rightarrow 0} \lim_{ n \rightarrow \infty} \frac{1}{n} H_{\mathrm{min}}^{\epsilon}(\roo^{\otimes n}|B^n) = H(\roo|B)$ and $\lim_{\epsilon \rightarrow 0} \lim_{ n \rightarrow \infty} \frac{1}{n} H_{\mathrm{max}}^{\epsilon}(\roo^{\otimes n}|B^n) = H(\roo|B)$. For the infinite-dimensional case we derive an upper (lower) bound to the conditional von Neumann entropy in terms of the smooth min-(max-)entropy. We then use these bounds to prove the above limits in the case where $H_A$ is finite-dimensional. To this end we need a well defined notion of conditional von Neumann entropy in the infinite-dimensional case. Here we use the definition introduced in [@Kuznetsova], which in turn is based on an infinite-dimensional extension of the relative entropy [@Klein31; @Lindblad73; @Lindblad74; @HolevoShirokov]. For $\rho , \sigma \in \tau_1^{+}(H)$ the relative entropy can be defined as $$\label{smdal}
H(\rho \Vert \sigma): = \sum_{jk} |\langle a_{j}|b_{k}\rangle|^{2}(a_j\log a_j - a_j\log b_k + b_k-a_j),$$ where $\{|a_j\rangle\}_j$ is an arbitrary orthonormal eigenbasis of $\rho$ with corresponding eigenvalues $a_j$, and analogously for $\{|b_k\rangle\}_k$, $b_k$, and $\sigma$. The relative entropy is always positive, possibly $+\infty$, and equal to 0 if and only if $\rho=\sigma$ [@Lindblad73]. For states $\roo$ with $H(\rho_A) < +\infty$, the conditional von Neumann entropy can be defined as [@Kuznetsova] $$H(\roo|B) := H(\rho_A) -H(\rho_{AB}\Vert \rho_A\otimes \rho_B).$$ For many applications it appears reasonable to assume $H(\rho_A)$ to be finite, e.g., in cryptographic settings it would correspond to restricting the states of the ‘legitimate’ users.
Similarly as for the min- and max-entropy, the conditional von Neumann entropy can be approximated by projected states [@Kuznetsova], i.e., for $\roo \in \mc{S}(H_A \otimes H_B)$ satisfying $H(\rho_A)<\infty$ with corresponding normalized projected states $\rooh^k$ it follows that $$\label{eq: limit von neumann entropy}
\lim_{k \rightarrow \infty } H(\rooh^k|B) = H(\roo|B).$$ In the finite-dimensional case it has been shown [@QuantumAEP] that the min-, max- and, von Neumann entropy can be ordered as $$\label{ordering}
\HminB \leq H(\roo| B) \leq \HmaxB.$$ A direct application of Proposition \[p:reduction of Hmin to finite dim\] and (\[eq: limit von neumann entropy\]) shows that this remains true in the infinite-dimensional case, if $H(\rho_A)<\infty$. Note, however, that the ordering between min- and max-entropy (\[orderingHminHmax\]) does not hold for their smoothed versions.
\[prop:AEP lower bound\] Let $\roo \in \mc{S}(\hahb)$ be such that $H(\rho_A)<\infty$. For any $\epsilon > 0$ it follows that $$\label{lowerbound}
\frac{1}{n} H_{\mathrm{min}}^{\epsilon}(\rho_{AB}^{\otimes n} | B^n) \geq H(\roo|B) - \frac{1}{\sqrt{n}}4\log(\eta) \sqrt{\log\frac{2}{\epsilon^2}},$$ $$\label{upperbound}
\frac{1}{n} H_{\mathrm{max}}^{\epsilon}(\rho_{AB}^{\otimes n} | B^n) \leq H(\rho_{AB}|B) + \frac{1}{\sqrt{n}}4\log(\eta) \sqrt{\log\frac{2}{\epsilon^2}}.$$ for $n \geq (8/5)\log(2/\epsilon^2)$, and $\eta = 2^{-\frac{1}{2}\HminB} + 2^{\frac{1}{2}\HmaxB} +1$.
Note that it is not clear under what conditions the limits $n\rightarrow \infty$, $\epsilon \rightarrow 0$ exist for the left hand side of equations (\[lowerbound\]) and (\[upperbound\]). If they do, Proposition \[prop:AEP lower bound\] implies $\lim_{\epsilon\rightarrow 0}\lim_{n\rightarrow\infty}\frac{1}{n}
H_{\mathrm{min}}^{\epsilon}(\rho_{AB}^{\otimes n} | B^n) \geq H(\roo|B)$ and $\lim_{\epsilon\rightarrow 0}\lim_{n\rightarrow\infty}\frac{1}{n}
H_{\mathrm{max}}^{\epsilon}(\rho_{AB}^{\otimes n} | B^n) \leq H(\roo|B)$. For the case of a finite-dimensional $H_A$ we show that these inequalities can be replaced with equalities (Corollary \[cor:AEP\]).
It should be noted that in the classical case a lower bound on the min-entropy and an upper bound on the max-entropy, analogous to Eqs. (\[lowerbound\]) and (\[upperbound\]), correspond [@Independent] to the AEP in classical probability theory [@CoverThomas]. Since in the finite-dimensional quantum case, the step from Proposition \[prop:AEP lower bound\] to Corollary \[cor:AEP\] is directly obtained [@QuantumAEP] via Fannes’ inequality [@Fannes], the limits in Corollary \[cor:AEP\] are usually referred to as ‘the quantum AEP’ [@QuantumAEP]. In the infinite-dimensional case the relation between Proposition \[prop:AEP lower bound\] and Corollary \[cor:AEP\] appears less straightforward, and it is thus not entirely clear what should be regarded as constituting ‘the quantum AEP’. We will not pursue this question here, but merely note that it is the inequalities in Proposition \[prop:AEP lower bound\], rather than the limits in Corollary \[cor:AEP\], that are the most relevant for applications [@RennerPhD]. However, for the sake of simplicity we continue to refer to Corollary \[cor:AEP\] as a quantum AEP.
We prove Proposition \[prop:AEP lower bound\] after the following lemma.
\[cor: eps dep smooth min-entropy\] Let $\roo \in \mc{S}(\hahb)$ and let $\{\rooh^k\}_{k=1}^{\infty}$ be a sequence of normalized projected states. For any fixed $1 >t>0$, there exists a $k_0 \in \mathbb{N}$ such that $$H_{\mathrm{min}}^{\epsilon}(\roo|B) \geq H_{\mathrm{min}}^{t\epsilon}(\rooh^k|B),\quad \forall k\geq k_0.$$
In the following let $t\in (0,1)$ be fixed. According to the definition of the smooth min-entropy in Eq. (\[def,eq1:smooth min-/max-entropy\]), it is enough to show that $\mc{B}^{t\epsilon}(\rooh^k) \subseteq \ball(\roo)$ for all $k \geq k_0$. Note that the purified distance is compatible with trace norm convergence, i.e., $\Vert \roo-\rooh^k\Vert_1\rightarrow 0$ implies that $P(\rooh^k,\roo) \rightarrow 0$. Hence, there exists a $k_0$ such that $P(\rooh^k,\roo) < (1-t)\epsilon$ for all $k \geq k_0$. For $k \geq k_0$ and $\root \in \mc{B}^{t\epsilon}(\rooh^k)$ we thus find $P(\root,\roo) \leq P(\root, \rooh^k) + P(\rooh^k,\roo) < \epsilon$, such that $\root \in \ball(\roo)$.
*(Proposition \[prop:AEP lower bound\])* Let $(P_k^A, P_k^B)$ be a generator of projected states. The pair of n-fold tensor products of the projections, $\big((P_k^A)^{\otimes n},(P_k^B)^{\otimes n}\big)$, is also a generator of projected states. If we now fix $1>t>0$ and $n \in \mathbb{N}$, it follows by Lemma \[cor: eps dep smooth min-entropy\] that we can find a $k_0\in \mathbb{N}$ such that $H_{\mathrm{min}}^{\epsilon}(\roo^{\otimes n}|B^n) \geq H_{\mathrm{min}}^{t\epsilon}((\rooh^k)^{\otimes n}|B^n)$ for every $k\geq k_0$. Since Eq. (\[lowerbound\]) is valid for the finite-dimensional case [@QuantumAEP], we can apply it to $H_{\mathrm{min}}^{t\epsilon}((\rooh^k)^{\otimes n}|B^n)$ to obtain $$\frac{1}{n}H_{\mathrm{min}}^{t\epsilon}((\rooh^k)^{\otimes n}|B^n) \geq
H(\rooh^k|B) - \frac{1}{\sqrt{n}}4\log(\eta_k) \sqrt{\log\frac{2}{(t\epsilon)^2}}$$ for any $n \geq (8/5)\log(2/(t\epsilon)^2)$, and $\eta_k = 2^{-\frac{1}{2}H_{\mathrm{min}}(\rooh^k|B) } + 2^{\frac{1}{2}H_{\mathrm{max}}(\rooh^k|B)} +1$. Hence $$\label{pf,lem:AEP lower bound,eq4}
\frac{1}{n} H_{\mathrm{min}}^{\epsilon}(\roo^{\otimes n}|B^n) \geq H(\rooh^k|B) - \frac{1}{\sqrt{n}}4\log(\eta_k) \sqrt{\log\frac{2}{(t\epsilon)^2}},$$ for all $k \geq k_0$. Since the left hand side of Eq. (\[pf,lem:AEP lower bound,eq4\]) is independent of $k$ we can use (\[eq: limit von neumann entropy\]) and Proposition \[p:reduction of Hmin to finite dim\], to find $$\begin{aligned}
\frac{1}{n} H_{\mathrm{min}}^{\epsilon}(\roo^{\otimes n}|B^n)
& \geq \lim_{k\rightarrow \infty} \Big\{H(\rooh^k|B) - \frac{1}{\sqrt{n}}4\log(\eta_k) \sqrt{\log\frac{2}{(t\epsilon)^2}} \Big\} \\
& = H(\roo|B) - \frac{1}{\sqrt{n}}4\log(\eta) \sqrt{\log\frac{2}{(t\epsilon)^2}}.\end{aligned}$$ We finally take the limit $t\rightarrow 1$ in the above inequality, as well as in the condition $n \geq (8/5)\log(2/(t\epsilon)^2)$ to obtain the first part of the proposition.
For the second part we use the duality of the conditional von Neumann entropy, i.e., $H(\rho_{AB}|B)= -H(\rho_{AC}|C)$ for a purification $\rho_{ABC}$ [@Kuznetsova]. This, together with the duality relation for smooth min- and max-entropy (\[eq:duality of smooth entropies\]) leads directly to (\[upperbound\]).
\[cor:AEP\] Let $H_A$ be a finite-dimensional and $H_B$ a separable Hilbert space. For all $\roo \in \mc{S}(\hahb)$ it follows that $$\label{cor:AEP min-entropy}
\lim_{\epsilon\rightarrow 0} \lim_{n \rightarrow \infty} \frac{1}{n} H_{\mathrm{min}}^{\epsilon}(\rho_{AB}^{\otimes n} | B^n) = H(\roo|B),$$ $$\label{cor:AEP max-entropy}
\lim_{\epsilon\rightarrow 0} \lim_{n \rightarrow \infty} \frac{1}{n} H_{\mathrm{max}}^{\epsilon}(\rho_{AB}^{\otimes n} | B^n) = H(\roo|B).$$
Let $\epsilon >0$ be sufficiently small, and let $(\operatorname{id}_A,P_k^B)$ be a generator of projected states $\roo^k$, with corresponding normalized projected states $\hat{\rho}_{AB}^k$. Let $\sigma_{AB}\in \ball(\roo)$, with projected states $\sigma^k_{AB}$, and normalized projected states $\hat{\sigma}^k_{AB}$. By $H_{\mathrm{min}}(\sigma_{AB}^k|B)=H_{\mathrm{min}}(\sih^k_{AB}|B)+\log\operatorname{tr}\sigma^k_{AB}$ and (\[ordering\]) we find $H_{\mathrm{min}}(\sigma^k_{AB}|B_k)
\leq H(\sih^k_{AB}|B)$, where $\sih^k_{AB}=(\operatorname{tr}\sigma^k_{AB})^{-1}\sigma^k_{AB}$. Since $H(\sih^k_{AB}|B_k)$ is finite-dimensional we can use Fannes’ inequality [@Fannes] to obtain (for $k$ sufficiently large) $H(\sih^k_{AB}|B_k) \leq H(\rooh^k|B_k) + 4 \Delta_k \log d_A + 4 H_{\mathrm{bin}}(\Delta_k)$, with $d_A=\dim(H_A)$, $\Delta_k = \Vert \rooh^k - \sih^k_{AB} \Vert_1$, and $H_{\mathrm{bin}}(t)= -t\log t - (1-t)\log(1-t)$. Due to the general relation $\Vert \rho-\sigma\Vert_1 \leq 2 P(\rho,\sigma)$ (see Lemma 6 in [@OnTheSmoothing]), we have $\Vert \roo-\sigma_{AB}\Vert_1 \leq 2\epsilon$ for all $\sigma_{AB}\in\ball(\roo)$, which yields $\lim_{k\rightarrow \infty} \Delta_k = \Vert \roo -\sih_{AB} \Vert_1 \leq 4\epsilon$, where $\sih_{AB} = \sigma_{AB}/\operatorname{tr}(\sigma_{AB})$. Combined with (\[eq: limit von neumann entropy\]) this leads to $H^{\epsilon}_{\mathrm{min}}(\rho_{AB} | B) = \ssupp\lim_{k\rightarrow\infty}H_{\mathrm{min}}(\sigma^k_{AB} | B) \leq H(\roo|B) + 16\epsilon \log d_A + 4 H_{\mathrm{bin}}(4\epsilon)$. Applied to an n-fold tensor product this gives $$\label{MinEntrUpperbound}
\frac{1}{n}H_{\mathrm{min}}^{\epsilon}(\roo^{\otimes n} | B^n) \leq H(\roo|B) + 16\epsilon \log d_A + \frac{4}{n} H_{\mathrm{bin}}(4\epsilon).$$ Equation (\[cor:AEP min-entropy\]) follows by combining (\[MinEntrUpperbound\]) with the lower bound in (\[lowerbound\]), taking the limits $n\rightarrow \infty$ and $\epsilon\rightarrow 0$. Equation (\[cor:AEP max-entropy\]) follows directly by the duality of the conditional von Neumann entropy [@Kuznetsova] together with the duality of the smooth min- and max-entropy (\[eq:duality of smooth entropies\]).
\[concl\] Conclusion and Outlook
================================
We have extended the min- and max-entropies to separable Hilbert spaces, and shown that properties and operational interpretations, known from the finite-dimensional case, remain valid in the infinite-dimensional setting. These extensions are facilitated by the finding (Proposition \[p:reduction of Hmin to finite dim\]) that the infinite-dimensional min- and max-entropies can be expressed in terms of convergent sequences of finite-dimensional entropies. We bound the smooth min- and max-entropies of iid states (Proposition \[prop:AEP lower bound\]) in terms of an infinite-dimensional generalization of the conditional von Neumann entropy $H(A|B)$, introduced in [@Kuznetsova], which is defined when the von Neumann entropy of system $A$ is finite, $H(A)<\infty$. Under the additional assumption that the Hilbert space of system $A$ has finite dimension we furthermore prove that the smooth entropies of iid states converge to the conditional von Neumann entropy (Corollary \[cor:AEP\]), corresponding to a quantum asymptotic equipartition property (AEP). Whether these conditions can be relaxed is an open question. In the general case where $H(A)$ is not necessarily finite, this would however require a more general definition of the conditional von Neumann entropy than the one used here.
For information-theoretic purposes it appears reasonable to require extensions of the conditional von Neumann entropy to be compatible with the AEP, i.e., that the conditional von Neumann entropy can be regained from the smooth min- and max-entropy in the asymptotic iid limit. This enables generalizations of operational interpretations of the conditional von Neumann entropy. For example, in the finite-dimensional asymptotic case the conditional von Neumann entropy characterizes the amount of entanglement needed for state merging [@StateMerging], i.e., the transfer of a quantum state shared by two parties to only one of the parties. An infinite-dimensional generalization of one-shot state merging [@Single; @Shot; @State; @merging], together with the AEP, could be used to extend this result to the infinite-dimensional case.
Some other immediate applications of this work are in continuous variable quantum key distribution, and in statistical mechanics, where it has recently been shown [@Oscar; @Lidia] that the smooth min- and max-entropies play a role. Our techniques may also be employed to derive an infinite-dimensional generalization of the entropic uncertainty relation [@Uncertainty]. Such a generalization would be interesting partially because it could find applications in continuous variable quantum information processing, but also because it may bring this information-theoretic uncertainty relation into the same realm as the standard uncertainty relation.
Acknowledgments
===============
We thank Roger Colbeck and Marco Tomamichel for helpful comments and discussions, and an anonymous referee for very valuable suggestions. Fabian Furrer acknowledges support from the Graduiertenkolleg 1463 of the Leibniz University Hannover. We furthermore acknowledge support from the Swiss National Science Foundation (grant No. 200021-119868).
\[section:Technical Lemmas\]Technical Lemmas
============================================
In the following, each Hilbert space is assumed to be separable. Let us define the positive cone $\mc{L}^+(H) := \{T \in \mc{L}(H)| \ T \geq 0\}$ in $\mc{L}(H)$. The next two lemmas follow directly from the definition of positivity of an operator.
\[lem:T geq 0 implies STS geq 0\] If $T \in \mc{L}^+(H)$, then for each $S \in \mc{L}(H)$ it follows that $STS^{\dagger} \in \mc{L}^+(H)$.
\[lem:pos weak op limit\] The positive cone $\mc{L}^+(H)$ is sequentially closed in the weak operator topology, i.e., for $\{T_k\}_{k\in\mathbb{N}} \subset \mc{L}^+(H)$ such that $T_k$ converge to $T \in \mc{L}(H)$ in the weak operator topology, it follows that $T\geq 0$.
The following lemma is a special case of a theorem by Grümm [@Grumm] (see also [@Simon], pp. 25-29, for similar results).
\[lemgrumm\] Let $A_{k},A \in\mc{L}(H)$, such that $\sup_{k}\Vert A_{k}\Vert<+\infty$, and $A_{k}\rightarrow A$ in the strong operator topology, and let $T\in \tau_1(H)$. Then $\lim_{k\rightarrow\infty}\Vert A_{k}T -AT\Vert_{1}=0$ and $\lim_{k\rightarrow\infty}\Vert TA_{k} -TA\Vert_{1}=0$.
\[dfnbkl\] If $P_{k}$ is a sequence of projectors on $H$ that converges in the strong operator topology to the identity, and if $\rho \in \tau_1^+(H)$, then $\lim_{k\rightarrow\infty}\Vert P_{k}\rho P_{k}-\rho\Vert_{1}=0$.
\[nvdakj\] If sequences of projectors $P_k^A$ and $P_k^B$ on $H_A$ and $H_B$, respectively, converge in the strong operator topology to the identity, then $P_k^A\otimes P_k^B$ converges in the strong operator topology to $\operatorname{id}_{AB}$.
\[lem:weakstar Tk implies weak op of id otimes Tk\] Let $\{T_k\}_{k\in \mathbb{N}} \subset \tau_1(H_B)$ be a sequence that converges in the weak\* topology to $T\in \tau_1(H_B)$. Then, the sequence $\operatorname{id}_{A} \otimes T_k$ in $\mc{L}(\hahb)$ converges to $\operatorname{id}_{A} \otimes T$ in the weak operator topology.
For each $\psi \in \hahb$ we find that $\langle \psi \vert \operatorname{id}\otimes T_k \vert \psi \rangle = \operatorname{tr}( T_k K^{B}_{\psi})$, where $K^{B}_{\psi}= \operatorname{tr}_{A}|\psi\rangle\langle\psi|$ is the reduced operator. Since $K^{A}_{\psi}$ is trace class (and thus compact) the statement follows immediately.
\[app:proofprop1\]Proof of Proposition \[p:reduction of Hmin to finite dim\]
============================================================================
In order to derive Proposition \[p:reduction of Hmin to finite dim\] we proceed as follows: In Section \[Section:reduction to finite\] we show that the min- and max-entropy of a projected state can be reduced to an entropy on a finite-dimensional space. In Section \[sec:monotonicity\] we show that the min- and max-entropies are monotonic over the sequences of projected states. Finally we prove the limits listed in Proposition \[p:reduction of Hmin to finite dim\]. Note that in what follows we mostly make use of the quantities $\Lambda(\rho_{AB}|\sigma_{B})$ and $\Lambda(\rho_{AB}|B)$, as defined in Eqs. (\[lambdadef1\]) and (\[lambdadef2\]), rather than the min- and max-entropies per se.
\[Section:reduction to finite\] Reduction
-----------------------------------------
Here we show that the min- and max-entropy of a projected state can be considered as effectively finite-dimensional, in the sense that restricting the Hilbert space to the support of the projected states does not change the value of the entropies.
\[I:id-P\] Let $P_A$, $P_B$ be projectors onto closed subspaces $U_A \subseteq H_A$ and $U_B \subseteq H_B$, respectively, $\root \in \tau_{1}^{+}(H_A \otimes H_B)$, and $\siit \in \tau^+_1(H_B)$. i) If $(P_{A}\otimes \operatorname{id}_{B}) \root (P_{A}\otimes \operatorname{id}_{B}) = \root$ it follows that $$\Lambda(\root|\siit) = \inf \{ \lambda \in \mathbb{R} | \lambda P_A \otimes \siit \geq \root \}.$$ ii) If $(\operatorname{id}_A\otimes P_{B})\root (\operatorname{id}_A\otimes P_{B}) = \root$ it follows that $$\label{prop,eq1:Lambda statement for reduction of Hmin}
\Lambda(\root|B) = \Lambda(\root|U_B),$$ where $\Lambda(\root |U_B)$ means that the infimum in Eq. (\[lambdadef2\]) is taken only over the set $\tau^+_1(U^B)$.
The proof is straightforward and left to the reader. In the particular case of projected states $\roo^k$ relative to a generator $(P_k^A,P_k^B)$, the evaluation of $\Lambda(\roo^k|\sii^k)$ and $\Lambda(\roo^k|B)$, where $\sii^k=P_k^B\sii P_k^B$, can be restricted to the finite-dimensional Hilbert space $U_k^A\otimes U_k^B$ given by the projection spaces of $P_k^A$ and $P_k^B$. Especially, we can conclude that the infima of Eqs. (\[lambdadef1\]) and (\[lambdadef2\]), and consequently the infimum in (\[def,eq1:min/max-entropy\]) and the supremum in (\[def,eq2:min/max-entropy\]), are attained for projected states, since these are optimizations of continuous functions over compact sets.
\[sec:monotonicity\]Monotonicity
--------------------------------
The next lemma considers the monotonic behaviour of the min- and max-entropies with respect to sequences of projected states.
\[l:mon incr\] For $\roo\in \mc{S}(\hahb)$, $ \sii \in \mc{S}(H_B)$, let $\{\roo^k\}_{k=1}^{\infty}$ and $\{\sii^k\}_{k=1}^{\infty}$ be projected states relative to a generator $(P_k^A,P_k^B)$.i) It follows that $\Lambda(\roo^k|\sii^k)$ and $\Lambda(\roo^k|B)$ are monotonically increasing in $k$, where the first sequence is bounded by $\Lambda(\roo|\sii)$ and the latter by $\Lambda(\roo|B)$.ii) For an arbitrary but fixed purification $\rho_{ABC}$ of $\rho_{AB}$ with purifying system $H_C$, let $\rho^k_{AC} = \operatorname{tr}_B\rho_{ABC}^k$ and $\rho_{ABC}^k = (P^A_k \otimes P_k^B \otimes \operatorname{id}_C)\rho_{ABC}(P^A_k \otimes P_k^B \otimes \operatorname{id}_C)$. Then it follows that $\Lambda(\rho_{AC}^k|C)$ is monotonically increasing and bounded by $\Lambda(\rho_{AC}|C)$.
Note that $\rho^k_{AC}$ as defined in the lemma is not a projected state in the sense of Definition \[def:projected states\]. Translated to min- and max-entropies, the lemma above says that $\Hmink$ and $\HminBk$ are monotonically increasing while $\HmaxBk$ is monotonically decreasing. But in general, the monotonicity does not hold for *normalized* projected states.
Set $P_k:=P_k^A \otimes P_k^B$ and recall that $\Lambda(\roo^k|\sii^k)= \inf \{ \lambda \in \mathbb{R}| \ \lambda P_k^A \otimes \sii^k \geq \roo^k \}$ according to Lemma \[I:id-P\]. To show the first part of i) note that for $k'\leq k$ the equations $$P_{k'}P_k (\lambda \operatorname{id}\otimes \sii - \roo ) P_{k'}P_k=P_{k'} (\lambda P_k^A \otimes \sii^k - \roo^k ) P_{k'} = \lambda P_{k'}^A \otimes \sii^{k'} - \roo^{k'}$$ hold, which imply via Lemma \[lem:T geq 0 implies STS geq 0\] that $\Lambda(\roo^{k'}|\sii^{k'}) \leq \Lambda(\roo^k|\sii^k) \leq \Lambda(\roo|\sii)$. For the second part, let $\siit \in \tau_1^+(H_B)$ be the optimal state such that $\Lambda(\roo^k|B)=\operatorname{tr}\siit$ and $P_k^A\otimes \siit \geq \roo^k$. But then we obtain that $P_{k'}^A \otimes P_{k'}^B\siit P_{k'}^B - \roo^{k'} \geq 0$ and therefore also $\Lambda(\roo^{k'}|B) \leq \Lambda(\roo^k|B)$. The upper bound follows in the same manner.
In order to show ii) we define the sets $ \mc{M}_k := \{\sitC \in \tau_1^+(H_C)|\ \operatorname{id}_A \otimes \sitC \geq \rho_{AC}^k\}$ such that $\Lambda(\rho_{AC}^k|C) = \inf_{\sitC \in \mc{M}_k}\operatorname{tr}\sitC$. To conclude the monotonicity we show that $ \mc{M}_{k'} \supset \mc{M}_{k}$ for $k'\leq k$. If $\mc{M}_{k}= \emptyset$, the statement is trivial. Assume $\tilde{\sigma}_{C}\in \mc{M}_k$. Using $P^B_{k'}\leq P^B_{k}$ we find $$\operatorname{id}_A\otimes \tilde{\sigma}_{C}\geq P_{k}^A \operatorname{tr}_{B}(P_{k}^B \rho_{ABC}P^B_{k}) P_{k}^A \geq P_{k}^A \operatorname{tr}_{B}(P_{k'}^B \rho_{ABC}P^B_{k'}) P_{k}^A.$$ Together with Lemma \[lem:T geq 0 implies STS geq 0\], this yields $P^A_{k'} \otimes \tilde{\sigma}_{C} \geq \rho^{k'}_{AC}$ and thus $\tilde{\sigma}_{C}\in \mc{M}_{k'}$. A similar argument provides the upper bound $\Lambda(\rho_{AC}^{k}|C) \leq \Lambda(\rho_{AC}|C)$.
\[sec:limits\]Limits
--------------------
After the above discussion on general properties of the min- and max-entropies of projected states we are now prepared to prove Proposition \[p:reduction of Hmin to finite dim\]. For the sake of convenience we divide the proof into three lemmas.
\[l=sup\] For $\roo\in \mc{S}(\hahb)$ and $\sii \in \mc{S}(\hb)$, let $\{\roo^k\}_{k=1}^{\infty}$ be the projected states of $\roo$ relative to a generator $(P_k^A,P_k^B)$, and let $\sii^k:= P_k^B\sii P_k^B$. It follows that $$\label{knbvx}
\Lambda(\roo|\sii) = \lim_{k \rightarrow \infty} \Lambda(\roo^k|\sii^k),$$ and the infimum in Eq. (\[lambdadef1\]) is attained if $\Lambda(\roo|\sii)$ is finite.
That the infimum is attained follows directly from the definition. To show (\[knbvx\]) we prove that $\Lambda(\roo|\sii)$ is lower semi-continuous in $(\roo,\sii)$ with respect to the product topology induced by the trace norm topology on each factor. Since this means that $\liminf_{k \rightarrow \infty} \Lambda(\roo^k|\sii^k) \geq \Lambda(\roo|\sii)$, the combination with Lemma \[l:mon incr\] results directly in (\[knbvx\]). To show lower semi-continuity recall that it is equivalent to say that all lower level sets $\Lambda^{-1}((-\infty,t]) =\{(\roo,\sii) | \ \Lambda(\roo|\sii) \leq t \}$, for $t\in \mathbb R$ have to be closed. But this follows by rewriting $\Lambda^{-1}((-\infty,t])$ as $\{(\roo,\sii) | \ t\operatorname{id}\otimes \sii \geq \roo \}$.
\[l:Lambda statement for reduction of Hmin\] For $\roo\in \mc{S}(\hahb)$, let $\{\roo^k\}_{k=1}^{\infty}$ be the projected states of $\roo$ relative to a generator $(P_k^A,P_k^B)$. It follows that $$\label{lakdsfn}
\Lambda(\roo|B) = \lim_{k\rightarrow \infty} \Lambda(\roo^k|B),$$ and the infimum in Eq. (\[lambdadef2\]) is attained if $\Lambda(\roo|B)$ is finite.
Let $\mu_k := \Lambda(\roo^k|B) = \Lambda(\roo^k|B_k)$, where the last equality is due to Lemma \[I:id-P\]. By Lemma \[l:mon incr\] this sequence is monotonically increasing, and we can thus define $\mu := \lim_{k \rightarrow \infty} \mu_k \in \mathbb{R} \cup \{+\infty\}$. In addition, Lemma \[l:mon incr\] also yields $\mu \leq \Lambda(\roo|B)$. Hence, the case $\lambda = +\infty$ is trivial, and it remains to show $\mu \geq \Lambda(\roo|B)$, for $\mu < \infty$.
For each $k\in \mathbb{N}$ let $\tilde{\sigma}_B^k$ be an optimal state such that $\Lambda(\roo^k|B)=\operatorname{tr}\siit^k$ and $\operatorname{id}\otimes\siit^k\geq \roo^k$. Note that due to positivity $\operatorname{tr}\siit^k=\Vert \siit^k\Vert_1
\leq \mu$, such that $\tilde{\sigma}_B^k$ is a bounded sequence in $\tau_1(H_B)$. Since the trace class operators $\tau_1(H_B)$ is the dual space of the compact operators $\mc K(H_B)$ [@SimonReeds], we can apply Banach Alaoglu’s theorem [@SimonReeds; @HillPhillips] to find a subsequence $\{\tilde{\sigma}_B^k\}_{k\in\Gamma}$ with a weak\* limit $\siit \in \tau_1(H_B)$, i.e., $\operatorname{tr}(K\siit^k) \rightarrow \operatorname{tr}(K\siit)$ $(k\in \Gamma)$ for all $K\in \mc K(H_B)$, such that $\Vert \siit\Vert_1 \leq \mu$. Obviously, $\siit$ is also positive. According to Lemma \[lem:weakstar Tk implies weak op of id otimes Tk\], $\operatorname{id}\otimes \siit^k$ (for $k\in \Lambda$) converges in the weak operator topology to $\operatorname{id}\otimes \siit$, and so does $\operatorname{id}\otimes \siit^k - \roo^k$ to $\operatorname{id}\otimes \siit - \roo$. But then we can conclude that $\operatorname{id}\otimes \siit - \roo\geq 0$ such that $\Lambda(\roo|B) \leq \operatorname{tr}\siit \leq \mu$.
\[l:Lambda statement for reduction of Hmax\] For $\roo\in \mc{S}(\hahb)$, let $\rho_{ABC}$ be a purification with purifying system $H_C$, and $(P_k^A,P_k^B)$ be a generator of projected states. It follows that $$\label{prop,eq1:Lambda statement for reduction of Hmax}
\Lambda(\rho_{AC}|C) = \lim_{k \rightarrow \infty} \Lambda(\rho^k_{AC}|C),$$ where $\rho^k_{AC} = \operatorname{tr}_B[(P^A_k \otimes P_k^B \otimes \operatorname{id}_C)\rho_{ABC}(P^A_k \otimes P_k^B \otimes \operatorname{id}_C)].$
Let $\nu_k := \Lambda(\rho_{AC}^k|C)$. Due to Lemma \[l:mon incr\] this sequence is monotonically increasing, so we can define $\nu := \lim_{k \rightarrow \infty} \nu_k\in \mathbb{R}\cup\{+\infty\}$, and conclude that $\nu \leq \Lambda(\rho_{AC}|C)$. Thus, the case $\nu=+\infty$ is trivial. It thus remains to show $\nu \geq \Lambda(\rho_{AC}|C)$ for $\nu <+\infty$. As proved in Lemma \[l:Lambda statement for reduction of Hmin\], the infimum in Eq. (\[lambdadef2\]) is attained even if the underlying Hilbert spaces are infinite-dimensional. Thereby there exists for each $k \in \mathbb{N}$ a state $\sitC^k$ such that $\operatorname{id}\otimes\sitC^k \geq \rho_{AC}^k$ and $\operatorname{tr}\sitC^k =\Lambda(\rho_{AC}^k|C)$. Now we can proceed in the same manner as in the proof of Lemma \[l:Lambda statement for reduction of Hmin\] to construct a weak\* limit $\sitC \in \tau_1^+(H_B)$ that satisfies $\operatorname{id}_A \otimes \sitC \geq \rho_{AC}$, and is such that $\Lambda(\rho_{AC}|C) \leq \operatorname{tr}\sitC \leq \nu \leq \Lambda(\rho_{AC}|C)$. This completes the proof.
Of course, Lemma \[l=sup\] and \[l:Lambda statement for reduction of Hmin\] can directly be rewritten in terms of min-entropies and yield the first two statements of Proposition \[p:reduction of Hmin to finite dim\]. The part for the normalized projected states follows via $H_{\mathrm{min}}(\roh^k_{AB}|\sih^k_B) = H_{\mathrm{min}}(\rho^k_{AB}|\sigma^k_{B}) - \log\operatorname{tr}\sigma_B^k + \log\operatorname{tr}\rho_{AB}^{k}$, and $H_{\mathrm{min}} (\roh^k_{AB}|B) = H_{\mathrm{min}}(\roo^k|B) + \log\operatorname{tr}\roo^k $.
In order to obtain the convergence stated for the max-entropy in Proposition \[p:reduction of Hmin to finite dim\], note that $(P^A_k \otimes P_k^B \otimes \operatorname{id}_C) \rho_{ABC} (P^A_k \otimes P_k^B \otimes \operatorname{id}_C)$ is a purification of $\rho_{AB}^k$, whenever $\rho_{ABC}$ is a purification of $\rho_{AB}$. Hence, $H_{\mathrm{max}}(\rho_{AB}^k|B) = -H_{\mathrm{min}}(\rho_{AC}^k|C) = \log\Lambda(\rho_{AC}^k|C)$. For normalized states use $H_{\mathrm{max}} (\hat{\rho}_{AB}^k|B_k) = H_{\mathrm{max}}(\roo^k|B_k) - \log\operatorname{tr}\roo^k$.
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S.: *Information-Spectrum Methods in Information Theory*. Springer-Verlag, New York (2002) Han, T. S., Verdu, S.: Approximation theory of output statistics. IEEE Trans. Inform. Theory, [**39**]{}, 752 - 772 (1993) Datta, N.: Min- and Max-Relative Entropies and a New Entanglement Monotone. IEEE Trans. Inf. Theor. [**55**]{}, 2816 - 2826 (2009) Brandão, F. G. S. L., Datta, N.: One-shot rates for entanglement manipulation under non-entangling maps. IEEE Trans. Inf. Theor. [**57**]{}, 1754 (2011). Buscemi, F., Datta, N.: Entanglement Cost in Practical Scenarios. Phys. Rev. Lett. [**106**]{}, 130503 (2011) Mosonyi, M., Datta, N.: Generalized relative entropies and the capacity of classical-quantum channels. J. Math. Phys. [**50**]{}, 072104 (2009) Dahlsten, O. C. O., Renner, R., Rieper, E., Vedral, V.: Inadequacy of von Neumann entropy for characterizing extractable work. New J. Phys. [**13**]{}, 053015 (2011) del Rio L., [Å]{}berg J., Renner R., Dahlsten O., Vedral, V.: The thermodynamic meaning of negative entropy. Nature [**474**]{}. 61 - 63 (2011) Scarani, V., Bechmann-Pasquinucci, H., Cerf, N. J., Du[š]{}ek, M., Lütkenhaus, N., Peev, M.: The security of practical quantum key distribution. Rev. Mod. Phys. [**81**]{}, 1301 - 1350 (2009) Bratteli, O., Robinson, D. W.: *Operator Algebras and Quantum Statistical Mechanics I*. Springer-Verlag, New York (1979) König, R., Renner, R., Schaffner, C.: The Operational Meaning of Min- and Max-Entropy. IEEE Trans. Inf. Theor. [**55**]{}, 4337 - 4347 (2009) Tomamichel, M., Colbeck, R., Renner, R.: Duality Between Smooth Min- and Max-Entropies. IEEE Trans. Inf. Theor. [**56**]{}, 4674 - 4681 (2010) Lieb, E. H., Ruskai, M. B.: A Fundamental Property of Quantum-Mechanical Entropy. Phys. Rev. Lett. [**30**]{}, 434 - 436 (1973) Lieb, E. H.: Convex trace functions and the Wigner-Yanase-Dyson conjecture. Adv. Math. [**11**]{}, 267 - 288 (1973) Lieb, E. H., Ruskai, M. B.: Proof of the strong subadditivity of quantum-mechanical entropy. J. Math. Phys. [**14**]{}, 1938 - 1941 (1973) Owari, M., Braunstein, S. L., Nemoto, K.,Murao, M.: Epsilon-convertibility of entangled states and extension of Schmidt rank in infinite-dimensional systems. Quant. Inf. and Comp. [**8**]{}, 30 - 52 (2008) Kraus, K.: Lecture Notes in Physics 190, *States, Effects, and Operations*. Springer-Verlag, Berlin Heidelberg (1983) Nielsen, M. L., Chuang, I.L.: *Quantum Computation and Quantum Information*. Cambridge University Press (2000) Uhlmann, A.: The transition probability in the state space of a \*-algebra. Rep. Math. Phys. [**9**]{}, 273 - 279 (1976) Holenstein H., Renner R.: On the Randomness of Independent Experiments. IEEE Trans. Inf. Theor. [**57**]{}, 1865 - 1871 (2011) Kuznetsova A. A.: Quantum conditional entropy for infinite-dimensional systems, Theory Probab. Appl. [**55**]{}, 782-790 (2010) Klein, O.: Zur quantenmechanischen Begründung des zweiten Hauptsatzes der Wärmelehre. Z. F. Phys. A [**72**]{}, 767 - 775 (1931) Lindblad, G.: Entropy, Information and Quantum Measurements. Comm. Math. Phys. [**33**]{}, 305 - 322 (1973). Lindblad, G.: Expectations and Entropy Inequalities for Finite Quantum Systems. Comm. Math. Phys. [**39**]{}, 111 - 119 (1974). Holevo A. S., Shirokov, M. E.: Mutual and Coherent Information for Infinite-Dimensional Quantum Channels. Probl. Inf. Transm. [**46**]{}, 201- 217 (2010) Cover, T. M., Thomas, J. A.: *Elements of Information Theory. 2nd ed.* Wiley, New York (2006). Alicki, R., Fannes, M.: Continuity of quantum conditional information. J. Phys. A [**37**]{}, L55 - L57 (2004) Horodecki, M., Oppenheim, J., Winter, A.: Partial quantum information. Nature [**436**]{}, 673 - 676 (2005) Berta M.: Single-shot Quantum State Merging. http://arXiv.org/abs/0912.4495v1 \[quant-ph\] (2009) Grümm, H. R.: Two theorems about $\mathcal{C}_{p}$ . Rep. Math. Phys. [**4**]{}, 211 - 215 (1973) Simon, B.: *Trace Ideals and Their Applications*. 2:nd Ed. Amer. Math. Soc. (2005) Reed, M., Simon, B.: *Methods of Modern Mathematical Physics, Vol.I: Functional Analysis*. Academic Press, New York (1978) Hille, E., Phillips, R. S.: *Functional Analysis and Semi-Groups*. American Mathmatical Society, Providence (1957)
[^1]: Max-entropy as we define it in Eq. (\[def,eq3:min/max-entropy\]) is related to the Rényi 1/2-entropy (see Section \[subsect:entropy of pure states versus unconditioned entropy\] or [@OperationalMeaning; @OnTheSmoothing]). In the original definition [@RennerPhD] max-entropy was defined in terms of the Rényi 0-entropy.
[^2]: With “projector” we intend a bounded operator $P$ such that $P^2 = P$ and $P^{\dagger} = P$, which in the mathematics literature usually is referred to as an “orthogonal projector”.
|
{
"pile_set_name": "ArXiv"
}
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---
author:
- 'Nirupam Gupta and Nitin H. Vaidya'
bibliography:
- 'ref.bib'
title: |
Randomized Reactive Redundancy for\
Byzantine Fault-Tolerance in Parallelized Learning
---
Introduction
============
[r]{}[0.45]{}
{width="40.00000%"}
We consider the problem of Byzantine fault-tolerance in synchronous parallelized learning that is founded on the parallelized stochastic gradient descent (parallelized-SGD) method.\
The system comprises a master, $n$ workers, and $N$ ($\gg n$) data points denoted by a set $\Z$. The system architecture is shown in Figure \[fig:sys\_arch\]. Let $d$ be a positive integer, and let $\R^d$ denote the set of $d$-dimensional real-valued vectors. For a global parameter $w \in \R^d$, each data point $z \in \Z$ has a non-negative loss function $\ell(w, z) \in \R_{\geq 0}$. The goal for the master is to [*learn*]{} a parameter $w^*$ that is a minimum point[^1] of the average loss evaluated for the data points. Formally, $w^*$ minimizes $$\begin{aligned}
\frac{1}{N} \sum_{z \in \Z} \ell(w, z) ~\end{aligned}$$ in a [*neighbourhood*]{} of $w^*$. Although $w^*$ may not be the only minimum point, for simplicity $w^*$ denotes a minimum point for the average loss throughout this report.\
The optimization framework forms the basis for most contemporary learning methods, including neural networks and support vector machines [@bottou2018optimization].
Overview of the parallelized-SGD method {#sec:p_sgd}
---------------------------------------
[Parallelized-SGD]{} method is an expedited variant of the stochastic gradient descent method, an iterative learning algorithm [@zinkevich2010parallelized]. In each iteration $t \geq 0$, the master maintains an estimate $w^t$ of $w^*$, and updates it using gradients of the loss functions for a certain number of randomly chosen data points at $w = w^t$. The details of the algorithm are as follows.
In each iteration $t$, the master randomly chooses a set of $m$ data points, denoted by $\Z_t \subset \Z$, and assigns $m_i$ data points to $i$-th worker for $i = 1, \ldots, \, n$, such that $\sum_{i = 1}^n m_i = m$. Let the data points assigned to the $i$-th worker in $t$-th iteration be denoted by $\{z^t_{i_1}, \ldots, \, z^t_{i_{m_i}}\}$. Each worker $i$ computes the gradients for the loss functions of its assigned points at $w^t$, $$g^t_{i_j} = \nabla \ell(w, z^t_{i_j}) \, |_{\,w = w^t}, ~ j = 1, \ldots, \, m_i ~,$$ and sends a [*symbol*]{} $c_i$, which is a function of its computed gradients $\{g^t_{i_1}, \ldots, \, g^t_{i_{m_i}}\}$, to the master. The master obtains the average value of the gradients for all the $m$ data points in $\Z_t$, $$g^t = \frac{1}{m}\sum_{z \in \Z_t}\nabla \ell(w, z) \, |_{\,w = w^t} ~ ,$$ as a function of the symbols $\{c_1, \ldots, \, c_n\}$ received from the workers. For example, if each worker $i$ sends symbol $$c_i\left(g^t_{i_1}, \ldots, \, g^t_{i_{m_i}}\right) = \frac{1}{m_i} \sum_{j = 1}^{m_i} g^t_{i_j} ~ ,$$ then $$g^t = \frac{1}{m}\sum_{i = 1}^n m_i c_i = \frac{1}{m}\sum_{z \in \Z_t}\nabla \ell(w, z) \, |_{\,w = w^t} ~ .$$ Upon obtaining $g^t$, the master updates the parameter estimate $w^t$ as $$\begin{aligned}
w^{t+1} = w^t - \eta_t \left( \frac{1}{N}\sum_{i = 1}^N g^t_i \right), \label{eqn:update}\end{aligned}$$ where $\eta_t$ is a positive real value commonly referred as the ‘step-size’. An illustration of the parallelized-SGD method is presented in Figure \[fig:sys\_arch\] for the case when $m_i = 1, \, \forall i$.
Vulnerability against Byzantine workers
---------------------------------------
The above parallelized-SGD method is not robust against [**Byzantine**]{} faulty workers. Byzantine workers need follow the master’s instructions correctly, and might send malicious incorrect (or [*faulty*]{}) [symbols]{}. The identity of the Byzantine workers remains fixed throughout the learning algorithm, and is unknown [*a priori*]{} to the master.\
We consider a case where up to $f$ ($< n/2$) of the workers are [Byzantine]{} faulty. [**Our objective**]{} is design a parallelized-SGD method that has [*exact fault-tolerance*]{}, which is defined as follows.
A [parallelized-SGD]{} method has [**[exact]{} fault-tolerance**]{} if the Master asymptotically converges to a minimum point $w^*$ exactly, despite the presence of Byzantine workers.
Proposed Solutions and Contributions
====================================
We propose two coding schemes, one of which is [**deterministic**]{} and the other is [**randomized**]{}, for guaranteeing exact fault-tolerance if $2f < n$. Obviously, the master [*cannot*]{} tolerate more than or equal to $n/2$ Byzantine workers [@chen2018draco]. Overviews of each these schemes are presented below. Before we proceed with the summary of our contribution and overviews of proposed coding schemes, let us define the [*computation efficiency*]{} of a coding scheme.
The [**computation efficiency**]{} of a coding scheme is the [*ratio of the number of gradients used for parameter update, given in , to the number of gradients computed by the workers in total*]{}.
For example, in each iteration of the parallelized-SGD method presented above, the total number of gradients computed by the workers is equal to $m$, and the master uses the average of all the $m$ gradients to update the parameter estimate . Therefore, the computation efficiency of a coding scheme (used for computing the symbols $c_1, \ldots, \, c_n$) in the traditional parallelized-SGD method is equal to $1$.\
\
Overview of the deterministic scheme {#sub:det}
------------------------------------
For each iteration $t$, after choosing the data points, the master assigns each data point to $f+1$ workers. Each worker $i$ computes gradients for all its data points, and sends a symbol $c_i$ to the master such that, the collection of symbols $\{c_1, \ldots, c_n\}$ forms an $f$ [*fault-detection*]{} code, i.e. the master can detect up to $f$ faulty symbols, and the average of the gradients (for all the data points) is a function of the non-faulty symbols. Upon detecting any fault(s), the master imposes [*reactive redundancy*]{} where each data point (or data point specific to the detected fault(s)) is assigned to additional $f$ workers. Each worker now computes gradients for the additional data points assigned, and send symbols $u_1, \ldots, \, u_n$ that enables the master to identify up to $f$ faulty symbols in $\{c_1, \ldots,c_n\}$. Upon identifying the Byzantine workers that sent faulty symbols, the master can recover the correct average of the gradients. Hence, the scheme guarantees [*exact fault-tolerance*]{}.\
A simple example illustrating the scheme is presented in Figure \[fig:coding\_scheme\]. A [*replication code*]{} for the generic case is presented in Section \[sec:scheme\].\
We note the following generalizations, and drawback of the scheme.
- **Generalizations:**
- The workers may send symbols that are function of [*compressed*]{} gradients, proposed for improved communication efficiency in the [*non-Byzantine*]{} case [@aji2017sparse; @bernstein2018compressed; @singh2019sparq; @tang2019doublesqueeze], instead of the original gradients.
- In general, any suitable fault detection code may be used in this scheme, we use a replication code as an example. The choice of the code will have impact on the communication and computation efficiency of the scheme. However, a deterministic scheme, that obtains [*exact*]{} fault-tolerance, cannot have computation efficiency greater than $1/(f+1)$ in all iterations.
- **Drawback:** In the deterministic scheme, each gradient is computed by $f+1$ workers even when all the $f$ Byzantine workers send non-faulty (or [*correct*]{}) symbols. In other words, $$\begin{aligned}
\textbf{computation efficiency} = \frac{\text{\# gradients used for update}}{\text{\# gradients computed in total}} = \frac{1}{f+1},\end{aligned}$$ even when all the workers send correct symbols. This unnecessary redundancy can be significantly reduced by using a randomized approach presented below.
Overview of the randomized scheme {#sub:rand}
---------------------------------
The master checks for faults only in intermittent iterations chosen at [*random*]{}, instead of all the iterations. Alternately, in each iteration, the master does a [*fault-check*]{} with some non-zero probability less than $1$. By doing so, the master significantly reduces the redundancy in gradients’ computations whilst [*almost surely*]{} identifying the Byzantine workers that send faulty symbols [*eventually*]{}[^2]. As in the deterministic scheme, upon detecting any fault(s) the master imposes reactive redundancy to identify the responsible Byzantine worker(s). However, correcting the detection fault(s) is optional. The identified Byzantine worker(s) are eliminated from the subsequent iterations.\
An illustration of the scheme is presented in Figure \[fig:rand\_code\]. Additional details for the generic case is presented in Section \[sec:r\_code\].\
[**Significant savings on redundancy:**]{} By reducing the probability of random fault-checks, the [*expected*]{} computation efficiency of the scheme can be made as close to $1$ as [*desirable*]{}. Note, a coding scheme that obtains exact fault-tolerance against a non-zero number of Byzantine workers have an expected computation efficiency of $1$.\
We note the following generalizations, and adaptation of the randomized scheme:
- [**Generalization:**]{}
- Obviously, as in the deterministic case, the randomized scheme can be easily generalized for [*compressed gradients*]{}.
- Instead of checking for faults for all the workers with equal probability, the master may use different probabilities for different workers. For doing so, workers can be assigned [*reliability scores*]{} as in the context of reliable crowdsourcing [@raykar2012eliminating]. Other generalizations are presented in Section \[sec:general\].
- **Adaptation:** A lower probability of fault-checks implies higher probability of using faulty gradients for parameter update, and vice-versa. Higher probability of faulty updates means higher probability of slower convergence of the learning algorithm. To manage the trade-off between the [computation efficiency]{} and the rate of learning, we present an adaptive approach in Section \[sec:adapt\]. Essentially, the master may vary the probability of fault-checks – depending upon the [*observed*]{} average loss at the current parameter estimate.
Related works
=============
There has been some work on coding schemes for Byzantine fault-tolerance in parallelized machine learning, such as [@chen2018draco; @data2018data; @rajput2019detox]. The scheme proposed by Data et al., 2018 [@data2018data], however, is only applicable for loss functions whose arguments are linear in the learning parameter. The scheme, named DRACO, by Chen et al., 2018 [@chen2018draco] relies on [fault-correction codes]{} and so, has a computation efficiency of only $1/(2f + 1)$. At the expense of exact fault-tolerance, the computation efficiency of DRACO can be improved using gradient-filters [@rajput2019detox]. Our randomized scheme has both; exact fault-tolerance, and favourable computation efficiency.\
The fault-tolerance properties of the known gradient filters – KRUM [@blanchard2017machine], trimmed-mean [@yin2018byzantine], median [@yin2018byzantine], geometric median of means [@chen2017distributed], norm clipping [@gupta2019bft], SEVER [@diakonikolas2018sever], or others [@mhamdi2019fast; @prasad2018robust] – rely on additional assumptions either on the distribution of the data points or the fraction of Byzantine workers. Moreover, the existing gradient-filters [*do not*]{} obtain exact fault-tolerance [*unless*]{} there are redundant data points.\
To the best of our knowledge, none of the prior works have proposed the idea of [*reactive redundancy*]{} for tolerating Byzantine workers efficiently in the context of parallelized learning. In other contexts, such as checkpointing and rollback recovery, mechanisms that combine proactive and reactive redundancy have been utilized. For instance, Pradhan and Vaidya [@DBLP:journals/tc/PradhanV97] propose a mechanism where a small number of replicas are utilized proactively to allow detection of faulty replicas; when a faulty replica is detected, additional replicas are employed to isolate the faulty replicas.
Generalizations of the Randomized Coding Scheme {#sec:general}
===============================================
Our randomized scheme can be generalized as follows.
- [**Variants of the parallelized-SGD method:**]{} We can use the randomized scheme even for different variants of the parallelized-SGD method where workers send [*compressed*]{} or [*communication-efficient*]{} gradients, as proposed in [@aji2017sparse; @bernstein2018compressed; @singh2019sparq; @tang2019doublesqueeze].
- [**Self-checks:**]{} Instead of imposing reactive redundancy, the master can compute the gradients on its own, and compare them with the gradients received from the workers to check for faults. Similarly as above, the master may optimize the additional workload by choosing the probability of fault-checks adaptively as presented in Section \[sec:adapt\].
- [**Selective fault-checks:**]{} Gradients (or symbols) that are outliers amongst the received gradients (or symbols) should be checked for faults with relatively higher probability. Additionally, the master can assign [*reliability scores*]{} to the workers, as done in the context of reliable crowdsourcing [@raykar2012eliminating]. Symbols from workers with lower reliability scores should be checked for faults with higher probability.
- [**Gradient-filters:**]{} The master can further improve on the computation efficiency by combining the randomized coding scheme with lightweight gradient-filters [@gupta2019byzantine; @mhamdi2019fast; @yin2018byzantine]. When using gradient-filters, the master does not have to identify all the Byzantine workers. This idea has been explored in Rajput et al., 2019 [@rajput2019detox] for a deterministic coding scheme.
- [**[*Distributed*]{} learning framework:**]{} Our randomized scheme can also be used for Byzantine fault-tolerance in [*distributed*]{} learning framework, where the data points are [distributed]{} amongst the workers, i.e. two workers may have different sets of data points [@chen2017distributed; @yin2018byzantine]. In this case, besides checking for faulty gradient(s), the master must also [*validate*]{} the data points used by the workers for computing the gradients in the first place. As most existing data validation tools are computationally expensive [@downs2010your; @JMLR:v18:15-650; @raykar2012eliminating; @vuurens2011much], the master may use our randomized scheme to optimize the trade-off between the cost of data validation and the convergence-rate of a distributed learning algorithm.
Summary {#sec:summ}
=======
In this report, we have presented two coding schemes, a deterministic scheme and a randomized scheme, for exact Byzantine fault-tolerance in the parallelized-SGD learning algorithm.\
In the deterministic scheme, the master uses a fault-detection code in each iteration. Upon detecting any fault(s), the master imposes reactive redundancy to correct the faults and identify the Byzantine worker(s) responsible for the fault(s).\
The randomized scheme improves upon the computation efficiency of the deterministic scheme. Here, the master uses fault-detection codes only in randomly chosen [*intermittent*]{} iterations, instead of all the iterations. By doing so, the master is able to optimize the trade-off between the [*expected*]{} computation efficiency, and the convergence-rate of the parallelized learning algorithm.
Acknowledgements {#acknowledgements .unnumbered}
================
Research reported in this paper was sponsored in part by the Army Research Laboratory under Cooperative Agreement W911NF- 17-2-0196, and by National Science Foundation award 1610543. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the the Army Research Laboratory, National Science Foundation or the U.S. Government.
[^1]: A local minimum point if the average loss function is non-convex, or a global minimum point if the average loss function is convex.
[^2]: As the parallelized-SGD method converges to the learning parameter regardless of the initial parameter estimate, a Byzantine worker that [*eventually*]{} stops sending faulty gradients poses no harm to the learning process. Hence, the master only needs to identify Byzantine workers that send faulty gradient(s) [*eventually*]{}.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'From the analysis of their interaction pseudopotentials, it is argued that (at certain filling factors) Laughlin quasiparticles can form pairs. It is further proposed that such pairs could have Laughlin correlations with one another and form condensed states of a new type. The sequence of fractions corresponding to these states includes all new fractions observed recently in experiment (e.g., $\nu=5/13$, 3/8, or 4/11).'
author:
- 'Arkadiusz Wójs,$^{1,2}$ Kyung-Soo Yi,$^{2,3}$ and John J. Quinn$^2$'
title: |
Pairing and Condensation of Laughlin Quasiparticles\
in Fractional Quantum Hall Systems
---
Introduction
============
In a recent experiment [@pan], Pan [*et al.*]{} observed the fractional quantum Hall (FQH) effect [@tsui; @laughlin] at novel filling fractions $\nu$ of the lowest Landau level (LL). Among other features, the FQH effect is a macroscopic manifestation of an incompressible character of the many-body ground state formed at a specific filling $\nu$. The new spin-polarized FQH states occur at filling factors outside the Jain sequence [@jain] of composite fermion (CF) states. Some of them, as $\nu=4/11$ or 4/13, appear in the Haldane hierarchy [@haldane] of quasiparticle (QP) condensates, and therefore at first sight it might only be surprising that they have not been observed earlier. However, the “hierarchical” interpretation of these states was questioned [@hierarchy] because of the specific form of the QP–QP interaction, which leaves their interpretation quite uncertain. Others, such as the $\nu=3/8$ or 3/10 states, do not belong to the Haldane hierarchy, and thus the origin of their incompressibility is puzzling in an even more obvious way. In the CF picture, these even-denominator fractions correspond to the half-filled first excited ($n=1$) CF LL. While for the electrons half-filling their $n=1$ LL (including the double degeneracy of the lowest LL, this amounts to the total electron filling of $\nu=2+1/2=5/2$) the FQH effect has been known for quite some time [@willet] (the incompressible $\nu=5/2$ state is known as a Moore-Read state [@moore]), the similar behavior of electrons and CF’s at this filling is rather unexpected. Pan [*et al.*]{} take their observations as evidence for residual CF–CF interactions, sufficiently strong to cause emergence of new FQH states (that, unlike virtually all observed fractions, cannot be predicted from the noninteracting CF model or standard Haldane hierarchy). However, they ignore the theoretical investigations in which these interactions were studied in detail [@hierarchy; @sitko; @lee].
In this paper we propose an explanation for these new states involving formation of QP pairs which display Laughlin correlations with one another. First, we explain the connection between the CF model and the QP hierarchy [@haldane; @hierarchy; @sitko]. Second, we recall two simple types of two-body correlations, Laughlin correlations and pairing, that may occur in an interacting system depending on the filling factor $\nu$ and on whether $V(\mathcal{R})$ is super- or subharmonic at the relevant range [@parentage]. Then, knowing the QP–QP pseudopotential $V_{\rm QP}(\mathcal{R})$, we apply the concept of Laughlin condensed states of (bosonic) pairs (used earlier for the electrons in the $n=1$ LL to describe such FQH states as $\nu=5/2$ or 7/3 [@fivehalf]) to the particles or holes in a partially filled CF LL, i.e., to Laughlin quasielectrons (QE’s) or quasiholes (QH’s). Finally, we propose the novel hierarchy of FQH states in which the incompressibility results from the condensation of QP pairs (QE$_2$’s or QH$_2$’s) into Laughlin correlated pair states. The series of FQH states derived from the parent $\nu=1/3$ state include all novel fractions: $\nu=5/13$, 3/8, 4/11, and 6/17 for the QE’s and $\nu=5/17$, 3/10, 4/13, and 6/19 for the QH’s.
Laughlin QP’s and CF’s
======================
Let us begin with recalling the connection between Laughlin QP’s [@laughlin] and the CF model (equivalent to the mean-field Chern–Simons transformation) [@jain]. Laughlin QP’s are the actual elementary excitations of a two-dimensional electron liquid filling a fraction of the lowest LL. They have well-defined and known wave functions (and thus also density profiles, size, etc.) and single-particle energy. They carry (fractional) electric charge, counted with respect to the uniformly spread charge of the underlying Laughlin state. The negatively and positively charged QP’s are called quasielectrons (QE’s) and quasiholes (QH’s), respectively. Moving in the external magnetic field, both kinds of QP’s follow cyclotron orbits, conveniently labeled by angular momentum.
Since the uniform-density Laughlin states only occur at a discrete series of filling factors ($\nu=1/3$, $1/5$, …), and Laughlin QP’s are simply the least-energy excitations able to carry charge in excess of the value corresponding to the nearest Laughlin state, their type and number depends on $\nu$. At $\nu$ precisely equal to any of the Laughlin values $\nu_p=(2p+1)^{-1}$ no QP’s exist in the ground state and they may only appear in form of excitonic, charge-neutral QE–QH pairs. At $\nu$ slightly smaller (or larger) than $\nu_p$, a number $N_{\rm QP}$ of the “$p$–type” QE’s (or QH’s) appear in the Laughlin liquid. The QP number is equal to the difference between the total magnetic flux through the sample and its value corresponding to the nearest “parent” Laughlin state, measured in units of elementary flux, $N_{\rm QP}=|\Phi-\Phi_p|/\phi_0$.
For electrons, the filling factor is expressed through the flux $\Phi$ and the electron number $N$ as $\nu=\Phi/N\phi_0$, For the QP’s of the parent $\nu_p$ Laughlin state, each carrying charge $\pm e/(2p+1)$, the LL degeneracy is reduced by a factor of $(2p+1)$ compared to the electron value. Consequently, the QP filling factor is equal to $\nu_{\rm QP}
=\Phi/(2p+1)N_{\rm QP}\phi_0$. Note that $\nu_{\rm QP}$ is a linear function of $\nu$ near each value of $\nu_p$, The situation with $\nu$ far away from the nearest $\nu_p$, corresponds to a large filling factor for appropriate QP’s whose mutual interactions and filling of higher QP LL’s both become important.
In the mean-field CF picture, the reduction of the QP LL degeneracy is attributed to a reduced magnetic field $B^*=B/(2p+1)$ rather than to a reduced QP charge. In the simplest formulation of the model, the effective field $B^*$ results from the capture of an even number of magnetic flux quanta $2p\phi_0$ by each electron (such bound state is called a CF), and the QE’s and QH’s are pictured as particles and vacancies in the otherwise empty (full) CF LL’s.
The CF picture turns out very useful for the description of many properties of the QP’s (e.g., size of cyclotron orbits or LL degeneracy) or of the FQH systems in general (e.g., the values of $\nu$ at which the incompressibility results from a complete filling of the QP/CF LL’s). However, the QP–QP interactions important for the present problem in the CF model arise as a combination of rather complicated two- and three-body gauge interactions between charges and fluxes, and because of this difficulty, are usually neglected. It is therefore important to realize that the QP–QP interaction really is a Coulomb interaction between a pair of charged particles. For example, for two identical QP’s it is repulsive, and at long range it is similar to the electron–electron repulsion, only reduced in magnitude by $(2p+1)^2$ because of a smaller QP charge, On the other hand, the exact form of the QP–QP interaction at short range (where it is different from the electron–electron repulsion because of the particular QP charge-density profile) has been quite accurately calculated numerically [@hierarchy].
Haldane Hierarchy and Jain Sequence
===================================
Knowing that the QE–QE and QH–QH interactions are generally repulsive, Haldane proposed [@haldane] condensation of QE’s and QH’s into the hierarchy of “daughter” states at the series of Laughlin values of $\nu_{\rm QE}$ or $\nu_{\rm QH}$. In these states, the appropriate QP’s correlate with one another in the same way as the electrons do in the Laughlin states, and their elementary excitations are simply a new generation of QP’s. Assumming such QP condensation at each level of the hierarchy one would predict incompressibility of the whole electron system at all filling factors given by any odd-denominator fraction. This is in striking disagreement with the experiments, and the reason for this discrepancy is that although Coulombic, the QP–QP interaction at short range is not quite identical to the electron–electron repulsion responsible for the Laughlin correlations. As a result, the QP’s form Laughlin liquids only at very few of the Laughlin fractions, which eliminates all but a few valid “hierarchy” fractions [@hierarchy], in good agreement with the experiment. The same series of fractions arise naturally in the CF picture. These are the states at $\nu=(2p+1/n)^{-1}$, corresponding to a number $n$ of completely filled CF LL’s each carrying flux $2p\phi_0$.
The new FQH states [@pan] occur at the values of $\nu$ from outside the Jain sequence, and thus corresponding to only partial filling of a CF LL. Hence, in the CF picture, their incompressibility implies role of CF–CF interactions. In the QP hierarchy picture, the new states either coincide with the “invalid” fractions (e.g., $\nu=4/11$) or are new fractions altogether (e.g., $\nu=3/8$). In both cases it is clear that the origin of observed incompressibility lies in the special form of QP–QP correlations, and that these correlations are of a new (non-Laughlin) type.
QP–QP Pseudopotential
=====================
The nature of QP correlations depends critically on the form of pseudopotential $V_{\rm QP}(\mathcal{R})$ describing their pair interaction energy $V_{\rm QP}$ as a function of relative pair angular momentum $\mathcal{R}$. We have shown earlier [@parentage; @fivehalf] that the correlations are of the Laughlin type (i.e., the particles tend to avoid pair states with one or more of the smallest values of $\mathcal{R}=1$, 3, …) only if $V(\mathcal{R})$ is “superharmonic” at the relevant values of $\mathcal{R}$ for a given filling factor $\nu$ (specifically, at $\mathcal{R}=2p-1$ for $\nu\sim(2p+1)^{-1}$, where $p=1$, 2, …). Laughlin correlations defined in this way justify reapplication of the CF picture to the QP’s to select the lowest states of the whole many-body spectrum, and lead to the incompressible QP “daughter” states of the standard CF hierarchy [@sitko]. The superharmonic repulsion is defined as one for which $V$ decreases more quickly than linearly as a function of the average particle–particle separation $\left<r^2\right>$ for the consecutive pair eigenstates labeled by $\mathcal{R}$. In spherical geometry [@haldane], most convenient for finite-size calculations, this means that $V$ increases more quickly than linearly as a function of $L(L+1)$, i.e., of the squared total pair angular momentum $L=2l-\mathcal{R}$, where $l$ is the single-particle angular momentum.
The qualitative behavior of the QP–QP interaction pseudopotential $V_{\rm QP}(\mathcal{R})$ at short range is well-known from numerical studies of small systems [@hierarchy; @sitko; @lee]. On the other hand, the repulsive character of the QP–QP interaction and the long-range behavior of $V_{\rm QP}(\mathcal{R})\sim
\mathcal{R}^{-1/2}$ follow from the fact that QP’s are charged particles (the form of QP charge density affects $V_{\rm QP}$ only at short range, comparable to the QP size). Combining the above arguments, it is clear that the dominant features of $V_{\rm QE}$ are the small value at $\mathcal{R}=1$ and a strong maximum at $\mathcal{R}=3$. Analogous analysis for the QH’s yields maxima at $\mathcal{R}=1$ and 5, and nearly vanishing $V_{\rm QH}(3)$.
QP Pairing
==========
It is evident that $V_{\rm QE}$ does not support Laughlin QE–QE correlations. Instead, we expect that at least some of the QE’s will form pairs (QE$_2$) at $\mathcal{R}=1$. A paired state would be characterized by a greatly reduced fractional parentage $\mathcal{G}$ [@parentage] from the strongly repulsive $\mathcal{R}=3$ state compared to the Laughlin correlated state, and have lower total interaction energy $E={1\over2}N(N-1)\sum_\mathcal{R}
\mathcal{G}(\mathcal{R})V(\mathcal{R})$. Let us stress that such pairing is not a result of some attractive QE–QE interaction, but due to a tendency to avoid the most strongly repulsive $\mathcal{R}=3$ pair state. At sufficiently high QE density this can only be achieved by having significant $\mathcal{G}(1)$, which can be interpreted as pairing into the QE$_2$ molecules. By analogy, the QH pairing is expected at $\mathcal{R}=3$. The range of $\nu_{\rm QP}$ at which pairing can be considered is limited by the condition that the separation between the pairs must exceed the pair size. While for the QE pairs this is satisfied at any $\nu_{\rm QE}<1$, the QH pairing can only occur at $\nu_{\rm QH}<1/3$.
Laughlin Correlations Between Pairs
===================================
Having established that the QP fluid consists of (bosonic) QP$_2$ molecules, the QP$_2$–QP$_2$ interactions need be studied to understand correlations. The QP$_2$–QP$_2$ interaction is described by an effective pseudopotential $V_{{\rm QP}_2}(\mathcal{R})$ that includes correlation effects caused by the fact that the two-pair wavefunction must be symmetric under exchange of the whole QP$_2$ bosons and at the same time antisymmetric under exchange of any pair of the QP fermions. This problem is analogous to that of interaction between the electron pairs in the $n=1$ LL [@fivehalf].
Although we do not know $V_{{\rm QP}_2}(\mathcal{R})$ accurately, we expect that since it is due to the repulsion between the QP’s that belong to different QP$_2$ pairs, it might be superharmonic at the range corresponding to the QP$_2$–QP$_2$ separation. Our preliminary numerical results for four QE’s seem to support this idea. However, in contrast to the $n=1$ electron LL [@fivehalf], the lack of accurate data for $V_{\rm QP}$ at the intermediate range makes such calculations uncertain.
Condensed Pair States
=====================
The assumption of Laughlin correlations between the QP$_2$ bosons implies the sequence of Laughlin condensed QP$_2$ states that can be conveniently described using the “composite boson” (CB) model [@fivehalf]. Let us use spherical geometry and consider the system of $N_1$ fermions (QP’s) each with (integral or half-integral) angular momentum $l_1$ (i.e., in a LL of degeneracy $g_1=2l_1+1$). Neglecting the finite-size corrections, this corresponds to the filling factor $\nu_1=N_1/g_1$. Let the fermions form $N_2={1\over2}N_1$ bosonic pairs each with angular momentum $l_2=2l_1-\mathcal{R}_1$, where $\mathcal{R}_1$ is an odd integer. The filling factor for the system of pairs, defined as $\nu_2=N_2/g_2$ where $g_2=2l_2+1$, equals to $\nu_2={1\over4}\nu_1$. The allowed states of two bosonic pairs are labeled by total angular momentum $L=2l_2-\mathcal{R}_2$, where $\mathcal{R}_2$ is an even integer. Of all even values of $\mathcal{R}_2$, the lowest few are not allowed because of the Pauli exclusion principle applied to the individual fermions. The condition that the two-fermion states with relative angular momentum smaller than $\mathcal{R}_1$ are forbidden is equivalent to the elimination of the states with $\mathcal{R}_2\le4\mathcal{R}_1$ from the two-boson Hilbert space. Such a “hard core” can be accounted for by a CB transformation with $4\mathcal{R}_1$ flux quanta attached to each boson[@theorem]. This gives effective CB angular momentum $l_2^*=l_2-2\mathcal{R}_1
(N_2-1)$, LL degeneracy $g_2^*=g_2-4\mathcal{R}_1(N_2-1)$, and filling factor $\nu_2^*=(\nu_2^{-1}-4\mathcal{R}_1)^{-1}$.
The CB’s defined in this way condense into their only allowed $l_2^*=0$ state ($\nu_2^*=\infty$) when the corresponding fermion system has the maximum density at which pairing is still possible, $\nu_1=\mathcal{R}_1^{-1}$. At lower filling factors, the CB LL is degenerate and the spectrum of all allowed states of the $N_2$ CB’s represents the spectrum of the corresponding paired fermion system. In particular, using the assumption of the superharmonic form of boson–boson repulsion, condensed CB states are expected at a series of Laughlin filling factors $\nu_2^*=(2q)^{-1}$. Here, $2q$ is an even integer corresponding to the number of additional magnetic flux quanta attached to each CB in a subsequent CB transformation, $l_2^*\rightarrow l_2^{**}=l_2^*-q(N_2-1)$, to describe Laughlin correlations between the original CB’s of angular momentum $l_2^*$. From the relation between the fermion and CB filling factors, $\nu_1^{-1}=(4\nu_2^*)^{-1}+\mathcal{R}_1$, we find the following sequence of fractions corresponding to the Laughlin condensed pair states, $\nu_1^{-1}=q/2+\mathcal{R}_1$. Finally, we set $\mathcal{R}_1=1$ for the QE’s and $\mathcal{R}_1=3$ for the QH’s, and use the hierarchy equation [@hierarchy], $\nu^{-1}=2p+(1\pm\nu_{\rm QP})^{-1}$, to calculate the following sequences of electron filling factors, $\nu$, derived from the parent $\nu=(2p+1)^{-1}$ state $$\nu^{-1}=2p+1\mp(2+q/2)^{-1},$$ where “$+$” corresponds to the QE’s and “$-$” to the QH’s. Remarkably, all the fractions reported by Pan [*et al.*]{} are among those predicted for the $\nu=1/3$ parent. Note also that the same values of $q=1$, 2, 4, and 8 describe both observed QE and QH states. This indicates similarity of the QE–QE and QH–QH pseudopotentials and suggests that both $V_{{\rm QE}_2}$ and $V_{{\rm QH}_2}$ may be superharmonic only at the corresponding four values of $\mathcal{R}$ (in such case, the remaining fractions could not be observed even in most ideal samples).
Conclusion
==========
We have studied the QP–QP interactions leading to novel spin-polarized FQH states in the lowest LL. Using the knowledge of QP–QP pseudopotentials and a general dependence of the form of correlations on the super- or subharmonic behavior of the pseudopotential, we have shown that QP’s form pairs over a certain range of filling factor $\nu_{\rm QP}$. Then, we argued that the correlations between the QP pairs should be of Laughlin type and proposed a hierarchy of condensed paired QP states. The proposed hierarchy of fractions agrees remarkably well with the recent experiment of Pan [*et al.*]{} [@pan].
The authors thank Jennifer J. Quinn and Josef Tobiska for helpful discussions and acknowlege support from US DoE grant Grant DE-FG 02-97ER45657. AW acknowledges support from KBN grant 2P03B02424. KSY acknowledges support from KOSEF grant R14-2002-029-01002-0.
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{
"pile_set_name": "ArXiv"
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author:
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Tomasz Maszczyk and Andrzej Weber[^1]\
Institute of Mathematics, Uniwersytet Warszawski\
ul. Banacha 2, 02–097 Warszawa, Poland\
e-mail:[email protected], [email protected]
title: Koszul Duality for modules over Lie algebras
---
\[12pt\][article]{} \[section\] \[df\][ Example]{} \[df\][ Proposition]{} \[df\][ Theorem]{} \[df\][ Corollary]{} \[df\][ Remark]{} \[df\][ Lemma]{} i[[**i**]{}\^\*]{} ø Ł[\^-4pt\^\*]{}
Let $\g$ be a reductive Lie algebra over a field of characteristic zero. Suppose $\g$ acts on a complex of vector spaces $\M$ by $i_\lambda$ and $\Ll_\lambda$, which satisfy the same identities that contraction and Lie derivative do for differential forms. Out of this data one defines the cohomology of the invariants and the equivariant cohomology of $\M$. We establish Koszul duality between each other.
Introduction
============
Let $G$ be a compact Lie group. Set $\Lambda_\x=H_*(G)$ and $S^\x=H^*(BG)$. The coefficients are in $\bf R$ or $\bf C$. Suppose $G$ acts on a reasonable space $X$. In the paper [@GKM] Goresky, Kottwitz and MacPherson established a duality between the ordinary cohomology which is a module over $\Lambda_\x$ and equivariant cohomology which is a module over $S^\x$. This duality is on the level of chains, not on the level of cohomology. Koszul duality says that there is an equivalence of derived categories of $\Lambda_\x$–modules and $S^\x$–modules. One can lift the structure of an $S^\x$–module on $H_G^*(X)$ and the structure of a $\Lambda_\x$–module on $H^*(X)$ to the level of chains in such a way that the obtained complexes correspond to each other under Koszul duality. Equivariant coefficients in the sense of [@BL] are also allowed. Later, Allday and Puppe ([@AP]) gave an explanation for this duality based on the minimal Brown-Hirsh model of the Borel construction. One should remark that Koszul duality is a reflection of a more general duality: the one described by Husemoller, Moore and Stasheff in [@HMS].
Our goal is to show that this duality phenomenon is a purely algebraic affair. We will construct it without appealing to topology. We consider a reductive Lie algebra $\g$ and a complex of vector spaces $\M$ on which $\g$ acts via two kinds of actions: $i_\lambda$ and $\Ll_\lambda$. These actions satisfy the same identities as contraction and Lie derivative do in the case of the action on the differential forms of a $G$–manifold. Such differential $\g$–modules were already described by Cartan in [@Ca]; see also [@AM], [@GS]. We do not assume that $\M$ is finite dimensional nor semisimple. We also wish to correct a small inaccuracy in the proof of Lemma 17.6, [@GKM]. The distinguished transgression plays a crucial role in our construction. This is a canonical identification between the space of primitive elements of $(\L)^\g\simeq H^*(\g)$ with certain generators of $(S\g^*)^\g$.
The results of the present paper were obtained with the help and assistance of Marcin Chałupnik. It is a part of joint work. Our aim is to describe Koszul duality in a much wider context. We thank the referee for careful corrections.
Category
=========
Let $\g$ be a reductive Lie algebra over a field $k$ of characteristic 0. We consider differential graded vector spaces $\M $ over $k$ equipped with linear operations $i_\lambda:\M
\rightarrow M^{\x-1}$ of degree $-1$ for each $\lambda\in\g$. We define $$\Ll_\lambda=di_\lambda+i_\lambda d:\M \rightarrow \M
\,.$$ We assume that $i_\lambda$ is linear with respect to $\lambda$ and for each $\lambda, \mu \in \g$ the following identities are satisfied: $$i_\lambda i_\mu = -i_\mu i_\lambda,$$ $$[\Ll_\mu,i_\lambda]=i_{[\mu,\lambda]}\,.$$ Then $\Ll$ is a representation of $\g$ in $\M $. The category of such objects with obvious morphisms will be denoted by $K(\g)$.
Let $G$ be a group with Lie algebra $\g$ and let $X$ be a $G$-manifold. Then the space of differential forms $\Omega^\x(X)$ equipped with the contractions with fundamental vector fields is an example of an object from $K(\g)$.
Another example of an object of $K(\g)$ is $\L$, the exterior power of the dual of $\g$. The generators of $\L$ are given the gradation 1. This is a differential graded algebra with a differential $\dl$ induced by the Lie bracket. The operations $i_\lambda$ are the contractions with $\g$.
For a representation $V$ of $\g$ define the invariant subspace $$V^\g=\{v\in V:\;\forall \lambda\in \g \;\; \Ll_\lambda
v =0\}.$$ Then $V^\g$ with trivial differential and $i_\lambda$’s is an object of $K(\g)$. In particular we take $V=\Sg$, the symmetric power of the dual of $\g$. The generators of $\Sg$ are given the gradation 2.
Suppose there are given two objects $\M $ and $N^\x$ of $K(\g)$. Then $\M \o N^\x$ with operations $i_\lambda$ defined by the Leibniz formula is again in $K(\g)$.
Note that the objects of $K(\g)$ are the same as differential graded modules over a dg–Lie algebra $C\g$ (the cone over $\g$) with $$C\g^0=C\g^{-1}=\g\qquad C\g^{\neq -1,0}=0\,.$$ The elements of $C\g^0$ are denoted by $\Ll_\lambda$ and the elements of $C\g^{-1}$ are denoted by $i_\lambda$. They satisfy the following identities: $$di_\lambda=\Ll_\lambda\,,
\qquad[\Ll_\lambda,\Ll_\mu]=\Ll_{[\lambda,\mu]}\,,
\qquad[\Ll_\lambda,i_\mu]=i_{[\lambda,\mu]}\,,
\qquad[i_\lambda,i_\mu]=0\,.$$
The enveloping dg–algebra of $C\g$ is the Chevaley–Eilenberg complex $V(\g)$ ([@Wei] p. 238) which is a free $U(\g)$–resolution of $k$. Thus the objects of $K(\g )$ are just the dg–modules over $V(\g)$.
Suppose that a Lie algebra $\g$ acts on a graded commutative $k$-algebra $A$ by derivations. Let $\Omega_k A$ be the algebra of forms; it is generated by symbols $a$ and $da$ with $a\in A$. The cone of $\g$ acts by derivations. The action on generators is given by $$i_\lambda a=0,\quad \Ll_\lambda a=\lambda a,
\quad i_\lambda da=\lambda a,\quad \Ll_\lambda da=d\lambda
a\,.$$
Then $\Omega_k A$ is in $K(\g)$.
Another point of view (as in [@GS]) is that the objects of $K(\g)$ are the representations of the super Lie algebra $\widehat\g=C\g\oplus k[-1]$ (where $k[-1]$ is generated by $d$) with relations: $$[d,d]=0,\qquad [d,x]=d x\qquad{\rm for}\; x\in
C\g\,.$$
The twist
=========
We will describe a transformation, which plays the role of the canonical map $$\begin{array}{ccccc}\Phi:& G\times X&\rightarrow&
G\times X\\ & (g,x)&\mapsto &(g,gx)\\
\end{array}$$ for topological $G$-spaces.
We define a linear map $\ii:\L\o\M\rightarrow\L\o\M$ by the formula: $$\ii(\xi\o m)=\sum_k\xi\wedge\lambda^k\o i_{\lambda_k}m\,,$$ where $\{\lambda_k\}$ is a basis of $\g$ and $\{\lambda^k\}$ is the dual basis. It commutes with $\Ll_\lambda$. The operation $\ii$ is nilpotent. Define an automorphism ${\bf T}:\L\o\M\rightarrow\L\o\M$ (which is not in $K(\g)$): $${\bf T}=\exp(-\ii)=\sum_{n=1}^{\infty}\frac{(-1)^n}{n!}\ii^n\,,$$ $${\bf T}(\xi\o m)=
\sum_{I=\{i_1<\dots<i_n\}}(-1)^{\frac{n(n+1)}2}
\xi\wedge\lambda^I\o i_{\lambda_I}m\,.$$ It satisfies ([@GHV], Prop. V, p.286, see also [@AM]) $$i_\mu({\bf T}(\xi\o m))={\bf T}((i_\mu\xi)\o m)\,,$$ $$d\,{\bf T}
(\xi\o m) = {\bf T}\left( d(\xi\o m)+
\sum_k\lambda^k\wedge\xi\o\Ll_{\lambda_k}m\right)\,.$$
Note that for the self-map $\Phi$ of $G\times X$ we have $$d\Phi^*\omega(1,x)=\left(d\omega+\sum_k
p^*\lambda^k\wedge\Ll_{\lambda_k}\omega\right)(1,x)\,,$$ where $p:G\times X\rightarrow X$ is the projection and $x\in X$.
The twist on the level of the Weil algebra has already been used by Cartan [@Ca] and later by Mathai and Quillen [@MQ]. From another point of view, for a d.g.vector space to be an object of $K(\g)$ is equivalent to having such a twist which satisfies certain axioms. We will not state them here. We just remark that understanding of this twist allows to develop a theory of actions of ${\cal L}_\infty$-algebras.
Weil algebra
=============
Following [@GHV], Chapter VI, p.223 we define the Weil algebra $$W(\g)=\Lambda^\x(C\g )^*=\Sg \o\L\,.$$ The generators of $\Sg$ are given the gradation 2 whereas the generators of $\L$ are given the gradation 1. The differential in $W(\g)$ is the sum of three operations: $$d_W(a\o b) =a\o\dl b+
\sum_k \lambda^k a\o i_{\lambda_k}b
+ \sum_k ad^*_{\lambda_k} a\o \lambda^k\wedge b\,,$$ where $\{\lambda_k\}$ is a basis of $\g$.
The differential $d_W$ satisfies Maurer–Cartan formula $$d_W(1\o \xi)-1\o \dl\xi = \xi\o 1\,,$$ for $\xi\in\g^*$.
The operations $i_\lambda$ are contractions with the second term. The resulting action $\Ll_\lambda=d_Wi_\lambda+i_\lambda
d_W$ is induced by the co-adjoint action on $\Sg \o\L$, [@GHV], rel. (6.5), p.226. With this structure $W(\g)$ becomes an object of $K(\g)$.
The cohomology of $W(\g)$ is trivial except in dimension 0, where it is $k$, [@GHV], Prop. I, p.228. There are given canonical maps in $K(\g)$:
- inclusion $(\Sg )^\g\simeq (\Sg )^\g\o 1\subset W(\g)$,
- restriction $W(\g)\epi \L$, which sends all the positive symmetric powers to 0.
The Weil algebra is a model of differential forms on $EG$ and the sequence of morphisms in $K(\g)$ $$(\Sg)^\g\mono W(g) \epi \L$$ is a model of $$\Omega^\x(BG)\mono \Omega^\x(EG) \epi \Omega^\x(G)\,.$$
It is easy to see that $W(\g)=\Omega_k\L$. Thus for any commutative d.g-algebra A $${\rm Hom}_{g-comm}(\L,A)={\rm
Hom}_{d.g-comm}(W(\g),A)\,.$$
The distinguished transgression
================================
The invariant algebra $(\L)^\g$ is the exterior algebra spanned by the space of primitive elements $P^\x$, whereas $(\Sg)^\g$ is the symmetric algebra spanned by some space $\widetilde P^\x$. The point is that $\widetilde P^\x$ can be canonically chosen.
\[trans\][[@GHV] Prop. VI, p.239]{}. Suppose $\xi\in P^\x$ is a primitive element. Then there exist an element $\omega\in W(\g)^\g$ such that $$\omega_{|\L}=\xi\,,$$ $$i_\lambda \omega= i_\lambda(1\o\xi)\quad for\, all\;
\lambda\in(\Lg)^\g\,$$ $$d_W(\omega)=\widetilde\xi\o 1\,.$$
The element $\omega$ is not unique, but $\widetilde\xi$ is. The set of $\widetilde\xi$ for $\xi\in P^*$ is the distinguished space of generators of $(\Sg)^\g$.
Example – ${\bf su}_2$
=======================
The algebra $ {\bf su}_2$ is spanned by ${\bf i}$, ${\bf j}$ and ${\bf k}$ with relation $[{\bf i},{\bf j}]=2{\bf k}$ and its cyclic transposition. In $\L$ we have $$\dl\i=2\j\wedge\k\quad{\rm
and\;cycl.}$$ In the Weil algebra we have $$d_W(1\o \i)=1\o
2\j\wedge\k +\i\o 1 \quad{\rm and\;cycl.}\,,$$ $$d_W(\i\o
1)=2(\k\o\j-\j\o\k) \quad{\rm and\;cycl.}$$ The primitive elements in $\L$ are spanned by $\xi=\i\wedge\j\wedge\k$. As $\omega$ of Proposition \[trans\] we take $$\omega=1\o\i\wedge\j\wedge\k+{\frac12}(\i\o\i+cycl.)$$ Then $$\widetilde\xi\o 1=d_W(\omega)={\frac12}({\i}^2 +{\j}^2
+{\k}^2)\o 1\,,$$ whereas $$d_W(1\o\xi)=\i\o\j\wedge\k+cycl.$$
Invariant cohomology and equivariant cohomology
================================================
Denote $(\Lg)^\g$ by $\Lambda_\x$. Let $D(\Lambda_\x)$ be the derived category of graded differential $\Lambda_\x$–modules. For $\M \in K(\g)$ the invariant submodule $(\M )^\g$ is a differential module over $\Lambda_\x$. We obtain an object in $D(\Lambda_\x)$. We call it the invariant cohomology of $\M$.
\[inv\] Let $X$ be a manifold on which a compact group $G$ acts. Let $\g$ be the Lie algebra of $G$. Then $\Omega^\x(X)$ is a $\g$-module. The invariants of $\Lg$ act on $\Omega^\x(X)$, but this action does not commute with the differential in general. We have $[d,i_\lambda]\omega={\cal L}_\lambda\omega$. To obtain an action which commutes with the differential one restricts it to $(\Omega^\x(X))^\g$. Fortunately the resulting cohomology does not change. We obtain a complex with an action of $\Lambda_\x=H_*(G)$, which is quasi-isomorphic to $\Omega^\x(X)$. The cohomology is equal to $H^*(X)$.
Denote $(\Sg)^\g$ by $S^\x$. Let $D(S^\x)$ be the derived category of graded differential $S^\x$–modules. Following [@Ca] we define: $$(\M
)_\g:=(\Sg\o \M )^\g$$ with differential $$d_{M,\g}(a\o m)=a\o
d_Mm-\sum_k\lambda^ka\o i_{\lambda_k}m\,.$$ It is a differential $S^\x$–module. We obtain an object $(\M )_\g$ in $D(S^\x)$. We call it the equivariant cohomology of $\M $.
For an object $N^\x$ of $K(\g)$ we define horizontal elements $$(N^\x)_{hor}=\{n\in N^\x:\;\forall \lambda\in\g\;\;i_\lambda
n=0\}\,.$$ Then define basic elements $$(N^\x)_{basic}=(N^\x)_{hor}^\g=\{n\in N^\x:\;\forall
\lambda\in\g\;\;i_\lambda n=0,\,i_\lambda dn=0\}\,,$$ which form a complex.
The following Lemma can be found in [@Ca], but we need to have an explicit form of the isomorphism, as in [@AM] §4.1.
\[Lemma\] [[@Ca]]{} The map $\psi_0$ is an isomorphism of differential graded $S^\x$–modules: $$\psi_0=1\o{\bf T}
_{|1\o\M}:(\M )_\g\rightarrow\left(W(\g)\o \M
\right)_{basic}\,.$$
[Proof.]{} The elements of $\left(\L\o \M \right)_{hor}$ are of the form $${\bf T}(1\o m)=1\o m -\sum_k\lambda^k\o i_{\lambda_k}m-
\sum_{k<l}\lambda^k\wedge\lambda^l\o
i_{\lambda_k}i_{\lambda_l}m\pm\dots\,,$$ thus they are determined by $m$. The conclusion follows since $$\left(W(\g)\o \M
\right)_{hor}=\Sg\o \left(\L\o \M \right)_{hor}\,.$$$\Box$
From the above description we see that $(\M )_\g$ is an analog of $\Omega^\x(EG\times_GX)$. Let $X$ and $\g$ be as in \[inv\]. The construction of the equivariant cohomology presented here is the so-called Cartan model of $\Omega^\x(EG\times_GX)$. We obtain a complex with an action of $S^\x=H^*(BG)$. The cohomology is equal to $H^*_G(X)$.
Koszul duality
===============
By [@GKM], §8.5 the following functor $h:D^{^+}(S^\x)\rightarrow D^{^+}(\Lambda_\x)$ is an equivalence of categories: $$h(A^\x)={\rm Hom}_k(\Lambda_\x,A^\x)=(\L)^\g\o A^\x$$ with differential $$d_h((\xi_1\wedge\dots\wedge\xi_n)\o a)=$$ $$=
\sum_j (-1)^{j+1}(\xi_1\wedge\dots
\vee^j\dots\wedge\xi_n)\o \widetilde\xi_j
a+(-1)^n(\xi_1\wedge\dots\wedge\xi_n)\o da\,,$$ where $\xi_j$’s are primitive.
\[Koszul Duality\] \[dual\] Let $\g$ be a reductive Lie algebra. Suppose $\M$ is an object of $K^{^+}(\g)$ then in $D^{^+}(\Lambda_\x)$ $$h((\M )_\g)\simeq (\M )^\g\,.$$
[Proof.]{} The action of $\g$ on $H^*(W(\g))$ is trivial and, since $\g$ is reductive, $W(\g)$ is semisimple. Thus by [@GHV], Th. V, p.172 the inclusion $$k\o (\M)^\g = W(\g)^\g\o
(\M)^\g \subset (W(\g)\o\M)^\g$$ is a quasi-isomorphism. We want to construct a quasi-isomorphism $\psi$ from $$h((\M )_\g)=(\L)^\g\o
(\Sg\o \M )^\g$$ to $$(W(\g)\o \M )^\g=(\Sg\o\L\o \M )^\g\,.$$ First we choose a linear map $\omega:P^*\rightarrow W(\g)$ satisfying the conditions of Proposition \[trans\]. We construct $\psi$ by the formula extending $\psi_0$ of Lemma \[Lemma\] with help of the distinguished transgression of §5. $$\psi((\xi_1\wedge\dots\wedge\xi_n)\o m)
=\omega(\xi_1)\dots\omega(\xi_n) \psi_0(m)\,.$$ It is well defined since $\left(W(\g)\o \M \right)^\g$ is $W(\g)^\g$–module and $$\psi_0(m)\in\left(W(\g)\o \M \right)_{basic}$$ $$\omega(\xi_1)\in W(\g)^\g\,.$$ The map $\psi$ commutes with $i_\lambda$ since the image of $\psi_0$ is horizontal. It commutes with the differential because $d_W(\omega(\xi))=\widetilde\xi\o
1$. This corrects an error in the proof of Lemma 17.6, [@GKM], where $\psi$ does not commute with the differential unless $\g$ is abelian.
We will check that $\psi$ is a quasi-isomorphism. Let’s filter both sides by $S^{ \geq i}(\g^*)$. Then the corresponding quotient complexes are $$Gr^S_i\psi:(\L)^\g\o (S^i\g^*\o \M
)^\g\longrightarrow (S^i\g^*\o\L\o \M )^\g\,.$$ The differential on the LHS is just $\epsilon\o 1\o d_M$ (where $\epsilon=(-1)^{{\rm
deg}\xi}$) and the differential on the RHS is $$1\o\dl \o 1
+ \sum_k ad^*_{\lambda_k} \o \lambda^k\wedge\cdot \o 1
+ 1\o \epsilon\o d_M\,.$$ The map $Gr^S_i\psi$ equals $1\cdot(1\o{\bf T}_{|1\o\M})$. When we untwist it (i.e. we apply $\exp(\ii)={\bf T}^{-1}$ to the RHS) the differential takes the form $$1\o\dl \o 1
+ \sum_k ad^*_{\lambda_k} \o \lambda^k\wedge\cdot \o 1
+ 1\o \epsilon\o d_M+\sum_k 1\o\lambda^k\wedge\cdot \o
\Ll_{\lambda_k}\,.$$ Since we stay in the invariant subcomplex the differential equals $$1\o\dl \o 1
- \sum_k 1\o \lambda^k\wedge ad^*_{\lambda_k} \o 1
+ 1\o \epsilon\o d_M\,.$$ Moreover $\sum_k 1\o \lambda^k\wedge
ad^*_{\lambda_k}=2\dl$, thus the differential on the RHS is $$-1\o\dl\o 1+1\o \epsilon\o d_M\,.$$ The cohomology of the LHS is $$H^*\left(\left(\L\right)^\g\o \left(S^i\g^*\o
\M\right)^\g\right) =\left(\L\right)^\g\o H^*\left(\left(S^i\g^*\o
\M \right)^\g\right)$$ and the cohomology of the RHS is $$H^*\left(\left(S^i\g^*\o \L\o \M \right)^\g\right)=
H^*\left(\left(\L\right)^\g\right)\o H^*\left(\left(S^i\g^*\o
\M\right)^\g\right)$$ again by [@GHV], Th. V, p.172, since the action on $H^*(\L)$ is trivial and $\L$ is semisimple. Thus cohomology of the graded complexes are the same. The conclusion of \[dual\] follows. $\Box$
Let $X$ and $\g$ be as in \[inv\]. Following [@GKM] let us explain the meaning of Koszul duality. The invariant and the equivariant cohomology of $X$ are defined on the level of derived categories. Theorem \[dual\] gives a procedure to reconstruct the invariant cohomology from the equivariant cohomology. Since this duality is an isomorphism of categories the invariant cohomology determines the equivariant cohomology as well. The corresponding statement on the level of graded modules over $\Lambda_\x$ and $S^\x$ is not true as an easy example in [@GKM] shows. One cannot recover $H^*_G(X)$ from $H^*(X)$ with an action of $H_*(G)$ even in the case $X=S^3$, $G=S^1$.
[GKM]{} A. Alekseyev, E. Meinrenken: [*The non-commutative Weil Algebra*]{}, Inv. Math. [**139**]{}, 135–172 (2000)
C. Allday, V. Puppe: [*On a conjecture of Goresky, Kottwitz and MacPherson*]{}, Canad. J. Math. [**51**]{} (1999) no. 1, 3–9.
J. Bernstein, V. Lunts: [*Equivariant Sheaves and Functors*]{}, Lecture Notes in Mathematics Vol. 1578, Springer Verlag N.Y. (1994)
H. Cartan: [*Notion d’algèbre differentielle; applications aux groupes de Lie et aux variétes ou opère un groupe de Lie*]{}, Colloque de Topologie (espaces fibrés) Bruxelles 1950 C.R.B.M, Paris (1951)
W. Greub, S. Halperin, S, Vanstone: [*Curvature, Connections and Cohomology*]{}, vol. III Academic Press New York. (1976)
M. Goresky, R. Kottwitz, R. MacPherson: [*Equivariant cohomology, Koszul duality and the localization theorem*]{}, Inv. Math. [**131**]{}, (1998) 25–83
V. Gulemin, S. Sternberg: [*Supersymetry and Equivariant de Rham Cohomology*]{}, Springer Verlag, Berlin-Heidelberg-New York 1999
D. Husemoller, J.C. Moore, J. Stasheff: [*Differential homological algebra and homogeneous spaces*]{}, J. Pure Appl. Algebra [**5**]{} (1974), 113–185.
V. Mathai, D. Quillen: [*Superconnections, Thom classes, and equivariant differential forms*]{}, Topology [**25**]{} (1986), no. 1, 85–110.
C. Weibel: [*An Introduction to Homological Algebra*]{}, Cambridge University Press. Cambridge (1994)
[^1]: Supported by KBN 2P03A 00218 grant.
|
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"pile_set_name": "ArXiv"
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|
---
abstract: 'We will be defining a type of perverse equivalence that always corresponds to a derived equivalence with two-term tilting complexes. We are going to show that the tilting considered by Okuyama in [@Okuyama] and Yoshii in [@Yoshii] for the proof of Broué’s conjecture for $SL(2,q)$ in defining characteristic is a composition of such perverse equivalences.'
author:
- William Wong
title: 'Perverse Equivalence in $SL(2,q)$'
---
Introduction
============
Broué’s Conjecture [@Broue]\[6.2, Question\] is a very important focal point of the block theory of finite groups:
Let $\operatorname{\mathbb{F}}$ be the algebraically closed field of characteristic $p>0$. Let $G$ be a finite group and $A$ be a block of $\operatorname{\mathbb{F}\mathit{G}}$. If $A$ has an abelian defect group $D$, then $A$ is derived equivalent to a block $B$ of $N_G(D)$, its Brauer correspondent.
This conjecture has been studied by many and it is proved when $D$ is cyclic. However, for the general abelian case it is still a case-by-case verification. For $SL(2,q)$ in defining characteristic, the principal block of Broué’s conjecture is proven by Okuyama [@Okuyama] using a construction that differs from most of the other cases. Yoshii generalise Okuyama’s method to the non-principal block case in [@Yoshii]. As time passes, tools such as mutation or perverse equivalence comes into play. In particular, perverse equivalence seems to gather some geometrical information of a certain derived equivalence in some surprising way. Some such example will be Craven’s application of perverse equivalence with Lusztig’s L-function in [@Craven].
This paper is to do the similar job for $SL(2,q)$, by showing the derived equivalence between full defect blocks of $SL(2,q)$ and its Brauer correspondent contemplated by Okuyama (and Yoshii) is a composition of perverse equivalences. As a consequence we extending the application of perverse equivalence to Okuyama’s construction, under some condition on the projective modules of algebras involved.
In section 2 we introduce some known machinery needed for this article. This includes an introduction to the cochain complex Okuyama is utilising in his proof, perverse equivalence in total order (filtration) form and partial order form, and various information on the representation of $SL(2,q)$ we use for this paper. In section 3 we introduce a particular type of perverse equivalence which will yield two-term tilting complexes, but yet very poorly understood. Then we utilize some extra facts deduced by Okuyama in his approach to construct our string of perverse equivalence, arriving at the following theorem:
The derived equivalence between full defect blocks of $SL(2,q)$ in defining characteristics and its Brauer correspondent introduced by Okuyama and Yoshii are compositions of perverse equivalences.
In section 4 we discuss some findings along the construction and we present $SL(2,9)$ as an example.
Acknowledgement {#acknowledgement .unnumbered}
===============
The author thanks the support by JSPS International Research Fellowship during the development of the project.
Background and preliminaries {#sec:Bkgd}
============================
We use right modules for the exposition of this article in order to facilitate our reference to Okuyama’s and Yoshii’s paper. When $A$ is a block, let $A\operatorname{\hyp\mathrm{mod}}$ be the category of finitely-generated (fg) right $A$-modules, $\operatorname{\mathrm{st}}(A)$ be the stable category of fg right $A$-modules and $D^b(A)$ be the bounded derived category of fg right $A$-modules. For complexes we use the cochain notation, that is, differential maps in a complex are of degree 1.
Rickard-Okuyama tilting complex {#sec:OKTilt}
-------------------------------
We introduce Rouquier’s construction of a two-term bimodule complex and Okuyama’s treatment to make such complex a tilting complex of symmetric finite-dimensional (fd) algebras. We start with a symmetric fd algebra $A$. Instead of just relying on information from $A$ we can utilise another symmetric fd algebra $B$ which is stably equivalent of Morita type. Using the simple-minded system in $A$ obtained from simple $B$-modules, under certain conditions this creates a tilting complex of $A$ with endomorphism algebra $C$ (which is symmetric fd). This algebra $C$, under some further conditions, would have all simple $C$-modules either ’inherited’ from $A$ or from $B$ (we shall make this clear in the exposition). The hope is this process can be iterated until all simple $A$-modules have been replaced by simple $B$-modules. Then we can apply Linckelmann’s theorem (quoted below) [@Linckelmann] to show that we have created a derived equivalence between $A$ and $B$. Effectively we have managed to ’lift’ the stably equivalence of Morita type to derived equivalence between $A$ and $B$.
\[thm:Linc\] Let $A$ and $B$ be two self-injective algebras with no simple projective summands. If $A$ and $B$ are stably equivalent of Morita type which the equivalence sends simple $A$-modules to simple $B$-modules, then $A$ and $B$ are Morita equivalent.
Let us start be defining our terminology.
Let $A$ and $B$ be two algebras that are stably equivalent of Morita type via ${}_BM_A$, a $(B,A)$-bimodule with ${}_BM$ and $M_A$ be projective $B^{op}$- and $A$-module respectively. Let $T_z$, $S_z$, $z \in Z$ be a complete set of mutually non-isomorphic simple $B$- and $A$-modules respectively. Let $\tau_z:Q_z \to T_z$ be a projective cover of $T_z$ and $\pi_z:P_z \to T_z \operatorname{\otimes}_B M$ be a projective cover of $T_z \operatorname{\otimes}_B M$ as an $A$-module.
It is only conjectured that when $A$ and $B$ are stably equivalent (of Morita type) they should have the same number of simple modules (Auslander-Reiten Conjecture). We will assume this holds throughout the article (and it holds for all cases considered here).
Now there exists an $A$-homomorphism $\rho_z:P_z \to Q_z \operatorname{\otimes}_B M$ such that $\pi_z=(\tau_z \operatorname{\otimes}id_M)\circ \rho_z$ by the projectivity of $P_z$. Through the natural isomorphism $$\operatorname{\mathrm{Hom}}_A(P_z, Q_z \operatorname{\otimes}_B M) \operatorname{\cong}\operatorname{\mathrm{Hom}}_{B^{op}\operatorname{\otimes}A}(Q_z^*\operatorname{\otimes}P_z, M)$$ $\rho_z$ corresponds to a homomorphism $\delta_z:Q_z^*\operatorname{\otimes}P_z \to M$. Then $$\bigoplus_{z\in Z}\delta_z:\bigoplus_{z \in Z}(Q_z^*\operatorname{\otimes}P_z) \to M$$ is a projective cover of $M$.
Let $I$ be a fixed subset of $Z$ and define a cochain complex $M_{I}^{\bullet}$ of (B,A)-bimodules: $$M_{I}^{\bullet}=\big(\dots \to 0 \to \bigoplus_{i \in I}(Q_i^* \operatorname{\otimes}P_i) \xrightarrow{\bigoplus_{i \in I}\delta_i} M \to 0 \to \dots\big).$$ where $M$ is in degree 0.
\[thm:condAtilt\] As a complex of projective $A$-modules, $M^{\bullet}_I$ is a tilting complex of $A$ if and only if the following conditions hold: For all $i \in I$, $j \notin I$,
1. $\operatorname{\mathrm{Hom}}_A(T_j\operatorname{\otimes}_B M, \Omega(T_i \operatorname{\otimes}_B M))=0$.
2. Any $A$-homomorphism from $P_i$ to $T_j \operatorname{\otimes}_B M$ factors through $\pi$.
\[def:Okutilt\] When the conditions in \[thm:condAtilt\] hold we set an algebra $$C=\operatorname{\mathrm{End}}_{D^b(A)}(M_{I}^{\bullet}).$$ We call such construction of a new algebra as an *Okuyama tilt* of $A$ with respect to $(B, I)$.
The algebra $C$ is symmetric, finite-dimensional, with a left-$B$ action induced from $M^{\bullet}_{I}$, and is derived equivalent to $A$. Furthermore,
\[lem:OY\] There exists a direct summand $N^{\bullet}_{I}$ of $(C \operatorname{\otimes}_B M_{I}^\bullet)$ as $(C,A)$-bimodule such that:
1. $N^{\bullet}_{I}$ is a split-endomorphism two-sided tilting complex for $(C,A)$.
2. $N_I^\bullet \operatorname{\cong}M_I^\bullet$ as the homotopy equivalent classes of $(B,A)$-bimodule complex.
3. $N^{\bullet}_{I}$ has the form of a two-term complex, i.e. $\big(\cdots \to 0 \to Q \xrightarrow{\delta}N \to 0 \to \cdots\big)$.
On the algebras $A$, $B$ and $C$:
1. $-\operatorname{\otimes}_B C: B\operatorname{\hyp\mathrm{mod}}\to C\operatorname{\hyp\mathrm{mod}}$ and $\operatorname{\mathrm{Hom}}_A(N,-):A\operatorname{\hyp\mathrm{mod}}\to C\operatorname{\hyp\mathrm{mod}}$ give stable equivalences (of Morita type).
2. There is an algebra monomorphism from $B$ to $C$.
3. $N$ has no projective summand as $(C,A)$-bimodule.
4. The algebra $C$ has no projective summands as a $(B,C)$-bimodule.
By Okuyama, there are two ways to trace simple $C$-modules (\[prop:simAC\] and \[prop:simBC\]).
[@Okuyama 1.3]\[prop:simAC\] Let $S$ be a simple $A$-module. If $$\operatorname{\mathrm{Hom}}_A(S, T_i\operatorname{\otimes}_B M)=0=\operatorname{\mathrm{Hom}}_A(T_i\operatorname{\otimes}_B M, S)$$ for all $i \in I$, then $\operatorname{\mathrm{Hom}}_A(N,S)\operatorname{\cong}S\operatorname{\otimes}_A N^*$ is a simple $C$-module. Furthermore,
1. $\operatorname{\mathrm{Hom}}_{D^b(A)}(N^\bullet_{I},S)$ is the stalk complex $\operatorname{\mathrm{Hom}}_A(N,S)$ concentrated in degree zero.
2. $\operatorname{\mathrm{Hom}}_A(N,S)\operatorname{\cong}\operatorname{\mathrm{Hom}}_A(M,S)$ as $B$-modules.
\[def:KwrtI\] Define the set $K$ to be the subset of $Z$ such that either $\operatorname{\mathrm{Hom}}_A(S_k, T_i \operatorname{\otimes}_B M)$ or $\operatorname{\mathrm{Hom}}_A(T_i \operatorname{\otimes}_B M, S_k)$ is nonzero for some $i$. Or equivalently, $$Z\setminus K:=\{z\mid \operatorname{\mathrm{Hom}}_A(S_z, T_i \operatorname{\otimes}_B M)=0=\operatorname{\mathrm{Hom}}_A(T_i \operatorname{\otimes}_B M, S_z) \text{ for all }i \in I\}.$$
So for a fixed $I\subset Z$ the set $K$ (depend on $I$) is defined such that $I \subset K \subset Z$. (In line with [@Okuyama] and [@Yoshii].) Proposition \[prop:simAC\] traced a correspondence of simple $A$-module indexed by $Z\setminus K$ with simple $C$-module (which we indexed via the correspondence). We need to obtain the rest of simple $C$-modules. The original Proposition applied in [@Okuyama] is as follows:
For $i \in I$, if $dim_{\operatorname{\mathbb{F}}}\operatorname{\mathrm{Hom}}_A(\Omega(T_{i}\operatorname{\otimes}_B M),\Omega(T_{i'}\operatorname{\otimes}_B M))=\delta_{ii'}$ for all $i' \in I$, then $T_i \operatorname{\otimes}_B M$ is a simple $C$-module.
For $j \notin I$, if $dim_{\operatorname{\mathbb{F}}}\operatorname{\mathrm{Hom}}_A(T_{j}\operatorname{\otimes}_B M,T_{j'}\operatorname{\otimes}_B M)=\delta_{jj'}$ for all $j' \notin I$, then $T_j \operatorname{\otimes}_B C$ is a simple $C$-module.
However, the condition in (2) above is not fulfilled in some of our situations. We instead use the following variation:
\[prop:simBC\]With simple $C$-modules known from \[prop:simAC\], assume $K\setminus I$ has only one element,
1. For $i \in I$, if $dim_{\operatorname{\mathbb{F}}}\operatorname{\mathrm{Hom}}_A(\Omega(T_{i}\operatorname{\otimes}_B M),\Omega(T_{i'}\operatorname{\otimes}_B M))=\delta_{ii'}$ for all $i' \in I$, then $T_i \operatorname{\otimes}_B M$ is a simple $C$-module.
2. For (the only) $j \in K\setminus I$, if $dim_{\operatorname{\mathbb{F}}}\operatorname{\mathrm{Hom}}_A(T_{j}\operatorname{\otimes}_B M,T_{j}\operatorname{\otimes}_B M)=1$, then $T_j \operatorname{\otimes}_B C$ is a simple $C$-module.
Part (1) is exactly the same as [@Okuyama]\[1.4\] and we now check (2). There is only one remaining simple $C$-module to be found. Consider $j \in K\setminus I$, $z \in Z\setminus K$, we first check the homomorphisms $$\operatorname{\mathrm{Hom}}_{C}(T_j\operatorname{\otimes}_B C, \operatorname{\mathrm{Hom}}_A(N,S_z)) \text{ and }\operatorname{\mathrm{Hom}}_{C}(\operatorname{\mathrm{Hom}}_A(N,S_z), T_j\operatorname{\otimes}_B C)$$are zero. By tensor-hom adjunction, $$\begin{aligned}
\operatorname{\mathrm{Hom}}_C(T_j\operatorname{\otimes}_B C, \operatorname{\mathrm{Hom}}_A(N,S_z))&\operatorname{\cong}\operatorname{\mathrm{Hom}}_A(T_j\operatorname{\otimes}_B C \operatorname{\otimes}_C N, S_z)\\
&\operatorname{\cong}\operatorname{\mathrm{Hom}}_A(T_j\operatorname{\otimes}M, S_z)=0, \text{ and }\\
\operatorname{\mathrm{Hom}}_{C}(\operatorname{\mathrm{Hom}}_A(N,S_z), T_j\operatorname{\otimes}_B C)&\operatorname{\cong}\operatorname{\mathrm{Hom}}_{C}(S_z \operatorname{\otimes}_A N^*, T_j\operatorname{\otimes}_B C)\\
&\operatorname{\cong}\operatorname{\mathrm{Hom}}_A(S_z, \operatorname{\mathrm{Hom}}_C(N^*, T_j\operatorname{\otimes}_B C))\\
&\operatorname{\cong}\operatorname{\mathrm{Hom}}_A(S_z, T_j\operatorname{\otimes}_B C \operatorname{\otimes}_C N))\\
&\operatorname{\cong}\operatorname{\mathrm{Hom}}_A(S_z, T_j \operatorname{\otimes}_B M)=0
\end{aligned}$$ because of $z \in Z\setminus K$. So the top and socle of $T_j\operatorname{\otimes}_B C$ is isomorphic to the missing simple $C$-module. Now the condition $dim_{\operatorname{\mathbb{F}}}\operatorname{\mathrm{Hom}}_A(T_{j}\operatorname{\otimes}_B M,T_{j}\operatorname{\otimes}_B M)=1$ shows there is only one copy of the said simple $C$-module. Hence $T_j\operatorname{\otimes}_B C$ is the missing simple $C$-module we are looking for.
In the situation we will encounter, all simple $C$-modules arises either from simple $A$-modules via \[prop:simAC\] or from simple $B$-modules via \[prop:simBC\], respecting the derived equivalence and the stable equivalence respectively. This observation is the basis of applying theorem \[thm:Linc\] to finish the proof for Okuyama and Yoshii. For this article, the same observation is the hint of applying alternating perverse equivalence, which we will define in \[def:simalt\].
Perverse Equivalence {#sec:perveq}
--------------------
To arrive at alternating perverse equivalence we first introduce perverse equivalence in general. The notion is introduced in [@ChuangRouquier]. A very brief summary would be ’equivalence of triangulated categories by shifted Serre subcategories’. For readers well-versed in mutation it can be understand as iterative simultaneous mutations with gradually smaller vertex set. We shall apply these to the equivalence between module categories of symmetric fd algebras. In this case, (assuming Krull-Schmidt property,) the Serre subcategories of such category are one-one correspondent with subset of simple modules. We first define some terminology.
Let $$\emptyset = {\mathbbmss{S}}_{-1} \subset {\mathbbmss{S}}_0 \subset \dots \subset {\mathbbmss{S}}_n={\mathbbmss{S}}$$be a chain of subsets of the set of non-isomorphic simple modules ${\mathbbmss{S}}$. Denote by ${\mathbbmss{S}}^-_i$ the set ${\mathbbmss{S}}_i-{\mathbbmss{S}}_{i-1}$. We say an element $S$ *belongs to filtrate-$i$* if $S \in S^-_i$.
We use the following definition for perverse equivalence, by Dreyfus-Schmidt in [@Leo].
\[def:perv\] Let $A$ and $B$ be two symmetric finite dimensional algebras, $${\mathbbmss{S}}_\bullet=(\emptyset={\mathbbmss{S}}_{-1}\subset {\mathbbmss{S}}_0\subset \dots \subset {\mathbbmss{S}}_n={\mathbbmss{S}}) \text{ and }\operatorname{\mathbbmss{T}}_\bullet=(\emptyset=\operatorname{\mathbbmss{T}}_{-1}\subset \operatorname{\mathbbmss{T}}_0 \subset \dots \subset \operatorname{\mathbbmss{T}}_n=\operatorname{\mathbbmss{T}})$$ be filtrations of the isomorphism class of simple $A$- and $B$-modules respectively. Let $\pi:\{0,\dots,r\} \to \operatorname{\mathbb{Z}}$ be a function. An equivalence $F:D^b(A) \xrightarrow{\sim} D^b(B)$ is *(filtered) perverse relative to $({\mathbbmss{S}}_\bullet, \operatorname{\mathbbmss{T}}_\bullet, \pi)$*, if for every $i$ with $0 \leq i \leq n$ the following holds.
- Given $S \in {\mathbbmss{S}}^-_i$, the composition factors of $H^m(F(S))$ are in $\operatorname{\mathbbmss{T}}_{i-1}$ for $m\neq -\pi(i)$ and there is a filtration $L_1 \subset L_2 \subset H^{-\pi(i)}(F(S))$ such that the composition factors of $L_1$ and of $H^{-\pi(i)}(F(S))/L_2$ are in $\operatorname{\mathbbmss{T}}_{i-1}$ and those of $L_2/L_1$ are in $\operatorname{\mathbbmss{T}}^-_i$.
- The map $S \to L_2/L_1$ induces a bijection ${\mathbbmss{S}}^-_i \xrightarrow{\sim} \operatorname{\mathbbmss{T}}^-_i$. Hence there is an induced bijection of simple modules $\beta:{\mathbbmss{S}}\to \operatorname{\mathbbmss{T}}$.
The filtration ${\mathbbmss{S}}_\bullet$ and the induced bijection will decide the filtration $\operatorname{\mathbbmss{T}}_\bullet$ (by $\beta({\mathbbmss{S}}_i)$, $0 \leq i \leq n$), thus we will not explicitly write down $\operatorname{\mathbbmss{T}}_\bullet$ thereafter.
Given an equivalence perverse relative to a certain filtration, the composition factors of the homology might belong to a smaller Serre subcategory. This correspond to a smaller subset of simple modules than the immediate subset in the filtration. This can be written into a partial order, coarser than the one given by filtration. If the homology of an equivalence is completely known, then the coarsest order available is ’the bare minimum of relations’ required for this equivalence. To take benefit of this we introduce the notion of poset perverse equivalence, defined also in [@Leo].
\[def:poset\] Let $A$, $B$ be two symmetric finite dimensional algebras, with ${\mathbbmss{S}}$ and ${\mathbbmss{S}}'$ the set of their non-isomorphic simple modules. A derived equivalence $F:D^b(A)\to D^b(A')$ is perverse relative to $({\mathbbmss{S}}, \prec, \pi)$, where $({\mathbbmss{S}}, \prec)$ is a poset structure on ${\mathbbmss{S}}$ and $\pi:{\mathbbmss{S}}\to \operatorname{\mathbb{Z}}$, if and only if
1. There is a one-to-one correspondence $\beta_F:{\mathbbmss{S}}\to {\mathbbmss{S}}'$.
2. Define $S_{\prec}=\{T \in {\mathbbmss{S}}\mid T\prec S\}$. The composition factors of $H^r(F(S))$ are in $\beta_F(S_{\prec})$ for $r\neq -\pi(S)$ and there is a filtration $L_1 \subset L_2 \subset H^{-\pi(S)}(F(S))$ such that the composition factors of $L_1$ and of $H^{-\pi(S)}(F(S))/L_2$ are in $\beta_F(S_{\prec})$ and $L_2/L_1$ is isomorphic to $\beta_F(S)$.
The two notion introduced care about different aspects of a derived equivalence (that is perverse). The filtered perverse equivalence is mainly about existence, as given some filtration and function one can certainly create an equivalence perverse relative to that filtration and function. This does not hold for poset perverse equivalence, which emphasis the actual interaction of Serre subcategories that is being involved in a known equivalence. We list out some properties of both notion. The merits we mentioned can be seen in \[prop:filtperv\](5) and \[prop:poperv\](3) respectively.
\[prop:filtperv\] Let $F: D^b(A) \to D^b(B)$ be filtered perverse relative to $({\mathbbmss{S}}_\bullet, {\mathbbmss{S}}'_\bullet, \pi)$.
1. (reversibility) $F^{-1}$ is perverse relative to $({\mathbbmss{S}}'_\bullet, {\mathbbmss{S}}_\bullet, -\pi)$.
2. (composability) Let $F':D^b(B) \to D^b(C)$ be perverse relative to $({\mathbbmss{S}}'_\bullet, {\mathbbmss{S}}''_\bullet, \pi')$, then $F' \circ F$ is perverse relative to $({\mathbbmss{S}}_\bullet, {\mathbbmss{S}}''_\bullet, \pi+\pi')$.
3. (refineability) Let $\tilde{{\mathbbmss{S}}}_\bullet=(0=\tilde{{\mathbbmss{S}}}_{-1} \subset \dots \subset \tilde{{\mathbbmss{S}}}_{\tilde{r}}$) be a refinement of ${\mathbbmss{S}}_\bullet$. Define the weakly increasing map $f:\{0,\dots,\tilde{r}\} \to \{0,\dots,r\}$ such that $\tilde{{\mathbbmss{S}}}_\bullet$ collapses to ${\mathbbmss{S}}_\bullet$ under $f$ (i.e. ${\mathbbmss{S}}_{f(i)-1} \subset \tilde{{\mathbbmss{S}}}_i \subset {\mathbbmss{S}}_{f(i)}$). Then $F$ is perverse relative to $(\tilde{{\mathbbmss{S}}}_\bullet, \pi\circ f)$.
4. If $\pi=0$ then $F$ restricts to a Morita equivalence of $A$ and $B$.
5. The information $({\mathbbmss{S}}_\bullet, \pi)$ determines $B$ up to Morita equivalence.
\[prop:poperv\] Let $F: D^b(A) \to D^b(B)$ be poset perverse relative to $({\mathbbmss{S}}, \prec, \pi)$.
1. (reversibility) $F^{-1}$ is perverse relative to $(\beta({\mathbbmss{S}}), \beta(\prec), -\pi)$.
2. (composability) Let $F':D^b(B) \to D^b(C)$ be perverse relative to $(\beta({\mathbbmss{S}}), \beta(\prec), \pi')$, then $F' \circ F$ is perverse relative to $({\mathbbmss{S}}, \prec, \pi+\pi')$.
3. (refineability) Let $\prec'$ be a finer partial order (i.e. $x \prec y$ induces $x \prec' y$). Then $F$ is perverse relative to $({\mathbbmss{S}}, \prec', \pi)$.
4. If $\pi=0$ then $F$ restricts to a Morita equivalence of $A$ and $B$.
where $\beta(\prec)$ is the partial order induced by $\beta$. That is, $\beta(S)\prec\beta(T)$ if and only if $S \prec T$.
Note that given a derived equivalence which is a poset perverse equivalence, refining a partial order to a compatible total order we can obtain a filtered perverse equivalence. On the other hand, if more information on the homology of a derived equivalence is known, one can find a coarsest order to describe this equivalence. We give an example here to illustrate the definitions. In particular this is an example of *elementary perverse equivalence*. See [@ChuangRouquier] for further details on this terminology.
\[ex:first\] Let $A$ be the principal block of $kA_5$ with $k$ algebraically closed of characteristic 2. There are 3 non-isomorphic simple $A$-modules: The trivial module $k$, two modules $V$, $W$ which are two-dimensional. Their corresponding indecomposable projective covers have Loewy series as follows: $$P_k=\begin{matrix} k \\ V \quad W \\ k \quad k \\ W \quad V \\ k \end{matrix} \qquad \qquad P_V=\begin{matrix} V\\k\\W\\k\\V \end{matrix} \qquad \qquad P_W=\begin{matrix} W\\k\\V\\k\\W \end{matrix}.$$ In fact this is a Brauer graph algebra. We consider mutation at $\{V, W\}$. The summands of the tilting complex are given by: $$P_k \qquad \oplus \qquad P_V \to P_k \qquad \oplus \qquad P_W \to P_k$$ where the rightmost term is in degree 0. It yields a correspondence of simple $A$-modules: $$k \mapsto \begin{matrix}k\\V\quad W\end{matrix}, \qquad V\mapsto V[1], \qquad W\mapsto W[1]$$ where \[1\] is the shift (in $D^b(A\operatorname{\hyp\mathrm{mod}})$). This derived equivalence is perverse with respect to the filtration $\big(\emptyset \subset \{V,W\} \subset \{k,V,W\}\big)$ and $\pi(i)=1-i$ for $i=0,1$.\
By the image of simple $A$-modules (or the projective summands), define a partial order $\prec$ on $\{k,V,W\}$ by only $V \prec k$ and $W \prec k$. Also a function $\pi':k\mapsto 0, V\mapsto 1, W\mapsto 1$. Then this derived equivalence is also perverse with respect to $(\{k,V,W\}, \prec, \pi')$.
In fact, due to Rickard, the heart of the new $t$-structure introduced by the equivalence is isomorphic to $kA_4$-mod.
This is also the first (non-trivial) example of our (forthcoming) construction, where $A_5 \operatorname{\cong}SL(2,4)$ and $A_4$ can be chosen as the normaliser of a Sylow-2 subgroup of $A_5$, which is the Klein-4 group.
Representation of SL(2,q)
-------------------------
First we lay down some well-known facts for the representation theory of $SL(2,q)$ in defining characteristics.
Let $G=SL(2,q)$, $q=p^n$, the base field $\operatorname{\mathbb{F}}$ is an algebraically closed field of characteristic $p$. Let $V$ be the natural $G$-module, $V^{(r)}=\mathrm{Sym}^r(V)$ be the $r^{th}$ symmetric product of $V$. Let $\sigma$ be the Frobenius map on $\operatorname{\mathbb{F}}$ and $V_m$ be the $m^{th}$ Frobenius twist of $V$. Then the tensor product $$S_z= \bigotimes_{i=0}^{n-1} V_i^{(z_i)}$$ where $0 \leq z \leq q-1$ and $z=\sum_{i=1}^{n}z_i p^{i-1}$ written as $p$-adic number, form the complete set of non-isomorphic simple $\operatorname{\mathbb{F}\mathit{G}}$-modules. (c.f. [@Jantzen II 3.17]) The simple $\operatorname{\mathbb{F}\mathit{G}}$-modules fall into 3 blocks when $p$ is odd:
1. A defect zero block consisting of the simple module $S_{q-1}$.
2. A (full defect) principal block consisting of simple modules $S_a$ with $a$ even.
3. A (full defect) non-principal block consisting of simple modules $S_a$ with $a$ odd.
When $p=2$ there are only two blocks, a defect zero block with $S_{q-1}$ and a full defect block with all the rest of simple modules. We take $A$ to be the direct sum of a copy each of the non-semisimple block(s).
Take the Sylow $p$-subgroup $P$ of $G$ consisting of elements of the form $\begin{pmatrix} 1 & * \\ 0 & 1 \end{pmatrix}$. It is a defect group of $A$. We have $H=N_G(P)$ is the Borel subgroup $\begin{pmatrix} \alpha^{-1} & * \\ 0 & \alpha \end{pmatrix}$. Let $T_b$ be the one-dimensional $kH$-module with non-zero vector $t$ such that $$t.\begin{pmatrix} \alpha^{-1} & * \\ 0 & \alpha \end{pmatrix}=\alpha^b(t).$$ These $T_b$ for $0 \leq b \leq q-2$ form a complete set of isomorphism classes of simple $kH$-modules (note $T_b \operatorname{\cong}T_{q-1+b}$).
If $p$ is odd, then $kH$ is split into two blocks: One with $T_b$’s of even subscript (principal block) and one with $T_b$’s of odd subscript (non-principal block). Again when $p=2$ these two blocks are merged into one. In any case we take the block $B$ to be $kH$, which is the Brauer correspondent of $A$. The restriction and induction between $A$-modules and $B$-modules induces a equivalence of their stable module categories. (This can be deduced from the Green correspondence. See, for example [@Alperin 10.1].) Okuyama and Yoshii then utilise some combinatorial properties of the composition factors of the induction and restriction, originated in [@HSW 6], to construct new algebras by tilting.
Data for $SL(2,q)$
------------------
Now we consider together Okuyama and Yoshii’s proof of derived equivalence between $A$ and $B$. We paraphrase and combine their definitions here. First of all, we list out some conditions that is satisfied by some projective modules of the non-semisimple blocks of $SL(2,q)$ (and later some projective modules of all the intermediate algebras).
\[def:TPC\][@Okuyama] Given an algebra $A$, stably equivalent to a fixed algebra $B$ of Morita type via $M$, a $(B,A)$-bimodule. Let $S_z$ and $T_z$ ($z \in Z$) be the complete set of non-isomorphic simple modules of $A$ and $B$ respectively. $I \subset Z$ are some chosen index. We say the pair $(A, I)$ satisfies *thin projective condition*, if the following holds for $A, I$ and $K$ (defined as \[def:KwrtI\]):
1. For $k \in K$, $T_k \operatorname{\otimes}_B M$ is not simple, and
1. $\operatorname{\mathrm{soc}}(\operatorname{\mathrm{Hom}}_A(M, S_k))\operatorname{\cong}T_k \qquad \operatorname{\mathrm{Top}}(\operatorname{\mathrm{Hom}}_A(M, S_k))\operatorname{\cong}T_{\tilde{k}}$.
2. $\operatorname{\mathrm{soc}}(T_k \operatorname{\otimes}_B M)\operatorname{\cong}S_{\tilde{k}} \;\;\qquad \qquad \operatorname{\mathrm{Top}}(T_k \operatorname{\otimes}_B M)\operatorname{\cong}S_k$.
2. Let $P_z$ be the (minimal, indecomposable) projective cover of $S_z$, then
1. For $i \in I$, $\dim_{\operatorname{\mathbb{F}}}\operatorname{\mathrm{Hom}}_A(P_i, P_i)=2$ if $\tilde{i} \neq i$\
$\dim_{\operatorname{\mathbb{F}}}\operatorname{\mathrm{Hom}}_A(P_i, P_i)=3$ if $\tilde{i} = i$
2. For $i, l \in I$, $\operatorname{\mathrm{Hom}}_A(P_i, P_l)=0$ if $l\neq i$ and $l \neq \tilde{i}$.
3. For $i \in I$, $\dim_{\operatorname{\mathbb{F}}}\operatorname{\mathrm{Hom}}_A(P_i, P_{\tilde{i}})=1$ if $\tilde{i} \neq i$.
3. For $i \in I$, $\operatorname{\mathrm{Hom}}_A(P_i, T_z\operatorname{\otimes}_B M)=0$ if $z \notin K$.
We called this ’thin projective condition’ as in our case, the $P_i$, $i \in I$ are the ones with ’relatively few (and controllable) composition factors’ in their Loewy layers.
\[def:orbits\] Let ${\mathbbmss{S}}$ and $\operatorname{\mathbbmss{T}}$ be the set of non-isomorphic simple $A$-modules and $B$-modules respectively. They share a common indexing set and we define such as $Z:=\{0,1,\ldots,q-2\}$. The *Frobenius action* $\sigma$ on $z\in Z$ is $$\sigma(z)=pz \mod{q-1}.$$ and the *sign action* $\tilde{.}$ on $z \in Z$ is $$\tilde{z}=-z \mod{q-1}.$$ The actions allow us to define *Frobenius orbits* and *signed Frobenius orbits* of $Z$.
The sign action and Frobenius action are commutative, hence each signed Frobenius orbits may split into two Frobenius orbits. We extend the use of $\sigma$ onto $Z$ since we have $$\sigma(S_z)=S_{pz}\text{ and }\sigma(T_z)= T_{pz}.$$ The sign action comes from the dual of simple $B$-modules $T_z$: $$T^*_z:=\operatorname{\mathrm{Hom}}_{k}(T_z,k)\operatorname{\cong}T_{\tilde{z}}.$$
\[def:IJK\] Define an indexing on signed Frobenius orbits as follows: Let $K_{-1},\ldots,K_r$ be the partition of $Z$ into signed Frobenius orbits, such that
1. $K_{-1}=\{0\}$.
2. $K_t$ is the set of all elements in a signed Frobenius orbit which contains the smallest number $z \notin \bigcup^{t-1}_{s=-1}K_s$. ($z \in K_t$)
3. $Z=\bigcup^r_{s=-1}K_s$.
We define a Frobenius orbit $I_t \in K_t$, $0 \leq t \leq r$ such that the largest number in $K_t$ belongs to $I_t$. We also define $J_t=K_t\setminus I_t$. We denote by $K_{\leq t}:=\cup_{s=-1}^{t}K_s$ the union of $K_s$ for $-1 \leq s \leq t$. Likewise for $I$ and $J$.
$J_t$ is empty if and only if $z$ and $\tilde{z}$ is in the same Frobenius orbit $I_t$ for any $z \in I_t$.
Let $p=3$, $q=9=3^2$. Then the partition into signed Frobenius orbit is: $$K_{-1}=\{0\}\qquad K_0=\{1,3,5,7\} \qquad K_1=\{2,6\} \qquad K_2=\{4\}.$$ The $I_t$, $0 \leq t \leq 2$ are $$I_0=\{7,5\}, \qquad I_1=\{6,2\}, \qquad I_2=\{4\}.$$
Okuyama and Yoshii use these data to define successive algebras as Okuyama tilt with respect to $(B, I_t)$.
\[def:OYString\] Let $A$ be a (sum of) full defect block(s) of $kSL(2,q)$ and $B$ its (their) Brauer correspondent(s). Define algebras $A_t$ for $0 \leq t \leq r$ such that
1. $A_0=A$.
2. $A_{t}=End_{D^b(A_{t-1})}(M_{I_t}^{\bullet})$ for $1 \leq t \leq r$.
For odd primes, each of the set $I_t$ contains only elements of the same parity. We regard the other algebra as unchanged.
The above definitions are well-defined since Okuyama and Yoshii show, in their respective papers, the following proposition:
\[prop:OY\][@Okuyama][@Yoshii] Let $t$ be any integer with $0 \leq t \leq r-1$.
1. $A_t$ is derived equivalent to $A$.
2. There exists a unitary $k$-algebra monomorphism from $B$ to $A_t$.
3. $A_t$ induces a stable equivalence of Morita type between $A_t$ and $B$. Moreover $A_t$ has no non-zero projective summands as $(B, A_t)$-bimodule.
4. $A\operatorname{\otimes}_B A_t$ is isomorphic to a direct sum of a nonprojective indecomposable module, denoted by $L_t$ and a projective module.
5. Set $$S_{t,z}=\begin{cases}
T_z \operatorname{\otimes}_B A_t &\text{ for } z \in K_s\text{ with }s \leq t-1\\
S_z \operatorname{\otimes}_A L_t &\text{ for } z \in K_s\text{ with }s \geq t.\\
\end{cases}$$Then $S_{t,z}$, $z \in Z$ is the complete set of mutually non-isomorphic simple $A_t$-modules.
6. The pair $(A_t, I_t)$ satisfies thin projective condition. (c.f. definition \[def:TPC\]).
Fact (6) further deduced that theorem \[thm:condAtilt\] holds hence it is valid to set $A_{t+1}$ as an Okuyama tilt of $A_t$ with respect to $(B, I_t)$. (c.f. definition \[def:Okutilt\])
To end this section, we remark that (5) gives a natural bijection between simple $A_{t_1}$-modules and simple $A_{t_2}$-modules ($0 \leq t_1, t_2 \leq r$) via their index (and their socles, in fact).
Main theorem and construction {#sec:main}
=============================
In this section we first define simply alternating perverse equivalence (in \[def:simalt\]) and characterise such equivalences. Then we show that, upon refining Okuyama’s construction for $SL(2,q)$ to either the set $I$ has one element or $K\setminus I$ is empty, we have each of these Okuyama’s tilt is simply alternating perverse. Then we compose these tilts to complete the proof of our main result (theorem \[thm:main\]).
Perversity of two degrees
-------------------------
Consider, for a certain filtration, the perverse function $\pi$ satisfies $\pi(i)=0$ when $i$ even and $\pi(i)=1$ when $i$ odd. We call this special case *alternating perverse equivalence*.
\[def:simalt\] Let $A$ and $A'$ be algebras with a derived equivalence $F:D^b(A) \to D^b(A')$. Let ${\mathbbmss{S}}$ be the complete set of non-isomorphic simple modules of $A$. If there exists filtration ${\mathbbmss{S}}_\bullet=(\emptyset={\mathbbmss{S}}_{-1}\subset {\mathbbmss{S}}_0 \subset \ldots \subset {\mathbbmss{S}}_r={\mathbbmss{S}})$ (of subsets of ${\mathbbmss{S}}$) and function $\pi$ that sends $i$ to $i$ modulo 2 for $0 \leq i \leq r$, then we say $F$ is an *alternating* equivalence. In particular, if $r=2$ (hence ${\mathbbmss{S}}_\bullet=({\mathbbmss{S}}_0 \subset {\mathbbmss{S}}_1 \subset {\mathbbmss{S}}_2={\mathbbmss{S}})$ and $\pi:\pi(0)=0; \pi(1)=1; \pi(2)=0$), then we say $F$ is a *simply alternating* equivalence. Or, $A$ and $A'$ is *simply alternating perverse equivalent* with respect to $({\mathbbmss{S}}_\bullet, \pi)$.
When $r=1$ this is just an elementary perverse equivalence. Usually the filtrations have to be reduced (i.e. ${\mathbbmss{S}}_i \neq {\mathbbmss{S}}_j$ for $i \neq j$). In this paper, for the convenience of presentation, when we consider simply alternating equivalence (i.e. $r=2$) we allow ${\mathbbmss{S}}_1={\mathbbmss{S}}_2$, and set the following convention:
\[conv:degen\] In the case ${\mathbbmss{S}}_1={\mathbbmss{S}}_2$, the simply alternating perverse equivalence degenerates to an elementary perverse equivalence. That is, if we say $F$ is perverse with respect to filtration ${\mathbbmss{S}}_\bullet={\mathbbmss{S}}_0 \subsetneq {\mathbbmss{S}}_1 = {\mathbbmss{S}}_2={\mathbbmss{S}}$ and function $\pi:\pi(0)=0; \pi(1)=1; \pi(2)=0$, then we mean $F$ is perverse with respect to filtration ${\mathbbmss{S}}_\bullet={\mathbbmss{S}}_0 \subsetneq {\mathbbmss{S}}_1={\mathbbmss{S}}$ and function $\pi:i \mapsto i$ for $i=0,1$.
We will only discuss simply alternating in this section. This is partly due to the description of the image of simple modules under general alternating equivalence is very complex to describe. One can build any alternating equivalence by composing simply alternating ones, see section \[sec:discussion\] for details. For symmetric algebras, simply alternating equivalence is characterised by the following:
\[thm:simalt\] Let $A$ and $A'$ be two symmetric algebras. Let ${\mathbbmss{S}}$ and ${\mathbbmss{S}}'$ be the sets of non-isomorphic simple $A$-modules and $A'$-modules, respectively. Let $${\mathbbmss{S}}_\bullet=({\mathbbmss{S}}_0 \subset {\mathbbmss{S}}_1 \subset {\mathbbmss{S}}_2={\mathbbmss{S}}) \text{ and } {\mathbbmss{S}}'_\bullet=({\mathbbmss{S}}'_0 \subset {\mathbbmss{S}}'_1 \subset {\mathbbmss{S}}'_2={\mathbbmss{S}}')$$ be filtrations of subsets of simple $A$-module and simple $A'$-module respectively. Suppose $F:D^b(A) \to D^b(A')$ is a simply alternating equivalence, with bijection of simple modules $S_z \leftrightarrow S'_z$. Then for $S_z \in {\mathbbmss{S}}^-_i$, $U'_z:=F(S_z) \in D^b(A')$ is:
1. $S'_z$ concentrated in degree 0 when $i=0$;
2. the largest quotient of $P'_z$ such that
- $\operatorname{\mathrm{Top}}(U'_z)=S'_z$ and
- all other composition factors are in ${\mathbbmss{S}}'_0$,
concentrated in degree $1$, when $i=1$;
3. the largest submodule $P'_z$ such that
- $\operatorname{\mathrm{soc}}(U'_z)=S'_z$.
- All other composition factors are in ${\mathbbmss{S}}'_1$.
- Composition factors of $\operatorname{\mathrm{Top}}(U'_z)$ are in ${\mathbbmss{S}}'^-_1$.
concentrated in degree 0, when $i=2$.
For $F^{-1}(S'_z)$, $S'_z \in {\mathbbmss{S}}'^-_i$ has image $U_z \in D^b(A)$ is
1. $S_z$ concentrated in degree 0 when $i=0$;
2. the largest submodule of $P_z$ such that
- $\operatorname{\mathrm{soc}}(U_z)=S_z$ and
- all other composition factors are in ${\mathbbmss{S}}_0$,
concentrated in degree $-1$, when $i=1$; ($U_z[-1]$ is the universal extension of $S_z$ by ${\mathbbmss{S}}_0$. See [@ChuangRouquier].)
3. the largest quotient of $P_z$ such that
- $\operatorname{\mathrm{Top}}(U_z)=S_z$,
- All other composition factors are in ${\mathbbmss{S}}_1$, and
- Composition factors of $\operatorname{\mathrm{soc}}(U_z)$ are in ${\mathbbmss{S}}^-_1$,
concentrated in degree 0, when $i=2$.
This perverse equivalence can be further decomposed as two (elementary) perverse equivalences: $$(\emptyset \subset {\mathbbmss{S}}_0 \subset {\mathbbmss{S}}),\qquad p_0:0 \mapsto 0, 1 \mapsto 1; \qquad \text{and} \qquad (\emptyset \subset {\mathbbmss{S}}_1 \subset {\mathbbmss{S}}), \qquad p_1:0 \mapsto 0, 1 \mapsto -1$$ The results of the theorem is essentially computed from this.
We will be using the inverse image later hence we shall just prove it for descriptions concerning $U_z$. First we prove the set $\{U_z\}$ is a simple-minded collection in $D^b(A\operatorname{\hyp\mathrm{mod}})$ using Rickard’s criterion, see [@Rickard3].
1. $\operatorname{\mathrm{Hom}}_{D^b(A\operatorname{\hyp\mathrm{mod}})}(U_x, U_y)=\delta_{xy}$:
1. If $S_x$ and $S_y$ belongs to same even(resp. odd)-numbered partition ${\mathbbmss{S}}^-_i$, then the top (resp. socle) of $U_x$ and $U_y$ has a unique composition factor $S_x$ and $S_y$. All other composition factors of $U_x$ and $U_y$ belongs to ${\mathbbmss{S}}_{i-1}$. Hence when $x \neq y$ there is no non-zero map since the factor $S_x\in {\mathbbmss{S}}^-_i$ does not exist in $U_y$ or vice versa. When $x=y$ it must induced an isomorphic map.
2. When $S_x \in {\mathbbmss{S}}^-_i$ is not in the same partition as $S_y \in {\mathbbmss{S}}^-_j$. Either
- both $i$ and $j$ are even, then $\{i,j\}=\{0,2\}$.
- either of $i$, $j$ is odd but not both.
For the first case, take $U=F(\bar{S})$ for the $\bar{S}$ in partition-2. Neither the top nor socle of $U$ have composition factors in ${\mathbbmss{S}}_0$. Hence given the $\tilde{S}$ in partition-0, we have $\operatorname{\mathrm{Hom}}_A(U, \tilde{S})=0=\operatorname{\mathrm{Hom}}_A(\tilde{S},U)$. For the second case, if there is a non-zero map, then it is only possible when $i$ is even and $j$ is odd (because of the degree of $U_x$ and $U_y$). Hence we have $j=1$. We consider the injective resolution $I^*_y$ of $U_y$, in particular the term $I^0_y$. If $i=0$, since $U_y$ is the universal extension of ${\mathbbmss{S}}_0$ at degree $-1$, $\operatorname{\mathrm{soc}}(I^0_y)$ have no composition factor in ${\mathbbmss{S}}_0$. Thus, no non-zero maps exist from $U_x=S_x \in {\mathbbmss{S}}_0$ to $I^0_y$. If $i=2$ and a non-zero map $f:U_x \to U_y$ in $D^b(A\operatorname{\hyp\mathrm{mod}})$ exists, we have a non-split short exact sequence $$0 \to U_y[-1] \to E \to U_x \to 0$$ in $A\operatorname{\hyp\mathrm{mod}}$. This is because $U_y[-1]$ is a stalk complex at degree 0 hence we have isomorphism $\operatorname{\mathrm{Hom}}_{D^b(A)}(U_x, U_y) \operatorname{\cong}\operatorname{\mathrm{Ext}}^1_A(U_x, U_y[-1])$. Let $E'$ be the largest module with composition factor only in ${\mathbbmss{S}}_0$ such that there exist non-split extension $$0 \to E'' \to E \to E' \to 0$$ in $A\operatorname{\hyp\mathrm{mod}}$.
Since $\operatorname{\mathrm{Hom}}_A(E', U_x)=0=\operatorname{\mathrm{Hom}}_A(U_x, E')$ ($E'$ has no composition factor in ${\mathbbmss{S}}_1^-$ and ${\mathbbmss{S}}_2^-$), the composite map $\varepsilon:U_y \to E \to E'$ maps surjectively into $E'$ while it is not bijective. Then we have (as picture) a non-split extension $0 \to \ker(\varepsilon) \to E'' \xrightarrow{\varepsilon} U_x$ with $\ker\varepsilon$ non-zero. Hence $E''$, while larger than $U_x$, also satisfy the description of (3), a contradiction to $U_x$’s assumed maximality.
2. $\operatorname{\mathrm{Hom}}_{D^b(A')}(U_x, U_y[l])=0$ for negative integer $l$: A non-zero map is only possible when $l=-1$ and $S_y \in {\mathbbmss{S}}^-_1$. When $S_x \in {\mathbbmss{S}}^-_0$ the socle of $U_y$ is not in ${\mathbbmss{S}}'_0$ hence no non-zero map. When $S_x \in {\mathbbmss{S}}^-_2$ there is no composition factor belongs to ${\mathbbmss{S}}_2$ in $U_y$ hence again no non-zero map is possible.
3. The set of all $U_x$ generates $D^b(A')$ as a triangulated category: It is obvious that we can generate all non-isomorphic simple $A'$-modules of each layer by triangulating $U_x \in {\mathbbmss{S}}'^-_i$ with the simple factors in ${\mathbbmss{S}}'_{i-1}$.
It is easy to check the generating criterion since every simple module can be iterated from the image with triangles involving lower filtration. Thus $U_x$’s are the image of simple modules under a derived equivalence. Since the composition factors of $U_x$ satisfy Definition \[def:perv\], hence the equivalence is perverse with function as given.
Note $\{U_x\mid S_x \in {\mathbbmss{S}}_1^-\}$ is a right-finite semibrick and $\{U_x\mid S_x \in {\mathbbmss{S}}\setminus{\mathbbmss{S}}_1^-\}$ is a left-finite semibrick.
We also describe the summands of the tilting complex of a simply alternating perverse equivalences.
\[thm:altpervproj\] Retain the notation of theorem (\[thm:simalt\]). Denote the minimal (indecomposable) projective cover of $S_z$, $z \in Z$ by $P_z$. Then the tilting complex $T$ of $A$ such that $\operatorname{\mathrm{End}}_{D^b(A)}(T)$ is Morita equivalent to $A'$ is given by the following summands for each $z$:
1. $P_{S_1^-} \to P_z$ for $z \in S_0$.
2. $P_z \to P_{S_2^-}$ for $z \in S_1^-$.
3. $0 \to P_z$ for $z \in S_2^-$.
where the second term is in degree 0, $P_X$ is a direct sum of some $P_x$’s for $x \in X$.
The summand $P_z \to P_{S_1^-}$ in (1) has no term involving $P_{S_2^-}$. Also, the term indicated by $P_{S_1^-}$ is not (necessary) the full approximation of $S_z$ by the additive closure of $P_{S_1^-}$. See Example \[ex:nonprincipal\] for such a case. The precise description of the factor $P_{S_1^-}$ should be the ’right $P_{S_1^-}$-approximation of the left-$P_{S_2^-}$-approximated’ factor of $S_z$.
We shall not prove this here since it is just an exercise of looking for the appropriate maps and cones in $K^b(\text{proj-}A)$ via left and right approximations.
A further breakdown of Okuyama’s string on SL(2,q)
--------------------------------------------------
Now we recall related notations from section 2.4. Our aim is to show, the algebra $A_t$ and $A_{t+1}$ in [@Okuyama] and [@Yoshii] is a composition of ’smaller’ Okuyama’s tilt. In this subsection we let $t$ be fixed.
Recall and continuing from definition \[def:IJK\]. Fix $t$, we further partition the set $K_t$ into $K_{t,0},\ldots,K_{t,d}$ for some $d$ (depending on $t$), such that $$K_{t,c}:=\{z,\tilde{z}\} \text{ for the largest } z \in I_t \setminus (\bigcup_{b=0}^{c-1} K_{t,b})\text{ for } 0 \leq c \leq d\text{, with } K_t=\bigcup_{c=0}^{d}K_{t,c}.$$ In other words, $K_{t,c}$ is the signed orbit that contains the largest number in $I_t$ yet to be partitioned. Define the sets $$I_{t,c}=K_{t,c} \cap I_t \text{ and }J_{t,c}=K_{t,c}\setminus I_{t,c}.$$ And define $$K_{\leq t,c}:=\bigcup_{s=1}^{t-1}K_s\cup\bigcup_{b=0}^{c}K_{t,b}.$$ Likewise for $I_{\leq t,c}$ and $J_{\leq t,c}$.
An example of this further partition of $Z$ is shown in \[ex:partition\]. Under the definition the sets $K_{t,c}$, $0 \leq c \leq d$, will contain at most two elements each. Furthermore, $I_{t,c}$ is non-empty and $J_{t,c}$ has at most one element.
\[def:WString\] Fix $t$, Construct inductively a string of algebra $A_{t,c}$, $0 \leq c < d$, such that
1. $A_{t,0}=A_t$.
2. $A_{t,c+1}=\operatorname{\mathrm{End}}_{D^b(A_{t,c})}(M_{I_{t,c}}^\bullet)$. That is, $A_{t,c+1}$ is the Okuyama’s tilt of $A_{t,c}$ with respect to $(B, I_{t,c})$.
Now we need to show this is a well-defined definition. This part is essentially a repetition of the approach by Okuyama in section 3 of [@Okuyama] or section 4 of [@Yoshii].
\[prop:W\] Fix an integer $t$. Let $c$ be any integer with $0 \leq c \leq d$. The algebra $A_{t,c}$ satisfies the following:
1. $A_{t,c}$ is derived equivalent to $A$.
2. There exists a (unitary) $k$-algebra monomorphism from $B$ to $A_{t,c}$ and we have $A_{t,c} \operatorname{\cong}B \oplus \text{ a projective (}B,B\text{)-bimodule}$. Hence $A_{t,c}$ induces stable equivalence of Morita type between $A_{t,c}$ and $B$. Furthermore $A_{t,c}$ as $(B,A_{t,c})$-bimodule has no nonzero projective summands.
3. $A \operatorname{\otimes}_B A_{t,c}$, as $(A, A_{t,c})$-bimodule, is isomorphic to a direct sum of a nonprojective indecomposable module, denoted by $L_{t,c}$, and a projective module.
4. Set $$S_{t,c,z}=\begin{cases}
T_z \operatorname{\otimes}_B A_{t,c}&\text{ if }z \in K_{\leq t, c-1}\\
S_z \operatorname{\otimes}_A L_{t,c}&\text{ if }z \in K_{\geq t, c}.
\end{cases}$$Then the set of $S_{t,c,z}$ for all $z \in Z$ is the complete set of non-isomorphic simple $A_{t,c}$-modules.
5. For $z \in J_{\leq t, c-1}$, every composition factor of $S_z \operatorname{\otimes}_A L_{t,c}$ is isomorphic to $S_{t,c,y}$ for some $y \in K_{\leq t, c-1}$.
6. $A_{t,c}$ and $I_{t,c}$ satisfies thin projective condition \[def:TPC\].
The proof is essentially a repetition of their respective proofs in loc. cit. with the appropriate alternation.
For $c=0$: (1,2,3,4,5) is directly from loc. cit. since $A_{t,0}=A_t$. Recall that $K_{\leq t, -1}=K_{\leq t}$. (Similarly for $J$.) Since we have a smaller set of $I$ (and hence $K$), the only non-trivial-to-check condition is (3) (in \[def:TPC\]) of (6). Let $l \in K_{t}\setminus K_{t,0}$ (i.e. $l \neq i, \tilde{i}$ for $i \in I_{t,0}$), if $l \in I_{t}$, consider a map in $\operatorname{\mathrm{Hom}}_A(P_i, T_l \operatorname{\otimes}_B M)$ must factor through the epimorphism $P_l \to T_l \operatorname{\otimes}_B M$. However, $l\neq i,\tilde{i}$ hence $\operatorname{\mathrm{Hom}}(P_i, P_l)=0$ (by (2b) in loc. cit.) forces $\operatorname{\mathrm{Hom}}_A(P_i, T_l \operatorname{\otimes}_B M)=0$. If $l \in J_{t}$, then a map in $\operatorname{\mathrm{Hom}}_A(P_i, T_l \operatorname{\otimes}_B M)$ extends to the injective hull of $T_l \operatorname{\otimes}_B M$, which is $P_{\tilde{l}}$ for $\tilde{l} \in I_{t}$. Apply (2b) in loc. cit. to see they are zero.\
Now the induction part: (1,2,3) comes naturally from the theory of Okuyama (c.f. section \[sec:OKTilt\]) (4,5,6) follows exactly as [@Yoshii].
Lastly we have to show this string of equivalence is essentially the same as a step constructed by Okuyama and Yoshii.
$A_{t,d+1}$ is Morita equivalent to $A_{t+1}$.
Consider the set of simple $A_{t+1}$-modules is being given as $$S_{t+1,z}=\begin{cases}
T_z \operatorname{\otimes}_B A_{t+1}&\text{ for }z \in K_{\leq t+1}\\
S_z \operatorname{\otimes}_A L_{t+1}&\text{ for }z \in K_{> t+1}
\end{cases}$$by \[prop:OY\]. Apply the functor $- \operatorname{\otimes}_{A_{t+1}}N^\bullet_{A_t}$ to see they are being correspondingly mapped from $$U_{t,z}:=\begin{cases}
T_z \operatorname{\otimes}_B A_{t}&\text{ for }z \in K_{\leq t+1}=K_{\leq t} \cup K_t\\
S_z \operatorname{\otimes}_A L_{t}&\text{ for }z \in K_{>t+1}
\end{cases}$$in $A_t$. We note the fact $U_{t,z}$ is simple for $z \in K_{\leq t} \cup K_{>t+1}$, by \[prop:OY\]. Now we send $U_{t,z}$ along the maps $- \operatorname{\otimes}_{A_{t,c}} N_{I_{t,c}}^\bullet$ for $0 \leq c \leq d$ to $A_{t, d+1}$. By \[prop:W\] all of these images of $U_{t,z}$ are simple $A_{t,d+1}$-modules. Hence we can conclude by \[thm:Linc\] that $A_{t,d+1}$ and $A_{t+1}$ is Morita equivalent.
In fact, consider each Okuyama’s tilt with respect to $(B, I_{t,c})$ in our case is one and the same as Okuyama’s tilt with respect to $(A_{t+1}, I_{t,c})$ we actually have an algebra isomorphism.
Now we are going to show that each of these ’smaller’ Okuyama tilt is a perverse equivalence. In fact, these tilts are simply alternating equivalences.
\[lem:main\] The derived equivalence $F_{t,c}:D^b(A_{t,c}) \to D^b(A_{t,c+1})$ for $0 \leq t \leq r$, $0 \leq c < d$ is a perverse equivalence with respect to $$I_{t,c,\bullet}=(\emptyset = {\mathbbmss{S}}_{t,c,-1} \subset {\mathbbmss{S}}_{t,c,0} \subset {\mathbbmss{S}}_{t,c,1} \subset {\mathbbmss{S}}_{t,c,2} = {\mathbbmss{S}}_{t,c}), \qquad \pi(i)=\begin{cases}
0 \text{ if } i \text{ is even} \\ -1 \text{ if } i \text{ is odd}.
\end{cases}$$where $$\begin{aligned}
&{\mathbbmss{S}}_{t,c,0}=\{S_{t,c,z}\mid z \in Z-K_{t,c}\}\\
&{\mathbbmss{S}}_{t,c,1}=\{S_{t,c,z}\mid z \in Z-J_{t,c}\}\qquad &({\mathbbmss{S}}_{t,c,1}^-=\{S_{t,c,z}\mid z \in I_{t,c}\})\\
\end{aligned}$$and when $J_{t,c}$ is empty, the equivalence degenerates as per \[conv:degen\].
We shall be showing $F_{t,c}^{-1}$ is perverse with respect to $I_{t,c,\bullet}$ (defined on $A_{t, c+1}$) and $\pi:\pi(0)=0; \pi(1)=1; \pi(2)=0$. Consider the simple $A_{t,c+1}$-modules $S_{t,c+1,z}$, as stalk complexes in $D^b(A_{t, c+1})$ we have $$F_{t,c}^{-1}(S_{t,c+1,z})=\begin{cases}
0 \to S_{t,c,z} &\text{ if }z \notin K_{t,c}\\
P_{t,c,z} \to T_z \operatorname{\otimes}_B M &\text{ if }z \in I_{t,c}\\
0 \to T_z \operatorname{\otimes}_B M &\text{ if }z \in J_{t,c}.
\end{cases}$$with the second term in degree zero. Recall that $A_{t,c}$, $I_{t,c}$ satisfy thin projective condition \[def:TPC\]. When $z \in I_{t,c}$, with (1b) of loc. cit., $H^{-1}(F_{t,c}^{-1}(S_{t,c+1,z}))$ has socle $S_{t,c,z}$. Given the structure of $P_{t,c,z}$ as in (2a), all composition factors of $H^{-1}(F_{t,c}^{-1}(S_{t,c+1,z}))/S_{t,c,z}$ is in ${\mathbbmss{S}}_{t,c,0}$. This concludes the case for a degenerated equivalence (i.e. if $J_{t,c}$ is empty, see \[conv:degen\]). Otherwise, let $z \in J_{t,c}$ hence $z \neq \tilde{z}$ and $\tilde{z}\in I_{t,c}$. Consider $T_z \operatorname{\otimes}_B M$ has injective envelope $P_{t,c,\tilde{z}}$. By (2a) there is only one copy of composition factor $S_{t,c,z}$ in $P_{t,c,\tilde{z}}$ hence the homology $H^0(F_{t,c}^{-1}(S_{t,c+1,z}))=T_z \operatorname{\otimes}_B M$ contains this copy of composition factor $S_{t,c,z}$ as its top. Also by (2a) this shows $T_z \operatorname{\otimes}_B M$ has no composition factors of either $S_z$ or $S_{\tilde{z}}$ except at top or socle. This concludes our argument since we have proved the homology of the required complexes satisfy definition \[def:perv\] with the given data.
In fact, (continuing the notation above,) for $z \in I_{t,c}$, $H^{-1}(F_{t,c}^{-1}(S_{t,c+1,z}))$ must be the universal ${\mathbbmss{S}}_{t,c,0}$-extension by $S_{t,c,z}$ (or else yields a contradiction on $\operatorname{\mathrm{soc}}(T_z\operatorname{\otimes}_B M)=T_{\tilde{z}}$). This satisfies (2) of theorem \[thm:simalt\]. Furthermore, $T_z \operatorname{\otimes}_B M$ must be isomorphic to the unique (indecomposable) module generated by $S_{t,c,z}$ and supported by $S_{t,c,\tilde{z}}$. This satisfies (3) of loc.cit.\
Gathering all the tilts together we arrive at our main theorem.
\[thm:main\] Let $A$ be a full defect block of $kSL(2,q)$ in the defining characteristic. Then $A$ is derived equivalent to its Brauer correspondent $B$ via algebras $A_{t,c}$ $(0 \leq t \leq r, 0 \leq c \leq d(t))$ where $A_{t,c}$ and $A_{t,c+1}$ is simply alternating (or elementary) perverse equivalent for $0 \leq c \leq d$, and stringed up by $A_{0,0}=A$, $A_{r+1,0}=B$ and $A_{t, d+1}=A_{t+1,0}$ for $0 \leq t \leq r$.
This follows from the induction we have in proposition \[prop:W\], lemma \[lem:main\] and incorporating inductions in proposition \[prop:OY\] by Okuyama and Yoshii.
Example of SL(2,9) and compositions of alternating perverse equivalences {#sec:example}
========================================================================
\[sec:discussion\]
This section demonstrates the case for $G=SL(2,9)$, $p=3$. After the example we make some observation about composing alternating equivalences.
Example:SL(2,9)
---------------
Recall this example has two full defect blocks, the subscript set is $Z=\{0,\dots,7\}$. We handle both blocks together hence the index will be slightly entangled.
\[ex:partition\] The partition belong to the principal block $B_0$ is $$K_{-1}=\{0\} \text{, } K_1 = \{2, 6\} = I_1=J_1=K_{1,0} \text{ and } K_2 = \{4\}=I_2=J_2=K_{2,0}.$$ and for the non-principal block $B_1$ is $$K_0 = \{1,3,5,7\} \text{, } J_0= \{1,3\} \text{ and } I_0=\{5,7\};$$ by which $K_0$ further split into $$K_{0,0}=\{7,1\} \text{ and } K_{0,1}=\{5,3\}.$$ with $$I_{0,0}=\{7\}, J_{0,0}=\{1\}; I_{0,1}=\{5\} \text{ and } J_{0,1}=\{3\}.$$
### The principal block of SL(2,9)
In $B_0$ all the signed Frobenius orbits coincides with Frobenius orbits (not true for general principal blocks of $SL(2,q)$ in prime $p$). So in $B_0$ all simply alternating equivalence degenerates to elementary perverse equivalence. Each $K_1$ and $K_2$ only yields one sub-$K$-set ($K_{1,0}=K_1$ and $K_{2,0}=K_2$), thus each Okuyama’s tilt (using set $K_1$ and $K_2$) is perverse by itself.
\[ex:principal\] (Continuing from above) The set $K_{-1}$ needs no handling (in line with [@Okuyama]). Now (using information of projective modules computed from [@Koshita]) the perverse equivalence corresponding to the set $K_1$ gives one-sided tilting summands $$Q_0= P_2 \oplus P_6 \to P_0 \qquad Q_2 = P_2 \to 0 \qquad Q_4=P_2 \oplus P_6 \to P_4 \qquad Q_6= P_6 \to 0$$ where the last term is in degree 0. The set $K_2$ further gives $$R_0=Q_4 \oplus Q_4 \to Q_0 \qquad R_2 = Q_4 \to Q_2 \qquad R_4=Q_4 \to 0 \qquad R_6=Q_4 \to Q_6$$ which compose into $$\begin{aligned}
R_0= P_2 \oplus P_6 \to P_4 \oplus P_4 \to P_0 & \qquad R_2 = P_6 \to P_4 \to 0 \qquad \\ R_4=P_2 \oplus P_6 \to P_4 \to 0 & \qquad R_6= P_2 \to P_4 \to 0
\end{aligned}$$
### The non-principal block of SL(2,9)
We will see the algorithm in full working in this case. The non-principal block has only one signed Frobenius orbit. It is also the smallest case that Proposition \[prop:intra-t\] does not hold for at least one orbit.
\[ex:nonprincipal\] We first produce the perverse equivalence with respect to the data given by $K_{0,0}$. Now using data from [@Koshita] the perverse equivalence gives one-sided tilting summands $$Q_1=0 \to P_1 \qquad Q_3 = P_7 \to P_3 \qquad Q_5=0 \to P_5 \qquad Q_7=P_7 \to P_1$$ where the last term is in degree 0. Note that in $Q_3$, $\operatorname{\mathrm{Hom}}_{B_1}(P_7, P_3)$ is two-dimensional, but the dimension induced from $\operatorname{\mathrm{Hom}}_{B_1}(P_1, P_3)$ is not included. The set $K_{0,1}$ further gives $$R_1=Q_5\to Q_1 \qquad R_3 = 0 \to Q_3 \qquad R_5=Q_5 \to Q_3 \qquad R_7=Q_5 \to Q_7$$ which yields (combined with Example \[ex:principal\]) all summands of a one-sided tilting complex $T$ of $A$ where $\operatorname{\mathrm{End}}_{D^b(A)}(T) \operatorname{\cong}B$. $$R_1 =P_5 \to P_1 \qquad R_3 = P_7 \to P_3 \qquad R_5= P_5 \oplus P_7 \to P_3 \qquad R_7= P_5 \oplus P_7 \to P_1$$
Composing alternating perverse equivalence
------------------------------------------
One main question is the possibility to compose perverse equivalences and the composition remain perverse. Some of these are inspired by the fact Okuyama can combine some of his tilting further [@Okuyama]. However such compositions in terms of perverse equivalence is not promising. In fact, Example \[ex:nonprincipal\] is such bad example. The best environment to discuss these is the notion of poset perverse equivalence, introduced in section \[sec:perveq\]. So from the perspective of composition factors, the question is whether the universal extensions (such as those consider in \[thm:simalt\]) contains certain factors or not. First we define a mechanism to turn inclusion of sets into partial orders.
Given a filtration/inclusion of sets $I:(\emptyset={\mathbbmss{S}}_{-1} \subset {\mathbbmss{S}}_0 \subset {\mathbbmss{S}}_1 \subset \ldots \subset {\mathbbmss{S}}_r={\mathbbmss{S}})$ on ${\mathbbmss{S}}$, we define a partial order on the set of ${\mathbbmss{S}}$, denoted by $\prec_{{\mathbbmss{S}},I}$ by imposing the filtration but no other relations. That is, given elements $S \in {\mathbbmss{S}}_i\setminus {\mathbbmss{S}}_{i-1}$, $S' \in {\mathbbmss{S}}_{i'}\setminus {\mathbbmss{S}}_{i'-1}$, $$S' \prec_{{\mathbbmss{S}}, I} S \text{ if and only if } i'<i.$$
We give a sufficient condition for $F_t$ to be a perverse equivalence for a fixed $t$.
\[prop:intra-t\] Using notation as in section \[sec:main\]. If for $z \in I_t$, the $A_t$-module $(\Omega T_z \operatorname{\otimes}_B M)/S_{t,z}$ does not contain any composition factor of $S_{t,z}$ for $z \in J_t$ then all $F_{t,c}$, $0 \leq c \leq d$ composes to one simply alternating perverse equivalence. This one simply alternating perverse equivalence is naturally isomorphic to $F_t$.
Translating filtered perverse equivalence to poset perverse equivalence, as stated in section \[sec:perveq\], $F_{t,c}$ is a poset perverse equivalence with respect to $({\mathbbmss{S}}_{t,c}, \prec_{{\mathbbmss{S}}_{t,c}, I_{t,c}}, \pi')$, where $\pi'$ is the function sending $S_{t,c,z} \in {\mathbbmss{S}}_{t,c,i}^-$ to $\pi(i)$. Then by definition, the homology $H^{-1}(F_{t,c}(S_{t,c,z}))$ for $z \in I_{t,c}$ and $H^0(F_{t,c}(S_{t,c,y}))$ for $y \in J_{t,c}$ does not involve any composition factors $S_{t,c,y'}$ with $y' \in J_t\setminus J_{t,c}$. Hence $F_{t,c}$ is also a poset perverse equivalence with respect to $({\mathbbmss{S}}_{t,c}, \prec_{{\mathbbmss{S}}_t, I^\bullet_t}, \pi')$ for all $0 \leq c \leq d$. As all $F_{t,c}$ is a poset perverse equivalence with respect to this same partial order $\prec_{{\mathbbmss{S}}_t, I^\bullet_t}$, we can compose them into one perverse equivalence. This equivalence is naturally isomorphic to $F_t$ since they possesses the same partial order and perverse function. Now translating to filtered perverse equivalence we have the (naturally expected) filtration coming from the inclusion $Z\setminus K_t \subset Z \setminus I_t \subset Z$ with the perverse function simply alternating.
We have mentioned in Section 3 that we can create a general alternating perverse equivalence using a composition of simple alternating ones. Consider any alternating perverse equivalence $F$ with a filtration of $r$ layers. This can be broken down as a composition of elementary perverse equivalence $F_i$ where each is perverse relative to $$(\emptyset={\mathbbmss{S}}_{-1} \subset {\mathbbmss{S}}_i \subset {\mathbbmss{S}}, p:0 \to 0, 1 \to (-1)^i).$$ for $1 \leq i \leq r$. (They are all elementary, in fact.) Then we can see that $E_j:=F_{2j} \circ F_{2j-1}$ with $1 \leq j \leq \lfloor r+1/2 \rfloor$ is a simple alternating perverse equivalence such that the composition of $E_j$ is naturally isomorphic to $F$ (define $F_{r+1}$ to be the identity if $r$ is odd to make $E_j$ well-defined, or use degeneration convention such as \[conv:degen\]). We use this to contemplate composition of equivalences among steps of Okuyama’s tilt (i.e. different $t$’s).
Let $F_s$, $F_{s+1}$,...,$F_{s'}$ be a string of Okuyama tilt such that for any $t$, $s \leq t \leq s'$, the $A_t$-module $\Omega (T_z \operatorname{\otimes}_B M)/S_{t,z}$, $z \in I_t$, has no composition factor isomorphic to $S_{t,y}$ for all $y \in J_{t'}$ with $t \leq t'$. Then all $F_s$, $F_{s+1}$,...,$F_{s'}$ composes into one alternating perverse equivalence.
Note that setting $t'=t$ for the condition will make $\operatorname{\mathbb{F}}_t$ satisfy the condition of \[prop:intra-t\], hence the condition secures each $F_t$ is a simply alternating perverse equivalence. Consider the set inclusion $$I^{\bullet}_{s,s'}=\big(Z\setminus\bigcup_{t=s}^{s'}K_t \subset Z\setminus(\bigcup_{t=s}^{s'}K_t\setminus I_s)\subset Z\setminus\bigcup_{t=s+1}^{s'}K_t\subset\ldots\subset Z\big)$$ and the hence defined partial order $\prec_{Z, I^{\bullet}_{s,s'}}$. For each $t$, the condition deduces that the composition factors involved in the homology of $H^{-1}(F_t(S_{t,z}))$ for $z \in I_t$ and $H^0(F_t(S_{t,y}))$ for $y \in J_t$ does not involve $S_{t, y}$ for any $y \in J_{t'}$ for all $t' \geq t$. So $F_t$ is a poset perverse equivalence with respect to $$\big( {\mathbbmss{S}}_t, \prec_{{\mathbbmss{S}}_t, I^\bullet_{s,s'}}, \pi_t\big),$$where $\prec_{{\mathbbmss{S}}_t, I^\bullet_{s,s'}}$ is defined by transferring the partial order on $Z$ to modules ${\mathbbmss{S}}_t$ via indexing, and $\pi_t(S_{t,z})=\begin{cases}
1 &\text{ if } z \in I_t\\ 0 &\text{ otherwise}.
\end{cases}$. Thus, all $F_t$, $s \leq t \leq s'$ composes into one alternate perverse equivalence.
Not every Okuyama’s tilt can be expressed as an alternate perverse equivalence. In particular, example \[ex:nonprincipal\] does not satisfy any proposition above. We know by the non-existence of stalk projective summands there is no way the example is any kind of perverse equivalence. Though it should be obvious to careful readers the projective summands are very ’perverse-alike’ in the sense all $P_5$ and $P_7$ are concentrated in degree $-1$ and $P_1$ and $P_3$ in degree 0. Although we have shown for the case $SL(2,q)$ in this paper, we do not know if every Okuyama’s tilt can be undoubtedly written as a composition of perverse equivalence. Our proof depends on thin projective condition, but it is an unnecessary strong requirement in considering the general case. This might be an interesting topic related to the geometry aspect of Okuyama’s tilting.
We end this section and the paper by giving an affirmative answer to a comment by Okuyama [@Okuyama p.15] in his first paper introducing his tilting complex. His method will lead to Chuang’s [@Chuang] and Holloway’s [@Holloway] constructions of derived equivalences in $SL(2,p^2)$ (in fact, the dual result) after a long but trivial calculation.
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---
abstract: 'The systems exhibiting quantum phase transitions (QPT) are investigated within the Ising model in the transverse field and Heisenberg model with easy-plane single-site anisotropy. Near QPT a correspondence between parameters of these models and of quantum $\phi ^4$ model is established. A scaling analysis is performed for the ground-state properties. The influence of the external longitudinal magnetic field on the ground-state properties is investigated, and the corresponding magnetic susceptibility is calculated. Finite-temperature properties are considered with the use of the scaling analysis for the effective classical model proposed by Sachdev. Analytical results for the ordering temperature and temperature dependences of the magnetization and energy gap are obtained in the case of a small ground-state moment. The forms of dependences of observable quantities on the bare splitting (or magnetic field) and renormalized splitting turn out to be different. A comparison with numerical calculations and experimental data on systems demonstrating magnetic and structural transitions (e.g., into singlet state) is performed.'
address: '620219, Institute of Metal Physics, Ekaterinburg, Russia'
author:
- 'V.Yu.Irkhin$^{*}$ and A.A.Katanin'
title: Quantum phase transitions and thermodynamic properties in highly anisotropic magnets
---
Introduction
============
The interest in quantum models of anisotropic spin and pseudospin systems is connected with that they describe miscellaneous magnetic and structural transitions. Examples of such transitions are transitions into singlet magnetic state in TbSb, Pr, Pr$_3$Tl (see Ref. [@CooperBook] and references therein), NiSi$_2$F$_6$ (see, e.g., Ref. [@FSiNi]), and orientational and metamagnetic phase transitions under magnetic field[@Coq; @Levitin].
The simplest model for the systems demonstrating a ground-state quantum phase transition (QPT) is the Ising model in the transverse field. This model is convenient for description of structural transitions in quantum crystals [@Samara; @Blinc; @Gehring]. It can be also applied to describe magnetic systems where both lowest and next energy levels are singlets. A more complicated first-principle model for spin systems in a strong crystal field is the Heisenberg model with an easy-plane single-site anisotropy; it is applicable in the case where next-to-lowest energy level is doublet. As well as transverse-field Ising model, this model also demonstrates QPT (the ground-state magnetization vanishes with increasing the anisotropy parameter).
A number of approximate methods were applied to study the transverse-field Ising model [@MF; @Bosons; @Grover; @RPA-TSCA; @HTSE; @GSPT; @c1/z; @Shender; @Numer] at $d>1$ and the Heisenberg model with easy-axis anisotropy [@QPT-Epl; @Fl-Epl; @HTSE-Pl; @Onufr; @Valkov]. However, all these methods (except for numerical ones) are applicable only not too close to QPT. In particular, they lead to Gaussian values of the QPT critical exponents. Thus analytical consideration of the ground-state and finite-temperature properties in the vicinity of QPT is still an open problem.
The case, where the system is close to QPT, is characterized by a small ground-state moment and low transition temperature. Such a situation is reminiscent of weak itinerant magnets[@Moriya]. An analysis of the ground-state QPT was performed in Refs.[@Hertz; @Millis]. It was shown that the upper critical dimensionality for such transitions is $d_c^{+}=4-z$ with $z$ being the dynamical critical exponent. This conclusion has a general character. In the present paper we consider only systems with $z=1,$ which holds for the transverse-field Ising model and anisotropy-induced QPT in the Heisenberg model. (Note that this is not the case for the QPT induced by magnetic field in degenerate systems with $n>1$ component order parameter since here $z=2,$ see Ref. [@XYh]). Thus for $d\geq 3$ the critical exponents are the Gaussian ones, while for $d<3$ they deviate from the corresponding mean-field values and can be calculated with the use of the $%
3-\varepsilon $ expansion. For the critical dimensionality $d=3$ the ground state properties contain logarithmic corrections.
Sachdev[@Sachdev] proposed a three-stage method of treating finite-temperature properties of the systems near QPT. At the first stage, ground-state renormalizations are performed. At the second stage, the non-zero Matsubara frequencies are integrated out to obtain an effective classical action. Finally, perturbation theory for the effective classical model is applied. This method ensures correct analytical properties of the resulting theory. While ground-state renormalizations are non-universal, finite-temperature properties, being expressed through quantum-renormalized ground-state parameters, turn out to be universal.
The approach of Ref. [@Sachdev] is based on a continuum model, namely, the quantum $\phi ^4$ model. This model is sufficient to express finite-temperature properties near QPT through the non-universal ground-state properties, but insufficient to obtain correct results for the latter. A convenient method to consider the lattice spin systems near their critical dimensionality is the expansion in the formal quasiclassical parameter $1/S$. Its applicability is connected with the fact that near $d_c$ the effective interaction of spin waves is small (except for a narrow critical region where the $\varepsilon $-expansion can be easily developed to correct the description of the critical behavior). For the Heisenberg model such a situation occurs for temperature transition near the lower critical dimensionality $d_c^{-}=2$. This provides success of the renormalization-group (RG) approach for the description of thermodynamics of $d=2$ (Ref.[@Chakraverty]) and $d=2+\varepsilon $ (Refs. [@Brezin; @Chakraverty]) Heisenberg magnets, and also quasi-2D and anisotropic 2D magnets [@OurRG] not too close to $T_c$.
For QPT in highly-anisotropic spin systems the $1/S$-expansion works well near the [*upper* ]{}critical dimensionality $d_c^{+}=3.$ In this case there are excitations, which are almost gapless near QPT (they are analogous to the spin-wave excitations in Heisenberg magnets). Besides that, for the ordered degenerate systems (with $n\geq 2$) there are always Goldstone modes with zero energy gap and the $1/S$-expansion becomes applicable at arbitrary anisotropy below its critical value. The situation is more complicated for finite temperatures, since close to temperature transition the system behaves as a corresponding classical magnet and therefore the picture of excitation spectrum differs from that at $T=0.$
The aim of the present paper is to apply the above-discussed concepts for calculating ground-state and finite-temperature properties of transverse-field Ising model ($n=1$) and Heisenberg model with strong easy-plane anisotropy ($n=2$). To this end we apply perturbation theory (which is in fact an expansion in $1/S$) to the original lattice models (not to their continuum analogs), which enables us to calculate non-universal ground-state quantum renormalizations. After that we combine perturbation results for short-wave fluctuations with the results of the $3-\varepsilon $ RG approach for the long-wave fluctuations to correct the results of perturbation theory. Finally, we consider finite-temperature properties within RG approach for the effective continuum classical model.
The plan of the paper is as follows. In Sect.2 we discuss the Ising model in the transverse field. We consider corresponding mean-field results, construct the perturbation theory in $1/S$ and apply a scaling approach to investigate ground-state and thermodynamic properties, in particular the influence of external magnetic field. In Sect.3 the Heisenberg model with easy-plane anisotropy is considered in a similar way. In Sect.4 we discuss the results obtained and compare them with experimental data on systems exhibiting structural and magnetic transitions. Some details of calculations are presented in Appendices.
Transverse-field Ising model
============================
The formulation of the model and the mean-field approximation
-------------------------------------------------------------
We consider the Hamiltonian of the Ising model in the transverse field $%
\Omega $ $${\cal H}=-\frac I2\sum_{\langle ij\rangle }S_i^xS_j^x-\Omega \sum_iS_i^z
\label{ITF}$$ where $I$ is the exchange parameter. This model can describe singlet magnetic systems. A derivation of such a model for Heisenberg magnets with strong single-site anisotropy is presented in Appendix A. The model (\[ITF\]) describes also structural transition in quantum crystals (cooperative Jahn-Teller effect, see Ref. [@Gehring]) where two lowest energy levels are singlets. In this case $I=\Delta _2,$ $\Omega =\Delta _1,$ where $\Delta
_{1,2}$ is the energy-level splitting at $T=0$ and $T>T_c$ respectively. For further purposes it will be useful to consider the model (\[ITF\]) for arbitrary values of (pseudo-) spin $S.$
At $\Omega =0$ the model (\[ITF\]) coincides with the Ising model and thus the order parameter $\overline{S}\equiv \langle S^x\rangle =S$ in the ground state. With increasing $\Omega $, the model (\[ITF\]) demonstrates a quantum phase transition where $\overline{S}$ vanishes. The one-dimensional $%
S=1/2$ transverse-field Ising model in the ground state can be solved rigorously [@one-dim]. In particular, it can be reduced to the two-dimensional Ising problem at finite temperatures [@Suzuki], so that critical exponents for both the phase transitions coincide. The transverse-field Ising model with $d>1$ requires approximate methods.
The mean-field theory [@Blinc; @MF] yields the critical field $\Omega
_0\equiv I_0S,$ and the equation for the order parameter at $\Omega <\Omega
_0$ reads $$\frac{\Omega _0}{H_e}B_S\left( H_e/T\right) =1$$ where $$\begin{aligned}
B_S(x) &=&(1+1/2S)\coth (1+1/2S)x-(1/2S)\coth (x/2S), \nonumber \\
B_{1/2}(x) &=&(1/2)\tanh (x/2)\end{aligned}$$ is the Brillouin function, $H_e=(\Omega ^2+\Omega _0^2\overline{S}%
^2/S^2)^{1/2},$ $I_0=2Id.$ Owing to the field $\Omega ,$ the value of $%
\langle S^z\rangle $ is finite in both ordered and disordered phase and reads $$\langle S^z\rangle =\frac{\Omega S}{H_e}B_S\left( H_e/T\right) .$$ It should be noted that at $\Omega <\Omega _0$ we have simply $\langle
S^z\rangle =S\Omega /\Omega _0.$ The critical temperature where $\overline{S}
$ vanishes is determined for the physically important case $S=1/2$ by $$T_c^{MF}=\frac \Omega {2\tanh ^{-1}(\Omega /\Omega _0)}\simeq \frac \Omega {%
\ln [2/(1-\Omega /\Omega _0)]} \label{TcMF}$$ (the last approximation is valid for $1-\Omega /\Omega _0\ll 1$). Thus the mean-field theory predicts a very weak inverse-logarithmic dependence for the critical temperature near QPT in arbitrary dimensionality. This contradicts to the results of the scaling approach [@Millis; @Sachdev] both above and below the upper critical dimensionality $d_c^{+}=3$.
To improve the mean-field theory, one has to take into account the collective excitations which are analogous to spin-wave excitations in Heisenberg magnets. The spectrum of these excitations in the random-phase approximation has the form [@Blinc] $$E_{{\bf q}}^2=\Omega [\Omega -I_{{\bf q}}\langle S^z\rangle ]+I_0^2\overline{%
S}^2 \label{EqRPA}$$ in both ordered and disordered phases. Near QPT (at $\overline{S}(T=0)\ll 1$), these excitations become almost gapless and give dominant contributions to physical properties.
The result of account of the collective excitations to first order in $1/%
{\cal R}$ (where ${\cal R}$ is the radius of exchange interaction) [@Shender] for $d=3$ reads $$T_c\sim {\cal R}^{3/2}\sqrt{1-\Omega /\Omega _0}. \label{TcR}$$ This has a correct square-root behavior (see, e.g., [@Sachdev])$.$ However, the logarithmic corrections, that occur for $d=3,$ are not reproduced by the result (\[TcR\]). Besides that, the $1/{\cal R}$ expansion does not enable one to determine correctly the coefficient in (\[TcR\]) for not too large ${\cal R}$.
Another approach used in [@RPA-TSCA] is to consider the excitations (\[EqRPA\]) self-consistently within the random-phase approximation (RPA) decoupling scheme for the sequence of equations of motion. Unlike the $1/%
{\cal R}$ expansion, this procedure gives a possibility to take into account the reaction of the RPA excitation spectrum (\[EqRPA\]) on the deviation of the critical field from $\Omega _0$. However, corrections to mean-field ground-state parameters turn out to be small enough, and at finite temperatures the RPA magnetization shows a double-value behavior with first-order temperature phase transition. Authors of Ref. [@RPA-TSCA] consider also a generalization of RPA, the two-site self-consistent approximation (TSCA) which gives a possibility to include partially correlation effects. This approximation gives more satisfactory results than RPA. However, it predicts first-order character not only for the temperature transition, but also for QPT.
One should mention also the papers[@HTSE; @GSPT] where high-temperature series expansions (HTSE) and ground-state perturbation theory (GSPT) were used. Although these expansions gives consistent results for the critical field, their applicability near QPT is questionable. Recently some results of GSPT and HTSE have been confirmed by numerical correlated-basis-function analysis [@Numer].
Below we use the $1/S$ expansion to treat ground-state and finite temperature properties of the transverse-field Ising model. Unlike the $1/%
{\cal R}$ expansion, it takes into account the “spin-wave” excitations already in zeroth order of perturbation theory. Contrary to RPA [@RPA-TSCA], this is a systematic expansion, and therefore scaling corrections can be easily calculated. It should be also noted that the $1/S$ expansion differs from the ground-state perturbation theory used in Ref. [@GSPT] where the expansions in powers of $\Omega /I$ and $I/\Omega $ are used for the ordered and disordered phases respectively. Indeed, the $%
1/S $ expansion treats both the terms in the Hamiltonian (\[ITF\]) on equal footing and thus yields physically correct results already in the first order in $1/S.$
\[Sec2\]Ground-state properties within the $1/S$ perturbation theory
--------------------------------------------------------------------
To construct perturbation expansion in a convenient form we use the spin coherent state approach[@Klauder]. The partition function is presented in terms of a path integral, $${\cal Z}=\int D{\bbox \pi }\exp [-({\cal S}_{\text{dyn}}{\cal +S}_{\text{st}%
})], \label{Zp}$$ where $$\begin{aligned}
{\cal S}_{\text{dyn}} &=&iS\sum_i\int\limits_0^{1/T}d\tau \left( 1-\cos
\vartheta _i\right) \frac{\partial \varphi _i}{\partial \tau } \nonumber \\
{\cal S}_{\text{st}} &=&-\int\limits_0^{1/T}d\tau \left[ \frac{IS^2}2%
\sum_{\langle ij\rangle }\pi _{xi}\pi _{xj}+\Omega S\sum_i\pi _{zi}\right] \end{aligned}$$ are static and dynamic parts of the action, ${\bbox \pi }_i=\{\pi _{xi},\pi
_{yi},\pi _{zi}\}$ is three-component vector field with ${\bbox \pi }%
_i^2=1+1/S$, $\vartheta _i$ and $\varphi _i$ are the polar and azimuthal angles of ${\bbox \pi }_i$ in an arbitrarily chosen coordinate system (which does not need to coincide with the $\pi _x$-$\pi _y$-$\pi _z$ coordinate system). Further we additionally rotate the coordinate system through the angle $\theta $ determined by $$\sin \theta =\langle \pi _x\rangle /\langle |{\bbox \pi }|\rangle$$ around $\pi _y$-axis (in the disordered phase $\theta =0$ and the rotated coordinate system coincides with the original one). Then $\langle \widetilde{%
\pi }_x\rangle =0$ in the new coordinate system.
The calculation of two-point vertex function $\Gamma ({\bf q},\omega )$ of the fields $\widetilde{\pi }_x,\,\widetilde{\pi }_y$ (the tilde sign is referred to rotated coordinate system), which is connected with matrix Green’s function $G$ of these fields by the relation $\Gamma ({\bf q},\omega
)=G^{-1}({\bf q},\omega )$, is performed for both ordered and disordered phases in Appendix B and yields to first order in $1/S$ the result $$\Gamma _{\pm }({\bf q},\omega _n)=\left(
\begin{array}{cc}
S^2(I_0-I_{{\bf q}}+I_0\Delta _{\pm }^2) & iS\omega _n\left(
w_S+X_0/2+Y_0/2\right) \\
iS\omega _n\left( w_S+X_0/2+Y_0/2\right) & I_0S^2D_{\pm }
\end{array}
\right) \label{Gamma}$$ where $w_S=(1+1/2S)^{-1},$ $$X_0=\frac 1{2S}\sum_{{\bf q}}\frac 1{\sqrt{1-I_{{\bf q}}/I_0}},\;Y_0=\frac 1{%
2S}\sum_{{\bf q}}\sqrt{1-I_{{\bf q}}/I_0},$$ $\Delta _{\pm }$ and $D_{\pm }$ are the dimensionless temperature-dependent energy gap and the renormalization factor for the exchange parameter in the disordered and ordered phases respectively, their concrete expressions being specified below. The matrix static uniform spin susceptibility in the rotated coordinate system is expressed through $\Gamma $ as $$\widetilde{\chi }^{ij}=S^2\Gamma _{ij}^{-1}(0,0) \label{Hij}$$ where $i,j=x,y.$ The renormalized spectrum of “spin-wave” excitations is determined by the condition $\det \Gamma ({\bf q,-}iE_{{\bf q}})=0$ and to first order in $1/S$ has the form $$\widetilde{E}_{{\bf q}}=S\left[ 1+1/2S-(X_0+Y_0)/2\right] \sqrt{I_0D_{\pm
}(I_0-I_{{\bf q}}+I_0\Delta _{\pm }^2)}. \label{Eqr}$$ The quantum-renormalized critical field $\Omega _c$ is given by $$\frac{\Omega _c}{\Omega _0}=1+\frac 1{2S}-\frac 1{4S}\sum_{{\bf q}}\frac{%
2I_0+I_{{\bf q}}}{\sqrt{I_0(I_0-I_{{\bf q}})}}. \label{Wc}$$ Last two terms in this expression yield the first-order $1/S$-correction to mean-field value of $\Omega _c.$ For $S=1/2$ numerical calculation of integral in (\[Wc\]) yields the result $\Omega _c=2.44I$ in the 3D case and $\Omega _c=1.10I$ in the 2D case. Thus the critical field is strongly renormalized by quantum fluctuations both in 3D and 2D cases. The critical field values obtained are considerably smaller than the corresponding RPA results[@RPA-TSCA], $\Omega _c=2.88I$ and $\Omega _c=1.83I$ and somewhat smaller than those obtained by HTSE [@HTSE] and GSPT [@GSPT], $%
\Omega _c=2.58I$ and $\Omega _c=1.54I.$ This demonstrates that considered first-order $1/S$ perturbation theory overestimates effects of quantum fluctuations (especially in the 2D case), but treats these fluctuations more correctly than RPA.
In the disordered phase with $\langle \pi _x\rangle =0$ ($\Omega >\Omega _c$) the expressions for the ground-state energy gap and factor $D_{+}$ has the form $$\begin{aligned}
\Delta _{+}^2(t_{+},0) &=&\frac{t_{+}}{1-t_{+}}\left( 1-X_0^{\prime }\right)
\left[ 1+A_{+}(t_{+})\right] \nonumber \\
\ A_{+}(t) &=&\frac 1{4St}\sum_{{\bf q}}\left\{ \frac{2I_0+I_{{\bf q}}(1+t)}{%
\sqrt{I_0[I_0-I_{{\bf q}}(1-t)]}}-\frac{2I_0+I_{{\bf q}}}{\sqrt{I_0(I_0-I_{%
{\bf q}})}}\right\} \label{DeltaP}\end{aligned}$$ and $$D_{+}(t_{+})=\frac{1+Y_0-X_0}{1-t_{+}}\left[ 1+t_{+}A_{+}(t_{+})\right]
\label{DP}$$ where $$t_{+}=1-\Omega _c/\Omega .$$ and $$X_0^{\prime }=\frac 1{2S}\sum_{{\bf q}}\frac{I_{{\bf q}}}{\sqrt{I_0(I_0-I_{%
{\bf q}})}}$$ In the ordered phase ($\Omega <\Omega _c$) we obtain $$\begin{aligned}
\Delta _{-}^2(t_{-},0) &=&t_{-}\left( 1-X_0^{\prime }\right) \left[
1+A_{-}(t_{-})\right] , \nonumber \\
A_{-}(t) &=&-2(1-t)A_{+}(t) \nonumber \\
&&\ \ \ -\frac{1-t}{8S}\sum_{{\bf q}}\frac{[2I_0+I_{{\bf q}}(1+t)]^2}{%
I_0^{1/2}[I_0-(1-t)I_{{\bf q}}]^{3/2}} \label{DeltaM}\end{aligned}$$ and $$D_{-}=1+Y_0-X_0 \label{DM}$$ where $$t_{-}=1-(\Omega /\Omega _c)^2.$$
Consider now the observable quantities. The expression for the order parameter $\overline{S}(t_{-},T)\equiv S\langle \pi _x\rangle $ at $T=0,$ $%
\Omega <\Omega _c$ reads $$\begin{aligned}
\overline{S}(t_{-},0) &=&St_{-}^{1/2}\left[ 1+B(t_{-})\right] ^{1/2}\left[
1+1/2S-(X_0+Y_0)/2\right] , \label{Sl} \\
B(t) &=&-2(1-t)A_{+}(t) \nonumber\end{aligned}$$ For the longitudinal susceptibility we have $$\chi ^{xx}=\cos ^2\theta \widetilde{\chi }^{xx}+\sin \theta \cos \theta
\left( \widetilde{\chi }^{xz}+\widetilde{\chi }^{zx}\right) +\sin ^2\theta
\widetilde{\chi }^{zz} \label{HiOrd}$$ where the tilde sign is referred to susceptibilities in the rotated coordinate system (recall that for the disordered phase $\theta =0$). For the ordered phase the first summand in (\[HiOrd\]) gives dominant contribution near QPT, and using the relation (\[Hij\]) yields in both the ordered phase near QPT and disordered phase the expression for the ground-state spin susceptibility through the gap in the excitation spectrum $$\chi ^{xx}=\frac 1{I_0\Delta _{\pm }^2(t_{\pm },0)}. \label{HiOrd1}$$
In the limiting case of zero transverse field we have the trivial result $%
\overline{S}=S,$ and the excitation spectrum reduces to its mean-field form, $\widetilde{E}_{{\bf q}}=\Omega _0.$ At very large $\Omega \gg \Omega _c$ we reproduce again the mean-field result $\widetilde{E}_{{\bf q}}=\Omega $. This is a consequence of the fact that in both the limits $\Omega =0$ and $%
\Omega \rightarrow \infty $ quantum fluctuations are absent. Thus the $1/S$-expansion gives a possibility to describe the ground-state properties for an arbitrary $\Omega \geq 0.$ However, as we shall see below, some difficulties arise in the region $\Omega \approx \Omega _c\;$where the quantum fluctuations are strong enough to modify considerably the results. A more detailed consideration of this region will be performed in Sect.\[Sec3a\].
\[Sec3\]Influence of longitudinal magnetic field
------------------------------------------------
To consider the influence of the external magnetic field we add to the Hamiltonian the term $$\Delta {\cal H}=-H\sum_iS_i^x.$$ The longitudinal magnetic field results in the appearance of nonzero $%
\langle S_i^x\rangle $ at any $\Omega /I.$ The influence of both transverse and external longitudinal fields is, of course, equivalent to applying one effective field which has the value $(H^2+\Omega ^2)^{1/2}$ and makes the angle $\arctan (H/\Omega )$ with the $\pi _x$-axis. However, it is useful to consider these fields as two independent ones.
Performing the calculations which are similar to those of Sect.\[Sec2\] and Appendix B we obtain to first order in $1/S$ the equation for the angle $%
\theta $ of coordinate system rotation $$\Omega -\Omega _0r(\theta )\cos \theta -H\cot \theta =0, \label{FEqA}$$ where $$r(\theta )=1+\frac 1{2S}-\frac 1{4S}\sum_{{\bf q}}\frac{2(I_0+I_{{\bf q}%
})\varphi (\theta )-I_{{\bf q}}\cos ^2\theta }{\sqrt{I_0\varphi (\theta
)[I_0\varphi (\theta )-I_{{\bf q}}\cos ^2\theta ]}},$$ with $\varphi (\theta )=\sin ^2\theta +(\Omega /\Omega _c)\cos \theta $. For a general $\Omega /\Omega _c$ the solution of this equation is rather cumbersome. However, near QPT (i.e. at $1-\Omega /\Omega _c\ll 1$), where the angle $\theta $ is small, one can expand (\[FEqA\]) in $\theta $ to obtain $$\theta =\left\{
\begin{array}{cc}
\theta _0+H/2[I_0Sr(0)-\Omega ] & \theta _H\ll \theta _0 \\
\theta _H+2\delta r/3\theta _H & \theta _0\ll \theta _H\ll 1
\end{array}
\right.$$ where $\theta _0=\sqrt{2(1-\Omega /\Omega _c)}$, $\theta _H=(2H/\Omega
_c)^{1/3}$ and $\delta r=r(\theta _H)-r(0)$.
For the magnetization we derive $$\overline{S}=\left\{
\begin{array}{lc}
\overline{S}(H=0)+\chi ^{xx}H, & \theta _H^2\ll 1-\Omega /\Omega _c, \\
\left( 1+\frac 1{2S}-\frac{X_0+Y_0}2\right) \theta _H\left[ 1+B^{\prime
}(\theta _H^2)\right] , & 1-\Omega /\Omega _c\ll \theta _H^2\ll 1,
\end{array}
\right. \label{FMagn}$$ where $\chi ^{xx}$ is determined by (\[HiOrd1\]), and $$B^{\prime }(\theta _H^2)=-\frac 1{6S\theta _H^2}\sum_{{\bf q}}\left[ \frac{%
2(I_0+I_{{\bf q}})(1+\theta _H^2/2)-I_{{\bf q}}(1-\theta _H^2)}{\sqrt{%
I_0[I_0(1+\theta _H^2)-I_{{\bf q}}(1-\theta _H^2/2)]}}-\frac{2I_0+I_{{\bf q}}%
}{\sqrt{I_0(I_0-I_{{\bf q}})}}\right] .$$ The ground-state energy gap is given by $$\Delta _{-}^2\simeq \left\{
\begin{array}{lc}
\Delta _{-}^2(H=0), & \theta _H^2\ll 1-\Omega /\Omega _c, \\
\frac 32\theta _H^2\left[ 1+A^{\prime }(\theta _H^2)\right] , & 1-\Omega
/\Omega _c\ll \theta _H^2\ll 1,
\end{array}
\right.$$ where $$A^{\prime }(\theta _H^2)=B^{\prime }(\theta _H^2)-\frac 1{12S}\sum_{{\bf q}}%
\frac{(I_0+2I_{{\bf q}})^2}{I_0^{1/2}[I_0(1+\theta _H^2/2)-I_{{\bf q}%
}(1-\theta _H^2)]^{3/2}}.$$
The longitudinal susceptibility in the presence of magnetic field is still determined by (\[HiOrd\]), and again the first term gives main contribution near QPT. Alternatively, the same result can be obtained by direct differentiation of $\overline{S}$ \[which is given by (\[FMagn\])\]with respect to $H.$
\[Sec3a\]Ground-state renormalizations near QPT
-----------------------------------------------
The results of $1/S$ expansion can be applied only at not too small $t_{\pm
} $. Indeed, at $d\leq 3$ the functions $A$ and $B$ contain terms which are divergent at $t_{\pm }\rightarrow 0$ as $t_{\pm }^{(d-3)/2}$ (at $d=3,$ logarithmic divergences are present). The same situation takes place for the functions $A^{\prime }$ and $B^{\prime }$ which are divergent as $\theta
_H^{d-3}$ at $\theta _H\rightarrow 0.$ Thus an $\varepsilon =3-d$ expansion can be developed within the RG approach to treat these divergences more correctly and to improve thereby the behavior of $\Delta _{\pm }$ and $%
\overline{S}$ near QPT. Further consideration of this section is related to the critical region $|1-\Omega /\Omega _c|\ll 1.$ However, as it will be clear below, the results can be extrapolated to arbitrary $\Omega ,$ since in the limits $\Omega \ll \Omega _c$ and $\Omega \gg \Omega _c$ they are smoothly joined with the results of the $1/S$ expansion of section \[Sec2\].
First we pick up the nonuniversal factors from $\overline{S},$ $\Delta _{\pm
}$ by introducing the quantities $$\begin{aligned}
\overline{S}_R(t,T) &=&\left[ 1+1/2S-(X_0+Y_0)/2\right] ^{-1}\overline{S}%
(t,T)/S \nonumber \\
\Delta _{\pm R}(t,T) &=&\left( 1-X_0^{\prime }\right) ^{-1}\Delta _{\pm
}(t,T)\end{aligned}$$ Consider the continuum limit of the above theory. The action ${\cal S}={\cal %
S}_{\text{dyn}}+{\cal S}_{\text{st}}$ in this limit takes the form $$\begin{aligned}
{\cal S}_{\text{cont}} &=&\frac 12\int d^dr\int\limits_0^{c/T}d\tau \left[
\,2i\widetilde{\pi }_x(\partial \widetilde{\pi }_y/\partial \tau )+%
\widetilde{\pi }_y^2+(\nabla \widetilde{\pi }_x)^2+m^2\widetilde{\pi }%
_x^2\right] \nonumber \\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ +\frac u{4!}\int d^dr\int\limits_0^{c/T}d\tau \,%
\widetilde{\pi }_x^4, \label{Seff}\end{aligned}$$ where the parameters $u$, $m^2$ and $c,$ determined in such a way, are given by $$\begin{aligned}
u_{\text{cont}} &=&6d\frac{c_0}{IS^2}\zeta , \nonumber \\
c_{\text{cont}} &=&c_0, \nonumber \\
m_{\text{cont}}^2 &=&2t_{+}d, \label{pcont}\end{aligned}$$ $c_0=(2d)^{1/2}IS$ being the bare spin-wave velocity,$\;\widetilde{\pi }%
_x^2=(IS^2/c_0)\pi _x^2,\;\widetilde{\pi }_y^2=(IS^2/c_0)\pi _y^2$ and the factor $\zeta $ ($\zeta =1-t_{-}$ in the ordered phase and $\zeta =1$ in the disordered phase) is introduced to extend the region of applicability of results obtained to arbitrary $t_{\pm }$. Note that the coefficients at first three terms of the quadratic part of (\[Seff\]) can be always chosen equal to their values in (\[Seff\]) by appropriate rescaling of $\pi
_{x,y} $ and $\tau .$ The model (\[Seff\]) is completely equivalent to the quantum $\phi ^4$ model. Indeed, integrating out the field $\pi _y$ we obtain $${\cal S}_{\text{cont}}=\frac 12\int d^dr\int\limits_0^{c/T}d\tau \left[
\,(\partial \widetilde{\pi }_x)^2+m^2\widetilde{\pi }_x^2\right] +\frac u{4!}%
\int d^dr\int\limits_0^{c/T}d\tau \,\widetilde{\pi }_x^4 \label{Seff1}$$ The continuum representation (\[Seff\]) determines the way in which the original lattice model can be renormalized. Following to the standard procedure [@Amit] we introduce the renormalization factors $Z_i^{\pm }$ for the ground-state parameters in the disordered and ordered phases by relations $$\begin{aligned}
\pi _x &=&Z_x^{\pm }\pi _{xR},\;\pi _y=Z_y^{\pm }\pi _{yR}\; \nonumber \\
t_{\pm } &=&(Z_2^{\pm }/Z)\,t_{\pm R},\;g=(Z_4^{\pm }/Z^2)g_R \label{Ren}\end{aligned}$$ where the indices $R$ denote quantum-renormalized quantities, $$g=K_{4-\varepsilon }L_\varepsilon \mu ^{-\varepsilon }u_{\text{cont}}$$ is the coupling constant, $\mu $ is a parameter with the dimensionality of inverse length,$\;K_d=[2^{d-1}\pi ^{d/2}\Gamma (d/2)]^{-1}$, and the factor $%
L_\varepsilon =\Gamma (1+\varepsilon /2)\Gamma (1-\varepsilon /2)$ \[$\Gamma
(z)$ is the Euler gamma function\] ensures the applicability of the one-loop order results for not small $\varepsilon $ [@MSS]. For further treatment it is useful to represent the renormalization factors as $$Z_i^{\pm }=Z_{Li}^{\pm }(g)Z_i^{\text{cont}}(g_R,\mu ) \label{ZZ}$$ where $Z_i^{\text{cont}}$ are the corresponding factors for the continuum model (\[Seff\]) that contain divergent terms (which are independent of lattice structure etc.) and $Z_{Li}$ all the others (lattice dependent) corrections. It is important that the factors $Z_{Li}$ do not contain divergences.
The expressions for $Z$-factors in the continuum model (\[Seff\]) are well known (see, e.g., Ref.[@Amit]). We use the cutoff scheme with cutting integrals over quasimomentum at $\Lambda .$ Then to one-loop order we have $$\begin{aligned}
Z_x^{\text{cont}} &=&Z_y^{\text{cont}}=1+{\cal O}(g_R^2),\,\, \nonumber \\
Z_2^{\text{cont}} &=&1+\frac{g_R}{2\varepsilon }\left( 1-\frac{\mu
^\varepsilon }{\Lambda ^\varepsilon }\right) ,\, \nonumber \\
\,Z_4^{\text{cont}} &=&1+\frac{3g_R}{2\varepsilon }\left( 1-\frac{\mu
^\varepsilon }{\Lambda ^\varepsilon }\right) . \label{Zi}\end{aligned}$$ For our purposes it is convenient to put $\Lambda =(2d)^{1/2}$ (the lattice constant is assumed to be equal to unity), rather than to pass to the limit $\Lambda \rightarrow \infty $ (as it is usual in the quantum field theory). The expressions for $Z_{Li}$ can be deduced by comparison the above results of perturbation theory for the original lattice model (Sections \[Sec2\] and \[Sec3\]) and standard perturbation results for the continuum model (\[Seff\]), see Ref. [@Amit]. We obtain $$\begin{aligned}
Z_{Lx} &=&Z_{Ly}=1\; \nonumber \\
(Z_{L2}^{\pm })^{-1} &=&1+A_{\pm }(t_{\pm })+\frac g{2\varepsilon }\left(
\frac 1{t_{\pm }^{\varepsilon /2}}-1\right) , \nonumber \\
\;(Z_{L4}^{-})^{-1} &=&1+A_{-}(t_{-})-B(t_{-})+\frac{3g}{2\varepsilon }%
\left( \frac 1{t_{-}^{\varepsilon /2}}-1\right) \label{ZLi}\end{aligned}$$ Note that, unlike the factors $Z_i^{\text{cont}}$, the quantities (\[ZLi\]) are defined only for integer $\varepsilon .$ As follows from (\[ZZ\]), the determination of factors $Z_{Li}$ enables one to consider the continuum model (\[Seff\]) with the parameters $$\begin{aligned}
m^2 &=&(Z_{L2}^{\pm })^{-1}m_{\text{cont}}^2,\;u=(Z_{L4}^{\pm })^{-1}u_{%
\text{cont}} \nonumber \\
c &=&c_0\left( 1+1/2S-X_0\right)\end{aligned}$$ instead of the original lattice one. Thus the factors $Z_{Li}$ represent the corrections owing to passing from the cutoff scheme in the original lattice model to that in the continuum model, cf. Ref. [@Chakraverty].
The flow functions for the coupling constant and energy gap have the standard form [@Amit] $$\begin{aligned}
\beta (g_R) &=&\mu \frac{\partial g_R}{\partial \mu }=-\varepsilon g_R+\frac
32g_R^2 \nonumber \\
\gamma (g_R) &=&\mu \frac{\partial \ln Z_2^{\text{cont}}}{\partial \mu }=-%
\frac 12g_R\end{aligned}$$ The effective-Hamiltonian parameters $g_\rho ,\,t_\rho $ at the scale $\mu
^{\prime }=\mu \rho $ as determined by these flow functions read $$\begin{aligned}
g_\rho &=&\left[ 1+\frac{g_R}{g^{*}}(\rho ^{-\varepsilon }-1)\right]
^{-1}\rho ^{-\varepsilon }g_R\,\, \nonumber \\
t_\rho &=&\left[ 1+\frac{g_R}{g^{*}}(\rho ^{-\varepsilon }-1)\right]
^{-1/3}t_{\pm R} \label{Scal}\end{aligned}$$ where $g^{*}=2\varepsilon /3$ is the stable fixed point to one-loop order.
We start the scaling procedure at $\mu =\Lambda $ and stop it at $\mu
^{\prime }=\Lambda t_{\pm }^{1/2}$ (thus $\rho =t_{\pm }^{1/2}$). For $%
\Delta _{\pm }$ and $\overline{S}$ we obtain the results $$\begin{aligned}
\Delta _{\pm R}^2(t_{\pm },0) &=&\frac 1{Z_{L2}^{\pm }}\frac{t_{\pm }}{%
\left[ 1+(3g_R/2\varepsilon )(1/t_{\pm }^{\varepsilon /2}-1)\right] ^{1/3}}
\label{dmrg} \\
\overline{S}_R(t_{-},0) &=&t_{-}^{1/2}\sqrt{\frac{Z_{L4}^{-}}{Z_{L2}^{-}}}%
\left[ 1+\frac{3g_R}{2\varepsilon }\left( \frac 1{t_{-}^{\varepsilon /2}}%
-1\right) \right] ^{1/3}\end{aligned}$$ where, according to (\[Ren\]), (\[Zi\]), $$g_R=(Z_{4L}^{\pm })^{-1}(2d)^{-\varepsilon /2}K_{4-\varepsilon }L_\varepsilon$$ In the 3D case $$\frac 1\varepsilon \left( \frac 1{t^{\varepsilon /2}}-1\right) \rightarrow
\frac 12\ln \frac 1t$$ so that the QPT critical exponents for the order parameter and the gap (inverse correlation length) are Gaussian one, $$\beta =1/2,\,\,\,\nu =1/2, \label{CE3}$$ and logarithmic corrections are present. At the same time, in the 2D case we obtain $$\beta =1/3,\,\,\,\nu =7/12 \label{CE2}$$ which are standard one-loop results for the one-component $\phi ^4$ theory in $d+1=3$ dimensions.
In a strong enough longitudinal magnetic field ($1-\Omega /\Omega _c\ll
\theta _H^2\ll 1$) we obtain $$\begin{aligned}
\Delta _{-R}^2(\theta _H,0) &=&\frac 2{3Z_{L2}^{\prime }}\frac 1{\left[
1+(3g_R/2\varepsilon )(1/\theta _H^\varepsilon -1)\right] ^{1/3}} \nonumber
\\
\overline{S}_R(\theta _H,0) &=&\theta _H\sqrt{\frac{Z_{L4}^{^{\prime }}}{%
Z_{L2}^{\prime }}}\left[ 1+\frac{3g_R}{2\varepsilon }\left( \frac 1{\theta
_H^\varepsilon }-1\right) \right] ^{1/3}\end{aligned}$$ where $$\begin{aligned}
Z_{L2}^{^{\prime }} &=&1+A^{\prime }(\theta _H^2)+\frac g{2\varepsilon }%
\left( \frac 1{\theta _H^\varepsilon }-1\right) , \nonumber \\
\;Z_{L4}^{^{\prime }} &=&1+A^{\prime }(\theta _H^2)-B^{\prime }(\theta _H^2)+%
\frac{3g}{2\varepsilon }\left( \frac 1{\theta _H^\varepsilon }-1\right)\end{aligned}$$ Thus, as well as for the dependences of ground-state properties on $t,$ in the 3D case one has the mean-field value $$\delta =3$$ and the logarithmic corrections are present. At $d=2$ we obtain the critical exponent $$\delta =9/2$$ Note that the scaling relations at $d=2$ are slightly violated since the corresponding value of $\varepsilon $ is in fact not small and the $%
\varepsilon $-expansion is applicable with a poor accuracy. However, this violation is not too large (the value $\delta =5$ can be calculated taking into account that the critical exponent $\eta =0$ to one-loop order), which indicates that the one-loop approximation gives adequate results even in this case.
The ground-state parameters at zero longitudinal magnetic field are shown and compared with RPA and GSPT results in Fig.1 for the 3D case and in Fig. 2 for the 2D case. The corrected values of $\Omega _c$ obtained from GSPT (see above), instead of those from first-order $1/S$ expansion, are used in the calculations. The $1/S$-results for $\Omega _c$ are marked by arrows. One can see that, unlike results of RPA and $1/S$-expansions, RG results have a correct critical behavior with critical exponents given by (\[CE3\]) and (\[CE2\]); besides that, they are very close to GSPT result for $%
d=3. $ For $d=2$ the difference between RG$^{\prime }$ and GSPT results increases, which demonstrates that the $\varepsilon $-expansion has a poor accuracy here. Far from the quantum phase transition, the RG results coincide with those of $1/S$ perturbation theory.
\[Sect.4\]Finite-temperature properties near QPT
------------------------------------------------
At finite temperature the situation is more complicated, since not only “spin-wave” excitations, considered in previous sections, contribute to thermodynamic properties. At $\Omega /\Omega _c\ll 1$ we have $T_c\sim IS^2$ and the phase transition occurs due to vanishing of $\langle |{\bbox \pi }%
|\rangle .$ The dominant excitations in this case are domain walls. Another situation occurs near QPT ($1-\Omega /\Omega _c\ll 1$) where the temperature phase transition is connected with the rotation of $\langle {\bbox \pi }%
\rangle $ in the spin space, while its absolute value is only slightly changed with temperature. The dominant excitations here are the “spin-wave” excitations, except for a narrow critical region close to $%
T_c. $ At intermediate values of $\Omega \ $both effects, the rotation of $%
\langle {\bbox \pi }\rangle $ and temperature variation of its absolute value, are important. Thus the $1/S$-expansion can be applied to describe the temperature phase transition only near QPT.
Being rewritten through the quantum-renormalized ground-state parameters, finite-temperature properties near QPT are universal. The finite-temperature order parameter and energy gaps obey the scaling laws
$$\begin{aligned}
\overline{S}^2(t_{-},T) &=&\overline{S}^2(t_{-},0)\,f\left( \frac T{c\Delta
_{-0}}\right) \label{Scal1} \\
\Delta _{\pm }(t_{\pm },T) &=&\Delta _{\pm }(t_{\pm },0)g_{\pm }\left( \frac
T{c\Delta _{\pm 0}}\right) \label{Scal2}\end{aligned}$$
where $\Delta _{\pm 0}=\Delta _{\pm }(t,0),$ $f$ and $g_{\pm }$ are universal scaling functions with $f(0)=g_{\pm }(0)=1$. The transition temperature is determined by zero of the function $f\left( T/c\Delta
_{-0}\right) $ or, equivalently, of $g_{-}\left( T/c\Delta _{-0}\right) .$ As discussed in Ref.[@Sachdev], the functions $g_{+}(x)$ and $g_{-}(x)$ are connected by the procedure of analytical continuation. Due to universality of scaling functions (\[Scal1\]), (\[Scal2\]), the continuum limit of developed theory, i.e. the action (\[Seff\]), can be used when treating the finite-temperature properties.
Now we pass to calculation of the functions $f$ and $g_{\pm }.$ Consider first the perturbation approach. We obtain
$$\Delta _{+}^2(t_{+},T)=\Delta _{+}^2(t_{+},0)+\frac u{2d}\left( \frac{2T}c%
\right) ^{d-1}F_d\left( \frac{c^2d}{4T^2}t_{+}\right)$$
for the disordered phase and
$$\begin{aligned}
\overline{S}_R^2(t_{-},T) &=&\overline{S}_R^2(t_{-},0)-\frac u{4d}\left(
\frac{2T}c\right) ^{d-1}F_d\left( \frac{c^2d}{2T^2}t_{-}\right)
\label{StPt} \\
\Delta _{-}^2(t_{-},T) &=&\overline{S}_R^2(t_{-},T)\left[ \frac{\Delta
_{-}^2(t_{-},0)}{\overline{S}_R^2(t_{-},0)}+\frac{3u}4\left( \frac{2T}c%
\right) ^{d-3}F_d^{\prime }\left( \frac{c^2d}{2T^2}t_{-}\right) \right]
\label{DmPt}\end{aligned}$$
for the ordered phase, where
$$F_d(x)=K_d\int\limits_0^\infty \frac{q^{d-1}dq}{\sqrt{q^2+x}}\left( \coth
\sqrt{q^2+x}-1\right) ,\,\,\,F_3(0)=\frac 1{24}$$ and $F_d^{\prime }(x)$ is the derivative with respect to $x$. Thus we have for the static susceptibility in the disordered phase at $I\Delta _{+}\ll
T\ll I$ [@Millis; @Sachdev] $$\chi ^{xx}=\frac 1{I_0}\frac 1{\Delta _{+}^2(t,0)+\gamma (2T/c)^{d-1}}
\label{HiDis}$$ with $\gamma =3I_0F_d\left( 0\right) /4c.$
At $T\geq I$ thermodynamic properties cannot be determined correctly from the above approach since in this temperature region higher-order terms in the $1/S$-expansion contribute to partition function and such an expansion becomes inapplicable. However, one can expect that at $T\gg I$ the thermodynamics is the same as for the well-studied Ising model. In particular, the susceptibility obeys the Curie law $$\chi ^{xx}=\frac{S(S+1)}{3T} \label{CurieLaw}$$ on both sides of QPT.
The equation for the transition temperature reads $$\overline{S}^2(t_{-},0)=\frac u{4d}\left( \frac{2T_c}c\right)
^{d-1}F_d\left( \frac{c^2d}{2T_c^2}t_{-}\right) \label{EqTc}$$ Since at small $t_{-}$ and $d\leq 3$ one has from (\[Sl\])$\,\overline{S}%
^2(t_{-},0)\propto (t_{-}{})^{(d-1)/2},$ we obtain $$T_c\propto \sqrt{t_{-}},\,\,\,\,1<d\leq 3 \label{TcSq}$$ where the coefficient of proportionality is determined by the solution of Eq. (\[EqTc\]). For $d>3$ we derive $$T_c\propto (t_{-})^{1/(d-1)},\,\,d>3$$ The mean-field logarithmic behavior (\[TcMF\]) is reproduced only for $%
d\rightarrow \infty $.
Consider now the renormalization of the finite-temperature properties at $%
d\leq 3$. To this end we use the approach of Ref. [@Sachdev], which treats the renormalization of the effective classical model. The disordered phase was considered in details in Ref. [@Sachdev]. Instead of analytical continuation of these results to ordered phase, we perform direct calculation of finite-temperature properties in the ordered phase. This gives a possibility to calculate correctly the value of $T_c$ not too close to QPT and also to describe finite-temperature properties at $T<T_c$. The generalization of the approach of Ref. [@Sachdev] to the ordered phase is trivial. We integrate out all the modes with nonzero Matsubara frequencies from finite-temperature partition function to obtain the effective action for the field $$\Pi =\int_0^{c/T}d\tau \,\pi _x,$$ which corresponds to $\omega _n=0$ mode, in the form $$\begin{aligned}
{\cal S}_{\text{cl}} &=&\frac c{2T}\int d^dr\left[ K(\nabla \widetilde{\Pi }%
)^2+R\widetilde{\Pi }^2\right] +\frac{\overline{\Pi }}{3!}\frac{cU}T\int d^dr%
\widetilde{\Pi }^3 \nonumber \\
&&\ \ \ \ \ \ \ \ +\frac 1{4!}\frac{cU}T\int d^dr\widetilde{\Pi }^4+...
\label{Scl1}\end{aligned}$$ Here $\widetilde{\Pi }=\Pi -\overline{\Pi }$, $\overline{\Pi }$ is determined by the condition of absence in (\[Scl1\]) of terms that are linear in $\widetilde{\Pi }$. The parameters of the model (\[Scl1\]) are given by $$R(T)=\frac 13U(T)\,\overline{\Pi }^2(T)$$ and for $d=3$$$\begin{aligned}
\overline{\Pi }^2(T) &=&\frac{18}u\left[ \overline{S}_R^2(t_{-},0)-\frac 13%
\frac{8\pi ^2g_R}{1+(3g_R/2)\ln (1/\Delta _{-0})}\left( \frac Tc\right) ^2%
\widetilde{F}_3\left( \frac{3\Delta _{-0}^2c^2}{2T^2}\right) \right] ,
\label{RR} \\
U(T) &=&\frac{8\pi ^2g_R}{1+(3g_R/2)\ln (1/\Delta _{-0})}\left[ 1+\frac{6\pi
^2g_R}{1+(3g_R/2)\ln (1/\Delta _0)}\widetilde{F}_3^{\prime }\left( \frac{%
3\Delta _{-0}^2c^2}{2T^2}\right) \right] , \label{RU}\end{aligned}$$ where $$\widetilde{F}_3(x)=K_3\int\limits_0^\infty q^2dq\left[ \frac{\coth \sqrt{%
q^2+x}-1}{\sqrt{q^2+x}}-\frac 1{q^2+x}+\frac 1{q^2}\right] ,$$ $\widetilde{F}_3^{\prime }(x)$ means the derivative with respect to $x,$ and we have represented (\[RR\]) and (\[RU\]) in the scaling form by replacing $t_{-}\rightarrow \Delta _{-0}^2$ in arguments of $F_3(x),$ $%
F_3^{\prime }(x)$. Near QPT (i.e. for small $\Delta _{-0}$) function $R(T)\ $coincides with that determined by continuation from paramagnetic phase, as it should be. The value of $K(T)$ will be needed only in zeroth-loop order, $%
K=1.$
The critical temperature is determined by the condition $\overline{\Pi }%
(T_c)=0$. Closely enough to the critical point (at $\ln (1/\Delta _{-0})\gg
1 $) we have $$T_c=\frac 3{2\pi }c\Delta _{-0}\sqrt{6\ln (1/\Delta _{-0})} \label{tcrg3d}$$ in agreement with Ref. [@Sachdev] (our definition of $\Delta _{-}$ differs $(2d)^{1/2}c$ times from that used in Ref.[@Sachdev]). At the same time, the expansion in the bare splitting (magnetic field) yields $$T_c\propto c\sqrt{t_{-}}\ln ^{1/3}(1/t_{-})\propto \overline{S}_R(t_{-},0)
\label{Tc1}$$ where the coefficient of proportionality can be determined numerically from (\[dmrg\]) and (\[tcrg3d\]). Thus, due to ground-state renormalizations, the dependences of $T_c$ on the bare and renormalized splittings turn out to be different.
The resulting classical action (\[Scl\]) is renormalized in a standard way [@Amit]. One can introduce the renormalization constants for finite-temperature theory in the form $$R=(Z_2^T/Z^T)R_r,\,\,\Pi =Z^T\Pi _r,\,\,\,U=\frac{\mu ^\epsilon }{%
K_{4-\epsilon }L_\epsilon }(Z_4^T/Z^{T2})U_r$$ where the index “$r$” stands for the quantities renormalized by temperature fluctuations, and $\epsilon =1+\varepsilon $. The expressions for $Z$-factors are the same as for the ground-state renormalization factors (\[Zi\]) with the replacement $\varepsilon \rightarrow \epsilon .$ Formulas of RG transformation also have the same form (\[Scal\]) as for the ground-state properties with $t\rightarrow R,$ $g\rightarrow U$ etc. However, now already at $d=3$ ($\varepsilon =0$) we have $\epsilon =1$ and thus the $\epsilon $-expansion can be used only approximately.
For the energy gap we obtain in this way the expression $$\Delta _{-}^2(t_{-},T)=\frac{R(T)}6\left[ 1+\frac{3TK_3L_1U(T)}{2cR^{1/2}(T)}%
\right] ^{-1/3}$$ where we have put $\epsilon =1$. For the temperature-dependent magnetization we obtain $$\overline{S}_R^2(t_{-},T)=\frac{uR(T)}{6U(T)}\left[ 1+\frac{3TK_3L_1U(T)}{%
2cR^{1/2}(T)}\right] ^{2/3}$$ The values of the temperature-transition critical exponents, $$\beta _T=1/3,\,\,\,\nu _T=7/12, \label{ExpT}$$ coincide with those of 2D quantum phase transition (\[CE2\]).
The calculated dependence $T_c(\Omega )$ is shown and compared with the mean-field and HTSE results in Fig. 3. One can see that near QPT the dependence $T_c(\Omega )$ calculated from (\[RR\]) is in excellent agreement with HTSE data. At the same time, far from QPT our approach gives much larger values of $T_c,$ as discussed in the beginning of the present section. The inflection point of the curve $T_c(\Omega ),\,\Omega
^{*}=0.35I_0,$ may be approximately related to the transverse-field value where the “non-spin-wave” excitations becomes important for description of finite-temperature properties.
For $d=2$ the system is far from its upper critical dimensionality ($%
\epsilon =2$) and $\epsilon $-expansion becomes inapplicable. Therefore we can perform only ground-state renormalizations in the results of perturbation theory (\[StPt\]) and (\[DmPt\]). In this case the critical exponents of the temperature phase transition still have their Gaussian values. However, universality hypothesis predicts that the temperature phase transition critical exponents coincide with those for the 2D Ising model, $$\beta _T=1/8,\;\nu _T=1.$$ With account of the ground-state renormalizations, the result (\[TcSq\]) for the critical temperature near the quantum phase transition ($1/\Delta
_{-0}\gg 1$) takes the form $$T_c\propto \Delta _{-0} \label{Tc2D-1}$$ while $$T_c\propto (t_{-})^{5/12} \label{Tc2D-2}$$ in terms of bare splitting (or external transverse magnetic field). The correct description of termodynamics below $T_c$ in the 2D case is still an open problem.
The Heisenberg model with strong easy-plane anisotropy
======================================================
Ground-state properties
-----------------------
We start from the general Hamiltonian of a spin system in crystal field which induces the single-site anisotropy, $${\cal H}=V_{\text{cf}}-\frac{{\cal I}}2\sum_{\langle ij\rangle }{\bf J}_i%
{\bf J}_j \label{Hcf}$$ where $V_{\text{cf}}$ is the crystal field potential, ${\bf J}$ are momentum operators, ${\cal I}$ is the exchange integral, and the direction of spin alignment will be supposed along $z$-axis. In this Section we consider the single-site easy-plane anisotropy which corresponds to $$V_{\text{cf}}=D\sum_i(J_i^x)^2 \label{E-pl}$$ where $D>0$ is the anisotropy parameter. For integer values of $J$ the lowest level is singlet. In this case with increasing $D$ the model (\[E-pl\]) demonstrates at some value $D_c$ a second-order phase transition from the phase with collinear ferromagnetic order $\langle J^z\rangle \neq 0$ to disordered phase. At the same time, the quadrupole order parameter $$Q\equiv 3\langle (J^x)^2\rangle -J(J+1)$$ is nonzero in both the phases. For half-integer values of $J,$ such transition is absent since the lowest state is two-fold degenerate. In the classical limit $J\rightarrow \infty $ with $J$ being integer, we have $D_{%
\text{c}}\sim J(J+1){\cal I}\rightarrow \infty ,$ so that integer and half-integer values of $J$ become indistinguishable.
For integer $J$ the ground state is $|A\rangle =|\widetilde{0}\rangle ,\;$and first excited state is doublet $\,|B_1\rangle =|\widetilde{1}\rangle $, $%
|B_2\rangle =|-\widetilde{1}\rangle $ where $|\widetilde{M}\rangle $ are the eigenstates of $J^x$. For $J=1$ passing to the eigenstates $|M\rangle $ of $%
J^z$ yields $$\begin{aligned}
|A\rangle &=&\frac 1{\sqrt{2}}(|1\rangle -|-1\rangle ), \nonumber \\
\,\,|B_1\rangle &=&|0\rangle ,\,\,|B_2\rangle =\frac 1{\sqrt{2}}(|1\rangle
+|-1\rangle )\end{aligned}$$
To consider the vicinity of QPT, we have to generalize the theory developed in previous Section on the singlet-doublet case. Further we restrict ourselves to the case $J=1$. In the initial spin space, QPT in the model (\[Hcf\]) with (\[E-pl\]) is not of orientational character: the spins always lie in the easy-plane. Thus the spin-wave theory in its standard form cannot properly describe the model (\[Hcf\]) near such a transition (see e.g. Ref. [@Chub]). However, as discussed in Refs. [@Onufr; @Valkov], this transition can be viewed as orientational one in the complete $SU(3)$ space which includes the $SU(2)$ spin subspace. Most convenient way to consider the rotations in the extended $SU(3)$ space is to rewrite the Hamiltonian (\[Hcf\]) with the crystal field (\[E-pl\]) in terms of the Hubbard operators $X_i^{mn}=|m_i\rangle \langle n_i|$, $$\begin{aligned}
{\cal H} &=&\frac D2\sum_i(X_i^{00}+X_i^{1,-1}+X_i^{-1,1}) \nonumber \\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -\frac{{\cal I}}2\sum_{\langle ij\rangle
}\left[
(X_i^{10}+X_i^{0-1})(X_j^{01}+X_j^{-10})+(X_i^{11}-X_i^{-1,-1})(X_i^{11}-X_j^{-1,-1})\right]
\label{He-pl}\end{aligned}$$ The rotation through “angle” $\theta $ in $SU(3)$ space is performed by the unitary transformation operator $U(\theta )$ (see Appendix C).
Following to strategy described in Sect.2, we define the ground-state critical value of $D_c$ from the condition $\sin \theta =1$ which yields $$\frac{D_c}{2{\cal I}_0}=1+\frac{3\lambda }2-\lambda \sum_{{\bf k}}\frac{6%
{\cal I}_0+{\cal I}_{{\bf k}}+{\cal I}_{{\bf k}}^2/{\cal I}_0}{2E_{{\bf k}}^0%
} \label{Dcr}$$ where $E_{{\bf k}}^0=2\sqrt{{\cal I}_0({\cal I}_0-{\cal I}_{{\bf k}})}%
,\;\lambda (=1)$ is the formal expansion parameter$.$ The critical value obtained from (\[Dcr\]) in the 3D case is $D_c/2{\cal I}_0=0.73$, which coincides with the result of HTSE [@HTSE-Pl]. For the ground-state magnetization we obtain $$\begin{aligned}
\langle J^z\rangle _{T=0}^2 &=&t_{-}\left[ 1+B(t_{-})\right] \\
B(t_{-}) &=&-\frac \lambda {t_{-}}\sum_{{\bf k}}\left[ 2\frac{2{\cal I}%
_0+(2-\eta ^2){\cal I}_{{\bf k}}}{E_{{\bf k}\alpha }}\right. \nonumber \\
&&\ \ \ \ \left. +\frac{(1+\eta ){\cal I}_0-{\cal I}_{{\bf k}}+\eta {\cal I}%
_{{\bf k}}^2/{\cal I}_0}{E_{{\bf k}\beta }}-\frac{6{\cal I}_0+{\cal I}_{{\bf %
k}}+{\cal I}_{{\bf k}}^2/{\cal I}_0}{E_{{\bf k}}^0}\right] \label{SD}\end{aligned}$$ where the excitation spectrum $E_{{\bf k}\alpha ,\beta }$ is given by (\[E-plEka\]), (\[E-plEkb\]), and $t_{-}=1-(D/D_c)^2,$ $\eta =(1-t_{-})^{1/2}$ The excitations of $\alpha $-type have a gap; they are analogous to the excitations in the transverse-field Ising model, considered in previous section. The excitations of $\beta $-type are gapless due to spontaneous breaking of rotational symmetry in the $y$-$z$ plane of spin space; these excitations are specific for $n\geq 2$ systems. Near QPT we have $E_{{\bf k}%
\beta }\approx E_{{\bf k}}^0$ and we return to the perturbation result (\[Magn\]) for the transverse-field Ising model with $E_{{\bf k}}=E_{{\bf k}%
\alpha }$ being the critical mode. However, the renormalization of (\[SD\]) is performed in a different way in comparison with the transverse-field Ising model because of another symmetry of the model (see below). The energy gap $\Delta _{-}$ in the ordered phase is given by [ $$\begin{aligned}
\Delta _{-}^2(t_{-},0) &=&t_{-}\left[ 1+A(t_{-})\right] , \\
A(t_{-}) &=&\frac 1{t_{-}}(A_0+A_1)-1 \label{DD}\end{aligned}$$ where ]{}$A_{0,1}$ are determined by (\[A0\]) and (\[A1\]).
In the presence of longitudinal magnetic field, i.e. of the field $H$, directed along $z$-axis, both modes $\alpha $ and $\beta $ becomes gapped, since this field breaks the rotational symmetry. As well as for the transverse-field Ising model, we can expect that at the intermediate magnetic field values$\,1-D/D_c\ll (H/D_c)^{2/3}\ll 1$ the ground-state properties near QPT will be determined by the magnetic field rather than by $%
t_{\pm }$. In this region we obtain the energy spectra $$\begin{aligned}
E_{{\bf k}\alpha }^2 &=&4{\cal I}_0\left[ {\cal I}_0(1+\theta _H^2)-{\cal I}%
_{{\bf k}}(1-\theta _H^2/2)\right] \nonumber \\
E_{{\bf k}\beta }^2 &=&\left[ 2({\cal I}_0-{\cal I}_{{\bf k}})+\theta _H^2(%
{\cal I}_0+{\cal I}_{{\bf k}})/2\right] \left[ 2{\cal I}_0+\theta _H^2({\cal %
I}_0-{\cal I}_{{\bf k}})/2\right]\end{aligned}$$ where $\theta _H=(4H/D_c)^{1/3}$. Performing the calculations which are similar to those for the transverse-field Ising model, we obtain the result $$\overline{S}=\theta _H\left[ 1+B^{\prime }(\theta _H^2)\right]$$ where $$\begin{aligned}
B^{\prime }(\theta _H^2) &=&-\frac \lambda {3\theta _H^2}\sum_{{\bf k}%
}\left[ 2\frac{2({\cal I}_0+{\cal I}_{{\bf k}})(1+\theta _H^2/2)-{\cal I}_{%
{\bf k}}(1-\theta _H^2)}{E_{{\bf k}\alpha }}\right. \nonumber \\
&&+\frac{(2+\theta _H^2/2){\cal I}_0-{\cal I}_{{\bf k}}+(1-\theta _H^2/2)%
{\cal I}_{{\bf k}}^2/{\cal I}_0}{E_{{\bf k}\beta }}\ \nonumber \\
&&\left. -\frac{6{\cal I}_0+{\cal I}_{{\bf k}}+{\cal I}_{{\bf k}}^2/{\cal I}%
_0}{E_{{\bf k}}^0}\right]\end{aligned}$$
A more complicated situation takes place in the case of the transverse field directed along $x$-axis [@Onufr; @Chub]. This field induces a deviation of spins from easy plane. With increasing the field value there occurs a cascade of $J$ second-order phase transitions from ferromagnetically ordered phases with $\langle J^z\rangle \neq 0,$ $\langle J^x\rangle \neq 0$ to phases which are ordered only along $x$-axis ($\langle J^z\rangle =0,$ $%
\langle J^x\rangle \neq 0$) and vice versa. The reason for this is the modification of level scheme in the magnetic field directed along the hard axis: in the case where lowest state is doublet the long-range order along $%
z $-axis is present, while in the case of singlet ground state it is evidently absent. We do not consider these transitions here (see discussion of such transition in Refs. [@Onufr; @Chub; @HTr]).
Ground-state renormalizations
-----------------------------
The above theory can be easily reformulated in the path integral formalism. The partition function has the form $${\cal Z}=\int D[a,a^{\dagger },b,b^{\dagger }]\exp \left\{ a^{\dagger }\frac{%
\partial a}{\partial \tau }+b^{\dagger }\frac{\partial b}{\partial \tau }-%
{\cal H}(a,a^{\dagger },b,b^{\dagger })\right\}$$ where ${\cal H}(a,a^{\dagger },b,b^{\dagger })$ is the average of the boson Hamiltonian over the coherent states $|a,b\rangle $ [@Klauder] (see also Ref. [@ArovasBook]). The continuum limit of the theory can be obtained if we introduce real variables $\pi _{x,y}$ and $Q_{x,y}$ instead of the complex ones $a,b$ by the relations $$\begin{aligned}
a &=&\pi _x+iQ_x \nonumber \\
b &=&Q_y+i\pi _y\end{aligned}$$ (Note that $\pi _x$ and $\pi _y$ correspond to $S^z$ and $S^y$ in the original spin space, and two additional variables $Q_{x,y}$ arise due to passing from $SU(2)$ to $SU(3)$ space). We obtain $$\begin{aligned}
{\cal S}_{\text{cont}} &=&\frac 12\int\limits_0^{c/T}d\tau \int d^dr\left[
-2i\widetilde{Q}_x(\partial \widetilde{\pi }_x/\partial \tau )+2i\widetilde{Q%
}_y(\partial \widetilde{\pi }_y/\partial \tau )\right. \nonumber \\
&&\ \ \ \ \ \ \ \left. +\widetilde{{\bf Q}}^2+(\nabla \widetilde{\bbox{\pi }}%
)^2+m^2\widetilde{\bbox{\pi }}^2\right] +\frac u{4!}\int\limits_0^{c/T}d\tau
\int d^dr\widetilde{\bbox{\pi }}^4 \nonumber \\
&&\ \ +h\int\limits_0^{c/T}d\tau \int d^dr\,\widetilde{\pi }_x \label{Seff2}\end{aligned}$$ where we have introduced the notations $\widetilde{\bbox{\pi }}^2=({\cal I}%
/c_0)\bbox{\pi }^2,\;\widetilde{{\bf Q}}^2=({\cal I}_0/c_0){\bf Q}^2$, $h=H/(%
{\cal I}c_0)^{1/2},$ the bare spin-wave velocity is given by $c_0=2\sqrt{2d}%
{\cal I}$ and we have included into (\[Seff2\]) the term connected with the external magnetic field $H$ along the $S^z$ axis. The parameters of this model, determined by the continuum limit, read $$\begin{aligned}
m_{\text{cont}}^2 &=&-t_{-}d \nonumber \\
u_{\text{cont}} &=&6d(c_0/{\cal I)}\lambda \zeta \nonumber \\
c_{\text{cont}} &=&c_0\end{aligned}$$ with $\zeta =1-t_{-}$ in the ordered phase under consideration. Proceeding in the same way as in the previous Section we integrate over $Q_{x,y}$. Then we obtain the action of the standard two-component quantum $\phi ^4$-theory in an external field, $$\begin{aligned}
{\cal S}_{\text{cont}} &=&\frac 12\int\limits_0^{c/T}d\tau \int d^dr\left[
\,(\partial \widetilde{\bbox{\pi }})^2+m^2\widetilde{\bbox{\pi }}^2\right]
\nonumber \\
&&\ \ \ \ +\frac u{4!}\int\limits_0^{c/T}d\tau \int d^dr\widetilde{\bbox{\pi
}}^4+h\int\limits_0^{c/T}d\tau \int d^dr\,\widetilde{\pi }_x. \label{S2}\end{aligned}$$
There is a crucial difference from the one-component model of the previous section, which is due to existence in the ordered phase of the gapless Goldstone mode at $H=0$. This mode changes the renormalization conditions since it leads to infrared divergences[@GIR]. To treat these divergences, we take the value of magnetic field $H$ finite, but small enough to satisfy $(H/D_c)^{2/3}\ll 1-D/D_c$. The renormalization of the action (\[S2\]) is considered in Appendix D. We obtain for effective Hamiltonian parameters at the scale $\Lambda \rho ,$ $\Lambda =(2d)^{1/2}$ the results ($d=3$) $$\begin{aligned}
g_\rho ^{-1} &=&g_R^{-1}\left[ 1+(3g_R/4)\ln (1/t_{-})+(g_R/6)\ln (1/\rho
)\right] \nonumber \\
t_\rho ^{-1} &=&t_R^{-1}\left[ 1+(3g_R/4)\ln (1/t_{-})+(g_R/6)\ln (1/\rho
)\right] \nonumber \\
&&\ \ \ \ \ \ \ \ \times \left[ 1+(5g_R/6)\ln (1/t_{-})\right] ^{-3/5}\Phi
_0(g_R,t_{-}^2) \label{scal2}\end{aligned}$$ where the function $\Phi _0(g,x)$ is given by (\[F0\]), $$g_R=K_4^{-1}Z_{L4}^{-}u_{\text{cont}}$$ is the renormalized coupling constant. For the non-universal $Z$-factors we have $$\begin{aligned}
Z_L &=&1\; \nonumber \\
(Z_{L2}^{-})^{-1} &=&1+\widetilde{A}(t_{-})+\frac g4\ln \frac 1{t_{-}},
\nonumber \\
\;(Z_{L4}^{-})^{-1} &=&1+\widetilde{A}(t_{-})-B(t_{-})+\frac{3g}4\ln \frac 1{%
t_{-}},\end{aligned}$$ where $A(t_{-}),$ $B(t_{-})$ are given by (\[SD\]), (\[DD\]), the tilde sign means that the contributions of the Goldstone mode $\beta $ should be excluded from $A(t_{-})$.
Putting in the above expressions $\rho =\widetilde{h}^{1/2}$ where $%
\widetilde{h}=H/[{\cal I}_0\overline{J}_R(H=0)]$ we have for the magnetization at $d=3$ the result $$\overline{J}_R(t_{-},0)=\sqrt{\frac{Z_{L4}^{-}t_{-}}{Z_{L2}^{-}}}\left[ 1+%
\frac{5g_R}6\ln \frac 1{t_{-}}\right] ^{3/10}\Phi _0^{1/2}(g_R,t_{-}^2)$$ (the terms divergent in $H$ are canceled in $\overline{J}_R$). The gap for $%
\alpha $-type excitations, which determines the longitudinal susceptibility, reads
$$\begin{aligned}
\Delta _{-}^2(t_{-},0) &=&12\Theta _0^2/\ln (1/\widetilde{h}) \label{dmrg3d}
\\
\Theta _0^2 &=&\frac{t_{-}}{Z_{L2}^{-}}\left[ 1+\frac{5g_R}6\ln \frac 1{t_{-}%
}\right] ^{3/5}\frac{\Phi _0(g_R,t_{-}^2)}{g_R} \nonumber \\
\ &=&Z_{L4}^{-}\frac{\overline{J}_R^2(t_{-},0)}{g_R} \nonumber\end{aligned}$$
Up to some nonuniversal factor $Z_\rho $ we have in the one-loop order $%
\Theta _0=Z_\rho (\rho _s/6dc)^{1/2}$ with $\rho _s$ being the ground-state spin stiffness. At $H\rightarrow 0$ the gap vanishes as $\ln ^{-1}({\cal I}%
_0/H)$ $,$ which is a consequence of degeneracy of the system.
For intermediate values of external magnetic field, i.e. at $1-D/D_c\ll
(H/D_c)^{2/3}\ll 1$, a characteristic scale for both types of excitations is $1/\theta _H$, and the expressions for renormalization factors $Z_i^{\text{%
cont}}$ have the form, which is standard in the two-component $\phi ^4$ model[@Amit]. Then we obtain for the magnetization
$$\overline{J}_R(\theta _H,0)=\theta _H\sqrt{\frac{Z_{L4}^{\prime }}{%
Z_{L2}^{\prime }}}\left[ 1+(5g_R/6)\ln (1/\theta _H^2)\right] ^{3/10}$$
with $$Z_{L4}^{^{\prime }}/Z_{L2}^{^{\prime }}=1-B^{\prime }(\theta _H^2)+g\ln
\frac 1{\theta _H},$$
Finite-temperature properties
-----------------------------
Using perturbation theory we obtain for the finite-temperature magnetization the result (see Appendix C) $$\langle J^z\rangle ^2=\langle J^z\rangle _{T=0}^2-\frac{{\cal I}_0\lambda }{%
2c_0}\left( \frac{2T}{c_0}\right) ^{d-1}\left[ 3F_d\left( \frac{c_0^2d}{2T^2}%
t_{-}\right) +F_d(0)\right] \label{JzT}$$ where $t_{-}=1-(D/D_c)^2.$ The first and second terms in the square brackets correspond to contributions of $\alpha $ and $\beta $-type excitations respectively. At extremely low temperatures $T\ll {\cal I}_0t_{-}$ the contribution of $\alpha $-excitations is exponentially small and the temperature-dependent part of magnetization is determined entirely by second term in square brackets of (\[JzT\]). At temperatures $T\gg {\cal I}%
_0t_{-} $ the situation changes and both type of excitations give the same temperature dependence, the contribution of the mode $\alpha $ being three times larger.
Consider now the renormalization of the finite-temperature theory. Integrating out the field $\pi (q,\omega _n)$ with $\omega _n\neq 0$ from the action (\[S2\]) we obtain the action of effective classical model $$\begin{aligned}
{\cal S}_{\text{cl}} &=&\frac c{2T}\sum_{{\bf q}}\left\{ q^2\widetilde{{%
\bbox \Pi }}_{{\bf q}}\widetilde{{\bbox \Pi }}_{-{\bf q}}+\left[ R(T)+\frac{%
3h}{\overline{\Pi }(T)}\right] \widetilde{{\Pi }}_{x{\bf q}}\widetilde{{\Pi }%
}_{x,-{\bf q}}+\frac h{\overline{\Pi }(T)}\widetilde{{\Pi }}_{y{\bf q}}%
\widetilde{{\Pi }}_{y,-{\bf q}}\right\} \label{Scl} \\
&&\ \ \ \ \ \ \ \ \ \ +\frac{\overline{{\Pi }}}{3!}\frac{cU}T\sum_{{\bf q}_1%
{\bf q}_2{\bf q}_3}\left( \widetilde{{\bbox \Pi }}_{{\bf q}_1}\widetilde{{%
\bbox \Pi }}_{{\bf q}_2}\right) {\Pi }_{x,{\bf q}_3}\delta ({\bf q}_1+{\bf q}%
_2+{\bf q}_3)+... \nonumber \\
&&\ \ \ \ \ \ \ \ \ \ +\frac 1{4!}\frac{cU}T\sum_{{\bf q}_1{\bf q}_2{\bf q}_3%
{\bf q}_4}\left( \widetilde{{\bbox \Pi }}_{{\bf q}_1}\widetilde{{\bbox \Pi }}%
_{{\bf q}_2}\right) \left( \widetilde{{\bbox \Pi }}_{{\bf q}_3}\widetilde{{%
\bbox \Pi }}_{{\bf q}_4}\right) \delta ({\bf q}_1+{\bf q}_2+{\bf q}_3+{\bf q}%
_4)+... \nonumber\end{aligned}$$ where the field $\widetilde{{\bbox \Pi }}={\bbox \Pi +}(\overline{\Pi },0)$ is now two-component one, and the dots stand for higher-order terms. For the parameters of the model (\[Scl\]) we have $$R(T)=\frac 13\overline{\Pi }^2(T)U(T)$$ and for $d=3$ $$\begin{aligned}
\overline{\Pi }^2(T) &=&\frac{18g_R}u\left[ \Theta _0^2-\frac{32\pi ^2}9%
\left( \frac Tc\right) ^2\widetilde{F}_3\left( 0\right) \right] \nonumber \\
U(T) &=&\frac{8\pi ^2g_R}{\ln (1/\widetilde{h})}\left[ 1+\frac{20\pi ^2g_R}{%
3\ln (1/\widetilde{h})}\widetilde{F}_3^{\prime }\left( \frac{\widetilde{h}%
^2c^2}{4T^2}\right) \right]\end{aligned}$$ Note that both $R(T)$ and $U(T)$ vanish at $H\rightarrow 0$ as $\ln ^{-1}(%
{\cal I}_0/H)$ due to quantum fluctuations, while $\overline{\Pi }(T)$ is finite in this limit. The value of $T_c,$ as determined by the condition $%
\overline{\Pi }(T_c)=0,$ read $$T_c=\frac{3\sqrt{3}}{2\pi }c\Theta _0 \label{Tc2}$$ where $\Theta _0$ is given by (\[DD\]). Thus the result (\[Tc2\]) coincides with that obtained in Ref.[@Sachdev] up to the nonuniversal factor $Z_\rho $ As well as for the 3D transverse-field Ising model, in one-loop order the transition temperature turns out to be proportional to the ground-state magnetization.
It should be noted that, owing to presence of the gapless Goldstone mode, the model (\[Scl\]) is applicable at $T=T_c$ only very close to QPT, unlike the corresponding model (\[Scl1\]) for the one-component case. Thus one can put $Z_{L2}=Z_{L4}=1$. The calculated dependence $T_c(D)$ is shown and compared with HTSE data in Fig.4. As well as for the one-component case, the result (\[Tc2\]) agrees well with HTSE data closely enough to QPT. A more complete treatment can be performed by considering quasimomentum-dependent vertices in (\[Scl\]). This is a complicated task which is not considered in the present paper.
By the same reason, the model (\[Scl\]) cannot be used for determining magnetization below $T_c.$ However, one can expect from the hypothesis of universality the standard value of the two-component three-dimensional $\phi
^4$ theory critical exponent $$\beta _T=\frac 12-\frac{3\epsilon }{2(n+8)}=7/20,$$ which is practically the same as in the one-component case ($\beta _T=1/3$). Unlike the one-component case, the logarithmic correction to $\overline{S}%
(t_{-},T)$ is expected near the temperature phase transition due to the gapless Goldstone mode.
For $d=2$ the contribution of the gapless mode is logarithmically divergent and therefore the long-range order at finite temperatures is absent, unlike the case of the transverse-field Ising model.
Conclusions
===========
In the present paper we have considered systems that demonstrate in the ground state a quantum phase transition (QPT). Near QPT the saturation moment $\overline{S}_0$ is small, but the Curie constant in (\[CurieLaw\]) is not suppressed. We have $T_c\propto \overline{S}_0$ which is determined by the value of dynamical critical exponent, $z=1$. The susceptibility (\[HiDis\]) in the intermediate temperature region $\Delta _0\ll T/I\ll 1$ is determined by the small ground-state energy gap $\Delta _0$ and demonstrates a $1/T^{d-1}$ behavior. In the strong enough longitudinal magnetic field $%
\Delta _0\ll (H/IS)^{1/3}\ll 1$ the ground state parameters are determined by magnetic field value rather than by closeness to QPT. The corresponding dependences have been obtained. Our approach gives a possibility to investigate both non-universal and universal renormalizations of the ground-state parameters. The ground-state renormalizations turn out to be important in the vicinity of QPT. Thus the results for thermodynamic quantities (e.g. transition temperature) have different forms as functions of renormalized splitting and bare transverse (external magnetic) field, see (\[tcrg3d\]),(\[Tc1\]) and (\[Tc2D-1\]),(\[Tc2D-2\]). This should be taken into account when treating experimental data.
The discussed class of magnets is similar in some respects to weak itinerant magnets. Note that for weak itinerant ferromagnets we have $T_c\propto
\overline{S}_0^{3/2}$ (see, e.g., Ref. [@Moriya]), which is due to that main contribution to thermodynamics comes from paramagnons ($z=3$). As well as for considered localized-moment systems, calculation of non-universal ground-state parameters for itinerant magnets is of interest, in particular for different forms of bare electron density of states.
Now we discuss the experimental situation for some systems exhibiting magnetic and structural transitions. The compound DyVO$_4$ demonstrates a structural phase transition at $T_D=14\,$K. The low–lying energy levels in the spectrum of this system are two Kramers doublets with the splitting $%
\Delta _1=27$cm$^{-1}$ at $T=0$ and $\Delta _2=9$cm$^{-1}$ at $T>T_D$. Neglecting the Kramers degeneracy one can describe this system by the transverse-field Ising model with $\Omega /I=1/3$ (see Ref. [@Gehring]). The corresponding point in $\Omega /I$-$T_c$ coordinates is marked in Fig.1d. This point lies exactly on the HTSE curve and therefore HTSE results are applicable in this region of $\Omega /I.$ One can see that DyVO$_4$ lies far from QPT, so that the above-developed theory is not applicable for this system.
Other systems, which are well described by the transverse-field Ising model, are the ferroelectric quantum crystals like KH$_2$PO$_4\;$(see, e.g., Ref.[@Samara]). However, to our knowledge, corresponding detailed data on ground-state order parameters are absent. To fit experimental data on $T_c$ of Ref. [@Samara], we need explicit dependence of the tunneling parameter $\Omega $ on pressure.
There are very few experimental data on singlet-singlet systems demonstrating magnetic phase transitions. The system LiTb$_x$Y$_{1-x}$F$_4$ [@LiTb] is usually assumed to be characterized by long-range exchange interactions and therefore well described by the mean-field theory. The singlet-doublet case is represented by the system NiSi$_2$F$_6$ which is an $%
J=1$ easy-plane Heisenberg magnet. The anisotropy constant is changed under pressure and thus the value of $D/D_c$ can be varied near unity in the experiment. The pressure dependence of anisotropy constant was measured experimentally [@FSiNi]. However, to our knowledge, the data on the pressure dependence of exchange parameter are absent, although it is supposed to be considerable [@HTSE-Pl]. There are also few experimental data on the ground-state magnetization near QPT. At $p=8.6$Kbar, one has the experimental values $\overline{J}(T=0)=0.3$ and $T_c=110$mK [@FSiNi]. The calculation according to (\[Tc2\]) yields ${\cal I}=60$mK which is larger than the $p=0$ value, ${\cal I}=40$mK[@FSiNi:Pars]. Praseodymium in the dhcp phase contains both “cubic” and “hexagonal” sites[@CooperBook; @Pr], so that separation of different contributions makes an additional problem. Generalization of our approach to the singlet-triplet case in connection with the Pr ions in cubic crystal field will be presented elsewhere.
Generally, the $1/S$ perturbation theory combined with field-theoretical scaling analysis enables one to obtain a description of ground-state properties of transverse-field Ising model, which in a good agreement with the results of the fourth-order ground-state perturbation theory [@GSPT] for all the values of $\Omega .$ The only fitting parameter used is the critical field value $\Omega _c.$ The finite-temperature properties are considered with the use of approach of Ref.[@Sachdev]. The same analysis for the $S=1$ easy-plane Heisenberg model is performed within the expansion in formal parameter $\lambda (=1)$ which plays the role of $1/S.$ In this case, besides the critical mode, there is a gapless Goldstone mode, which considerably modifies the conditions of renormalizations. The consideration of QPT in degenerate systems induced by the external magnetic field within the approach used is of interest. In particular, in the case of single-site anisotropy oscillations of the effective moment with increasing magnetic field or temperature are expected in such systems with $S>1.$
It is of interest to apply the approach used to various 3D and 2D systems demonstrating orientational and metamagnetic phase transition with changing the external magnetic field or anisotropy, e.g. for yttrium garnets[@Levitin] and magnetic films[@films]. Depending on a concrete physical situation, such systems can be described by the strongly anisotropic transverse-field Ising model or Heisenberg model with small anisotropy.
Mapping of the anisotropic Heisenberg model onto the transverse-field Ising model
=================================================================================
In this Appendix we discuss a possibility of mapping procedure of anisotropic Heisenberg model (\[Hcf\]) onto the transverse-field Ising model. We consider only one important case where the lowest level of $V_{%
\text{cf}}$ is singlet and there is QPT to disordered phase at strong enough $V_{\text{cf}}.$ Provided that the first excited state is also singlet, neglecting all energy levels except lowest and first excited states we can introduce the pseudospin-$1/2$ operators ${\bf S}$[**,**]{} to obtain [@CooperBook; @Grover] $$V_{\text{cf}}=-\Delta \sum_iS_i^z,\,\,\,\,J_i^z=2\alpha S_i^x \label{JS}$$ where $\alpha =\langle A|J^z|B\rangle $ is the matrix element of ${\bf J,}$ $%
|A\rangle $ and $|B\rangle $ are the lowest and first excited states, $%
\Delta $ is the energy gap between these states. (It should be noted that left-hand sides of Eqs. (\[JS\]) act in real-spin space, while right-hand sides in pseudospin space. Thus the equality signs are used only in the sense of identity of averages.) Then we obtain the transverse-field Ising model with $I=4\alpha ^2{\cal I}$ and $\Omega =\Delta .$ The order parameter of Heisenberg model $\langle J^z\rangle $ is connected with the order parameter in the transverse-field Ising model by $$\langle J^z\rangle =2\alpha \langle S^x\rangle$$
Consider now the case where the excited state is a multiplet with the states $|B_m\rangle ,$ $m=1...N-1.$ Neglecting the degeneracy of upper energy level, one can use the same mapping (\[JS\]) if we choose $\alpha
^2=\sum_{m=1}^{N-1}\langle A|J^z|B_m\rangle ^2.$ However, in this case the original $SU(N)$ spin space is projected onto $SU(2)$ pseudospin space, and thus $N-2$ degrees of freedom are neglected. Thus this approach does not give a possibility to take into account properly the symmetry of the original model and therefore can be applied only outside the critical region. To obtain a correct description of such systems in the critical region one should consider the transition in complete $SU(N)$ space.
The above consideration shows that, in principle, the transverse-field Ising model (\[ITF\]) can qualitatively describe singlet magnets even in the case where the exchange interactions in the true momentum space are isotropic, as in model (\[Hcf\]).
Calculation of spin Green’s function and order parameter of the transverse-field Ising model within the $1/S$ expansion {#app:B}
=======================================================================================================================
Consider first the disordered phase where $\langle \pi _x\rangle =0.$ Representing $\pi _{zi}=(1+1/S-\pi _{xi}^2-\pi _{yi}^2)^{1/2}$ and assuming $%
\langle \pi _{x,y}^2\rangle \sim 1/S$ (validity of this statement will be checked below) we expand square root to second order in $1/S$ to obtain $$\begin{aligned}
{\cal S}_{\text{dyn}} &=&\frac{iSw_S}2\sum_i\int\limits_0^{1/T}d\tau \left(
\pi _{xi}\frac{\partial \pi _{yi}}{\partial \tau }-\pi _{yi}\frac{\partial
\pi _{xi}}{\partial \tau }\right) +\frac{iS}8\sum_i\int\limits_0^{1/T}d\tau
\left( \pi _{xi}^2+\pi _{yi}^2\right) \left( \pi _{xi}\frac{\partial \pi
_{yi}}{\partial \tau }-\pi _{yi}\frac{\partial \pi _{xi}}{\partial \tau }%
\right) \label{Ldyn} \\
{\cal S}_{\text{st}} &=&-\frac 12\int\limits_0^{1/T}d\tau \left[
IS^2\sum_{\langle ij\rangle }\pi _{xi}\pi _{xj}+\left( T{\cal P}-\Omega
Sw_S\right) \sum_i\left( \pi _{xi}^2+\pi _{yi}^2\right) -\frac{\Omega S}4%
\sum_i\left( \pi _{xi}^2+\pi _{yi}^2\right) ^2\right] \label{Lst}\end{aligned}$$ where $w_S=(1+1/2S)^{-1}$ and ${\cal P}=\sum_{\omega _n}1$ ($\omega _n$ being the Matsubara frequencies) is formally divergent quantity which comes from the measure of integration, this divergence will be canceled in final results [@Div]. To first order in $1/S$ (we suppose that $\Omega _c\sim
I_0S$) we obtain by standard perturbation theory methods the matrix two-point vertex function of $\pi _x$,$\,\pi _y$ fields in the form $$\Gamma ({\bf q},\omega _n)=\left(
\begin{array}{cc}
\Omega S\left( w_S+\frac{3X+Y}2\right) +S\Upsilon -T{\cal P}-I_{{\bf q}}S^2
& iS\omega _n\left( w_S+\frac{X+Y}2\right) \\
iS\omega _n\left( w_S+\frac{Y+X}2\right) & \Omega S\left( w_S+\frac{3Y+X}2%
\right) +S\Upsilon -T{\cal P}
\end{array}
\right)$$ where $$\begin{aligned}
X &=&\langle \pi _{xi}^2\rangle =T\sum_{{\bf q},\omega _n}\frac{\Omega /S}{%
\omega _n^2+E_{{\bf q}}^2}, \nonumber \\
\,\,Y &=&\langle \pi _{yi}^2\rangle =T\sum_{{\bf q},\omega _n}\frac{\Omega
/S-I_{{\bf q}}}{\omega _n^2+E_{{\bf q}}^2} \label{XY}\end{aligned}$$ and $$\,\Upsilon =i\langle \pi _{xi}(\partial \pi _{yi}/\partial \tau )\rangle =%
\frac TS\sum_{{\bf q},\omega _n}\frac{\omega _n^2}{\omega _n^2+E_{{\bf q}}^2}%
,$$ with $E_{{\bf q}}=\sqrt{\Omega (\Omega -SI_{{\bf q}})}$ being the bare “spin-wave” spectrum in the disordered phase. (One can easily verify that $%
X,Y$ are of the order of $1/S,$ as it was supposed in the beginning). Using the identity $$\Omega SX+S\Upsilon -T{\cal P}=I_0S^2X^{\prime } \label{Id1}$$ where $$X^{\prime }=\langle \pi _{xi}\pi _{xj}\rangle =\frac{T\Omega }{I_0S}\sum_{%
{\bf q},\omega _n}\frac{I_{{\bf q}}}{\omega _n^2+E_{{\bf q}}^2} \label{X'}$$ to eliminate the divergences, we derive $$\Gamma ({\bf q},\omega _n)=\left(
\begin{array}{cc}
\Omega S\left( w_S+\frac{X+Y}2\right) +I_0S^2X^{\prime }-I_{{\bf q}}S^2 &
iS\omega _n\left( w_S+\frac{X+Y}2\right) \\
iS\omega _n\left( w_S+\frac{Y+X}2\right) & \Omega S\left( w_S+\frac{3Y-X}2%
\right) +I_0S^2X^{\prime }
\end{array}
\right) \label{GammaD}$$ The value of $\Omega _c$ is determined by the condition $\Gamma _{xx}(0,0)=0$ However, the averages (\[XY\]) and (\[X’\]) which determine $1/S$-corrections to $\Gamma ({\bf q},\omega _n)$ are $\Omega $-dependent itself. For consistency, we have to calculate these averages with the zeroth-order value $\Omega _c=I_0S$. Then we obtain the result for $\Omega _c$ (\[Wc\]) of main text. At an arbitrary $\Omega $ we perform the replacement $\Omega
\rightarrow I_0S(\Omega /\Omega _c)$ in (\[XY\]) and (\[X’\]), which changes $\Gamma ({\bf q},\omega _n)$ in the order of $1/S^2$ only and gives a possibility to take into account consistently the shift of $\Omega _c$ owing to quantum fluctuations. Substituting the result for $\Omega _c$ into (\[GammaD\]), we obtain the results (\[Gamma\]), (\[DeltaP\]) and (\[DP\]) of the main text.
In the ordered phase we rotate the coordinate system through the angle $%
\theta $ around $\pi _y$-axis and again, in new coordinates, expand $%
\widetilde{\pi }_z$ in powers of $\widetilde{\pi }_x,\widetilde{\pi }_y\;$($%
\widetilde{\pi }_y=\pi _y$)$.$ Then we obtain to fourth order $$\begin{aligned}
{\cal S}_{\text{st}} &=&-\frac{IS^2}2\sum_{\langle ij\rangle
}\int\limits_0^{1/T}d\tau \left[ -\widetilde{\pi }_{xi}w_S^{-1}\sin 2\theta +%
\widetilde{\pi }_{xi}\widetilde{\pi }_{xj}\cos ^2\theta -(\widetilde{\pi }%
_{xi}^2+\widetilde{\pi }_{xj}^2)\sin ^2\theta \right. \nonumber \\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left. +\frac 12\widetilde{\pi }_{xj}(%
\widetilde{\pi }_{xi}^2+\widetilde{\pi }_{yi}^2)\sin 2\theta +\frac{(%
\widetilde{\pi }_{xi}^2+\widetilde{\pi }_{yi}^2)(\widetilde{\pi }_{xj}^2+%
\widetilde{\pi }_{yj}^2)-(\widetilde{\pi }_{xi}^2+\widetilde{\pi }_{yi}^2)^2}%
4\sin ^2\theta \right] \nonumber \\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -\Omega S\sum_i\int\limits_0^{1/T}d\tau
\left[ \widetilde{\pi }_{xi}\sin \theta -\frac{\widetilde{\pi }_{xi}^2+%
\widetilde{\pi }_{yi}^2}2w_S\cos \theta -\frac{(\widetilde{\pi }_{xi}^2+%
\widetilde{\pi }_{yi}^2)^2}8\cos \theta \right] \nonumber \\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\frac{T{\cal P}}2\sum_i\int\limits_0^{1/T}d%
\tau \left( \widetilde{\pi }_{xi}^2+\widetilde{\pi }_{yi}^2\right) ,
\label{SOrd}\end{aligned}$$ ${\cal S}_{\text{dyn}}$ having the same form (\[Ldyn\]) as in the disordered phase with the replacement $\pi \rightarrow \widetilde{\pi }$. Determining the angle $\theta $ from the condition $\langle \widetilde{\pi }%
_x\rangle =0$ we obtain $$\cos \theta =\frac \Omega {I_0S}\left[ 1+\frac 1{2S}-\frac{X+2X^{\prime }+Y}2%
\right] ^{-1} \label{TetITF}$$ where $$\begin{aligned}
X &=&\langle \widetilde{\pi }_{xi}^2\rangle =T\sum_{{\bf q},\omega _n}\frac{%
I_0}{\omega _n^2+E_{{\bf q}}^2}, \nonumber \\
X^{\prime } &=&\langle \widetilde{\pi }_{xi}\widetilde{\pi }_{xj}\rangle
=T\sum_{{\bf q},\omega _n}\frac{I_{{\bf q}}}{\omega _n^2+E_{{\bf q}}^2},
\nonumber \\
\,\,Y &=&\langle \widetilde{\pi }_{yi}^2\rangle =T\sum_{{\bf q},\omega _n}%
\frac{I_0-\eta I_{{\bf q}}}{\omega _n^2+E_{{\bf q}}^2}. \label{XX'Y}\end{aligned}$$ $E_{{\bf q}}=S\sqrt{I_0(I_0-\eta I_{{\bf q}})}$ and $\eta =(\Omega /I_0S)^2.$ Besides the averages (\[XX’Y\]), we introduce the quantity $$\,\Upsilon =i\langle \widetilde{\pi }_{xi}(\partial \widetilde{\pi }%
_{yi}/\partial \tau )\rangle =\frac TS\sum_{{\bf q},\omega _n}\frac{\omega
_n^2}{\omega _n^2+E_{{\bf q}}^2}.$$ For the two-point vertex function of the fields $\widetilde{\pi }_x,%
\widetilde{\pi }_y$ we have $$\Gamma ({\bf q},\omega _n)=\left(
\begin{array}{cc}
I_0S^2\left( 1+X-\eta X^{\prime }\right) -I_{{\bf q}}S^2W+F_{{\bf q}%
n}+S\Upsilon -T{\cal P} & iS\omega _n\left( w_S+\frac{X+Y}2\right) \\
iS\omega _n\left( w_S+\frac{Y+X}2\right) & I_0S^2\left( 1+Y-\eta X^{\prime
}\right) +S\Upsilon -T{\cal P}
\end{array}
\right)$$ where $$W=\eta (w_S+X+2X^{\prime }+Y)+(1-\eta )X^{\prime },$$ The term with $$\begin{aligned}
F_{{\bf q}n} &=&-\frac{S^4}{2T}\eta (1-\eta )\sum_{{\bf k},\omega _m}\left[
(2I_{{\bf k}}I_{{\bf k+q}}+4I_{{\bf q}}I_{{\bf k}}+2I_{{\bf k+q}}^2+I_{{\bf q%
}}^2)M_{xxxx}(k,q)\right. \nonumber \\
&&\ \ \ \ \ \left. +I_{{\bf q}}^2M_{yyyy}(k,q)+2I_{{\bf q}}(I_{{\bf q}}+2I_{%
{\bf k}})M_{xyxy}(k,q)\right]\end{aligned}$$ where $k=({\bf k,}\omega _m)$, $q=({\bf q,}\omega _n)$ and $$M_{\alpha \beta \gamma \delta }(k,q)=\langle \widetilde{\pi }_\alpha (k)%
\widetilde{\pi }_\beta (-k)\rangle \langle \widetilde{\pi }_\gamma (k+q)%
\widetilde{\pi }_\delta (-k-q)\rangle$$ ($\alpha ,\beta ,\gamma ,\delta =x,y$) arises due to the contribution of the cubic term in (\[SOrd\]) in the second order of perturbation theory. This term has the same order in $1/S$ as other terms and should be retained. Using the identity $$I_0S^2X+S\Upsilon -T{\cal P}=I_0S^2\eta X^{\prime },$$ which is an analog of (\[Id1\]) for the ordered phase, we obtain $$\Gamma ({\bf q},\omega _n)=\left(
\begin{array}{cc}
S^2(I_0-WI_{{\bf q}})+F_{{\bf q}n} & iS\omega _n\left( w_S+X/2+Y/2\right) \\
iS\omega _n\left( w_S+X/2+Y/2\right) & I_0S^2\left( 1+Y-X\right)
\end{array}
\right) \label{GG}$$ Performing again the replacement $\Omega \rightarrow I_0S(\Omega /\Omega _c)$ in (\[XX’Y\]) and reexpressing (\[GG\]) in terms of $\Omega _c,$ we obtain the results (\[Gamma\]), (\[DeltaM\]) and (\[DM\]) of the main text. For the temperature-dependent order parameter we obtain $$\begin{aligned}
\overline{S} &\equiv &S\langle \pi _x\rangle =S\sin \theta \langle
\widetilde{\pi }_x\rangle \nonumber \\
\ &=&S\left\{ 1-\eta -\frac \eta {2S}\sum_{{\bf q}}\frac{2I_0+\eta I_{{\bf q}%
}}{\sqrt{I_0(I_0-\eta I_{{\bf q}})}}\coth \frac{S\sqrt{I_0(I_0-\eta I_{{\bf q%
}})}}{2T}\right. \nonumber \\
&&\ \ \ \ \ \ \ \ \left. +\frac \eta {2S}\sum_{{\bf q}}\frac{2I_0+I_{{\bf q}}%
}{\sqrt{I_0(I_0-I_{{\bf q}})}}\right\} ^{1/2}\left( 1+\frac 1{2S}-\frac{X+Y}2%
\right) \label{Magn}\end{aligned}$$ Note that near QPT the last multiplier in this expression is close to unity and only slightly temperature-dependent. Thus it can be replaced by its zero-temperature value.
Rotation in $SU(3)$ space for the easy-plane Heisenberg model {#app:C}
=============================================================
Following to Refs. [@Fl-Epl; @Onufr; @Valkov] we perform in the Hamiltonian (\[He-pl\]) the unitary transformation $$\widetilde{X}_i^{pq}=U^{\dagger }(\theta )X_i^{pq}U(\theta )$$ with $$\begin{aligned}
U(\theta ) &=&\exp [\theta (X^{-1,1}-X^{1,-1})/2] \nonumber \\
\ &=&1+[\cos (\theta /2)-1](X^{-1,-1}+X^{1,1})+\sin (\theta
/2)(X^{-1,1}-X^{1,-1})\end{aligned}$$ Then the Hamiltonian takes the form $${\cal H}={\cal H}^{(1)}+{\cal H}^{(2)}$$ with $$\begin{aligned}
{\cal H}^{(1)} &=&\frac 12\sum_i\left[ (D\sin \theta +2{\cal I}_0\cos
^2\theta )(2X_i^{11}+X_i^{00})+DX_i^{00}+(D-2{\cal I}_0\sin \theta )\cos
\theta (X_i^{1,-1}+X_i^{-1,1})\right] \nonumber \\
{\cal H}^{(2)} &=&-\frac{{\cal I}}2\sum_{<ij>}\left[ \cos \theta
(2X_i^{11}+X_i^{00})-\sin \theta (X_i^{1,-1}+X_i^{-1,1})\right] \left[ \cos
\theta (2X_j^{11}+X_j^{00})-\sin \theta (X_j^{1,-1}+X_j^{-1,1})\right]
\nonumber \\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -\frac{{\cal I}}2%
\sum_{<ij>}\sum_{\sigma =\pm 1}\left[ 2X_i^{0\sigma }X_j^{\sigma 0}+\sigma
\sin \theta (X_i^{0\sigma }X_j^{0\sigma }+X_i^{\sigma 0}X_j^{\sigma 0})+\cos
\theta (X^{-\sigma 0}X^{\sigma 0}+X^{0,-\sigma }X^{0\sigma })\right]\end{aligned}$$ where ${\cal I}_0=2d{\cal I}$ and we have dropped the tilde sign at the $X$-operators$.$ Further we represent Hubbard operators via boson ones[@Onufr; @Valkov] $$\begin{aligned}
X^{1,-1} &=&a^{\dagger }(1-\lambda a^{\dagger }a-\lambda b^{\dagger }b)^{1/2}
\nonumber \\
X^{0,-1} &=&b^{\dagger }(1-\lambda a^{\dagger }a-\lambda b^{\dagger }b)^{1/2}
\nonumber \\
X^{10} &=&a^{\dagger }b,\,\,\,X^{00}=b^{\dagger }b,\,\,\,X^{11}=a^{\dagger }a
\label{HubBose1}\end{aligned}$$ where $\lambda (=1)$ is the parameter introduced to construct perturbation theory (cf. the Holstein-Primakoff expansion in the case of a Heisenberg magnet). Then we obtain the Hamiltonian of the bosons $${\cal H}={\cal H}_1+{\cal H}_2+{\cal H}_3+{\cal H}_4+... \label{H1234}$$ where $$\begin{aligned}
{\cal H}_1 &=&\frac 12\cos \theta (D-2{\cal I}_0\sin \theta )\sum_i\left(
a_i^{\dagger }+a_i\right) \\
{\cal H}_2 &=&(D\sin \theta +2{\cal I}_0\cos ^2\theta )\sum_ia_i^{\dagger
}a_i+[D(1+\sin \theta )/2+{\cal I}_0\cos ^2\theta ]\sum_ib_i^{\dagger }b_i
\nonumber \\
&&\ \ \ \ \ \ \ \ -\frac{{\cal I}}2\sum_{\langle ij\rangle }\left[ \sin
^2\theta \left( a_i^{\dagger }+a_i\right) \left( a_j^{\dagger }+a_j\right)
+2b_i^{\dagger }b_j-\sin \theta \left( b_i^{\dagger }b_j^{\dagger
}+b_ib_j\right) \right] \\
{\cal H}_3 &=&-\frac 12\cos \theta (D-2{\cal I}_0\sin \theta )\sum_i\left[
a_i^{\dagger }\left( a_i^{\dagger }a_i+b_i^{\dagger }b_i\right) +\left(
a_i^{\dagger }a_i+b_i^{\dagger }b_i\right) a_i\right] \nonumber \\
&&\ \ \ \ \ \ \ \ +{\cal I}\cos \theta \sum_{\langle ij\rangle }\left[ \sin
\theta \left( 2a_i^{\dagger }a_i+b_i^{\dagger }b_i\right) \left(
a_j^{\dagger }+a_j\right) -\left( b_ia_j^{\dagger }b_j+b_i^{\dagger
}b_j^{\dagger }a_j\right) \right] \\
{\cal H}_4 &=&-\frac{{\cal I}}2\sum_{\langle ij\rangle }\left[ \sin ^2\theta
\left( a_i^{\dagger }a_i+2b_i^{\dagger }b_i\right) \left( a_j^{\dagger
}a_j+2b_j^{\dagger }b_j\right) +2a_i^{\dagger }a_jb_j^{\dagger }b_i\right.
\nonumber \\
&&\ \ \ \left. -\cos ^2\theta \left( b_i^{\dagger }+b_i\right) \left(
a_j^{\dagger }a_jb_j^{\dagger }+b_j^{\dagger }b_j^{\dagger }b_j+\text{h.c.}%
\right) +\cos 2\theta \left( a_ia_jb_i^{\dagger }b_j^{\dagger }+\text{h.c.}%
\right) \right. \nonumber \\
&&\ \ \ \left. -\left( a_i^{\dagger }a_j^{\dagger }a_ja_j+a_i^{\dagger
}b_j^{\dagger }b_ja_j+\text{h.c.}\right) +\cos 2\theta \left( a_i^{\dagger
}a_i^{\dagger }a_ia_j^{\dagger }+a_i^{\dagger }b_i^{\dagger }b_ia_j^{\dagger
}+\text{h.c.}\right) \right]\end{aligned}$$ and we have omitted terms containing more than four Bose operators. Diagonalizing the quadratic part ${\cal H}_2$ of the Hamiltonian (\[H1234\]) we obtain $${\cal H}_2=\sum_{{\bf k}}\left( E_{{\bf k}\alpha }\alpha _{{\bf k}}^{\dagger
}\alpha _{{\bf k}}+E_{{\bf k}\beta }\beta _{{\bf k}}^{\dagger }\beta _{{\bf k%
}}\right)$$ where the spectra of excitations are given by $$\begin{aligned}
E_{{\bf k}\alpha } &=&2\sqrt{I_0(I_0-\eta ^2I_{{\bf k}})}, \label{E-plEka}
\\
E_{{\bf k}\beta } &=&\sqrt{(1+\eta )(I_0-I_{{\bf k}})[I_0-I_{{\bf k}}+\eta
(I_0+I_{{\bf k}})]}, \label{E-plEkb}\end{aligned}$$ $\eta =\sin \theta =D/D_c$. The angle $\theta $ is determined by the condition $\langle a_i\rangle =0.$ To first order in the formal parameter $%
\lambda $ we have $$\begin{aligned}
\sin \theta &=&\frac D{2I_0}\left[ 1-\lambda \left( R_1+R_2\right) \right]
^{-1} \label{E-plTet} \\
R_1(T) &=&2\sum_{{\bf k}}\frac{2I_0+(2-\eta ^2)I_{{\bf k}}}{2E_{{\bf k}%
\alpha }}(1+2N_{{\bf k}\beta })-1 \nonumber \\
R_2(T) &=&\sum_{{\bf k}}\frac{(1+\eta )I_0-I_{{\bf k}}+\eta I_{{\bf k}}^2/I_0%
}{2E_{{\bf k}\beta }}(1+2N_{{\bf k}\alpha })-\frac 12 \label{R1R2}\end{aligned}$$ The gap in the excitation spectrum for the mode $\alpha $ to first order in $%
\lambda $ is given by $$\begin{aligned}
\widetilde{E}_{{\bf k}=0,\alpha } &=&2\sqrt{(1+B_0+B_1)}{\cal I}_0\Delta _{-}
\nonumber \\
\Delta _{-}^2 &=&(1+A_0+A_1)(1-\eta ^2)\end{aligned}$$ where $$\begin{aligned}
A_0 &=&\sum_{{\bf k}}\left[ \frac{\eta ^2\gamma _{{\bf k}}-4\gamma _{{\bf k}%
}-4}{E_{{\bf k}\alpha }}+\frac{\gamma _{{\bf k}}(1+\eta +\eta ^2)/(1+\eta
)-(\eta +1)}{E_{{\bf k}\beta }}\right] \label{A0} \\
B_0 &=&-\frac 12\sum_{{\bf k}}\left[ \frac{8(1+\gamma _{{\bf k}})-20\eta
^2\gamma _{{\bf k}}+11\eta ^4\gamma _{{\bf k}}}{E_{{\bf k}\alpha }}+\frac{%
2(1+\eta )-2\gamma _{{\bf k}}+2\eta ^3\gamma _{{\bf k}}^2}{E_{{\bf k}\beta }}%
\right] \\
A_1 &=&2\eta ^2\sum_{{\bf k}}\frac{(5/2)(2-\eta ^2\gamma _{{\bf k}%
})^2+4(2-\eta ^2\gamma _{{\bf k}})\eta ^2\gamma _{{\bf k}}+(5/2)(\eta
^2\gamma _{{\bf k}})^2}{E_{{\bf k}\alpha }^3} \nonumber \\
&&\ \ \ \ \ \ +\frac 12\sum_{{\bf k}}\frac{(\eta ^2+1)(1+\eta -\gamma _{{\bf %
k}})^2+(\eta ^2+1)(\eta \gamma _{{\bf k}})^2+4\eta (1+\eta -\gamma _{{\bf k}%
})(\eta \gamma _{{\bf k}})}{E_{{\bf k}\beta }^3} \label{A1} \\
B_1 &=&\frac 12(1-\eta ^2)\sum_{{\bf k}}\left[ \frac{4\eta ^2(2-\eta
^2\gamma _{{\bf k}})}{E_{{\bf k}\alpha }^3}+\frac{(1+\eta -\gamma _{{\bf k}%
})^2-(\eta \gamma _{{\bf k}})^2}{E_{{\bf k}\beta }^3}\right]\end{aligned}$$ The mode $\beta $ has Goldstone type and is gapless to arbitrary order in $%
\lambda $.
Renormalization of the two-component $\phi ^4$ model with spontaneously broken symmetry {#app:D}
=======================================================================================
In this Appendix we consider the renormalization of the action (\[S2\]) in the ordered phase, $m^2<0.$ Introducing the quantity $\kappa ^2=-2m^2>0$ and performing the shift $\pi _x\rightarrow \pi _x+\pi _0$ we obtain $$\begin{aligned}
{\cal S} &=&\frac 12\int\limits_0^{c/T}d\tau \int d^dr\left[ \,(\partial %
\bbox{\pi })^2+(\kappa ^2+3\widetilde{h})\pi _x^2+\widetilde{h}\pi
_y^2\right] \nonumber \\
&&\ \ \ \ \ \ \ \ \ \ \ \ +\frac u{4!}\int\limits_0^{c/T}d\tau \int
d^dr\,\left( 4\pi _x+\bbox{\pi }^2\right) \bbox{\pi }^2 \label{S1}\end{aligned}$$ where $\widetilde{h}=h/\overline{\pi }_0$, $\overline{\pi }_0=(3\kappa
^2/u)^{1/2}$ and $\pi _0=\overline{\pi }_0(1+\widetilde{h}/\kappa ^2)$ is determined by the requirement of absence in the action of terms that are linear in $\pi _x$. In these notations the condition of smallness of magnetic field is $\widetilde{h}^{1/2}\ll \kappa $ (the condition of closeness to QPT $\kappa \ll \Lambda $ is also assumed). Under these conditions, the action (\[S1\]) has two characteristic lengths, $1/%
\widetilde{h}^{1/2}$ and $1/\kappa $. This situation is the same as in the theory of crossover phenomena [@Cross]. Further we follow to Ref.[@Cross] to include exactly the smaller characteristic length into $Z$-factors. Then we obtain to one-loop order $$\begin{aligned}
Z^{\text{cont}} &=&1+{\cal O}(g^2), \nonumber \\
\;Z_2^{\text{cont}} &=&1+\frac g{2\varepsilon }\frac 1{(1+\kappa ^2/\mu
^2)^{\varepsilon /2}}+\frac g{6\varepsilon }-\frac{2g}{3\varepsilon }\frac{%
\mu ^\varepsilon }{\Lambda ^\varepsilon }, \nonumber \\
\;Z_4^{\text{cont}} &=&1+\frac{3g}{2\varepsilon }\frac 1{(1+\kappa ^2/\mu
^2)^{\varepsilon /2}}+\frac g{6\varepsilon }-\frac{5g}{3\varepsilon }\frac{%
\mu ^\varepsilon }{\Lambda ^\varepsilon }.\end{aligned}$$ (note that the Ward identities guarantee that the structure of the interaction term is preserved by renormalizations, and one renormalization constant is sufficient to renormalize all four-particle vertex functions, see, e.g., Ref.[@Amit]). The flow functions for the effective-Hamiltonian parameters are $$\begin{aligned}
\beta (g,\kappa /\mu ) &=&-\varepsilon g+\frac{3g^2}2\frac 1{1+\kappa ^2/\mu
^2}+\frac{g^2}6 \nonumber \\
\gamma (g,\kappa /\mu ) &=&-\frac g2\frac 1{1+\kappa ^2/\mu ^2}-\frac g6\end{aligned}$$ Putting in these expressions $\varepsilon =0$, performing the integration and supposing that scaling starts at $\mu \gg \kappa $ we obtain the effective-Hamiltonian parameters at the scale $\mu \rho $ $$\begin{aligned}
\frac 1{g_\rho } &=&\frac 1g-\frac 34\ln (\rho ^2+\kappa ^2/\mu ^2)-\frac 16%
\ln \rho \nonumber \\
\kappa _\rho ^2 &=&\kappa ^2\exp \left[ \frac g2\int\limits_1^\rho d\rho
^{\prime }\left( \frac{\rho ^{\prime }}{\rho ^{\prime 2}+\kappa ^2/\mu ^2}+%
\frac 1{3\rho ^{\prime }}\right) \right. \nonumber \\
&&\ \ \ \ \ \ \ \ \ \ \ \ \left. \times \frac 1{1-(g/6)\ln \rho ^{\prime
}-(3g/4)\ln (\rho ^{\prime 2}+\kappa ^2/\mu ^2)}\right] \label{gk2}\end{aligned}$$ Our plan now is to use these scaling formulas to reach the scale $\mu \rho
\sim \widetilde{h}^{1/2}\ll \kappa .$ For these values of $\rho $ the formulas (\[gk2\]) are simplified: $$\begin{aligned}
g_\rho ^{-1} &=&g^{-1}\left[ 1-(3g/2)\ln (\kappa /\mu )-(g/6)\ln \rho \right]
\nonumber \\
\kappa _\rho ^{-2} &=&\kappa ^{-2}\left[ 1-(3g/2)\ln (\kappa /\mu )-(g/6)\ln
\rho \right] \nonumber \\
&&\ \ \ \ \ \ \ \ \ \ \ \ \times \left[ 1-(5g/3)\ln (\kappa /\mu )\right]
^{-3/5}\Phi _0(g,\kappa ^2/\mu ^2) \label{gt}\end{aligned}$$ where $\Phi _0(g,x)$ is given by $$\begin{aligned}
\ln \Phi _0(g,x) &=&\frac g{12}\int\limits_0^1\frac{dy}y\left[ \frac{4y+x}{%
y+x}\frac 1{1-(g/12)\ln y-(3g/4)\ln (x+y)}\right. \nonumber \\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \left. -\frac{\theta (x-y)}{1-(g/12)\ln
y-(3g/4)\ln x}-\frac{4\theta (y-x)}{1-(5g/6)\ln y}\right] \label{F0}\end{aligned}$$ and $\theta (x)$ is the step function. At extremely small $x$ we have $\Phi
_0(g,x)\simeq \exp (1/\ln ^2x)\simeq 1$. The result (\[gt\]) with $\Phi
_0(g,\kappa ^2/\mu ^2)=1$ can be obtained more directly if we perform the scaling procedure in two steps: at the first step $\rho \gg \kappa /\mu $ and the flow functions are the same as for the two-component isotropic $\phi
^4$ model, while at the second step $\rho \ll \kappa /\mu $ and the flow functions include only contributions of the Goldstone modes. The scaling formulas are joined at $\rho =\kappa /\mu $. However, this procedure does not give a possibility to describe correctly the contribution of the crossover region $\rho \sim \kappa /\mu $. Putting in the above results $\mu
=\Lambda =(2d)^{1/2},$ we obtain the result (\[scal2\]) of the main text.
At finite but small $\varepsilon $ we obtain in a similar way $$\begin{aligned}
g_\rho ^{-1} &=&g^{-1}\left[ 1+(3g/2\varepsilon )(\kappa ^{-\varepsilon
}/\mu ^{-\varepsilon }-1)+(g/6\varepsilon )(\rho ^{-\varepsilon }-1)\right]
\\
\kappa _\rho ^{-2} &=&\kappa ^{-2}\left[ 1+(3g/2\varepsilon )(\kappa
^{-\varepsilon }/\mu ^{-\varepsilon }-1)+(g/6\varepsilon )(\rho
^{-\varepsilon }-1)\right] \nonumber \\
&&\ \ \ \ \ \ \ \ \ \ \ \ \times \left[ 1-(5g/3\varepsilon )(\kappa
^{-\varepsilon }/\mu ^{-\varepsilon }-1)\right] ^{-3/5}\Phi _\varepsilon
(g,\kappa ^2/\mu ^2)\end{aligned}$$ with some function $\Phi _\varepsilon (g,x)\simeq 1.$
Electronic address: [email protected]
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One can obtain the same results in a more correct way (which is free from divergences) by keeping along the calculation the imaginary time to be discrete, $\tau =\tau _i=i/{\cal N}T\;$with $i=0...{\cal N}$, and passing to the limit ${\cal N}\rightarrow \infty $ only at the end.
[Figure captions]{}
Fig.1. Ground-state energy gap $\widetilde{E}_0(\Omega )$ ($\Omega >\Omega
_c,$ a) and order parameter $\overline{S}(\Omega )$ ($\Omega <\Omega _c$, b) for the 3D transverse-field Ising model in different approaches. The value $%
\Omega _c=2.58I$ is used for calculating the $1/S^{\prime }$ and RG$^{\prime
}$ curves. Arrow shows the value of the critical field $\Omega _c,$ obtained by $1/S$-expansion
Fig.2. Ground-state energy gap $\widetilde{E}_0(\Omega )$ ($\Omega >\Omega
_c,$a) and order parameter $\overline{S}(\Omega )$ ($\Omega <\Omega _c$, b) for the 2D transverse-field Ising model in different approaches. The notations used are the same as in Fig.1
Fig.3. Transition temperature as a function of $\Omega /I_0$ for the 3D transverse-field Ising model in different approaches.
Fig.4. Transition temperature as a function of $D/{\cal I}_0$ for an easy-plane ferromagnet in different approaches
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Hot line lists for two isotopologues of water, [H$_2$$^{18}$O]{} and [H$_2$$^{17}$O]{}, are presented. The calculations employ newly constructed potential energy surfaces (PES) which take advantage of a novel method for using the large set of experimental energy levels for [H$_2$$^{16}$O]{} to give high quality predictions for [H$_2$$^{18}$O]{} and [H$_2$$^{17}$O]{}. This procedure greatly extends the energy range for which a PES can be accurately determined, allowing accurate prediction of higher-lying energy levels than are currently known from direct laboratory measurements. This PES is combined with a high-accuracy, [*ab initio*]{} dipole moment surface of water in the computation of all energy levels, transition frequencies and associated Einstein A coefficients for states with rotational excitation up to $J=50$ and energies up to 30 000 [cm$^{-1}$]{}. The resulting HotWat78 line lists complement the well-used BT2 [H$_2$$^{16}$O]{} line list (Barber et.al, 2006, MNRAS, [ **368**]{}, 1087). Full line lists are made available in the electronic form as supplementary data to this article and at [www.exomol.com](www.exomol.com).'
author:
- |
Oleg L. Polyansky$^{1,2}$, Aleksandra A. Kyuberis$^{2}$, Lorenzo Lodi$^{1}$, Jonathan Tennyson$^{1}$[^1], Sergei N. Yurchenko$^{1}$, Roman I. Ovsyannikov$^{2}$, and Nikolai F. Zobov$^{2}$\
$^1$Department of Physics and Astronomy, University College London, London WC1E 6BT, UK\
$^{2}$Institute of Applied Physics, Russian Academy of Sciences, Ulyanov Street 46, Nizhny Novgorod, Russia 603950.
date:
-
- 'Accepted XXXX. Received XXXX; in original form XXXX'
title: 'ExoMol molecular line lists XIX: high accuracy computed hot line lists for H$_2$$^{18}$O and H$_2$$^{17}$O'
---
\[firstpage\]
molecular data; opacity; astronomical data bases: miscellaneous; planets and satellites: atmospheres; stars: low-mass; stars: brown dwarfs.
Introduction
============
Water spectra can be observed from many different regimes in the Universe, several of which are discussed further below. The spectrum of water, particularly at elevated temperatures, is rich and complex. A few years ago @jt378 presented a comprehensive line list, known as BT2, which used well-established theoretical procedures to compute all the transitions of [H$_2$$^{16}$O]{} of importance in objects with temperatures up to 3000 K. BT2 contains about 500 million lines. A similar line list for HD$^{16}$O, known as VTT, was subsequently computed by @jt469.
The BT2 line list has been extensively used. It forms the basis of the most recent release of the HITEMP high-temperature spectroscopic database [@jt480] and for the BT-Settl model [@BT-Settl] for stellar and substellar atmospheres covering the range from solar-mass stars to the latest-type T and Y dwarfs. BT2 has been used to detect and analyse water spectra in objects as diverse as the Nova-like object V838 Mon [@jt357], atmospheres of brown dwarfs [@10RiBeMc.H2O] and M subdwarfs [@14RaReAl.H2O], and extensively for exoplanets [@jt400; @13BiDeBr.exo]. Within the solar system BT2 has been used to show an imbalance between nuclear spin and rotational temperatures in cometary comae [@jt330; @jt349] and assign a new set of, as yet unexplained, high energy water emissions in comets [@jt452], as well as to model water spectra in the deep atmosphere of Venus [@Jeremy09].
Although BT2 was developed for astrophysical use, it has been applied to a variety of other problems including the calculation of the refractive index of humid air in the infrared [@07Mathar.H2O], high speed thermometry and tomographic imaging in gas engines and burners [@07KrAnCa.H2O; @10ReSa.H2O], as the basis for an improved theory of line-broadening [@jt431], and to validate the data used in models of the earths atmosphere and in particular simulating the contribution of weak water transitions to the so-called water continuum [@jt463].
There are several water line lists published in the literature [@jt197; @97PaScxx.H2O; @jt378; @spectra]. Two linelists have also been computed specifically for the isotopologues: @jt438 created the 3mol room-temperature line lists for [H$_2$$^{16}$O]{}, [H$_2$$^{17}$O]{} and [H$_2$$^{18}$O]{} based on the PES of @jt375; Tashkun created a number of line lists based on the work of @97PaScxx.H2O, see @spectra. These are considered further below.
At present hot line lists are only published for [H$_2$$^{16}$O]{} and HD$^{16}$O. However isotopically-substituted water containing $^{18}$O or $^{17}$O provides important markers for a variety of astronomical problems [@12NiGa.H2O]. For example @14MaYaBa.H2O recently detected [H$_2$$^{18}$O]{} in the emission-line spectrum of the luminous M-supergiant VY CMa. Astronomical spectra of water isotopologues [@13NeToAg.H2O] and their direct analysis in cometary dust particles [@10FlStMe.H2O] and carbonaceous chrondrites [@84ClMa.H2O; @08VoHoBr.H2O] have been used to determine formation mechanisms and constrain formation models. Water isotope ratios are also used to monitor stellar evolution [@12AbPaBu.H2O] and to probe the atmosphere of Mars [@15ViMuNo.H2O]. The seemingly minor isotopologues of water can be important species in their own right with, for example, [H$_2$$^{18}$O]{}being the fifth largest absorber of sunlight the earth’s atmosphere.
There is therefore a need for line lists equivalent to BT2 for [H$_2$$^{17}$O]{}and [H$_2$$^{18}$O]{} to aid spectroscopic studies, and it is these that are presented here. These lists form part of the ExoMol project [@jt528] which aims to provide a comprehensive set of molecular line lists for studies of molecular line lists for exoplanet and other hot atmospheres.
Although our new line lists in some way mimic BT2, they also take advantage of a number of recent theoretical developments. In particular a IUPAC task group [@jt562] used a systematic procedure [@jt412] to derive empirical energy levels for all the main isotopologues of water [@jt454; @jt482; @jt539; @jt576]. These levels are combined with a newly-developed procedure for enhancing the accuracy of calculations on isotopically substituted species, which is used for the first time here. This ensures that most of the key frequencies in our new line lists are determined with an accuracy close to experimental, even though many of them are yet to be observed. Furthermore, theoretical work on improving the accuracy and representation of the water dipole moment [@jt424; @jt509] has improved the accuracy with which water transition intensities are predicted [@jt467]. Some of these advances have already been used to create improved room temperature line lists for [H$_2$$^{17}$O]{} and [H$_2$$^{18}$O]{} [@jt522] which were included in their entirety in the 2012 release of HITRAN [@jt557s].
The paper is structured as follows: section 2 outlines our overall methodology and presents the derivation of potential energy surfaces (PES). The details of the calculation of the new line lists, along with comparison with previous line lists, are given in section 3. Section 4 discusses further improvement of the line list by the substitution of calculated energy levels with empirical ones, together with the procedure used to label energy levels with approximate vibrational and rotational quantum numbers. Our results are discussed in section 5.
Potential Energy Surfaces
=========================
The fitting of water ([H$_2$$^{16}$O]{}) PESs to experimental spectroscopic data has a long history. The first fitted PES giving near to experimental accuracy was PJT1 [@jt150]. @97PaScxx.H2O constructed a fitted PES starting from a highly accurate [[*ab initio* ]{}]{} calculation; all subsequent water potentials followed this procedure and have been based on [[*ab initio* ]{}]{} studies of increasing sophistication. As a result there are several very good water PESs available [@jt308; @jt438; @11BuPoZo.H2O].
Here we need a PES which satisfies two criteria. First, it should be at least as accurate as the PES used for the BT2 line list with the calculated energies ranging up to 30 000 [cm$^{-1}$]{}. Second, the PES should be adapted to the calculation of energy levels of the two water isotopologues [H$_2$$^{17}$O]{} and [H$_2$$^{18}$O]{}. This second requirement is harder to fulfill, as the characterisation of the experimental energy levels of both [H$_2$$^{17}$O]{} and [H$_2$$^{18}$O]{} is significantly less extensive than for [H$_2$$^{16}$O]{}[@jt562].
To take advantage of the accumulated knowledge on the spectrum [H$_2$$^{16}$O]{}in constructing a PES for [H$_2$$^{17}$O]{} and [H$_2$$^{18}$O]{} and following previous work [@jt186; @jt469; @11BuPoZo.H2O], we decided to fit a Born-Oppenheimer (BO) mass-independent PES to the available data for [H$_2$$^{16}$O]{} and fix the adiabatic BO diagonal correction (BODC), mass-dependent surface to the [[*ab initio* ]{}]{} value of @jt309. Obviously this procedure requires the accuracy of predictions for [H$_2$$^{17}$O]{} and [H$_2$$^{18}$O]{}to be verified. This is done by comparing the calculated [H$_2$$^{17}$O]{} and [H$_2$$^{18}$O]{}energy levels to the available experimentally-determined ones [@jt454; @jt482].
We used the same fitting procedure as @11BuPoZo.H2O. Nuclear motion calculations were performed with <span style="font-variant:small-caps;">DVR3D</span> [@jt338]. As elsewhere, in the fit the experimentally derived energies of [H$_2$$^{16}$O]{}for the $J=0,2$ and 5 rotational states by @jt539 were used.
In the following our new empirical PES obtained using the fitting procedure described above will be referenced to as PES1, while the PES by @11BuPoZo.H2O will be referenced to as PES2. Tables \[PES17\] and \[PES18\] present a comparison between the $J
= 0$ energy levels calculated using PES1, PES2 for [H$_2$$^{17}$O]{} and [H$_2$$^{18}$O]{}respectively. For comparison as a third column we present the $J = 0$ levels and corresponding discrepancies using the PES (called PES3 in the tables) due to @97PaScxx.H2O taken from the linelist calculated by Dr. S.A. Tashkun and summarised by @spectra. The line list based on PES3 was calculated for three temperatures: T=296 K, 1000 K and 3000 K. For all versions the highest value of the rotational quantum number $J$ considered is 28 and the spectral range is 0-28500 [cm$^{-1}$]{}. The number of lines for [H$_2$$^{18}$O]{} is 108 784 and for [H$_2$$^{17}$O]{} 109 083.
Indeed, one can see that the agreement with the experiment is very good. Although the results obtained using PES2 are somewhat better than those for PES1. However employing PES1 gives us the opportunity to use the information on [H$_2$$^{16}$O]{} experimental energy levels to predict very accurately energy levels of [H$_2$$^{17}$O]{} and [H$_2$$^{18}$O]{}. We call these predicted levels pseudo-experimental energies for the reasons explained below. Table \[sd\] illustrates the unprecedented accuracy of the prediction of the [H$_2$$^{17}$O]{} energy levels for those states whose energies are known experimentally. The slightly less good, but still very accurate, energy levels predicted for [H$_2$$^{18}$O]{} are shown in the column 2 of Table \[sd\]. We might expect a similar level of accuracy for predictions of the [H$_2$$^{17}$O]{} and [H$_2$$^{18}$O]{} energy levels for states yet to be measured for these isotopologues, but known for [H$_2$$^{16}$O]{}. We note that the standard deviations given in Table \[sd\] are rather systematic suggesting that further improvement in the predictions may be possible. This and details of our final pseudo-experimental energy levels are discussed in section 4.
$v_{1}$ $v_{2}$ $v_{3}$ Observed PES1 Obs.-Calc. PES2 Obs.-Calc. PES3 Obs.-Calc.
--------- --------- --------- ----------- ----------- ------------ ----------- ------------ ----------- ------------ --
0 0 1 3748.318 3748.334 -0.02 3748.326 -0.01 3748.463 -0.15
0 0 2 7431.076 7431.103 -0.03 7431.059 0.02 7431.467 -0.39
0 0 3 11011.883 11011.936 -0.05 11011.860 0.02 11012.268 -0.38
0 1 0 1591.326 1591.297 0.03 1591.342 -0.02 1591.413 -0.09
0 1 1 5320.251 5320.241 0.01 5320.251 0.00 5320.378 -0.13
0 1 2 8982.869 8982.868 0.00 8982.844 0.03 8983.118 -0.25
0 1 3 12541.227 12541.267 -0.04 12541.207 0.02 12541.614 -0.39
0 2 0 3144.980 3144.934 0.05 3144.993 -0.01 3145.085 -0.10
0 2 1 6857.273 6857.260 0.01 6857.266 0.01 6857.476 -0.20
0 7 1 13808.273 13808.224 0.05 13808.371 -0.10 13809.171 -0.90
1 0 0 3653.142 3653.147 0.00 3653.121 0.02 3653.193 -0.05
1 0 1 7238.714 7238.773 -0.06 7238.726 -0.01 7238.932 -0.22
1 0 2 10853.505 10853.545 -0.04 10853.504 0.00 - -
1 0 3 14296.280 14296.340 -0.06 14296.265 0.01 14296.584 -0.30
1 1 0 5227.706 5227.691 0.01 5227.704 0.00 5227.881 -0.18
1 1 1 8792.544 8792.578 -0.03 8792.546 0.00 8792.816 -0.27
1 2 0 6764.726 6764.747 -0.02 6764.722 0.00 6764.905 -0.18
1 2 1 10311.202 10311.247 -0.05 10311.199 0.00 10311.421 -0.22
1 3 1 11792.822 11792.861 -0.04 11792.834 -0.01 11793.172 -0.35
2 0 0 7193.246 7193.265 -0.02 7193.257 -0.01 7193.394 -0.15
2 0 1 10598.476 10598.550 -0.07 10598.483 -0.01 10598.763 -0.29
2 1 1 12132.993 12133.056 -0.06 12132.984 0.01 12132.365 0.63
2 2 1 13631.500 13631.542 -0.04 13631.489 0.01 13631.650 -0.15
3 0 1 13812.158 13812.215 -0.06 13812.170 -0.01 13812.394 -0.24
3 2 1 16797.168 16797.182 -0.01 16797.177 -0.01 16797.011 0.16
4 0 1 16875.621 16875.662 -0.04 16875.643 -0.02 16875.474 0.15
: Comparison of calculated $J=0$ term values for [H$_2$$^{17}$O]{}using three potentials with experimental data. Experimental (obs) data is taken from @jt454. []{data-label="PES17"}
$v_{1}$ $v_{2}$ $v_{3}$ Observed PES1 Obs.-Calc. PES2 Obs.-Calc. PES3 Obs.-Calc.
--------- --------- --------- ---------- ----------- ------------ ----------- ------------ ----------- ------------ --
0 0 1 3741.57 3741.581 -0.01 3741.567 0.00 3741.575 -0.01
0 0 2 7418.72 7418.741 -0.02 7418.693 0.03 7418.759 -0.03
0 0 3 10993.68 10993.734 -0.05 10993.659 0.02 10993.689 -0.01
0 1 0 1588.28 1588.240 0.04 1588.271 0.00 1588.299 -0.02
0 1 1 5310.46 5310.443 0.02 5310.438 0.02 5310.388 0.07
0 1 2 8967.57 8967.552 0.01 8967.519 0.05 8967.491 0.07
0 1 3 12520.12 12520.153 -0.03 12520.089 0.03 12520.068 0.06
0 2 0 3139.05 3138.999 0.05 3139.038 0.01 3139.031 0.02
0 2 1 6844.60 6844.580 0.02 6844.566 0.03 6844.539 0.06
0 2 2 10483.22 10483.264 -0.04 10483.202 0.02 10483.212 0.01
0 3 0 4648.48 4648.435 0.04 4648.469 0.01 4648.452 0.03
0 3 1 8341.11 8341.109 0.00 8341.086 0.02 8341.114 -0.01
0 3 2 11963.54 11963.580 -0.04 11963.507 0.03 11963.615 -0.08
0 4 0 6110.42 6110.408 0.02 6110.433 -0.01 6110.410 0.01
0 4 1 9795.33 9795.354 -0.02 9795.324 0.01 9795.329 0.00
1 0 0 3649.69 3649.688 0.00 3649.649 0.04 3649.667 0.02
1 0 1 7228.88 7228.934 -0.05 7228.883 0.00 7228.888 0.00
1 0 2 10839.96 10839.986 -0.03 10839.942 0.01 - -
1 0 3 14276.34 14276.389 -0.05 14276.318 0.02 14276.229 0.11
1 1 0 5221.24 5221.233 0.01 5221.227 0.02 5221.298 -0.05
1 1 1 8779.72 8779.747 -0.03 8779.707 0.01 8779.722 0.00
1 1 2 12372.71 12372.723 -0.02 12372.679 0.03 - -
1 2 0 6755.51 6755.528 -0.02 6755.483 0.03 6755.501 0.01
1 2 1 10295.63 10295.673 -0.04 10295.616 0.02 10295.524 0.11
1 3 0 8249.04 8249.063 -0.03 8249.023 0.01 8249.073 -0.04
1 3 1 11774.71 11774.742 -0.03 11774.701 0.01 11774.670 0.04
2 0 0 7185.88 7185.894 -0.02 7185.879 0.00 7185.880 0.00
2 0 1 10585.29 10585.357 -0.07 10585.292 -0.01 10585.300 -0.01
2 0 2 14187.98 14188.069 -0.09 14187.985 0.00 - -
2 1 0 8739.53 8739.530 0.00 8739.520 0.01 8739.589 -0.06
2 1 1 12116.80 12116.851 -0.05 12116.778 0.02 12116.833 -0.04
2 2 0 10256.58 10256.604 -0.02 10256.569 0.02 10256.537 0.05
2 2 1 13612.71 13612.745 -0.04 13612.688 0.02 13612.468 0.24
2 3 0 11734.53 11734.543 -0.02 11734.517 0.01 11734.625 -0.10
3 0 0 10573.92 10573.955 -0.04 10573.927 -0.01 10573.898 0.02
3 0 1 13795.40 13795.455 -0.06 13795.410 -0.01 13795.280 0.12
3 1 0 12106.98 12107.025 -0.05 12106.974 0.00 12107.006 -0.03
3 2 1 16775.38 16775.396 -0.01 16775.385 0.00 16774.779 0.60
4 0 1 16854.99 16855.126 -0.14 16855.099 -0.11 16854.534 0.46
: Comparison of calculated $J=0$ term values for [H$_2$$^{18}$O]{}using three potentials with experimental data. Experimental (obs) data is taken from @jt454. []{data-label="PES18"}
Recently, highly lying energy levels of [H$_2$$^{18}$O]{} have been measured using multiphoton spectroscopy [@15MaKoZo]. These levels lie at about 27 000 [cm$^{-1}$]{} and therefore provide a stringent test of our procedure. The highest upper energy level considered in this work, as for BT2, is 30 000 [cm$^{-1}$]{}; Table \[makarov\] illustrates the high quality of our calculations over the whole range considered. In fact recent studies confirm that BT2 is not so accurate for these high energy states [@jt645].
$J$ $ N $ [[H$_2$$^{17}$O]{}]{} $ N $ [[H$_2$$^{18}$O]{}]{}
----- ------- ----------------------- ------- -----------------------
0 27 0.0058 39 0.0092
1 93 0.0056 124 0.0093
2 161 0.0071 212 0.0109
3 199 0.0074 254 0.0090
4 236 0.0118 316 0.0147
5 232 0.0103 335 0.0141
6 263 0.0100 401 0.0116
7 222 0.0138 385 0.0140
8 182 0.0146 381 0.0130
9 138 0.0123 335 0.0174
10 116 0.0130 288 0.0176
11 72 0.0080 232 0.0168
12 47 0.0111 188 0.0201
13 26 0.0083 135 0.0179
14 9 0.0096 106 0.0198
15 3 0.0150 73 0.0176
16 1 0.0066 46 0.0184
17 1 0.0015 19 0.0156
18 11 0.0187
: Standard deviation in [cm$^{-1}$]{} with which our pseudo-experimental energy levels the of [H$_2$$^{17}$O]{} and [H$_2$$^{18}$O]{} predicted the observed ones compiled by @jt482 as a function of rotational state, $J$, $N$ is number of levels used for calculation of the standard deviation.[]{data-label="sd"}
------- ---------- ------------ ------------ -- -- --
$ J $ Observed Calculated Obs.-Calc.
0 27476.33 27476.24 0.09
1 27497.03 27496.92 0.11
1 27510.64 27510.31 0.33
1 27517.09 27517.44 -0.35
2 27537.12 27536.96 0.16
2 27546.82 27546.45 0.37
1 27509.55 27509.19 0.36
2 27545.66 27545.28 0.38
------- ---------- ------------ ------------ -- -- --
: Prediction of experimental energy levels of [H$_2$$^{18}$O]{}. Experimental (obs) data is taken from @15MaKoZo.[]{data-label="makarov"}
Thus, the line lists, details of whose calculations are given in the following section, are computed using a higher quality PES than that used to compute BT2. Three sets of energy levels are provided as part of this line list. The first set is the variationally calculated energy levels obtained using PES2. The second set comprises these energy levels substituted by the experimental values [@jt454] where available. The third set is further with pseudo-experimental energy levels substituted whenever [H$_2$$^{16}$O]{} experimental energy levels [@jt539] are available (see below). This third set is the one we recommend for creating spectra with HotWat78 because of its increased accuracy.
![The distribution of the [H$_2$$^{18}$O]{} transitions per $J$ in the line HotWat78 list. []{data-label="f:18:lines:J"}](Nlines_18.pdf){width="70.00000%"}
![Comparison between BT2 and HotWat78 for [H$_2$$^{18}$O]{} at the temperature $T$=2000 K, and comparison of HotWat78 with 3mol [@jt438] and HITRAN at $T$=296 K for [H$_2$$^{18}$O]{} and [H$_2$$^{17}$O]{} respectively.[]{data-label="Figure01"}](./183molv2.pdf "fig:"){width="107.00000%"} ![Comparison between BT2 and HotWat78 for [H$_2$$^{18}$O]{} at the temperature $T$=2000 K, and comparison of HotWat78 with 3mol [@jt438] and HITRAN at $T$=296 K for [H$_2$$^{18}$O]{} and [H$_2$$^{17}$O]{} respectively.[]{data-label="Figure01"}](./17Hv2.pdf "fig:"){width="114.00000%"}
Line list calculations for [H$_2$$^{17}$O]{} and [H$_2$$^{18}$O]{}
==================================================================
The line list calculations were performed with the <span style="font-variant:small-caps;">DVR3D</span> program suite [@jt338] using the PES1 and PES2 discussed above, and the [*ab initio*]{} dipole moment surfaces LTP2011S of @jt509. As for BT2, the highest rotational state, $J$, in the calculation was taken as $J=50$ and the limiting energy as 30 000 [cm$^{-1}$]{}. Analysis using the [H$_2$$^{16}$O]{} partition function [@jt263] performed in BT2 suggests that these parameters are sufficient to cover all transitions longwards of 0.5 $\mu$m for temperatures up to 3000 K.
Wavefunctions were obtained by solving the nuclear Schrödinger equation using two-step procedure of calculation of rovibrational energies [@jt46]. The calculations benefitted from recent algorithmic improvements [@jt626], in particular in the method used to construct the final Hamiltonian matrices for $J>0$ due to @jt640. Transition intensities were computed for $\Delta J = 0$ and 1 for all four symmetries and every $J\le 50$. The matrix elements of the DMS were calculated using the program <span style="font-variant:small-caps;">Dipole</span> of the suite <span style="font-variant:small-caps;">DVR3D</span> and the actual spectrum for both isotopologues was generated with the program Spectra. About 500 million transitions were calculated for each isotopologue.
Figure \[f:18:lines:J\] shows the distribution of the [H$_2$$^{18}$O]{} lines in HotWat78.
Using our calculations we provide the values of partition function for both isotopologues for wide range of temperatures, which are presented in the Table \[pf\] as well as in the supplementary data on a grid of 1 K. We use the HITRAN convention [@03FiGaGo.partfunc] and include the nuclear statistical weights $g_{\rm ns}$ in to the partition function explicitly [@jt631]. The nuclear statistical weights for [H$_2$$^{18}$O]{} are the same as for the main isotopologue, 1 and 3 for the para- and ortho- states, respectively. In case of [H$_2$$^{17}$O]{}, $g_{\rm ns}$ are 6 (para) and 18 (ortho). For calculation of partition functions for [H$_2$$^{18}$O]{} and [H$_2$$^{17}$O]{} we used all available energy levels with applying the cut-off at 30000 [cm$^{-1}$]{}.
$T(K)$ [[H$_2$$^{17}$O]{}]{} [[H$_2$$^{18}$O]{}]{}
-------- ----------------------- ----------------------- --
10 7.97970859 1.33135007
20 20.1629004 3.37074465
40 56.7292812 9.48860674
60 101.331587 16.9509639
80 153.237432 25.6357152
100 211.822453 35.4382143
200 587.053283 98.2237727
296 1052.12202 176.043783
300 1073.45356 179.613285
400 1654.78625 276.895547
500 2328.51505 389.655412
600 3099.26294 518.674912
800 4966.65892 831.352302
1000 7346.85187 1230.02825
1200 10357.5304 1734.46724
1400 14140.2160 2368.43292
1500 16371.1820 2742.40404
1600 18857.9004 3159.29345
1800 24694.5428 4137.93895
2000 31855.8230 5338.90908
2200 40570.4778 6800.61746
2400 51091.7815 8565.59949
2500 57116.1119 9576.29200
2600 63698.8388 10680.7274
2800 78697.3411 13197.3344
3000 96419.4218 16171.1873
3200 117222.299 19662.2543
3400 141485.523 23734.2409
3500 155038.487 26008.8411
3600 169606.832 28453.8904
3800 201996.792 33890.0829
4000 239072.534 40112.7834
4200 281250.969 47191.9028
4400 328941.890 55196.1417
4500 354979.000 59566.0429
4600 382541.321 64191.8753
4800 442425.403 74242.1299
5000 508945.054 85405.6885
5200 582421.516 97736.3470
5400 663142.877 111282.333
5500 706300.716 118524.515
5600 751361.549 126085.883
5800 847292.676 148990.861
6000 951113.377 159603.233
: Partition Function of [H$_2$$^{17}$O]{} and [H$_2$$^{18}$O]{}.[]{data-label="pf"}
Improved pseudo-experimental energy levels
==========================================
The series of IUPAC papers on the various isotopologues of water [@jt454; @jt482; @jt539; @jt576] used measured transition frequencies to derive ro-vibrational energy levels using the so-called MARVEL (measured active rotation-vibration energy levels) procedure [@jt412; @12FuCs.method]. These energy levels can be used to generate pseudo-experimental values of the line frequencies in our line lists when the calculated energy level is substituted by the corresponding (pseudo-)experimental one. The comparison of these generated line frequencies with actual experimental ones demonstrate near-perfect coincidence. The number of generated pseudo-experimental lines is significantly higher than the number of the directly observed lines because line frequencies between pseudo-experimental levels can be predicted to high accuracy even when the lines have not been measured, as demonstrated by @jt539. Less than 200 000 experimentally observed [H$_2$$^{16}$O]{} lines give rise to about 5 000 000 lines with pseudo-experimental frequencies generated in this way. Use of such a procedure provides significantly more accurate line lists than just the calculated ones. We therefore substituted our computed energy with those of @jt454 where possible.
However as described in section 2, the procedure for fitting PES using [H$_2$$^{16}$O]{} data opens the way for us to further improve the accuracy of the calculated line lists. Looking at Table \[oc\], we can see that the obs$-$calc residuals for a particular [H$_2$$^{16}$O]{} vibrational state are very similar to the residuals for the same states of [H$_2$$^{17}$O]{} and [H$_2$$^{18}$O]{}. The following procedure can be used to exploit this. First let us consider the idealised situation when all the residuals for energy levels of [H$_2$$^{16}$O]{}, $R_{v,J}(16)$, are exactly equal to those of [H$_2$$^{18}$O]{}, $R_{v,J}(18)$, where $(v,J)$ represent the vibrational and rotational quantum numbers. In this case we can predict the precise “estimated” value of an [H$_2$$^{18}$O]{} level, $E_{v,J}^{\rm est}(18)$, from the empirically-determined levels of [H$_2$$^{16}$O]{}, $E_{v,J}^{\rm obs}(18)$ $$E_{v,j}^{\rm est}(18) = E_{v,J}^{\rm calc}(18) + R_{v,J}(18) = E_{v,J}^{\rm calc}(18) + R_{v,J}(16)$$ where $E_{v,J}^{\rm calc}(18)$ is the corresponding calculated [H$_2$$^{18}$O]{}energy level. So even if the level of the [H$_2$$^{18}$O]{} isotopologue has yet to be observed, its pseudo-experimental value can be retrieved from the calculated level of [H$_2$$^{18}$O]{} using our calculations plus the residual for [H$_2$$^{16}$O]{} provided the experimental level of [H$_2$$^{16}$O]{} is known.
--------------- ------------------- -------- ------------------- -------- ------------------- --------
$(v_1v_2v_3)$ [H$_2$$^{16}$O]{} [H$_2$$^{17}$O]{} [H$_2$$^{18}$O]{}
obs o–c obs o–c obs o–c
(010) 1594.75 0.019 1591.33 0.028 1588.28 0.036
(020) 3151.63 0.040 3144.98 0.046 3139.05 0.051
(100) 3657.05 -0.007 3653.14 -0.005 3649.69 -0.002
(110) 5234.97 0.005 5227.71 0.014 5221.24 0.010
(120) 6775.09 -0.028 6764.73 -0.022 6755.51 -0.018
(200) 7201.54 -0.024 7193.25 -0.019 7185.88 -0.016
(012) 9000.14 -0.009 8982.87 0.001 8967.57 0.013
(102) 10868.88 -0.049 10853.51 -0.040 10839.96 -0.030
(001) 3755.93 -0.017 3748.32 -0.015 3741.57 -0.014
(011) 5331.27 -0.002 5320.25 0.010 5310.46 0.019
(021) 6871.52 0.004 6857.27 0.012 6844.60 0.019
(101) 7249.82 -0.063 7238.71 -0.059 7228.88 -0.051
(111) 8807.00 -0.044 8792.54 -0.034 8779.72 -0.027
(121) 10328.73 -0.055 10311.20 -0.045 10295.63 -0.039
(201) 10613.36 -0.074 10598.48 -0.075 10585.29 -0.072
(003) 11032.40 -0.061 11011.88 -0.053 10993.68 -0.053
(131) 11813.20 -0.041 11792.82 -0.039 11774.71 -0.034
(211) 12151.25 -0.072 12132.99 -0.064 12116.80 -0.054
(113) 12565.01 -0.050 12541.23 -0.041 12520.12 -0.030
(221) 13652.66 -0.045 13631.50 -0.042 13612.71 -0.035
(301) 13830.94 -0.062 13812.16 -0.057 13795.40 -0.057
(103) 14318.81 -0.069 14296.28 -0.061 14276.34 -0.053
--------------- ------------------- -------- ------------------- -------- ------------------- --------
: Vibrational band origins, in [cm$^{-1}$]{}, for [H$_2$$^{16}$O]{}, [H$_2$$^{17}$O]{} and [H$_2$$^{18}$O]{}. Observed (obs) data is taken from @jt539 and @jt454; calculated results are given as observed minus calculated (o–c).[]{data-label="oc"}
Table \[oc\] shows that residuals for [H$_2$$^{16}$O]{} and [H$_2$$^{18}$O]{}are slightly different, we can therefore improve this procedure. We notice from the Table \[oc\], that the [H$_2$$^{16}$O]{} and [H$_2$$^{18}$O]{} residuals differ by similar amounts. If we average this value: $$\Delta R(18) = \frac{1}{N}\sum_{v=1}^N R_{v,0}(18) - R_{v,0}(16).$$ where $N$ runs over the number of vibrational states for which $J=0$ levels are known, which corresponds to 40 for [H$_2$$^{17}$O]{} and 24 for [H$_2$$^{18}$O]{}. Then we can use this average difference to further correct our estimated [H$_2$$^{18}$O]{} energy levels using the revised formula: $$E_{v,j}^{\rm est}(18) = E_{v,J}^{\rm calc}(18) + R_{v,J}(16) + \Delta R(18).$$ Calculating the observed values of energies of [H$_2$$^{18}$O]{} using Eq. (1) gives a standard deviation for $E_{v,j}^{\rm est}(18)$ levels from the known experimental values, $E_{v,j}^{\rm obs}(18)$, of 0.009 [cm$^{-1}$]{}. However, $\Delta R(18)$ is 0.006 [cm$^{-1}$]{}. If instead we use Eq. (3), then the standard deviation reduces to 0.003 [cm$^{-1}$]{}. Although $\Delta R(18)$ is evaluated for $J=0$ only, this procedure still works for higher $J$ values. For example it also results in a standard deviation of 0.003 [cm$^{-1}$]{} when applied to the $J=10$ levels of the (010) state.
This procedure, which can clearly also be applied to [H$_2$$^{17}$O]{}, leads to the generation of about 5 million transitions which involve the pseudo-experimental levels of [H$_2$$^{17}$O]{} and [H$_2$$^{18}$O]{}. It therefore provides a line list with much more accurate values of the frequencies of these transitions: in general better by about 0.005 [cm$^{-1}$]{}for [H$_2$$^{17}$O]{} and somewhat worse for [H$_2$$^{18}$O]{}, but still much more accurate than possible with variational calculations.
The reason this procedure can be applied to the construction of the pseudo-experimental values of the energy levels of minor isotopologues is that for the major water isotopologue [H$_2$$^{16}$O]{} the number of energy levels known experimentally is significantly higher, then that for [H$_2$$^{17}$O]{} and [H$_2$$^{18}$O]{}. For example the assignment of weak [H$_2$$^{16}$O]{} lines in various regions is available [@jt360; @jt218; @jt285], where isotopologues data are not known. As a result very highly-excited bending [@jt203; @jt362] and stretching energy levels [@07MaMuZoSh; @jt467; @jt472] are known, which form the basis upon which our pseudo-experimental energy levels are constructed.
Results
=======
The newly constructed [H$_2$$^{17}$O]{} and [H$_2$$^{18}$O]{} line lists are named HotWat78. The new HotWat78 line lists are calculated for J $\leq$ 50 and for the spectral range 0-30000 [cm$^{-1}$]{}. HotWat78 contains 519 461 789 lines for [H$_2$$^{18}$O]{} is 519 461 789 and 513 112 779 lines for [H$_2$$^{17}$O]{}. The new linelist is both the most complete and the most accurate one, see Tables \[PES17\] and \[PES18\]. They are stored in the ExoMol format [@jt548] which uses the compact storage of results originally developed for BT2. This involves using a states file (`.states`), see Table \[states\], and a transitions file (`.trans`), see Table \[trans\]. The energy levels in the states files are marked as ‘observed’ if the results are taken from the IUPAC compilation, ‘estimated’ if they are generated using Eq. (3) or as ‘calculated’, for which the results of the PES2 calculation are used.
$i$ $\tilde{E}$ [g$_{tot}$]{} $J$ Ka Kc v1 v2 v3 $S$
----- -------------- --------------- ----- ---- ---- ---- ---- ---- -----
1 0.000000 6 0 0 0 0 0 0 A1
2 1591.322876 6 0 0 0 0 1 0 A1
3 3144.980225 6 0 0 0 0 2 0 A1
4 3653.145752 6 0 0 0 1 0 0 A1
5 4657.115211 6 0 0 0 0 3 0 A1
6 5227.703125 6 0 0 0 1 1 0 A1
7 6121.557129 6 0 0 0 0 4 0 A1
8 6764.726562 6 0 0 0 1 2 0 A1
9 7193.246582 6 0 0 0 2 0 0 A1
10 7431.093262 6 0 0 0 0 0 2 A1
11 7527.489258 6 0 0 0 0 5 0 A1
12 8260.781250 6 0 0 0 1 3 0 A1
13 8749.905273 6 0 0 0 2 1 0 A1
14 8853.288086 6 0 0 0 0 6 0 A1
15 8982.860352 6 0 0 0 0 1 2 A1
16 9708.538086 6 0 0 0 1 4 0 A1
17 10068.091797 6 0 0 0 0 7 0 A1
18 10269.661133 6 0 0 0 2 2 0 A1
19 10501.353516 6 0 0 0 0 2 2 A1
20 10586.049805 6 0 0 0 3 0 0 A1
: Extract from the final states file for [H$_2$$^{17}$O]{}.[]{data-label="states"}
\
[$i$: State counting number.\
$\tilde{E}$: State energy in [cm$^{-1}$]{}.\
$g$: Total state degeneracy.\
$J$: Total angular momentum\
$K_a$: Asymmetric top quantum number.\
$K_c$: Asymmetric top quantum number.\
$\nu_1$: Symmetric stretch quantum number.\
$\nu_2$: Bending quantum number.\
$\nu_3$: Asymmetric stretch quantum number.\
$S$: State symmetry in C$_{2v}$.]{}
$f$ $i$ $A_{fi}$
-------- -------- ------------ -- --
142344 150189 5.6651e-05
2235 2362 1.7434e-03
34497 35342 5.7700e-09
125681 114596 5.5394e-10
135143 128340 6.3329e-08
24055 16736 1.5208e-03
147918 137719 1.3405e-04
45027 45537 8.0306e-07
37457 31884 9.0168e-08
39192 43632 7.3676e-07
25153 26085 4.3393e-05
131146 124272 8.5679e-04
134840 128287 8.5680e-04
88744 94220 1.2221e-03
102017 106580 2.4131e-04
193489 187074 2.7697e-06
202910 204558 7.0571e-03
53725 50906 1.8345e-06
142862 135857 2.5908e-05
: Extract from the transitions file for [H$_2$$^{17}$O]{}[]{data-label="trans"}
\
[$f$]{} : Upper state counting number.\
[$i$]{}: Lower state counting number.\
$A_{fi}$ : Einstein-A coefficient in [s$^{-1}$]{}.\
The states file lists all the ro-vibrational levels for each $J$ and for four C$_{2v}$ symmetries. It is common to further label the every level with (approximate) vibrational quantum numbers $(v_1, v_2, v_3)$ which correspond to the symmetric stretch, bending and asymmetric stretch modes, respectively and the Rotational levels within each vibrational state by $J,K_a,K_c$, where again the projection quantum numbers $K_a$ and $K_c$ are approximate. <span style="font-variant:small-caps;">DVR3D</span> does not provide these approximate labels but there are several methods available for labeling water energy levels [@97PaScxx.H2O; @12SzFaCs.method; @jt438]. Here we label levels with $J\leq20$ and energies below 20 000 [cm$^{-1}$]{}. As our energy levels differ by less than 1 [cm$^{-1}$]{} from those of @jt438, transferring the labels from this previous study proved to be straightforward. We note that the labels we use are based on the normal modes from a harmonic oscillator model. It is well know that the higher stretching states of water are better represented with a local-mode model [@84ChHoxx]. However, there is a one-to-one correspodance between the two labelling schemes [@jt242]; the use of normal mode labels are used for simplicity.
The accuracy of the present line lists can be established by the comparison with the previous line lists calculations. Two types of comparison could be made. The overall picture for the high temperature is that the coverage the HotWat78 [H$_2$$^{17}$O]{} and [H$_2$$^{18}$O]{} line lists should be very similar to BT2, but that both the predicted intensives and the line positions should be significantly better. Furthermore lines may shift by between a few [cm$^{-1}$]{} to a few tens of [cm$^{-1}$]{} between isotopologues. Figure 1 demonstrate that, as expected, the overall picture is very similar for BT2 ([H$_2$$^{16}$O]{}) and HotWat78 ([H$_2$$^{17}$O]{} and [H$_2$$^{18}$O]{}). Here we provide the comparison only for [H$_2$$^{18}$O]{} but for the [H$_2$$^{17}$O]{} it looks the same.
Figures 2 and 3 illustrate the similarity of the HotWat78 line lists with the previous high accuracy [H$_2$$^{17}$O]{} and [H$_2$$^{18}$O]{} line lists (called 3mol) of @jt438 for these molecules at the room temperature. Figures 4 and 5 also provide a comparison with the HITRAN data for the room temperature for [H$_2$$^{17}$O]{} and [H$_2$$^{18}$O]{}. These figures only provide an overview, but a detailed line by line comparison confirms that all the calculations we present here are done correctly.
The present line lists are significantly more complete, but this is only apparant at higher temperatures, see Fig. 3. For the room temperature the previous line lists should look similar, as they indeed do, see Figures 2.
Conclusions
===========
This paper reports hot line lists for [H$_2$$^{17}$O]{} and [H$_2$$^{18}$O]{}. These line lists represent significant improvement on both coverage and accuracy of the previous [H$_2$$^{17}$O]{} and [H$_2$$^{18}$O]{} line lists [@spectra; @jt438]. The predicted frequencies in these line lists have been significantly improved using information obtained from the corresponding [H$_2$$^{16}$O]{} empirical energy levels. This procedure can be adapted to give improved predictions of energy levels and transition frequencies for isotopologues of molecules for whom the empirical energy levels of the parent molecule are well-known.
The complete HotWat78 line lists for [H$_2$$^{17}$O]{} and [H$_2$$^{18}$O]{} can be downloaded from the CDS, via <ftp://cdsarc.u-strasbg.fr/pub/cats/J/MNRAS/>, or via <http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/MNRAS/>. The line lists together with auxiliary data including the potential parameters, dipole moment functions, and theoretical energy levels can be also obtained at [www.exomol.com](www.exomol.com), where they form part of the enhanced Exomol database [@jt631]. The BT2 [H$_2$$^{16}$O]{} line list [@jt378] is already available from these sources.
Finally we note that pressure-broadening has been shown to have a significant effect on water spectra in exoplanets [@jt521]. ExoMol, in common with other databases, assumes that pressure-broadening parameters for [H$_2$$^{17}$O]{} and [H$_2$$^{18}$O]{} are the same as those for [H$_2$$^{16}$O]{}. This assumption is built into the recently updated structure of the ExoMol database [@jt631]. @jt669 have recently presented a comprehensive set of pressure-broadening parameters for [H$_2$$^{16}$O]{} lines which form the basis for the ExoMol pressure-broadening diet for water [@jtdiet]. These parameters, which are available on the ExoMol website, are also suitable for use with the HotWat78 line lists.
Acknowledgements {#acknowledgements .unnumbered}
================
This work is supported by ERC Advanced Investigator Project 267219 and by the Russian Fund for Fundamental Studies.
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[^1]: Email: [email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In this paper we provide six different equivalent sets of axioms for affine $\Lambda$-buildings, by providing different types of metric conditions, exchange conditions, and atlas conditions. The work on atlas conditions builds on the work of Anne Parreau on equivalence of axioms for Euclidean buildings. The new axiom systems provide for potentially easier proofs that spaces are $\Lambda$-buildings. Moreover, we apply our result to show that the definition of a Euclidean building depends only on the topological equivalence class of the metric on the model space of the building. In an appendix a class of examples is constructed to illustrate the sharpnes of the axioms dealing with metric conditions. These examples show that a space $X$ defined over a given model space (with metric $d$) is possibly a building only if the distance function induced on $X$ (by $d$) satisfies the triangle inequality.'
address: 'Curtis D. Bennett, Department of Mathematics, Loyola Marymount University, 1 LMU Drive, Suite 2700 Los Angeles, CA 90245 Petra N. Schwer, Mathematical Institute, University of Münster, Einsteinstrasse 62, 48149 Münster, Germany Koen Struyve, Department of Pure Mathematics and Computer Algebra, Ghent University, Krijgslaan 281, B-9000 Ghent, Belgium '
author:
- 'Curtis D. Bennett and Petra N. Schwer With an appendix by Koen Struyve'
bibliography:
- 'literaturliste-axioms.bib'
date:
title: 'On axiomatic definitions of non-discrete affine buildings'
---
.
[^1]
Introduction {#sec_introduction}
============
Euclidean buildings, also referred to as non-discrete affine or ${\mathbb{R}}$-buildings, form one of the prime examples of ${\mathrm{CAT}}(0)$-spaces and were defined by Bruhat and Tits in [@BruhatTits] and [@TitsComo] to study Lie-type groups over local fields, as well as fields with non-discrete valuations.
The first author introduced the more general class of affine $\Lambda$-buildings (or more simply $\Lambda$-buildings) in [@BennettThesis] and [@Bennett], allowing for groups over Krull-valuated fields, that is fields having a valuation taking its values in an totally ordered abelian group $\Lambda$. Initially, the two main examples that were considered were the case of Lie type groups over the field $F(x,y)$ rational functions in two variables, with two possible valuations, namely, one into ${\mathbb{Z}}\times{\mathbb{Z}}$ lexicographically ordered, and the second onto the subgroup $\{x+y\pi\mid x,y\in{\mathbb{Z}}\}$ of ${\mathbb{R}}$. Care should be taken in the latter case that while geometrically the building can be naturally embedded in a traditional Euclidean building (by thinking of the valuation as landing in ${\mathbb{R}}$), the different topology of the underlying group can wreak havoc on many of the properties of and standard proofs on Euclidean buildings. Thus the $\Lambda$-building definition generalized both the notion of non-discrete ${\mathbb{R}}$-buildings and that of $\Lambda$-trees.
Kramer and Tent made use of $\Lambda$-buildings in their study of asymptotic cones and their short proof of the Margulis conjecture [@KTcones], [@KSTT]. Recently these $\Lambda$-buildings have been studied by the second author in [@Convexity2] and [@BaseChange], the latter showing that $\Lambda$-buildings are functorial in the underlying field and providing an alternative proof of Kleiner and Leeb’s result [@KleinerLeeb] that asymptotic cones of ${\mathbb{R}}$-buildings are again ${\mathbb{R}}$-buildings.
One short-coming of the definition of non-discrete buildings and affine $\Lambda$-buildings is that often in the interesting cases, the axioms can be very hard to verify. This complication led Parreau [@Parreau] to find equivalent axioms for Bruhat-Tits buildings. In our work we undertake to generalize the work of Parreau to $\Lambda$-buildings, as well as to present other equivalent axiom systems that may prove easier to work with depending on the situation. As alluded to above, while some of Parreau’s proofs in the Bruhat-Tits situation carry over to $\Lambda$-buildings, any proof that uses compactness or connectedness properties of ${\mathbb{R}}$ must be reworked as $\Lambda$ need not have these properties.
A second complication arises when looking at the definition of $\Lambda$-buildings. Namely, in the case that $\Lambda={\mathbb{R}}$ the metric on ${\mathbb{R}}^n$ (which is a model for an apartment of the building) used to define the $\Lambda$-building is not the usual Euclidean metric. This begs the question of what importance does the choice of metric play. One consequence of our main theorem is that so long as the metrics are equivalent, there will be no change in the definition of a $\Lambda$-building. Moreover, in this case, the induced metric will necessarily satisfy the triangle inequality.
It is mentioned in [@Ronan] that a certain subset of the classical axioms of Tits’ is enough to guarantee that a certain space $X$ modeled on a $\Lambda$-apartment ${\mathbb{A}}$ is in fact a $\Lambda$-building. However, it appears that there was a further assumption that the distance function induced on the space $X$ by the metric on the model space ${\mathbb{A}}$ satisfies the triangle inequality as we can see by the construction given in the appendix, i.e. Section 9. See also Remark 3.3 in this regard.
The present paper is organized as follows: {#the-present-paper-is-organized-as-follows .unnumbered}
------------------------------------------
In Section \[Sec\_Def\] we define affine $\Lambda$-buildings and list the properties and axioms in consideration. After that we will present our main results in Section \[Sec\_mr\] where we also give an outline of their proofs. The detailed proofs are then given in Sections \[Sec\_LA\] through \[Sec\_A4\]. Further details concerning the content of these sections are given after the statement of the main theorem on equivalent axiom sets in Section \[Sec\_mr\].
Finally, Section \[Sec\_example\] is devoted to the construction of certain examples of spaces emphazising the sharpnes of axiom (A5).
Definitions and axioms {#Sec_Def}
======================
The model apartment of an affine $\Lambda$-building is defined by means of a (not necessarily crystallographic) spherical root system ${\Phi}$ and a totally ordered abelian group $\Lambda$. Just as the apartments of Euclidean buildings are isomorphic copies of ${\mathbb{R}}^n$, the *model space* ${\mathbb{A}}$ of an affine $\Lambda$-building is isomorphic to $\Lambda^n$. We define $${\mathbb{A}}({\Phi},\Lambda) = \mathrm{span}_F({\Phi})\otimes_F \Lambda,$$ where $F$ is a sub-field of the real numbers containing all evaluations of co-roots on roots.
The spherical Weyl group ${\overline{W}}$ associated to ${\Phi}$ acts on ${\mathbb{A}}$ by naturally extending its action on ${\Phi}$. We define a *hyperplane* $H_\alpha$ in the model space as a fixed point set of a reflection $r_\alpha$ in ${\overline{W}}$ which separates ${\mathbb{A}}$ into two half-spaces, called *half-apartments*. An *affine Weyl group* ${{{W}}_T}$ acting on ${\mathbb{A}}$, is the semi-direct product of ${\overline{W}}$ by some ${\overline{W}}$-invariant translation group $T$ of the model space. In the case that $T$ is the entire space ${\mathbb{A}}$ we will write ${W}$ instead of ${{{W}}_T}$.
Associated to a basis $B$ of the root system ${\Phi}$ there is a *fundamental Weyl chamber* ${\mathcal{{C}}_{f}}$. The chamber ${\mathcal{{C}}_{f}}$ is a fundamental domain for the action of ${\overline{W}}$ on ${\mathbb{A}}$. Its images in ${\mathbb{A}}$ under the affine Weyl group are *Weyl chambers* (sometimes called *sectors*). If two Weyl chambers contain a common sub-Weyl chamber we call them *parallel*. A *Weyl simplex* is a face of a chamber. The smallest face of dimension 0 is called *basepoint*, and a *panel* is a Weyl simplex of co-dimension one.
One can endow ${\mathbb{A}}$ with a natural ${W}$-invariant metric taking its values in $\Lambda$, and thus making ${\mathbb{A}}$ a $\Lambda$-metric space in the following sense: A map $d:X\times X \mapsto \Lambda$ on a space $X$ is a *$\Lambda$-metric* if for all $x,y,z$ in $X$ the following conditions are satisfied
1. $d(x,y)=0$ if and only if $x=y$
2. $d(x,y)=d(y,x)$ and
3. the triangle inequality $d(x,z)+d(z,y)\geq d(x,y)$ holds.
While in principle there exist many different potential $\Lambda$-metrics for $X$, depending on $\Lambda$, the definition of one may be somewhat complicated. In the case where $\Lambda=\mathbb{R}$, the standard Euclidean metric works, however, in the case where square roots may not exist ($\Lambda=\mathbb{Q}$ or $\mathbb{Z}\times\mathbb{Z}$ for example) things can be more difficult. One such solution for all $\Lambda$ is to use a modified Minkowski metric (see [@Bennett] for details). We now define generalized non-discrete affine buildings. Throughout the following fix a model space ${\mathbb{A}}$ and an affine Weyl group ${{{W}}_T}$.
\[Def\_LambdaBuilding\] Let $X$ be a set and ${\mathcal{A}}$, called the *atlas* of $X$, be a collection of injective charts $f:{\mathbb{A}}\hookrightarrow X$. For each $f$ in ${\mathcal{A}}$, we call the images $f({\mathbb{A}})$ *apartments* and define *Weyl chambers, Weyl simplices, hyperplanes, half-apartments, etc. of $X$* to be images of such in ${\mathbb{A}}$ under any $f$ in ${\mathcal{A}}$. The pair $(X,{\mathcal{A}})$ is a *(generalized) affine building* (or $\Lambda$-building) if the following conditions are satisfied
1. The atlas is invariant under pre-composition with elements of $ {{{W}}_T}$.
2. Given two charts $f,g\in{\mathcal{A}}$ with $f({\mathbb{A}})\cap g({\mathbb{A}})\neq\emptyset$. Then $f^{-1}(g({\mathbb{A}}))$ is a closed convex subset of ${\mathbb{A}}$ and there exists $w\in {{{W}}_T}$ with $f\vert_{f^{-1}(g({\mathbb{A}}))} = (g\circ w )\vert_{f^{-1}(g({\mathbb{A}}))}$.
3. For any pair of points in $X$ there is an apartment containing both.
Given a $\Lambda$-metric $d_{{\mathcal{A}}}$ on the model space, axioms (A1)–(A3) imply the existence of a $\Lambda$-valued distance function on $X$, that is a function $d:X\times X\mapsto \Lambda$ satisfying all conditions of the definition of a $\Lambda$-metric but the triangle inequality. The distance between points $x,y$ in $X$ is the distance between their pre-images under a chart $f$ of an apartment containing both.
1. Given two Weyl chambers in $X$ there exist sub-Weyl chambers of both which are contained in a common apartment.
2. For any apartment $A$ and all $x\in A$ there exists a *retraction* $r_{A,x}:X\to A$ such that $r_{A,x}$ does not increase distances and $r^{-1}_{A,x}(x)=\{x\}$.
3. Let $f, g$ and $h$ be charts such that the associated apartments pairwise intersect in half-apartments. Then $f({\mathbb{A}})\cap g({\mathbb{A}})\cap h({\mathbb{A}})\neq \emptyset$.
By (A5) the distance function $d$ on $X$ is well defined and satisfies the triangle inequality.
One problem raised by the definition is that condition (A5) relies on the choice of $\Lambda$-metric $d_{{\mathbb{A}}}$. However, in the case of $\Lambda=\mathbb{R}$ there are two natural $\Lambda$ metrics to choose from, namely the standard Euclidean metric and the modified Minkowski mettric of [@Bennett]. More generally, there are potentially many possible metrics to choose from, and it would be helpful to have a better understanding of the role of the choice of metric in the definition (i.e., at what level does the definition depend on the choice of the metric). To keep our notation manageable, we will assume the metric on ${\mathbb{A}}$ is given as part of ${\mathbb{A}}$.
The main goal of the present paper is to prove equivalence of certain sets of axioms. From this equivalence, we will be able to show that for any two topologically equivalent metrics on ${\mathbb{A}}$ the definitions of a $\Lambda$-building for the given metrics are equivalent. Let us therefore collect all properties which are necessary to state the main result. These properties break into three categories, metric conditions, exchange conditions, and atlas conditions.
We begin with the two *metric conditions*. We note that (A5) is one such condition since it implies that the global distance function is a metric on the space $X$. In [@Bennett], this condition is used to prove the existence of a building at infinity for a $\Lambda$-building. However, in [@Parreau] (for $\Lambda=\mathbb{R}$) and [@Convexity2] (for general $\Lambda$), proofs of the existence of the building at infinity[^2] are given that do not require the full power of condition (A5). Consequently, there is benefit to having the weaker condition necessary, namely that the induced distance function on $X$ is a metric.
- (Triangle inequality) The distance function $d$ on $X$, which exists assuming (A1)–(A3), is a metric, i.e. satisfies the triangle inequality.
We next move to the *exchange type conditions*. One difficulty with generalizing the definition of an $\mathbb{R}$-building to the $\Lambda$-building case was that the totally ordered group $\Lambda$ might not be topologically connected. In the case of trees (the lowest dimension affine buildings), the move from $\mathbb{R}$-trees to $\Lambda$-trees required the introduction of a $Y$-condition, which is essentially a condition that says when two paths diverge, that the symmetric difference of those paths together with the point of divergence itself forms a path. For the higher dimensional $\Lambda$-building context, this condition was encapsulated in (A6). For our purposes, we have two other exchange conditions, each of which is slightly stronger than (A6). The first is most naturally an exchange condition, and hence we give it that name.
- (Exchange condition) Given two charts $f_1,f_2\in{\mathcal{A}}$ such that $f_1({\mathbb{A}})\cap f_2({\mathbb{A}})$ is a half apartment, then there exists a chart $f_3\in{\mathcal{A}}$ such that $f_3({\mathbb{A}})\cap f_j({\mathbb{A}})$ is a half apartment for $j=1,2$. Moreover, $f_3({\mathbb{A}})$ is the symmetric difference of $f_1({\mathbb{A}})$ and $f_2({\mathbb{A}})$ together with the boundary wall of $f_1({\mathbb{A}})\cap f_2({\mathbb{A}})$.
Note that the exchange condition can be restated in “apartment language” as: Given two apartments $A$ and $B$ intersecting in a half-apartment $M$ with boundary wall $H$, then the set $(A\oplus B)\cup H$ is also an apartment, where $\oplus$ denotes the symmetric difference of $A$ and $B$.[^3]
We will also consider the following even stronger exchange condition for which Linus Kramer suggested the name *sundial configuration*.
- (Sundial configuration) Suppose $f_1\in{\mathcal{A}}$ and $S$ is a Weyl chamber of $(X,{\mathcal{A}})$ such that $P=S\cap f_1({\mathbb{A}})$ is a panel. Letting $M$ be the wall of $f_1({\mathbb{A}})$ containing $P$. Then there exist $f_2 \neq f_3\in{\mathcal{A}}$ such that $f_1({\mathbb{A}})\cap f_j({\mathbb{A}})$ is a half-apartment and $(M\cup S)\subset f_j({\mathbb{A}})$ (for $j=2,3$).
The sundial configuration can be restated as: Given an apartment $A$ of $X$ and a chamber $c$ in the building at infinity such that $c$ shares a co-dimension one face $p$ (a panel) with $\partial A$ (the boundary of $A$ or the apartment at infinity associated to $A$) but is not contained in the $\partial A$. Then there exist two apartments $A_1 \neq A_2$ such that $c\in\partial A_i, i=1,2$ and such that $A_i\cap A$ is a half apartment with bounding wall spanned by a panel in $p$.
The last set of conditions are the *atlas conditions*. These conditions all state properties of the atlas set ${\mathcal{A}}$ in terms of containing subsets of $X$. Thus conditions (A3) and (A4) are atlas conditions. These atlas conditions typically correspond to statements about objects (two points or Weyl chambers for example) being contained in an apartment (with one exception). To be more precise, we need some terminology.
We say that two Weyl simplices $F$ and $G$ *share the same germ* if both are based at the same vertex and if $F\cap G$ is a neighborhood of $x$ in $F$ and in $G$. It is easy to see that this is an equivalence relation on the set of Weyl simplices based at a given vertex. The equivalence class of an $x$-based Weyl simplex $F$ is denoted by $\Delta_x F$ and is called the *germ of $F$ at $x$*. The germs of Weyl simplices at a vertex $x$ are partially ordered by inclusion: $\Delta_x F_1$ is contained in $\Delta_xF_2$ if there exist $x$-based representatives $F'_1, F'_2$ contained in a common apartment such that $F_1'$ is a face of $F_2'$. Let $\Delta_xX$ be the set of all germs of Weyl simplices based at $x$. We note that since the definition of a germ is dependent on the definition of a neighborhood of a point, the notion of germs are necessarily dependent on the (equivalence class of the) metric on ${\mathbb{A}}$ with which we start.
A germ $\mu$ of a Weyl chamber $S$ at $x$ is *contained* in a set $Y$ if there exists $\varepsilon\in\Lambda^{+}$ such that $S\cap B_\varepsilon(x)$ is contained in $Y$ where $B_\varepsilon(x)$ denotes the usual $\varepsilon$-ball around $x$. We are now ready to state the first three of our new atlas conditions.
- (Large atlas) Any two germs of Weyl chambers are contained in a common apartment.
- (Almost a large atlas) For all points $x$ and all $y$-based Weyl chambers $S$ there exists an apartment containing both $x$ and $\Delta_yS$.
- (Locally a large atlas) Any two germs of Weyl chambers based at the same vertex are contained in a common apartment.
Note that both (LA) and (ALA) imply (A3).
We say two $x$-based germs are *opposite* if they are contained in a common apartment $A$ and are images of one another under the longest element of the spherical Weyl group (which acts on the set of (germs of) $x$-based Weyl chambers in $A$).[^4] Two Weyl chambers are *opposite at $x$*, if their germs are opposite. This leads us to our fourth new atlas condition,
- (Opposite chambers) Two opposite $x$-based Weyl chambers $S$ and $T$ are contained in a unique common apartment.
For our last atlas condition, we need a metric notion of a “convex hull” like object. Thus, we define the *segment* ${\mathrm{seg}}_X(x,y)$ of points $x$ and $y$ in a metric space $X$ to be the set of all points $z\in X$ such that $d(x,y)=d(x,z)+d(z,y)$.
- (Finite cover) For all triples of points $x,y$ and $z$ in $X$ and all apartments $A$ containing $x$ and $y$ the segment ${\mathrm{seg}}_A(x,y)$ is contained in a finite union of Weyl chambers based at $z$.
Both, the existence of a large atlas (LA) and its local analog (GG) were introduced by Parreau [@Parreau]. Condition (LA) was called (A3’) in [@Parreau] according to its proximity to axiom (A3) and the abbreviation (GG) probably stood for “germ - germ”. The opposite chamber property (CO) also appeared in [@Parreau], where (CO) stood for “chambre opposées”. The condition (ALA) to almost have a large atlas is ’in between’ (A3) and the existence of a large atlas and suffices for one of the implications in \[MainThmB\].
Main results {#Sec_mr}
============
The purpose of this section is to state our main results.
Recall that we say that $(X,{\mathcal{A}})$ is *a space modeled on ${\mathbb{A}}$* if $X$ is a set together with a collection ${\mathcal{A}}$ of injective *charts* $f:{\mathbb{A}}\hookrightarrow X$ such that $X$ is covered by its charts. That is $X=\bigcup_{f\in{\mathcal{A}}} f({\mathbb{A}})$. Throughout the remainder of the paper we will assume that conditions $(X,{\mathcal{A}})$ satisfies conditions (A1)–(A3).
Using explicit constructions and combinatorial properties of links and the building at infinity we prove in Proposition \[Pr:A6 A6’\] that (A6) and the exchange condition (EC) are equivalent asuming (A1)–(A5) are also satisfied. By similar arguments we obtain in Proposition \[Pr:A6’ A6”\] that (EC) is implied by (A4) and the sundial configuration and that on the other hand (SC) follows from (A4), (A5) and (EC), that is (EC) and (SC) are equivalent. Hence we have
\[MainThmA\] Let $(X,{\mathcal{A}})$ be a space modeled on ${\mathbb{A}}={\mathbb{A}}({\Phi},\Lambda)$ such that axioms (A1)–(A5) are satisfied. Then $$\mathrm{(A6)} \Leftrightarrow \mathrm{(SC)} \Leftrightarrow \mathrm{(EC)}.$$
Summarizing all results proved in this paper we obtain the following theorem.
\[MainThmB\] For a space $(X,{\mathcal{A}})$ modeled on ${\mathbb{A}}={\mathbb{A}}({\Phi},\Lambda)$ which satisfies axioms (A1)–(A3), the following are equivalent:
1. \[e1\] $(X,{\mathcal{A}})$ is an affine $\Lambda$-building, that is axioms (A4), (A5) and (A6) are satisfied.
2. \[e2\] $(X,{\mathcal{A}})$ satisfies (A4), (TI) and (A6).
3. \[e3\] $(X,{\mathcal{A}})$ satisfies (A4), (TI) and (SC).
4. \[e4\] $(X,{\mathcal{A}})$ satisfies (A4), (A5) and (SC).
5. \[e5\] $(X,{\mathcal{A}})$ satisfies (A4), (A5) and (EC).
6. \[e6\] $(X,{\mathcal{A}})$ satisfies (GG) and (CO).
7. \[e7\] $(X,{\mathcal{A}})$ satisfies (LA) and (CO).
8. \[e8\] $(X,{\mathcal{A}})$ satisfies (ALA), (A4), (FC) and (EC).
Before discussing the proof, it is worth saying a few words about the relationship of the various conditions. In particular, statements (\[e2\])–(\[e5\]) correspond to various options on metric and exchange conditions. Statements (\[e6\]) and (\[e7\]) are the stronger versions of the atlas conditions (each having one on germs and one on Weyl chambers) that imply $(X,{\mathcal{A}})$ is an affine $\Lambda$-building, allowing us to ignore both metric and exchange conditions. Finally, statement (\[e8\]) lets us use the weakest of the atlas conditions (in that (ALA) is weaker than (LA) and (A4) is often easier to show than (CO)) at the cost of the finite cover condition and the exchange condition. Also recall that both (LA) and (ALA) imply (A3).
To prove the above theorem we will show the following implications: $$\xymatrix{
(\ref{e4}) \ar@{<=>}[rr]^{\text{Prop.}~\ref{Pr:A6' A6''}} & &(\ref{e5}) \ar@{<=}[rrr]^{\text{Sec.}~\ref{Sec_retractions}} & & & (\ref{e8}) \ar@{<=}[d]^{\text{Sec.}~\ref{Sec_ECSC}, \ref{Sec_retractions}, \ref{Sec_A4}} &\\
(\ref{e3}) \ar@{<=>}[rr]_{\text{Thm.}~\ref{Th:main}} & &(\ref{e1}) \ar@{=>}[r]\ar@{<=>}[u]^{\text{Prop.}~\ref{Pr:A6 A6'}} & (\ref{e2}) \ar@{=>}[rr]_{\text{Sec.}~\ref{Sec_localStructure}} & & (\ref{e6}) \ar@{<=>}[rr]_{\text{Sec.}~ \ref{Sec_LA}} &&(\ref{e7}) &\\
}$$
By Proposition \[MainThmA\] we have that (\[e1\]), (\[e4\]) and (\[e5\]) are equivalent. Equivalence of (\[e1\]) and (\[e3\]) is shown in Theorem \[Th:main\]. That (\[e1\]) implies (\[e2\]) and (\[e7\]) implies (\[e6\]) are obvious. Assuming (\[e2\]) we obtain (GG) and (CO) as discussed in Section \[Sec\_localStructure\]. For details see Corollaries \[Cor\_GG\] and \[Cor\_CO\]. Hence item (\[e2\]) implies (\[e6\]).
In Section \[Sec\_LA\] we prove Proposition \[Prop:sec sec\], which implies that (\[e7\]) follows from (\[e6\]).
By Proposition \[Pr:EC\] the axioms listed under (\[e6\]) imply that (EC) holds. In Proposition \[Prop\_FC\] it is shown that the finite cover condition (FC) follows from (A1)–(A3) and (CO). Finally we prove in Section \[Sec\_A4\] that axiom (A4) follows from (\[e6\]). This completes the proof of the fact that (\[e6\]) implies (\[e8\]).
Axiom (A5) is verified in Section \[Sec\_retractions\] using (A1), (A2) and condition (ALA) and (FC). Therefore item (\[e8\]) implies (\[e5\]). This completes the proof of our main result.
\[Rem\_history\] The original axiomatic definition of affine ${\mathbb{R}}$-buildings is due to Jaques Tits, who defined the “syst[è]{}me d’appartements” in [@TitsComo] by listing five axioms. The first four of these are precisely axioms (A1)–(A4) as presented above. His fifth axiom originally reads different from ours but was later replaced with what is now axiom (A5) as written in Definition \[Def\_LambdaBuilding\]. One can show that in case of ${\mathbb{R}}$-buildings axioms (A6) follows from (A1)–(A5). In appendix 3 of [@Ronan] it is stated that (A5) and (A6) are equivalent, which would imply that condition (TI) can be dropped in (\[e2\]) of Theorem \[MainThmB\]. It is however possible to construct examples of spaces satisfying (A1)–(A4) that vacuously satisfy (A6) but satisfy neither (A5) nor (TI). See Section \[Sec\_example\] for details.
An application
--------------
In this section we explain a simple yet interesting consequence of Theorem \[MainThmB\] alluded to earlier. The class of affine $\Lambda$-buildings is a generalization of Euclidean buildings, which themselves generalize the (geometric realizations of) simplicial affine buildings. The affine $\Lambda$-buildings can be endowed with a $\Lambda$-metric.
Concerning the metric structure the following difficulty arises when viewing Euclidean buildings as the subclass of affine $\Lambda$-buildings where $\Lambda={\mathbb{R}}$ and where the translational part $T$ of the affine Weyl group equals the co-root-lattice spanned by a crystallographic root system, or is the full translation group of an apartment in the non-crystallographic case. When studying Euclidean buildings one usually uses the Euclidean metric on the model space. Compare for example [@Parreau] or Kleiner and Leeb [@KleinerLeeb].
The natural metric on the model space of an affine $\Lambda$-building is however defined in terms of the defining root system ${\Phi}$, compare [@Diss], and is a generalization of the length of translations in apartments of simplicial affine buildings. This length function on the set of translational elements of the affine Weyl group is defined with respect to the length of certain minimal galleries. Hence this natural metric used for $\Lambda$-buildings is different from the Euclidean one in case $\Lambda={\mathbb{R}}$.
The question arising is the following: Let us assume that $X$ is an affine building with metric $d$, which is induced by a metric $d_{\mathbb{A}}$ on the model space. Let $d'_{\mathbb{A}}$ be a metric on the model space different from $d_{\mathbb{A}}$ and does hence induce a second distance function $d'$ on $X$. Does $d'$ satisfy the triangle inequality? And is $(X,d')$ an affine building? To be able to answer these questions one has to understand whether the retractions appearing in (A5) do exist and are distance diminishing. Theorem \[MainThmB\] answers these questions with “yes”, so long as the metrics are topologically equivalent on ${\mathbb{A}}$.
\[thm\_cor\] Let $(X,{\mathcal{A}})$ be an affine building with metric $d$ on $X$. Then every ${W}$-invariant metric $d'$ topologically equivalent to $d$ which can be defined on the model space extends to a metric on $X$. In particular “being a building” only depends on the equivalence class of $d$, not on the metric itself.
Let $(X,{\mathcal{A}})$ be a building equipped with a metric $d$. Then for any topologically equivalent metric $d'$ axioms (A6) and (A1) to (A4) are still satisfied, since these axioms do not contain conditions on the metric. In addition, since the metrics $d$ and $d'$ are topologically equivalent on the model space, they define the same germs of Weyl chambers. In particular, conditions (GG), (LA), and (ALA) are all satisfied for $d'$ if they are satisfied for $d$. Since condition (CO) is independent of the metric, it follows that both statements (6) and (7) of \[MainThmB\] are true for $d'$ also. Consequently $(X,{\mathcal{A}},d')$ is also a $\Lambda$-building and the metric induced by $d'$ satisfies (TI) by statement (2).
Thus whether or not a pair $(X,{\mathcal{A}})$ modeled on ${\mathbb{A}}$ is an affine building only depends on the topological equivalence class of the metric imposed on ${\mathbb{A}}$.
Having a large atlas (LA) {#Sec_LA}
=========================
Assume that $(X,{\mathcal{A}})$ is a pair satisfying axioms (A1) to (A3). Recall that the germs of Weyl simplices based at a vertex $x$ are partially ordered by inclusion. A germ $\Delta_x S_1$ of a Weyl simplex $S_1$ is contained in $\Delta_xS_2$ if there exist $x$-based representatives $S'_1, S'_2$ contained in a common apartment such that $S_1'$ is a face of $S_2'$. The *residue* $\Delta_xX$ of $X$ at $x$ is the set of all germs of Weyl simplices based at $x$.
\[Thm\_residue\] Assume that $(X,{\mathcal{A}})$ has property (GG) in addition to (A1)–(A3). Then $\Delta_xX$ is a spherical building of type ${\Phi}$ for all $x$ in $X$. Furthermore if in addition (A5) holds $\Delta_xX$ is independent of ${\mathcal{A}}$.
We verify the axioms of the definition of a simplicial building, which can be found on page 76 in [@Brown]. It is easy to see that $\Delta_xX$ is a simplicial complex with the partial order defined above. It is a pure simplicial complex, since each germ of a face is contained in a germ of a Weyl chamber. The set of equivalence classes determined by a given apartment of $X$ containing $x$ is a subcomplex of $\Delta_xX$ which is a Coxeter complex of type ${\Phi}$. Hence we define those to be the apartments of $\Delta_xX$. Therefore, by definition, each apartment is a Coxeter complex. Two apartments of $\Delta_xX$ are isomorphic via an isomorphism fixing the intersection of the corresponding apartments of $X$, hence fixing the intersection of the apartments of $\Delta_xX$ as well. Finally due to property (GG) any two chambers are contained in a common apartment and we can conclude that $\Delta_xX$ is a spherical building of type ${\Phi}$.
Let ${\mathcal{A}}'$ be a different system of apartments of $X$. We will denote by $\Delta$ the spherical building of germs at $x$ with respect to ${\mathcal{A}}$ and by $\Delta'$ the building at $x$ with respect to ${\mathcal{A}}'$. Since spherical buildings have a unique apartment system the buildings $\Delta$ and $\Delta'$ are equal if they contain the same chambers. Let $c\in\Delta'$ be a chamber; we will show $c\in\Delta$. Let $d$ be a chamber opposite $c$ in $\Delta'$ and $a'$ the unique apartment containing both. Then $a'$ corresponds to an apartment $A'$ of $X$ having a chart in ${\mathcal{A}}'$ and there exist ${\mathcal{A}}'$-Weyl chambers $S_c$, $S_d$ contained in $A'$ representing $c$ and $d$, respectively. Choose a point $y$ in the interior of $S_c$ and let $z$ be contained in the interior of $S_d$. By (A3) there exists a chart $f\in {\mathcal{A}}$ such that the image $A$ of $f$ contains $y$ and $z$. By [@Convexity2 Prop. 6.2], which uses (A5), the point $x$ is contained in $A$. And by construction the unique $x$-based Weyl chamber in $A$ which contains $y$ has germ $x$ and the unique $x$-based Weyl chamber in $A$ containing $z$ has germ $d$. Thus $c\in\Delta$. Interchanging the roles of $\Delta$ and $\Delta'$ above we have that they contain the same chambers. Hence $\Delta=\Delta'$.
\[Lm:secgm sec\] Assume in addition to (A1)–(A3) that $(X,{\mathcal{A}})$ satisfies (GG) and (CO) or, alternatively, that (A4), (TI), and (SC) are satisfied. Let $S$ and $T$ be two $x$-based Weyl chambers. Then there exists an apartment containing $S$ and a germ of $T$ at $x$.
In case that (GG) and (CO) are satisfied, the proof is as in [@Parreau Prop 1.15].
Assume (A4), (TI), and (SC). We write $d(S,T)$ for the Weyl group-valued Weyl chamber-distance of $\partial S$ and $\partial T$ in the spherical building at infinity and we similarly write $\delta(\mu, \nu)$ for the Weyl chamber-distance of germs $\mu$ and $\nu$ in a residue $\Delta_xX$. Taking $\ell$ to be the usual length function on Coxeter groups, we use $\ell(\delta(\mu,\nu))$, and $\ell(d(S,T))$ to denote the length of a minimal gallery connecting the germs $\mu$ and $\nu$ and respectively of the one connecting the chambers $\partial S, \partial T$ in the building at infinity. Further, let $A_0$ be an apartment containing $S$.
By axiom (A4) there exists an apartment $A'$ containing sub-Weyl chambers $S'$ of $S$ and $T'$ of $T$, and $\ell(d(S',T'))\geq \ell(d(S,T))$. Replace $S'$ and $T'$ by Weyl chambers in $A'$ based at a common vertex $x'\in S$, consider a minimal-length sequence $$S'=S_0,\dots,S_n=T',$$ of $x'$-based Weyl chambers. By construction $A_0$ contains $S'$. Let $j$ be minimal such that $S_{j+1}$ contains no sub-Weyl chamber in $A_0$. If $j=n$ (i.e., $T'$ has a sub-Weyl chamber $T''$ contained in $A_0$), then $T''$ is Weyl chamber of $A_0$ and, as $x\in A_0$, by convexity it follows that $T\subset A_0$ and there is nothing to prove. We will induct on $n-j$. The basis step having been proved, assume $S_{j+1}$ has no sub-Weyl chamber contained in $A_0$ but $S_0,\dots,S_j$ all have sub-Weyl chambers in $A_0$. In this case, there exists a Weyl chamber $S_{j+1}'$ parallel to $S_{j+1}$ (in $A'$) such that $S_j'\cap A_0$ is a panel (parallel to a panel of $S_j$). By (SC) there exists an apartment $A_{j+1}$ containing $S_{j+1}'$ and the germ $\Delta_yS$ of the Weyl chamber $S$ (since for any wall $\Delta_yS$ must lie on one side or the other of the wall). If $S$ is contained in $A_{j+1}$, then we replace $A_0$ with $A_{j+1}$ and by induction on $n-j$ we have the result. On the other hand, if $S\not\subset A_{j+1}$, let $S''$ be the Weyl chamber of $A_{j+1}$ with germ $\Delta_yS''=\Delta_yS$. Then $$\ell(d(T',S_{j+1}))=\ell(d(T,S_{j+1}))-1.$$ Moreover, considering the case of $S''$ and $T$, together with $A_{j+1}$ as our new $A_0$, by induction there exists an apartment $A$ containing $\Delta_yS''$ and $T$, with $\ell(\delta( \Delta_yS'',\Delta_yT))\le \ell(d( S'', T))$. However, $\Delta_yS=\Delta_yS''$ and $$\ell(d( S'', T))\le\ell(d( S, T))-1.$$ Hence $$\ell({\delta}(\Delta_yS,\Delta_yT))\le\ell(d(S,T))$$ as desired. Note that if equality holds, then in each case, the apartment $A_j$ contains $S$ (where we take $A_j$ as the apartment containing $S_j$ and $S_y$ in the proof), and in particular, $A_n$ contains both $S$ and $T$ as desired.
\[Lm:opp sec\] Let $(X,{\mathcal{A}})$ be a pair satisfying axioms (A1)–(A4), (TI), and (SC). If $S$ and $T$ are Weyl chambers of $X$ based at $y$, and $\delta(\Delta_yS,\Delta_yT)$ is maximal, then $S$ and $T$ are contained in a common apartment.
Since $\delta(\Delta_yS,\Delta_yT)$ is maximal, the proof of Lemma [\[Lm:secgm sec\]]{} implies that $\ell(\delta(\Delta_yS,\Delta_yT))=\ell(d(S,T))$. However, in this case the lemma implies the existence of an apartment $A$ containing $S$ and $T$.
Another proof of the above corollary can be found in 5.23 of [@Diss]. The following proposition was already proved in [@Diss 5.15] using similar techniques.
\[Prop:sec sec\] Assume that $(X,{\mathcal{A}})$ satisfies either (GG) and (CO) or, alternatively, axioms (A4), (TI) and (SC). Then (LA) is satisfied as well.
Let $\Delta_xS$ and $\Delta_yT$ be two germs of Weyl chambers $S$ and $T$. We begin by showing that $\Delta_xS$ and $y$ are contained in a common apartment $B$. By (A2), there exists an apartment $A$ containing $x$ and $y$. Let $C$ be a Weyl chamber of $A$ based at $x$ containing $y$. By Lemma \[Lm:secgm sec\] there exists an apartment $B$ of $X$ containing $\Delta_xS$ and $C$. Take $S'$ to be the Weyl chamber of $B$ based at $y$ containing $\Delta_xS$. Again by Lemma \[Lm:secgm sec\], there exists an apartment $A'$ containing $\Delta_yT$ and $S'$. Since $\Delta_xS\subset S'$, it follows that $A'$ contains $\Delta_yT$ and $\Delta_xS$ as desired.
Exchange axioms {#Sec_ECSC}
===============
In this section, we prove equivalence of the sundial configuration (SC), the exchange condition (EC) and axiom (A6) given that (A1)–(A5) are satisfied. Recall that (A1)–(A5) are enough to show that there exists a spherical building at infinity ([@Diss]). Given $(X,{\mathcal{A}})$, we will use the standard notation of $(\partial X,\partial{\mathcal{A}})$ to denote the associated spherical building at infinity, and given an apartment $A$ of ${\mathcal{A}}$, we will write $\partial A$ to denote the associated apartment at infinity. For notational convenience we will also use lower case letters for apartments in the building at infinity when not using the $\partial$-notation.
\[Pr:A6 A6’\] Let $(X,{\mathcal{A}})$ be a pair satisfying conditions (A1)–(A5), then condition (A6) and the exchange condition (EC) are equivalent.
First assume $(X,{\mathcal{A}})$ satisfies (A6). Suppose $A_1=f_1({\mathbb{A}})$ and $A_2=f_2({\mathbb{A}})$ are two apartments of $X$ with $A_1\cap A_2=H$ a half-apartment. Then $\partial A_1$ and $\partial A_2$ are apartments of $\partial X$ that intersect in a half-apartment with bounding wall $m$. By spherical building theory, it follows that there exists an apartment $a_3$ whose chambers are the chambers of $(\partial A_1\oplus \partial A_2) \cup\, m$, further there exists an apartment $A_3$ of $X$ with $\partial A_3 = a_3$.
Since $\partial A_1\cap a_3$ is a half-apartment and $A_1\cap A_3$ is convex by (A2), it follows that $A_1\cap A_3$ is a half-apartment. Similarly $A_2\cap A_3$ is a half-apartment. Condition (A6) now implies that $A_1\cap A_2\cap A_3$ contains some element $x\in X$. Since $x\in H$ and $a_3$ contains the chambers of $\partial A_1$ that are not in $\partial A_2$, it follows that $A_3$ contains $A_1-{H}$. Similarly $A_2 - {H}\subset A_3$. By convexity the bounding wall $M$ of ${H}$ is contained in $A_3$. But now the convexity of $A_3$ implies that $x \in M$ as otherwise the wall parallel to $M$ through $x$ would not separate points of $A_1\cap A_3$ and $A_2\cap A_3$. This implies that the exchange condition (EC) holds.
Now assume that (A1)–(A5) and (EC) are satisfied, and let $A_1$, $A_2$, and $A_3$ be half apartments of $X$ such that any two intersect in a half-apartment. By way of contradiction, suppose $A_1\cap A_2\cap A_3=\emptyset$. Let $H_{i,j}=A_i\cap A_j$ for $i,j\in\{1,2,3\}$. Since $H_{1,2}\cap H_{1,3}=\emptyset$, it follows that if $H$ is a half-apartment of $A_1$ with $H_{1,2}\cap H$ contained in the boundary $M$ of $H$, then $H_{1,3}\cap H$ is again a half-apartment. Now the exchange condition (EC) implies that there exists an apartment $A_4$ such that $A_4=(A_1\oplus A_2) \cup M_{1,2}$, where $M_{1,2}$ stands for the bounding wall of $H_{1,2}$. Note that $H_{1,3}\subseteq A_4$, so that $\partial A_4$ consists of the same Weyl chambers as $\partial A_3$. However, the apartments of $X$ are in one-to-one correspondence with the apartments of $\partial X$. Therefore, $A_3=A_4$.
\[Pr:A6’ A6”\] Let $(X,{\mathcal{A}})$ be a pair satisfying conditions (A1)–(A3). Then the following implications hold:
- if (A4), (A5) and (EC) are true, then (SC) is satisfied, and
- provided (A4) and (SC) hold, then (EC) does as well.
In particular, given (A1)–(A5) conditions (EC) and (SC) are equivalent.
Suppose $(X,{\mathcal{A}})$ satisfies conditions (A1)–(A5) and (EC). Let $A_1$ be an apartment and $S$ a Weyl chamber such that $S\cap A_1$ is a panel of $S$. Then $\partial A_1\cap \partial S$ is a panel in ${\partial_{\mathcal{A}}}X$. Therefore there is an apartment $a_2$ of ${\partial_{\mathcal{A}}}X$ such that $\partial S\in a_2$, and $\partial A_1\cap a_2$ is a half-apartment. Let $A_2$ be the apartment of $X$ with $\partial A_2 = a_2$. Since $\partial A_1\cap a_2$ is a half-apartment the intersection $A_1\cap A_2$ is a half-apartment. We now apply condition (EC) to obtain (SC).
Conversely, suppose $(X,{\mathcal{A}})$ satisfies conditions (A1)–(A4) and (SC), and let $A_1$ and $A_2$ be apartments of $X$ intersecting in a half-apartment $H$. Let $S$ be a Weyl chamber of $A_2$ such that $S\cap A_1$ is a panel $P$ of $S$, and let $M$ be the wall of $A_1$ containing $P$. By (SC), there exists an apartment $A_3$ containing $M$ such that $A_1\cap A_3$ is a half-apartment and $A_2\cap A_3$ is a half-apartment (as $A_2$ must be one of the apartments whose existence is guaranteed by (SC)) containing $M\cup S$. By convexity, it follows that $A_3=(A_1\oplus A_2)\cup M$ as desired.
The following proposition is used in the proof of Theorem \[MainThmB\] in order to show that item (\[e6\]) implies item (\[e8\]).
\[Pr:EC\] Let $(X,{\mathcal{A}})$ be a pair satisfying axioms (A1)–(A3) and (CO) and assume that the germs at each vertex form a spherical building. (By \[Thm\_residue\] this is true assuming (GG), for example.) Then the exchange condition (EC) is satisfied.
Let $A$ and $B$ be apartments intersecting in an half-apartment $M$. Let $x$ be a point contained in the bounding wall $H$ of $M$. By assumption $\Delta_xX$ is a spherical building. Therefore the union of $\Delta_x(A\setminus M)$, $\Delta_x(B\setminus M)$ and $\Delta_xH$ is an apartment in $\Delta_xX$, which we denote by $\Delta_xA'$.
We choose two opposite germs $\mu$ and $\sigma$ at $x$ which are contained in $\Delta_x(A\setminus M)$ and $\Delta_x(B\setminus M)$, respectively. Let $T$ be the unique Weyl chamber in $A$ having germ $\mu$ and let $S$ be the unique Weyl chamber in $B$ with germ $\sigma$. By construction $S$ and $T$ are opposite and thus condition (CO) implies that the Weyl chambers $S$ and $T$ are contained in a common apartment $A''$. Since two opposite Weyl chambers contained in the same apartment determine this apartment uniquely we can conclude that $\Delta_xA''=\Delta_xA'$. We conclude that $A''\cap ((A\oplus B)\cup H)$ contains $S$, $T$ and $\Delta_xA'$. Axiom (A2) says that apartments intersect in convex sets. Therefore $A''\cap (B\setminus M) = B\setminus M$ and $A''\cap (A\setminus M) = A\setminus M$ which implies that $A''\cap ((A\oplus B)\cup H) = A''$.
Local structure {#Sec_localStructure}
===============
Recall that we say a germ $\mu$ of a Weyl chamber $S$ at $x$ is *contained in a set $Y$* if there exists $\varepsilon\in\Lambda^{+}$ such that $S\cap B_\varepsilon(x)$ is contained in $Y$.
In the following let $(X,{\mathcal{A}})$ be a pair satisfying (A1)–(A4), (TI) and (A6).
\[Prop\_tec16\] Let $c$ be a chamber in ${\partial_{\mathcal{A}}}X$ and $S$ an $x$-based Weyl chamber in $X$. Then there exists an apartment $A$ such that $\Delta_xS$ is contained in $A$ and such that $c$ is a chamber of $\partial A$.
The proof of the proposition above is as in [@Parreau Prop. 1.8]. As a direct consequence we obtain:
\[Cor\_GG\] Any such pair has property (GG).
By the previous corollary $(X,{\mathcal{A}})$ satisfies the assertion of Theorem \[Thm\_residue\], i.e. the germs at a fixed vertex form a spherical building.
\[Prop\_LA\] $(X,{\mathcal{A}})$ has a large atlas (LA).
We need to prove that if $S$ and $T$ are Weyl chambers based at $x$ and $y$, respectively, then there exists an apartment containing a germ of $S$ at $x$ and a germ of $T$ at $y$.
By axiom (A3) there exists an apartment $A$ containing $x$ and $y$. We choose an $x$-based Weyl chamber $S_{xy}$ in $A$ that contains $y$ and denote by $S_{yx}$ the Weyl chamber based at $y$ such that $\partial S_{xy}$ and $\partial S_{yx}$ are opposite in $\partial A$. Then $x$ is contained in $S_{yx}$. If $\Delta_yT$ is not contained in $A$ apply Proposition \[Prop\_tec16\] to obtain an apartment $A'$ containing a germ of $T$ at $y$ and containing $\partial S_{yx}$ at infinity. But then $x$ is also contained in $A'$.
Let us denote by $S'_{xy}$ the unique Weyl chamber contained in $A'$ having the same germ as $S_{xy}$ at $x$. Without loss of generality we may assume that the germ $\Delta_yT$ is contained in $S'_{xy}$. Otherwise $y$ is contained in a face of $S'_{xy}$ and we can replace $S'_{xy}$ by an adjacent Weyl chamber in $A'$ satisfying this condition. A second application of \[Prop\_tec16\] to $\partial S'_{xy}$ and the germ of $S$ at $x$ yields an apartment $A''$ containing $\Delta_xS$ and $S'_{xy}$ and therefore $\Delta_yT$.
Propositions \[Prop\_A6\] to \[Prop\_liftGallery\] below are due to Linus Kramer. A proof of \[Prop\_A6\] can be found in [@Diss Prop. 5.20].
\[Prop\_A6\] Let $A_i$ with $i=1,2,3$ be three apartments of $X$ pairwise intersecting in half-apartments. Then $A_1\cap A_2\cap A_3$ is either a half-apartment or a hyperplane.
\[prop\_sc\] $(X,{\mathcal{A}})$ satisfies (SC).
Let $A$ be an apartment in $X$ and $c$ a chamber not contained in $\partial A$ but containing a panel of $\partial A$. Then $c$ is opposite two uniquely determined chambers $d_1$ and $d_2$ in $\partial A$. Since any pair of opposite chambers is contained in a common apartment, there exist apartments $A_1$ and $A_2$ of $X$ such that $\partial A_i$ contains $d_i$ and $c$ with $i=1,2$. The three apartments $\partial A_1,\partial A_2$ and $\partial A$ pairwise intersect in half-apartments.
Axiom (A6) together with the proposition above implies that the three apartments of the sundial configuration intersect in a hyperplane.
For any point $x\in X$ one can define a natural projection $\pi:{\partial_{\mathcal{A}}}X \to \Delta_xX$ from the building at infinity to the residue at $x$ as follows. Let $c$ be a chamber at infinity. Then there exists a unique Weyl chamber $S$ based at $x$ such that $\partial S = c$. Let $\pi (c) =\Delta_xS$.
\[Prop\_liftGallery\] Let $(c_0, \ldots, c_k)$ be a minimal gallery in the building at infinity, $x$ a point in $X$ and for all $i$ denote by $S_i$ the $x$-based representative of $c_i$. If $(\pi_x(c_0), \ldots, \pi_x(c_k))$ is minimal in $\Delta_xX$, then there exists an apartment containing $\bigcup_{i=0}^k S_i$.
This follows by induction over $k$ using Proposition \[prop\_sc\] as in [@Diss Prop. 5.22].
\[Cor\_CO\] If $(X,{\mathcal{A}})$ satisfy properties (A1)–(A4), (TI), and (A6) then it also satisfies property (CO).
Let $S$ and $T$ be Weyl simplices opposite at $x$. Choose a minimal gallery $(c_0,c_1,\ldots, c_n)$ from $c_0=\partial S$ to $c_n=\partial T$ and consider the representatives $S_i$ of $c_i$ based at $x$. With $S_0=S$ and $S_n=T$ Proposition \[Prop\_liftGallery\] implies the assertion.
Retractions based at germs {#Sec_retractions}
==========================
In this section unless otherwise otherwise stated let $(X,{\mathcal{A}})$ be a pair satisfying axioms (A1), (A2) and assume there is an almost large atlas (ALA) (implying (A3) also). Further fix an apartment $A$ in $X$ with chart $f \in {\mathcal{A}}$.
\[Def\_vertexRetraction\] Let $\mu$ be a germ of a Weyl chamber and $y$ a point in $X$, then, by (ALA), there exists a chart $g\in{\mathcal{A}}$ such that $y$ and $\mu$ are contained in $g({\mathbb{A}})$. By axiom (A2) there exists $w\in{W}$ such that $g\vert_{g^{-1}(f({\mathbb{A}}))}=(f\circ w)\vert_{g^{-1}(f({\mathbb{A}}))}$. Hence we can define $$r_{A,\mu}(y) = (f\circ w\circ g^{-1} )(y).$$ The map $r_{A,\mu}$ is called *retraction onto $A$ centered at $\mu$*.
\[Prop\_r\] Let $\mu$ be a germ of a Weyl chamber in $A$. Then $r_{A,\mu}$ is well defined and the restriction of $r_{A,\mu}$ to any apartment $A'$ containing $\mu$ is an isomorphism onto $A$.
By (A2) the map $r_{A,\mu}$ is well defined. Since each map $w\in W$ preserves the distance on the model space ${\mathbb{A}}$, we have that $d(y,z)=d(r_{A, \mu}(y),r_{A, \mu}(z))$ for all $y,z\in X$ such that $y$, $z$, and $\mu$ are contained in a common apartment. Hence the second assertion follows.
We now introduce finite covering properties which will allow us to prove that under certain conditions the defined retractions based at germs are distance diminishing.
\[Lem\_cover\] Assume $(X,{\mathcal{A}})$ is a pair satisfying (A1)–(A3) and property (CO). Let $A$ be an apartment of $X$ and $z\in X$ a point. Then $A$ is contained in the (finite) union of all $z$-based Weyl chambers parallel to a Weyl chamber in $A$, that is all Weyl chambers $S$ based at $z$ with $\partial S\in\partial A$.
In case $z$ is contained in $A$ this is obvious. Hence we assume that $z$ is not contained in $A$. For all $p\in A$ there exists, by (A3), an apartment $A'$ containing $z$ and $p$. Let $S_+\subset A'$ be a $p$-based Weyl chamber containing $z$. We denote by $\sigma_+$ its germ at $p$. There exists a $p$-based Weyl chamber $S_-$ in $A$ such that its germ $\sigma_-$ is opposite $\sigma_+$ at $p$, i.e. these two germs are images of one another under the longest element of the spherical Weyl group interpreted as the point stabilizer of $p$ in $A$. By property (CO) the Weyl chambers $S_-$ and $S_+$ are contained in a common apartment $A''$. Let $T$ be the unique $z$-based translate of $S_-$ in $A''$. Since $z\in S_+$ and $\sigma_+$ and $\sigma_-$ are opposite we have that $S_-\subset T$. In particular $p$ is contained in $T$.
\[Prop\_FC\] Assume $(X,{\mathcal{A}})$ is a pair satisfying (A1)–(A3) and properties (GG) and (CO), then the finite cover condition (FC) is satisfied. Further, if $\mu$ is a $z$-based germ of a Weyl chamber, then the segment between two points $x$ and $y$ in an apartment $A$ is contained in a finite union of apartments containing $\mu$.
Condition (FC) follows by Lemma \[Lem\_cover\]. To show the rest let $I$ be the (finite) index set of the $z$-based Weyl chambers $S_i$ with equivalence class in $\partial A$. By \[Lem\_cover\], we may conclude that ${\mathrm{seg}}_A(x,y)\subset A\subset \bigcup_{i\in I} S_i$. We fix $i$ and deduce from (GG) that there is an apartment ${A}_i$ containing $\mu$ and $\Delta_zS_i$. Let $S_i^{op}$ be a Weyl chamber in ${A}_i$ whose germ is opposite $\Delta_zS_i$. Then (CO) implies that there is a unique apartment $A'_i$ containing the union of $S_i$ and $S_i^{op}$. Hence $A$ and therefore ${\mathrm{seg}}_A(x,y)$ is contained in $\bigcup_{i\in I} A'_i$. Hence the proposition.
Next we show a local version of the sundial configuration.
\[cond\] From now on assume that $(X,{\mathcal{A}})$ is a pair satisfying conditions (A1)–(A4), (TI) and the sundial configuration (SC).
\[Lm:sup exch\] With $(X,{\mathcal{A}})$ as in \[cond\] let $A$ be an apartment of $X$ and $\Delta_xS$ a germ of a Weyl chamber such that $\Delta_xS\cap A$ is a panel-germ $\Delta_xP$. Then there exist apartments $A'$ and $A''$ such that $\Delta_xS \in A'\cap A''$ and $A\subset A'\cup A''$.
By (SC) it suffices to show that there is a Weyl chamber $S'$ of $X$ intersecting $A$ in a panel such that $\Delta_xS'=\Delta_xS$. Let $T$ be a Weyl chamber of $A$ based at $x$ having a panel containing the panel-germ $\Delta_xP$. By Lemma \[Lm:secgm sec\] there exists an apartment $B$ containing $T$ and $\Delta_xS$. Let $S'$ be the Weyl chamber of $B$ having the panel-germ $\Delta_xS$. Then $S'$ has $P'$ as a panel. Moreover, by convexity, if $S'\cap A\ne P'$, then $\Delta_xS=\Delta_x{S'}\subset A$ contrary to our hypothesis. Therefore $S'\cap A=P'$ and by (SC) there exists apartments $A'$ and $A''$ such that $S'\subset A'\cap A''$ and $A\subset A'\cup A''$.
This exchange condition allows us to work with germs based at a common point, much as in the simplicial buildings case one works with chambers in a spherical residue. The assertion of the following proposition is similar to the finite cover condition (FC) shown in Proposition \[Prop\_FC\]. However, the assumptions made in \[Prop\_FC\] differ from the ones here.
\[Pr:opps\] Suppose $(X,{\mathcal{A}})$ is as in \[cond\]. Let $A$ and $B$ be apartments of $X$ and $\Delta_xS$ a germ of a Weyl chamber in $A$. Then for every point $y\in B$ there exists a Weyl chamber $T$ of $B$ such that
1. the $x$-based Weyl chamber $T'$ parallel to $T$ contains $y$, and
2. there exists an apartment $A'$ of $X$ containing $T$ and $\Delta_xS$.
By Proposition \[Prop:sec sec\], for every Weyl chamber $T$ of $B$ based at $y$, there exists an apartment $B'$ of $X$ containing $\Delta_yT$ and $\Delta_xS$. Let $S'$ denote the Weyl chamber of $B'$ based at $y$ containing $\Delta_xS$. Choose $T$ such that $\ell(\delta(\Delta_yT,\Delta_yS'))$ is maximal.
If $\Delta_yT$ and $\Delta_yS'$ are not opposite (that is $\delta(\Delta_yT,\Delta_yS')$ is not the longest element of $\overline{W}$) then let $\Delta_yP$ be a panel germ of $\Delta_yT$ such that the wall $M$ of $B'$ through $\Delta_yP$ does not separate $\Delta_yT$ and $S'$. In the apartment $B$ there exists a Weyl chamber $R$ such that $\Delta_yR$ shares $\Delta_yP$ with $\Delta_yT$.
Lemma \[Lm:sup exch\] implies, that there exists an apartment $B''$ containing $S'$ and $\Delta_yR$. Hence, since $\Delta_yT$ and $S'$ lie on the same side of $M$, by convexity the apartment $B''$ also contains $\Delta_yT$. In $B''$, we then have $\ell(\delta(\Delta_yR,\Delta_yS'))=\ell(\delta(\Delta_yT,\Delta_yS'))+1$, contradicting the choice of $T$.
Hence we may assume that $\Delta_yT$ and $\Delta_yS'$ are opposite. By Corollary \[Lm:opp sec\], there exists an apartment $A'$ of $X$ containing $S'$ and $T$. But $\Delta_xS\subset S'$, so that $\Delta_xS\subset A'$. Moreover, since $A'$ contains $T$, take $T'$ to be the Weyl chamber based at $x$ parallel to $T$ (in $A'$). Since $T$ and $S'$ were opposite Weyl chambers and $x\in T$, it follows that $y\in T'$, completing the proof of the proposition.
\[Lm:fin cover\] Suppose that either $(X,{\mathcal{A}})$ is as in \[cond\] or is a pair satisfying (A1), (A2), (ALA), and (FC). Let $A,B$ be apartments of $X$ and $\Delta_xS$ a germ of a Weyl chamber in $A$. Then there exist closed convex sets $X_1$, …, $X_n$ in $B$ such that
1. $B=X_1\cup\dots\cup X_n$ and
2. and for each $i$ there is an apartment containing $X_i$ and $\Delta_xS$.
In the first case Proposition \[Pr:opps\] implies that there exist $x$-based Weyl chambers $S_1$, $S_2$, …, $S_n$ and apartments $A_i, i=1,\ldots,n$ such that $S_i$ and $\Delta_xS$ is contained in $A_i$ for all $i$. In the second case let $S_1, \ldots S_n$ be the Weyl chambers provided by the finite cover condition (FC). Set $X_i = S_i\cap B$ and the corollary follows.
We are now ready to prove that the retractions in consideration are distance diminishing.
\[Prop\_retraction\] Let $(X,{\mathcal{A}})$ be as in \[cond\] or a pair satisfying axioms (A1), (A2), (ALA), and (FC). Then for all apartments $A$ and germs $\mu$ of Weyl chambers contained in $A$ the retraction $r_{A,\mu}$ defined in \[Def\_vertexRetraction\] is distance non-increasing. In particular we conclude that the pair $(X,{\mathcal{A}})$ satisfies axiom (A5).
Assuming $X$ is as in \[cond\] condition (LA) is satisfied by Proposition \[Prop:sec sec\]. Hence using Proposition \[Prop\_r\] in both cases we can conclude: if $B$ is an apartment containing $\mu$, $y$, and $z$ then $d(y,z)=d(r_{A,\mu}(y),r_{A,\mu}(z))$, so the result holds true. Now, suppose $y$ and $z$ are arbitrary. By (A2) there exists an apartment $B$ containing $y$ and $z$. By Corollary \[Lm:fin cover\] there exists closed convex sets $X_1$, …, $X_n$ such that $B=\bigcup_{i=1}^n X_i$ and each $X_i$ is contained in a common apartment with $\Delta_xS$. Since each $X_i$ is convex and closed, there exists a sequence of points $y=y_0,y_1,\dots,y_k=z$ such that $y_{i-1},y_i\in X_{j_i}$ for some $j_1$, …, $j_k$ and $y_i$ is in the convex hull of $y_{i-1}$ and $y_{i+1}$ for $i=1,\dots,k-1$. Then $$\begin{aligned}
d(y,z)&=&\sum_{i=1}^k d(y_{i-1},y_i) \\
& = & \sum_{i=1}^k d(r_{A,\mu}(y_{i-1}),r_{A,\mu}(y_i)) \\
& \ge & d(r_{A,\mu}(y_0),r_{A,\mu}(y_k))
= d(r_{A,\mu}(y),r_{A,\mu}(z)),\end{aligned}$$ where we use the triangle inequality for $d$ restricted to $A$ in the next to the last step. Thus, $r_{A,\mu}$ is distance diminishing and hence a retraction with the required properties of (A5).
We now wish to show that condition (SC) can replace conditions (A5) and (A6) in the definition of an affine $\Lambda$-building.
\[Th:main\] Suppose $(X,{\mathcal{A}})$ satisfies axioms (A1)–(A4). Then conditions (TI) and (A6) together are equivalent to the sundial configuration (SC). In other words: in Theorem \[MainThmB\] item (\[e1\]) is equivalent to (\[e3\]).
Since we have already shown in Section \[Sec\_ECSC\] that (SC) is satisfied by an affine $\Lambda$-building $X$, and that (A6) can be replaced by (SC), it remains to show that axioms (A1)–(A4), (TI) and (SC) imply the retraction condition (A5). This follows from Propositions \[Prop\_r\] and \[Prop\_retraction\], which completes the proof.
Verifying (A4) {#Sec_A4}
==============
Assume that $(X,{\mathcal{A}})$ is a pair satisfying axioms (A1)–(A3) and properties (GG) and (CO). By Theorem \[Thm\_residue\] these assumptions are enough to conclude that the germs at a given vertex form a spherical building.
\[Prop\_A4\] Under the above assumptions, the pair $(X,{\mathcal{A}})$ satisfies (A4).
Let $S$ and $T$ be two Weyl chambers in $X$. We will show that by passing to sub-Weyl chambers $S'$ and $T'$ we will find an apartment containing both $S'$ and $T'$.
Given a point $x\in T$ we denote by $T_x$ and $S_x$, the unique $x$-based Weyl chambers parallel to $T$ and $S$, where the latter exists by [@Diss Cor 5.11]. We denote by $\delta(x)$ the length of a minimal gallery from $\Delta_xS$ to $\Delta_xT$ in the spherical building $\Delta_xX$. Since the number of possible values for $\delta(x)$ is finite we may without loss of generality (by choosing different sub-Weyl chambers of $C'$ if necessary) assume that $x$ is chosen such that $\delta(x)$ is maximal.
Now replace $S$ by $S_x$ and $T$ by $T_x$ where $x$ is chosen such that $\delta(x)$ is maximal. By Lemma \[Lm:secgm sec\] there exists an apartment $A$ containing $T$ and a germ of $S$ at $x$ and we denote by $S'$ the $x$-based Weyl chamber in $A$ which is opposite $S$ at $x$. Property (CO) implies that there is an apartment $A'$ containing $S$ and $S'$. By (A2) the intersection $A\cap T$ is a convex subset of $T$. Let $z$ be a point in this intersection. The unique $z$-based Weyl chambers $S_z$ and $S_z'$ parallel to $S$ and $S'$, respectively, are both contained in $A'$. By construction the length of a minimal gallery from $\Delta_zS_z$ to $\Delta_zT_z$ is not greater than $\delta(x)$. On the other hand, since $T$ and $S'$ are both contained in the apartment $A$, we can conclude $$\delta_z(T_z, S_z') = \delta_x(T, S') = d-\delta_x(S,T) = d-\delta(x)$$ where $d$ is the diameter of an apartment of $\Delta_xX$, that is the diameter of the spherical Coxeter complex associated to the underlying root system ${\Phi}$. The function $\delta_x$ assigns to two $x$-based Weyl chambers the length of a minimal gallery connecting their germs in $\Delta_xX$.
The germ $\Delta_zT_z$ lies on a minimal gallery connecting the opposite germs $\Delta_zS_z$ and $\Delta_zS_z'$. Such a minimal gallery is contained in the unique apartment containing $\Delta_zS_z$ and $\Delta_zS_z'$, which is $\Delta_zA'$. Therefore $\Delta_zT_z$ is contained in $\Delta_zA'$ as well. This allows us to conclude that $A'\cap T$ contains a germ of $T_z$. One can observe that $A'\cap T$ is a convex subset of $T$ containing $x$ which is open relative to $T$. Hence the Weyl chamber $T$ is contained in $A'$. Thus (A4) follows.
Appendix: Sharpness of axiom (A5) {#Sec_example}
=================================
by Koen Struyve
This section is devoted to the following question: Let $(X,{\mathcal{A}})$ be a space modeled on ${\mathbb{A}}= {\mathbb{A}}(\Phi,\Lambda)$, which satisfies axioms (A1)–(A4) and (A6). Is $(X,{\mathcal{A}})$ again an affine $\Lambda$-building?
We will construct examples of such spaces which vacously satisfy (A6) but do not satisfy either (A5) nor (TI), answering the above question in the negative.
Some additional definitions
---------------------------
The images in the model space ${\mathbb{A}}$ of the fundamental Weyl chamber ${\mathcal{C}}_f$ under the spherical Weyl group $\overline{W}$ will be called *vector Weyl chambers*. The unique fixed point in ${\mathbb{A}}$ of this group (so the basepoint of all the vector Weyl chambers) is denoted by $o$. The image of a vector Weyl chamber under a chart will again be called such.
We call the closed ball $\{x \in {\mathbb{A}}\vert d(o,x) \leq \lambda \}$ and their images under charts *centered balls with radius $\lambda$*.
Recall that if two Weyl chambers contain a common Weyl chamber we call them *parallel*. If an atlas satisfies axiom (A2), then this relation forms an equivalence relation.
A space modeled $(X,{\mathcal{A}})$ modeled on ${\mathbb{A}}= {\mathbb{A}}(\Phi,\Lambda)$ will be called *$\lambda$-admissible* if the following conditions are satisfied.
- No two injections in ${\mathcal{A}}$ have the same image.
- $(X,{\mathcal{A}})$ satisfies the axiom (A2).
- If two apartments intersect, then they either intersect in a single point contained in the centered balls with radius $\lambda$ of both apartments, or they intersect in a Weyl chamber contained in the interior of vector Weyl chambers in both apartments.
- All the vector Weyl chambers in one parallelism class contain a common sub-Weyl chamber.
Fix a sequence $(\lambda_i)_{i=1,2,\ldots}$ with $\lambda_i \in \Lambda$ such that $ 0< \lambda_1 < \lambda_2 < \dots$ and the sequence converges to infinity.
Extension procedure
-------------------
In this section, we will construct from a given $\lambda_i$-admissible space $(X_i,{\mathcal{A}}_i)$ ($i\in \mathbb{N} \backslash \{0\}$) a $\lambda_{i+1}$-admissible space $(X_{i+1},{\mathcal{A}}_{i+1})$, extending the previous one.
### Step 1: Covering pairs of points
Let $P$ be the set of pairs of points (up to order) in centered balls with radius $\lambda_{i+1}$ in $X_i$ not yet covered by a common apartment. For each pair $p := (x,y)$ in $P$ we choose distinct points $x_p$ and $y_p$ in the centered ball of radius $\lambda_{i+1}$ of a copy ${\mathbb{A}}_p$ of the model space ${\mathbb{A}}$. Let $\pi_p$ be the canonical isometry from ${\mathbb{A}}$ to ${\mathbb{A}}_p$.
We now define $X'_i$ to be the union of the sets $X_i$ and ${\mathbb{A}}_p \setminus \{x_p,y_p\}$ for each $p:= (x,y)$ in $P$. The set of charts ${\mathcal{A}}'_i$ is defined as the set of charts ${\mathcal{A}}_i$ together with a chart $$f_p: a \in {\mathbb{A}}\to \left\{\begin{array}{cl} \pi_p(a) & \mbox{if } x_p \neq \pi_p(a) \neq y_p \\
x & \mbox{if } \pi(a_p) = x_p \\
y & \mbox{if } \pi(a_p) = y_p
\end{array}\right.$$ for each $p:= (x,y)$ in $P$.
It is straigthforward to verify that the newly obtained space $(X_i',{\mathcal{A}}_i')$ satisfies conditions (T0) up to (T3) (with $\lambda =\lambda_{i+1}$), and hence is an $\lambda_{i+1}$-admissible space.
### Step 2: Covering pairs of sectors
We will now extend $(X_i',{\mathcal{A}}_i')$ to a $\lambda_{i+1}$-admissible space $(X_{i+1},{\mathcal{A}}_{i+1})$.
Let $Q$ be the set of parallelism classes of Weyl chambers in $X_i'$. For each class $q\in Q$, consider all vector Weyl chambers in this parallelism class. We know that these vector Weyl chambers contain a common sub-Weyl chamber by condition (T3). Condition (T2) together with the definition of vector Weyl chambers then implies that there exists a sub-Weyl chamber which has no points in common with the centered balls of radius $\lambda_{i+1}$ in any apartment. Fix such a sub-Weyl chamber $S_q$.
Let $R$ be the set of pairs (up to order) of parallelism classes of sectors not yet covered by a common apartment (meaning that there are no two elements, one of each class, contained in a common apartment). For each such pair $r := (q_1,q_2)$ we pick two disjoint sub-Weyl chambers $S^1_r$ and $S^2_r$, both contained in vector Weyl chambers of a copy ${\mathbb{A}}_r$ of the model space ${\mathbb{A}}$, and such that $S^1_r$ and $S^2_r$ are disjoint with the centered ball of radius $\lambda_{i+1}$ in ${\mathbb{A}}_r$. Let $\pi_r$ be the canonical isometry from ${\mathbb{A}}$ to ${\mathbb{A}}_r$. Let $\pi^1_r$ and $\pi^2_r$ be the canonical isometries from $S^1_r$ and $S^2_r$ to respectively $S_{q_2}$ and $S_{q_1}$.
The set of points $X_{i+1}$ of the space we want to construct is the union of the sets $X_i'$ and ${\mathbb{A}}_r \setminus (S^1_r \cup S^2_r)$ for each $r$ in $R$. The set of charts ${\mathcal{A}}_{i+1}$ is ${\mathcal{A}}'_i$ extended with a chart
$$g_r: a \in {\mathbb{A}}\to \left\{\begin{array}{cl} \pi_r(a) & \mbox{if } \pi_r(a) \notin S^1_r \cup S^2_r \\
\pi^1_r(a) & \mbox{if } \pi_r(a) \in S^1_r \\
\pi^2_r(a) & \mbox{if } \pi_r(a) \in S^2_r \\
\end{array}\right.$$ for each $r$ in $R$.
We now claim that $(X_{i+1},{\mathcal{A}}_{i+1})$ is $\lambda_{i+1}$-admissible space. Conditions (T0) and (T3) are automatically satisfied. In order to see conditions (T1) and (T2) note that for any two points $a\in S_q$ and $b \in S_{q'}$, where $q,q'$ are two distinct parallelism classes in $Q$, one has that $a$ and $b$ are not contained in a common apartment by condition (T2) for $(X_i',{\mathcal{A}}_i')$.
Direct limit and conclusion
---------------------------
By repeating the extension procedure laid out in the previous sub-section one obtains sets of points $X_i \subset X_{i+1} \subset X_{i+2} \subset \dots$ and sets ${\mathcal{A}}_i \subset {\mathcal{A}}_{i+1} \subset {\mathcal{A}}_{i+2} \subset \dots$ of injections. Let $$X_\infty = \bigcup_{j=i}^\infty X_{j} \text{ and } {\mathcal{A}}_\infty = \bigcup_{j=i}^\infty {\mathcal{A}}_{j} .$$
This direct limit yields a space $(X_\infty,{\mathcal{A}}_\infty)$ modeled on ${\mathbb{A}}= {\mathbb{A}}(\Phi,\Lambda)$. In order to satisfy axiom (A1) we replace ${\mathcal{A}}_\infty$ by the set ${\mathcal{A}}_\infty' = \{f \circ w \vert f\in {\mathcal{F}}, w \in W\}$. The space $(X_\infty,{\mathcal{A}}_\infty')$ satisfies axiom (A2) by condition (T1) for the intermediate steps. The repetition of the first and second step of the procedure implies that $(X_\infty,{\mathcal{A}}_\infty)$ satisfies axioms (A3) and (A4) as well.
If the dimension of ${\mathbb{A}}$ is at least 2 then no two apartments intersect in a half-apartment by condition (T2) for the intermediary steps, so axiom (A6) is satisfied vacuously. So in this case we obtain a space which satisfies axioms (A1)–(A4) and (A6). However it cannot consists of more than a single apartment and satify axiom (A5) at the same time because if it would, then there would be apartments intersecting in half-apartments (see for example [@Parreau Prop. 1.7]). By Theorem \[MainThmB\] it cannot satisfy (TI) either then.
A more direct way to obtain an example which does not satisfy (TI) is to start from a $\lambda_1$-admissible space (for a suitable choice of $\lambda_1$) which does not satisfy (TI). An example would be three apartments glued pairwise together at a (distinct) point. The three glueing points form a triangle, for which the side lengths can be chosen such that they violate the triangle inequality.
[^1]: The second author was financially supported by the SFB 478 “Geometric structure in mathematics” at the University of Münster and the DFG Project SCHW 1550/2-1. Her last name recently changed from Hitzelberger to Schwer. The third author is supported by the Fund for Scientific Research – Flanders (FWO - Vlaanderen).
[^2]: In order to show that a given pair $(X,{\mathcal{A}})$ has a spherical building at infinity it is enough to assume that said pair satisfies axioms (A1)–(A4) and (TI).
[^3]: By definition $A\oplus B =\{x\in A\setminus B \}\cup \{x\in B\setminus A \}$.
[^4]: It is easy to see that in the case where we know that the germs at $x$ form a spherical building, then opposite germs are also opposite in the usual spherical building sense.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In this study, a phase-field lattice Boltzmann model based on the Allen-Cahn equation with a filtered collision operator and high-order corrections in the equilibrium distribution functions is presented. Here we show that in addition to producing numerical results consistent with prior numerical methods, analytic solutions, and experiments with the density ratio of 1000, previous numerical deficiencies are resolved. Specifically, the new model is characterized by robustness at low viscosity, accurate prediction of shear stress at interfaces, and removal of artificial dense bubbles and rarefied droplets, etc.'
address: |
Dassault Systémes, 55 Network Drive, Burlington, MA 01803, USA\
[email protected]
author:
- 'Hiroshi Otomo, Raoyang Zhang, Hudong Chen'
title: 'Improved phase-field-based lattice Boltzmann models with a filtered collision operator'
---
Introduction
============
The modified Allen-Cahn (AC) equation, in which curvature driven dynamics is eliminated [@2007_Sun; @2011_Chiu], is widely studied as an efficient interface-tracking solver in multiphase flow with high density ratio due to its robustness and handiness of the conserved second-order partial differential equation.
In previous studies[@2016_Fakhari; @2017_Liang], a second set of the distribution functions is introduced to solve the AC equation for the interface dynamics, which is coupled with the hydrodynamic lattice Boltzmann (LB) based momentum solver. In order to improve stability at low viscosity and accuracy around interfaces, the multiple relaxation time (MRT) scheme and the biased difference scheme for the gradient of the order parameter were implemented [@2016_Fakhari]. The MRT scheme, however, suffers from high computational costs and a large number of model parameters. Also, due to using the biased difference scheme, the requirement of information from sites next to the nearest neighbors causes difficulties in near boundary regions, which may involve nontrivial extrapolation, complex code vectorization and parallelization, and additional computational cost.
In the present study, our LB model is formulated with a filtered collision operator[@2006_Chen; @2006_Zhang; @2006_Latt; @2006_Shan; @2013_Chen; @2014_Chen] with a single relaxation time. The central difference scheme, which only requires information at nearest neighbor sites, is applied because of its high computational efficiency. It is known that the truncation error of the central difference scheme may cause inaccurate shear stress on interfaces [@2017_Fakhari]. Also, the original AC equation may inevitably produce artificial dense bubbles and rarefied droplets. We propose a single solution to both of these unphysical effects and present numerical results from our improved model where these undesirable features are mitigated. All validation cases are conducted with a single source code containing our newly proposed algorithm.
Lattice Boltzmann models {#LB_model_pot_issue}
========================
Two lattice Boltzmann (LB) equations, one for the order parameter $\phi$ and the other for hydrodynamic quantities such as pressure $P$ and momentum $\rho \vec{u}$ are solved. For both equations, the D3Q19 or D2Q9 lattice model is adopted with index $i \in \left\{ 1, \cdots 19 \right\}$ or $i \in \left\{ 1, \cdots 9 \right\}$.
Formulation of the LB equation for $\phi$ follows the previous study[@2016_Fakhari], $$\label{eq:LB_phi_filter_col}
{h}_{i} \left( \vec{x}+\vec{c}_{i} \Delta t, t+\Delta t \right) =
{h}_{i} \left( \vec{x}, t \right) - \frac{{h}_{i} - {h}^{eq}_{i}}{\left(M/T\right)+ 0.5} \vert_{\vec{x}, t},$$ where $M$ is the mobility and ${h}^{eq}_{i}$ is the equilibrium state defined as, $$\begin{aligned}
{h}^{eq}_{i} &=& \phi \Gamma_i + \theta w_i \left( \vec{c}_i \cdot \vec{n} \right), \\
\Gamma_i &=& w_i \left\{ 1+ \frac{\vec{c}_i \cdot \vec{u} }{T} + \frac{\left( \vec{c}_i \cdot \vec{u} \right)^2}{2 T^2} - \frac{ \vec{u}^2}{2T} \right\}, \\
\theta &=& \frac{M}{T} \left\{ \frac{1- 4 \left( \phi - 0.5 \right)^2 }{W} \right\}. \label{theta_phai_LBeq}\end{aligned}$$ Here $W$ is the interface thickness and $ \vec{n}$ is the unit vector normal to the interface calculated by $\vec{\nabla} \phi / \left( |\vec{\nabla} \phi| + \epsilon \right)$ where $\epsilon$ is a tiny parameter such as $1.e-10$ in order to avoid division by zero. The value of $\phi$ is evaluated by $\sum_i {h}_i$.
Formulation of the hydrodynamic LB equation is based on the previous study[@2016_Fakhari]. One with the BGK collision operator is, $$\begin{aligned}
\label{BGK_LBeq_hydroeq}
\bar{g}_{i} \left( \vec{x}+\vec{c}_{i} \Delta t, t+\Delta t \right) =
\bar{g}_{i} \left( \vec{x}, t \right) - \frac{\bar{g}_{i} - \bar{g}^{eq}_{i}}{\tau_{mix}} \vert_{\vec{x}, t} + K_i \left( \vec{x}, t \right),\end{aligned}$$ where $$\begin{aligned}
\bar{g}^{eq}_{i}&=& \rho \Gamma_i + w_i \left( \frac{P}{T} - \rho \right)- \frac{K_i}{2}, \\
K_i &=& \left\{ \left( \Gamma_i - w_i \right) \rho_{dif} + \frac{\Gamma_i \mu_{chm} }{T} \right\} \left( \vec{c}_{i} - \vec{u} \right) \cdot \vec{\nabla} \phi + \Gamma_i \frac{\left( \vec{c}_{i} - \vec{u} \right) \cdot \vec{F}_{ex} }{T}.\end{aligned}$$ Here $K_i$ is a force term responsible for phase separation and the external force $\vec{F}_{ex}$. $\rho_{dif}$ is difference between the light and heavy characteristic density $\rho$. $\mu_{chm}$ is the chemical potential defined as, $$\begin{aligned}
\mu_{chm}=\frac{48 \sigma}{W} \phi \left( \phi -1 \right) \left( \phi -0.5 \right) - \frac{3 \sigma W}{2} \vec{\nabla}^2 \phi,\end{aligned}$$ where $\sigma$ is the surface tension.
The relaxation time $\tau_{mix}$ is approximated using the harmonic interpolation, $$\label{tau_interpolate}
1/ \tau_{mix} = \left( 1 /\tau_{air} \right) + \phi \left\{ \left( 1/\tau_{water} \right) - \left(1/\tau_{air} \right) \right\},$$ with relaxation times of water and air, $\tau_{water}$ and $\tau_{air}$, which correspond to their kinematic viscosities, $\nu_{water}$ and $\nu_{air}$.
The right hand side in Eq. (\[BGK\_LBeq\_hydroeq\]) is filtered as, $$\label{eq:LB_filter_col}
\bar{g}_{i} \left( \vec{x}+\vec{c}_{i} \Delta t, t+\Delta t \right) =
\bar{G}_{i}^{eq} + \left( 1 - \frac{1}{ \tau_{mix} } \right) \Phi_i : \Pi,$$ where $\Phi_i$ is a filtered operator that uses Hermite polynomials and $\Pi$ is the nonequilibrium moments of the momentum flux, $$\begin{aligned}
\Phi_i = \frac{w_i}{2 T^2} \left( \vec{c}_{i} \vec{c}_{i} - T I \right), \label{momentum_flux0} \\
\Pi = \sum_{l} \vec{c}_{l} \vec{c}_{l} \left( \bar{g}_{l} - \bar{G}_{l}^{neq} \right). \label{momentum_flux}\end{aligned}$$ Here $I$ is the identity matrix. The equilibrium and nonequilibrium parts $\bar{G}_{i}^{eq}$ and $\bar{G}_{i}^{neq}$ are naturally determined via correspondence with Eq. (\[BGK\_LBeq\_hydroeq\]) and $\tau_{mix}$ dependence. More details of filtered collision procedure can be found in previous studies [@2006_Chen; @2006_Zhang; @2006_Latt; @2006_Shan; @2013_Chen; @2014_Chen; @2016_Otomo; @2018_Otomo]. Projection from the state-space to moment-space is performed in Eq. (\[momentum\_flux\]) only for 9 moments in the case of D3Q19. Also, since the spatially dependent relaxation time $\tau_{mix}$ is factored out from the projected term in Eq. (\[eq:LB\_filter\_col\]), the calculation of $\Phi_i : \Pi$ can be simplified down to the multiplication between a $19\times19$ matrix [^1] and a $1 \times 19$ matrix. On the other hand, in the MRT scheme the projection is performed for the 19 moments on D3Q19 via the multiplication of the matrix $M^{-1}DM$ where $M$ is the $19\times19$ conversion matrix from the state-space to the moment space and $D$ is the 19th-rank diagonal matrix involving the relaxation time. This multiplication of matrices is known to cause deterioration of computational efficiency. As a result, the filtering discussed in the present study can possess much higher computational efficiency compared to the MRT scheme.
After Eq. (\[eq:LB\_filter\_col\]) is solved, pressure and momentum are evaluated by $T \sum_i \bar{g}_i+\left(T \rho_{dif}/2 \right) \vec{u} \cdot \vec{\nabla} \phi$ and $\sum_i \vec{c}_i \bar{g}_i + \left( \mu_{chm} \vec{\nabla} \phi +\vec{F}_{ex} \right)/2$, respectively.
In this study, the gradient and Laplacian of $\phi$, that are used for calculation of $\vec{n}$, $P$, $\rho \vec{u}$, and $K_i$, are approximated with the central difference (CD) scheme, $$\vec{\nabla} \phi =\frac{ \sum_i \left\{ \phi \left( \vec{x} + \vec{c}_i \right) -\phi \left( \vec{x} - \vec{c}_i \right) \right\} \vec{c}_i w_i}{2T},
\: \: \: \: \: \:
\vec{\nabla}^2 \phi = \frac{2 \sum_i \left\{ \phi \left( \vec{x} + \vec{c}_i \right) -\phi \left( \vec{x} \right) \right\} w_i}{T}. \nonumber$$ This discretization scheme requires information only from the nearest neighbor sites, whereas existing implementations utilizing biased differencing use next-nearest neighbor sites. For all results shown in this manuscript, unless specifically mentioned, the density ratio $\rho_{ratio}= \rho_{water}/ \rho_{air}$ is 1000, consistent with air - water mixtures. The dynamic viscosity ratio $\mu_{water}/\mu_{air}$ is denoted as $\mu_{ratio}$.
Pathological cases
==================
In spite of high computational efficiency, the LB models described in the previous section are inadequate for some benchmark cases. In this section, a few simple cases are chosen as pathological cases and the remedies are proposed.
First, with the LB models in Section \[LB\_model\_pot\_issue\], shear stress on the interface is not evaluated accurately. In Fig. \[fig:Poise\_Cou\], numerical results for two-phase Poiseuille flow and Couette flow are shown. In both cases, water occupied the left half domain and air is on the right half side. The domain sizes $L$ are 64 and 100 for each case. $\mu_{ratio}=100$, $\nu_{water}=0.17$, $M=0.1$, and $W=2.5$ for both cases. Gravitational acceleration $g=1.0e-6$ is applied in the Poiseuille flow. The analytic solutions are derived while density profiles with non-zero interface thickness and viscosity in the mixture are taken into account. Results of the LB models in Section \[LB\_model\_pot\_issue\] show peaks on the interface and deviate from analytic solutions obviously. Since the similar behavior is observed even with the MRT collision operator in the previous study [@2017_Fakhari_JCP] if they use the pure CD scheme, it is not mainly due to the filtered collision operator but likely due to the CD scheme. Indeed, our analysis reveals that irregular effects at the interface originated from some high-order terms such as $\partial_{x} \left( v_y \partial^3_{x} \rho \right)$ in the Navier-Stokes (NS) equation where the CD scheme is used. Although using the biased scheme can solve the issue as the study [@2017_Fakhari] indicated, such an approach requires information from sites farther than the nearest neighbors, resulting in computational complexities. Instead, a high order correction in the equilibrium distribution is added as the following in the present work together with the CD scheme and modification of definition of velocity so that the term $\partial_{x} \left( v_y \partial^3_{x} \rho \right)$ is removed, $$\begin{aligned}
\delta f^{eq}_i = \frac{1}{4T} w_i \left( \vec{c}_i \cdot \vec{u} \right) \vec{\nabla}^2 \rho.\end{aligned}$$ Fig. \[fig:Poise\_Cou\] shows results of the modified model marked as “Present” matching with analytic solutions very well.
It is worth mentioning that this issue may be sometimes hidden in pressure or force driven flow [@2017_Liang] because the higher density side has relatively lower velocity. Due to the same reason, if velocity is assigned on a wall of the air side in the Couette flow, the velocity with the original models apparently matches with the analytic solution as shown in the right figure of Fig. \[fig:Poise\_Cou\].
Here we would like to emphasize that the purpose of comparisons in Fig. \[fig:Poise\_Cou\] is to check the local balance of shear stress in each lattice by taking account of the non-zero interface thickness and fixing the interpolation of viscosity on the interface. This is different from a recent paper [@2017_Fakhari], in which the interpolation method of $\tau_{mix}$ is studied. In our study, if the formula in Eq. (\[tau\_interpolate\]) is changed, both of analytic solutions and numerical results are changed in Fig. \[fig:Poise\_Cou\].
In the previous study [@2017_Fakhari], this issue was improved with the so-called velocity-based LB model, in which the first moment of the distribution function is velocity instead of momentum. Therefore, in contrast to the momentum-based LB model used in the present work, density and velocity are conserved independently, but there is no guarantee of momentum conservation during the particle advection even at zero surface tension.
Furthermore, this modified LB model also improves the Galilean invariance. In the one-dimensional domain of $L$=100 bounded by periodic boundaries, the droplet is sitting in the center and a value of 0.025 for lattice velocity is homogeneously assigned initially. $\nu_{water}=1.7e-2$ and $W=3.0$. The rest of settings are the same as in the above cases of Couette flow. In Fig. \[fig:Galilean1D\_dens\], profiles of $\phi$ and velocity $v_x$ are presented in terms of the cycle period during which the droplet comes back to the original position. As seen here, though droplets’ movements are similar with both models, the velocity with the modified model maintains a constant value and preserves Galilean invariance during the entire evolution, while the velocity with the original model described in Section \[LB\_model\_pot\_issue\] fails.
Second, with the LB models for $\phi$ from Section \[LB\_model\_pot\_issue\], a large number of unphysical droplets/bubbles whose $\phi$ is close to 0/1, can be produced. In order to clarify this issue, a simple two-dimensional case is set as shown in the left figure of Fig. \[fig:Shock\_dens\_bubble\]. Here, the lattice velocity of 0.05 is assigned in the thin layer of water domain. $\mu_{ratio}=1$, $\nu_{water}=1.7e-1$, $W=3.0$, and $M=1.7e-1$. The shock created by initial velocity is reflected on the top and bottom edges of domain and bounces back and forth for a while. It produces a lot of bubbles which stay in the water domain even after the steady state is reached as shown in the center figure of Fig. \[fig:Shock\_dens\_bubble\]. Because of the steady state, it is likely that the value of the mobility does not depend on the solution of these bubbles in the AC equation. As the contour’s range indicates, these are dense bubbles, whose $\phi$ is $0.90-0.99$. They are totally different from the “decent” bubbles, whose $\phi$ goes down to zero in their centers. Hence such dense bubbles are insensitive to buoyancy force and sometimes distort streamlines. This issue seems to be originated from the Allen-Cahn model, that prompts the nucleation of droplets and bubbles no matter how much $\phi$ they have. In order to solve this problem, a minor diffusion correction is added to the Allen-Cahn model via $\theta$ in Eq. (\[theta\_phai\_LBeq\]) so that such bubbles are diffused. Specifically, $\theta$ is corrected as, $$\begin{aligned}
\delta \theta = C \frac{M}{T} F \left( \phi, \vec{\nabla}^2 \phi \right) \mid \vec{\nabla} \phi \mid,\end{aligned}$$ where $C$ is a constant value. There are various choices of function $F$ but it is determined so that this correction is turned off for the decent bubbles and droplets. As a result, dense bubbles disappear as seen in the right figure of Fig. \[fig:Shock\_dens\_bubble\]. In the test cases discussed in the next section, no obvious issues resulting from this modification are observed. It indicates that the main interface dynamics is insensitive to this minor diffusion correction. The similar problem of rarefied droplets in the air domain can be improved with a similar modification.
A set of regular validation cases
=================================
A static droplet
----------------
Through the simulation of a static droplet in free space, consistence with the Laplace law is examined and the spurious current is compared to the other multi-phase LB models. A two-dimensional static droplet with variable initial radius, $R= \left\{ 8,
12, 16 \right\}$, is put in the center of domain, whose size is five times of $R$ and periodic boundaries are assigned on each pair of domain’s edges. $\nu_{water}= \left\{ 3.3e-4, 1.7e-1 \right\}$, $\mu_{ratio}=60$, $M=0.1$, and $W=2.5$. First, with a small viscosity of $\nu_{water}=3.3e-4$, a droplet with $\sigma = \left\{ 1.0e-2, 6.0e-3, 1.0e-3 \right\}$ is simulated. The left figure of Fig. \[fig:2D\_droplet\_cntLap\] shows the relation between $1/R$ and the pressure difference across the interface, $dP$. Lines with slopes of inputted $\sigma$ are presented. All cases comply with the Laplace law, $dP=\sigma /R$, very well and output the consistent value of $\sigma$. Next, spatially-averaged spurious current of a droplet of $R$=40 is measured for various $\sigma$ in the periodic domain of $250 \times 250$. In the right figure of Fig. \[fig:2D\_droplet\_cntLap\], results with $\nu_{water}= \left\{ 1.7e-1, 3.3e-4 \right\}$ are compared to the previous study [@2016_Lycett] in which the recent pseudo-potential model for $\rho_{ratio}=1000$ is used. While more diffusion with higher viscosity leads to less spurious current, the difference between two viscosity options becomes small as $\sigma$ is increased. The phase-field LB model in this study shows much improved spurious current than the recent pseudo-potential model by factor of $10^3 - 10^4$ for this droplet case of $\rho_{ratio}=1000$.
Droplet collision
-----------------
Binary droplet collisions are often simulated in order to check the interface dynamics and robustness of computational models for multi-phase flows with high-density ratio [@2016_Lycett; @2004_Inamuro; @2016_Inamuro]. In this study, a binary droplet collision under two flow conditions are compared to experimental results [@2009_Pan]. For each case, the droplet diameter is $\left\{ 32, 50 \right\}$, $\sigma= \left\{ 5.5, 0.64 \right\}$, and $\nu_{water}= \left\{ 1.9e-3, 6.0e-4 \right\}$. $\nu_{ratio}=60$, $M=0.1$ and $W=2.5$. The relative velocity $U$ is set as $0.1$ and $0.75$ for the case with coarse and fine resolution, respectively. Hence Reynold number $Re$, $UD/ \nu_{water}$, is $\left\{ 1700, 6300 \right\}$ and Weber number $We$, $\rho_{water} U^2 D / \sigma$, is $\left\{ 58, 440 \right\}$, respectively. As shown in Fig. \[fig:Droplet\_collisionII\], the simulated droplets’ deformation and splashing patterns depending on $Re$ and $We$ seem to be comparable with Fig. 2 and Fig. 4(a) in the paper[@2009_Pan]. Compared to a previous study with the recent pseudo-potential LB model for $\rho_{ratio}=1000$ [@2016_Lycett], similar accuracy can be achieved using half interface thickness and quarter resolution approximately in this study.
Rayleigh-Taylor instability
---------------------------
Rayleigh-Taylor instability induced by heavy fluid’s penetration into light fluid with gravity is simulated as a benchmark problem for the interface dynamics. Numerical results are compared to previous studies [@2017_Fakhari; @2000_Guermond]. Two cases using $\rho_{ratio}= \left\{ 3, 1000 \right\}$ are tested in two-dimensional domains of $\left\{ 150 \times 600, 256 \times 1000 \right\}$, whose horizontal lengths are denoted as $L$. For each case, $\sigma= \left\{ 8.6e-4, 4.7e-1 \right\}$, $\mu_{ratio}= \left\{ 1, 100 \right\}$, $W= \left\{ 2.5, 5.0 \right\}$, $ M = \left\{ 5.8e-3, 1.3e-2 \right\}$, and $\nu_{water} = \left\{ 1.9e-3, 4.1e-3 \right\}$, respectively. Gravitational acceleration $g$ is set as $1.e-5$ for both cases. Accordingly, if the characteristic velocity $U$ is estimated as $\sqrt{g L}$, $Re $ = $UL/\nu_{water}$= 3000 and the Péclet number, $UL/M$, is 1000 approximately. Capillary number, $\rho_{water} U \nu_{water} / \sigma$, is $\left\{0.26, 0.44 \right\}$, respectively. In the left and center figures of Fig. \[fig:Ray\_tay\_inst\] contours of $\phi$ and histories of bubble/liquid front normalized by $L$ where $\rho_{ratio}=3$ are presented. The non-dimensional time $t^{*}$ is defined as $t \sqrt{g/L}$. The penetration pattern and its quantitative positions are consistent with previous studies[@2000_Guermond], in which a finite element method was used. In the right figure of Fig. \[fig:Ray\_tay\_inst\], contours of $\phi$ with $\rho_{ratio}=1000$ are shown. Compared to results of the previous study[@2017_Fakhari] with the velocity-based phase-field LB models, the same-level robust results are captured while $W$ seems to be smaller and the finger shape is slightly sharper in the present study.
Dam-breaking case
-----------------
In a recent study [@2014_Lobovsky], dynamics of the dam-breaking wave is investigated experimentally by measuring the water heights and the impact pressure on the downstream vertical wall. This case is simulated with the LB models in this study and measurements are compared with the experiment.
The computational domain of $537 \times 198 \times 50$ is bounded by friction walls except for the top boundary of the pressure condition. The initial height of water column $H$ is 100. $g=4.0e-6$. $W=4.0$, $M=0.02$, and $\nu_{air}=8.3e-3$. $\sigma = 5.2e-3$ and $\nu_{water}=4.9e-6$ so that the Weber number is $7.7e3$ and the Reynolds number of water is $4.1e5$. The sensors’ positions for the pressure measurement are shown in Fig. \[fig:Dam\_breaking\]. The water heights are detected on three lines, D1 through D3, located in the center of the z-coordinate.
In Fig. \[fig:Dam\_breaking\], the measured pressure, the surge-front position of water, and water heights are compared with the experimental results. Results for pressure, front position, and water height are nearly consistent with observations. We note that although our proposed model lacks treatment for turbulence and the simulations are underresolved for the Reynolds number used, the physical structures captured by our model are nearly consistent with experiments. In the beginning stage, the surge-front position is slightly underestimated due to the initial condition. We believe the prediction of the pressure peak may be improved with increased spatial resolution.
Summary
=======
An improved phase-field lattice Boltzmann model based on the Allen-Cahn equation with the filtered collision operator and the equilibrium distribution with a high-order correction in the LB equations for $\phi$ and hydrodynamic quantities is introduced without sacrificing the simplicity of algorithm and computational cost significantly. These modifications improve solver robustness at low viscosity, accuracy of shear stress at interfaces, the Galilean invariance, and production of artificial dense bubbles and rarefied droplets. In addition to extremely low spurious current, the same accuracy as the recent pseudo-potential model is achieved using half of the interface thickness and a quarter of resolution. Across various cases, consistent results with other numerical methods, analytic solutions and experiments are obtained with the efficient choices of parameters such as the resolution and the interface thickness.
Acknowledgments {#acknowledgments .unnumbered}
===============
H.O. would like to thank Takaji Inamuro and Taehun Lee for effective and enlightening discussions. We would also like to thank Luca D’Alessio for discussing setups of test cases. Also, we express gratitude to Casey Bartlett and Ilya Staroselsky for valuable comments on this study.
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[c]{}
![ Inverse droplet radius $1/R$ vs pressure difference $dP$ across the droplet interface (left). Surface tension $\sigma$ vs spatial averaged spurious current $<|v|>$ with $\nu_{water}= \left\{ 3.3e-4, 1.7e-1 \right\}$ (right). Results are compared to a recent study of the pseudo-potential model for $\rho_{ratio} =1000$.\[fig:2D\_droplet\_cntLap\]](Galilean1D_mix){width="12cm"}
![ Inverse droplet radius $1/R$ vs pressure difference $dP$ across the droplet interface (left). Surface tension $\sigma$ vs spatial averaged spurious current $<|v|>$ with $\nu_{water}= \left\{ 3.3e-4, 1.7e-1 \right\}$ (right). Results are compared to a recent study of the pseudo-potential model for $\rho_{ratio} =1000$.\[fig:2D\_droplet\_cntLap\]](Shock_dense_bubble_setting_2){width="12cm"}
![ Inverse droplet radius $1/R$ vs pressure difference $dP$ across the droplet interface (left). Surface tension $\sigma$ vs spatial averaged spurious current $<|v|>$ with $\nu_{water}= \left\{ 3.3e-4, 1.7e-1 \right\}$ (right). Results are compared to a recent study of the pseudo-potential model for $\rho_{ratio} =1000$.\[fig:2D\_droplet\_cntLap\]](Laplacelaw_new){width="13cm"}
![ In the dam-breaking case, settings (left top), comparisons of the surge-front position (center top), pressure on the wall (left bottom), and water heights (others) between the simulation displayed as ’Sim’ and the experiment displayed as ’Exp’ \[fig:Dam\_breaking\]](Droplet_collisionII_2){width="14cm"}
![ In the dam-breaking case, settings (left top), comparisons of the surge-front position (center top), pressure on the wall (left bottom), and water heights (others) between the simulation displayed as ’Sim’ and the experiment displayed as ’Exp’ \[fig:Dam\_breaking\]](Rayleigh_Taylor_instability_newpg){width="13cm"}
![ In the dam-breaking case, settings (left top), comparisons of the surge-front position (center top), pressure on the wall (left bottom), and water heights (others) between the simulation displayed as ’Sim’ and the experiment displayed as ’Exp’ \[fig:Dam\_breaking\]](Dambreaking){width="13cm"}
[^1]: Simply following Eq. (\[momentum\_flux0\]) and Eq. (\[momentum\_flux\]), we can decompose this $19\times19$ matrix to the multiplication between a $19\times3$ matrix and a $3 \times19$ matrix so that the operation count is saved further.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Synchronization is of central importance in power distribution, telecommunication, neuronal, and biological networks. Many networks are observed to produce patterns of synchronized clusters, but it has been difficult to predict these clusters or understand the conditions under which they form, except for in the simplest of networks. In this article, we shed light on the intimate connection between network symmetry and cluster synchronization. We introduce general techniques that use network symmetries to reveal the patterns of synchronized clusters and determine the conditions under which they persist. The connection between symmetry and cluster synchronization is experimentally explored using an electro-optic network. We experimentally observe and theoretically predict a surprising phenomenon in which some clusters lose synchrony while leaving others synchronized. The results could guide the design of new power grid systems or lead to new understanding of the dynamical behavior of networks ranging from neural to social.'
author:
- 'Louis M. Pecora'
- Francesco Sorrentino
- 'Aaron M. Hagerstrom'
- 'Thomas E. Murphy'
- Rajarshi Roy
bibliography:
- 'SymsClusterSyncIsoDesync-v7.bib'
title: 'Symmetries, Cluster Synchronization, and Isolated Desynchronization in Complex Networks'
---
Synchronization in complex networks is essential to the proper functioning of a wide variety of natural and engineered systems, ranging from electric power grids to neural networks [@Motter2013]. Global synchronization, in which all nodes evolve in unison, is a well-studied effect, the conditions for which are related to the network structure through the master stability function [@Pecora1998]. Equally important, and perhaps more commonplace, is partial, or cluster-synchronization (CS), in which patterns or sets of synchronized elements emerge [@Allefeld2007; @Ji2013; @Zhou2006]. Recent work on cluster synchronization has been restricted to networks where the synchronization pattern is induced either by tailoring the network geometry or by the intentional introduction of heterogeneity in the time delays or node dynamics [@Do2012; @Dahms2012; @Fu2013; @Kanter2011; @Rosin2013; @Sorrentino2007; @Williams2013; @Belykh2008]. These anecdotal studies illustrate the interesting types of cluster synchronization that can occur, and suggest a broader relationship between the network structure and synchronization patterns. Recent studies have begun to draw a connection between network symmetry and cluster synchronization, although all have considered simple networks where the symmetries are apparent by inspection [@D'Huys2008; @Nicosia2013; @Russo2011]. More in-depth studies have been done involving bifurcation phenomena and synchronization in ring and point-symmetry networks [@GolubitskyBOOKII; @GolubitskySewartBOOK]. Here we address the more common case where the intrinsic network symmetries are neither intentionally produced nor easily discerned. We present a comprehensive treatment of cluster synchronization, which uses the tools of computational group theory to reveal the hidden symmetries of networks and predict the patterns of synchronization that can arise. We use irreducible group representations to find a block-diagonalization of the variational equations that can predict the stability of the clusters. We further establish and observe a generic symmetry-breaking bifurcation termed *isolated desynchronization*, in which one or more clusters lose synchrony while the remaining clusters stay synchronized. The analytical results are confirmed through experimental measurements in a spatio-temporal electro-optic network. By statistically analyzing the symmetries of several types of networks, as well as electric power distribution networks, we argue that symmetries, clusters, and isolated desynchronization are commonplace and important in many complex networks.
The general dynamical equations to describe a network of $N$ coupled identical oscillators are $$\label{eq:1}
\dot{\bf x}_i(t)={\bf F}({\bf x}_i(t))+\sigma\sum_j A_{ij} {\bf H}({\bf x}_j), \; i=1,...,N,$$ where ${\bf x}_i$ is the $n$-dimensional state vector of the $i$-th oscillator, ${\bf F}$ describes the dynamics of each oscillator, $A$ is a symmetric matrix of 1’s and 0’s that describes the connectivity of the network, $\sigma$ is the coupling strength, and ${\bf H}$ is the output function of each oscillator. Eq. (\[eq:1\]) can be extended to discrete-time systems or more general coupling schemes[@Fink2000].
The symmetries of the network form a (mathematical) group ${\cal G}$. Each symmetry of the group can be described by a permutation matrix $R_g$ that re-orders the nodes in a way that leaves the dynamical equations unchanged (i.e., each $R_g$ commutes with $A$). The set of symmetries (or automorphisms) [@GolubitskyBOOKII; @TinkhamBOOK] of a network can be quite large, even for small networks, but they can be calculated from $A$ using widely available discrete algebra routines [@Stein; @GAP4]. Figure \[fig:1\]a shows three graphs generated by randomly removing 6 edges from an otherwise fully connected 11-node network. Although the graphs appear similar and exhibit no obvious symmetries, the first instance has no symmetries (other than the identity permutation), while the others have 32 and 5,760 symmetries, respectively. So for even a moderate number of nodes (11) finding the symmetries can become impossible by inspection.
Once the symmetries are identified, the nodes of the network can be partitioned into $M$ clusters by finding the “orbits” of the symmetry group: the disjoint sets of nodes that, when all of the symmetry operations are applied, permute among one another [@GolubitskyBOOKII]. Because Eq. \[eq:1\] is essentially unchanged by the by the permutations the dynamics of the nodes in each cluster can be equal, which is exact synchronization. Hence, there are M synchronized motions $\{{\bf s}_1,...,{\bf s}_M\}$, one for each cluster. In Fig. \[fig:1\]a, the nodes have been colored to show the clusters. For the first example, which has no symmetries, the network divides into $M=N$ trivial clusters with one node in each. The other instances have 5 and 3 clusters, respectively. Once the clusters are identified, Eq. (\[eq:1\]) can be linearized about a state where synchronization is assumed among all of the nodes within each cluster. This linearized equation is the variational equation and it determines the stability of the clusters.
Equation (\[eq:1\]) is expressed in the “node” coordinate system, where the subscripts $i$ and $j$ are identified with enumerated nodes of the network. Beyond identifying the symmetries and clusters, group theory also provides a powerful way to transform the variational equations to a new coordinate system in which the transformed coupling matrix $B = TAT^{-1}$ has a block-diagonal form that matches the cluster structure. The transformation matrix $T$ is not a simple node re-ordering, nor is it an eigendecomposition of $A$. The process for computing $T$ is non-trivial, and involves finding the irreducible representations (IRR) of the symmetry group. We call this new coordinate system the “IRR coordinate system.” A more detailed description of this process is given later in this article.
Figure \[fig:1\]c shows the coupling matrix $B$ in the IRR coordinate system for the three example networks. The upper-left block is an $M \times M$ matrix that describes the dynamics within the synchronization manifold. The remaining diagonal blocks describe motion transverse to this manifold and so are associated with loss of synchronization. Thus, the diagonalization completely decouples the transverse variations from the synchronization block, and partially decouples the variations among the transverse directions. In this way the stability of the synchronized clusters can be calculated using the separate, simpler, lower dimensional ODEs of the transverse blocks to see if the non-synchronous transverse behavior decays to zero.
![[**Three randomly generated networks with varying amounts of symmetry and associated coupling matrices.**]{} (a) Nodes of the same color are in the same synchronization cluster. The colors show the maximal symmetry the network dynamics can have given the graph structure. (b) A graphic showing the structure of the adjacency matrices of each network (black squares are 1, white squares are 0). (c) Block diagonalization of the coupling matrices $A$ for each network. Colors denote the cluster, as in (a). The $2 \times 2$ transverse block for the 32 symmetry case comes from one of the IRRs being present in the permutation matrices two times. The Supplementary Information displays the matrices. \[fig:1\]](fig1)
The general form of the transformed variational equations for $M$ clusters is, $$\label{eq:2}
\dot{{\pmb \eta}}(t)=\left[\sum_{m=1}^{M} E^{(m)} \otimes D{\bf F}({\bf s}_m(t)) + \sigma B\otimes I_n \sum_{m=1}^{M} J^{(m)} \otimes D{\bf H}({\bf s}_m(t)) \right] {\pmb \eta}(t),$$ where we have linearized about synchronized cluster states $\{{\bf s}_1,...,{\bf s}_M\}$, ${\pmb \eta}(t)$ is the vector of variations of all nodes transformed to the IRR coordinates, $D{\bf F}$ and $D{\bf H}$ are the Jacobians of the nodes’ vector field and coupling function, respectively, and $B$ is the block diagonalization of the coupling matrix $A$. Further details are given in a later section. We note that this analysis holds for any node dynamics, steady-state, periodic, chaotic, etc.
Figure \[fig:2\]a shows the optical system used to study cluster synchronization. Light from a 1550 nm light emitting diode (LED) passes through a polarizing beamsplitter (PBS) and quarter wave plate (QWP), so that it is circularly polarized when it reaches the spatial light modulator (SLM). The SLM surface imparts a programmable spatially-dependent phase shift $x$ between the polarization components of the reflected signal, which is then imaged, through the polarizer, onto an infrared camera[@Hagerstrom2012]. The relationship between the phase shift $x$ applied by the SLM and the normalized intensity ${\cal I}$ recorded by the camera is ${\cal I}(x)=\left( 1-\cos x \right)/2$. The resulting image is then fed back through a computer to control the SLM.
![**Experimental configuration.** a) Light is reflected from the SLM, and passes though polarization optics, so that the intensity of light falling on the camera is modulated according the phase shift introduced by the SLM. Coupling and feedback are implemented by a computer. b) An image of the SLM recorded by the camera in this configuration. Oscillators are shaded to show which cluster they belong to, and the connectivity of the network is indicated by superimposed gray lines. The phase shifts applied by the square regions are updated according to equation (\[eq:3\]). \[fig:2\]](fig2)
The dynamical oscillators that form the network are realized as square patches of pixels selected from a $32 \times 32$ tiling of the SLM array. Figure \[fig:2\]b shows an experimentally measured camera frame captured for one of the 11-node networks considered earlier in Fig. \[fig:1\]. The patches have been falsely colored to show the cluster structure, and the links of the network are overlaid to illustrate the connectivity. The phase shift of the $i$-th region, $x_i$, is updated iteratively according to: $$x^{t+1}_i=\left[ \beta {\cal I}(x^{t}_i) + \sigma \sum_j A_{ij} {\cal I}(x^{t}_j) + \delta \right] \text{ mod } 2\pi
\label{eq:3}$$ where $\beta$ is the self-feedback strength, and the offset $\delta$ is introduced to suppress the trivial solution $x_i=0$. Eq. (\[eq:3\]) is a discrete-time equivalent of Eq. (\[eq:1\]). Depending on the values of $\beta$, $\sigma$ and $\delta$, Eq. (\[eq:3\]) can show constant, periodic or chaotic dynamics. There are no experimentally-imposed constraints on the adjacency matrix $A_{ij}$, which makes this system an ideal platform to explore synchronization in complex networks.
![**Experimental observation of isolated and intertwined desynchronization.** a) Cluster synchronization error as the self-feedback, $\beta$ is varied. For all cases considered, $\delta = 0.525$ and $\sigma = 0.6\pi$. Colors indicate the cluster under consideration and are consistent with Fig. \[fig:1\]. b) MLE calculated from simulation. c-e) Synchronization error time traces for the four clusters, showing the isolated desychronization of the magenta cluster and the isolated desychronization of the intertwined blue and red clusters. \[fig:3\]](fig3)
Figure \[fig:3\] plots the time-averaged root-mean square (RMS) synchronization error for all four of the non-trivial clusters shown in Fig. \[fig:2\]b, as a function of the feedback strength $\beta$. The RMS synchronization error was calculated for each cluster as $\Delta x_{\rm RMS} \equiv \left(\overline{\bigl<(x_i^t - \overline{x}^t)^2\bigr>}_T\right)^{1/2}$ where $\left<\bullet\right>_T$ indicates an average over a time interval $T$ (here taken to be 500 iterations) and $\overline\bullet$ denotes a spatial average over the nodes within the cluster. In Fig. \[fig:3\]c-e, we plot the observed intra-cluster deviations $x_i^t - \overline{x}^t$ for three specific values of $\beta$ indicated by the vertical lines in Fig. \[fig:3\]a-b, showing different degrees of partial synchronization that can occur, depending on the parameters. Together, Fig. \[fig:3\]a and Figs. \[fig:3\]c-e illustrate two examples of a bifurcation commonly seen in experiment and simulation: isolated desynchronization, where one or more clusters lose stability, while all others remain synchronized. At $\beta=0.72 \pi$ (Fig. \[fig:3\]c), all four of the clusters synchronize. At $\beta=1.4 \pi$ (Fig. \[fig:3\]d), the magenta cluster, which contains four nodes, has split into two smaller clusters of 2 nodes each, while the other two clusters remain synchronized.
Between $\beta=0.72 \pi$ and $\beta=1.76 \pi$, two clusters, shown in Fig. \[fig:1\] as red and blue, undergo isolated desynchronization together. In Fig. \[fig:3\]a, the synchronization error curves for these two clusters are visually indistinguishable. The synchronization of these two clusters is intertwined: they will always either synchronize together or not at all. While it is not obvious from a visual inspection of the network that the red and blue clusters should form at all, their intertwined synchronization properties can be understood intuitively by examining the connectivity of the network. Each of the two nodes in the blue cluster is coupled to exactly one node in the red cluster. If the blue cluster is not synchronized, the red cluster cannot synchronize because its two nodes are receiving different input. The group analysis treats this automatically and yields a transverse $2 \times 2$ block in Fig. \[fig:1\]c.
The isolated desynchronization bifurcations we observe are predicted by computation of the maximum Lyapunov exponent (MLE) of the transverse blocks of Eq. (\[eq:2\]), shown in Fig. \[fig:3\]. The region of stability of each cluster is predicted by a negative MLE. While there are four clusters in this network, there are only three MLEs: the two intertwined clusters are described by a 2-dimensional block in the block-diagonalized coupling matrix $B$. These stability calculations reveal the same bifurcations as seen in experiment.
The existence of isolated desynchronizations in the network experiments raises several questions. Since the network is connected why doesn’t the desynchronization pull other clusters out of sync? What is the relation of ID to cluster structure and network symmetry? Is ID a phenomenon that is common to many networks? We provide answers to all these questions using geometric decomposition of a group which was developed in [@MacArthur2009; @MacArthur2008]. This technique enables a finite group to be written as a direct product of subgroups ${\cal G}={\cal H}_1 \times ... \times{\cal H}_{\nu}$ where $\nu$ is the number of subgroups and all the elements in one subgroup commute with all the elements in any other subgroup. This means that the set of nodes permuted by one subgroup is disjoint from the set of nodes permuted by any other subgroup. Then each cluster (say, ${\cal C}_j$) is permuted only by one of the subgroups (say, ${\cal H}_k$), but not by any others. There can be several clusters permuted by one subgroup. This is the case of the red and blue clusters in the 32 symmetry network in Fig.\[fig:1\], because the associated ${\cal H}_k$ cannot have a geometric decomposition, but may have a more structured decomposition such as a wreath product [@Dionne1996].
One can show (see the Supplementary Information) that the above decomposition guarantees that the nodes associated with different subgroups all receive the same *total* input from the other subgroups’ nodes. Hence, nodes of each cluster do not see the effects of individual behavior of the other clusters associated with different subgroups. This enables the clusters to have the same synchronized dynamics even when another cluster desynchronizes. If that state is stable we have ID.
How common is such an ID situation we outlined above? We have examined statistics for some classes of random and semi-random graph types that suggest that when symmetries are present the opportunity for ID dynamics will be common although the stability for such will depend on the dynamical systems of the network nodes.
We examined 10,000 realizations of three random and semirandom networks: (1) randomly connected nodes (random graphs) similar to Erdos-Renyi graphs [@BollobasBOOK], (2) scalefree tree graphs following Barabasi and Albert [@Barabasi1999; @Albert2002], and (3) scalefree using the construction to give predetermined degree distributions characterized by exponent $\gamma$ with same number of connections as in (1) [@goh2001universal]. In the Supplementary Information we detail how we generated the realizations and tested for duplicates and statistical relevancy.
![ **Symmetry, cluster, and subgroup statistics for three types of networks.** The networks are random, Barabasi and Albert (BA in the figure), and the fixed exponent case ($\gamma$ in the figure). The statistics are (a) the cumulative distribution of the number of symmetries (the dashed line is the median), (b) the counts of the number of nontrivial clusters, and (c) the counts of the number of subgroups in the decomposition. \[fig:4\]](fig4){width="\textwidth"}
Figure \[fig:4\]a shows the cumulative distribution of symmetries for each type of network. The $\gamma=2.5$ scalefree graphs generally have fewer symmetries than the other two types. The Barabasi and Albert scalefree graphs often have many orders of magnitude more symmetries than the others which is a result of their hub and tree structure. All have similar distributions overall, but on different scales of symmetries. Cases of no symmetry are extremely rare for all graphs in these parameter ranges.
As shown in Fig. \[fig:4\]b,c, almost all graphs for each type have several nontrivial clusters and more than 1 subgroup with the Barabasi and Albert distribution skewing toward somewhat larger numbers. The median numbers of clusters for the random, Barabasi and Albert, and $\gamma=2.5$ networks are 3, 5, and 3, respectively. The median numbers of subgroups are 3, 4, and 3, respectively. The percents of cases where the number of subgroups is less than the number of clusters (intertwined cases) are 33%, 59%, and 33%, respectively. Thus, the scenario is present for almost all of these networks to experience ID.
Finally, we examined two existing networks: the Nepal power grid [@NepalElectReport2011] and the Mesa del Sol electrical grid [@abdollahy2012pnm]. We show the Nepal grid since its small size is easier to display in Fig. \[fig:5\]. Also shown is the block diagonalization of the coupling matrix. Here we treat the grid analogous to [@Motter2013] in which all power stations are identical with the same bidirectional coupling along each edge. This man-made network has 86,400 symmetries, three nontrivial clusters (plus two trivial ones), and three subgroups (one for each nontrivial cluster). This implies it is possible for this network to split into three sets of synchronized clusters and one of those could lose stability while the others remain synchronized which is ID. The Mesa del Sol grid has 4096 symmetries, 20 nontrivial clusters, and 10 subgroups. The network has three intertwined clusters, two with 4 clusters and one with 5 clusters, making ID a possibility.
![**Geographical diagram of the Nepal power grid network.** Colors are used to indicate the computed cluster structure. The matrix (inset) shows the structure of the diagonalized coupling matrix, analogous to Fig. \[fig:1\]a. The diagonal colors indicate which cluster is associated with each column. \[fig:5\]](fig5)
Many other networks were studied for symmetries in [@MacArthur2008; @MacArthur2009] for the purpose of finding motifs and redundancies, but not dynamics. Those networks were Human B Cell Genetic Interactions, C. Elegans genetic interactions, BioGRID data sets (Human, S. cerevisiae Drosphila, and Mus musculus), the internet (Autonomous Systems Level), and the US Power Grid. All the networks had many symmetries ranging in number from on the order of $10^{13}$ to $10^{11,298}$, and could be decomposed into many subgroups (from 3 to more than 50). The subgroups were 90% or more made up of basic factors (not intertwined) consisting of various orders $n$ of the symmetric group $S_n$. Hence, viewed as dynamical networks, all could show ID in the right situations.
The phenomena of symmetry-induced cluster synchronization and ID appear to be possible in many model, man-made, and natural networks. We’ve show that ID is explained generally as a manifestation of clusters and subgroup decompositions. Furthermore, computational group theory can greatly aid in identifying cluster synchronization in complex networks where symmetries are not obvious or far too numerous for visual identification. It also enables explanation of types of desynchronization patterns, and transformation of dynamic equations into more tractable forms. This leads to an encompassing of or overlap with other phenomena which are usually presented as separate. This list includes (1) remote synchronization [@Nicosia2013] in which nodes not directly connected by edges can synchronize (this is just a version of cluster synchronization), (2) some types of chimera states [@Abrams2004; @Hagerstrom2012] which can appear when the number of trivial clusters is large and the number of nontrivial clusters is small, but the clusters are big (see [@Laing2009] for some simple examples), (3) partial synchronization where only part of the network is synchronized (shown for some special cases in [@Belykh2000]). We note that although we have concentrated mostly on the maximal symmetry case, we can also examine the cases of lower symmetry induced by bifurcations that break the original symmetry and the same group theory techniques will apply to those cases. Some of this is developed for simple situations (rings or simple networks with point group symmetry) in Ref. [@GolubitskyBOOKII], but we now have the ability to extend this to arbitrary complex networks. Finally, we note that it is possible to extend this approach to systems with nonidentical oscillators and weighted and/or directional coupling or to hypernetworks [@Irving2012].
Symmetries, Synchronization Clusters, and Block Diagonalization of the Variational Equations. {#symmetries-synchronization-clusters-and-block-diagonalization-of-the-variational-equations. .unnumbered}
=============================================================================================
Here we outline the steps necessary to determine the symmetries of the network, obtain the clusters, find the irreducible representations (IRRs), and the most crucial part, calculate the transformation $T$ from the node coordinates to the IRR coordinates that will block-diagonalize $A$, since $A$ commutes with all symmetries of the group [@SaganBOOK].
Using the discrete algebra software it is straightforward to,
\(1) Determine the group of symmetries of $A$.
\(2) Extract the orbits which give the nodes in each cluster and extract the permutation matrices $R_g$
\(3) Using the character table of the group and the traces of the $R_g$’s determine which IRRs are present in the node-space representation of the group. *Remark:* This step is discussed in any book on representations of finite groups (e.g. Ref. [@TinkhamBOOK])
\(4) Put each $R_g$ into its appropriate conjugacy class.
The next steps are to generate the transformaion $T$ from the group information and they require writing code on top of the discrete algebra software.
\(5) For each IRR present construct the projection operator $P^{(l)}$ [@TinkhamBOOK] from the node coordinates onto the subspace of that IRR, where $l$ indexes the set of IRRs present. Thus, $$P^{(l)}= \frac{d^{(l)}}{h} \sum_{\cal K} \alpha^{(l)}_{\cal K} \sum_{g \in \cal K} R_g$$ where $\cal K$ is a conjugacy class, $\alpha^{(l)}_{\cal K}$ is the character of that class for the $l$th IRR, $d^{(l)}$ is the dimension of the $l$th IRR and $h$ is the order (size) of the group. *Remark:* The trivial representation (all IRR matrices=1 and $\alpha^{(l)}=1$) is always present and is associated with the synchronization manifold. All other IRRs are associated with transverse directions.
\(6) Use singular value decomposition on $P^{(l)}$ to find the basis for the projection subspace for the $l$th IRR.
\(7) Construct $T$ by stacking the row basis vectors of all the IRRs which will form an $N \times N$ matrix.
Once we have $T$ we can transform the variational equations as follows. Let ${\cal C}_m$ be the set of nodes in the $m$th cluster with synchronous motion ${\bf s}_m(t)$. Then the original variational equations about the synchronized solutions are (in vectorial form),
$$\label{linv}
\delta \dot{{\bf x}}(t)=\left[\sum_{m=1}^{M} E^{(m)} \otimes D{\bf F}({\bf s}_m(t)) + \sigma A \sum_{m=1}^{M} E^{(m)} \otimes D{\bf H}({\bf s}_m(t)) \right] \delta {{\bf x}}(t),$$
where the $Nn$-dimensional vector $\delta {\bf x}(t)=[\delta {\bf x}_1(t)^T,\delta {\bf x}_2(t)^T,...,\delta {\bf x}_N(t)^T]^T$ and $E^{(m)}$ is an $N$-dimensional diagonal matrix such that $$\begin{aligned}
E^{(m)}_{ii}=\left\{ \begin{array} {ccc} {1,} \quad \mbox{if} \quad {i \in {\mathcal C}_m,} \\ {0,} \quad \mbox{otherwise,} \end{array} \right.\end{aligned}$$ $i=1,...,N$. Note that $\sum_{m=1}^{M} E^{(m)}=I_N$, where $I_N$ is the $N$-dimensional identity matrix.
Applying $T$ to Eq. (\[linv\]) we arrive at the variational matrix equation shown in Eq. (\[eq:2\]),where ${\pmb \eta}(t)= T \otimes I_n \, \delta {\bf x(t)}$, $J^{(m)}$ is the transformed $E^{(m)}$, and $B$ is the block diagonalization of the coupling matrix $A$. We can write the block diagonal $B$ as a direct sum $\bigoplus_{l=1}^L I_{d^{(l)}} \otimes C^{l}$, where $C^{l}$ is a (generally complex) $p_l \times p_l$ matrix with $p_l=$ the multiplicity of the $l$th IRR in the permutation representation $\{ R_g \}$, $L=$ the number of IRRs present, and $d^{(l)}=$ the dimension of the $l$th IRR, so that $\sum_{l=1}^L d^{(l)} p_l=N$ [@Vallintin2009; @GoodmanBOOK]. For many transverse blocks $C^{l}$ is a scalar, i.e. $p_l=1$. However, the trivial representation which is associated with the motion in the synchronization manifold has $p_1={M}$. The form of the variational equation for the first examples is shown in Fig. \[fig:1\]c. Each block in Fig. \[fig:1\]c is governed by a separate variational ODE as given in Eq. (\[eq:2\]). Note that the vector field ${\bf F}$ can contain a self-feedback term $\beta {\bf x}_i$ as in the experiment and other feedbacks are possible, e.g. row sums of $A_{ij}$, as long as those terms commute with the $R_g$.
[**Acknowledgements**]{} We acknowledge help and guidance with computational group theory from Prof. David Joyner of the US Naval Academy and information and computer code from Ben D. MacArthur and Rubén J. Sánchez-García both of the University of Southampton to address the problem of finding group decompositions. Thanks to Vivek Bhandari for providing us a copy of the Nepal Electricity Authority Annual Report [@NepalElectReport2011]. Thanks to Shahin Abdollahy and Andrea Mammoli for providing us data on the Mesa Del Sol electric network [@abdollahy2012pnm].
Supplementary Information {#supplementary-information .unnumbered}
=========================
[**Adjacency, Transformation, and Block-Diagonal Matrices and orbits.**]{} Here are the adjacency matrices, the clusters (group orbits), the transformation, and block-diagonalized coupling matrices for Fig. 1 in more detail.
The 0-symmetry case:
$$A= \begin{pmatrix}
0&1&1&1&1&1&0&1&1&1&1\\
1&0&1&0&1&1&1&1&0&1&1\\
1&1&0&1&1&1&0&1&1&1&0\\
1&0&1&0&1&1&1&1&1&1&0\\
1&1&1&1&0&1&1&1&1&0&1\\
1&1&1&1&1&0&0&1&1&0&1\\
0&1&0&1&1&0&0&1&1&1&1\\
1&1&1&1&1&1&1&0&1&1&1\\
1&0&1&1&1&1&1&1&0&1&1\\
1&1&1&1&0&0&1&1&1&0&1\\
1&1&0&0&1&1&1&1&1&1&0
\end{pmatrix}$$
There are no nontrivial clusters, the transformation matrix $T$ is just the identity matrix $I_{11}$, and $B=A$.
The 32-symmetry case:
$$A= \begin{pmatrix}
0&1&1&1&1&1&0&1&1&1&1\\
1&0&1&1&1&1&0&1&1&1&1\\
1&1&0&1&1&1&1&1&0&1&1\\
1&1&1&0&1&1&1&1&1&1&1\\
0&1&1&1&0&1&1&1&1&1&0\\
1&1&1&1&1&0&1&1&1&1&1\\
1&0&1&1&1&1&0&1&1&1&1\\
1&1&1&1&1&1&1&0&1&0&1\\
1&1&0&1&1&1&1&1&0&1&1\\
1&1&1&1&1&1&1&0&1&0&0\\
1&1&1&1&0&1&1&1&1&0&0
\end{pmatrix}$$
The nontrivial clusters are the nodes \[1, 7\], \[2, 3, 7, 9\], \[4, 6\], \[5, 10\] (the numbering of nodes matches the row and column numbers of $A$). The transformation matrix is,
$$T= \begin{pmatrix} 0.000&0.000&0.000&0.000&0.000&0.000&0.000&0.000&0.000&1.000&0.000\\
0.000&0.000&-0.707&0.000&-0.707&0.000&0.000&0.000&0.000&0.000&0.000\\
0.000&0.000&0.000&0.707&0.000&0.000&0.000&0.000&0.707&0.000&0.000\\
0.000&0.000&0.000&0.000&0.000&0.000&-0.707&0.000&0.000&0.000&-0.707\\
-0.500&-0.500&0.000&0.000&0.000&-0.500&0.000&-0.500&0.000&0.000&0.000\\
0.000&0.000&0.000&-0.707&0.000&0.000&0.000&0.000&0.707&0.000&0.000\\
0.000&0.000&0.000&0.000&0.000&0.000&-0.707&0.000&0.000&0.000&0.707\\
0.000&0.000&-0.707&0.000&0.707&0.000&0.000&0.000&0.000&0.000&0.000\\
-0.500&0.500&0.000&0.000&0.000&-0.500&0.000&0.500&0.000&0.000&0.000\\
-0.707&0.000&0.000&0.000&0.000&0.707&0.000&0.000&0.000&0.000&0.000\\
0.000&0.707&0.000&0.000&0.000&0.000&0.000&-0.707&0.000&0.000&0.000\\
\end{pmatrix}$$
And the block diagonal coupling matrix is, $$B= \begin{pmatrix}
0.0&-1.41&0.0&-1.41&-2.00&0.0&0.0&0.0&0.0&0.0&0.0\\
-1.41&1.00&-2.00&2.00&2.83&0.0&0.0&0.0&0.0&0.0&0.0\\
0.0&-2.00&1.00&-1.00&-2.83&0.0&0.0&0.0&0.0&0.0&0.0\\
-1.41&2.00&-1.00&1.00&2.83&0.0&0.0&0.0&0.0&0.0&0.0\\
-2.00&2.83&-2.83&2.83&2.00&0.0&0.0&0.0&0.0&0.0&0.0\\
0.0&0.0&0.0&0.0&0.0&-1.00&1.00&0.0&0.0&0.0&0.0\\
0.0&0.0&0.0&0.0&0.0&1.00&-1.00&0.0&0.0&0.0&0.0\\
0.0&0.0&0.0&0.0&0.0&0.0&0.0&-1.00&0.0&0.0&0.0\\
0.0&0.0&0.0&0.0&0.0&0.0&0.0&0.0&-2.00&0.0&0.0\\
0.0&0.0&0.0&0.0&0.0&0.0&0.0&0.0&0.0&0.0&0.0\\
0.0&0.0&0.0&0.0&0.0&0.0&0.0&0.0&0.0&0.0&0.0\\
\end{pmatrix}$$
The 5670-symmetry case:
$$A= \begin{pmatrix} 0&1&1&1&1&1&1&1&1&1&1\\
1&0&1&1&1&1&1&1&1&1&1\\
1&1&0&0&1&1&1&1&1&0&1\\
1&1&0&0&0&1&1&1&0&0&1\\
1&1&1&0&0&1&1&1&0&1&1\\
1&1&1&1&1&0&1&1&1&1&1\\
1&1&1&1&1&1&0&1&1&1&1\\
1&1&1&1&1&1&1&0&1&1&1\\
1&1&1&0&0&1&1&1&0&1&1\\
1&1&0&0&1&1&1&1&1&0&1\\
1&1&1&1&1&1&1&1&1&1&0\\
\end{pmatrix}$$
The nontrivial clusters are the nodes \[2, 6, 1, 7, 8, 11\], \[3, 5, 9, 10\]. The transformation matrix is, $$T= \begin{pmatrix} 0.000&0.000&-1.000&0.000&0.000&0.000&0.000&0.000&0.000&0.000&0.000\\
-0.408&0.000&0.000&0.000&-0.408&-0.408&-0.408&0.000&0.000&-0.408&-0.408\\
0.000&0.500&0.000&0.500&0.000&0.000&0.000&0.500&0.500&0.000&0.000\\
0.000&0.000&0.000&0.000&0.643&-0.448&-0.114&0.000&0.000&0.390&-0.471\\
0.000&0.000&0.000&0.000&-0.522&-0.522&0.224&0.000&0.000&0.596&0.224\\
-0.913&0.000&0.000&0.000&0.183&0.183&0.183&0.000&0.000&0.183&0.183\\
0.000&0.000&0.000&0.000&-0.332&0.568&-0.138&0.000&0.000&0.472&-0.570\\
0.000&0.000&0.000&0.000&-0.066&-0.066&0.847&0.000&0.000&-0.264&-0.451\\
0.000&-0.500&0.000&0.500&0.000&0.000&0.000&0.500&-0.500&0.000&0.000\\
0.000&-0.707&0.000&0.000&0.000&0.000&0.000&0.000&0.707&0.000&0.000\\
0.000&0.000&0.000&-0.707&0.000&0.000&0.000&0.707&0.000&0.000&0.000\\
\end{pmatrix}$$ And the block diagonal coupling matrix is, $$B= \begin{pmatrix}
0.0&2.45&0.0&0.0&0.0&0.0&0.0&0.0&0.0&0.0&0.0\\
2.45&5.00&-4.90&0.0&0.0&0.0&0.0&0.0&0.0&0.0&0.0\\
0.0&-4.90&2.00&0.0&0.0&0.0&0.0&0.0&0.0&0.0&0.0\\
0.0&0.0&0.0&-1.00&0.0&0.0&0.0&0.0&0.0&0.0&0.0\\
0.0&0.0&0.0&0.0&-1.00&0.0&0.0&0.0&0.0&0.0&0.0\\
0.0&0.0&0.0&0.0&0.0&-1.00&0.0&0.0&0.0&0.0&0.0\\
0.0&0.0&0.0&0.0&0.0&0.0&-1.00&0.0&0.0&0.0&0.0\\
0.0&0.0&0.0&0.0&0.0&0.0&0.0&-1.00&0.0&0.0&0.0\\
0.0&0.0&0.0&0.0&0.0&0.0&0.0&0.0&-2.00&0.0&0.0\\
0.0&0.0&0.0&0.0&0.0&0.0&0.0&0.0&0.0&0.0&0.0\\
0.0&0.0&0.0&0.0&0.0&0.0&0.0&0.0&0.0&0.0&0.0\\
\end{pmatrix}$$\
[**Subgroup decomposition and cluster dynamics.**]{} To start let ${\cal H}_k$, a subgroup of ${\cal G}$, permute only cluster ${\cal C}_m$ and $\pi$ be the permutation on the indices of nodes in ${\cal C}_m$ for one permutation $R_g,\; g \in {\cal H}_k$. Assume ${\bf x}_i$ is not in ${\cal C}_m$ so it is not permuted by $R_g$ and recall that $G$ commutes with all permutations in ${\cal G}$, then we have (just concentrating on the terms from ${\cal C}_j$),
$$\label{decompcplg}
\begin{split}
[R_g \dot{\bf x}(t)]_i=\dot{\bf x}_i(t)=...+ \sigma[R_g A {\bf H}({\bf x})]_i =...+ \sigma[A R_g {\bf H}({\bf x})]_i = ...+ \sigma\sum_{j \in {\cal C}_m} A_{ij} {\bf H}({\bf x}_{\pi(j)}),
\end{split}$$
where $\pi(l)$ is, in general, another node in ${\cal C}_m$ and the sums over other clusters are unchanged. This shows that all nodes in ${\cal C}_m$ are coupled into the $i$th node in the same way (the same $A_{ij}$ factor). Similarly, if we use a permution $R_{g'}$ on the cluster ${\cal C}_{m'}$ containing ${\bf x}_i$ we can show that all the nodes of ${\cal C}_{m'}$ are coupled in the same way to the nodes in ${\cal C}_m$. Hence, nodes of ${\cal C}_{m'}$ each receive the same input sum from the nodes of ${\cal C}_m$ whether the nodes of ${\cal C}_m$ are synchronized or not. This explains how the cluster ${\cal C}_m$ can become desynchronized, but the nodes of ${\cal C}_{m'}$ can still be synchronized – they all have the same input despite the ${\cal C}_m$ desynchronization, thus making the ${\cal C}_{m'}$ synchronous state flow invariant. If it is also stable, this is the case of ID. This argument is easily generalized to the case when ${\cal H}_k$ permutes nodes of several clusters as this will just add other similar sums to Eq. (\[decompcplg\]). The latter case explains the intertwined desynchronization in the experiment and is a more general form of ID.
[**Statistics of random graphs.**]{} Random graphs were generated by starting with 25 nodes completely connected and randomly deleting 20 edges. Scalefree Barabasi and Albert graphs were based on the original Barabasi and Albert preference algorithm [@Barabasi1999] using the `SAGE` routine `RandomBarabasiAlbert`. These had 25 nodes with 24 edges and a tree structure. Scalefree graphs with a specific power-law distribution were generated according to [@goh2001universal] using $\gamma=2.5, 3.0,$ and $3.5$. 10,000 realizations of each graph type were generated. We tested several 10,000 realizations and we see very little variation in statistics between realizations of the same class leading us to believe that we are sampling fairly and enough to trust our results. We also checked for equivalent (isomorphic) graphs to see how much repetition we had. The random systems yielded on average 1 equivalent pair per 10000 realizations. The scalefree cases yielded about 5 to 10% equivalent graphs. Apparently we are not near the maximum number of inequivalent graphs for any of the classes although the results suggest that the scalefree classes are much smaller than the random class. Even with just 100 realizations the main trends in number of symmetries and other statistics are evident although such small samples occasionally miss those symmetry cases that are not too common in the class.
[**The scalefree $\gamma$ model.**]{} The model generates a scale free network with $N$ nodes and $E$ edges and a specified power law degree distribution exponent $\gamma$. Start with $N$ vertices, assign to each vertex $i =1, 2, ...,N$, a weight $w_i = i^{-\mu}$, where the exponent $\mu$ lies in the range \[0, 1). Assume that initially no edges are present among the network vertices, then edges are added one by one until $E$ connections are created. For each new edge, two vertices are randomly selected, each one with probability proportional to its weight, and they are connected unless a link already exists or the two selected nodes are the same. By following this procedure, the resulting network is scale free and the power law degree distribution $P(k) \sim k^{-\gamma}$, with exponent $\gamma =(1+\mu)/\mu = 1/\mu+1$.
[**SmallWorld networks**]{} We also studied symmetries, clusters, and subgroup decompositions in smallworld graphs. Smallworld graphs [@Watts1998; @BollobasBOOK] were generated by starting with a ring of nearest neighbor connected nodes, then adding a fixed number of edges to give the same number of edges as the random graphs in the text. We found we had to add many edges beyond the usual few used to generate the smallworld effect because adding only a few edges beyond the ring rarely resulted in any symmetries. As a result the smallworld examples approached being a like the random graphs so we do not display their results although the two systems each have symmetries that the other does not so they appear to not be exactly identical.
[**The Mesa del Sol power grid.**]{} In Supplementary Fig. \[Suppfig:1\] we show a circle plot of the Mesa del Sol network which, because of the network size (132 nodes), exposes the cluster structure much better. There are 20 nontrivial clusters and 10 subgroups in the decomposition. There are a large number of trivial clusters with only about 1/3 of the nodes being in a synchronizable cluster. However, in that subset of clustered nodes the subgroup decomposition shows that ID is dynamically possible.
![**Network and cluster structure of the Mesa del Sol electric grid.** Colors are used to denote clusters. Nodes colored white are trivial clusters, containing only one element. \[Suppfig:1\]](fig6){width="\textwidth"}
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